 Grazie. Lo streaming è partito? Sì, sto guardando, vediamo ancora un secondo. Perciò, eccolo. Bene, siamo pronti. Grazie, benvenuti a tutti per questo nuovo giorno con molte lezioni excitanti. Perciò, prima di iniziare, solo un ricordatore dei alcuni regolari su come chiamare questioni. Quindi, se stai seguendo da YouTube, potete chiamare questioni nel chat e io li vedo per te. Se sei connettato a Zoom, potete chiamare le questioni nel chat o usare la risa e il feature di Zoom. Grazie, è il mio piacere di introdurre l'introduzione di Andrea Rinaldo, che è giocando il secondo di tre lezioni. Grazie, Andrea, per essere con noi. Perciò, potete share la screen. Grazie molto, Iacopo. Andiamo alla scena, mi raccomando che, se non vedete, mi raccomando che la mia scena fuorità sia visibile a te. Perciò, andiamo alla seconda di tre lezioni sui netori e ecologici. Siamo intitoli di specie, popolazione e passaggi. Poi puoi ricordare, se hai che essenzialmente l'unico motivo di la mia parola è che ci chiediamo che i substrati indietrici, come l'unico netto, con certe proprietà di recurso che abbiamo analizzato, ma questi substrati indietrici per le interaccioni ecologiche, sono importanti conseguenze per molte processi che avrebbero analizzato, che sono essenzialmente pattenze per la diversità, che non ho fatto, che ho fatto oggi in parte, l'esercito della specie per le interaccioni biologiche, e l'esercito della disease di waterborne, che vedete nel mio ultimo classico, riferendo all'epidemica di colore, e alla disease pericolose di disease pericolose di fischia. La idea è che il sistema è così constretto dal substrati, dalla natura del substrati, le proprie proprietà, le proprie locali impredate in questo, ma in realtà, avrebbero indagato il sistema un certo grado di predictabilità, beyond ciò che era origenale pensato. Quindi, cosa che abbiamo visto durante il primo classico, è che, quando si mette queste constretti in strada, e si usa anche il modello più semplice, in questo caso, era il modello neutral di biodiversità, se usando un modello di interaccione di meta, o un modello di comunità di meta, se usando le interaccioni dei neighbori, o se prendete un cerno, che si tratta da una certa lengtha discreta, o un approccio minimo, in cui si avverde l'ensemble per rilassare, per ottenere le rilassi, o le specie che si rilassano randomemente da un single site, il sistema tende a avere certe indicazioni, che è che la vera natura del substrati e delle interaccioni con la nettora d'endritica ha importanti conseguenzi, che abbiamo, dopo, verificato da un numero di differenti tool. L'ultimo argomento che ho messo, e è una cosa che ha detto, che le lecce che ho provato a fare, ho messo alcune linee di argomenti, per suggerire che l'interaccione di un framework di ecohydrologia che inizia il laboratorio, il filo e l'evidenza terratica ha continuato il sostanzio alla nostra comprensione della funzione di la nettora di rilassi. Alla fine di oggi, iniziamo con le invasioni biologiche, possibilmente le invasioni biologiche sui supporti fractal, come il 3, che è l'unico che abbiamo visto in caso della nettora, quindi quello che hai visto qui è una mappa di invasione del massimo zibram d'alongi il sistema di misura del Missouri, anche all'Università, e la coda di diversi colori che è successiva per una specie che è completamente aliena per l'accidente al sistema. Stiamo parlando di hidroquiri e per fare un primo esempio stiamo parlando di espansioni di rilassi huma e di migliori migliori nel passaggio, che è un'interessante idea del gruppo di fisi spanish che ho messo per oggi. Per prima, cominciamo con delle cose standardi che molte di voi hanno visto in un numero di context e è un fissile spanish tra l'appropriazione del sistema. Se hai un modulo in cui hai l'esperimento dispersa in questo caso è come una densità della specie o se lo chiedete o ogni tipo girando a supporto di una dimensione di un dimensione, in caso di netto della rida, fa un senso di pensare di un sistema di dimensione in un sistema in interesse perché la velocità e la coda del ribero sono muchi più smalleri che la lungitudinale lente che hai avuto, quindi l'absumptione di fare l'opera cannellata e l'absumptione in piper, etc., è una cosa dimensionale, è un'absolutamente standard, l'absumptione verificato per 100 anni di pratica. Quindi essenzialmente, cosa diremmo, supponiamo che il sistema avrebbe una diffusione semplice, con l'effettione di costa, quindi il termine d², ρ² dx², coppere a un significativo della scale di diffusione, il sistema di diffusione. In questo caso è un'equilibrio logistico in cui essenzialmente la diffusione della densità segua la fase iniziale, che è la fase di diffusione di modusia e di dinamica, poi incerti per ottenere un asymptote. Quindi il termine di diffusione è nil, che è l'esercito della densità 0, e la rosa è una capacità di carina. Perché questo è interessante? L'esercito è stato studiato per un'antica volta, e non soltanto per l'ultima, per l'ultimo semplice applicato per la matematica dell'ultimo senso, Colmogorov, in realtà, e R.A. Fischer, l'inventor, o tali risultati che sono cruciali ora a modi genetici e in medici moleculi, etc. Sono studiati under diversi angosi per aver visto il stesso risultato. Se supponiamo che, per esempio, si sta generando un punto singolo in spazio, su una strada lenta, e un solo, si injecta un solo punto di un certo numero di organi, di questa specie, in cui stiamo imparando a analizzare. Il sistema qui è conosco per generare un solito, che è un after a short while che propaga senza la solarità constanta con questa equazione, che è soltanto dictata dal valore del growth rate in verso di tempo e la grafica diffusione in strada in strada di tempo per generare una velocità caratteristica che è sempre la strada strada di A times D. Come provate questo risultato? Beh, è un punto interessante in cui essenzialmente prende l'equazione di diffusione sotto questo termo limitato e inizialmente vedete che questo termo è negativo. E quello che hai è che puoi calcolare l'espansione radiosa di la più massima o di una fraccia assignata della massima che hai nel vostro sistema e come questa espansione in tempo. Poi, quello che stai imparando è il comportamento che ha un feature importante per le risorse e la capacità che scrivono l'effetto di netto. C'è la capacità di una densità low quando questo sta propagando, in realtà. La idea è che se calcolate l'espansione di radiosi che in un regime in cui questo log della quantità o qualunque che vuoi avere è iniziale comparedi al termo in cui hai la strada dipende della strada di radiosi. Quindi la velocità della velocità è essenzialmente due d x r che è il growth rate nel modello in cui si eliminano in cui si eliminano l'effetto della capacità. E non mi sorpreso l'effetto della capacità anche se ti accendete può essere mostrato che la velocità fronte è essenzialmente in caso della velocità metodologica ancora cosa succede nella velocità di questo termo di reazione per cui la densità approccia zero. E e che non è la possibilità perché lo che stiamo dicendo è che essenzialmente il stesso risultato della velocità metodologica in cui se prendete una propria competizione della velocità metodologica per la velocità metodologica e attraverso la velocità qui e la soluzione del sistema in cui vedete qualsiasi il sistema lo comporta esattamente in the same manner. Quindi che è interessante è che nel sistema due dimensioni per esempio o in un migrazione isotropica sistema 1, 2, 3 dimensioni cosa che hai il sistema potrebbe essenzialmente la velocità della velocità o la velocità generata che arriva a un plateau e stai a propagare la formazione è proporzionale per queste due coefficienti ignorando l'effetto della capacità di carina che è in materiale ma solo dependendo della coefficiente di diffusione e della reproduzione la reproduzione in una volta. Questo perciò in un direttore supporto che è in 1, 2 dimensioni ecc. E questo avrà alcune importazioni che seguono. Che è interessante perciò è che cosa è importante e che è importante per il particolare substratto che stiamo studiando che è un strato che è un unico modo per alcun no a alcun altro no è un graffo direttore in cui c'è un senso di direzione che è impredo nella struttura del sistema di diffusione e quindi essenzialmente puoi mappare l'evoluzione della densità degli organismi in spazio e in tempo come un variabile di un profondo di underform che è il seccio tipico del caratteristico di Giorgio Dei. Quindi, in realtà che ciò che interessa è quello che sta passando nella vicinità della densità zero è importante quando hai a guardare i sistemi come questa questa è la network di piano che ho introduito durante l'ultima domanda perché mi sono usando il piano ho usato il piano perché è un frattore deterministico che ha riusciuto Giuseppe piano che ha scritto un pittore un pittore che riusciuto nel piano che è considerato come un trastore per il matematicio di questo terreno e è ancora cominciato 300 anni dopo il tempo quando un fondatore di fondatore ha riusciuto per supportare questa idea che gli montaggi non sono coni costali non sono bloccati non troverle in linei dopo 2.000 anni, dopo l'euclidee, abbiamo iniziato a guardare la geometria di natura con differenti occhi, era impredato in una curva che è semplicemente deterministica e self-simile, perché è creata da un processo, o è dall'originatore, che essenzialmente ha il sistema che è avvenuto a mezzo del segmento qui, aggiungendo a un sistema di questo tipo. E poi ti permette di calcolare esattamente un po' di proprietà. Perché questo è un feature interessante? Perché se guardate le proprietà topologiche, ovviamente se guardate l'esercito, questo non è un network, ovviamente non è, ma ti permette di calcolare alcune quantità esattamente. Per esempio, se puoi calcolare il numero, supponere il graffetto direttamente. E si può calcolare, e se supponere si ha un punto secco, come qui, per esempio, o in questo caso qui, se ti assumerà la luce di linea proporzionale all'area di total contribuzione, è una città, quindi in qualsiasi posto, puoi calcolare il numero di noti che ti hanno abstrutte. E quello che è interessante è che puoi calcolare esattamente il numero di città ugualmente distante da l'outlet. La distanza è calcolata in cosa è chiamata normalmente la distanza chimica che è la struttura di un network. Che è molto importante in hydrologica, la quantità, perché se supponere che si arrabbi, una quantità giusta in ogni noto, la distanza, se il sistema lo comporta in modo dinamico, significa l'arrivalo di tempo e l'outlet. Quindi essenzialmente il risponso del sistema è molto relato a questa quantità. E in caso del piano, questo è un interessante tema, perché questo è un processo dinamico multiplicativo soltanto esattamente. È un esatto multifrattal che è stato studiato da molte lezze in maniera. In questo caso, cosa specifica di questo è che, anche se, in questo caso, questo non è un network di river. Però, le proprietà topologiche che sono non distinte, in molti risponsi, quasi statisticamente inevitibili, la struttura topologica è esattamente indistinguistica, come i numeri di bifurcazione, lezze, etc. Perciò l'uno è esibito da network river. Quindi il messaggio che ho convinto è che in questo caso si può studiare esattamente il comportamento di la struttura di questo bifurcazione o propagazione, per esempio, perché si può calculare esattamente il numero di bifurcazioni che il sistema va ad esperienza nel suo viaggiamento in Amsterdam. Quindi se vedete questo numericamente, questo è, vi mostreremo direttamente cosa che vedete, è una diffusione e un termo di reazione un termo logistico che sta passando in questo senso che vedete qui. E questo è come il sistema propaga. Che è interessante che se prende la direzione longitudinale che è che è, per esempio, di andare da una struttura a un'altra struttura, lo metto per clarità horizontalmente, ma può essere anche alcun termo di struttura che ha esattamente il stesso comportamento. Che è che se se fai questo, quindi questo è come il pao propaga al sistema dependendo di differenti feature. In questo caso hai una struttura e poi, per esempio, in questo caso, avrete una struttura di un significativo di un determinato struttura. E quello che è che è che hai un sistema con un paio di struttura che è attraverso la heterogeneità generata dalla struttura, ma genera un struttura di struttura. E quello che è fenomenale è che un walker che è a struttura su una struttura può sparare un certo tempo di una struttura secondaria se andete qui e avrete un struttura in questa direzione. Spende un po' di tempo sul sistema prima di che puoi avventualmente farla e continuare il tuo struttura verso l'outlet. Quindi il punto principale è che se andate a un sistema che è una dimensione ma con significanti struttura perché del struttura di struttura essenzialmente produza la struttura della struttura del fronte di struttura. Quindi se sei solo interessato nel comportamento del sistema tra la struttura puoi interpretare le strutture secondarie come essenzialmente un produttore del struttura tra le strutture del fronte. Quindi le strutture potrebbero essere chemicali, physicali, biologici e reazioni che generano essenzialmente strutture di struttura perché è quello che sono interessati al sistema. E si può farlo con una tecnica chiamata Continuous Timber Underwalk. Ho solo lasciato un po' di struttura qui, ma hai la densità di un punto che è qualcosa di cui si prende in account la proprima che hai di stare hai una struttura di struttura in una struttura un po' di struttura in una struttura a questo punto è il momento in cui si prende se si è chiamato in una struttura di struttura come lungo si prende in un po' di struttura prima di andare all'instituto e è una distribuzione di un'attima vita qui. E poi hai anche un termine che descrive il giumpo su ogni struttura che in questo caso si chiamano Reactive Random Walk perché si chiamano ogni struttura che hai con probabilità una settimana che si chiamano in una struttura di struttura in una struttura di struttura o se hai zi nei nostri neighbori hai probabilità di 1 over zi di un'altra struttura di un'altra struttura Quindi una struttura lunga può essere tritata da le trasformi di Laplace che si chiamano Hamilton-Jakobi meccanismi e si può calcolare infatti molto bene la Place Transforms come il sistema quindi si può calcolare di creare il sistema di Fischer di un homogeneo e dimensionale per in particolare un sistema dimensionale il tempo di propagazione di un solito che si generano da la diffusione e una reagizione logistica in questo caso particolarmente e in caso di un metodo in particolare si può farlo esattamente quindi una struttura lunga se qualcuno è interessato ha detto che nel libro che Marino Gatto, Ignacio Rodriguez e io hanno già pubblicato un metodo di drivazione nel metodo che spiegherà in realtà sembra essere qualcosa che è se guardate la size di un giump che assurde che il sistema che ha in spazio discretto e in tempo continuo è delta square diventato da tau il tempo di tempo del sistema potete vedere che questo è l'equivalente di un metodo di l'altro e questo è l'equivalente di una coefficiente quindi è il struttura di r times o a times d ma diventato da coefficiente che è essenzialmente la convoluzione del tempo di spazio che hai l'ultimo di cui si propaga che è l'ultimo quindi la predicata theoretical su un modello di struttura su una particolare struttura è dirlo che in principio dovete che in piano puoi calcolare esattamente e puoi fare numericamente in alcuni altri casi la heterogeneità generata dalla struttura bifurcata che si faceva propagando in una direzione la struttura di fronte e è quello che sta succedendo quindi se prendete un sistema di due dimensioni questo è quello che hai nel sistema di due dimensioni di questo tipo come se questo è che si injecta in un particolare punto oops e vedete come si propaga in spazio isotropico e la struttura è sia la struttura di a times d e quasi non è una struttura particolare per piano hai un soluzione esattivo per occhi che hai visto l'altro tempo hai qualcosa di il stesso tipo hai systematicamente una una struttura di propagazione hai una struttura che lo so e per fare questo ho risultato una struttura che ho trovato assolutamente fascinante un gruppo di fisici spagnoli che ha scritto la popolazione teoretica nel 2006 che ha iniziato il nostro interesse nel soggetto ha studiato un modello quantitativo per la colonizzazione americana di la struttura verso il west nel XIXe ha visto le lenzze del trasporto della struttura struttura quindi l'idea è che le struttura di heterogeneità e il bisogno di acqua forzano i settelari sui corsi di flughi ovvero ogni settelar migliora una struttura infatti perché miglioriamo il rive e nemmeno trattano le strutture in piccoli tempi quando trattano le strutture come i barocchi o qualcosa perché bisogna le strutture per le corsi? perché bisogna le strutture di water resources per la struttura di drink energia trasportazione o qualcosa e e l'idea dell'esercito del modello di diffusione sembra un modo sensibile per un network rivello e una struttura che ha la struttura di rivello geologistica con i parametri di rivello A per la struttura di rivello in un single node che è meraviglioso che l'archivazione del Congresso USA ti permette di evaluare infatti come è arrivata la struttura e che questo ragazzo è che hai il valore di struttura di rivello che è la distribuzione della struttura di rivello della struttura da i colonizzatori in diversi direzzi e perciò uno dei parametri che hai per calcolare il sistema che è la distribuzione della struttura di rivello e che è fatto e non ha bisogno di calibrare tutto e che è totalmente ricordato se hai l'archivazione della struttura di rivello hai una distribuzione della le feature della struttura di rivello in spazio con la distribuzione delle strutture che i settelni avrebbero in un marzo isotropico all'esterno del west e quando hai la struttura di rivello che è poi hai i parametri la struttura di rivello di A times T sarà in ordine di 40 km per anno che è totalmente ricordato che se hai simulazioni ogni tipo delle assumptioni etc. si avvergono a spazio di ordine di 30 km per anno e a obiettivo che l'attuale spazio della struttura fosse l'ultima 1.3 13.5 quindi l'idea è che l'ultimo suggerirò che indi anche in questo caso la sua natura del substrage in questo caso della popolazione della popolazione dinamica era la struttura dominata da le feature della struttura di rivello in modo perché sono le strutture che hanno il sistema questo può essere quindi questa è la struttura quindi la struttura che abbiamo impostato sulla propria di la popolazione delle strutture di rivello facendo le bifurcazioni che hanno la struttura che vedete inizialmente le proprietà topologiche che sono le proprietà delle bifurcazioni che si avvergono accanto il tuo strutturo su ogni particolare della struttura sarebbe lo stesso in questa struttura che è statisticamente identica alla struttura della struttura di rivello replicabile eccetera come abbiamo visto nel caso di OSEAN e nel reto piano che in realtà comporta molto in il stesso modo quindi ha un'altra indicazione che il strutturo fa un grande ruolo nella ecologia del sistema quindi come ho fatto come ho detto all'altro la prima cosa che dobbiamo dire ok, è una strong suggestion e possiamo studiare nel laboratorio cosa che sta passando quindi abbiamo sviluppato la viabilità nel mio lab per lavorare con i protisti e quindi abbiamo essenziali protistodromi se vuoi quindi abbiamo fatto queste stesse e vediamo come possiamo mesurare da più o meno progressi i tools eccetera come e injeccere la popolazione nel 1N può invadere la velocità che invadere il sistema quindi questo sembra come una piccola canzina ma essenzialmente la struttura che ha preso con la microscopia che ho fatto nel mio lab l'individuale il tragetto è di ogni protista in questa direzione e mi piace che l'un dei due dimensioni che sono più piccoli che il lungiduzionale che mi rende una dimensione più semplice quindi sono interessati come la heterogeneità sceglie la fronte e una sarà per testare come la demografia di stochasticità sceglie con il rispetto del modello fisico-mogorov dell'invasione quindi essenzialmente che abbiamo preso il modello di questo tipo e abbiamo decattato il senso di spazio dal l'intensic noio che hai in un sistema l'individua le differenze realizzazioni di un protista attraverso il sistema sono cose di questo tipo quindi l'applicazione che è assolutamente necessaria per vedere come il sistema comporta ma il punto è che se fai questo e si applica questo normalmente che ha un 95% range che vedete ancora in un context in cui hai non c'è nulla in questo non c'è ma le demografiche di stochasticità induzono la biologia del sistema e ancora cosa è fenomenale è che il sistema fisico-mogorov può riuscire con un'accursione ricorda cosa sta passando nel laboratorio quindi interessante enough ci vediamo ok, ora possiamo generare altri tipi di heterogeneità nel sistema e come puoi farlo? una idea è usare i fototaxi nella invasione biologica e e l'idea è usare la generalizzazione della lezione perché hai un sistema in cui ogni node essenzialmente hai un reaccio nel node e hai un transporte tra i nodes quindi l'idea è usare le fototaxi nel sistema di questo tipo questo è il il il il runner il drone il stadio per questo proti per riuscire il punto d'injectione e il punto d'injectione e quello che abbiamo nel sistema fototaxico è usare le lezze per generare le risorse distributate di heterogeneità nel spazio che non è stato fatto prima e la invasione nel ambiente heterogeneo è una delle tendenze in cui ho tentato di ottenere l'interesse e come l'ho visto quindi l'idea è per esempio la heterogeneità potrebbe essere la heterogeneità in cui per esempio puoi mettere un drift nel sistema puoi mettere un bias nel sistema non semplice diffusione ma puoi aggiungere l'idea che fa che è più heterogenea ovviamente perché l'idea in questo caso puoi assumere l'idea e la probabilità in ogni noto che avete è una probabilità che è un'una probabilità di spazio nel caso di diffusione e non è anche se tu hai un drift di alcun tipo quindi essenzialmente quello che hai nel sistema è un parametro che unifica il drift cosa che hai è che questo è la velocità in un caso quindi essenzialmente in un caso in cui tu hai come zero drift nel strato di A T come tu hai nel sistema up to a system che è completamente dominato da avvezione che è importante perché io sono interessato nel studiare la persistenza delle specie in una scena ecologica quindi un drift non può essere importante depending on ecologicamente quantità viva come la natura delle organi che abbiamo discussi quindi comunque cosa che vediamo è che il metodo Jacoby che abbiamo usato e le similazioni numeriche si mettono molto bene e se invece per drifze inizia ad avere negativi e posthi posthi fissure strati e puoi avere le approximazioni che sono importanti in questo caso che ci sono briefi che parli di un equatrici telegraphi Piano poi sta in il sistema con feature che sono concurati esacamente ovvero non presti con i dettagli sono abbastanza invogliati ma in modo strato e analiticamente solvabili e in modo significativo perché Piano topologico è indistinguisciabilo da un netto real ma se guardate vedete che semplicemente non è l' equatrici telegraphi è anche qualcosa di interesse ovvero no per esempio questo è se guardate come il sistema propagato che ha chiuso tre differenti spazi per colonizzare il sistema quindi questo è come il sistema propagato lo comporta e c'è un certain degree di colonizzazione come un degree di colonizzazione del sistema che ora perché la reazione telegraphi contro la diffusione quando cosa sta passando con la drifte cosa che sta passando è che ah dovete assurda che se usate una reazione di modulare essenzialmente se usate alcune provvisioni per l'e lo assurde the speed of propagation of the jump will be at infinite velocity which is in fact it's a complicated thing to digest long story short if you put the proper model which overcomes the inconsistency well known inconsistencies of a diffusional model in the case of bias in this case long story short the computation nothing happens so a reaction around the wall describe what happens in the system even in the case of the drift fairly well so back to the system which heterogeneity is imposed by me by choosing a certain distribution of resources with a certain correlation length for instance you have like silence resources silence you see it's a personal process the one we modeled with in the system here so we had the parts at the bottom of the channel for changing the distance from the lead the angle of light propagation of the light to use something which is called the Keller Siegel framework for calculating how the resource acts like sort of an advezione in the system and I won't get into the taste for first if you're interested I'd be delighted to see what happens but what what is interesting in this case is that we do have a framework so you turn off a light the system behaves and that's very clear from the spectral behavior of the system behaves like a linear e to the minus the wave number square times dt so so it turns out to be a normal diffusion if you turn off the light and it comes up into something different if you generate a light field which aggregates and acts as a drift in the system thereby disturbing the overall behavior of the system so individual trajectories here are much larger and as you see in the different realizations and different kinds and yet the coherent picture appears so this is a light intensity profile we're using the experiment and you see the one spread experiment was done for each landscape and and thereby you have a a significant way of starting the system what is the punch line of these relevant to our cases that the mean front propagation that we can compute theoretically through the Keller-Siegel framework and the experimental one in fact they show that the speed has a new actor it is the autocorrelation length of the resource field so heterogeneity appears in the system and affects the propagation of the speed a propagation of the traveling wave and these autocorrelation does slow the speed as you've seen here in the case you're having there is which have a mean invasion speed computer directly without any light field included the square root of 2D system of that which is the slope of the system here so heterogeneity whether given intrinsic to the structure of the river network whose architecture recurrent characters were long studying and which I have introduced in my first class have an effect and to jump into something connected like believe it or not I'm going to a giant of a field the late Ilka Hansky craft food prize few years back salva died too early and whom whom we miss sorely in fact had invented a very important quantity that was key to metapropolation analysis called the metapropolation capacity of a fragmented landscape that easy found out that the ecological measure super sound and quite well established experimentally from fieldwork and theoretical work that characterizes the suitability of the substrate to effect of heterogeneity in his case was a fragmentation of a landscape and there was a main motivation for the ecological report so technically metapropolation capacity is the leading eigenvalue of an appropriate landscape matrix which I'll explain in a minute and essentially the punchline was that a species is created to persist in a landscape which is the matrix for ecological interaction in my case I would put this in in the other in the case of the other network is larger than some threshold value which is developed by properties of a species and which is essentially the capability to disperse and the capability to to to the dispersion capability and the whatever the mortality and reproduction rate so metapropolation capacity is a number that can be conveniently be used to rank landscapes or for ecological interactions in terms of their capacity to support metapropolations metapropolation of course is another one of those bold statements not unlike the neutral model of biodiversity in that metapropolation essentially ignore interspecific interactions among species that means there's no predator pre-relations etc so essentially it is intrinsic capacity of one species that determines its ability to survive on this I shall not spend issues time because of the limited time I have that we have discussed at length for the implications so what happens is that the key to that is essentially a focus on the probability to be at point I in space in time of a focus species the one whose ability to survive your study which is the probability of being present in patch I at time t which is a balance of colonization excision forces and we just kill out the interspecific interactions over which ecologists spent lifetimes in the past there was a greatness of one of the most in fact gifted field ecologist of our times Ilka Hanskin said so the parties that follow me rate of change of this probability in time is essentially a balance of the probability of being colonized by different places at distance at places site j at distance d i j measured along the system and and that multiplies the colonization rate and then you have the extinction rate which is birth minus death ratios in the two so the ratio e over c is what determines the teacher that you're having there now the main result is that what is lambda and the the the eigenvalue of the system well it's a landscape it's a the maximum eigenvalue of a matrix which is a landscape matrix which essentially this is the dispersal ability of a species and this is the distance that you're having the system itself so it's a it's a positive matrix it's a reducible matrix in fact so ground for various theorem applies in second and that's what he found out and if it turns out that if the maximum eigenvalue of this particular landscape matrix which is very easy to calculate for landscape like ours like I show in a minute the landscape matrix for us it's easier and more constraining than in say a savanna ecosystem like we had seen the other day so what happens is the following if you take a path along which an open channel network develop so we started from an any particular system and we essentially streamline a simplified network by assuming that the system would tends to minimize its energy dissipation approximating the structures that and the statistics that the river network inevitably has well like it or not it's some sort of an unintended consequences within decreasing energy the second line of energy you have as a bonus in fact the best better viability in abstract terms generated by the river network so the ecosystem in a sense benefits from the physical process that determines the substrate that we are talking about and then well there are other issues but I realize it's I've been chatting too much so I'll be skipping fields but anyways this is a significant result in terms of the probability and if you take several realizations of the same network etc. you have a band a narrow band and if you have like different kind of mean field versus OCN same you see that this is a significant result from a statistical view point so I've the last 10 minutes of my talk will be devoted to can we use the same concept to study river networks and biodiversity and I will leave to the third and last class how this in fact is key to evaluate how pathogens spread along river networks psychological corridors thereby propagating deadly disease deadly or chronic disease so what am I seeing is that in a river network is a set of is an oriented graph my my notes and edges edges are physical edges obtained offline by digital terrain maps everything last time you do have reactions which would be physical chemical or or biological reactions in this case biological ecological reactions and essentially the links are what act as transport models between nodes and this applies to individuals this applies to species or to populations so essentially we can talk about rather than meta relations meta communities and the first example I want to show it's a paper in which we have Marina and I care very much for Lorenzo Mai is the first author on the meta population persistence in in river networks what happens is a following so the to study meta population dynamics can be done by if by looking at the system in which you may have like species that live on a substrate of this kind which eventually might in fact get off a move off of the network itself so the species we are talking about the meta population and that somewhat forces what's gonna happen with more complex compartmental models for a number of different phenomena would be like you have like in every node you have two ecologically distinct developmental stages so we split the population living in the node which is a rich if you please and in the young non reproductive individuals why and the adult reproductive individuals movement from one node to the other can occur for different pathways but along the structure of the network either along the same network if you are a fish or even overland if you are an amphibian for instance in certain stages of your life you take into account local demographic processes birth growth and death and the dispersal dynamics in each node of a river network which again using the full constraining power of the system of this kind now this is important to explain what's called the drift paradox because to explain the long term dispersal in system in which the drift could be very significant in fact there's there's something which ecologically puzzled ecology for a long time for instance to explain the long term persistence of populations for instance empirical documented examples came in for instance for Scandinavian freshwater ecosystems insect species compensate the larval drift that is transported passively along the river network with upstream directed flight of adults prior to OV position and it's called the Mueller's colonization cycle otherwise an excess production hypothesis has been forth in which drifting organisms would be exceeding the balance of demography of a local scale which implicitly assume that the drift essentially represents an extra mortality in terms of these kinds but anyways all I'm saying is that what is new here is not the ecology which is well known but is the fact that you're using a subset of this kind and to give an example of how coupled are these in this case two for every node at node i you have the young the number of young non reproductive individuals and the number of adult reproductive individuals they had a certain mortality term possibly density dependent you have them in what you have the most important you have the probability of dispersing from any j site so you may have adults at site i can be taken because of the connectivity matrix for any site j by simple proportion and transport that you're having in the system whereas in the case of the adults in addition to that you have reproduction terms it's not important actually how you do that of the specificity of that which is very well known system that we having in this place here but what I'm showing you is the fact that how do you study those systems is two equations per n nodes of a river network so you can actually assume that those matrices dispersal matrices are probability which depends on the connectivity structure something which you assign and give offline and which you can study by studying the stability of the the metafolation persistence is actually related to its stability of a population that is if a state x0 is stable the population cannot persist in any of the river network nodes if it is unstable Juvenile and adult abundances grow and they grow the multiviralation persistence so you can study the global equilibrium of those matrices which is essentially it's a super Jacobian in which you have two n by two n matrices that can be done exactly and calculate the survival and the non survival of the different ranges of the system and what for instance one particular case which was particular interest was the study of salamanders for which you had in critical data from I think it was in Virginia someplace in which you can relate the conditions for metafolation to persist and metafolation not to persist counting on the fact that the juvenile and the adults have different ability to jump off the river network itself I am almost done regretfully so I will compensate next thing but give me three minutes to complete with another example the invasion of the zebra muscle which paves the way of something which I'll be showing you Friday that is what you're seeing here it was the zebra muscle was by accident introduced in the Mississippi Missouri river basin by introduction it was taken from the veligers that are the the larvae were introduced in ballast water of cargo ships that came in from Europe where it was native and it started spreading through the system reaching the DTAs even invaded hydrologic units in the system have been growing dramatically over the years so up to a point what is totally remarkable about this is the fact that at a certain point it didn't even simply diffuse or disperse downstream all of a sudden you started having places where you started having flare ups of invasions in completely unrelated places far away from it that introduces the fact that we can study river networks fairly well in those systems and essentially the local reaction equation is the local age growth model in four stages in which a lover production transport of larvae is a passive larvae are small the veligers are essentially passive scholars so they diffuse and they are vetted to the river network sector or the river network but most important you have to introduce what is called modernly a multiplex network why because in those recreational ponds in which you put some boats what happens is that if a guy picks up a boat doesn't empty its ballast water properly put it on the trailer and brings a thousand kilometers away then in as much as COVID-19 spread through human mobility so did the spread of this system here so the long story short on which I shall stop in time regretting that I have dedicated too much time to the early phases but I will be saving what I miss from here next Friday so here is the computed and simulated values in like in this case that is a measure that is a computed as you're seeing here only by our ability to introduce those long range flights generated in the injection by a different network which is not related to the topology of a river network but they're related to the road infrastructure in a certain probability of making a distance in the system you are able to calculate one of the most devastating because zebra mussel grew at the level of density that generated significant damage not only ecological because it displaced every other native species of that kind but it also generated clogging of hydropower plants production and the likes and this is an example of how in fact biological invasions reach the place yeah this is a good point to stop in here so the point line is that and I will be taking on the next class to see how this in fact becomes the essence for studying for the same environmental matrix and possibly multiplex network in which pathogens disperse waterborne I mean through the network and human mobility spreads the disease how in fact they can use oriented graphs no other reactions and hydrologic distract the hydraulic transport to tackle individuosi metac relations o metac communities now I'm ready for your questions great thanks a lot Andrea for the very very nice lecture so we have time for questions so please if you want to ask one you can use the raise and feature or you can type it in the in the chat yes please on the um hey monday can you hear me yes i do yeah okay yeah can you hear me sir I do okay um okay so I I'm looking at the last the last points you presented in this slide is it or do I join the class not to uh at the point to start it is it that the the network of of of the the model that you brought was validated before you presented the result of the data or is it a a direct impulse without validation but it's a good question actually monday what happens is the following okay there are networks and networks in this case if you're talking about the transport of it's in you see the main point right you have nodes in which you have physical chemical biological reactions right and then and you have transport along the network structure now if you're talking about the dispersal of the organism in the larval form for instance they are small seeds they behave like a passive scalar so they diffuse and they are affected and diffuse along the river network right but if you get a bunch of them and you take them from a node and you transport that completely artificial into another node like you do with the with the veligas that get trapped in the ballast water unclean of the boat then is the mechanism that can explain why you can actually have a flare up of a of a of a proliferation of the zebra muscle hundreds of kilometers away without any in between why this is interesting because it's like receding the infection if you want or the density of a population somewhere else now in one case the network is essentially the physical substrate which we know directly right because we that's unavoidable digital terrain maps allow you to extract those connectivity matrices that is how I know to connect to any other node in the system and you can actually distinguish whether you for instance you can also go you have a preferential transport downstream but you may even go upstream with a probability which difference we had a bias transport I'll be talking about that Friday but in the case of it of a mobility network generated by the other we have tried something else so in that case what we did was to evaluate the kernel of dispersion but it was calibrated on the data actually which is the displacement of commercial boats into the place through some proxy which is data on the number of boats parked in different positions when Mississippi, Missouri were basing so when whereas in the case of a river network it's a given the connectivity matrix is given once and for all because it's extracted objectively it's added remotely and objectively manipulated if you have a case that you may have to make assumptions when we'll be talking about human mobility for instance to spread disease which is not unlike these you will have two possibilities one assume one of the models for like normally used like gravity models radiation models or you may use cellular phones and track individual mobility of large numbers now you can now in that case you may wonder whether the use of a telephone in certain places where color spreads is socially biased or not if you're looking only in certain segments of a population but mind you my experience is that's not the case that's actually socially neutral the ownership of a phone even in the most dilapidated place on earth and I've seen it in my eyes and I'll tell you next week okay thank you so much for the explanation but I will appeal if there's maybe a kind of a reference material then I would like to to read more in two minutes I'd be delighted well actually if I'm not I'm not selling anything because of the money we go to Cambridge University Press but the book that I just published that with Marino Gatto in Ignacio Rodriguesi Turbe it's called the river networks as ecological corridors species population pathogens had everything where we have done the past 15 years on the subject and and a reference hopefully a good review of the literature as well thank you Monday great thanks a lot for the question and the answer so we have time for more questions if any also on YouTube you can type the question if you want otherwise I think we are actually on time so otherwise thanks again very much Andrea for the for the lectures and looking forward for the next one thanks a lot if you want to listen to Marino can I stay on here? absolutely yes so what we are going to do now is that we are taking a a break before the before Marino Gatto's last lecture and we are going to be splitted in breakout rooms so feel free to use them to chat freely with other participants if you have a version of Zoom which is five or higher you can switch room to chat with whoever you want so we'll be back at 2.30 for Marino Gatto last lecture thank you very much great I think we are almost ready to start again so we are just to close the breakout rooms and when people join we'll start with Marino Gatto's last lecture and please see eccoci eravamo la breakout room ma siamo usciti adesso ho chiuso la breakout room mi senti? grazie grazie great so I think that great I think that in a few seconds people should join back to the main room and we can start again so just the usual announcements if you just joined us so if you are following from youtube you can ask question in the chat I'm sure you have heard this many times and if you are on Zoom you can either use the raise and button or ask a question in the chat and I read it for you so I think that everyone is back in the main meeting room so please Marino if you can share your screen you can start thank you very much ok good afternoon to everybody so let me share my screen and then optimize screen view here it is ok so this is my third lectures ok and the topic I'm going to to speak today is unfortunately fashionable and that is covid-19 again I stress that in a way my lectures and professor Andrea Rinaldo's lectures are coordinated in a way but covid-19 however is not included in the book that Andrea Rinaldo has been advertising and I have also been advertising ok so very briefly the summary first I will talk about the context and the main epidemiological characteristics of covid-19 then I will talk about our model that we developed for Italy which is a spatially explicit model and then I will illustrate the model results up to the end of March 2020 which was the first month well the first one month and a half in Italy and then I will show what might happen after the lockdown might have happened after the lockdown and I will talk about the scenarios and the possible containment measures and then a few conclusions now this is something I have already shown to you I want to stress again that if you look at the statistics of communicable diseases infectious diseases around the world you see that the share of infectious diseases has been shrinking in a way along years of course the number of deaths have been increasing because the population of the world has been increasing but if we look at the share the red share of communicable and also maternal neonatal and nutritional diseases the share has been shrinking and so around 2017 and probably the same for 2019 about 8 million and now if you look at the statistics of covid-19 it is about 1.55 million deaths you know that it is possible to link to Johns Hopkins University site and day by day you see the said statistic global cases which is probably a large underestimation of the real global cases global death is also possible possibly an underestimation but not such a large underestimation at the global cases and so we are now to a figure of more than 1.5 million deaths which is really really a very large share and which mean that covid-19 is really a very important disease in a way it is not like other infectious diseases that might ravage the globe every year like influenza it is much more important now let me stress that what I am going to present is actually a teamwork a team of people with whom I've been working for a long time and who are located in different universities and in different places although I must admit that they share one common feature they are Italian anyway now let me connect to the lecture by Professor Rinaldi on next Thursday 11th if I'm not wrong to say that we have a lot of experience with spatial temporal dynamics and that is something that Professor Rinaldi already stressed in the past lecture and so for instance he will show the progress of cholera and Haiti and you see data on the left and model on the right just to let you understand that there is a spatial signature in many many many cases that is something that you should account for and that is common to many diseases so for instance the Spanish flu probably you may remember that the Spanish flu was probably the largest pandemic I mean with the exception of COVID-19 I mean if we go back in time the Spanish flu claimed more lives than the first world war and actually my grandfather was detained in a concentration camp in in Austria in Kathnau because he was an Italian citizen and unfortunately he was in the Austrian Hungarian Empire in 1915 so he was sent a concentration camp but he didn't die there because that's a lack of food or anything he died of Spanish flu and if you look for instance at the spread of Spanish flu in the United States you see how fast oh I'm sorry how fast it was so it started in the eastern coast and then very rapidly in a few weeks spread to the whole United States now let me describe the characteristics of the COVID-19 first of all the pathogen of COVID-19 is SARS-CoV-2 so why two because it is at the same family that actually caused the SARS so it is an RNA virus it has a crown-like appearance it's actually due to spikes on the surface so corona in Latin and in Italian means crown so it's a beta coronavirus like SARS-CoV-1 and again to stress the ecological importance of what I am going to say it is a zoonosis and the host host of several species of bats and rodents and particularly pangolin maybe other rodents and what are the main characteristics of the disease now first of all the main way the virus spreads is by respiratory droplets among people who are in close contact with each other now let me show something now this is an example of the aerosol emission when you breathe now in this case this is a study which was conducted together with a very famous orchestra you know I'm very fond of classical music sorry for that and so the symphony orchestra there's Bayerian runesongs and here you see a way for visualizing the droplets and the aerosol that one can emit when singing or playing an instrument now more or less you can get the virus via contaminated surfaces but it is possible so you I should be careful about that now another main characteristic that we know now and now let me provide a few technical terms which are shown here so if you consider one individual who gets infected he at first is not infectious this is called the latent period you may remember when we described micro parasitic models that we were also talking about susceptible exposed infected recovered exposed latent or exposed just the same same same term which means the same thing that you are infected but not infectious then at a certain point you will become infectious but that does not mean that you show symptoms symptoms actually in general show after a while now that oops I'm sorry this period is called incubation and then you become sorry then you become symptomatic now of course when you're infectious you can infect someone else and again the infectee will have a latent time incubate the disease becomes symptomatic and the cell is going to become infectious now we call generation time the interval of time between the moment when the infecter was infected and the moment when the infectee was infected we call serial interval instead the time between when the infectee was became infectious and when the infected becomes sorry the time when the infectee gets symptomatic and the time when the infected gets symptomatic okay this called the serial interval now the generation time the average generation time which under some independence hypothesis is equal to the serial interval is about six seven days so it means that that the the scale the time scale for the infection is the order of one week that is important so from one infecting and one infected okay in the average there's a week into one week interval six days six seven days as probably all of you know the asymptomatic fraction is quite high larger than 50% it may vary between countries you probably know that it is more likely that old people show symptoms than than young people so the asymptomatic fraction might be higher in countries where there are a lot of young people and let's say smaller in countries like Italy with a lot of old people and one important message is that the maximum infectiousness occurs during the pre symptom transmission so that little time when you end to be latent and you are already infectious but you've not developed symptoms ok e' che è chiaro da molti da molti studi quindi è circa 5 giorni dopo essere infecti poi cosa sta succedendo? che il pre-symptomatico in realtà deve essere che noi dovremmo chiamare questo periodo postlatento perché il pre-symptomatico alcuni di loro non avranno sviluppato il symptom quindi rimane il semptomatico mentre alcuni avranno semptomatico questo è importante per cosa vorrei dire dopo un'altra cosa è la mortalia mortalia è circa 1-2% perché si dipende su una condizione su un sistema senso su un fatto che c'erano molte persone e ovviamente le altre persone sono più mortali e così e così ma noi sappiamo che più o meno è circa 1-2% e questo può essere compare per esempio per mortali dall'influenza è qualcosa di 10 volte più grande quindi ci sono due combinazioni che rendono una Covid-19 in un modo perfetto epidemica l'epidemia che, infortunatamente, tutti i epidemiologi e gli ecologi che stanno parlando un grande fraccio di semptomatici persone e la mortalia che non è solo come per farle a loro non molto rispondere diciamo ma su un'altra mano non è così come per esempio Abala per l'influenza dell'influenza l'influenza e l'influenza dell'influenza perché puoi ricordare che quando introduziono anche il semplice il modello mi ha stressato che la fraccia di che la prevenza dell'influenza e anche l'influenza erano in un modo diminuendo con la mortalia la mortalia di l'influenza quindi e è una sorta un'intermedia mortalia diciamo una grande fraccia di semptomatici rende questa pandemia così aggressiva e poi neanche c'è una signazione e questo disease a livello globale quindi andrò a mostrare te questa mattina e questo video che mostra come è stato inizia in China e poi in Francia e in Germania e poi in Italia e poi all'Europa e poi ovviamente in strada per le Filippi e Korea e poi arriva i United States e poi arriva Canada South America Africa Australia e so on and so on so again the it is clear that the reason very large important part played by space by how the disease develops in time and space and therefore it is very important to take into account space and in fact that was which in a way hit our interest ok because we we have been starting many diseases and in space and time and when the disease rich Italy and that was at the beginning of January it is now clear that it was January maybe even December but that was not clear until the end of February and if you look at again at the development the spatial temporal development of the Italian epidemic that this is March March 15 March 16 March 17 and so on and so on so there is a clear spatial signature of the epidemic and that's why we decided to work as soon as we can very rapidly on building a model for the spatial temporal spread of COVID-19 in Italy and that actually what then materialize in this paper which appeared on the end of the end of April maybe on the precision National Academy of Sciences and that I'm going to to illustrate to you now first of all the epidemiological compartments now the subscript I indicates the nodes the nodes of the network that we're going to do to consider and then we have the compartments of susceptible the compartment of exposed but then we had to introduce with respect to the usual single compartment of in infectious three compartments because we have the pre-syntomatic infectious the symptomatic infectious and the asymptomatic or mindless symptomatic infectious in each location I and then of course if you are infectious and symptomatic you might be hospitalized or if you're not too symptomatic let's say you might be quarantined or maybe you might die without even being hospitalized or you are hospitalized and you might die or you might recover and if you are asymptomatic usually you recover okay so that the basic let's say engine of our model with all these compartments and here are the equations and basically the again the core of this local model in each node is actually the force of infection now first of all we can assume frequency dependent contact rates now it should be clear that in an unpopulated area then you should assume density dependent contact rate but that's not the case of Italy wherever you are in Italy basically you are in a populated area and therefore it is reasonable to think that every day you cannot have more than a certain number of contacts so usually people about say 10, 12 closed contacts every day so you can assume frequency dependent contact rates now so at the denominator you have the sum of susceptibles exposed etc etc so the total number of individuals and then at the numerator you have who the people who can infect so the pre-syntomatic the symptomatic infection and the asymptomatic infections and they might have a different transmission rate different transmission rate because for instance we were expecting from other studies that the transmission rate of pre-syntomatic might be higher than those of asymptomatic and or symptomatic infections so that really the core now if you go to the spatially explicit model now you have the local model at each note say the province of Milan where leave or the province of Padua professor Ronaldo leaves but then these notes are connected of course they are connected by mobility there is mobility within each note and there is mobility connecting the different notes so now the first of the infection when you go to the spatially explicit model is much more complicated in a way you have to now introduce those mobility matrices I was talking about when I showed to you the model of schistosomiasis in Senegal and so the probability that individuals who are susceptibles pre-syntomatic etc etc move from side i to side j and contact individual people from from side i who usually live inside i will contact individual inside j and so on and so on we even consider the probability that one individual living inside i and one individual living inside j will actually meet inside k and one is infected the other one is not infected and so you have a close contact and the one who is not infected becomes infected ok and then of course we have the again this transmission rate which depend on the stage pre-syntomatic or symptomatic infections or symptomatic infections and in principle might also depend on the side because you might think that there are different behaviors in different sides and so on and so on so this is the general structure of the spatially explicit model now for this spatially explicit model you can calculate that basic index which is very much utilized and that i have introduced to you for very simple model which is the generalized the reproduction number which is however called now generalized because it is generalized to a model with a network so again let's consider the initial phase with no containment enforced so we can calculate the basic reproduction number and to do that we can as usual start from the disease free equilibrium in the spatial model the no infection infection anywhere in all over Italy and then we introduce a little bit of infection initially the susceptible prevalence is one and then again at r not equal to one we have a transcritical bifurcation so you can run a bifurcation analysis on the model and so you you can find and on a transcritical bifurcation which occurs at r not equal to one another equivalent way is to use the next generation matrix which was introduced by dickman esterbeck and man and one can show that r not is the spectral radius of the next generation matrix and in practice this generate next generation matrix is actually built from the connection of the matrix representing the contact probabilities and then you can recognize some specific times one divided delta p is the residence time in the precentomatic compartment one divided by eta plus gamma a plus alpha is the residence in the symptomatic infectious and one divided by gamma a is the residence time in the asymptomatic so in this case because you can get infected from precentomatic from symptomatic or from asymptomatic actually you have to sum this matrices find the spectral radius of the sum of those matrices and that will provide the generalized reproduction number actually when you calculate when you do the do the stability condition of the disease free equilibrium and you go through the bifurcation analysis you are considering the Jacobian of that complicated system at the disease free equilibrium and the dominant tag and value of the Jacobian is the initial exponential increase rate on the other hand elements of the next generation matrix describe the main roots of infection and the dominant eigenvector of the Jacobian which is the unstable manifold of the disease free equilibrium provides the geographic distribution the initial geographic distribution of infections and in fact this is the way that we got this beautiful picture and you know which is quarta z by one by two of our co-authors and equal but two terms different amicably which represent the main pathway the main roots of infection in Italy at the beginning of the infection and you can see Milano, Turin, Rome here okay now of course we have to calibrate the model and here you see an logarithmic scale semi logarithmic scale days and first patient in Codogno in February 19 and then the development of the disease up to the end of March and these are the the data that we've been using to calibrate the model now the parameter estimation procedure I want to devote say one or two minutes to that first of all when you have to calibrate the model which is quite complicated quite complex there is always a trade-off between presumably on one hand and realism so you cannot use too many parameters too few parameters so first of all we add mobility from an international census at the municipal level but we upscale mobility to the second administrative level 107 provinces and metropolitan areas the epidemiological parameters are not space dependent and these first are in the first paper that we that we wrote the transmission parameters beta pre-syntomatic beta infectious symptomatic and beta asymptomatic are not space dependent and then of course we had to take into account that there were containment measures so we should expect that these transmission rates would decrease so we consider a sharp decrease within two days after the measurements announced announced on February 24 and March 8 and so we consider also the step reduction um then okay we had red areas and we're going to do that of course there was a reduced fraction of traveling people how could we account for that through uh mobile applications data that were collected by some colleagues from uh voluntary from voluntary mobile data collection and then there's an important thing true spread of the disease from some initial foci but there was not only one initial focus although it was clear that the main infection for foci were located in northern in the northern part of Italy but however there were other foci so we had to estimate also and away the initial condition in the in the different in each in each province and also the starting time of the epidemic also we made another decision the number of cases is not reliable what is more reliable in terms of statistics is the number of hospitalized people unfortunately the number of deaths and the number of patients discharged from the hospitals so we used the Bayesian framework we gave priors for the parameters we sample the posterior parameter distribution via Marco chain Monte Carlo algorithm and uh okay technically we use the likelihood according to negative binomial distribution and okay sorry and here are the results the results that I show for the whole Italy these are the hospitalized people these the number of deaths but also for the hardest it regions now should be clear that uh these the results are shown with reference to regions for convenience only not because regions are isolated from one another because that's you know that's a common problem with many other problems that they focus on the data of that region as if that region were disconnected from other regions and the same is true for countries at the global level but okay that's another thing that we should discuss and here you see a pictorial representation of the spread these are the data these are the calibrated model at the level of the second administrative level and here is a fancy projection you know made a simulation at the municipal level but of course he okay it is just in a way a fancy simulation now one important thing of having a model that it is possible to make retrospective scenario because one question that we asked were the containment measures effective and the answer is yes and you can estimate how effective they were by using the model because suppose for instance that no restriction had been taken okay without any restriction so no change of the transmission race no change in the mobility and of course no change in the people's behavior that would have been the number of uh no sorry sorry let me first describe hospitalized cases that would have been the number of hospitalized cases and then you can consider another scenario February restriction but no March restriction and that is the second scenario now these are the reality so we had about 40 000 hospitalized people a huge number sufficient enough to send all our hospitals into big problem but the number of averted hospitalized cases were about 180 000 so we would have had so much more without any containment measures another thing that you can estimate is the people who were infected because the people that you measured the number of cases the number of cases that you saw and that John's hopping inside of course these are the cases that are discovered people taking a swab and the swab being positive or let's say oh they using these antigen antigen tests ok I'm sorry this is happening again ok thank you ok so you can estimate how many people were infected and possibly infectious at the end of March where there were they were about 700 000 10 times more than the official number of cases but if the containment measures had not been taken then the accumulated number of infected cases would have probably run to six million cases so you see now that a model is really useful in this way because you can also estimate variables that are not measured directly so main epidemiological result we calculated basic reproduction number it is about 3.60 well very similar 3 let's say 3 3.2 2.8 something like that that is a common number all over the work we add a confirmation that the pre-syntomatic are extremely infectious because we estimated the beta the transmission range of the pre-syntomatic and in order to fit the Italian data that beta had to be much larger than the transmission rate of the asymptomatic or the symptomatic infectious however there was a large negative correlation between the fraction of asymptomatic cases and their transmission rate ok so that's a problem containment measures had reduced the transmission rates by 45 percent not enough at that time to make the reproduction numbers smaller than one that will be achieved later and again the model allowed the estimation of in apparent infections and the prevalence of susceptible prevalence of infected I'm sorry then after after that we went on Italy came out of the lockdown on May 3 so we started thinking what is going to happen after May 3 and of course everywhere in Italy too there was a concern about the economic consequences of enforcing a lockdown and of course the lockdown is a mess in the social and the economic terms so we wanted to understand what might happen after the end of lockdown now of course the first thing we had to do we had to recalibrate the model between March 25 and May 3 the end of the lockdown which we did and so we updated the model and here again you see at the end of sorry at the end of February beginning of May the data the cumulative hospitalization and the model and then we also calculated the transmission reduction the transmission what down to say 0.30 0.4 depending on the regions and and promises okay and in fact you see you can see here not about us again I think you should stop annotate somewhere annotate let me stop annotate where is it I think if you have the annotation you can do clear and it should clear because I don't see my mouse now there's something wrong this light looks nice though so there is no sign or okay in fact that I can now move my mouse to annotate well anyway let me go on anyway so here you see the calibration and then we try to have future scenarios so first of all we showed that was a further decrease of transmission after March 25 the transmission rate fortunately and then we tried to estimate some scenarios so the blue line is the baseline scenario suppose that the transmission rate after the end of lockdown stays the same as it was during the lockdown or green and purple represents scenarios with transmission rates that increase by 20% and 40% now why should they increase well because of course mobility increases the end of the lockdown on the other hand you know mobility increase might be effectively being mitigated by personal protective equipment now people are free to go around but they wear masks and so on and so on and so let me show how effective that might be so yeah personal protective equipment oh i can show again i can show again i can show again the area say the researchers two and a half meter distance to the neighbor to the front and one and a half meter to the side best with Plexiglas the gap in between and constantly blowing masks provide additional security and these gates there's a surgical mask and a barrier okay then of course you might consider social and physical distancing then again you might implement again a lockdown that's more a large spatial scale and then one important containment measure of course testing tracing quarantine hospitalization isolation of people and then of course you might have a combined implementation of all these containment so so you know we sent the paper we had reviews and of course the reviewers first of all they're saying well well you all know that review takes time and first of all they say oh it might be nice if you could update the model so to answer the reviewer we updated the model up to June 15 and from here you can see that Italy was actually quite let me say virtues that in a way Italy although the lockdown had been relieved were following the blue line so the baseline scenario as if the transmission rate remained the same as they were during the lockdown while with the exception maybe of Lombardy the big the highest populated region Italy where by the way they happened to leave we also ran a sensitivity analysis to understand whether the fraction of asymptomatic which is very unclear in a way was say 10% 25% 50% sorry the fraction of symptomatic in this case 10% 25% in 50 50% that famous study and and boy of gamma blue and it turned out that 25% is the most likely scenario because that was confirmed by the estimated the zero prevalence of the infected a statistic which was available in Italy on July 15 and for instance came out at Lombardia 7.5% July 15 had been infected with more or less you know support the green line the 25% that we were considering as a possible scenario of course another containment measure that you can do is testing and tracing on a large scale and again what we did we estimated the for instance the daily percentage of exposed and pre symptomatic people that should be isolated now it's very difficult to isolate exposed of course because many cases when you're exposed the swap test is not yet positive so it probably more realistic to consider the infection generated by symptomatic cases and trying to trace to test the symptomatic case then look at the context of the symptomatic cases positive symptomatic cases and then look at the context the context and so on okay just to finish one of the reviewers asked are there possible scenarios after June 15 and okay so and what about the problem suppose that there is immunity loss so within scenarios when in September 1st the transmission rate were back to the initial transmission rate March and with and without immunity loss I would say that it is now clear that our immunity lost at least for a few months not clear yet that there's not enough evidence and hopefully we can hope that it is long so these were the scenarios that we ran let's say without immunity loss the one you see here for Italy for Lombardy this is what is going on we were assuming that there would have been a new lockdown on October 1st instead the lockdown has been implemented one month later and that is what is going on now to conclude so a spatial model at least according to our opinion including mobility is fundamental if we want to project in real time the epidemic trajectories at least at the very beginning when it might be useful to implement immediately containment measures in order to stop the spread of the disease in these ways it is possible to estimate the demand for critical care the hospitalization that you should expect in order to avoid the big problem for hospital to estimate how much testing and tracing we should do on the other hand integrated models are also necessary age structure should be included social counter structure should be included many other models that have been developed around the world and even in Italy also consider age structure and social counter structure but unfortunately I must say that they do not include space unfortunately lack of available data is a problem they are not made public so in some cases and in our case too there was insufficient spatial granularity for some compartments and if we look at the problem really at the global scale then it is clear that there should be nested models from global I mean all the world like for the problem climate change for instance where you have global modeling and then nested you have country models and then nested regional model so that you can fine grain the strategies because these strategies might differ from location to location and one thing that is unfortunately common to many countries around the world that although the data were made available in relatively short time and you have seen the Johns Hopkins University site and so on and so on however natually and in other countries most of the data were not made available to the scientific community the scientific community were requesting a wide availability of these data in many cases these data only part I would say a small part of these data of course anonymized data were made available and that unfortunately well I'm sorry it takes such a long time I'm sorry and I stop here no I mean it was very interesting there is no way to no reason to be sorry thank you very much Marina for the the very nice lecture so we have time for questions so there are a few in the chat so in one from Miguel Rodriguez in the local model is the parameter H which I think represent the hospitalized individuals capped by the maximum hospital capacity in each region or is it assumed to be unlimited? no I don't remember H no H no I don't remember what I indicated with H in the local model let me see oh the hospitalized ok the hospitalized people no no no no ok the hospitalized people or a varying number it is number of people who are in the hospital at a given time yes so of course it was capped by the hospital capacity by definition in a way although you know ah the people were hospitalized and and put everywhere everywhere though it was real amass and and it is now clear that at that time many people were actually dying even if they were not without even being taken to critical care units yeah I guess this question alludes to the to the fact that the admission for instance to ICUs depends on the occupancy of ICUs so the criteria change with with the capacity then there is a question by Silvia asking what is the interpretation of the large negative correlation between the fraction of asymptomatic cases and their transmission rate yes okay the interpretation as follows that if the number of symptomatic if the asymptomatic case is large very large then the transmission rate can be in a way lower because it is the product let's say of the beta a and the transmission rate of a and the number of asymptomatic because that will be let me say the viral load that goes into infecting the susceptible people now I'm you know a good thing of by easy and of by easy and not only of by easy and sorry statistical approach is that you cannot only estimate the the confidence of each parameter but you can also look at the correlation matrix and in fact that you shouldn't if you inspect the correlation matrix then you find out that some parameters are not so much correlated without parameters which are good things means that they are estimated that they are well estimated if you see negative or positive correlation that means that the model is not so parsimonious in a way e' that's why we were concerned and that's why we made a sensitivity analysis in that in the second paper with respect to the fraction of symptomatic and the fraction of asymptomatic people and then the serological test confirm that the fraction 25 percent of symptomatic and 75 percent of asymptomatic or mildly symptomatic so that they don't even care they don't don't go to the hospital they don't take to up that thing and so on is reasonable at least for Italy great so there is a question about vaccines good news of course yes so after in four two thousand and one and we hope that we are doing that and Ronaldo the thing is is still joining maybe the no no longer I am here I'm here I'm here in a marina oh I you're here here no the the question yeah so the question is I think that yes yes yes we are working on that yes so can this model be used to optimize the special distribution of vaccination campaign or do you think that it is of small importance relative to the demographic groups well well of course there are some things that that pertain to the common sense and actually these are the rules that in a way would be enforced by the european union to distribute the vaccine you first vaccinate people in the hospitals medical doctors nurses and so on so on and people who are at grayscott people in would say retirement houses and so on and so on and but then of course you can anyway optimize the vaccination campaign given some constraints so the constraints are say the rules of the european union happens for Europe now the or the number of the batch of vaccines and of course the number of vaccines well of course that's what the idea if i may oh well let's say 200 for Europe something like that no but the the question is whether even a batch of vaccines and set of rules finding out the best distribution in space and time of that batch is still and i think which are precisely what we are working on now and it's constantly not trivial it's not trivial great so there is a question by Luca that says that well he really enjoyed your talk and he found very interesting the Bayesian approach and he's asking according to your model which parameters are the one that changed the model results the most well certainly uh let's say the the beta of the pre-syntomatic make a lot of difference you know i i told you that there is evidence that is experimental evidence that the pre-syntomatic are very much infections and that's why we decided to include a compartment of pre-syntomatic but it's not our invasion there were other people who had done that and but we found that that was fundamental but that we didn't fix the transmission rate of the pre-syntomatic and it naturally came out from calibration that the beta of the pre-syntomatic must be quite large compared to the beta of the asymptomatic and the infected symptomatic and again it came out of calibration that the other two betas apart from that problem of the negative correlation of the beta of the asymptomatic are similar in terms of the ornamentation which makes sense say even from studies on the viral shedding so viral shedding is actually decreasing after you get the symptoms or if you are asymptomatic and you are followed during the asymptomatic development of your disease viral shedding is similar kind of similar to that of symptomatic but it is the pre-syntomatic with the highest viral shedding more or less of course there are a lot of individual variations I don't know whether I answered Luca whether he answered your question let's see if answering the chat so is there any other question yes okay the question is answered if not well next week we are gonna have actually one of the round tables they should appear soon in the program if they are not yet appeared one of the round tables is gonna be precisely about COVID and the pandemic and Marino Gatto will be one of the panelists so there will be more discussion about it and more possibility to interact about the yes the most important topic of 2020 so thank you very much Marino for the lectures you thank you all and okay so I'll see you at the round table yes so thanks for Andrea for the first thing with us oh thank you and so now we are gonna break in the breakout rooms oh sorry there is actually a question that I didn't see so I don't know if Andrea my marino is still here yeah all day please ask the question if you yeah yeah um please I I am asking um can the generation generalize reproductive number of influence on the generation train of the model function to you is it possible sorry because you know at my age I cannot hear very well he understood that better than he I think it's better if you if you type it because the communication is a little bit disturbed I think you have a 49 probably okay okay I can type it right yeah thank you okay Marino I think at Monday was asking about generalize the production number applied to the covid model yes but again uh you know which kind of question I don't know you want to know more in mathematical terms no I think was a very precise question I think he's typing it so is he using chat yes I think he's typing because I don't think optical connection was possibly writing okay but I can ask a question where oh no it's here please can the generalize okay so uh if the generalize a reproductive number have influence on the generational trend of the model function q the model function q now what is q now wait a minute but I'm sorry but what is q the q function because I don't I thought I didn't remember but I don't I don't see any q function in my my presentation what what do you mean by oh quarantine ah on the generational trend of the model function q the quarantine so it's not function you mean a compartment so this yeah the quarantine compartment no it is just the it is it is just the opposite that the rate at which you quarantine people influences the are not of course because if if you quarantine people so there was now let me share again my free maybe okay so there's a you see a rate at which you quarantine people that are symptomatic infected because at the beginning at least in the first at first the first paper you know at that time Italy was not able to discover any asymptomatic so it was a miracle if they did swap testing for the symptomatic so you are symptomatic and then you can be quarantine because the the medical doctors might decide that you do not have enough symptoms to be hospitalized and so you introduce a rate and of course the larger the rate and the battery in terms of decreasing the are not because these people are quarantine and they are no longer in infectious at least they are not so infectious they are still infectious in the household of course unless they can isolate they have a very large apartment and in the family you can isolate from one another okay but anyway in any case you are constrained to stay at home and so you cannot go around and infect people so the larger that and the battery in terms of the are not so so of course if you quarantine a lot a lot of people that that is a containment measure that will have a positive effect on the reproductive number I hope that was clear okay I see that you're doing so okay so let me stop yes sharing okay man is satisfied is satisfied so thank you very much Marino again for you again for answering all the questions so we will take a five minute break before Jonathan Levin question and if you're following from youtube the for the next lecture is not going to be live stream but we'll be back on youtube for the lecture of Daniel Segre at 5pm so we're going to be splitted in breakout rooms see you in five minutes thanks marino and then then ciao ciao ciao bio okay I'll come back everybody we are live again and this is Antonio Cianani speaking and it's my pleasure to chair the final part of today's session and we have another first of three lectures by Daniel Segre from Boston University and you can read the title by yourself from genome scale to a consistent level modeling of metabolism thank you Daniel thank you Antonio welcome everyone It's a great pleasure to be talking to all of you I'm talking from Boston where it's mid morning I'm going to start by showing you the agenda so we're going to talk today mostly about the logic of the cell how to think about metabolism as a resource allocation problem and we'll see next how to scale from genomes to ecosystems in looking at spatio temporal modeling and the long term history of metabolism so my hope for today is really to convey the way I think it's interesting to think about metabolism and some you know addressing some of these questions hopefully motivating deeper insight and I'll keep this fairly broad so to make sure that everybody's on the same page and we'll address questions such as what is metabolism and why does it matter why should we be interested in starting metabolism and why do we need mathematical models to understand it and we'll see also we'll start seeing why we can think of it as an ecosystem property and I'll start by sharing something that I always find stunning which is that microbes despite being so small can really change the destiny and fate of a planet and in fact they did this is what you see here is the line of the amount of oxygen present in the atmosphere on on our planet throughout the history of life from about 3.8 billion years ago until today and one thing that is clear here is that early on there was no oxygen in the atmosphere barely any and the reason oxygen started rising and becoming what it is today is due to the activity of bacteria such as this one oxygenic cyanobacteria that photosynthesize and in doing so produce oxygen and it is because of this microbes that we have oxygen today and it is because of this microbes that the planet changed completely allowing the rise of multicellular complex systems complex biological systems so it's really important to understand how the metabolism of these organisms can affect such as global scale other systems another reason to study metabolism is of course all of you know is that we are largely made of microbes that is many of the cells in our body are microbial cells this is an estimate from Ron Milo and colleagues showing that there is a little bit more bacterial cells inside and around us than there are our own human cells so these are mostly in our gut but there are microbes everywhere and we're just starting to understand what they're all they play and how they interact with our body and with each other and this also happens largely through metabolism as we'll see and there are many other reasons to be interested in metabolism and by microbial metabolism in particular here are some examples some of you may be familiar with this but there is a lot of interest in trying to understand how microbes can help reduce biofuels and useful molecules from plant biomass how they can affect global biogeochemical cycles whether they can help plant and crop production and there is a rising interest in how microbiomes in the built environment affect human life at different scales now if you zoom inside and try to understand what is it that happens and that makes all of this processes happen this is what it is this is metabolism many of you have seen this chart hanging on the walls of biochemistry labs in metabolism what is one of the things that is beautiful about it is that really spans all scales of biology from individual cells again to metabolism is happening in the biosphere each line in this graph is a chemical reaction we see we'll see some examples soon and this is a global biochemical chart that is the collection of all chemical reactions happening across all living system these are all now available on databases but it's you know one of the questions how do we start understanding the history of this system and how it translates into all these processes that microbes are involved in and I like to think of the hierarchy of biology in a way that maybe is a little bit unusual we tend to think of how you go from molecular reactions this would be a simple chemical reaction and how this builds complex networks of for example the intracellular circuits in a cell and this in terms leads to cellular dynamics growth and division and so on and then scaling up to ecosystem level processes but what I hope we'll appreciate is that actually the this is process can go back and forth and in fact a lot of the modeling we'll discuss we'll have to do with starting from you know determining the molecular structure but also looking at the ecosystem dynamics dynamics and cellular processes and trying to understand based on what we know at the higher level how those functions constrain the way molecular reactions should work so somehow we can navigate this hierarchy back and forth and not necessarily just building upward more and more complex the systems by the way I should say feel free to interrupt any time I don't know if I'll be able to see in the chat but feel free to step in if you have any question or anything is unclear and I have an agenda for today but if we don't cover everything it's fine and I'd rather be happy for everyone to have these concepts clear so one of the questions we'll delve in between today and the next two hours is whether and how we can predict ecological interactions there is this green arrows based on intracellular circuits that is what happens inside each cell and how do we do that I will start by actually stepping back into something much simpler what I like to think of as a sandbox for playing with toy metabolic networks and this is a I think a really useful exercise to start thinking about what is metabolism all about in a much much simplified way but it also raises a lot of interesting questions as you'll see so this is was motivated by work we did a few years ago with Sid Redner Paul Karpiskin Bill Reel and and it's inspired by other work on the string chemistry and this is a artificial chemistry a toy chemistry that is very simple in this case there are just monomers A that can be combined for example in this way 1A plus 2A will be right to 3As you can express it in this way and in general you have this joining processes where a polymer of length i joins a polymer of length j to give rise to the combined polymer and you can write the diagram for the complete chemistry for example all the way to polymers up to length 4 and you can see that this is fairly simple and if you ask at this level for this kind of chemistry question such as what is the most efficient way of producing A4 from A1 this is a pretty straightforward problem which we can solve manually and we want to have no waste of products and just use the minimal number of reactions and the solution of course this logarithmic growth from 2A1s you produce A2 and from 2A4 you produce from 2A2s you produce A4 but this can get complex quite quickly for example if I show you the chemistry up to A7 and ask you how would you produce in an optimal way A7 from A4 this is something that would require a bit of thinking you may have to break A4 into 2A2s and so on and so forth and gradually build up to A7 and I'm pointing you here if anybody's interested in playing with this toy chemistry you can create now there is this tool we generated to produce arbitrary steering chemistry with multiple monomers in different lengths and you can play with this with this kind of scenarios but I want to show you how this connects back to our real metabolism and in order to do so I will start by showing how one can find this optimal pathways for going from any initial molecule to any final molecule in this simplified chemistry and just to give you an example again if you want to go from molecule again of this string chemistry of length for one to molecule of length 6 you can make the two two ones produce three and so on and this is easy but there are some interesting patterns that emerge for example for some of these processes the optimal solution such as going from A the 7 to 8 includes having cycles like this where you need to have as an input a molecule that is generated by the process itself so this is a little bit like an auto catalytic cycle where you need some of the internal molecules in order to bootstrap the activity of the cycle and this is interestingly similar to a cycle that is present in real biochemical networks and this is the representing the carbon backbone of the TCA cycle which we'll see shortly it's a fundamental biochemical pathway present in almost all living systems and it has this interesting property that is very similar you have an input of a certain molecules these are molecules with two carbons but it needs the cyclic behavior in order to sustain itself so this was interesting but let's see you know if we go back to real metabolism why this is so important and how we translate this analysis from our artificial chemistry to real chemistry and of course I don't have time and you know I'm covering here in a superficial way material that could take all courses but I wanna just give you an idea for those that haven't looked at the chemistry in recent times what are the kind of molecules we think about and we'll take into account when looking at metabolism for microbes this is methane and it turns out there are microbes that can survive on methane as the only carbon source and the only energy source which is quite striking given the simplicity of this molecule and this of course contains just carbon and hydrogen but the chemistry of life of course requires a lot more types of atoms this is glucose that includes oxygen a main source of carbon and energy for our own metabolism and many microbial cells and you can add to it nitrogen which of course is the essential on essential atom of all amino acids such as tryptophan and I'm pointing this out because sometimes the chemistry is so complex that one gets easily lost like myself but it's always useful to remember that just by looking even at what atoms are present in different molecules there is a lot you can figure out what are the demands and the needs of the cells in order to produce a certain category of molecules so nitrogen is essential for the production of amino acids and proteins and there are molecules such as ATP that contain phosphorus this is the three phosphate groups that can be hydrolyzed to release energy and in fact ATP as you probably know is a fundamental molecule that stores energy so this is the energy currency of the cell and it's very important because that's how the cell transfers energy between processes and allows the driving of reactions that would be otherwise thermodynamically infeasible so the other atom that I want to point out is sulfur it hasn't appeared yet and it appears in this molecule which is called quinzim A and you'll see by the way similarities this molecule is similar to ATP and there's this chain that contains now a sulfur molecules and here we really pretty much completed the main elements that are essential for living systems there's of course more there are metals and so on but by and large these are some of the atoms one worries about when thinking about basic metabolism e quinzim A è un cofattore che è usato per trasferire i gruppi tra le reazioni e è, insieme, una molte molte molte attraverso il sistema e avremo un diverso scalo per mostrare un diverso tipo di molte molte questo è un protein di una grande molte molte enzyme complex chiamato ATP synthase questo è una macinata ed io sono sempre amato quando vedo questo è un molte molte che è fatto di molte molte atoms come puoi vedere e è una di le enzime che catalizzano eservano le reazioni che trasformano le molte molte molte che abbiamo visto prima in questo caso questo molte molte è quello che eserva la produzione di ATP attraverso il membro in un processo chiamato respirazione che vedete in una molto breve overview molto pronto questo è in i celli mammaliani ma quello che è meraviglioso è che ogni single cell che contiene queste molte molte ha per produrre questi protini in sufficient amount per carrire le reazioni e le reazioni che si sono necessari per ottenere la produzione di queste molte molte in un modo molto complex un set di luci di feedback noto, per esempio, che i protini non contengono fosferi quindi, per esempio, è interessante guardare la composizione elementale di diversi classi di molte molte protini del massimo dei nostri celli e non contengono fosferi quindi uno può iniziare a chiedere questioni che vedremo dopo quando e come i differenti elementi hanno preso i differenti stati nella storia della vita to potete farlo per i systemi viventi per iniziare a iniziare queste processi metaboliche quindi voglio ora iniziare da questo per iniziare a cercare come e perché queste processi metaboliche davvero può effettare l'ecologia vogliamo iniziare a ecologi molto rapidamente altrimenti oggi ci parleremo di metabolismo su un lungo organismo ma voglio iniziare a iniziare alcune cose di perché e come questo è così importante ecologicamente quindi queste sono due basic metaboliche e io iniziare a iniziare per iniziare a iniziare in questa molto semplificata modo che è la carbonata di carbonata quindi queste sono il numero di molte molte molte coinvolte in questo primo molte molte che è iniziare a iniziare a i lungo organismo è iniziare ultimamente iniziare a iniziare a iniziare a iniziare a tre molte molte molte molte e in questo caso c'è un altro cleavage lasciare un molte molte che ci vedete è una fermentazione byproduct e in questo processo i soldi possono produrre due 80p quindi c'è l'energia produzione è due 80p per glucose che è stato bloccato e questo è il processo di fermentazione che ci serve per esempio a ethanol in iniziare e etc ora c'è un diverso quando molte soldi possono carriere i due processi infatti se continui a fissare alcuni di queste carbonate due molte molte in un numero di differenzi attraverso il ciclo che è il ciclo che abbiamo visto before come semi auto catalytic cycle che che abbiamo visto prima questo processo in un modo molto più complicato che non posso andare in questo modo può diffusare la produzione di 32 80p per glucose che si consuma quindi questo è un'edizione che fa un grande differenza in termini di produzione di 80p e ancora molte soldi hanno l'optione di justi arrampare il tabulizio fare la fermentazione o continuare e respirare le molte ora ovviamente cose che sono molto più complesse in reale vita ma questo è solo per dare un'idea per quei che non sono familiari con il tabulizio la quale di questioni che si può rispondere e quali sono le implicazioni quindi quali sono le implicazioni per l'ecologina come pensate di questo? quindi una cosa che può essere già ovvio da queste varie differenze è che in realtà che può essere una rada di tradotto che può essere molto importante per la competizione e la cooperazione in across different bacteria ovviamente questo tipo di tabulismo è molto più efficiente si produza molto più energie per glucus consumi ma c'è evidente che questo è più cumbersome potenzialmente più lentamente in termini di rada e certamente necessita molto più proteine per dare queste processi infatti questo molte il sintetese di l'ATP che ho mostrato è esattamente che permette la produzione del 32 ATPs e che è interessante è anche ovviamente questo necessita l'oxygen come si può immaginare il respeto è questo è l'ultimo final l'elettronecceptor per questo processo ma che è importante è che senza l'oxygen o altri l'elettronecceptor questo il metabolismo non può occorre i celluli può rimanere questo metabolismo nell'absenzio di l'oxygen e se vuolte che è molto interessante e ancora purmente capito alcuni celli decidere a usare anche in presenza dell'oxygen il l'elettronecceptor e infatti questo è uno dei concili di cancer per quelli che sono interessati nel metabolismo in mammalia e c'è anche alcune domande di l'ecologia di come diversi celluli interattano con l'altro e due più i celluli ma c'è questo fenomeno chiamato l'effetto di warburg dove i celluli anche in presenza dell'oxygen decideremo a fermentare c'è un piatto se qualcuno è interessato in ricordare più di questo su questo piatto di warburg tradotto e possibili conseguenze per la competizione tra faste ma inefficiente e slow ma efficiente organismo e questo è questo piatto da 5 per Schuster in Bonn-Hoffer c'è un altro importante possibile impatto ecologico impatto dell'ecologia tra questi differenti metabolici e questo è il fatto che come abbiamo detto questo molecule che è un carbon si si 6 molecules può essere glucose il c1 molecule che è il c2 e tipicamente la fermentazione by prodotti sono organiche acidi questi sono carbon 2 molecules come l'acetate l'acetate etnol l'acetate è la fermentazione by prodotto prodotto by human cells per esempio e by cancer cells acetate è prodotto by color è prodotto by yeast perciò quindi che è interessante è che se i soldi siano a usare questo fermentare di metabolismo che si secrere questi by prodotti e questi by prodotti sono perfetto usabile carbon sources per altri organi quindi si può immaginare che la decisione dell'individuoso specie per portare un metabolismo verso un altro può avere un consistente importante in termi di la capacità di interacciare con altri organismi evitando crescenti in exchange di molecules across different organismi quindi guarda in mind e ci vediamo per questo dopo quindi la cosa che ho detto un po' di un po' di in questo primo slide è come metabolismo è importante generare l'energia currenti l'ATP e anche l'equilibrio redox che non abbiamo parlato di ma c'è un altro funzione che il metabolismo interessa che non è più importante infatti molto complicata e è la produzione di tutte le differenze le differenze che sono usate per produrre proteine la DNA la RNA e tutte le componenti del cello quindi cosa vedete qui è un histogram della proporzione delle differenze componenti di biomasse in un cello nicole queste sono le differenze aminoacidi di differenze aminoacidi fosso di nucleotide e questo è se per fare un snapshot del cello dry mass di un cello e mescolare quanto c'è l'impasto di questa compound questo è quello che avete questo in millimoles per gramma di mass dry mass quindi è vero che la stessa patria che vi ho mostrato quindi potete recognizzi qui la fermentazione o la patria della glycolysis è andato da Glucos a pyrovate e poi raffreddare nel ciclo di TCA quindi questa stessa patria in attimo di produrre l'ATP può essere imparata along the way per produrre altre componenti un po' di aminoacidi e precursori per nucleotide e lipidi sono all'interno di queste differenze differenze quindi nella stessa patria che ha l'energia ha bisogno di carriare la produzione di tutte queste altre moleculi e tutto questo ha di essere balanciato in un modo molto delicato perché, ovviamente, bisogna avere l'amore giusto di queste moleculi che non ti aiuta se sei molto buono a produrre gliscine ma non puoi produrre allanine tutti questi sono necessari in le otte proporzioni e tutto questo ha di essere accomplicato durante la stessa alla stessa volta la stessa også produrre l'amore giusto di l'ATP perché l'ATP è usato per degradere moleculi e per creare moleculi quindi è una complessiva balanza di diverse reazioni che hanno a a poter in order produrre tutti i componenti biomascari quindi potete iniziare a vedere il sapore di un problema di reserzo dell'allocazione per la stessa e un problema che, in realtà, è un minimo per l'analisi matematica quindi stiamo parlando di modelli di scale genome ma, in realtà, mi non lo so, posso pausare qui solo per dare un'opportunità se non ci sono alcune domande così farci non mi continuo ok quindi stiamo a iniziare a introduire modelli di scale genome in strano modelli di metabolismo e voi finirei questi modelli neanche alcuni potete già sapere di queste queste sono conosciati in diverse mani per esempio sono conosciati modelli di scale genome a volte modelli di strano o modelli di stocchiometri e sono tutti sui above normalmente modelli di scale genome perché uno trova a modellare l'ultimo metabolic network di una stessa strano perché, come vedete, ci rilieveranno fortemente sui strani soprattutto per mass conservazione stocchiometri perché sono basate sui strani di stocchiometri delle reazioni differente nella stella quindi uno dei primi problemi nel tentare di fare un modello scale genome di metabolismo è per construire da questo metabolismo universal che ho mostrato prima che contiene neanche tutte le reazioni conosciati tra tutti i sistemi i sistemi e ora in realtà i numeri sono grandi questo è un numero c'è probabilmente più di 20.000 moleculi che sono parte di questi database ora ma bisogna filtrare questo nel geno dell'organismo individuale e per esempio che, tra tutti queste reazioni differente è il colore capito di fare e questo è ritenuto nel geno e bisogna ripetere e vedere che reazioni sono presenti e encotevano i mezzi nel geno del questo organismo e trasferire questo into a smaller network che è il metabolismo metabolico che è specifico per questo organismo un tipico materiale organismo ha l'ordero di 1.500 reazioni e di above the same number of metabolites e questo è che è chiamato metabolico network riconstruzione è questo è in itself in now a whole a few process particolarmente perché non conosco la funzione del geno quindi quando tentare di ricordare il geno dell'organismo c'è un po' dei geni con le funzioni o particolarmente le funzioni alcune funzioni questo è molto molto complex un po' di steppe e le le you know è molto interessante che è involta literatura curazione a volte curazione manuale c'è un po' di esercizzi ora per tentare di ottenere questo processo ma noi non avremmo i dettagli di questo e solo assumere che abbiamo un geno scalo network il network di tutte le metaboliche che occorre in un specifico organismo e poi chiamo la prossima questione di come modulare questo quindi se sei interessato in dove trovare questo network questi sono alcuni poveri Bernard Paulson del CST ha un listo di modulare modulati e è uno di il primo a portare questo campo a veramente il campo biologico modulare c è un database di automatiche riconstruzioni modulati da genomi queste sono alle presenti in la base k also an open database da del Departement of Energy che ha un numero di modulati per riconstruire modulati da genomi e io voglio solo ricordare se sei in questo che hai a ricordare che c'è un modo di completità di completità e l'accurità di questo modulato alcuni potrebbero undergone per l'esperimentale testo per molti anni e potrebbero molto bene testato e accurato altri sono strappi da il geno annotato e non potrebbero essere accurato e completato ma è ancora sempre interessante di avere qualcosa da iniziare da e da iniziare da iniziare quindi cosa è che stiamo parlando di come andiamo ora da questa nettora e come rappresentare la nettora e come si trasforma questo in una prediczione di cosa un organismo può e non può non fa e io iniziare da illustrare cosa costa o è ancora come l'approve per adrezzare la questione di cosa un organismo può fare che è un modulato quindi again se un po' tipico procurato accel può avere circa 1.500 reazioni in metabolica questo involve circa 10.000 kinetic parameters potete scrivere differenziali equation descrivendo il cambiamento di ogni metabolica nella nettora come funzione delle rate della reazione che produzione consuma quel metabolico e io assurdo molti di voi potete essere familiari con questo questo è il Mica Elysement un equasione raccontando che il reazione della reazione dipende su l'amplice presenta in il molto e questo Vmax che è il maximale attenibile reazione che dipende su l'amplice di enzine presenta per quella reazione e cosa importante è che questo è una funzione non-linear della reazione quindi se vedete la concentrazione della reazione potete calcolare i flexi ma non è una reazione di reazione tra le due e a un momento se continui ad ad un reazione la reazione non continui a crescere in reazione perché potete essere limitati per l'amplice e la cosa importante è che potete scrivere un equasione come questo per ogni single flex e ogni single reazione in questa netta e le reazioni reali le reazioni di reazione avrebbero avuto più di questo di questo perché potete avere due molte iniziali e due molte iniziali molte prodotti quindi hanno un po' di differenti parametri e complexi non-lineari che rendono questo modello modello molto complicato ma non va a andare in questo e andiamo ad abbandonare il modello kinetico approccio per questa molto semplificata versione di modello metabolico che è che va a venire in questa esplosione esplosione ma vediamo prima iniziali con questo esempio come rappresentano la netta insieme questo è un'illustrazione di una netta semplificata dove hai un metabolico A per esempio in una cella che è stato importato in questa reazione B1 e prodotto da una reazione consuma per la reazione B1 e poi anche consuma da B2 prodotto da B3 da questo metabolico C quindi potete scrivere per metabolico A questa equazione differenziale dove c'è un termine per ogni reazione che consuma o produttura questo metabolico e questo è un'illustrazione lineari tra questi differenti flexi ma ricorda che each di questi flexi ha questa dependenza sullo substituto questo è un'illustrazione non-lineari dependenze basi sullo di la reazione di Michalis in questa reazione quindi questo è questo è semplice ma è un'illustrazione un'illustrazione differenziale potete scrivere questa equazione differenziale per ogni metabolico e potete semplicemente representare questo in formato di una matrizia quindi potete avere un vettore di tutti i flexi un vettore di tutti i metabolici e i loro derivativi in tempo e e questa matrizia è un'illustrazione che converte il set di flexi in le change di metabolici è quello che è che è la matrizia di il netorto e è davvero un' rappresentazione molto valibile della struttura di metabolici quindi questa matrizia è essenzialmente quello che bisogna avere come un'outcome del netorto metabolico di riconstruzione per iniziare a modellare metabolici usando l'analisi di flexi quindi come ho detto questo inizia come un il problema di allocazione di risorse è inizia in il fielde di ingegneria da Terri Pappuzzakis e altri e portato ora allo il biologico di forfanto di Bernardo Palsone e i colleghi e ora è una approccia in che è often called flux balance analysis come vedete vedete in un secondo perché questo è una rappresentazione di metabolici per i colli ci sono nutrienti e ricorda nutrienti hanno almeno una sorsa di carbono nitro e fosfora solfora e così tutti i elementi che sono necessari per produrre i differenti moleculi e poi queste moleculi sono le proteine e la DNA e l'RNA e le lipide attraverso la construzione del precursor il moleculio che è essenziale per per imparare queste moleculi e la cellula per mettere queste moleculi in le proporzioni giusti come abbiamo visto quindi prima produzione cosa chiamiamo biomass nuove cellule che hanno l'organizzazione giusti dei componenti di biomass che potrebbero essere produtti di produtti e cosa che dobbiamo fare ora è provare il modo per semplificare questo processo quindi se zoomo in una delle reazioni questo è glucose 6-phosphate l'inizio della fermentazione o di glycolysis che ho mostrato prima con glucose avvenendo e c'è una molto semplice conservazione massima che si imposano attraverso che il sistema è a stile state e questo è l'approximazione della balanza che è molto molto semplice ovviamente se potrebbe se non questo è riusciuto possiamo parlare di questo più dopo se qualcuno è interessato ma per ora immagino che stiamo mantenendo una popolazione di cellule in un bioreactor in condizioni stati e davvero sembra riusciuto ad assumere che l'overall ad avveraggio tra tutte le diverse cellule c'è l'amore net di ogni metabolica l'overall in la popolazione sta constantemente non c'è l'accumulazione net o la deputazione di compound e questo tradizia nel fatto che ora l'amore net di tutte le cellule produttando e consumando questo molecule è ora balanza per provare zero quindi questa è la parte balanza di cellule e ora ricorda di nuovo ogni di queste cellule dependerà del concentrazione ma questo è dove abbiamo abbandonato le concentrazione metabolica e riusciuto solo sulle cellule e se riusciutiamo sulle cellule se tutto che è interessato è sapere quali sono i riusciuti di queste reazioni poi stiamo davvero riusciuto con la equazione vennera e questo è il mondo della balanza di balanza è il mondo delle flezze dove l'amore di metabolica le concentrazione metabolica interno e stiamo riusciuto a capire le flezze di queste reazioni e ovviamente questo fa questo problema molto semplice c'è una constrazione come questo per ogni metabolica in la cellule ma c'è anche un più constrazione che puoi aggiungere in particolare c'è constrazione di la capacità di cosa viene into la cellule e queste sono molto importanti constrazioni perché definono le condizioni specifici sotto cui running a certaine esperimenti quindi per esempio se una cellula è crescita se la cellula di popolazione cresce in un bioreactor con le flezze disponibili e sai quanto le flezze che puoi aggiungere nel bioreactor le flezze non potrebbero prendere più flezze che le provvedere quindi questo farà un'inqualità nel flasco delle flezze di le flezze in tecno quindi questo è una relazione e vedete in un secondo come tutto questo la linearità delle constrazioni per le flezze lo farà possibile per noi per avere un problemi matematicamente trattabile there are other constraints for example some reactions all reaction or metabolic reactions are supposedly reversible but effectively some reactions may be so unbalance thermodynamic that are effectively irreversible at physiological concentrations and if this is known one can impose additional constraints on some reactions going only in one direction so for example this flux for this irreversible reaction would be set to be positive and this is again another constraints and another linear constraint on the flux so all of these constraints together the linear constraints for the conservation of mass at each node at each metabolism trains for each molecule coming inside the network possible irreversibility constraint they form they define a space in the multidimensional space of fluxes which is called the feasible space and as you can already see this is a convex polyhedron why is this a convex polyhedron if this is not obvious well because you have just hyper planes right reaction constraints like this are hyper planes of dimension n minus one in the n dimensional space of fluxes and of course it's difficult to represent this so I am representing here a projection of the space on two dimensions two arbitrary fluxes so you have hyper planes and you have one hyper plane for each constraint for each metabolite that is conserved this hyper plane will intersect each other and form subspace whose dimensionality depends on on the whether or not these constraints are linearly dependent or not and when you add this capacity constraints you take half spaces and you end up having these polyhedron structures convex polyhedron that represent the feasible spaces for the cell ok so this is again a simple ideal projection into the dimension of this space so this is in itself interesting and there is a lot of work now on just sampling this space so if you know nothing more but you have these constraints for intercellar metabolism you know what the capacity constraints are related to the specific conditions under which you're running an experiment you can already have this characterization of what the cell can and cannot do and this is already quite intriguing and quite interesting and again if you think about this as in contrast to kinetic models where you have differential equations you solve the differential equations you have a specific solution this has a very different flavor here we don't have a specific solution but we have this algebraic representation of a space of where the cells can be found so it's an interesting geometrical object that allows, enables a lot of subsequent analysis and one thing that has become kind of the standard approach in stoichiometric modeling is the idea of using optimization you can you know you might imagine why optimization might be helpful but if you think of a cell as a system that has undergone long evolutionary um selection towards you know for for being efficient but producing its own biomass growing efficiently, growing fast you can imagine that objective functions such as the maximization of the growth rate might be a reasonable hypothesis for what a cell might be trying to do and the advantage of having an objective function is of course that now you can look within this space of possible fluxes you can look for a flux that is optimal for a given objective function and for example if you were to find with trying find within this visible space the point that maximizes vj this would be the point up here if you are have if you have in general a general objective function represented by hyperplane that you can imagine sliding along the space and when this encounters an extreme of the space this will be the optimum for the function and you can in this way find your maximum for this growth rate which will tell you among all the feasible points for the cell that balance all this consumption in production of molecules you can find the point that allows the cell to grow in an optimally efficient way in this point what is important about this specific point is that this is now a prediction that one can test experimentally you'll have as an outcome vector of all the fluxes of the cell for every single reaction as well as a prediction of this growth rate so we'll have a value for each of the fluxes as well as the value for the growth rate which will tell you how fast given all this constraints how fast you expect a cell to be able to grow and by the way something that again we'll see more next time but you can also predict whether a cell will produce by products and those by products if you were to somehow embed this organism into an ecosystem with other organisms that by product could now be the source of a cross-feeding interaction between multiple species which is why you know one at some point realize that this kind of models can be really helpful for modeling the ecology of microbes I want to point to some practical resources if you have never seen this of course I have really enough information to get started right away but I want to point to a couple of things that might be helpful this is a paper from Jason Papin's lab that I think is a really nice overview and a practical way of getting started with doing these flux balance models I think it has some python scripts that you can start using right away for doing first simple models for simple networks and then going into increasingly complex networks there is a very nice python toolbox called CobraPy Cobra stands for constraints-based reconstruction and analysis so this is one of the names by which you'll find this flux balance models and CobraPy is a freely available resource for doing all sorts of things with flux balance modeling uploading models running optimizations and so on and it's quite convenient so this would be a good starting point I also put here in a github on our lab webpage some basic scripts that I've been using for doing simple FBA and some models including the human cell model that is that is now available and there is it's a big world so there are a lot of possibilities out there there is a MATLAB toolbox there are different resources feel free to ask me later on if you need pointers to specific resources but this should be a good starting point now okay in the next few minutes I want to point out some of the applications of flux balance modeling before going back to the ecological side and you'll see actually that there is interesting connection between looking at what happens inside individual cells and scaling this up to the ecosystem level so one of the typical applications of flux balance modeling in the past has been to try and understand what happens if you delete the gene from a network so imagine having E. coli you can predict the growth rate using flux balance modeling and ask what happens if you remove one gene one reaction say from the organism will the organism be able to survive and one thing I realized now I forgot to mention is that the reason one of the reasons this method is so valuable and efficient some of you may be already aware of this is that let me go back here for a second solving this problem is really in itself a very efficient process this can be done through a number of linear programming packages and algorithms starting from the simplex algorithm to now more advanced models that algorithm that use heuristics but essentially in a fraction of a second I think a hundredth of a second or so you can have a solution to a single flux balance model so imagine now yes there are caveats and there are things to be careful about the assumptions we made the simplifying assumption we made but on the other hand in a fraction of a second you get a prediction of all the fluxes in the cell and again this is why this is because it's so fast one can use it to address question such as doing all possible perturbations of the environment or the internal circuits of the cell to see how the cell responds and we're going to go back here to this slide where the process of finding what is how a cell responds to a perturbation can be viewed as a problem of reducing the space and finding again a point that is physiological relevant on the reduced space let me just give an intuition for why that is the case when you have so this green region represents again the feasible space for the wild type unperturbed organism where you can find its own objective function if you remove and let's leave aside the fact that there may be a complex mapping between genes and reactions but let's assume for now you just remove a reaction from the network a reaction is made impossible because of the lack of a mutation into a given gene so that flux will suddenly only have the option of zero flux there is no flux through that reaction anymore so you have an additional constrain in this multi-dimensional space so we'll reduce this space to a subspace represented here in yellow and now you can find within this subspace what is for example again the optimal the maximal capacity for the cell to grow and in this way you can find first of all whether or not the cell can still grow after doing that perturbation and how fast and you can compare and predict all the different knockouts in the genome of an organism relative to each other in relative to the wild pet and again this enables a lot of different downstream applications from metabolic engineering to evolutionary start about evolution of metabolic pathos and so on there is one thing that Salviu may be asking yourself which is a question we were interested many years ago which is whether really metabolism should be optimal also for knockouts and you can think of first of all this question of whether or not we know what the objective function is is in itself an interesting open question but it's particularly tricky when you think of a knockout organism if you remove a gene from an organism that maybe did undergo long-term evolutionary optimization for being efficient at growing there is no reason for a perturbed organism to be efficient in its own subspace right so it's entirely possible that this perturbed organism will not be able to perform in its optimal way given that you just perform this perturbation so what is interesting is that you can look at alternative points in this space that may better represent what you expect a perturbed organism to do and one possibility is to look at the projection of this wild type point optimal on the original space onto the space of the knockout why might this a good better prediction for the knockout if you think about this what the implication of assuming that the wild type organism is tending towards this optimum is that its internal regulatory path was really allowed it to upregulate and downregulate the different genes to achieve this optimal production of the biomass components in a balanced way but once you remove a gene the organism still have that same regulatory circuit so it still try to go towards this optimum and you can ask what is the point that is as close as possible to this wild type optimum but still constrained onto the space of the knockout and this would be the point that is at minimal distance on the yellow region of the knockout space that is as close as possible the wild type within this yellow space and you can solve this by minimizing this distance there is no obvious reason whether one should choose euclidean distance or L1 norm or other distances all of this has been tried you can use quadratic programming for minimizing the euclidean distance and you can find this prediction of the knockout which turns out to be in many cases a little bit more accurate than the prediction of the what the wild type or what the knockout optimum would do and of course one could imagine that throughout evolutionary processes mutated organismi could go from this suboptimal initial point to the optimal point in evolutionary steps I think I have a few more minutes so this method is called minimization metabolic adjustment and is one of many methods now that people use to probe metabolic networks or under different scenarios one thing that I haven't told you and I want to give you a glimpse of whether and how one can test these models and and also the you know the caveats that one has to keep in mind in making these models and this is an example from a comparison I did many years ago based on data from Uwe Sauer's lab at ETH so this is E. coli growing in a chemostat in a bioreactor kept a constant flow and you see by the way you can recognize here again the path we saw before like colisus and the PCA cycle and what was done here was to compare experimentally measured fluxes with fluxes predicted with FBA and first of all I want to mention that measuring fluxes experimental is a very tricky and laborious process typically this is done with carbon 13 labeled metabolites that go through the network and the carbon 13 atoms are dispersed through the network in different ways and one can then figure out the actual fluxes by mapping where the carbon 13 went and but this is very complicated and by the way we know to do this and not we experts in metabolic measuring can do this for individual organisms but it's still an open challenge to try and measure fluxes for communities and a very important one as we'll see later on so but what I want to highlight here is that there is a good agreement overall if you compare experimentally measured fluxes with predicted fluxes the fluxes for this equal agro under carbon limited conditions so this is one mode of running this chemostat carbon limited and there is very high agreement which was for me when when I first saw this and now there is a lot of testing of different models some work better some not as well but this allows me to illustrate the fact that even the same organism under different conditions can have very different degrees of agreement with the flux balance prediction so for example if you take the same organism and just compare experimental fluxes with predicted fluxes under nitrogen limited conditions you can see there's still some correlation but there is clearly something that we don't understand here or there is something that the model doesn't predict correctly and one could speculate of on you know why and what could be going wrong here why is the model working under one condition and not the other and you know one possibility for example is that the assumption of maximal growth rate is reasonable for carbon limited ecology but not for nitrogen limited ecology perhaps evolution adaptation and the regulator circuits in ecology are really compatible with his idea and hypothesis of maximal growth rate when there is abundant carbon sorry when carbon is limiting resource but not when nitrogen is limiting e there may be other strategies that the cell may choose to pursue so this is one example and just to illustrate again that I view flux balance modeling as a hypothesis testing tool as a way of asking interesting biological questions sometimes it can be used for valuable predictive modeling for metabolic engineering application but one has always to keep in mind that some of these assumptions we may may not be true under all conditions I think I have a few more minutes I will point just to other papers that I think are representative of how people tested these models there is a very nice work on showing how adaptive evolution can lead for from organisms that are initially suboptimal under a certain condition to gradual optimization there are other work showing that this is again not always unnecessarily the case this is from Bernard Paulson this is from Chris Marks and Will Harkom this is actually based on data from the Lensky evolved E. coli lines very very interesting work so this is there is now a lot of work using these models and you'll see more and I bet you'll see exciting work from Alvo Sanchez group that also has more of an evolutionary flavor I will last conclude just by mentioning some of the good and bad aspects of flex balance so I think I want to highlight again that there are some really valuable and good reasons for using this modeling approach for a number of applications including as we'll see looking at ecological interactions it's very fast it's very scalable you can easily look at larger organism multiple organisms together and very importantly you do not need the kinetic parameters once we made this transition from the world of metabolic concentrations metabolic concentration to metabolic fluxes we really forgot about the metabolic concentrations and therefore the steady states we compute do not really we don't really need to know the kinetic parameters it gets more complicated as we'll see later when you want to go to communities but we'll leave this for next time but one thing I want to highlight which again is on the positive side here is that concentrations are obviously important but there is something really unique about fluxes and I think the cells care about the fluxes we care about the fluxes if you want to know how much is produced of a given compound how much is consumed this is going to be super important for ecological interactions so it's somehow fortunate that these models through these simplifications are good at predicting fluxes because they will be very helpful for embedding the single organism model into ecological models but there are some limitations and sometimes you really would like to know metabolite concentrations inside the cell because that's what is more easily measurable now with metabolomics approaches and because of the lack of metabolite concentrations intracellularly you cannot really explicitly model regulation because regulation in particular allosteric regulation where small molecules bind to the enzyme for example this is strongly dependent on the internal metabolite concentrations and this is really beyond what flux balance models can do there is very interesting work now being explored where if you incorporate some thermodynamic constraints in these networks it's possible to put back concentration of metabolites but I think that's where you know the field needs a lot of creative energy for people trying to think how to go beyond the kinetic paradigm and the flux the hybrid models that have the best parts of both there are other limitations you cannot easily model fast dynamics because of this inherent dynamic steady state approach and what you predict is really population and time average is not single cell fluxes the other thing that is important to remember is that how accurate these models are will depend also on how well you know the boundary condition for example the uptake rates of different nutrients the some of the challenges some of the open directions which again will be relevant for ecology as well you know we I showed you the snapshot of the biomass composition of a cell and this is often treated as a fixed vector of numbers but in real life this is a condition dependent composition right the E. coli and all living cell will change their biomass composition as a function of the environment there are striking examples of marine bacteria that instead of phospholipids use sulfolipids when they're under phosphorus limited conditions so this is really important and we barely know how the biomass composition of cells change so this is very interesting we know very little about the maintenance how much energy is spent by metabolism in doing non metabolic processes and there is interest again in as I was saying mapping the global effects of thermodynamics of this network and but this is an ongoing challenge partially because we don't know the chemical potential or a lot of these molecules and the last and not least important challenge is that as I said we know quite well the models for some organism even despite the limitations but when you start thinking about extending this approaches to modeling a whole gut microbiome with thousands of different species or the ecology of microbes in soil this becomes a much bigger challenge and what a lot of people struggle with is how to efficiently build models for all these different species and scale up these modeling approaches to really start looking at interaction between species in complex communities and this is where we'll start from next time so I will pause here and I think we have time for questions so thank you thank you thank you very much Daniel yeah I think we have a raised hand from Ashish you know Josh you you can ask the question hi hey Daniel that was a very interesting talk I had a question actually about the first part in terms of like this kind of acetate crossfeeding in E. coli or like the respiration fermentation kind of thing what's so special about I guess acetate or I guess starting stopping in this point just before the TCA cycle like why not crossfeed somewhere else why not in between the TCA cycle a little further above is there something special about this point that E. coli likes or like you know cells like to so that's a that's an excellent question and I think I think this is this is still a highly debated and hot area of research I'll give you an idea for example one of the and you know there is one of the classical examples which is true both for yeast metabolism and for cancer metabolism that we really don't know why certain cells despite having the possibility of doing the TCA cycle in despite having oxygen around choose to ferment so for for yeast for example one of the hypothesis is that in a competition for survival with other organisms it may actually be beneficial to stop here secret ethanol as you know ethanol is a good disinfectant right so it can kill bacteria around so it may be a good strategy for yeast to first secret the ethanol kill competing organism and in fact yeast is capable of taking up the ethanol again doing the respiration of ethanol after consuming all the glucose so this is called a dioxic shift and so there is a lot of complexity in this processes where cells may decide to do first the first half to the pathway and then retake up the compound that was secreted and use it through the complete okay so like respiration so like acetate and glycerol or something would also be could have inhibitory effects like I guess like E. coli shows like excretes acetate and glycerol or something right yeah so let me tell you something else that happens so again this can vary very much from system to system and and you know people have come up with different reasons for why organisms times ferment even if they could respire but in this let's say in the case of acetate for E. coli for example what might happen is that E. coli might be actually oxygen limited and if you're oxygen limited right you cannot run this pathway or you can run it only partially so it is possible in principle that when you're limited by oxygen you know you have no option but to run this pathway to keep going and then you produce acetate and if oxygen becomes available you could in principle respire but in other cases such as the lactate production for cancer cells is really highly debated and there are many different reasons for you know people believe this might be happening one of which has to do with efficiency and again this tradeoff between rating and yield I think this paper I was mentioning by Piper Shusten-Borhofer has some really interesting hypothesis about the fact that really maybe there are some deep thermodynamic reasons and there are some old papers proposing this that fermentation might be indeed inherently faster than respiration so really there is a tradeoff between rate and yield and you can imagine that if there is a population you know the fast but inefficient will take over typically and that what cancer cells might do whereas in the slow but inefficient the slow and inefficient sorry the slow but efficient organisms o cells would be able to survive in the competition with the fast and inefficient if there is spatial structure so this is one of the hypothesis of this paper and it's potentially a hypothesis that could explain for example the competition between planktonic cells, cancer cells versus the structure of the body that is based on this efficient well organized metabolism but also about the rise of multi-cellular organism and this all seems to match in the sense that when oxygen becomes available you can do this more efficient metabolism there is a more cells are more thoughtful about using resources efficiently enabling the rise of complex multi-cellular system so I don't know if this address completely your question but the complete landscape can be very complicated and some of this has to do with deeper biochemistry reasons which we cannot go into now and you know they have to talk about thank you that was good okay we have a question from Martina Arbello thank you for your talk and so could you come back to the slide that we were talking about about the usable by products no this one yeah so my question is it so here it seems that the cells releasing the environment usable by products that can be used by other cells only if they are ferment but isn't it a bit of a huge assumption thank you yes so this is definitely not the case so this is one way in which cells could secrete something that is usable by other cells but by no means the only option I think this is observed often but there may be many many many different ways in which cells secrete by products that are used by other organisms and we'll see this extensively next time but there is a lot of I mean there are if you measure the molecules that are spilled out of a cell they can be very different and complex there is also a whole really interesting and again fairly open question of how much of this cross-feeding happens through by products that are secreted by live cells as opposed to cells dying and spilling out everything they have inside so in that case everything that a cell has inside becomes usable by product or other cells with huge ecological implications definitely in the ocean cycling and so on so again I think you point to a very interesting questions you know in some cases we know but in most cases we really are just starting to scratch the surface of mapping this usable by product okay, thank you you have one more question from Mande Salle Adirha yeah please so you talked about limited oxygen in the in the two padway of the fermentation and the respiration but I'm looking at something se there is a limit in oxygen would this pathway still exist or there can be a reverse or a distortion okay great question so the pathway itself right if an organism has this pathway this pathway is there meaning that the proteins for making this reaction happen they are in the genome they are staying there and of course the organism in absence of oxygen could decide not to express those proteins proteins that may not be expressed although as you remember from this slide right there are other reasons for running the TCA cycle in fact the cycle itself can run even in absence of oxygen what needs oxygen is some of the downstream processes but in order to produce these by products the cell may need to run some of all of the reaction in this cycle but it's also true and I think this is what you're hinting to that this pathways can also run in reverse the diversity of metabolic pathways across microbes is incredible and some organisms will have some portions of this cycle some organism and for example if you feed an organism ethanol or acetate they will not have the glucose that is necessary for building other molecules so they will run this pathway backwards so I think different organisms will have different options and they could depending again on whether there is the two aspects of this one is what capabilities they have in their genome and some may or may not have all of these capabilities but even if they have the capabilities based on the presence in the absence of oxygen and through sensing and signaling and so on the organism may decide and figure out which ends up to express does this answer your question? Yeah thank you thank you very much I'm okay I don't see any more questions so maybe it's time to take a short break before we start with our new lecture thanks again Daniel and see you soon thank you and we'll make a a five minutes break more or less and we can meet again at quarter past six CET time May I open the breakout room? Antonio? Yes I just wanted to check you can you can hear me the audio I can hear you well okay thanks sure and I can share yeah if you want to check share the screen sharing yeah yeah I the one difficulty is that you can't my face that you're seeing is not the one that's talking unless people pin the video but to do what I'm doing for an iPad and so on so if I'm near it it should it should just appear say if I'm muted no but it'll show me on my with just a picture not not my me talking because I'm on there twice and the other one is the one that you can see but because it's some feedback issues I can't have that one being the one that picks up the microphone so anyway it doesn't matter the can I I'm not sure if I can see the chat if things in the chat right now it's empty okay you see the messages chat now yeah but I somehow moved to a breakout room when I tried to send oh okay okay I'll leave room okay I can do leave room in the breakout room and stay in the main return to main session okay okay yeah so I'll try to keep track of um if there's questions that come up in the chat so I can address right away so sure hey welcome back everybody we are live again and uh it's my great pleasure to introduce to you uh Verstein official from Stanford who will give three lectures on evolution in a college in high dimensions what should not be surprising thank you very much Stan okay thank you so I hope you can hear me so I want to just start with some caveats I'm a theoretical physicist not a not a necologist I've worked only a tiny bit in things that are relevant for this for this school so I would ask people if there's things that I say that are blatantly wrong facts to correct me however I reserve the right to dispute what's a what's a fact if you could please ask questions in the in the chat and I'll try to keep an eye on it so if there's questions that are of immediate clarification then I'll try to answer them right away and the other one's going to leave till later if you something needs seems to need clarifying and you want to unmute and and ask quickly you can also do can do that the second just apology is because of set up and feedback issues there's two pictures of me one of them is the one which is has a picture of me actually talking the other one which is the one you'll see if you're on the speaker which is of me just a snapshot if you'd rather see my my mouth moving you can put that one on and you can pin pin that one so I'm going to do a mixture of sort of preprepared things and not be prepared and so I hope that will go more slowly than if I just write things write things up so this is very much a theoretical talk except for some things I'll show at the beginning for for motivation and I'll explain what I mean by the subtitle of what should not be what should not be surprising so the general focus is going to be thinking of biology in real sense of being high dimensions and the role of a randomness in modeling modeling can you note can anyone see my screen now I think your audio cut out for the last if you can you can you hear me in five seconds sorry you can hear you can hear me now that's okay yes it's okay for some reason my my computer one seems to have cut out cut out okay there sorry okay sorry so that I'm gonna add ecology and then the second and third lecture I'm gonna come to something which has already been introduced by many speakers particularly by I'm Stefano Alicina of talking about the ecology in high dimensions in the sense of local terror models with large numbers of the species and then I'm gonna add a crucial part to that which mostly raising questions about the evolution on on that okay so first some of the questions and motivation okay so one of the key questions why is there so much biological diversity as obviously a big general theme of this of this school and you know the classic example of the Beatles there are various factors that contribute a lot to being able to have that there are many different niches there are many different environments and they geographically separated what does that mean that means the nodal competing with with each other okay well one of the things that's become clear in recent in recent years and becoming more and more clear than more people dig into it is that the diversity really goes down to much smaller scales and I'm gonna show one of the most dramatic examples so this is a bacterial diversity within a species and this is a species of perchlorococcus it's a cyanobacteria that dominates the synthesis in the tropical tropical oceans and this is showing a tree named from 96 single cells in a very closely related subclade of this species and there's tremendous amount of diversity in here and more data with more and more cells so that even from the same sample and these are all taken from a few samples even from the same sample you see basically every cell is different genetically and the time since the most common ancestor of this whole group is about a million years of the smaller subclades of this is about tens of thousand years and these are all mixed together they're free floating in the ocean and in fact they're mixed together over the whole ocean in about a hundred years or globally in about a thousand years so this is extensive work of Penichism's group in this particular paper by Kashton and collaborators so that's really dramatic and that's something which we'd like to try to understand surely there are not so many niches here that we can explain every possibility with a niche so we'd like to ask about general questions about the diversity ok so a question we'll want to ask is whether or not sufficient organismic and environmental complexity in a general sense implies that one should expect this very kind of extensive diversity which is an enormous amount of diversity all the way down to within species, subspecies and so on and particularly in the most dramatic case where everything is mixing together and competing ok so that part I'm really going to talk about in the next two lectures and today I'm going to most start off with the other part which is about evolution in simple environments and so this is really intentional experiments done in the lab where they're trying to make a very simple environment where there's no ecology everything is well mixed together it's E. coli in low glucose so the selective pressure from the low glucose and this is a very long and beautiful experiment carried out by Rich Slansky over I guess is now more than 30 years and what they find is that the E. coli evolve there are many different genetic roots to higher fitness even in this very simple environment the E. coli find many ways to do better now when this experiment was started there was essentially no DNA sequencing at all and the kind of things that have become possible with this experiment in recent years from the enormous advantages of the sequencing have meant that one can go back and look at the fossil record and try to look at the fossil record and track the evolution that's gone on so the particular thing I'm showing here is the dynamics of the frequencies of the mutations so this is the fractions of the population that each mutation has and the mutations as they arise and generally it's over here at this end the mutations a lot of mutations arise quickly and take over the population so this is one line this is one flask it's always shaken so there's no geographical structure he's done many different flasks in parallel and they all do somewhat differently and some aspects quite different so these mutations come in quite fast and then not surprisingly it's sitting in a fixed environment gradually the evolution starts slowing down the evolution comes up more slowly they tend to come up in groups we have theoretical understanding of that but then after 10 years or so something sort of funny happens you get things coming up and coming back down again and then some of those sort of swerve up and down over here and there's a whole sense in here as to what's going on and it's clear that there's something in there which is that they're interacting with each other there's more than one type at a time and so this is really indication of the development of ecology even in this very simple conditions and that's seen that the ecology that develops is different in the different runs of the experiment but then this is the bit that I want to just focus on now then something happens later on it seems to go back to being relatively simple here with sweeps coming up but the weird thing that notices is that the sort of rate at which mutations come up and take over here is not that much different than what it was much earlier on it's slower than it was at the beginning but it's not different much later on if you try to compete these with the original ancestor they may be doing slightly better but not much and it's not clear if that can account for the rate at which they come up and how fast they come up how come up they there so this is a real question mark as to what's going on here I'm not going to claim to really understand it but I want to use that as a motivation because this raises the possibility of whether even in a constant environment so in a constant environment here whether or not the evolution can continue forever okay so the question I want to ask here is whether evolution in a constant environment can continue forever without slowing and I really want to put a can in here oh sorry I put a can at the beginning can evolution in a constant environment continue forever without slowing okay so this is the thing which we're going to address from theoretical considerations today and makes among about whether or not we should expect this or not expect this and therefore is it something which is intrinsically surprising or something which is not so not so surprising so I want to before going into that I really want to sort of think about the sense in which I mean by high dimensional high dimensional biology so if I think of within the within the cell then we've got what we could call the the nano phenotype so the nano phenotype which is the properties of all the proteins and how the proteins bind to DNA and so on okay and that's some very high dimension a very high dimension at least tens of thousands even for a simple cell and already substantial for a bacteria so that's a sort of dimension the phenotype that's all the different ways in which the organism can change now of course that's driven by the changes in the genotype but the change in the genotype there's something which doesn't directly coupled to the environment or the properties so we're going to want to work entirely in terms of the phenotype in order to try to understand what goes on and then of course one can try to connect that to the genotype genotype as well so then this nano phenotype and this codes for the organismic phenotype which is still something very high dimensional it's the way all the ways in which it interacts with the environment affects the environment how it's affected by the environment how it interacts with other organisms and so okay so we've got organismic phenotype still very high of dimensions but the space in which the evolution is working is really of this nano phenotype of the microscopic level of the proteins and then of course we've got the environmental and the simplest way to think about this would just be say the number of chemicals which affect the affect the organism and the number of chemicals that with it can affect the organism now Daniel Segre just said in the question at the end of the talk that if you took all each cell and liced the cell and so all the chemicals in the cell came into the environment which others could use or could be affected by could be poisoned by and so on that's something which already is very is very complicated that surely is going to give all kinds of roots for evolution potential evolution and is something which unfortunately no one seems to have really done that experiment carefully of the time to evolve in those conditions but the one thing we can be sure of is that the environment is very high dimensional the very large number of chemicals and the phenotype is very high dimensional okay so what we want to what I want to do is try to be motivated by this and think about the consequences of the this high dimensional environment so I want to think about the consequences of this high dimensional high dimensional environment so what we want to consider is organisms that are already well adapted they've been around for a long time making artificial organisms usually okay and so then we have to think about the fact that everything is conditioned on the evolutionary history okay so the evolutionary history of the organism and of course that includes the history of the environment what is that going to imply well if they're really well adapted that's going to imply any changes so changes which can be genetic changes or changes in the environment okay genetic and environmental changes their effects will be a sum of positive and negative parts okay well if you have a large number of positive and negative parts why do I say that well supposing the genetic chains always had positive contributions then it would already affect the population something which is universally positive so whether it's positive effects or negative effects will depend on the environment it will depend on the genetic background the rest of the evolution is the same before so this is something in which we can generally expect as a consequence of the evolutionary history so then we can take a just a sort of guess well if it's sort of a sum of positive and negative parts it makes it very unpredictable and this will give rise to something which we can try to approximate in a way it's approximating by some kind of randomness and then look at the consequences of that okay so this already you've seen some of thinking about the interactions has been complicated they can sort of be effectively random but I'm going to really take that and try to run with it or run with it somewhat okay so what is the hope okay well the hope comes from the fact that we have high dimensions so this very general quote of Philandesons more is different that when you've got a large number of things interacting together there's something the general character of high dimensions then things are different than you would expect from looking at small numbers interacting interacting together there are certain kinds of behaviors some of the behavior is independent of details so we want to know what the behaviors are that can occur behaviors in a loose general sense or the possible things that can occur and one of the things that we've learned from physics is that not all the details are going to matter so we can hope to get things from simple models so not all the details matter in particular we're interested in behaviors that are sort of robust they don't depend upon all of the details and that's a sort of loose term and I'll say what it means in some particular particular context okay so it would like to try to get a robust robust understanding okay so this really comes to the sort of goal of these of these talks so what we're going to try to do is look at some very simple simple models where a key feature is that they're high dimensional and the interactions and things are complicated enough that we can approximate those as being random and of course we then have to go beyond them okay so those simple models we want to then what can exist what things can occur okay now this doesn't mean that it applies in biology at all or that it implies in ecology in any particular system but it's asking what can occur if we see something that can occur in a simple model that suggests that seeing it in nature should not be surprising okay now we have to be a little careful there and that's where this robust part comes in if we can sort of argue that can sort of robustly happen into the class of models a range of models then we can say some more about and well okay that wouldn't be surprising if we saw that we don't fully understand it but we get some general sensitive okay now this is in a context which is going to be loosely we can sort of think of phases meaning phases like a solid or liquid or a superconductor so the analogy with phases in physics and we would like to ask what kind of phases can occur and obviously we can't ask about that generally but we're going to talk about particular I want to talk about particular examples okay so that's the general general scope and sort of what the goal is going to to be and now I'm going to go to some specifics but before I do that maybe I can pause if there are any questions on sort of where we can try to try to go okay Antonio can you just say something so I make sure at least we can hear sure I can hear I don't think there was a question so not questions at this point no okay okay there's one question just now oh yeah can you read that yes this is just to clarify when you say what can occur do you mean behaviors or patterns are rising out of the nano phenotypes so yes I mean things that can come from the interaction between the nano phenotypes as manifested in the organismic phenotypes and the environment and I'm what you can occur like for example do we expect extensive diversity with very large number of closely related strains that can can coexist okay and then also the question of things can evolution continue forever if the environment is is constant so that's the that's the kind of thing that I want to be able to want to be able to do okay so we're gonna now consider an idealized idealized system and a system we're gonna consider evolution in the phenotype space okay so this is gonna be of all the properties of the organisms and I'm gonna say this is D dimensional which like D was the dimension of phenotype I'm gonna drop the phenotype for now and my phenotype then is going to be given by some characterized by some vector which is all the D properties of the organisms that will will matter okay and then I'm gonna make some things incredibly simple I'm gonna make there's just at any given time there's just one strain okay no spatial structure no spatial structure or variations in the environment everything is mixed together everything can competes with each other and then my fitness in that case I can make well defined it's just gonna be some function phi which is gonna be a function of the phenotype and the environment so this is a vector describing the environment okay it's gonna be a function of the phenotype environment and this is just gonna be the relative growth rate okay now I personally I hate the word fitness it seems like it's a singular term it's usually used as D fitness this is fitter than that okay well it's even though as two S's it really is a very parallel quantity and I've considered trying to get into the literature the idea of the of a fit now I hate the word no things ending in ohm generally but I think this is actually illustrative of the complexities going on and because at the very least even when the environment is fixed of course everything depends on the environment if I change the environment then things are going to are going to change okay so that's that's the I'm going to go up but we're gonna fix the the environment at least for now okay so then we get a small mutation a smaller fact mutation and that mutation will take x to x plus dx e i'll try to keep doing the vectors at least for a for a while I'll get lazy and I'll get lazy shortly and then if the phi of x plus dx in this environment is bigger than phi of x okay then that mutation fixes so the mutation takes over takes over so what does this mean it means that I can treat my dynamics as being approximately deterministic if the dx is very small so that the evolution is now approximately deterministic okay and where will it go well it would just go uphill so it just goes simply uphill in this fitness of fitness landscape okay so my dx dt the way in which it changes with evolution okay that's just gonna be the gradient with respect to x so the gradient with respect to x of the phi I'm fixing the environment and so that's that's not changing so I've got a very simple thing and then of course if we're doing this for the fixing environment then this phi is just the the landscape that I've got is a really evolution in a continuous landscape is the simplest simplest possible a possible model okay so then of course things are trivial or at least seem to be trivial okay so I can of course draw the behavior in the case where dimensions are two some people can draw things in three dimensions I can't and so what I've drawn here is contours of the of the landscape so these are contours of phi and then in those contours I will get get dynamics so if I start at some point if I start at some point here I'll come come up I'm here I'm going up the gradient so I'll bend around like this and I'll come to a fitness fitness maximum so I'm going to go up like that of course if I started over here then I'll come up like this I'll come up here and go over to this maximum if I came here I would come up and go and by mistake erase my contours which is cheating and I would go over in this way there okay so here I've got two possible maxima and these are separated by a saddle which is sort of the decision the decision point of what goes on and depending on whether the saddle it comes to one side of that or to the port of the other then it will go to one of these of the maximum model and of course I can have many maxima but the thing which we know is that it's a long times as we get the long times the evolution will go to one of the maximum okay um there was a question when I say relative growth what is it relative to is genetic drift important so the first one what I mean is that it's growing faster it's higher fitness will mean it goes faster with something with lower fitness okay so the thing to the only things that really matter are the fitness differences so it's really the difference between this fitness and this fitness that matters that means this will grow faster and we'll take over I'm going to ignore genetic drift one can consider the effects of genetic drift if we're in bacterial populations particularly if they live in the ocean the populations are very large drift is not very important even though a possibility of extinction is important okay, so that I'll talk about next time okay so this is the behavior in two dimensions and the general assumption in the way in which most talk about evolution in simple conditions goes is that there are some number of fitness maxima and long times you go towards the towards the maximum so the question is that what is that what really happens okay so we're now going to want to consider things going in higher dimensions but let me first show a bit of the the behavior the complex is going to happen in two dimensions so in two dimensions I can look at all of the stationary points all of the points in which the there's no no dynamics so those are the points at which x dx dt is equal to zero so those stationary points they can be maxima as I've already shown there's a maxima they can be saddles like this one here well they can be saddles of index two and the index of these points is just the number of the unstable eigenvalues so a maxima is zero and a one which is a minimum this is the minimum in two dimensions is two and the intermediate one is one okay now if we go to to high dimensions so we now go to high to your general d okay so then we can get all possible indices all possible indices of the saddles from i equals zero which will be a maxima all the way up to d and of course the expectation would would be that one would go towards these maxima at um at long times okay now what am I going to consider what kind of land escapes I'm going to consider I'm going to consider landscapes where they have lots of maximum it's going to be the complexities of the um of the biology the complexity of the evolutionary history evolutionary constraints and so on I'm going to assume that fine is a complex some complex landscape okay and I'm approximate this is some random landscape and I'll be specific about example shortly approximate this is random with some um statistics okay so this is meaning that the number of extrema um this is the number stationary points sorry I was calling that number stationary points goes exponentially with the um with the dimension and that coefficient then will depend on i if I look at the number with a given index the exponential dimension and all possible indices will occur generally the maximas will occur at higher phi and the minimum will occur at lower phi but of course I can have local maxima and local minima which occurred in immediate immediate effects okay so one of the general features which is known from so mechanics is you'll get exponentially many um maxima and a minimum so okay one would still expect that the behavior would go to the towards one of the maxima okay so what actually happens what happens in the limit of um high d is if we look at the function of time so we can look at function of time if we look at function of time and we can look at the phi so what happens is initially phi will go will go up fast then phi will go along and it'll saturate so it'll go very slowly very slowly upwards so that's um uh that's phi okay I can look at the index I can look at the index um coming um uh coming here the index will start up being um being of order d over two and the index will gradually come come down okay and what happens here is that the behavior is that it wanders around the saddles so it wanders around the saddles all the saddles it's generally getting to lower and lower index saddle mean they're getting closer to maximum but they always have some unstable directions and the crucial part here is that it never never in the limit of infinite d commits to a maximum okay so there is no sense in the limit of very high dimensions that it approaches the maximum it always keeps wandering around the um the wandering around the saddles it's as if in two dimensions which of course I can't get this it went it came from this saddle it wandered that saddle it wandered to another saddle after that wandered over here went to some other saddle and so on and just kept wandering around the saddles and of course it can't do that in two dimensions but in high dimensions it can do that and that's the generic behavior so this is the generic behavior that happens in in high dimensional complex landscapes this is the generic behavior it's robust under a specific assumptions about what the dynamics what the landscape from what the landscape looks like okay so that's already a surprise is that one's intuition about low dimensions is wrong when it comes to high dimensions okay so here the evolution continues forever it doesn't commit to a saddle but it still slows down right so it's definitely still slowing down here right the fitness is going up slowly the index is going down slowly if I think in terms of the mutations the mutations will fix less and less often it's harder and harder to find the find the mutations the evolution will slow down okay so of course not surprisingly the answer to my question of whether evolution can continue forever without slowing down is no in a constant environment but now what we have to do is we have to add in the ecology so we have to add a little bit of ecology what does that mean will any organism changes its environment okay so if you have a phenotype x so that's the whole population it's just one strain right with phenotype x well that will do it'll take the environment to being an environment will be equal to some in some function of that phenotype of the organism which is there and the external conditions the things that you're holding fixed okay so these you're holding fixed in the experiment but you can't hold fixed the effects of the organism because of course if the organism mutates if x changes then it will change okay so it's simple then to think in terms of the fitness of the organism when it's the fitness as a function of x okay so that's the function of phenotype okay and then it's going to be the e which is this e here so it's now going to be e I'm given by y okay and then I'm this I'm going to call a different a different quantum I'm just going to call psi of x and y okay and this x here this is the resident organism it's the one that controls the environment this is the one that controls the environment and this one can be any so any other organism that comes in can have this fitness if this is larger than the resident right so if this is larger than the resident the fitness then the that can invade and come so again I look at a mutant okay so I start off with x equal to at the y it's just there but then I look at a mutant which is x plus dx and I ask if the if this is higher so if psi psi x plus dx and this fixed on y okay is greater than psi at x and y and of course this is at y equal to x right this is a y equal to x is the resident that's there then it invades then it comes in and this takes over okay and my environment changes and my fitness changes so it takes over the environment now goes to I get now the environment becomes e of this x plus dx okay so what does that imply that implies that the the dynamics the dynamics is now going to be driven by the by these changes and so the dynamics is dx dt this is now the evolution is dx dt is going to be gradient with respect to x of psi at x and y but I evaluate that at y equal to x right because that's the thing that's already there okay so that's going to be my my dynamics and this is my dynamics here the put that in red that's my dynamics okay now an important thing about about this is this is not gradient flow okay so this dynamics here is not gradient flow it doesn't just go up here okay if you like the physics thing you can think of it has some curls to it it doesn't doesn't intrinsically take psi up what it does is it takes psi up at fixed a value here right so it's increasing here the fitness is going the fitness is going up but then that'll change it so it's then going up a different okay so what this is going to look like move this okay oh damn actually it's not working okay so let me look at what happens here I get flows coming around a around a saddle so if I if I for example have my flow here so now I'm sitting at fixed at fixed phi I now say get a flow which comes like this so I'm starting getting a flow it starts gets near the saddle it has to decide right so it bends over in this this direction because it's that side of the saddle okay but now what can happen of course is the environment can change as it's going along the environment is changing so if I do this correctly if I do this correctly my environment is now changing so my environment changes here my saddle has moved okay I'm doing it of course as a discrete step there it isn't really a discrete step it's continuous so what does that mean that means that this now goes a different direction okay so the environment changes can qualitatively change where it's where it goes so I've got my environment change as it's evolving driven by the evolution okay gives you qualitative changes you go in different directions so this is a picture in in two dimensions again I can't do things they don't get very interesting because I've only got small number of a maximum that I can go to and so on okay so this is a question did we assume separate timescales for the ecology evolution ah thank you very much okay I should have certainly said that we're assuming all the way through through this and I should have said it right the beginning is that the evolution is very slow evo is very slow okay I mean environmental changes a fast with the evolution a fast this means evolution slow in the sense of rare mutations only one mutation is coming in at a time okay and the crucial bit here is this idealized the fact that I said it's always one strain it's one strain except in transient when I get a new mutation coming yeah thanks for that question okay so what we want to ask is what will happen in high dimensions we've now got something which is more complicated than this because as we're going up here in phi then the whole function is changing so I'm going up somewhere somewhere different okay so we want to ask what what happens when we've got on this okay so what happens if we're in high dimension in particulate we would ask like to ask what happens with minimal ecology okay so what does the minimal ecology meaning I'm going to put small small changes in eco small changes in the environment in the environment okay and the specific thing that's going to mean is that if I look at the sort of scale of the gradients with respect to x of psi so this is the change with the phenotype okay this is going to be much much larger than the typical scale of the gradients with respect to y of the psi okay so this was the environment part environmental change and this was the phenotypic change and in particular I can make a ratio between these so I'm going to say this is much much bigger this is going to be oops this is going to be here say of order some parameter here which I'm going to call delta okay so it's going to be of order that where I'm going to be considering delta much less than one okay so what I want to ask if I want to ask a minimal effect minimal effect of ecological changes and I want to ask what happens with delta small okay well in this two-dimensional example it's clear what happens when the when delta is small if delta is small then the amount that I'm going to move the saddle by as this comes up is going to be very small so only if it was extremely close initially it was extremely close initially and this changes a bit will go in a different direction okay so here it can change it it'll make little changes where it goes but not much okay so I think this is again that sort of conventional view okay of course they change the environment particularly bacteria they're going to change it by a little bit but it's not going to be very not going to be very interesting okay so what will happen if in high dimensions so what happens is that there the phase that comes in I'm going to call red queen phase okay and this red queen phase occurs for any arbitrarily small delta this occurs for any any delta no matter how small in a limit of d to infinity in high dimensions any delta will change it any delta will make it that I have the following following behavior if I again plot things versus time on these evolutionary evolutionary timescales okay so if I look at the psi which is the fitness so I'm going to look at the psi what how that how that behaves this this will come up it'll come up come up slow down get slower and slower then it'll roughly saturate now if this saturates you can say wait a minute that means the evolution stopped but no it doesn't because evolution is keeping going because the environment is changing even though the overall fitness is staying is staying the same okay so the fitness of course now is operating well defined since the environment is changing but this quantity which I've defined the psi is staying the roughly the same now we can ask how far is it how far has it got okay so we can draw a line up here okay and this is the line of the typical maxima the fitness at which I would find the maxima as a function of psi a function of x okay okay so if I fix the environment I would find the I find the maxima and it's not getting to that there's some gap there so there's some gap as to where it gets to there's a little gap gap here okay there's a little gap there and this gap is going to be proportional to some power of some power of delta okay the smaller delta raise the closer it gets but the it never quite gets to where the maxima so this is really going to be very much is the wandering around the settles okay how could I see that how can I see that well I could plot the index so I can plot the index here the index the index will come down and saturate so this is now the index so this means I'm getting closer to the settles if this was zero every maxima and this will saturate at a value here which goes as some other power of delta so it doesn't go to zero so this is where I goes to I goes to I goes to infinity okay if I'm more precise as to what I meant by over here I meant that the psi will go to so psi at infinity will be I'm approximately the psi at which the maxima I first find maxima first my maxima minus something which is delta v okay so I'm never seeing the maxima at all I'm just keeping wondering around and what this behavior is it's deterministic chaos so this behavior is that I get deterministic chaos it's deterministic because I'm assuming no stochasticity I'm just going uphill but uphill in this weird way about pulling this gradually changing environment okay so of course the effects of small delta in the limit of small delta this effect is small it's going to get very close to this it's going to slow down so if I look at the typical size of the the dx dt if I look at the dx dt typical magnitude of it with the evolution this typical magnitude is going to go to zero and it'll go as as t goes to infinity as t goes to infinity this will go as delta to some other power some other power okay and you can work out what those what those powers are in the concrete example which I'm now going to ensure okay so this is all very abstract so this is but however it's a result of at least semi honest calculations and so I want to at least say what the model is which will give this okay but the claim is that this is going to be very robust there's a whole class of models that will will give this behavior okay so I'm now going to write down and say a concrete concrete model okay for convenience it's convenient to put the x's on a sphere a hypersphere with x squared say equal to d that gives you high symmetry and you can do various and analytical analytical things and a lot is known about such landscapes random landscapes on spheres but where to fix landscape okay so a lot is known about about that and all kinds of things about the statistics of the landscapes and the uphill dynamics on that landscape the effects of fluctuations temperature or fluctuations like drift are are not okay so I'm going to now write down a particular model I'm going to write down my side this is going to be a sum of all of the all of the components so my vector x has now components x i with i equals 1 to d and this is some j i j k x i x j x k so that's my landscape in the absence of the ecological feedback and the j are i i d gaussian they're independent gaussian gaussian variables um with some variance with mean it means zero okay so that's my that's my landscape it's a simple form of a complex landscape and is known to be one of the generic classes of landscapes but now I'm going to add a part to to this I'm going to add a part here okay plus a part here which is now going to have my parameter delta in front of it it's going to be small plus delta and now I'm going to put again the sum on the i j k of some other set of random variables um i j k and this is of course is going to depend on the x that's the way it depends on the um on the phenotype okay but then it's going to have instead of the x coming in there it's going to have y k okay so this is the way it depends on the environment that's the way it depends on the environment okay okay and my double use are also i i d gaussier okay so i've got a gaussian random random potential on the sphere the random landscape coming from this part and then i'm putting a modification that comes from this which depends on the um on the environment okay and then what is my what is my dynamics my dynamics is that the x i d t we've got one component of it and i t is d by d xi psi okay evaluated it's evaluated at y equals to x right that's the environment that's there it's now changing coming from the mutations right so this is the evolutionary dynamics on this landscape so devaluated there and then i needed a piece which has a Lagrangian multiplier to keep the this bit keeps the keeps it on the sphere tecnica okay so this is my um this is my dynamics and then this is a model which one can analyze by methods i'm going to talk about tomorrow so this is analyzed by methods for dynamical mean field theory um sorry about that okay um dynamical mean field theory and i'm going to talk about how to do that tomorrow it's analogous to things done for spin glasses for those of you who are familiar with that and i sound going to explain it um explain it tomorrow okay so this is where the source of these um these predictions the source of these um these understandings um and the one thing which we know from analysis of this is the kinds of behavior so particularly the phase this red queen phase is generic you can meaning you can change the form you can make this having you know quartic terms you can put other things in here and the small other changes won't change the change the behavior and this is really then in the sense of the physics is like a generic phase i shouldn't mention why am I calling it red queen it's red queen because the um you have to keep going running very fast just to stay still you have to keep up with the changes in the environment by by changing and that's what driving the evolution and with deltage small you know to go very fast it's not very red queen but it's i'm always have to keep racing and never really getting anywhere so there's no sense in which is an overall improvement statistically you can't tell a difference between whether you're at this time or whether you're much longer times once you've got these initial transience things will saturate and it'll get to a statistical steady state out here at long at long time ok um yeah so the question is was the saturation of psi over long times on on average um well this is one of the advantages of high dimensions this is a bit like thermodynamics is the quantities will fluctuate but the quantities will fluctuate by something which is much less than the amount in which the average so this will go to essentially a constant amount it'll have fluctuations of relative magnitude something like one over square root of d coming around this that'll depend on delta as well ok so it won't it will it will saturate roughly the index will come around the index also fluctuate around a little bit but not by not by much ok and this is the big advantage of the large dimensions or one of the big advantages um for doing the other is the ability to use these um these methods ok so what I just want to um I'm gonna say I'm gonna explain how you do those kinds of problems a bit tomorrow but what I want to do today is just ask some questions and extensions about this so one question is is there also a phase with different random potential without red queen meaning that um five slows everything slows down um it always just keeps slowing down um everything just keeps slowing down so that's one um possibility I think I know the answer to this but I'm not sure um that's one one possibility um the other another one is if I'm in a simple environment to the extent that that exists a simple environment so there I would sort of think well in some sense there's not going to be very many peaks so there's a modest number of peaks or at least fewer peaks but I have a lot of evolutionary constraints on how y can change in other words it can only go in certain directions whether it can go in one direction or not will depend on where it's gone before so that'll introduce what's like the settles and will introduce something which has a large number of possible places that it could get to even though they're the phenotypically sort of the same there's fewer peaks in the organismic phenotype but in the level of the nano phenotype here the changes are constrained different changes could have similar effects and so I could change in this way or I could change in that way there's similar effects but a way say in between this way I'm changing like that this would be forbidden by the by the constraints okay so I'd have a large number of constraints and so I would like to ask what happens if I have a large number of constraints so there are many of these many of these in particular actually I would like to ask about what happens if there are order D constraints okay then we'll analyze some things as well and again I have some sense of what then happens but not certainly not fully understand okay and the last one and the most important one is about diversification in particular I can get what's called evolutionary branching oh and I should mention this whole framework um this whole framework is called adaptive dynamics this framework of where the environment changes due to the um evolution and particularly Michael um Dobley um Michael Dobley has a enormous amount of work on this including finding chaos in some models of low dimensions they doing it miraculously primarily so can't do high dimensions but one of the important phenomena is that happens is you can get as evolutionary branching um so say I've got some again I've got some some saddle in the um here and I'm coming along and so my I'm coming along here and I'm coming along say getting near the saddle okay but then I get a mutant my mutant can go over to here my mutant is now not infinitesimal okay so this mutant then can go off in another direction it comes here it'll go off say another direction another direction this um this will go this way and these can coexist so these two then can coexist so now I've got two types okay and the system will be quite generically will tend to be unstable to this to this to this branching meaning it'll be unstable to having coexisting types okay so the question here is if we ask what is the question can you get large number types strains coexisting now of course some will go extinct other ones will be the driven extinct deterministically other ones will um branch and so you can get a continuing turnover from um from this and this is I think the most interesting question as to whether or not this can occur because that starts bringing together the questions about the evolution and the ecology so I don't know what the answer to this is possibly you could get maybe order D different strains that would be I think one of the most interesting interesting possibilities this I don't know I've somewhat some thoughts on it but not very not very advanced okay this I'm going to talk about now tomorrow this I'm going to talk about tomorrow in the general question of many strains coexisting non in this framework but in in local terror one and then come back to this sort of question at the at the end of on on friday okay so the main messages here is the evolution in high dimensions is very different than in low dimensions okay it's very different than one's intuition in particular that any small amount of ecological feedback any small amount of that will make the evolution continue forever so in some sense if evolution does continue forever even in a simple environment we shouldn't be too surprised okay so I'm going to say stop there and take take questions and sorry I suspect I've gone on a bit longer than longer than I should have already so questions waiting for questions yeah if you want to unmute and ask the questions directly can I make a comment hello hello Daniel thank you very much I'm going to say very interesting to try to understand you I think there is a fascinating question which is when you get this interaction of ecology and evolution how how does the the same evolutionary forces that have to do with coexistence yes the dynamismal phenotypes will determine the dimensions of the nano phenotype so how big the variation the pool of variation from which you assemble the high resistance at the higher level I think those things are connected and the way they are connected may be super important for thresholds that we may cross when we set up conditions where we lose the capacity of keeping a diverse nano phenotype so I think this is a fascinating question I think where you have hyper diverse systems in biology evolution over long times and large space has set up a very high dimension of nano phenotype yes but that's not my chance and I think in that connection is critical to maintaining diversity so that's a great point or I guess many great points in there so most of the interesting properties of environment that organism fields are properties made by the rest of the biology and so they said simple example when you take a cell which is complex you lice it you've got all those chemicals in it all kinds of potential of course with geographical structure the biology are also set up gradient set up different conditions and you'll get the development of the all the further complexities which enable the more evolution and more diversification so the way I want to sort of try to ask that is is that something which is sort of we should expect that once the biology has got complicated enough they're much less complicated than this now once it's got complicated enough do we generally expect that one will get in this state or this phase of continuing diversification the environment in some sense continue to get complex as well but even if we sort of saturate how complex the environment is bacteria not that much more complex now than they were a billion years ago and we can still have everything continuing once we get to such a state but there's a question as you say about sort of threshold as some threshold one has to go over where this sort of picture might start to apply okay and in particular there's in some of the parameters which I had here of course d isn't infinite it's large for any finite d there's a delta but if it's not big enough if you don't give enough feedback then things will just slow down and you'll go to a maximum okay so you sort of need the d to be big enough to get this going and then the question is once it gets going does it sort of continue and should we be you know be surprised that it that it continues I'm going to say a little bit tomorrow about thresholds in the diversification in what give a terror a lot of terror models in absolute simplest situation again I don't know whether any of this has anything directly to do with the the biology it's really the goal is to sort of make some conceptual things and give them some sort of mathematical footing in the sense of knowing this is something which could happen with sufficient complexity but a really interesting question and to some extent maybe it makes some comments on this Friday is that one can ask if you start with something which is not very complex and get the complexities added by the as the evolution occurs can you sort of get into some kind of high diverse state yeah I'll give an example tomorrow perhaps in the malaria case and they're very high transmission of the existence of such threshold it may be of interest ah great well so I was going to make a point I listened to your lecture yesterday and really enjoyed it and I was going to make a point of listening to it tomorrow and one of the advantages of doing things on whiteboard rather than slides is I can modify in real time things based on the other to know that is a connection and I'd be interested in that connection yeah so I think these are good questions I guess for the round table next week also so yes are there some yeah questions from students or postdocs yes okay um from Roberto is it physically reasonable to think about a rotating saddle environment like the electric field in a pole trap for ions so that x is trapped in periodic dynamics ah okay so that's interesting so one of the things that um Michael um doble in collaborators um find is that you can get things in these kinds of conditions where you get um you actually do get you get a limit cycle you get some rotating around as it's going around it changes the environment and it just goes on a cycle you can also get where it branches you can get where it branches and you have two of them going on both going on cycles so something like this this would go on a cycle this would go on a cycle and they'd go around and do that um together so you definitely can get cycles generally and i'll say something about this tomorrow generally in high dimensions cycles tend to be sort of unstable you generally tend to have either sort of static things or chaos okay and so in simple situations you can get cycles but you probably if you go for a while one of the organisms will find a way to sort of get out of hat um by doing something different and it'll go off in that um in that direction have the questions so when you show the um you had these uh plots of the the index saturating the index of the of fixed point saturating at some level of value so are these are these saturations usually um like do they happen fast on is there some time scale like exponential okay so there's a time scale so there's a time scale associated with this okay so there's a time scale associated with this coming down and this going up depending on your initial conditions so if you thought of going into a new environment externally controlled environment then initially wouldn't be very fit you would get you go up here this would be pretty much independent of delta small changes you're just doing better in an environment but these small ecological changes and they would only really start to matter when you've sort of gotten near to what would be maximum in that environment that wasn't changing okay so there's a time scale associated with this which is just sort of basic time scales in the in the model when you go out over here you don't converge this exponentially you converge this as a parallel so you converge this power law convergence towards towards these at this end and so you get you get slow convergence and that's even that's the case also if you look at the slowing down here this is sort of a power law slowing down not an exponential exponential slowing down okay but there are characteristic scales I sort of set them all to one to make life to make life simpler so so does that imply that if you could so is the rate at which new mutations fix would also go down as a power law I guess is that yeah so this is so this is the rate of which is this is this is in fact this is I should have written it down this is the rate of mutations fixing okay so that'll go that'll go down and how big the effects of those mutations are so how fast they fix will also will also go down but it won't continue to grow down right as you go to reward for infinite for infinite times this will still this won't slow down completely it'll keep going at some rate depends on itself if we turned off the ecological feedback it would just get slower and stop okay thank you so maybe I have a question hi Daniel so I'm sorry I lost the beginning of your lecture so probably you already addressed this but so the question is this is based on the deterministic picture of the dynamics if I understand so are you thinking about so and the question is what is the effect of a noise on this so are you thinking in your species I mean that quasi species picture where essentially your variables are robust with respect to noise or so how does so I'm not I'm for now today I'm not talking about quasi species because at any given time I have one strain and then a mutant comes in and the mutant can take over so I have two strains temporarily I only have one strain at a time okay certainly the fluctuations there'll be demographic fluctuations there'll be fluctuations in the mutations and particularly this thing I showed here about the evolutionary branching this mutation actually has to occur some distance away it can't be infinitesimal otherwise you don't get this so it occurs some small distance away so that's a stochasticity associated with mutations so any stochasticity will just add to it looking more chaotic if it's deterministically chaotic then a bit of stochasticity just sort of adds to that if it's not chaotic so if I'm in a situation like I get in two dimensions or low dimensions where I have a small number of I'm small numbers of maxima and I go I'll just go to those and just turn by the saddles then when I get near a saddle of course the fluctuations are very important and so which direction I go here will depend on the mutation okay so there's an analogy with about evolution which I give when giving talks to to physics some audience or general ones which is the analogy is of a qualifying exam in physics in which some departments it's traditional to ask one general question or sometimes even a thesis defense and the professor asks the student how do you measure the height of a building with a barometer so the student answered I throw it off the roof and measure how long it takes to get to the ground so he was failed on that part but the professor realized that was kind of unfair so he asked one of his colleagues to re-examine the student and she did and she wrote the story down so it's actually based on a true story I'm sure it's been embellished so she asked him well what other way would you do it she said I would go up the stairs and use it as a ruler and measure the height of the building because what if you didn't have access to the building she goes well then I would measure the shadow of the barometer and the shadow of the building and I would get the height from that so she said what about something it uses more interesting physics so he said I would make a very good pendulum with the barometer and I would measure the period of the pendulum on the ground the period of pendulum on the top of the building and being able to get its height from the change in gravity from the bottom to the top she goes what about something that uses its value as a barometer so he says well I would go to the superintendent of the building and I would say to him if I give you this really nice barometer I'll give you this very nice barometer if you tell me how tall the building is so at that point the professor said you know you pass and as they're leaving the room she goes but surely you know what the right answer is and he says well I know what answer you want but I see no reason that I should give it and I would say all of the lessons from evolution experiments in the lab is that you try to select on one thing and the organisms do something different the very first chemist at experiments Leo's lord early ones where he was trying to select on faster growth rate the bacteria were getting all flushed out what did they do they evolved to stick to the walls instead they just played a different game ok so this enormous number of possibilities and I think this relates goes back to Mercedes question is if you think about the number of sort of possible ways in which organisms can do better in some conditions that way he gets more and more the more complicated the biology and the ecology meaning how many other species and things are around gets ok so you so there's really there a you get into the more and more possibilities the stochesity will will certainly be important will change things in detail but in big populations the stochesity doesn't have to change things intrinsic ok and I think that's say related a bit to what I'll talk about tomorrow well mostly talk about deterministic things so we understand a little bit about putting in stochastic effects so I'm a believer that for you know microbial populations except possibly pathogens in some extent drift doesn't matter extinctions matter drift matters when you have only one of you but it doesn't that otherwise the effects of the drift are very important the stochesity as far as mutations is certainly important but even that there are some aspects of it which don't have to be such as say the statement that I got once you know you're over here you will tend to go there you're not so likely to jump all the way over to here you know small mutations can have big big effects I don't really answered your question but that was well thanks so I said tomorrow I'm gonna be much more much more concrete and deal and work through things with concrete models and so on and be more precise about what the kind of questions and things that I ended with ended with here can we have just one more question yeah some kind of traveling way of effects so there's very interesting things in this I will say something about as to what happens if you put in spatial structure and so by spatial structure I'm gonna mean the simplest possible the external environment is the same everywhere things can move around or get you know carried around and ask what the effects of that are okay and there are very big effects of that in some circumstances they can sort of look like wave effects but when you get lots of traveling waves they tend to again get more chaotic like in the ocean and so there are again sort of chaotic things can dominate but the spatial traveling waves is certainly important waves in phenotype space that's like a question that I've previously about sort of cycles in the phenotype phenotype space so all kinds of things are possible which things are sort of generic in the sense of not special but particular models that one has to get at and that's really you know sort of deep methods in developed in physics in the last 50 years enable one to sort of think about that to talk about which things are more generic and again I'm going to do example tomorrow with Lotka Volterra of some very interesting behavior that's completely non-generic but I will use that to get some more understanding of some more generic more robust behavior that shouldn't depend on all of the all of the details I'm a question I imagine evolutionary branching in two or three dimensions as a possible spherical wave I'm not sure I understand that I think here there's a discrete number of strains at any time and so it isn't that there's a whole sphere of them branching out it would be two points that would be separating and then of course one of them branch again and you can get you can get more you know a radiation you're going in a new environment completely a new environment you know you can get a lot of radiation into different into many different directions you know all happening simultaneously and of course the mutation can be fast enough and things that you'll get a lot going on at once okay so it appears that we don't have any further question thank you thanks very much beautiful if you have further questions or you want clarifications of things that come up overnight then please please email me and I'll try to address those on what I probably generally tomorrow thank you thank you very much and see you all tomorrow for a new session goodbye olá al mún ciao oh perdón I just saw that I just saw you oh we are still there so I don't know when you appear in my screen people are still there I'm leaving right now I'll see you sometime no no we should not talk to this channel