 ...start up perfectly on time. Ja, ja. Ok. So, yeah. Next speaker is Tuan Fan. Ok, so thank you for the organizations. So today I am going to explain about my recent work with professor Kunik Kanenko on some kind of theory for the genotype phenotype co-evolutions. We call it double replica, but maybe at the end of the talk you will have a better name and suggest it to me, maybe. So, my talk consists of three parts. The first part is just to introduce you with the broad context of the questions, and then we will move to a more technical part, and then the final one will be about the biological relevance of the model and the method. So, the motivation is following. I think you have seen a lot during the last two lectures by Kunik, that the main questions of interest is in biological system you have two components, so the phenotype and the genotype, and they can both influence each other. So, you have one arrow going from the genotype to phenotype, representing the influence of genotype on phenotype, but then you also have the feedback effect from phenotype to genotype. So, this form feedback loop between these two entities. And we also know from more traditional theory established in biology that the genotype can undergo some mutations, and the selections based on fitness actually act on phenotype. And under the influence of this feedback loop, and under the effect of mutation and selections, you may ask the question about the robotness of the system as an emergent behavior. So, an emergent property is coming out from this system of many degree of freedom interacting with each other. So, this is the main question here. Try to show that robotness is some emergent property in the, how to say, more familiar to you, like condensed matter physics style of emergent behavior. So, this is the main question, but how we need to approach this. First of all, we need to remind ourselves that the system are located in some environment. So, the set it area here just to represent that everything happened under some given environmental conditions. And the next thing, also a bit of a reminder is that in the traditional field of population genetics, you have the one-to-one mapping from the genotype to the fitness. So, fitness here is defined as a function on the sphere of genotypes. But actually, one, we think about this feedback loop. The feedback loop between two components, phenotype and genotype. We think that this traditional point of view does not work in general, because this means that you are missing the intermediate component of the phenotype. So, in our opinions, we think that it may be better to consider this kind of sandwich structure. So, the sandwich structure is that you go from genotype to phenotype and then to fitness. And then this will capture the interrelations between phenotype and genotype here, through the genotype-phenotype mapping. So, the aim of my talk is just to first formulate this kind of sandwich structure, and then solve how this kind of sandwich structure can be used, can be applied in that context to answer the question of the robotness of the biological system. Can I ask a question that perhaps I should ask at the end? So, here, I mean, clearly, if the mapping between genotype and fitness is deterministic, we are in population genetic serial, and there is no problem. Now, the fact that there is a stochasticity, it's in both of the two arrows. We will show that we take in two kinds of stochasticity in both of these arrows. So, from genotype to phenotype, one stochasticity, and from phenotype to fitness, another stochasticity. Is there any other questions? If not, then I can move on. So, first we have two arrows here, right? Genotype to phenotype and phenotype to fitness. First, let's take a look at the mapping from phenotype to fitness. So, fitness, one interesting observation, and I think Kuni had emphasized in his last two lectures, is that quite typical in many biological systems such as protein. You have two parts. One is a functional part, and the other is a remaining non-functional one. And he calls the functional part the target part. For example, the binding side, the protein, and the non-target one, the non-functional one is some residual, so in active side, for instance. And here, just as a cartoon, this guy has a parkman shape. So, with the mouth of the parkman is a functional, the target part, and owns the remaining, representing the non-target one. And so, with this kind of structure, very naturally to ask the following questions that how does this vision, so the composition of the system into two components will affect robotness. And as a matter of fact, so in this recent paper, they published a few data set about the protein and actually it had been shown that the relative size of the functional part is around 10 to 20%. So, it is kind of relatively, how to say, smaller than the remaining one. And in the next few slides, we also focus on this kind of small fraction of the functional part as a demonstration of the approach and the theory. So, now we go back next to the, maybe most, how to say, complicated mapings. So, they are the one from genotype to phenotype. So, from the fitness of an integral carrying phenotype p, so p is just a mapping from the space of phenotype p here to r plus. If there are multiple phenotype p corresponding to the same given genotype g, then the fitness can be defined as a expectation value of this fitness over the distribution of the phenotype. Biological literature, for instance, you can look at this review for more detail. But what you can take as a take home message at the end from this kind of structure is that you will have the fitness for given phenotype, but you have a distribution of the phenotype and that distribution is conditional on the phenotype genotype mapping here. So, this is the genotype phenotype mapping. That plays a very important role in connecting different biological systems at different scales. So, you have the scale of the genotype. So, this is kind of a micro scale and then you have the scale of the phenotype which is the observable biological chain of the organism, right? So, this mapping, this phenotype genotype mapping actually connects the things that happen at two different scales. And then the fitness, the more traditional fitness just like the right fitness landscape of the genotype actually can be understood as the mean value with respect to the distribution of the phenotype here. So, according to this genotype phenotype mapping argument and so, this function here, so, this genotype phenotype mapping here, right? It's very general. If you look at many review in biological literature, everyone almost have one realization of that mapping. So, it's not uniquely defined. It can have multiple forms. It can be stochastic function. It can be deterministic. It can be whatever you come up with. But the aim of my talk today is to talk about one specification of that function. So, like 15 years ago, Guni have introduced a model, a simple statistical physics type model which give you one specification of this genotype phenotype mapping. Just to stress it, we are not aiming to capture the more general function RGP here. We will only show one specification of that and hope that this even is a toy model with some simplification, but it can give you some meaningful insight. So, let me call that model the SHK model because it's introduced by Guni and his collaborator in this PL 2009. So, all the components here are represented by something that you are more or less familiar in physics. So, you have the spin. So, you have a vector of spin, because you have a system with many spins. So, each component, each spin corresponds to one component of a spin vector. So, the full spin vector represents the phenotype and all the spin configurations, all the possible spin configurations represent the phenotype space. And then you have the gene. So, this is quite interesting that so, I asked Guni why he come up with that idea. Is that maybe because the spin are interacting, so they are determined by the rule given as the gene here, the couplings between the two spins dictate how the spin need to behave. So, this is why, as a matter of metaphor, the genes can be encoded, represented by this coupling between two spins. So, this is the phenotype, this is genotype, here is the phenotype. And then, because the system are embedded into environment, then you need a few more parameters which represent. Now, we come to Djakopo question. We have two sort of stochastic city. One is the noise to the dynamic of the phenotype, which is the T-S here. And then you have another kind of noise, but you can call it also not noise, because of the selection pressure which acting on the dynamic of the genes. And then, you have quite a standard external condition represented by this external fuel h here. So, these are on the basic component of the model. But of course, we want to use this sandwich structure. We need to define two functions. So, the first one is the one that more or less clear from the previous slide up, you have the phenotype to fitness. So, because you have the division of the spin into two groups, so the target and the non-target, then based on the configuration of the target spin, you can define the fitness. So, the fitness is a mapping from the spin configuration the phenotype here to the fitness. This is one function you need. And another function you need is how the dynamic of the spin is influenced by the matrix of the couplings, the J-I-J here. So, the phenotype phenotype mapping in the context of this model is basically the Hamiltonian. So, some kind of spin glass Hamiltonian. So, go back to one slide. In that model, in that general structure, we can open how you specify this mapping and the specifications we have introduced to use this Hamiltonian to specify the mapping because the assumption is that the mapping is also stochastic mapping. So, with that Hamiltonian you can run Monte Carlo simulation. You can introduce the TS here. So, the noise to the phenotype. Yeah? I don't understand what is the environment here. The environment here is just some kind of bias. If you think about this as a B string second, plus one or minus one or on and off for the gene. So, the external field just bias some gene need to be on, some gene need to be off. So, it enters in the Hamiltonian. Yeah, it will be the external field enter to Hamiltonian. But it doesn't enter in the fitness. No, no, it does not affect that fitness. It does not enter the fitness explicitly, but seeing it affects the spin implicitly. So, everything is taking into account into two different ways, sometimes explicitly and sometimes implicitly, because everything is wrapped up inside that function. And as soon as you see the expression for that fitness function it will be clear for you. Is there any other question? So, just to summarize again the dictionary of the SHK model. You have two different types of degree of freedom two different kind of dynamical variable. So, the couplings this is very important to stretch here because it is different from the more spin glass type model. Here the couplings are also dynamical variable. So, you can see on different timescale assumed to be slower than the timescale of the phenotype. But then just you have the Hamiltonian to give the rule of the dynamic for the phenotype and then you have a fitness function and own this sort of parameter. So, this is the dictionary of the SHK model. Now we come to the more detail of the model. So, enjoy the spring college for the last three weeks. You may be quite familiar with two well-established concepts one in physics and one in biology. On the left hand side you have the free energy landscape defined as a function of the spring configurations under a given set of the couplings. So, if you fix the couplings you have some kind of free energy landscape structure and then you also know that you have a fitted landscape which basically define in the space of the genotype and they are uncoupled in the sense that they emerge in different field you don't put them into any connection and the SHK model I think this is the most innovative feature of the SHK model is they have tried to combine and to connect these two picture into a single model. They integrate the two landscape into a couple manner. So, now we have the dynamic on the free energy landscape affects the things that happen on the fitted landscape and then the fitted landscape in turn we also deform the free energy landscape because under the selection the genotype will change and whenever the genotype changes the phenotype on this free energy landscape will be deformed. So, this is the most important structure that you need to at least recognize that what we are talking is different from physics or biology because it's a combination of the two in the integrated manner and now we are so one other thing just to emphasize again so this dynamic of the spin we call it the fast evolution and here the dynamic of the genes fast evolution and for the spin there are two set of spin target and non target and now the dynamic of the spin is quite standard many of you have seen it maybe this is just a global update dynamic of the spin so that you will try to minimize the energies over time and each step you just randomly pick one spin so you do that evolutions until the spin configuration reach to equilibrium and yeah please go ahead Is the Jij symmetric? Jij is symmetric here so before any kind of evolution the Jij see draw from the random distribution like Gaussian distribution for instance lies the same line in the SK model so we see fully connected spin glass model without the evolution of the couplings this is the standard SK model and now it is coupled to the slow evolution so in the slow evolution now I am really happy to show you for the first time this explicit expression of the fitness function so the fitness functions is a thermal average and the magnetization of the absolute value because we want to take into account the Z2, the mirror signature between the up and out configuration so we take the absolute value here but what we want to say is that in order for the system to be functional the spin needs to be quite cooperative so cooperative in that context means they own line up they rely to maximize this sum so we take the thermal average with respect to the bond distribution given by this exponential of this minus beta H and then you get this fitness as the thermal average and once you get that fitness you will update the couplings but now instead of the normally you choose the change in the energies to update the spin you will choose change in the fitness to update the coupling so this is you can change that for just to make it as an analogy but maybe not the perfect analogies so instead of the energy now the role of the energies is the fitness and this will be the fitness that determines how the JJJ should involve yeah so when you say fast low, fast low okay so I need to explain that first so now we go to the realization of the dynamic so you first you first implement the fast evolution of the spin one needs to reach the equilibrium you go to the slow dynamic and then once the slow dynamic is also supposed to reach the equilibrium you go back to the fast dynamic of the spin because now you have a new coupling constant matrix and then once it's over, you go back to the slow you iterate that consecutively until the entire system relax to the equilibrium so this is the picture I'm sorry maybe this is not the best way to describe the dynamic but it's really like so you do the spin dynamic then you do the coupling dynamic and then you go back to spin and then to coupling so on and so forth and then you reach to equilibrium yeah the field is inside the tau what variable is this sorry, this is the index of the spin in the target set yeah and like because at the beginning you writing a field h i like it's just present in the this is the external field it can be added here so you don't have in some version gu ni consider also external field but in the baseline model there is no external field so now I just don't talk about external field because I want to make things simple first okay and also when you perform the dynamics you first do the fast then the slow how do you know that it equilibrates to the right system if you so when you do the multi-couple simulation what you need to measure is the correlation time once you know that the correlation time you can know how much how many multi-couple steps you need to run the system to at least get some some sort of equilibrium both the fast and the slow at the same time maybe you reach a different equilibrium system yeah it could be true but one point is so this is a simplification description of the model actually in order to speed up the dynamic to converge to the whole system equilibrium here gu ni and his collaborator use a technical parallel tampering so maybe you heard about that so parallel tampering is just a trick to swap the temperature between different system in order to get the entire system relaxed to the equilibrium so I'm not so sure I agree with you maybe it has not reached to total equilibrium state but let's take it as something granted because actually they check it quite carefully normally because they simulate the system of 15 spins only so with 15 spins you can run million multi-couple steps so with million multi-couple steps I think you can trust your simulation that it reach to equilibrium yeah is there any other question ok let me continue so just to again just to summarize structure because I think it's need to be go over and over to this structure otherwise it's not clear right so you have a system of spin you divide them into target and non-target so target is a left hand side here and non-target is a right hand side and then you have the green connection here which represent the interaction between two different group of spin this is what we call JTO so JTO is a coupling between target and non-target spin and inside each group you have for instance JTT which is interaction among the target spin and the JOO which are the interaction among the non-target of spin and finally the fitness way to distinguish between the target and non-target spin because it's only act as a function that is dependent on the configuration of the target spin so this is the basic structure of the model I think it's very important to emphasize this before we move to the analysis and some result of the model ok so now when you have seen that picture move to the next part so first what have been done in the simulation what have been observed in GUNI paper so he first consider to quantity which is in the physics so we call it auto parameter measurable that is out of the great important to learn about the system behavior so on the left hand side you have the fitness side here with respect to the dj and ds dj is the inverse of the selection pressure for the genotype ds is the standard spin temperature the thermal noise and that things have the high value of course the low value of the ds as expected at low temperature you should have higher fitness because there is smaller thermal fluctuation and but in term of the in term of the energy the thing that he observe is quite peculiar in my opinion because actually the regions in the phase diagram in which you have minimal energy is not at the lowest temperature ds some of you may know it is the condom effect in condensed matter like the energy is not at the global minimum at the zero temperature but at some intermediate value of the temperature it is quite peculiar something interesting and when he plot it is at a 3D plot you see deep minima is here so corresponding to ds around 1.2 does this depends on the fact that there is not enough time to equilibrate no, I don't think so because as I mentioned to answer some of the question from the audience the system he simulate here I think I raise a number ok, I did not put it here but he have only 15 spins and he have 5 targets no, 3 targets and they run the motor couple simulation for 1 million time step and for each of these like switch from slow to fast they also use up enhancement with this parallel tempering so parallel tempering is a speeding up technique for equilibrate the spin last system and one of the co-author is actually the inventor of this parallel tempering so I have no doubt in term of they really equilibrate the system so and ok, now we can go to the next slide leg plot, sorry so apart from the energy, the fitness Kuni also measures what he called Jin-Jin correlation so the Jin-Jin correlation is a product of the two couplings which have the carbon node i here so you can call them adjacent connection in the sand that they share the same node i but they are having different node k and k price so this is the Jin-Jin correlation and he observe in the simulation that this correlation also become quite positive and close to one in this intermediate value of the T-S but apart from that neighborhood is almost zero so this quantity is quite correlated to the structure in what you observe in the energy so when that correlation is maximal it's also kind of at the global minimum of the energy function so this is what have been observed in the simulation with 15 speed and 3 of them are target and by combining all these 3 quantity, 3 observable he draws this phase diagram structure of the model so you can have this what we call local martistate why the local but not global because the local martistate just ensure that there is no frustration among the target spin but among the non-target spin there still are a lot of frustration so this is just the local in that sand martistate only for target spin so we have this as a phase spin glass phase and the para phase and with his collaborator he also try to apply some standard quench limit approach in spin glass to have some picture of the phase diagram it's representative but it's also some discrepancy with respect to what obtain in the simulation for instance you see that here the local martistate only emerge for non-zero temperature but here it's already immersed at zero temperature for instance and another thing is that the quench limit assume that the Jijs are fixed they are quenched they don't invone over time but the model setting actually invone the Jijs because of the mutation so this work only as a then there are a lot of things that need to be build up on and improve so any question so the martistate by definition is equivalent to the ferromagnetic state so the ferromagnetic state you have on the spin ally and there is no frustration among the spin and the martistate can happen if you have for instance two up and down here but in the triangle the spin that is up here are linked by positive couplings and here they are linked by the negative coupling so the martistate is the one by definition can be obtained from the ferromagnetic state by a gauge transformation so if you do some gauge transformation you can transform the martistate to the ferromagnetic state so they are related by the gauge symmetry okay other question no question, okay so this is the description of the model and some of the previous result in Baikunian his collaborator and just because some of you may not know this type spin glass quencher plots so this is what I just try to make one on the same page so I will spend just too many to explain that so in this kind of where we start with field research pioneering by Giorgio Parisi and Fin Anderson they consider spin model which is the kind of random couplings and then this means that the coupling constant are specified by some distribution way and then what people can learn about this system is basically they take this the observation of the existence of some kind cell averaging behavior so cell averaging just to say that for any random instant of the system of coupling the property of the system are much not different from the average system the one that you obtain by taking the average of the free energy with respect to the distribution of the coupling here so and then the famous well-known replica chick to analyze the free energy of this system so this is just to explain you how these people have been working on this quencher limit approach for this to try to get some analytical argument for this simulation so I will not keep more detail on that ok, maybe it's also something I can skip but now I will focus on the aim of the new approach why we need the new approach this is the first question because maybe it's the quencher approach it's already very powerful it's like were created by many great scientists better than me like thousands of time so maybe we don't need the quencher extension approach but one thing we want to emphasize that the coupling constantly are involving it's very contradictory to the quencher approach when you assume that the Jij are fixed so at least this is one motivation you need to have something that capture the dynamic of the Jij and then the second thing that is important is as you see in the description of the model many of you ask fast, slow, fast, slow so you have some kind really hybrid description of the model you don't have a single dynamic which can describe both the evolution of S and J in a single framework so the question is can we formulate a combined dynamic if we can formulate such a combined dynamic we don't need to do fast, slow, fast, slow anymore we have only one single equation describe both of them and in that combined dynamic the Jij will become explicitly a degree of freedom and the second thing is that as a last aim is hopefully with this new kind of idea and argument we can extend to some non-equilibrium system so this is the aim of why do we need a new approach so this is for the sake of motivation now we go to the detail I will try to be minimalistic here because there are many paper, many previous work that have inspired me to come up with this approach so first of all there is one important paper written by Serrington Kulin in 1994 so they introduced this kind of long-term dynamic for the coupleings in their model, in their original paper is work for neural network so basically for neural network this is the dynamic of Jij is what they call learning dynamic and then the dynamic of the Jij have to apart from the null term here this is actually the derivative term coming from some potential and you can call that effective potential so this is the kind of framework to describe the evolution of the Jij within the context of the neural network we are not doing neural network here but I got inspiration by this approach but in this effective potential Kulin and his group only consider two term so V here is a sum of two term first term is V0 so V0 is just basically a decay term you need something that makes a thing about it if you don't have the decay term everything grows infinitely and then there is some term, another term which represents the effect of the spin on the Jij and for many of you it may be quite familiar it is represent to some extent the happy learning rule so the state of the Jij need to be adapted to the state of the two spin at the two end of the link so basically which is two components do you have the first description for learning dynamic in neural network context but in our model it's very important to say that we have two other components that the old paper had not consider first is the fitness so there is no fitness in this neural network model and secondly there is no decomposition to target and non-target spin so the model here that we consider first time here have only two term, no fitness no decomposition this is very important so why we need to extend and generalize this approach because otherwise this model will only work for neural network but not for the evolution genotype, phenotype question that we are interested in by the way just to make connection to some reason like kind of empirical paper so in this PNAS last year paper this group of author consider something they call frustration of course their definition of frustration in not exactly frustration in spin glass physics but it's rather be called frustrated edges so the fraction of frustrated edges is what they call frustration but anyway this quantity they observe that quite minimal in many empirical data set so they consider different biological gene regulatory network and the observation they found is that the fraction of these frustrated edges is quite minimal which means that if you take a connection from left and right here you see which means the couple links Jij and the state of the two spin Sijj should keep a positive product so the meaning of this term is quite similar it's some kind of adaptation of the link to the state of the two end of the links so just one example if file away context but just somehow explain some empirical evidence because otherwise you will just look at this equation ok, some people introduce this form that may work only for neural network but actually it's quite reasonable form of interaction between two spin that affect the dynamic of the link between them as they are also observing other context ok, so this is what have been done and this is what can be done like back to 1994 now if we have the fitness and the decomposition what should we change first we need to consider a new term which will represent the fitness so as you may remember the fitness is the local marketization of the target spin so this is the sum here over the target spin and then it's the can be represented in this form so that it can contribute to the derivative of the effective potential here in the Langevin equation and apart from the fitness if you take a look back at the cartoon of the model you have one group of spin target and you have another group of spin non target if there is no interaction intergroup interaction then the frustration can only happen within each of the group so you can have the frustration in this group of spin you can have frustration among the other group of spin but there is no frustration between for instance the two non target spin and one target spin here so when you consider the structure of the model so the decomposition into two group of spin you will have this kind of triangle so this is the kind of triangle that connects two different group of spin and then that triangle can be frustrated and then it introduce some frustration into the dynamic of the group of spin so just to clarify it's why it's important because without intergroup interaction so without greenling here you don't need system so the last term what we call frustration here but in the presence of this greenling you need to take into account this kind of gene-gene interaction so basically if you multiply this app with the Jij you will have Jij iK and Jki so it's actually a product of the three term and with these frustration effect feedback effect these are the two new ingredient in our framework so in different from the neural network context we need to have two more term one is a fitness and one is a frustration and the frustration we induce say again by the greenlings so with all these four term now I just summarize because maybe it's not clear so basically this is V and V is the sum of the four term and all of these term I look here explicitly how they look like and then you can just do a very standard statistical physical approach you know that this long-term dynamic will relax to this boneman distribution when the term go to infinite and then you can calculate the partition function and now the structure of that partition function is something that can be interesting to remark so apart from the more or less standard to view from the spin the last literature so you have the term which represents the spin replica you will have another term which represents this JTO so the greenling in the previous slide so the greenling because it also have the plus and minus one value here of course with some rescaling so you can reinterpret this term as a sum chi of two indexes the upper index is the replica index of that variable and apart from the more conventional spin replica you will have some chi deconit coupling sigma replica to present the greenling so just need to be clear here you have the structure with the spin so the spin one type replica but one you consider this greenling they become the second type second species so the first species of replica and then the second species of replica is the greenling and one we consider that the second chi of replica species we can compute the partition function explicitly using sister saddle point approximation and you can get some set of self consistency equation and then you can solve it when you solve it you will have some feeling of the structure like for those who know a bit more technical details so you can go even beyond the replica species solution you go to the one step replica species breaking basically everything is just following some standard recipe one you know that you just apply that so there is no logic here but this is all about physics we want to talk about the last part biological part so the physics is just to give you some framework to understand that you want to formalize a co evolution genotype genotype mapping and you can achieve that by formulating a statistical physics problem with two different types of replica species and now with that framework you can analyze the model and to answer again the question that I emphasize in the beginning how the interrelationship between genotype and genotype affects the robot of the system and just to make life simple because maybe it will become too complicated as you have seen a bit this kind very big equation in this part we restrict ourselves to the replica symmetric solution and learn about the biological behavior from that solution of course maybe if you do more in term of the hierarchy of the replica you can understand more but we try to be modest so does your replica symmetric ansets give you negative entropy when you solve it? No, we will analyze that thing so it will give the negative entropy if you cross the 80 line we analyze that let's look at it in the next few slides we analyze that there is some negative entropy as a standard we observe in the SK model so let go first to that so thank you for your comments basically for the replica symmetric solution we neglect the dependent on the indexes so instead of this quantity depend on one index and that quantity depend on two index what people call overlap you can just now take it as some number instead of a matrix so with that you have the first type of replica the spin which have the finite number because the number of replica here is between the two temperature so it's also again different from the quench limit when you need to take this n as an integer at the beginning and then take the limit n go to zero and the second type replica the coupling one it have similar structure it's also again you need to consider the overlap but now the overlap becomes the overlap between two coupling constants but here that replica have an infinite number so this is a picture of two different species of replica one with finite number and another with infinite number and then you can consider this order parameter in the replica symmetric answer which is one thing that they need to remark so nt is the number of target spin n is the number of total spin in the system np here is the ratio between the number of target to the total number of spin and that one will be fixed so the thermodynamic limit here if you make that one and that one both go to infinite but the ratio is fixed ok, too much into detail I afraid about that so now you will get some solution of this equation you can do it in the computer you make it a iterative solution and then you can find for instance if you fix the fraction in 0.1 in the following picture write the phase diagram for the m for instance so the local marketization and sometimes you can also call it fitness because this is basically the average marketization of the spin in the target set so far so good it's just a few pictures to show you and this is different from the non-target because for the non-target most of the time on the quantity are just 0 for instance and of course you can have some spin last phase with that non-zero value of q but just to say that for the non-target they are not under any fitness effect so the behavior of the non-target are the ramastically different from the target one so for the target one you can have some phases in which the value of m is high the value of q is high the correlation between the two links so oh, I forgot that that may be interesting for you relating to your morning question what can be the structure of this network actually the link between two targets spin the jij here so their mean value is 5 if you plot it as a function of tj and ts you will see that it's almost 0 out there it's positive here but it's maximal in this intermediate value matching to the simulation result from Kuni for instance so this mean that the system inside these smaller regions tend to become a field magnet because the link become positive outside from that it's type random system as a much random as a spin graph system here it's also slightly positive but here it's just really the cases where we believe in the field phase and as you can suspect it if you have all the couple links constant field magnetic then you can have just a very simple type of using system inside that phase so this mean you can imagine that the target spin so you can imagine that very simple structure so the target spin are linked by the field magnetic interaction here and it's some isolated island and it is surrounded by the non-target spin and here the interaction can be plus minus randomly distributed so this is some analogies here and ok, so this is the behavior of the target and it's very different from the non-target and now if you come by on this sort of information we can plot this phase diagram before I go to your question let's focus on this phase diagram so this phase diagram have five different regions in total two of them are not how to say biologically interesting because in the first p1 p1 here is actually the short name for paramagnetic one so in that paramagnetic phase you have no fitness because fitness is equal to this no magnetization you have also no overlap and you also have no Jin-Jin correlation so in this p1 regions, nothing interesting happen in the sorry in the sp1 in the sp1 so the very small neighborhood here do you have the something maybe very interesting for spin last physics for biologist for instance because in this regions you have non-zero overlap between two spin replica so easy kind of spin in the last phase but again here the magnetization is zero so no fitness so both of these p1 and sp1 they are physically meaningful interestingly for some spin the last physics but for biophysics for instance it may not be so important now we move to the next two phases which is the sp2 and the p2 so the sp2 is now becoming interesting because for the first time you see non-zero fitness so the fitness magnetization here is positive indicating that the system is functional it has some sort of fitness and then it also have a little bit of overlap here but still the correlation between the two in are zero so in this regions the system is not robust because by our definition the system can only robust if all of these three order parameters are non-zero and actually they need to have high value so non-zero is not sufficient it need to have quite sufficiently high value and now we in that our regions what we call robust region here you actually fulfill this requirement you have all the other parameters having the high value and we call it functional and robust phase with respect to this phenotypic noise and the genotypic noise too and now to your questions what happen in this border if you cross this border you end up having the replica symmetry solution losses stability so actually this kind of line is really the 80 line in the context of this portal so replica symmetry breaking on this border between these SP2 and R you can show that by the stability analysis so negative entropy happens there and of course if you want to make the full comprehensive treatment model you need to go to full replica here to analyze the transition between these two regions but we are happy with things that we got so biologically meaningful here with the replica solution and from that point onward I only focus on these robot regions I will just focus on these are phases and analyze the structure of that are phases because it's not my task to make full complex physics treatment of the model just first to give you the feeling of the model how the phases are located and now we can move on to the so one thing is that the y axis here go to one which mean that we consider from very small very high selection pressure to kind of small selection pressure but now since we are only interested in these are regions we will focus in the much lower genotypic noise neighborhood so I will cut the phase diagram until that point to analyze this in more detail to take a closer look so let's take a closer look when we fix the TRTJ up to 0.1 now we can ask the first question is that all of the result I have showed you so far it forces p so p say again is the ratio between target to the total number of spin and there should be some p dependent of the model behavior so the first question to be asked is how the value of p affect the selection pressure in such a way that the robot phase only emerge below some tj for instance and this is how you can see I take here 3 representative value of p 0.1, 0.5, 0.7 you see that for all of the value of p of course you always observe these are robot regions but it seems that the higher you have the p the lower tj so the higher beta j so the inverse of tj is the pressure you need to achieve this robot phase so once you plot the inverse of the pj as a function of the p you see that it increase with the function p meaning that now we go back to the cartoon that I really love the Pacman shape so the Pacman mean that the p represents the relative size of the model of the Pacman the higher selection pressure unit which mean that the lower pj that unit for achieving the robot region so this mean maybe that this is why as I saw in the beginning the optimal fraction that you observe in the not numerical in the empirical data set is p around 20 to 10 to 20 percentage so we see that if p is small you can use relatively low selection pressure if the p is large it require much higher selection pressure so at least maybe biologically relevant answer and the second thing which we also observe when we analyze the model in more detail we watch the we call the type of chat of relationship between the genotype and phenotype this is quite interesting so now you plot two things and you compare so on the left hand side is the marketization on the right hand side this is the gene correlation Q you see that if you increase the TS you increase the Q so the Q zoom to close to one if you increase the TS you decrease the fitness so the behavior of these two quantity are different one goes up, the other goes down and so it's quite interesting first of all you can define two critical type of temperature one is the temperature at which the correlation between the gene go to one so we call it Tc1 here and the other critical temperature Tc2 which is a prior tweet the marketization here vanishes so there is no fitness and we still don't get maybe full comprehensive answer for this type of relationship but there is some argument maybe the argument is quite heuristic not sufficiently complicated but maybe it's sufficiently explainable in some sense so if you consider the number of non-target spins so n-nt is the number of non-target spins and the frustration index is something like one minus the correlation then you see I can say it's a phase transition like behavior if you analyze also the scaling equivalent you can do whatever thing you want here but the point is that you see a proposonality whenever this one over Q goes up non-target spin also goes up and the number of non-target spins represents the redundancy of the genotype so this redundancy increase with the number of the non-target spins and Q is homogeneity between the gene so one minus Q is redundancy and this means that if you want to increase the number of ways to change the fitness with the number of non-target spins then you will have higher redundancy and at the end this will decrease the fitness so this is just an intuitive explanation for why this kind of chat of relationship happened and ok, so it's another point we also want to explain here is that so far I just explained the dependence of the fitness and the Q as a function of the TS so the phenotypic noise actually you can also look at that as a function of the genotypic noise and do you see some kind of non-monotonic behavior here this is quite interesting so when you have vehicle 0.2 it is here 0.3 it is here 0.4 it is here 0.5 it is here but then 0.5 here 0.6 it is here 0.7 it is here so actually if you plot it so the temperature at which the M becomes non-zero it behaves like that so it suggests that the relationship between the fitness and the function of the genotypic noise is non-monotonic with regard to P so just to show that it is not trivial in this system that you expect the dependence on P is the case here and I think this is the final point from our approach which is related to the morning lecture by Kuni so in the morning lecture Kuni emphasized the relationship between robotness to noise and robotness to mutation so within this context mutation means a chain in the network structure a chain in the genotype so some small infinitesimal chain of the genotype you can consider what would be called mutational susceptibility so you take this infinitesimal chain of the genotype and you consider the corresponding chain in the fitness you take the derivative basically you get this quantity and on the other hand the susceptibility by perturbation of some kind of external field and then you can show that the two types susceptibility are proportional so it is maybe it is not as a general as what could be obtained in some dynamical system context but just to show you that at least within this approach you can observe the same phenomenon of proportionality between the susceptibility to the environmental effect and the susceptibility to the mutational effect and by the way just to commend that this type relationship between two different types susceptibility also have been obtained and analyzed in Kuli as a model like in the context of gene regulatory or some more recent paper I just mentioned it here okay so I think I come to the end of my presentation I just scan this QR code for the paper so just to wrap up the thing we want to understand how the interrelationship the co-evolution of the genotype effect its robotness and we observe that it can give rise to a robot system robot biological system but only within an intermediate range of the phenotypic noise so which means that to no noise is not good only in some intermediate value of noise it's good so it's quite similar to the model to the conclusion from the model in the modeling lecture of Kuli even though these two models are in quite different context so just to say something speculative in the future work so this model that we use is the SHK model with some Hamiltonian dynamic we think that this quite possible to extend the idea not the technique because the technique you need to come up with some more technical detail but the idea of considering two different types of degree of freedom and then consider how phenotype are represented by one type degree of freedom and phenotype are represented by another type degree of freedom and then how these two different type degree of freedom affect each other is something we want to extend it and actually this kind of system with co-evolution are quite popular nowadays so it can be observed in social system with involving relationship between people again from the original paper that I cited you see it have been applied in the context of neural network with learning it can also go to the ecosystem involving inter-species interaction so one example is that the paper and so finally it can be applied and extended to the gene expression dynamic and actually I am working on that extension now so just to give you some pictures of this KUNY model you saw in the morning this model can be phrased in term of the setup two different type of system with the stochastic city for the gene expression and the stochastic city for the updating of the gene and then you have two different time scale and we are developing some theory for this system too thank you for your attention we have time for questions in the model you did not use the fact that also the environment can be changing and can be noisy the noise is phenotypic temperature and the environment actually in that last slide you see in order to consider how the system respond to perturbation we consider that quantity we did calculate it at the end just in the formulation to skip many detail I did not solve that but we use that at some point no but when you define the fitness you just use a target but you can use also for the environment for example a Hamiltonian and in this way you can get different results for example a field instead of the just the target set and the non target set do you think that this could lead to different results and this can enrich the model behavior because for instance actually kuni is working on some model with three different group of spin now and then the model behavior become richer so one so let thing about is as a layer you have one layer is a target another is a non target you can put some like intermediate layer for instance or you can put some group of spin as a censoring spin so ZR spin which are directly in contact with the environment so if you alter the state of these censoring spin you affect the rest of the system for instance so thank you for the question external field in this model and this external field changes in time or something environmental change so that maybe he can do that but it's not here and for the gene regulation network model I did a little bit in that case so variance somehow remains I had a very similar question so I have a question in the sense that in general here you have like a map between genotype so DJ and fitness which is stochastic and you have in principle two very different sources of stochasticity one is the fact that the environment is fluctuating so the target function is changing and one is the fact that you are mapping between genotype and phenotype and therefore fitness is stochastic by itself so I think I have a different flavour of the question is can you distinguish between the two or you expect that also in the other case namely that you have an environment which is fluctuating this relation between robustness and stochasticity still hold I think it's the matter of belief so I believe that they are correlated but maybe that does not sound as a good answer but the point is that when you formulate that map everything in tango so in the biological like review paper they always try to make things like convoluted so that you never see the effect of the two maps particularly the way we work it out here is like approximation that put it that way so you have one stochasticity here and another stochasticity here but in most of the review paper I read they consider only one stochasticity but for both of the processes so it somehow convoluted and it's not clear how to deal with that type system so this is why we formulate it in such a way so just to say that it's just an approximation we don't have the full answer to your question this is the limitation of the approach I don't understand this if they are equivalent or not they may not be equivalent because how to say because actually I also skip some of the detail so in order to get to that for instance some kind of calculation here right you need to take some assumption you need to respect this time scale separation or assumption so that the idea is that the stochasticity play on one scale does not affect the stochasticity play on another scale but if you combine the two then maybe it's different because now everything are two how to say interrelated I'm sorry if I don't have the good answer other questions