 E-team wants to start recording whenever that happens. Yeah. OK. OK, everyone. I think we're already being recorded and starting for the day. So good morning. This is the second lecture. My screen says the meeting is going to be recorded. Is there something going on with the mic? I've turned it on. It seems OK. It seems to be amplifying pretty well. Can you hear me? I want to be able to speak in a normal voice. And it should be amplified. Good. OK. I'm going to pick up where I left off yesterday, continuing these foundational lectures on viral ecology. And I know I did a little bit of a rush at the end, only a tiny bit of a rush, but I want to pick up from where I left yesterday in part because I've been told that I'm supposed to give some exam. And then I ask you what you do in the afternoons and you say you study for the exam. So I want to give some example of what I might want to ask on an exam. And I even, maybe I don't know if today or tomorrow, you have an exam this afternoon, right? Exactly. And just before I begin, because I want to get a sense of what I want to do later in the week, there doesn't seem to be a laboratory component. How many of you can code autonomously? Like you can open up your computer and start programming. Right? Almost everyone, which means that even if some of you feel as comfortable, you can probably work in pairs. Right? And how many of you work in Python, for example? Most of you work in Python. Okay. Any R or MATLAB? One or two. Okay, so it's mostly a pipe, which means that I will talk about that maybe tomorrow instead of having a normal lecture Friday. I'm strongly considering turning around the classroom and giving some assignments that we could work on in an hour and a half period to let you actually implement some of the ideas that we've been building. And that might be more productive or interesting than me continuing to tell you stuff and you being a bit passive. So I will decide. We might have a discussion of that tomorrow. Okay. So where I left off the other day was the following. We had a model in which we have these uninfected cells that can become infected because virus particles inject the genetic material into the host. And then these infected cells obviously can lice at some rate eta producing plus beta new viruses. And this interaction has some adsorption rate psi, phi, phi, not psi, phi, not psi, phi, just phi. And all of these can be washed out at some rate omega, omega, omega, and this reproduces at some rate r. Okay. I just want to make sure everyone is on the same page that r is a growth rate. And you can think of this as even having some negative feedback because there's this carrying capacity. There's a washout rate because remember there's a chemostat and everything is flowing out. We have an adsorption rate, a few more, lysis rate, and beta is birth size. Now I realize that since you're probably preoccupied with your exam for this afternoon, you may not read something that I were to share, but I will share one of the introductory chapters from my book that just talks broadly, what is a virus? Where are we here in this space? I'm going to show you some typical traits, give you a little bit more background, and I'll post that as a PDF. It should be a light read on to the Slack channel. I don't really use Slack typically because I already can, you can only imagine how many emails I already get, so I can't take 50 new notifications. Here I am, I'm here for the next nine days, so if you see me, you can ask me a question at a coffee break. And if you have an urgent question, you can always send me an email. But I will share something that gives you a sense of scales and units. Just to have a little pause here, things like birth size, we often think about 50, or 100, something like that, meaning that's how many new virus particles come out, and it's a dimensionless number. This washout rate may be something on the order depending of one over 10 hours, depending on if slow, or maybe one over two hours if you're running a fast chemostat that you're really turning it over quickly, or it can even be slower. The carrying capacity could be something like 10 to the eighth per mil, or even 10 to the ninth per mil. So it could be a large number. This growth rate could be something like one over hour, or even two over hour, or even a higher number depending on how fast things are replicating. Obviously it can be much slower. This lysis rate is often similar. Maybe it's something like two per hour. And this could be some weird parameter like 10 to the seventh, and I have to remember, my unit's milliliters per hour. So these are the sort of parameters that we need to make any of this stuff work so that our output ends up being something reasonable that we might observe, like the dynamics in which there's something like 10 to the fifth, or 10 to the sixth, microbes and something like 10 to the seventh viruses, again in the units of per milliliter. I don't expect you all to memorize these numbers. But just like when you work in a physics domain, there are some key constants, and often there are very few, and you'll realize that when you start to work in biology there are many, but you're going to have to have some intuition for them if you're going to operate. So I just wanted to review this a little bit to orient ourselves, because some of the questions yesterday which are good, and I encourage more, and sometimes I'll ask you questions, and don't be afraid. Some place called that the Socratic Method. I'm not trying to call people on the hot seat. I just am trying to ask to make sure we have a dialogue. I'm not worried if you don't know the answer to that question, and I'm also not worried if you speculate in your answer. That's fine. It's just a dialogue. We're having a conversation. This is the structure of the dynamical system that I wrote there. The fact that I have three compartments notes the fact that there are three variables. Each one of these arrows, I have to have a corresponding rate in my equations, and if they enter one place, there should be one minus and one plus going in and going out. We're just moving, flow around, and sometimes it gets magnified here by this factor. The host has spotlighted your video for everyone. Okay, great. Thank you. It's been spotlighted. Okay. So once we establish this, the other thing I want to keep in mind for an exam is that you should be able to do a procedure. One which is you should be able to check or find fixed points. And I went over that yesterday. Okay? And what I'm going to do now is just to remind us and just make sure that this is clear. Was there a question there in the middle? No, it's okay. To find the fixed points in which all of these state variables, their rates are set equal to zero. And in this case, when I ask this question, what life history traits enable viral invasion and persistent with their microbial hosts, I'm implying that there already was a fixed point that was virus free. So first of all here, there's a disease free. Actually, I should be even more careful. There's a sterile media fixed point. What is the value of the state variables if I have just prepared my media? There's nothing in it. There should be three numbers. All zeros. That's a fixed point. But that's not the one I'm interested in because if I add viruses to that, they won't persist. So in fact, if I were to do a stability analysis here, I should find a direction in which the system would relax. A perturbation would not grow in magnitude. And that is if I add more viruses. Or in fact, if I add more infected hosts in the absence of anything else, because they would produce viruses, there would be nothing for them to grow on, and then they would wash out. So it's not as if just because there can be viral invasion, if I don't have the right ingredients, I need the disease free equilibrium. This is just bacteria. And reading off there, you can see it is something like k-wiggle-zero-zero, and I don't want to have to keep writing k-1-omega over r, so that's what I mean by k-wiggle. It is slightly less than the carrying capacity because of this wash-out term. It also says that for this to be true, my growth rate must be greater than my wash-out rate. If it were not, then the bacteria would be dividing, but they'd be going out more frequently than they'd be dividing, and my bacteria would not persist in my chemostat. So I can't set my chemostat wash-out rate too fast. So if the growth rate is like 1 or 2, I can't set this anywhere near 1 or 2 because then the bacteria won't persist. Now, when I'm asking this question, what viral life history rates enable viral invasion and persistence, what I'm asking is what happens if I went k-wiggle-zero, epsilon? I added a small number of virus particles. What would happen, given this dynamical system, n dot i dot v dot, and where would we go? And what I'm trying to make a claim here is that one of the options is I might... No, I didn't want to go there. I might go back up there. I might end up here. I might go back with the disease for equilibrium, even though I added viruses. Or I might get to a point where I have n star, i star, v star, where I have a persistent infection that's sustained in this chemostat. And that's the sort of example I have there. We're actually the dynamics. I have to be careful here because it could go to an equilibrium or there's nothing to say that it can't be some n of t, i of t, v of t, and you can think of this as an appropriate time average, that there may be a steady state rather than conversion to a fixed point equilibrium. So I could end up in a place where there's coexistence because it's coexistence at a stable fixed point, or maybe because there's some oscillatory or limit cycle-like behavior. This procedure of finding fixed points, first identifying them and potentially even doing some linear stability work should be accessible. And I want to point out though this is a three-dimensional system and that's kind of annoying. It would be nice to look just at the infected subsystem. And so I just want to point out here that when we're doing this linear stability analysis, in some sense it turns out that what we're assuming that n is like n star. We're not worried about the change in that host density at the start when we're looking at the small perturbation. And so in fact this is really a system in which I can think of it in terms of i dot v dot and just looking at that component of the system and assuming n is fixed. I get the fact that I should treat the full three-dimensional system and do my Jacobian and 3D, etc., but it looks operationally like the following which if you can see now I have phi n star v minus eta i minus omega i beta eta i minus phi n star v minus omega v. This little 2 by 2 system is our infected subsystem. And in fact as far as I can tell it looks linearized because it's just a product now all this n star stuff has gone away and so I could go through the, turn the crank and calculate in this system the eigenvalues and find the condition in which one is positive. If one is positive I'm going to go here. If all are negative I'll go back there. But I don't want to do that today. I'll let you all do that on your own time because it's laborious and algebraic and annoying. I just want to point out again what I finished up with yesterday, the intuition that the viruses invade when one virus or an infected cell produce greater than one virus or infected cell in their life cycle. Yes. One virus or infected cell invade when. The IT people didn't haven't told me I turned off the whole machine by doing that so I think it's okay. I was told not to go too low. I don't even know I'm going right to left but I'm starting near the board and moving away from it rather than getting closer to the board. So I just want to reiterate again these cycles because this is a point that I will share a paper that explains this in laborious detail and really goes through all the details which I'm giving you the highlights right now and should give you enough of the intuition in a good sense. If I have this is me sketching out some bad capsid. Okay? Capsid. Virus particle. Its life cycle involves injecting genetic material into a host cell but remember there's this alternative that it might be washed out. This happens at a rate omega. This happens at a rate phi K wiggle. Right because we have absorption to these cells that have a particular density. Right larger. Yes. My drawings are for the Greek symbols. Everything. If I do this that will help remind me. Virus particle. Not drawn to size anymore. Much bigger than host cell. Should be much smaller but it's okay. Exorbs to host cell because there are a lot of host cells around and you can see that this competition between these processes depends on how well the hosts are doing in the chemostat. So if you wash them out too quickly, don't get enough nutrients, this number will go down and this route will be favored. If it gets in again it could be washed out. Again a rate omega. It can also lice with a rate eta. Producing. I can't draw them that big. I'm not even drawing them correctly anymore. That one is just a little house. When I was in elementary school I got like the worst. I can't even draw within a line straight. A rate eta of these virus particles. Okay? We start with a virus particle. Not a giant one but you get the idea. And this is its life history so that by the end we're back to virus particles. We've completed a life cycle. Are there more of these than this? Because it could be on average we don't even get one. We get half of one. Obviously it doesn't mean we get half a virus. It means sometimes we get zero and sometimes we get two. Maybe rarely we get three. How do we figure out the probability here? Well this should be phi k tilde over phi k tilde plus omega. This is the probability of infection before wash out. Do you even get in? And this is true for all those virus. Obviously you don't just add one. You add a relatively small number but still this is true for all of you. If you do then what's the chance that infected cell lices? Well that is just eta over eta plus omega. And if it lices it produces beta on average. And I claim which you can explore that if you were to look at this system and find the condition for one of the eigenvalues to be positive you will see that this whole thing and let me put the beta on the other side just to make it easier should be greater than one. And to make a connection with tomorrow I'll just sort of foreshadow this that this result is equivalent to saying something like greater than one. Has everyone been following this pandemic? Have you heard about R naught before? Which I will go in and explain more tomorrow in a microscopic context and then next week I will hit it again from the epidemiological context. This is called the basic reproduction number in epidemics. It says the average number of new infections caused by a single infectious individual now we're talking about humans in an otherwise susceptible population. It is a threshold criteria. We'll revisit it next week we'll revisit tomorrow. You can use this same concept and it's effectively the average number of new virus particles caused by a single virus particle at the disease for equilibrium. It turns out that you can also use this concept to ask a life cycle question starting with an infected cell if instead of adding epsilon here I put my epsilon there I could ask the same question and say what is the average number of new infected cells caused by a single infectious cell in an otherwise susceptible population I get the same threshold criteria. Okay? And this probably suggests why I didn't need to rush this at the end of yesterday's lecture there was a lot more to say. This then helps to answer this checkpoint question. Which traits enable invasion all of those that drive this up? Higher birth size higher absorption rate faster lysis keeping in mind that it's all environmentally dependent and I will elaborate on that tomorrow. I'm not going to get into all that today but I'll elaborate on tomorrow. So it says that just because you're a virus with all these good traits if the environment is bad for your host you as a parasite of that host an obligate intracellular parasite may not necessarily make it. Okay? If I do this for the next 65 slides we will have a very long week. So I should move on unless there are questions. Yes. So it depends on time, right? So this is a threshold criteria that is true it's independent of time because I'm assuming that in that time in which I am doing my linearization or essentially assuming that the viral impacts remain small because they remain small the host essentially is relatively constant. Now it could be that the host itself is going through some dynamics or maybe that's what you mean even before I add the viruses. Often then there is a way to do a time average of the invasion over that orbit and the appropriate time average can be used to make an equivalent kind of condition. And I don't know if that was the question you were going to ask and there's if you want to read more about that you can look up something called flow k theory and that's nice and I'm not going to talk about that today but that's cool so you can look into the ways in which you might think about invasion in a time varying system that's still disease free but maybe you meant what happens to this value as time changes and in that case you'll notice I can't say k wiggle anymore in fact this basic reproduction number becomes an effective reproduction number and the cool thing is if I want to look for where the equilibrium is the equilibrium should be for a time invariant system here obviously there's a limit cycle that on average I can't make more than one because the thing would take off I can't make less than one so in fact this goes to a system where my effective reproduction number gets close to one and I think it's probably what you were also thinking about. Yep and the ends turn into i's at a rate 5v and in fact v's go away at a rate phi n and if I think of it as a per capita obviously the combined rate is phi nv so I have to think about them both as density dependent rates I put the phi as a placeholder because it's really the total rate is phi nv per capita on either side is different this k what I'm trying to say is that this reproduction rate is not just n dot equals rn but there's a carrying capacity which limits that growth rate I was using sort of a short hand to say that this is self limiting I could have done something where that's what I was trying to imply that the growth rate itself is limited by its density and the parameter that limits it's not a rate anymore it's a parameter maybe I'll go like that just to remind you it's not a rate but there's some limitation and that's the carrying capacity whereas before I had this explicit resources here I'm making it implicit other questions and this would be a sort of a foundation simple enough model that contains enough of the pieces that would be a good thing for you all to know and whether we do this on Friday computationally but certainly it could be the exam itself can't have computation on it but there could be some questions about this kind of structure okay and hopefully accessible questions any more questions about this good I'm going to move on last call yes here we go last one I'm happy to talk about this for a while yes go ahead that's right because should I be repeating the questions for the folks yes I forgot about that what I don't understand is at point C you say that it's now the system is infected because you put the epsilon right correct but what I don't understand is when you write the equations epsilon doesn't appear so the epsilon is just some initial condition for this small perturbation so if we were to if we were to think about this this is saying that our v0 is equal to epsilon and our i0 is equal to epsilon it doesn't appear anywhere here because this is a dynamical system equation it would appear because the value of v when you're doing your dynamics would be epsilon I'm just saying I was trying to note the fact that it's small that's all I'm trying to say there okay and then second question is you say that these sets of initial conditions they can evolve to either the set with the stars or the the perfect one with just the k-tiles or the dependent in time so it will be different and but they obey the same equation I don't really get let me try to answer the question I think you want to know the conditions by which we avoid the loss of the viruses and what these two outcomes so the first thing that I'm focusing on is just whether or not we avoid the viral extinction and maybe that's good or bad could be bad depending on your perspective on these viruses but when this number is greater than one then one of these two outcomes will take place I haven't said which one when it's less than one, this criteria then we'll go back to this condition so the viruses will wash out now which of these happens depends on other parameters that I haven't talked about that could lead to oscillations in a stable limit cycle or to convergence to a fixed point and I don't have time to go through the details of that other conditions so I'm only telling you whether the virus is invading and yet there is a half-fiffication which allows an oscillatory solution to emerge on another set of criteria and it's more technical and not that intuitive to understand at least not in this format but nonetheless you are right to point out that I haven't told you which of those two outcomes happens, you are correct I've just told you which class whether it goes away or persists but in the set of equations right here you introduced N star on the right the I and V I introduced N star to remind you that invasion has a biological environmental context so it depends on how many hosts are available keeping in mind that the host this N star is the disease free equilibrium value of the bacteria which is equal to the carrying capacity modulated a little bit by the relative growth rate and the wash out rate so if you were for example to simulate this initially you would plug in this value or if you were to calculate the Jacobian you have to evaluate it at a fixed point V and I are all small they are already implicitly small this is fixed so you would have your constants in your Jacobian you calculate your eigenvalues and my claim is that you would end up getting a threshold criteria where one of the eigenvalues is greater than zero when that condition holds it would provide some context just when I evaluate and linearize the nonlinear system around a fixed point I need a fixed point so that's what I'm trying to say the fixed point is when there are no virus around and that one is unique so that's why I can write N star there would be of size 2 correct correct which is also convenient if I can say something I think part of the question comes from the fact that on the right you use N star to calculate the fixed point with the virus got it let me is that better yes I was using in my mind that I can go yes that's what I meant got it, thank you now we really have last call this is like popcorn popping and I heard the last popcorn pop so I'm going to move ahead okay let's see if my screen even remembers that I'm still there with this very long preface in mind I want to go back to what I kind of rushed through at the end and unpack one of these experimental time series in which experimentalist Brendan Bohanen, Rich Lensky looked at the dynamics between phage T4 and E. coli B so we have a bacteriophage that could obliquely infect and then lice cells releasing new viruses over something like a week plus about eight day or so experiment and I would be nice of course to have even more data points but each one of these data points takes a lot to do you have to measure the bacteria by looking maybe at cell counts and colonies in order to count the viruses then you also have to take them and then put them on bacterial lawns and count those plaques that I talked yesterday about which are sort of the whole version the inverse of the colonies and you end up getting these large scale oscillations even though this is a chemostat it's being provided constant input there's no oscillatory input there's no light dark cycle day night seasons it's just everything the same but nonetheless we end up getting these very large scale oscillations two to three ores of magnitude and densities driven by the fact that viruses infect cells driving them down and then as they're down you'll notice that these lines I know would be better to have more resolution they look pretty parallel because at that point the viruses are facing this dilemma when this value of n going back to your point about time if the n gets very low here no longer the equilibrium then this probability gets much less than one and you end up having largely the fates of these virus particles be that they're washed out before they ever find a cell and because of that they're effectively decaying exponentially at the washout rate of the chemostat as they drop down the host can recover but as the host recover now this is no longer the case and they can start to find these host cells and lyse them leading to recovery that cycle repeats again and again which is cool I think so this is what I tried to get to the end of yesterday and obviously in the last three minutes I had like a 30 minute sub lecture in my mind which I assume you all understood but now you do and that's fine because I haven't prepared all five lectures yet I wanted to see how things were going and so I'm not going to try to accelerate to make up for whatever the 30 minutes I just did I'd rather get through I have some core material we'll get through the core stuff this week and whatever is extra we won't get to so these original models of virus host interactions presuppose a simple relationship that's very much prayer or pray like viruses which are parasites and I'll elaborate on the parasitic nature more tomorrow or Thursday lead to cyclical dynamics in which viral peaks follow host peaks driving host down and then the viruses follow leading to host recovery and we get these counterclockwise like cycles and as I tried to elaborate today invasion and persistence are not inevitable they depend on properties of the host context so conditions that don't necessarily favor the host are also going to disfavor viral invasion okay in all of this this wasn't already enough of a mess or fun depending on your taste I've made this very strong assumption that we have fixed properties we have these two players host and virus but evolution can rapidly change the number and relative abundances of both the host and viral strains and I'm going to try in this lecture on packet I'll see how far I go and continue tomorrow so this next part I want to ask the question of how does evolutionary change alter virus host dynamics and do you usually take pictures of the board or no we're just going to let it slide it's recorded wonderful can I erase all this because I might need the board again or do you some people want to take pictures of the board is aren't any this is just sort of it wasn't what I expected to do so there's no piece of paper that says that this is what was going to happen today this is what happened okay erase fine good and I even have a wet towel today this is awesome I like the wet towel makes this much easier good in case I need to go and do stuff and probably we'll get you a new wet towel for your lecture this one is giving messier by the second okay so recall I just told you so it's not a long break this was meant to be maybe the start of today so here I am I just told you that I just told you that but look there's more they continued the experiment they didn't just stop at 8 hours excuse me they kept going 600 hours about 3 weeks plus of these experiments and I think you should notice there's been a change in the dynamics now we have still the host in blue and the viruses in red and I think it should be apparent that after about 200 hours or so maybe about 10 days into this experiment you see that the host flat line and the viruses start to oscillate how can this be how can this be how can this be say it loudly and then I'll repeat it's okay I'll just repeat instead of you having to chase because then for the listeners online that would be easier just go ahead so maybe the infected cells are bursting in some different way over here maybe if they were it's curious though the host don't seem to mind so yes there is some bursting maybe that explains one of these virus bumps so the question is maybe the common is maybe the host are bursting there's some infected cells but then it wouldn't explain the fact that the host don't seem to change in their abundance other comments I'm just going to keep unpacking them maybe we've selected for some resistant cells could be makes it curious though that the viruses seem to be able to keep going up at various times by the way I might just be correcting you maybe all these answers are correct so I'm just giving you a counter as my natural instinct any other comments or people are like I don't want to do this he's just going to tell me I'm wrong some of these are actually correct so I'm just trying to was there a comment here in the middle on the chat there's a chat comment Rayner said the host achieved some kind of immunity after exposed to some of the mutated viruses okay so now we've already jumped to the mutated viruses maybe and maybe there's immunity and that's a different word than resistance and we'll actually talk about that usually in these biological systems we think about resistance meaning the inability often of a parasite to infect and lice often from the outside can't even get in or it could be resistant because it gets in and it's not effective immunity might imply something like this CRISPR-Cas mechanism that actually can identify the foreign invader specifically and even learn from it I think that's often what we think about with an adaptive immune system and there's no evidence in this particular case but it's a good suggestion that there might be that particular kind of adaptive immunity I think that's enough ideas on the table but I want to point out some features what makes us curious is that the host look relatively stable if I had stopped it right here I think it would be fair to say that some resistance evolved and the viruses were going to be washed out but they didn't and the viruses can't replicate on their own, they need a host they need a susceptible host so clearly there's still some susceptible hosts around but yet also there seems to be a problem if these were all susceptible hosts why isn't it doing the dynamic from before so there are cells that in fact are being affected in lice and there are resistant cells so neither one of you are both partially correct there's actually a mix at this point what happened was there was a mutation to resistance but somehow not all of the cells are resistant if they were then this viral population might go extinct in the absence of a counter resistance on the side of the virus when they actually looked at the very end of the experiment there were different kinds of colonies and so they realized that what had happened there were two different types which they then isolated marked and instead of just counting the total number of bacteria because when you count colonies remember I told you what these colonies are you take bacteria, you put them on an agar plate dilute them down so that individual colonies, these little worlds of bacteria are growing up from a single potential bacterium and you can count how many bacteria there are by diluting it down until it's visible on the plate they can't tell the difference between what's the characteristic of that particular bacteria or not, they're just counting colonies but the colonies look different so then they could pick those colonies isolate them, grow them up and realize that in fact there were resistant and susceptible bacteria in the same time in the chemostat and instead of waiting for that to happen at random they put the resistant kinds back in and noted that they went to relatively speaking a flat line level even as susceptible hosts remained in the chemostat if you add up the solid and the open circles on a log scale it looks like a flat line, those oscillations are in the background noise of the resistant type so resistance was there but cells were being infected and laced and if you notice where the peaks are the virus peaks are following the susceptible hosts just as they had been before so now there's a different question we've explained a little bit why we can get these cryptic dynamics but we haven't necessarily explained why the susceptible hosts are sticking around you have this pressure you have wash out to both and yet somehow resistant hosts which aren't being laced by viruses are not excluding the presence of susceptible hosts so how could that be why are the susceptible hosts sticking around why haven't the resistant hosts essentially eliminated them and therefore the viruses should be totally wiped out any speculation so it could be that this resistant susceptible property is some sort of phenotypic switching property maybe I'm just bouncing back and forth and every time I replicate I make more susceptible hosts and that would just always keep things around so that could be in fact that kind of phenotypic switching does happen so that definitely does happen there's other mechanisms so that's one that could actually enable a coexist between two types right genetically the same phenotypically different and they would coexist and maybe because they coexist then they get driven by these other dynamics that's possible any other suggestions let me hear if I can hear from a few times let me see if I have any other voice and then I will turn to you in a second any other voices I haven't heard from yet yes okay so we have a chemostat in which all the things are being washed out at rate omega so from a perspective of whose favor neither one host are not being killed by these viruses which are can rapidly eliminate these hosts and so I have one host which is not being killed one host that is but the one that's being killed somehow is not being washed out but now I have something that has a growth advantage over the other right relatively speaking these resistant hosts remember because we have one minus N plus we have R plus S over K they're competing and yet I have a growth rate that it has an advantage over you potentially I'm kind of pointing towards an answer they're in the same chemostat right but I'm kind of giving away the answer there in asking your questions and answering your questions so let me turn around and say when can resistant hosts invade instead of saying why aren't susceptible hosts washed out let me even just ask the question when can they invade and give a little preview by thinking of the fact that it is not necessarily guaranteed resistance comes with no cost I believe you had the no free lunch theorem they've been learning about the no free lunch theorem in this course recently you haven't been learning about didn't you have a lecture by Wolpert I didn't talk about no free lunch okay do you get free lunches here see there's no free lunch it even applies to ICDV I don't get free breakfast you guys don't get free lunch so you know we're all in the same boat here you can look that up later here nor there with respect to my lecture today but there could be a cost of this resistance okay so if we think about how a virus might get in and there may be some receptor on the outside it is often the case that these receptors have some relevance beyond letting viruses in all they were doing were letting viruses in they would have been ruled by selection probably a long time ago they might provide some structural integrity they might allow for export of molecules or they might allow for the import of molecules including things like sugars that the bacteria might need so either by getting rid of them all together or reducing the density of those receptors so some of there but less this might change the properties of this bacteria with respect to the virus but it could also come with some cost to this growth rate or maybe even the carrying capacity because the efficiency of uptake which is setting that carrying capacity has now changed so it could be that the susceptible hosts have some property r and k and the resistant hosts have some new properties and I'm just going to change the growth rate for now rather than both the growth rate and the carrying capacity but I certainly could do that does everyone understand what I'm trying to say here I'm trying to say that the life history traits of this resistant host need not be exactly the same as that of the original host there's evolution and it doesn't just affect necessarily one trait so I would now like to introduce a different kind of model and when I'm doing these you'll probably both to some level which I was thinking back to my days as a physics student and when I got annoyed with my professors and sometimes when they made a move that I felt was not totally formal and rigorous I'm about to make such a move because now that I'm here I realize that if I make the totally rigorous move I can't do any of the things that get you the essential idea and so I'm going to have to make a simplification you recognize that when I started I had these resources and then I made a resource implicit and yes if you want to read about how I do it how I can do that and you can do it, it's in a book and you can make a reasonable approximation a moment ago I had infected cells I'm about to eliminate them because it would take me so long to explain all that with the infected cells and I lose the intuition I'm going to make a model in which I have susceptible cells viruses and resistant cells yes there is a question from the chat about the assumption that why do both susceptible hosts and resistant hosts have the same carrying capacity they don't have to but I'm going to just do it for convenience as a way so they certainly don't have to in this particular example I'm going to show you one consequence just by looking at a growth rate cost there could also be an efficiency of uptake which could lead to a different carrying capacity that question is totally correct but I'm going to do one such assumption if I want to build such a model I would have some growth rate r and I'm going to use my little bracketed k to remind you that it's not a rate but it's just a limitation and I'm going to have susceptible hosts being washed out and I'm going to make even a simpler way of denoting the fact that there's some adsorption rate phi and you realize what that means now these can also be wiped out washed out and the resistant hosts just get to replicate but at some different rate r prime and they get washed out and I'm going to use my little k to remind me that they also are limited okay and I'm going to try to ask this question when can the resistant hosts invade well let me now try to write down this new system in which we have coupling between the susceptible hosts the resistant hosts and the viruses and I'm not going to worry right now for the infected cells yes so the chemostat is just pulling everything out at the same rate it could be if this were a model of a marine system that's sort of what's going on in a bioreactor or even in our gut and maybe that even the resistance has some cost upon the residence time or a intrinsic mortality rate but because here I'm assuming most of this loss is just dominated by the chemostat I'm going to assume is the same okay so I can write this as following that we have some logistic growth that looks the same for susceptible resistance hosts except I've put a little prime there denoting the fact that there may be a different growth rate and yet the susceptible hosts can be infected and liced by viruses whereas the resistant hosts only get this extra wash out term the viruses infect cells leading here because I've eliminated this late period and made it soon to be very small immediately burst of the burst size beta they also can get wiped out okay so we want to answer this question when can resistant hosts invade what would be the procedure I just explained this a moment ago find these fixed points look potentially at whether or not the new type might then increase in abundance if it starts off at a very small value so let me first of all point out that we have to think of none of the disease free equilibrium but the disease equilibrium right we are starting not in the absence of the virus the virus is there so we have to find these fixed points and here I'm going to rewrite this as beta tilde phi sv minus omega v where beta tilde is just beta minus one because it's just the discounted burst size I have to have one to make my burst and I'll just call that beta tilde so you can see that our initial conditions should be s star equal to omega over beta tilde phi and I should point out that this should be less than k tilde the level of the susceptible host that the viruses draw down should be less than that of the initial carrying capacity or the viruses couldn't have invaded if the viruses needed more than the carrying capacity to invade they couldn't have done so so we can only have the viruses present if they have drawn down their resources the host to a level less than that set by the chemist that itself I get some nods that that makes some logical sense good that's one point we can look at this s dot equation to find the value for v star and we can see that it should be something like r over phi one minus omega over beta tilde phi k minus omega over phi I think that is right this is our initial condition and we can ask the question what is r dot well we have r prime r yet initially we can think about the limitation is not being caused I'll now put an approximation there as caused by the impact and competition with other resistant host because there are so few caused by the presence of these susceptible host that are there minus omega r and now if I were to rewrite this in line which I know is annoying because you have to write it twice but I don't if I write this again as just r prime is the original growth rate one minus some cost right so I just want to make the cost explicit this r prime could be less than r because there is some cost times this stuff what you can see is we can take certain limits let's say that omega over beta tilde phi is much much less than k tilde meaning this virus is really wiping out the host drawing them down to very low levels if that ratio is very small right then we get a limit where r dot is approximately equal to r one minus cost minus omega r in other words as long as the host the resistant host can replicate faster than the washout late it could invade so when the virus the more efficient the virus is the cost of resistance can be very high and yet the resistant host can still make it let me just finish the thought so I can get to both sides of this and then I'll take your question another limit could be that omega over beta tilde phi is nearly k less than but nearly k in other words the viruses are only making a small dent in drawing down the host population in which case this number gets to be very small right and so if that becomes very small then the only way we have r one minus cost times some let me call this something near to zero I know there's a terrible notation I don't want to use epsilon I'll use epsilon again I'm going to use epsilon whenever I want something to be small epsilon minus omega r so here a large cost is going to be deeply problematic in fact it might not even matter what the cost is because it may be even if there's no cost right obviously if there was no cost it would be back to this we'd have a zero we'd have a zero invasion but if there's any reasonable cost then we're going to have this less than the growth rate because remember at this point the susceptible host has a net growth rate of zero so this suggests that if I wanted a when can host invade it depends on the cost and how efficient the original virus is did you have a question? because I'm assuming that at the beginning when resistant hosts are invading they are a small population and r squared is very small so I'm essentially linearizing without linearizing question from the chat? yes it's a minus omega not plus omega yeah this is a minus it just hit a parentheses minus yep yep two questions make them loud so the people on the online can hear this one is the first that is K tilde or just K? K tilde K tilde okay and the other could you repeat how did you find the initial conditions on S star and V star? these initial conditions are ones in which I'm evaluating the emergence of resistance and I've reached the fixed point but you are right that technically speaking the question was how do I choose these initial conditions I've assumed I've already reached an effectively a stable point in the original system mutations could have risen before that so if we were to actually do this as an experiment or if we do this as a stochastic model representing the experiment the time when this resistant hosts emerge is not going to be the same every time and you can see here also we have oscillations so I'm using something in which it's a time average effectively but you're right that in fact what I should be doing is evaluating over the whole cycle but by illustrating this I'm trying to show you what are the levers by which even resistant hosts can invade and the levers are first of all how efficient viruses lice the more efficient they are the more the resistant can be costly the less efficient they are than less costly the resistant is permitted for that resistant host to invade and obviously if there's no cost to resistance then we're going to get invasion because then we have two equivalent types and one of them bears this extra cost and essentially then we have this selection of the resistant type over the septal which has this extra additional burden any other questions? yep equation because in the term in the parenthesis we got one minus omega divided by beta 5k but that's s plus r divided by k I don't get how we get from there to there considering the initial condition I'm interested that the beginning invasion implies the population itself is near zero so I'm ignoring the r squared terms ok so again the point that I'm trying to make in this case is context matters before for the viruses it was the host context here for the resistant host it's actually something about the previous virus host relationship that determines whether or not whatever the costs are involved with this resistance are they permitted with respect to the ecological circumstance and can that resistance take off? ok so now in the next 20 minutes or so and I'll stop a little early my tendency is to go straight through right to the end without a break and then maybe I'll stop a little bit earlier today because I have a third part but I think the third part will probably be tomorrow rather than today this is going to be a lot to do today it's my feeling ok so these ideas go back a long time to Lacombe Volterra even in the late 70s by studying these virus host systems Bruce Levin Lin Chow, Frank Stewart together recognized that intrinsic feedback between this predator, this bacteriophage this obliquely litic bacteriophage and this susceptible host, the prey could lead to cycling for exactly the same reasons that Lacombe Volterra identified these counterclockwise cycles because there was outside pressure but because of this non-linear feedback could lead to oscillations in which you had this predator peak, followed by the host decline and then the predator decline host recovery and the cycle again and we get this loop in the prey predator phase plane Bohanin and Lenski and others identified in the 90s something called cryptic cycles where if you were to go out and either in a field setting where you're measuring things because you don't have a way to put them into culture back in your laboratory you might measuring some hallmark gene some feature that you think is associated with this type you might see things like that, cryptic cycles if you observe time series like this there's a little dip there but when you have measurement noise you probably see a flat line oscillations and I know some of you may be interested in things like Granger causality and inferring relationships from time series data it's very hard to see a relationship if one is flat very hard to see that there's a fundamental relationship these are called cryptic cycles you can even get things like anti-phase cycles things that don't look at all like the canonical predator prey cycles you can't get those from the predator prey system there are a number of features you can't get in part because you can imagine if it was a two-dimensional system the phase cycle, cryptic cycle goes back on itself in the phase plane which is not permitted the extra dimension is provided by evolution there's in fact more than one type there we're just looking at this three-dimensional system in the two-dimensional plane and realizing something is wrong it turns out you can get even wackier stuff when both the prey and the predator evolve and the wackier stuff is noted here where if you see I've labeled the red and blue and they're not incorrectly labeled where the predator peak precedes the prey peak in other words the prey seem to do best when there are the most number of predators around this is exactly the opposite it implies clockwise cycles rather than counterclockwise cycles so what I'm going to try to do today in the last bit here and I won't even try to do part three is explain this when I go from ecology to evolution to co-evolution good here is a particular example of how I got that and you should not go and try to write down all these systems of equations very quickly first of all I'm going to give you the slides and second of all you can get the Gestalt diagram we have two kinds of bacteria two kinds of viruses one of the bacteria is more susceptible to infection, one is more resistant one of the phages is more virulent and able to infect at a higher rate and lice at a higher rate these cells one is less able to do so you can imagine then we could write down four systems of equations that have growth rates, some infection and some loss if you were to then simulate it and look at the total in the solid or the individual strains in the dash you can begin to see it's possible to construct examples in which the virus peak precedes that of the host peak but I've shown the dash lines to give you a sense that although if you looked at the total you might see clockwise cycles the reality is that it's underneath that is something else and that something else our strain level changes in the frequency of the different kinds of genotypes, the host or the virus genotypes ok, so I want to unpack it this is an example of such a situation in this phase plane where I have prey and predator but now they're going around the clockwise way, the wrong way where we basically have a peak in predators the peak time in predators where the prey seem to take off it's exactly the wrong thing from a lack of ulterra perspective what I'm going to do is highlight four points and I'm going to use the following kind of notation to provide an intuition what I'm going to try to do is to use the blue bars what are the color, blind colors it's not blue red I hope blue red, what is the color what are color of blind colors anyway, the hosts are on the left I think it's blue green I don't think blue red is a color of blind color on the left are the hosts and on the right are the viruses the solid denotes the fraction of those hosts that have low vulnerability in other words are these resistant types the open fraction denotes the fact that these are high vulnerability easily infected in lice by the viruses on the right side with the red we have the solid being a low offense type and the open being a high offense type that is more likely to infect in lice so let's look for example at this peak point we have the most viruses around but they tend to be these low offense types how can the prey take off what is happening is as the total prey is going up what's really going on is that they're shifting to a low vulnerability type at a time in which the viruses are dominated by low offense viruses so they can essentially escape that viral infection pressure at the moment when the world is dominated by these high vulnerability types it's exactly the right moment that viruses don't need to pay a cost for these high offense types and the viruses are shifting to the low offense type just as the hosts are shifting to the low vulnerability types and that explains why the prey can take off but in a context in which we now have a lot of low vulnerability types and we have a low number of viruses there's a lot of hosts around and you better switch to the high offense types and when I'm saying better switch I'm anthropomorphizing what is happening autonomously then there's a shift to the high offense viruses but because there aren't many viruses around a host that doesn't pay all that cost for being well defended will invade so we get a high vulnerability type invading amongst the host just as the viruses start to switch and that leads to the prey taking off and this cycle repeats the wrong way these clockwise population dynamics which you can't get from an evolutionary system alone you actually need co-evolution to get the system to go the clockwise direction letting that sink in I spent a lot of time building up some of the foundations and then showing you examples both of these oscillatory dynamics and even things like cryptic cycles so I now want to ask the question are there actually clockwise cycles in phage bacteria data sets this is an image of what a chemist actually looks like I know it's a little bit small there's a sampling port, these vessels which are shaking, there's some line to a vacuum to pull everything out that ray omega and this is an interaction between a phage and vibrio cholera vibrio cholera is bad you don't want to get cholera it can be infected by a phage so that's interesting people have been thinking about phage based terrapheria for cholera on the other hand it's often moved and transported by phage so it has a very interesting relationship for some other day when they put this system into a chemistat this was what the empirical data looked like it's again about a 25 maybe 3 plus week experiment and it looks probably to you a bit like noise one of the things that we noticed and I guess this is one of the things that happens when you're in a field for a long enough time you start to notice things that look like aberrations and in the data itself the dynamics here would look like clockwise counterclockwise block of ulterra like cycles but they didn't seem to in fact it seemed to be the fact that when the viruses peaked were precisely when you see these big jumps the viruses seem to be abundant and all of a sudden the hosts take off so I want to focus on those two parts and take them into a phage plane and point out I know that not much data here but it's the data we have and I'll do some statistics in a moment these things seem to go around in a clockwise way rather than a counterclockwise way now it turns out that in this way at all paper they found that there was more than just this time series evidence but there seemed to be when they looked at the size of these plaques remember the plaques I keep talking about there were T for turbid plaques they didn't look like they were doing that well and B for big plaques which seemed to be very efficient at lysing like wise they seemed to be already identified multiple levels of resistant hosts so things that were relatively not going to be infected by the turbid kind of producing virus and some things that seemed to be less likely to be infected by both so it seemed to have all the right ingredients I think you can probably raise a question of is this enough statistical evidence for concluding that this is a clockwise cycle rather than a counterclockwise cycle so what we did is to take these time series the full length find the point-to-point variations and construct an ensemble of synthetic data sets that had the same point-to-point characteristics so we're creating an artificial time series so there are time series not just moving the points at random we're taking the differences adding them up making a time series and then in a cycle of that length asking the question what is the winding angle going around in a short period of time and how close does it come back to where it started this is the winding angle this is the distance from where the little orbit started this would be an idealized clockwise cycle and this is where the data sits and this is where the ensemble sits all these black points I want to give you an example we applied the same idea to the classic links-hair data the data that is used canonically to demonstrate the lockable pterolite dynamics have this counterclockwise cycle and you can see that we have similar sort of support to the extent to which we can conclude that links-hair data is counterclockwise we also are concluding that this virus host data is clockwise okay so we have some evidence here that there is in fact a very fundamental difference in the output of these interactions okay we also looked at a number of other studies with the same sort of method finding that there were other examples in which things seem to go the wrong way in the phase plane and for another day if you're interested there's a whole bunch of literature has come out showing that many of the cases that were supposedly because we assume they're prey and therefore they were lockable pterocycles when we didn't find them we tend to think the data was messy in fact there are many examples in which people have found anti-phase dynamics cryptic cycles and even these clockwise cycles so it turns out it's not just because you have a prey system that you get inevitably this lockable pterolite dynamic that only happens when there's not evolution taking place but evolution is always taking place so we often see exceptions okay I'm going to now sum up what I've done in the first two lectures with this one picture to the extent to which you take a system or at least sum up parts of it we expect there to be in the absence of evolution counter-clockwise cycles and I've given you the conditions even when there's invasion that could happen obviously these cycles could spiral into a fixed point when you have prey or prey evolution you get things like anti-phase cycles which things are going like this and clockwise cycles which I just showed you and again this should already point out we have this point in the phase plane in which there are two directions that it can go that clearly can't happen if we have actually a two-dimensional system it implies that we're taking this more complex system and projecting it down so evolution can undo this counter-clockwise cycle of lockable ptero and co-evolution can actually make it run the other way around so I think I'm going to stop there and conclude with these ideas that rapid change in the frequency of genotypes can have effects on ecological dynamics we saw that in the Bohan and Lensky experiment and I've also unpacked it when there's co-evolution that can have a fundamental change including leading to clockwise cycles and there are many other questions to ask but I won't ask them now I can continue part three tomorrow just to conclude before I wrap up I know you have an exam today so I'm not going to post other than the slides I won't post any reading but if you feel actually tomorrow I'm going to post both some background material, light background material that you can read in advance if you want to but it's not required but I will post a more detailed thing tomorrow which will get you ready for Thursday and tomorrow we can also have a discussion on what you all would like to get out of Friday I'd almost be inclined to do something hands-on and I'll talk to you about that as well which I think could be fun any other questions for today? yes why a clockwise cycle is so strange from a physical perspective when people first started to see clockwise cycles one of the earliest examples were people mislabeled data and the question was asked do hair eat links because that's what it seems to imply do bacteria eat phage that's why it seems unusual because it seems at the moment when the predator is most abundant the prey are delighted taking off so typically this is not what one expects in a predator or prey system and it takes appealing not just evolution but to call evolution or plasticity different kinds of types going back to your question about whether or not this is a genotype effect or a phenotype effect it actually could be plasticity or behavior change or phenotypic differences that could drive these same kinds of things it's actually hard to tell the difference here I'm attributing it to co-evolution but that's why it seems so paradoxical ok I think probably everyone needs a coffee break so I will continue tomorrow picking up here in part 3 thank you very much and good luck on your exam have fun with Fabio in about 20 minutes or so so we'll be back at 11 with the next lecture by Fabio Cicconi oh wait I started how far was I supposed to go today well in principle to 10.45 oh I can go for 20 more minutes yes ah I thought it was 10.30 aha a plot twist wait but maybe they want to take the time off now actually you know what I'm fine I'm gonna stop now anyway it's very hard to oh god I forgot I was looking at the clock it was like I have to finish by 10.30 no no no I don't know if you want to give ah how are you all feeling you were very excited that I was about to let you all go 15 minutes early ok you know what I can't undo that because then it's very cruel there is no coffee oh there's no coffee now ah interesting no there is coffee there is a coffee break but it's not now I'm all mixed up am I online still can everyone see this well ah we can do two things I could continue I'll make it by popular vote usually ah in fact this is supposed to be not a democracy but I will turn to a popular vote we could have a discussion a little bit what we want on Friday or I can continue to part 3 who would like more stuff I think they're already very tired you would no I can tell they're very tired can we talk about Friday then I'm going to give an extra 15 minutes and it'll be fine this is actually like intellectually for me I'm not going to finish this and we added up a lot of stuff here so on Friday and for those of you who are listening online I stopped 15 minutes early today whoops that was just me misreading the clock ah we can do another lecture or we can do a hands-on material in which you I've built some laboratories which usually take two and a half hours to run but you can get them started and then some of your afternoon time which my impression is free you can continue them in which they are presented as code bases that you begin to replicate some of these dynamics that I have elaborated here meaning you actually start to build models and ask in sort of a question answer there's a student version I have the instructor version and if you're willing to do that it would mean that everyone has to come with a computer on Friday and I realize for the folks at home it's a little trickier so I've talked about that does everyone have a computer with a python preloaded on it if anyone doesn't there's enough people with yes and I assume the folks at home the answer can be yes I won't be able to provide as much support for that but I will go over the concepts has anyone ever done these kind of simulations before like two or three of you so almost all of you this would be a new experience okay online is there a way for me to do a pole a slack pole later yes on the luck yes I think we can do it okay so I will talk to Matteo about doing a slack pole and figuring out if maybe we can do that on Friday I think there's some inclination would that be something you all would be interested in and actually some a practicum so I think we're going to I will go in the direction of a practicum for Friday which I will distribute and we may even start that and that can be something that you all can do if we don't get done and revisit it in the next week okay so even if we don't finish it we can get it started and revisit it Jaco I'm sorry to say that I missed her at the time and I've gone so much further with this derivation on the board that I'm there's no way for me to start and finish part 3 you all get 15 minutes in the Trieste Sun and you have half an hour break instead of 15 great