this is a hard chord so this cannot be so it's fixed this cannot change in time because these are the minimal condition for for for the for bacteria to work okay then we have two other region two other families one is the proteome allocated for nutrient uptake and the metabolism that is the one related to our case and we call this with by 5 p and then there is the allocation of proteome for biomass synthesis that are ribosomes okay now 5 p and 5 r as i will show can vary but are in trade in a trade off okay and this important relation between these different proteome allocation and the growth rate so this interdependence of cell growth and gene expression is a very similar work of a similar work important work of the group of tennis one that was published in science in 2010 so in this work they found phenomenological slow okay that described the relation between the proteome allocation for example here of the r sector and the growth rate of the species g okay so in this case you can see that so these are data from the experimenter in the y-axis this is basically a proxy for this proteome allocation for for the for the sector r 5 0 you see is the basically a hard core so this is the mean you cannot devote less than 5 0 because these are needed to leave okay so it's a compressible part of the 5 r and then you can see that increasing the the amount of of of proteome dedicated to this sector you increase the growth rate okay so it's a linear relation okay kt is basically a measure of translational capacity so how fast the microbial species expresses genome while raw is just a conversion factor okay so these are the details on the other way he also for the proteome allocated in the p fraction so the one of the metabolize for the metabolize also this is in a linear relation with the with the growth rate okay so these are the two phenomenological law that they observed it can be synthesized in this way so we have that the fraction of proteome dedicated to the mutual uptake is proportional to the growth rate the similar is for the r sectors that is proportional to the growth rate while the the housekeeping function is is uncompressible it's constant and we have this condition of course the fraction the total fraction should be one okay so what we have done is to generalize first the this phenomenological laws for n res nr resources and np species okay so now we have nr resources np species what we consider is that the p sector is subdivided okay for the different resources okay so phi sigma p one is the metabolize is the proteome allocated to uptake resource of type one phi sigma two p is the proteome allocated to uptake resource two and so on okay okay so in this case we just generalize the basically the same definition but considering the growth rate contribution due to the resource i this is this g sigma i okay so because we are focused on the on the on the proteome dedicated to the p sector we will for simplicity just denote this fight phi so when i there is no uh superscript superscript this is of the p sector okay so an important assumption that we are doing is that the sum of the different contribution of the growth rate of the different resources is basically this is the sum gives the total growth rate okay so the total growth rate is simply the sum of the growth rate contribution for each single resource and if you do this and you put this condition in this constraint the normalization constraint you go you obtain this condition for the uh for the proteome uh allocation for uh species sigma to all the resources i okay here and you can see that this sum this is constrained by this capital phi okay so this capital phi is the total proteome that can be allocated for the p sector basically for growth for me for me for resource update okay so this is just come from this uh water and logical law so now the model is quite easy to generalize in this sense because we have that okay for each resources we can have a proteome allocation of this sigma one that will be proportional to the uptake rate of the species and again and in in term this uptake rate that will contribute to the growth rate of species one following the the the laws that i just showed okay and the total growth rate will be just the sum of the different uptake rate for all the different resources and then we have a maintenance cost that is called the IQ that is similar to the that rate okay so the new equation are simply this coupled equation here so it's the same as before but now g sigma directly depends on the proteome allocation of the species so this is an explicitly all the all the equation so so now we'll go a little bit quick because i have not much time but i want to stress that this is really similar structure of the equation that i before presented the the change this now we have actually a more microscopic understanding of the different parameters okay because we are working at the at the cellular scale and also the new there is the new constraint the totally budget allocation now takes just the form of this proteome allocation constraint okay and very interestingly you see that this proteome allocation for the p sector is fixed but this is equal to this left part this is the third equation and actually you can see that because r changing time also phi must be variable so you see that the proteome allocation for growth so for for these metabolic strategies must be dependent on time so we don't have to suppose it we just came out from this cellular description of the proteome allocation so at this point this is just should be so if this phi must be variable then we need to write some dynamic dynamic equation for the phi that correspond to the metabolic dynamic metabolic strategies and again we have optimization of the growth rate but with these new constraints okay so the mathematics is the same as before so i'm not going to too much detail but you just need to optimize it through the gradient by imposing the constraint that is given by this equation and then you end up with this final equation here that is the same as before for the alpha okay but now the phi are the proteome allocated for the p sector so again you can study the result of condition you can look at the stationary solution of the system okay these are the stationary solution of the system from this you find that one solution is given by this r star and again you can call the ratio that ratio they are with the capital um theta and then you have that here you can write r star in this way and then you can actually the the the constraint the constraint the biological constraint now i mean this is a constraint that has a biological meaning is that you can see that maintenance cost is proportional to the proteome allocation total proteome allocation of the p sector and in this sense if phi is phi sigma increases we have to spend more energy to synthesize catalytic and ribosomal protein and therefore the maintenance cost increases okay so this makes sense and this is just a condition so this condition must be fulfilled for the equation for the stationary solution to make sense so if this is not fulfilled you don't have a stationary solution out there this is not enough for having coexistence for having coexistence again you do the similar as before you rescaling m s and phi and you obtain again that species coexist if the supply vector is within the convex hull okay and now phi is the proteome at the stationality phi star is the proteome allocation stationality of the species i to resource to the three sigma to resource i so again now you can see now you have that through the hope we start with our given initial proteome allocation in the first case if the supply vector is outside the convex hull you would have extinction but now proteome allocation change so that the new metabolic strategies are the supply vector is within the convex hull and you find and you find coexistence so this is what I was mentioning Leonardo used two engineering strains of the collie and completed demo on one common resources and one of the strain was engineered so to we can change the proteome allocation experimentally and we can evaluate the outcome of the competition as we do so and in fact we predict the outcome of the competition in this case okay so we can really do experiment engineering the proteome allocation on species and and and try to predict the outcome of the competition and again you can see that uh the the the velocity of the adaptation is again a fundamental parameter if they as you can see here this is a the condition where the velocity where the adaptation is high so the species can uh adapt and coexist all the species can coexist in this case adaptation velocity is uh is low and in fact you can have uh in this case you you you you don't observe coexistence okay so in this case you can see if ipsion is a ipsion is large enough you start to have extinction it is to say if adaptation is low enough you start to observe adaptation so depending on on the adaptation velocity again we have we have coexistence or not i wanted to align and i'm i'm i'm going very few slides and i concluded uh because this i there was a previous talk was mentioning this so depending on on the amount of of of supply rate even if adaptation is low but if supply rate is very strong then we can have a species to coexist so if we have a strong influx of of resources then you have that in this case more species than resources can coexist for example here you can see that if the supply rate is smaller the adaptation velocity is not enough and so only two species coexist yeah we have always three resources only the yellow and the and the blue species coexist but now if we increase two times uh the the the supply rate yet uh only two species coexistent rationality but the extinction take more much more time and then if we instead increase the file time the supply rate then you see that uh all the species now start to exist okay so also this the amount that the magnitude of the supply rate is an important control parameter for coexistence in this case so the conclusion of this part and overall talk is that including problem including problem allocation models of competitive communities give us new insight on the on the dynamics of our bacterial communities and the relationship between a problem allocation and population dynamics seems relevant to understand the origin of the high level of biodiversity so here we have like a microscopic let's say uh explanation or not explanation microscopic ingredients microscopic process that have an impact on the on the on the macroscopic coexisting pattern of the community so it's just a starting point but it's a starting point that bridge physiology of micron to a community ecology so we are very excited about this uh and because we think that this is really an opportunity because basically all the questions that probably we will uh we'll ask now are open questions because this was just uh really uh done in in the few in the last few months and but these consumer problem resource models suggest that coexistence could be reached by self-organization adjustment of cellular properties of this of the species itself okay and with that with that I take for the remaining time questions and I want to thanks of course all the collaboration again especially uh I really want to thank Leonardo for his great work and these are the two references of the what I have presented the one is already published in plus computational biology and one was just accepted like a week ago in the ISME journal so it will be available soon in the ISME but you can find the work in the bio archive so thank you very much thank you so much so we can start accepting questions there's one in the chat if you want to yes so now I can look at the chapter okay can adaptation velocity uh be interpreted as the lag phase to change the metabolic machinery this is a very very interesting question and actually this is one of the key missing aspect so because we know I mean not an expert of this but we have our whole machinery to to model metabolic fluxes so the metabolic networks through metabolic networks so it would be very interesting but yes I mean I don't have a quantitative answer to this but for sure uh qualitatively adaptation velocity must depend on the lag phase to change the metabolic machinery and sorry can I add something so I made this question I'm asking hi so do you think that this could originate trade-offs like so let's say that one species is faster and another species is lower and this could let's say uh change the outcome of the final composition because of these differences in the velocity of adaptation absolutely yes this is the understand this is the overall understanding so all the time that we observe the coexistence basically we see that there are trade-off in place and this is the main let's say attempt that we are doing in order to understand this trade-off we think that we need to go at the cellular scale for example at the plot location but also for example this is a very good path to look at different uh uh uh lag phases in the metabolic machinery and understanding in this way the emergent trade-off between species and and so to understand uh basically that this trade-off emergent from the from the cellular and and the metabolic scale will have a for a strong impact on the species coexistence this is exactly the point okay thank you yeah i have i have a related question um so i mean related to to to the adaptation velocity but i mean is there a way that we can quantify the you know adaptation velocity by comparing that with the data but i think that's important because um you know by quantifying the velocity we know you know if it's under out of the magnitude of the days or weeks or you know hours then we can know if that adaptation in the metabolic model actually reflects the physiological change without the mutations or on the you know more longer evolutionary you know time scale with a lot of mutations i mean do you have any thoughts on that yeah thank you yes i can offer a thought again i'm really i'm really not the guy going to the lab so i might say so i don't know if Leonardo is connected want to correct me or suggest something but my i think that my understanding this is a way to to to to to test adaptation velocity can actually be done through uh the oxy shift uh uh pools okay so if for example you take different species you cultivate you cultivate independently into into different medium but with two different resources and you can actually have an idea of the velocity of of adaptation for for each of the species so i think that basically this was i mean this is the main uh uh suggestion here so to to have a model which parameter can actually be measured in the lab so in this in this way you can actually perform the first experiment with only single species and then try to see when you put together if you want to put together you have expectation of what is going to happen and so test that maybe basically the the mechanism that you are thinking are important okay so that's what we have done in the experiment with with with Leonardo where basically we have we have uh cultivate uh yes so we have basically cultivate two strains and one of the strains was producing basically useless proteomes was allocating proteome in not useful for for the resources for the medium in which the species growth and in fact we saw that this leads to a competitive advantage in that quantitative way as expected by the model yeah yeah thank you thank you thank you okay i don't think i see any other no operation uh so if that's the case thank you again samir okay thank you it's very interesting in research talk thank you very much uh i will and have a good continuation thank you thank you bye bye we will have the next lecture in about uh 10 minutes so everybody can take a break bye welcome back everybody uh we can start again with our Friday session the next lecture that is Daniel Segre from New York Steel Boston with his second lecture about metabolism from his genomic skills to existence please Daniel thank you slides work yes okay great um hi everyone again and so we're gonna continue talking about metabolism in microbial communities so this is a perfect segue from the previous talk and i just want to remind you um in the previous time we were talking about the logic of the cell the metabolism as a resource allocation problem and just to remind you briefly briefly um the idea was to look at the complete metabolic network in an organism with knowledge of the nutrients that are coming in the biomass components that are needed to produce a new cell and new biomass and we introduce flux balance analysis where there are constraints on the concentration of each metabolite not changing in time so steady-state constraint there are assumptions of capacity limits for the nutrients that are coming in and for some reactions that may be known to be irreversible and we showed that one can use optimization for example finding the the state that is optimal for the cell to efficiently produce its biomass through max maximization of the growth rate and in general one can actually write this problem in the following way where there is a steady-state this expressed as this relationship between the vector of the fluxes multiplied by the stoichiometric matrix we saw last time and there are general capacity constraints in general what can put upper and lower bound to each flux and optimize any linear combination of the fluxes using linear programming we also showed the geometrical interpretation of this where there's a feasible space and we're really looking for an edge and vertex in this polyhedral cone this convex structure that will represent the point that maximizes our objective function now we want to dive straight into how we can apply this to the study of microbial consortia and I will first show why metabolism really matters for microbial consortia you saw some examples already but I want to walk you through the way we and others started thinking about this from the perspective of the complete knowledge of the metabolic capability of organisms and one slide I always like to show about this is the following so this is one of few instances I think where a lot of the interaction between different microbes are well characterized so this is a picture from a review by Colin Brander and colleagues showing different shapes these are different microbes that colonize in this case the human teeth so this is part of the human oral microbiome and what is stunning about this image is that really a lot of these links are known metabolic interactions between microbes some of these are the early colonizers and then there is this growing community that is we try to get rid of by brushing our teeth and what is known about this interaction some of these are known as contact interaction between different species and what is known also is that some of these interactions are related to metabolic exchange and as you heard before this is probably a very common way in which microbes can cooperate with each other or exchange with each other material in many different ways in this case for example a non-pathogen for pyromonas gingivalis can exchange different metabolites with another microbe in this biofield so this was you know this is interesting the question is how common are these interactions between microbes based on metabolic exchange can we model them using flex balance modeling and so on and I'll show you first how we um early on tested this idea that metabolism really plays a role in the formation of biofilms and microbial communities using exactly this uh known structure of the biofilm um this is an idea from a former student in my lab Barou Mazumdar who took this map and asked the following question so you can imagine having looking we knew the genomes of all of these microbes these were well characterized microbial strains so we could look at each of these strains and their intracellular internal capabilities in terms of what metabolic functions they had and at this stage we weren't ready to do yet genome scale modeling this flux balance analysis for each of these organisms but we took a much simpler approach which as I'll show you soon will was nevertheless quite quite insightful um and the idea was the following for each pair of microbes we could ask we could compute a metabolic distance so this is a pair way pairwise metabolic distance just based on the profiles of which reactions each organism contained so for example if you have two organisms here a and b um you know simple version of this reaction vector reaction content vector so for example organism a contains reaction one and reaction four organism b contains reactions three and four and based on these strings that are just binary strings which you can obtain from the annotated genomes of this organism you can compute a distance we computed a jacquard distance um to to quantify how different metabolically these two organisms are and of course you can do this for any pair of organisms and you'll have this matrix of similarity or dissimilarity based on the metabolic capabilities of this organism um so what was done then was to compute the average metabolic distance for different kinds of paths through this community so you can imagine taking paths that are we called order preserving so paths that only go upwards in the biofilm in what is known to be uh the layer structure of the biofilm from the early colonizers to the late colonizers so this would be an order preserving path because it only goes upwards and of course there are many other random paths that do not preserve the order of colonization that can jump up and down um between different species in the biofilm so in doing so we could then compute the average distance um between all pairs of organisms in both the random paths and the order preserving paths and what we observed was that there was a clear difference between the distribution of metabolic of average metabolic distances between the order preserving paths and the random paths and in particular the order preserving paths had a pairwise average pairwise distance between subsequent organism that was significantly smaller than the average pairwise metabolic distance for any possible random uh non-order reserving path um so this was an interesting indication that somehow if you look at the correct order of colonization there is something unique about the species species similarity in terms of metabolic functions and um as you may imagine one possible interpretation of this is that organisms that are um metabolically more similar to each other will have metabolites to share and will be able to gradually connect to each other metabolically building this biofilm now um this somehow is a there was an early indication that metabolism matters in microbial communities and in particular in this case in the order of colonization of the biofilm we observed by the way that if you look at the same um property of in pairwise average distance for non-metabolic genes you don't see this clear distinction between the two distributions um and of course there are different possible interpretation of what was found right this could reflect really the fact that organisms build on top of each other but could also reflect at least partially an environmental gradient for example maybe there is a more anaerobic environment at the bottom of the biofilm and increasingly aerobic as you uh as the biofilm grows and in that case perhaps the similarity in the similarity reflects uh adaptation to this gradient uh but there was something else that came out of this which uh felt was quite interesting this kind of paradox if you take this idea that metabolically seemingly similar organisms will tend to stay close to each other and build the biofilm um what would prevent this to collapse into a you know absolutely minimal distance boredom where all the organisms are really clustering in in a structure by uh by their similarity of course in that situation competition might dominate the capacity to exchange metabolites to synergy but it raises this interesting question of whether there is an optimal metabolic distance for metabolic synergy between uh complete difference between two metabolic networks and complete similarity and I don't I don't think this is a resolved problem I think there are interesting papers coming out recently on this question I want to show you how we start addressing this early on by using a concept that is called elementary flux modes and I don't have time and I'm going to hold the details of that but I'll just mention what is essential here which is this elementary flux modes are a way of um enumerating all the possible pathways in a metabolic network all the possible minimal unique pathways in a metabolic network um and you can take an organism um and um duplicated and ask the following question if I put two organisms together and just count the number of pathways that are possible when I have these two organisms together uh relative to the elementary modes that are present in organism one plus the number of elementary modes the number of these pathways that are present in organism two and if the two organisms are completely non overlapping and you can build artificial metabolic network to um engineer to have an arbitrary degree of overlap between these two networks so if these two networks have zero overlap then if you come compute this quantity the number of pathways of the system all together divided by the sum of each of the two individually then of course the sum of the pathways exactly the the sum of the pathways present in each organism and this quantity is one and at the opposite extreme if you take two organisms that are exactly the same have exactly the same metabolic capabilities uh when you compute this quantity each organism the two organs are the same so the elementary modes of the uh junction between these two organism is really the same of each organism alone and this quantity will have a value of half and what is interesting and you cannot really uh quantify this unless you actually do the calculations and we did these calculations you can find that there is and this is almost like an um analysis theorem right there there is a maximum here uh as one might expect but it's interesting that you can find this sweet spot of how much metabolic overlap will lead to the maximal number of new metabolic pathways that are embedded in this combined systems uh system of organs one organism two so this is just based on the topology it's a very simplistic calculation but it points to the possibility that there may be out there some um a sweet spot or some kind of uh ideal level of metabolic similarity that will lead to maximal enhancement of metabolic capabilities and of course this doesn't take into account uh competition it's just about how much new metabolism can be done by bringing two organisms together and as I mentioned last time I'm gonna pause occasionally but feel free to interrupt and if I'm not monitoring the chat but there is a question just someone please stop me and I'm happy to uh pause and address any question so from this early very um simple um presence absence analysis we really want to move to a stage where we can model computationally the dynamics of communities and as we saw in in multiple talks including the previous one one can look at uh a community as a set of entities right that can represented between you know in different different different levels of description this would represent maybe more like a lot of altera model where you have individual variables representing individual organisms you can try and model the community as um an ecological system in this way um there is the possibility which is really what flux balance can do and what we'll focus on where you can think of uh the circuits within each organism and try to predict the interactions between different species based on what you know about the intracellular circuits of each organism and as we'll discuss probably uh next time there is also the possibility of thinking but I want to hint to this now there is the possibility of thinking of a community as a soup of enzymes where perhaps uh for a complex community what really matters is what functions are present overall in the community and we can ask the question of whether or not compartment compartmentalization matters so is it important to know which functions are performed by which organism or is it possible and useful to think of a community as this overall conglomerate of metabolic functions so for now we will focus on this type of modeling where we really uh know the intracellular wiring we know the environment and we try to predict ecological interactions in the dynamics and the structure of the community and we'll also talk a little bit about design of the community and I see there is something in the chat if it's a question yeah do you want to ask this question uh yeah yeah thank you so I type in the box so my question is that for the first example of the order of the colonization in the bathroom so I saw the result is that the bacteria in the order of the past has smaller metabolic distance so my question is that does that mean that you know if there are two types of the the effects which is one is the environmental gradient another one is the ecological interactions could could change the dynamics so would that result mean that the uh environmental ingredients dominate over the ecological interactions uh because in my mind you know maybe I'm wrong just think that if the ecological interactions are more importantly to you know preserve the order then there might be a lot of cross-fitting between the metabolically more different bacteria that might be prevalent which is not the case in your data so I just want to add some some comments on that question yeah that's a very interesting question and I guess I think from this early analysis we don't don't really have enough information to to determine this what I thought was interesting here is that if you practically look at what organism what like the evidence we found there was that if you look at the organs that are close to each other in the biofield there seems to be a tendency toward a smaller metabolic distance and I agree that we don't really have enough information to you know to know whether to interpret this as okay there is a just driving force that is the gradient of environmental oxygen or nutrients and so on but there could be also a situation where each organism right modifies the environment for the next organism to occur and and then the question is and the possibility what I what I think might be happening is that uh organisms that are too different from each other they may not have enough um cross-feeding opportunities and and of course if organisms are too close they will compete but there may be some sweet sweet spot in between and we can get back to this I think uh the answer to this will really come from looking at the more advanced dynamical models will which uh analyze but I'm happy to go back to this question which is very interesting so um so we want to move to to these dynamical models and I want to show you how there was a different direction early on and now there is an explosion of different experiments of this kind but this is really these early days of trying to think of synthetic cooperation and now the idea of building synthetic microbial communities as a way of testing hypothesis and checking what is really happening when you put the organisms together can microbes cross feed and so on and so forth and there were these early attempts that were really interesting and a motivation for a lot of the things we did later on so this was a paper from um Winnin Shu and collaborators in 2007 and this was an engineered cooperation between two yeast strains one of which could not produce adenine and the other could not produce lysine and the idea was that only when grown together they could really uh would be able to survive and in fact this was indeed the case so this was an engineered cross feeding interaction that made each of these two strains uh completely dependent on the other it turns out and Winnin has continued working on this doing beautiful work showing for example that it's not clear um that the metabolites that you would expect being exchanged but it's the terminal portions of um of these pathways are the one being exchanged and there is a lot of interesting aspects of these dynamics and I'll mention more later on the other example um more focused on uh just trying to find different types of interactions as opposed to engineering them this was done with E. coli strains uh library of mutants and the idea was to put these mutants together first I mean if you grow them individually they grow fine in rich medium uh individually they would grow very poorly in minimal media because they're mutants that lack the capacity to synthesize I think these are only amino acids but occasionally when you put them together you could see synergistic growth so this was a way of trying to detect new interactions and there are a number of new interactions that were detected this was worked by Ed Wintermuth and Tom Silver so when we started thinking about this we thought it could be interesting to um try and mimic some of these ideas using stoichiometric models this was work that a former student in my lab Neil's Clipcord pioneered and we tried to do things in a slightly different way so rather than tweaking the internal circuits of the cell as done in these previous examples or you can do mutations and try to induce interactions based on changes in the circuits in particular auxotrophies or removal of genes that are essential for producing essential compounds we thought that perhaps it would be interesting to tweak the environment and so take two organisms that are natural occurring microbial strains and ask whether we can induce an interaction not by changing the circuits inside but changing the environment and the idea was that well first of all this is you know simpler to test potentially because if you want to test in particular a high throughput interactions of this kind it's much simpler to just provide different nutrients experimentally than having to do mutations to the strains and the other aspects of this which turned out to be really the beginning of a new line of research is that and a lot of people are obviously interested in this the environment clearly have a has a strong effect on modulating interactions so this is a way of starting to look at how the environment can really you know kind of environment environmental changes induce interactions and what is the role and how much variability there is in these interactions as a function of environmental composition you can change in principle the carbon the nitrogen the sulfur phosphorus source and so on so there is an endless combinations of different nutrients that can be used to try and induce these interactions of course one could use a rational approach and we'll see more of this but for now what we did was just simply use flux balance modeling to try and find in a large space of possible compounds some that would induce interactions and I need to tell you a little bit more about how this is done in practice because it's non-trivial right when you look at this we saw how to model an individual organism but how do you know how do you go from a stoichiometric model of an individual organism to a stoichiometric model of a community where you have two organisms together and the answer in the end is really something that existed already in the flux balance world but was used for different purposes and the idea is to use compartments so you can build a compartmentalized model and I'll illustrate this with this very simple example where you have two organisms one and two and they have a very minimal network organism one can produce b from a organism two can produce c from a but each of them has a biomass that depends on a b and c so each of them needs all all of these three compounds to survive and if a is the only compound provided in the environment the only possibility for these two organisms to survive is to exchange b and c so this is a you know minimal example of cross feeding if you wish but is all it also illustrates how you can build a model of a community using stoichiometry and flux balance modeling and the idea is that you can define you'll have multiple versions of each metabolite so you'll have an environmental metabolite a and you'll have an environmental sorry a metabolite a that is in organism one you can label it as a one and you have a metabolite a that is in organism two you can label it as a two and so on and so forth and you can write the system of reactions just labeling the organism the metabolites based on which organism they occur in and and you'll have essentially essentially a block diagonal matrix representing this system of two species interacting so this is um you know in a very superficial way that you know the way this stoichiometric models for communities can be built based on this multi compartment model this was first proposed by uh stoner and um david stahl in a very nice molecular systems biology paper in 2007 so we took this approach and used it to scan systematically systematically the space of possible environmental metabolites um and just to illustrate briefly the way this algorithm was designed you can first take two organisms and ask under what conditions can these two organisms grow both a growth rate that is above a minimal threshold and you can search all the possible carbon sources you have all the possible nitrogen sources and you can do the same for other elemental sources but you'll have you find those that provide growth to the pair of organisms together you'll have a set of putative media that support growth of the whole ecosystem and now what is interesting once you take this media you can ask for each of them whether it also supports growth of each organism on its own and all of this again to remind you you do you can do easily using flux balanced analysis so you have by definition right we've had many many different environments all these different combination of carbon and nitrogen that all support growth of species one and two together in this joint stoichiometric model but then you can take a given environment and ask will it that environment also support growth of organism one alone and organism two alone and if that the answer is yes then you found an environment right that supports the pair together but supports also each individual organism and this is a case where the organs are not really interacting they can grow on their own they can grow together nothing interesting about it but you can start finding then environments where for example the two can grow together that's again how this were originally found organisms one can grow but organisms two cannot grow and then what this means is that the two organisms grow together but two cannot grow by itself this means that one must be providing an essential component to organism two same situation here so if you find environments that satisfy these conditions these would be environments that would support a commensal interaction where one organism depend on the other and if none of the two organisms can grow on its own then again because the pair by definition was growing then what you find is a set of conditions that imposes induces a mutualistic two-way cross-feeding interaction between the two species so what is nice is that you can easily make this list of many many environments and for each of them you can test whether which of this is the case and ask how many times will you find environments that induce for example these mutualistic interactions and I should say when we started doing this we really didn't know what to expect how often would this happen and the idea was to start getting an idea of how frequent how prevalent how large is the space of this possible cross-feeding interactions in metabolism and what Niels found was actually oh yes sorry just a quick question maybe I missed this but in the joint FBA what's the objective function is it the sum of both great questions thank you yeah thanks for asking yeah so there are different flavor flavors of this joint FBA in this early so I'll we'll get back to this because that's actually this very question motivated a lot of other things I'm gonna talk about but compartment models you have to choose an objective function and the very first case was based on maximizing a linear combination of the two biomasses so you can create a new reaction that builds a linear combination of these two biomasses with a fixed proportion but then you determine in advance right what is the proportion of the two species you can do this by scanning many different proportions and seeing which one seems to be most growing faster but it's it's a little bit tricky so you can see already that this question of what is the objective function of a community turns out to be a really interesting question but also a tricky one and if you choose an objective function for these communities you know you this is a little bit like testing a hypothesis what we did in this case because all well all we wanted to know is whether we could find an environment that supports growth of each organism so what all we did was in this case ask that the growth rate of each organism has to be above a certain threshold so we asked that each of them grows at least a certain amount doesn't matter you know they don't have to grow optimally they have to grow above a certain threshold and then what we did we used mixed integer linear programming to minimize the number of exchange reactions so we asked what is the minimal way for these two organisms to potentially exchange something so that they can go grow both above a certain threshold which is why if the minimal number is zero that's totally fine maybe the two organisms grow together without you know having to exchange anything and in this case is there will be a non-zero number of exchange reaction but thanks for asking these questions because I forgot to mention it so that's clarified yes yes thanks okay great so so let's go back to the results here which again were quite interesting and this exemplifies some of what we found so these are seven species for which we run all these pairwise interactions and there is organism one and organism two here but these are the same organisms you can look for example at the interaction between E. coli and Salmonella and what you see in this pie chart the overall size of the chart pie chart represents the number of media of different combination of nutrients that we found that could support the two organisms together so for example if you look at E. coli and Salmonella there are millions of different nutrient combinations that can support growth of the pair whereas for example E. coli and H. pylori have a very small number of sets of nutrients that can support both of them together and then out of all these possible nutrients you can look how many of these are of this neutral kind that is environments that support also growth of each organism on its own and Salmonella and E. coli are very similar in their metabolic capability so as expected you find a large proportion of this green portion of the pie chart of nutrients nutrient combinations media that are essentially good minimal media both for E. coli and Salmonella and of course they support growth of them together but there is nothing interesting about this but there was also a lot of interactions that are commensal of one organism providing something for the other organism and what was most stunning and we really didn't expect that there are a lot of opportunities for this cross feeding all the yellow portions in this pie chart are cases where really this would be environment such that if you feed those nutrients to those two species they really need each other in order to survive and I should say that this is only based on stoichiometry right if we were to do exactly the experiments in the lab I wouldn't expect all of this interaction to occur because the fact that this stoichiometry have this property doesn't mean necessarily that the organism will have the right regulatory program to express the right genes to induce that interaction and so on so this is a purely theoretical flux balance stoichiometry based diagram but what it illustrates right is that there is out there in the microbial world there are millions of opportunities for cross feeding and that they're strongly dependent on the environments in which the organisms grow and also illustrates that in principle if you learn how to manage these possibilities there could be a lot of opportunities for engineering communities where you by designing the environment you could decide whether or not two organisms will depend on each other so this was promising but as I was hinting to and you know the question asked was hinting to and there are some underlying assumptions in this type of model that are that are somehow tricky and will so in particular right the require assumption on this ecosystem level objective of what you know how do you manage these two biomasses how do you know there there is some interesting hypothesis on the possibility that maybe you could use thermodynamic base objective function so I think this is a fascinating question but there are other limitations of this approach for example in the same way as flux balance will not allow you to predict intracellular metabolite concentrations because those are factor out when you assume the steady state for similar reasons with this type of compartmentalized based approaches you cannot predict the amount of each species in the community which is often one of the main main things you would like to be able to know so you cannot predict how much there is of each species at steady state and this is a major limitation of this approach another limitation is that it's very difficult to do spatial temporal dynamics you cannot really do dynamical models based on this because it's all steady state approximations and you can just compute one steady state and as we'll see later the solution to this is going to be what is called dynamic fba or dfba which is an extension of flux balance analysis which will simultaneously solve a lot of these issues and I think open up a lot of new opportunities now before we go there I want to pause for a second and think about this question of why would microbes exchange metabolites right and there are many different angles for this and we'll see this from different perspectives and this came up and we'll come up again I'm sure in other talks but you know metabolites are part of this strategy that microbes develop to you know grow and produce their own biomass why should they give out metabolites to someone else and as we saw last time right there are some pathways such as fermentative pathways that inherently give rise to secretions and these secretions might be helpful to other organisms but it's not clear how prevalent these secretions are and whether indeed these secretions are typically you know something that would be very costly for microbes to produce and then give rise to questions about stability cheaters and so on and we'll kind of get more into this so so one way we started thinking about this was first by quantifying really the cost of metabolic secretions and this is work by Gardison and the lab Alan Pacheco in the first observation we made that motivated this this was initiated by Niels Klickler before it was asking the following question if you take a flux balance model say for E. coli and you can grow it on different combination of nutrients and ask the following question if you impose a secretion flux will you induce a reduction in growth rate so how much how much will you have to pay in terms of the growth rate if you impose that that organ is secretes a certain metabolite and as you might expect right there are secretions in this case succinate such that if you ask the cell you know before maximizing growth you say okay there has to be this amount of flux of succinate going out of the cell and then you maximize growth and the growth rate you obtain in this case for example on glucose glucose and glycerol as carbon sources the growth rate you obtain is significantly smaller than the maximal growth rate when you don't impose a secretion flux so this would be a case of a metabolite by this costly and kind of corroborating what we're saying earlier whether or not metabolite secretion is costly depends strongly on what are the nutrients on a different set of nutrients succinate production is not not very costly until you produce a lot of it but then what is interesting and let's look at this first actually because that's an example we already illustrated before there are metabolites such as acetate if you're growing under carbon lean sorry oxygen limited conditions this is something that will spontaneously happen and in fact if you impose a secretion flux of acetate that is small or zero right the cell will not be able to grow optimally and in fact it will grow better when you impose that there is a high secretion of acetate possible okay so in this case the secretion is actually beneficial and there are some some cases in between such as format where apparently for the two environments explored here secretion doesn't change the objective function doesn't change the growth rate so these are kind of neutral secretions so we we will call both you know these two kinds of secretions for the purpose of this up we'll call them costless that is metabolic secretions that do not impose a cost or at least a reduction in the growth rate under this assumption and I want to remind you this is particularly relevant since we just heard about the cost of protein production and the capacity of embedding into flux balance model the or consumer resource model that is also possible in flux balance models the cost of protein production so this doesn't include any of that this was done with regular flux balance models but one could extend this kind of approach to models that also include protein production and the cost of protein production so let me show you what we found so what Alan did he wanted to I find how frequent are these costly costless secretions in the microbial world and again the idea was to scan many different flux balance models looking for how often would we find we find costless secretions so I designed the following experiment the idea was to have enough variability of environments to explore systematically a large space so again this was done only in silico and I'll mention later there is a follow-up work that is being done done experimentally now on this but the idea here was to give two different carbon sources chosen out of a pool of different carbons and also choose whether or not to provide oxygen so we could do this in silico experiments aerobically or anaerobically and as hinted to before of course this can have a big consequence on whether or not there are secretions or what secretions being produced and then we computed the possible the maximum growth rate of two organisms of a pair of chosen organism under these conditions and what we did was estimate if any of the organisms could grow what metabolite could be secreted in a costless way so we asked is there any metabolite that upon maximizing growth each of these organisms could produce and now we did iterations where these costless metabolites was added to the medium rather than medium and we repeated the experiment growing the same organism but now on the original medium plus the costlessly produced metabolites so this the idea of this analysis was that we could get inside both into what costless metabolites could be produced but also whether these costless metabolites were really useful for facilitating growth of a second organism so we did this for these two different conditions of oxygen availability 108 carbon sources 14 different species for a total of over a million unique simulations and I'm not going to show all the details there is a lot of data that came out of this that is available in this nature communications paper but I just gonna illustrate what you know summarize what we found which is that there are many different type of secretions I know much more than what we originally thought and not just the organic acid the organic acids are this light brown portion so there is a large portion of organic acids but there are a lot of other molecules carbohydrates and your organic compounds the metals are not necessarily so interesting because they're just coming in and out of the models but there are some peptides some phosphate that are being exchanged and for an overall total of about 60 metabolites and a little bit more when you have no oxygen and as expected somehow there is a little bit a larger number of secretions when oxygen is unavailable which is interesting in itself and and again it's interesting to think of this in terms of ecological niches and whether really you know that's something that would be testable whether indeed there is prevalence or additional metabolic interactions in anaerobic conditions and the other thing that was interesting is that these secretions could really induce interactions across different species so for example overall all the different species if you focus first or for example the oxygen dependent one you can count how many see how many simulations both organisms could initially grow and there is a certain number and there is this is what is interesting this is the number of combinations of microbes and environments where growth could occur after cross-feeding so after at least one round of costless lipidus metabolites being fed back into the medium so there is a large increase in the possible growth capabilities induced by this costless production and these are the proportions of which either zero or one of the two organisms grow so the interesting part is that you can almost double the the number of combination of nutrients in an organism in which there is growth of a pairs of organisms because of this cross-feeding interaction so again this was based only on flaxbanas modeling based only on stoichiometry but it's a different illustration and points out the fact that in this case right the cross-feeding can be induced by metabolites that are really not inducing not causing a decrease in the growth rate of each individual organism and the idea that right where there's a lot of interesting work and I'm sure there is many cases in which interactions are due to metabolites that are costly and this could be evolved traits and we're gonna talk about this soon but it would be evolved traits where an organism produces a metabolite that is costly because it leads to an advantageous interaction but there is the idea that is merging from this analysis that there are a lot of opportunities out there for costless interactions things that are induced just by organising what is best for themselves and at the same time in doing so throwing out there something that someone else can use I view this as a little bit like as recycling right in social community social you know human societies you know there are things you know when you are finished with with your milk you know we throw away the bottle and if you can actually recycle it instead it doesn't cost anything to you but it actually can be valuable for someone else and the idea is that this kind of interaction may be very abundant in the microbial world and I this is a map of the specific metabolites that can be exchanged I'm not gonna go into the details and to this but if anybody's interested you can actually look at what specific metabolites are being secreted under what conditions and there is a lot of interesting follow-up analysis one can do on this but I want to summarize this just by showing that the emerging network that we you know we we looked at by analyzing this costless interaction so this is a network of what organism could feed which other organism in this set of organisms we analyze based on costless secretions and the picture that emerged here is that there is a dense network of possible interactions that emerge spontaneously between different species that may not require organisms to give up anything anything valuable but are just emerging property of the system and somehow this was similar to the result that in parallel Alvaro Sanchez and when you hear about this found in this organ in this community grown from on simple carbon sources from plants and soil and and this is one of the illustration from that work which you'll hear I'm sure more but what is interesting again the same picture emerged that on each organism could grow on the spent medium of each other organism somehow suggesting again that there is really there are a lot of dense networks of exchange out there and and this is somehow I must have been very different from what I was expecting initially that is that it's not necessarily individual targeted interaction but there is probably a dense network of possibilities now um and as we started thinking about this with these models um yeah I think yeah 50 moments as we started thinking of whether in addition to looking at the natural interactions in communities we could use flux balance models to also purposely design cross feeding interaction between species um and the idea was that you know when we look at this natural interactions through stoichiometry we really look at many different organisms many environments and and we look at what are the possible outcomes but we wanted to do this in a more targeted way and also potentially get insight into what what I'd like to think of as deep symbiosis where the exchange metabolites are not necessarily byproducts end of end results of byproducts of specific pathways such as amino acids but more convoluted interactions that you may not be able to look at or find intuitively such as exchange of two amino acids again so let me show you you'll see in a second what I mean so the idea and this is sorry I worked by Megan Thomas another from a student in the lab with the annus plus the annus plus colitis and others and the idea here was the following we took E. coli and you can ask the following question you have a certain number of reactions in E. coli about a thousand but you can force E. coli to use a smaller number of either internal reactions or exchange reactions so imagine the way I think about this is you have a knob you can say okay instead of using a thousand internal reactions now you're allowed to only only use 900 and you can ask how well can you do and you can tweak also the number of exchange reaction how many metabolites you can transport from the external environment and at some point you can imagine let's say focus on the internal reaction if you turn this knob too much to the left right you you decrease the number of possible reactions too much at some point the organism will not be able to grow you might see at some point a decreasing growth if you limit the number of reactions and at some point there is no way for the organism to grow but then one can ask the same question for a pair of organisms so you can start with two E. coli that are initially exactly the same and you can impose these constraints on the internal reactions on both of these but now each of them could choose a different set of reactions to use and now you say okay I limit to let's say seven percent of the original number of reactions they can use but now this you know top organism could choose one set of reactions these other organisms could choose another set of reactions and what we were wondering was whether we could find a constraint that would not allow an individual organism to grow but would make it possible for the pair of organisms to grow together again in an obligate synergistic cross-feeding interaction that would be now induced by our arbitrarily tuning the number of possible reactions and again this was done using classical fba in this multi-compartment model and I'll show you a couple of outcomes of this analysis this was done using mixed integer programming linear programming where in addition to the variables representing each flux we had Boolean variables representing whether or not each reaction is present in each of the two organisms and I'm not going to go into the detail of the optimization it doesn't complete a trivial and it gets pretty time consuming as you go to full genome scale models so it's there is I think interesting computational work to be done in terms of trying to improve this kind of algorithms but let me show you first the outcome that we got from looking at the core metabolism of E. coli so this is a simplified model of E. coli metabolism and what was fascinating is that one of the solutions that the algorithm found was these two E. coli I would say subspecies right so these are species that are limited in their capabilities and you may recall here this is glycolysis this is the TCA cycle but here each of the two two organisms uses a different half of the TCA cycle this species uses one half this other species using this using this other portion and the exchange with each other multiple metabolites including pyruvate here but also some of the byproducts of this you know this intermediate byproducts of the TCA cycle and what is interesting here is you know there's there's a few things one is that it would have been very difficult to come up with such a scheme for a possible cross feeding without the algorithm this is again not the exchange of two amino acids this is what again they're called deep symbiosis where two organisms exchange interest uh metabolites that are uh are known to be uh transportable but are part really really of central carbon metabolism and there are multiple exchanges that are required for this to happen the other part that is interesting is that in nature there are indeed organisms that do half of the TCA cycle the incomplete TCA cycle as we hinted to um some of these are related to the capabilities of producing amino acid through these reactions but it's quite interesting that one of the solutions of this algorithm really resembles some of these half TCA cycle strategies that are found in submarine bacteria now when you go to uh genome scale models this is much more complicated than it's really impossible to visualize the whole network so this is another way of visualizing what happens so what you see here again is the number of exchange reaction the the limit on the exchange reactions and the limit on intracellular reactions so you can imagine right you start from this is where you have this corner top right all the reactions are possible and you gradually can decrease the number of allowed exchange reactions or internal reactions and these uh uh areas shaded areas in green and blue represent the feasibility and the growth rate that is possible um as you do this and as you can see if you start with one organism right one organism is feasible in this region so what this means is that right if you decrease the number of reactions to below about 250 one E. coli cannot grow anymore on its own uh same if you decrease uh through this it seems like a Pareto frontier this exchange reaction if you decrease them too much the single organisms cannot grow but this is where you can see again what we're hinting to before that if you have two organisms in each of the each of them has these limitations then there is a much larger space where the two organisms can grow if they grow together in the exchange metabolites even under constraints uh below this 250 thresholds up to 210 and so or so where uh yeah individual organisms would not be able to grow but the two organisms can grow together and then there is a lot of data again imagine for each of these cases you have this genome scale models of the E. coli networks and you can look at what is the structure of this networks what metabolites are exchange exchange and so on and this is just to exemplify the kind of insight you can get you can see that there are regions in this space as shown here where acetate is one of the key uh exchange metabolites which is not surprising again acetate comes back again but there are regions when you go to the extreme uh you know you push this pair of organisms to the limit then turns out succinate uh again one of the TCA cycle intermediate uh is one of the metabolites you would exchange you expect to be exchanged in order for these two organisms to coexist and there are areas where some um amino acids need to be exchanged so probably the two organisms will do will perform complementary metabolic functions in exchange amino acids and again there is much more one thing that I you can't really see here but I just want to um highlight is that there is an interesting very thin layer here between the one organism the two organism uh regions where one organism is still possible and two organisms of course are possible but there is a an area here where the two organisms growth rate is faster than the one organism single organism growth rate under those conditions so what this would imply is that if you were to put a chemostat experiment and force this organism to grow at a certain rate there would be a situation where the two organisms would outcompete the single organisms hinting to the possibility right that this could be a transition where even if a single organism could grow on its own but cross feeding could be evolutionary advantages and and give an advantage to a pair of synergistic organisms it's you know this is not not saying anything about the details of how this could happen in real life but it's showing that in principle there is this um overlap that would give an advantage to two organism solution rather than the single organism solution okay we have uh five more minutes and I will um tell you a little more about another aspect of this genome scale model so right we dealt so so far only with organisms that are very well characterized you know well-built models such as E. coli either organisms for which we had very good models but um as we hinted to right we want to start understanding complex microbiomes or more complex microbial communities and one of the limitations that we have to be aware of is that in many of these cases the knowledge about the metabolic capabilities of this organism is much more limited than what we have for E. coli or yeast and so on so the question is can we get around some of these limitations and um David Bernstein another former student in the lab did some really nice analysis using what we call metabolic percolation in order to address this question and this I think is a really interesting area because if you think about this when you have a metabolic network that is not working if you do flux balance analysis and you cannot produce a certain amino acid the network will just not grow and will will give you zero information of why it's not growing so it's a little bit like I think of it as a broken computer and if you don't know you know where to start you know that one element in the computer is missing but the computer is not working doing diagnostic could be very challenging and so on and and this is all with the situation we face when you build a flux balance model and it's not working it's very challenged to find out why and of course there are methods for doing this you can look at pathway by pathway but the idea that David started thinking about is that one could think of the problem of biomass production as a percolation problem where metabolites that are present in the environment could be present and chosen with a certain probability and then you can ask about the probability of producing a given metabolite and you can ask this for any metabolite that is part of the biomass of an organism and systematically choose many many different random environments based on these probabilities and ask what is the chance that by giving as inputs the different metabolites I'll be able to produce one of these biomass components and the advantage of this is that even if the network has some holes you'll occasionally add some of these metabolites by chance and you'll get an overall picture of how producible a metabolite is given all these different chances of different metabolites from the environment being available and this is illustrating for example how much more robust this algorithm is regular relative to regular FBA so for example if you remove randomly reactions from a network with the you know the fraction being let's say one in 100 or one in 10 FBA will soon not be able to give you really accurate prediction of the growth rate but the productivity will tell you whether or not an organism can grow even after you remove a pretty large number of reactions from the network so it doesn't give you accurate prediction of the growth rate but it will tell you what is the productivity of each metabolite in the network in a way that is very robust and I'll illustrate one of the applications of this this was done again circling back to the human oral microbiome and I'm showing here just a big heat map of 456 strains from the human oral microbiome and 88 metabolites that are part of their biomass and the darkness of the red shade here indicates how producible each metabolite was for each of these strains so this is basically a map of the metabolic capabilities of all these different organisms based on this percolation algorithm and I'll there is a lot of information here that could be compared then to for example co-currents of different organisms in microbiomes and we started doing this but I want to illustrate one specific example that was quite interesting one of these oral microbes is what is called TM7 you can really see it from here but there is a set of organisms that are not are uncultivated bacteria so these are organisms that are cannot be grown in the lab on a medium like E. coli and many other bacteria they depend on something else that is unknown and they can only be cultivated in cooperating together with another organism in this case octino bacteria and there is a lot of laborious work at the foresight and other places to find these partnerships and there is also very recent efforts that allow to sequence this TM7 organisms through metagenomic sequencing or single cell sequencing something that was unthinkable a few years ago but now we have the four genomes of this organism we could compute this productivity and what we found analyzing the reproducibility and the capabilities of this TM7 uncultivated bacteria and the host the octino bacteria we found putative metabolites that are complementary between these two so for example the host the octino bacteria could provide vitamins and amino acids to the TM7 and what is interesting is that there are cell wall components that could be exchanged potentially also from the TM7 to the host producing a two-way interactions that gives rise to a lot of testable predictions and that gave rise to some follow-up studies but this is just to illustrate how one can expand these ideas of flux balance modeling beyond just computing the detail growth rate and using it to start analyzing real complex microbiomes and their metabolic capabilities and I think it's time to stop and I'll just hint to the fact that what we'll talk about Monday is about how you extend you know go beyond this compartmentalized model and look at dynamic flux balance modeling where you can start thinking of really not just a more realistic way of how to model exchange with an organism but this will allow us to look at the dynamics and also the spatial structure of communities and a lot more so I'll stop here and see if there is any question. Thank you Daniel. For the last example I mean the crossfading between TM7 and the host have you tried to validate the finding here in Rachel? So I haven't this is I mean we're doing experiments but this is well beyond the kind of capabilities we have I think there are a couple of labs that can do this kind of experiments because and this is just really fascinating work that is being done for example the Forstner Institute but just being able to isolate this TM7 organs which are very small to do with the host is a whole lot of itself. So there is I think listed here there is some evidence from preliminary studies for example that there is gene expression studies where it's really it was found that some of this for example enacetylbicocosamine is really implicated in the expression of the host in co-pultivation with TM7 so there are now a lot of efforts to try and characterize some interactions and and I think it will be very interesting to see if you know this is valuable. So yeah we're not doing this but it is being done and that there is a lot of interest. These TM7 are a fascinating very big clade in the tree of life that has just starts being characterized. Yeah thank you yeah I think it's definitely very interesting to delineate which is the metabolite among all of this potential carcinometrolite is playing the key role of the interaction that would be really fascinating. Yeah thank you. That was a raised hand by Miguel Rodriguez please. Yes thank you. I have a question about well I don't know if it might be more relevant for the DFBA but I will ask anyway so you explicitly model you have been showing how to explicitly model syntrophy and sharing some of these metabolites but is there a way to also explicitly model the fact that competition has a very strong attractor for one dominating species? Yes yes and absolutely and as you hinted to the answer is going to be dynamic FBA in dynamic FBA this comes very naturally as we'll see because I can just hint to this right in dynamic FBA do this step-wise approximation of the growth curve where you solve flux balance time by time at each time point you predict the growth rate and you know how much nutrient is being depleted so if you have multiple organisms in the same environment this is a little bit like a consumer resource model where you keep track of the nutrient extracellularly and each organism will try to use it in its own best capabilities but they will all compete for the same nutrient and therefore competition will come as a very natural outcome of this simulation which is why now basically most of what we do and I think dynamic FBA is really the way to go for modeling community. If I can I don't see any raise hands so if I can ask another question are these is there a way to explicitly model also toxic by-products the accumulation I don't know yeah at which by-products become toxic? Yeah that's another very good question the short answer is no I mean the you can think of right reactions if you block internal reactions and there is no way for a metabolite to go this will give an infeasible solution in FBA so that is a little bit similar to toxicity but not really I mean toxicity if you think about this is really about a metabolite having a very high concentration and starting to do stuff that it shouldn't be doing inside the cell but by definition then FBA regular FBA does not have any notion of concentration inside the cell so it really cannot do that you know there are kind of you know just hint to the fact that if you look at things like shadow prices that give you sensitivity of the biomass production rate to changes in the constraints and including the mass conservation constraints you start having a notion of concentration inside the cell you can do this also with thermodynamic flux balanced models where you can put back the concentration so in principle I think it will be possible to do this but I think it's we're not there yet and I think it's a super important question but the other thing I'll say is that even if you have the concentrations you need to know somehow what compounds do to proteins why they're toxic so if you have knowledge in advance you can model the kind of rising of a toxicity in a certain situation but where we can predict the noble the toxicity I think that's even more challenging and I think that's a beautiful question but I don't think I think that's really beyond FBA it would require structural modeling or other approaches I don't seem to see any further question I am sorry I have a question but I cannot I cannot find the thing to raise my hand thank you very much for the very nice talk my question is to be related to what was asked before and I wanted to ask like if I understood well in this in the in the balance analysis like you use genomic data like so already sequenced organisms I was wondering just if this could be applied to non-cultivated organisms but let's say if this if it would be too much of a stretch to apply this to max or to differently sequenced organisms I'm thinking of the unculturable extreme environments my community yes yes beautiful question I think you know again the answer until a few years ago was no because all of these FBA models really are based on a knowledge of the genome and without that you know there there may be other ways to you know typically characterization but I don't think I think you're really in the genome but what is amazing now is that with the new technology of single cell sequencing right if you can see the sequence single cell and I think that starts being possible with with bacteria but the other thing that I think will really change this dramatically and there are some beautiful example already out there is that from a deep metagenomic sequencing now you can get enough resolution to close individual genomes so even you know when in a metagenome if you have an organism that is uncultivated you may have enough information to determine its its genome so I think that will open up huge possibilities in terms of looking at this unculturable two-flex planas yeah I was and I was thinking perhaps help guiding like the the culture ability of these organisms right right I think that would be absolutely fascinating again you know this is what we we're trying to do here the problem I just want to put a caveat there is that we know nothing about these unculturabilities right and there are many different hypotheses if metabolites is what determines those then yes fba can help but there could be you know signaling molecules like protein factors you know all sort of other things that are beyond fba and if those you know for those cases actually we will not be able to say much thank you very much yeah I have a really good question then you so you mentioned so there's a question from Martina okay raise that right and then it's okay I mean it's okay if it's related I can wait no problem okay please go ahead yeah so me can I ask or I yeah sure yeah so yeah my question is about reconstructing model I mean which is related to the previous question about the reconstruction model from metagenomics I mean even if the we can't identify the the genome the specific genome from the metagenomics is that possible or is there any work to you know reconstruct the model from the metagenome you know by using you know the conception of a super bus you know not compartmentalized you know we probably don't have the enough information about the genome of each bacteria in the metagenome but it's possible to construct a super bus like model you know solving the metagenomics is that possible yeah thank you very much so yeah very great question the it is possible and we'll talk a little bit about this next time we there are some approaches that essentially do a little bit of that but it's not clear to me that I think yeah it's not clear to me that they you really miss something and destroy something so I'll tell you very briefly one analysis really a few years ago was looking at yeast that has compartments and we compared FBA of yeast with compartments to an FBA of yeast where you destroy all the compartments so it's a little bit like a simulation of exactly what you said where you decompartmentalize the model and what we found is that the decompartmentalized yeast you make a lot of mistakes in the predictions basically because anything that depends on energy production across the membrane oxidative phosphorylation doesn't work anymore so I think it might be possible still to do ecosystem flux balance models I don't know that anybody has actually done this but it could be possible if you keep track maybe also of meta compartments but there are other approaches network expansion based which we'll talk about that actually do a little bit of that so the answer is that it's not clear it's a very interesting area and more to come on this yes that's very much yes Martina yeah hi um so my question is so for example uh I don't know pH and temperature can change uh the availability of the byproducts these kind of things is it possible to integrate these things in the flux balance analysis the dynamic one I don't know in general yeah uh it's a it's a very good question the the it's hard I mean it's not technically it's not impossible I think there's there's been a little bit of work out there on temperature um I mean the tricky part of temperature is that it changes so many things right you could in principle try to put the some constraints on um right the rate of reaction based on the Arrhenius equation and try to figure out some way of putting the the temperature in there but but you know if you think about this there is denaturation of proteins or you know suboptimal so there's so many things that could be happening and I think it will it would be very difficult it's it's a very painful thing for me because I and many others I think because there is a lot of important applications uh of how micro communities will change with temperature for example climate change and so on but unfortunately I don't think there is a very effective way of incorporating temperature and similar for pH uh we actually we had did some work which is unpublished but on trying to simulate exactly what are the molecules that are being secreted and how this will induce changes in pH in the medium so this is in principle possible but but it it's very complicated and uh one of the things is that let's say proton exchange and so on is very hard to keep track of um so I think for pH there is hope or temperature is much harder I would love you know if anybody had ideas of how to do this I think that would be hugely important but so far I think we you know it's not been possible okay thank you great I think we had a very lively and interactive uh lecture uh with Daniel uh more to come uh over the next phase um and so thanks thanks again Daniel and thank you thank you everyone and we'll basically move directly to the next lecture by Mercedes Pasquale Mercedes I think you're here already um yes I'm here good morning or good afternoon whatever yeah so I guess that we are I just want to check that you can hear me yeah can you hear you well and I can see your slides great great okay thank you see you in a short while welcome back everybody for the last lecture uh of today's session we welcome back again Daniel Fischer who will lecture on uh the college revolution in high dimensions okay thank you um thank you so again please um put questions in the chat and I will try to keep an eye on them and because Antonio or probably if I haven't seen uh I haven't seen one of them um so um yesterday talked about some and assembled communities which a lot of speakers have talked about and particularly talked about the effects of having many islands with migration um uh between them and what the effects of that are so what I'm going to talk about first today is how one gets to some of the results that I talked about the dynamical wind field theory um and unfortunately I'm not going to be able to go to that in in huge detail it's it's quite technical and difficult to actually solve the the dynamical wind field equations so I will give what they are and sort of motivate it and then sort of give some heuristics about those but if you're interested in that in in detail it's in this paper in um uh PNAS from Michael Pierce and Atisha Agaruala who'll be involved in all of this um all of this work the um some of the recent things that I'm gonna um uh mention or come to wars I won't really get to his current work which is also involves Aditya Mahadevan who where if we were all in Trieste together you would um meet since he's at the at the workshop um at the school okay so after talking about um that I'll then talk about some of the robustness of this phase that comes that I showed last time this spatiotemporal chaotic um chaotic phase and then I'll turn to things that are really um uh open and ongoing um asking about the question of whether communities like this can evolve or how they evolve in the future and then makes some brief comments about phenotype um um models um and where particularly in the focus in the context of bacteria and phage um um and phage interactions okay so just um first remind of what the um um of what the model is so we have k strains okay closely related strains labeled i equals one to k these all exist on i islands alpha equals one to the total amount of islands the islands are all identical um the populations of on island i sorry an island alpha of strain i is n and the total per island n alpha is is equal to big n which is roughly a um a constant kept by the overall limits and resources the frequencies which will use or the fractional abundances are the new i alphas which are just the ratio of the on that island to n so those are the basic dynamical variables that we'll um work with okay then there is a small migration rate between islands it's small compared to the typical growth rates on the islands which are of order one or actually order one of a root k but um um uh the m is small compared to um uh compared to those um and that's actually it's important to be relatively small for the um for the behavior for this phase okay then there um can be some selective differences between the types they just overall grow faster than others or slower than others the si and that's going to have some variance sigma s squared and mostly i'm going to ignore this but i'm going to come back and say some things about it towards the end okay the interactions they're only within the islands they don't depend on the island so there's matrix vij we also can have an interaction of a type um with itself a strain with itself which is much stronger potentially with a um which would be then minus a q but since we're interested in closely related strains where there's no particular reason for that to be stronger we're going to mostly set this to be equal to um zero okay well then the crucial parts with the v's instead the v's are random so i have to tell you the um statistics of the v's so the average of the v's is um zero as is the average of the s's and then the um the variance just v1 and this just sets the um uh sets the time scale um uh here so this basically just gives you the time um scale okay but then and then the v's are independent except for correlations across the um across the diagonal so in particular the correlation of the effect of what j does to i with the effect of what i does to j and that has this parameter gamma and we're going to particularly focused on gamma negative which is sort of motivated by particularly by the predator prey um um uh uh context but i'll say something about i'm thinking more generally so we're looking at these anti-symmetric correlations then gamma in this um um in this range and the canonical value for the simulations and things is gamma of minus 0.8 um a good value okay so what is the basic um um the basic dynamics then so here's my dynamics so there's the overall growth rate that'll depend on ghost minus death that'll depend on the selective differences it'll be a niche interaction if we include it but we're mostly not we mostly ignore that and then there's the interaction with all the others on the same um on the same island um and then there is this piece which is the Lagrange multiplier whose role is to keep the total n on that island being um being fixed at um n so it acts on each island separately and it's going to be in you know transient things at least it's going to be dependent on time okay and then there is the part from the migration and the crucial feature in the migration is that the migration comes from all islands to all other islands so the total migration in them will come from all of the other islands be the sum over the same strain or all of the other islands which is basically going to be the island average in the limit that i is very large okay we're going to take entirely deterministic equations with no stochasticity there is a possibility of local extinctions we'll sort of add that in if the news since they're the fraction of population become less than one over n so this is less than one um individual but they can get repopulated by migration from the um uh from the others and so we can understand the effects of this but we're initially not going to include that and take n to infinity okay so i showed last time from the simulations that what the system goes into is it goes into a spatio-temporally chaotic phase some fraction of the um all the strains go globally extinct it turns out to be a small fraction um usually a small fraction depends on parameters um they go globally extinct but the surviving ones that persist they form a they go into a chaotic steady state after some initial transients and the crucial part of that chaos is it's de-synchronized across the islands and the reason is if you have chaotic dynamics on two islands there's a positive gap on an exponent and the coupling between them from the migration is relatively small then they um um you will they will tend to um uh de-synchronize and that happens all except the migration is quite large so we've got this this phase and i'll just show the one of the figures that showed from last um um from last time so this is one type one strain across 10 of the um islands and this is plotting on a log scale and that's because the natural bouncing around is on the log scale because the growth rates um and death rates um vary so this is bouncing around they can bloom up to high abundances here so they all have these um um each of them has a bloom um they all bloom up to high abundance but then mostly go down and sort of hang out down here um down here here now what stops them going too low well is the input here from the migration there's a called a migration floor which is this curve here it's bouncing around because that's an average over all the islands and this is this new bar coming in so that stops them going to um too low they can go extinct but i put the extinction threshold down here so that's the condition that the population's migration is big enough the product of those so that's the extinction threshold then go they can go below and you know it's here actually um the the global population goes below that but some of them are surviving and they uh um repopulate this island that actually went um um went extinct there locally okay so the crucial part here as far as the you know the qualitative features is that the fluctuations of each island are on a log scale they're fluctuating over the log scale um um log scale here and that log scale is set by the size of the um the size of the migration so the range this goes over here these fluctuations is a range of log one over m will basically be the um the range of the fluctuations so if m is small they can go over a big um a big range okay um and the um oftentimes in fact if many of the strains most of the time they sort of hang out near the migration um near the migration floor um but they occasionally then bloom up to high um abundances now the blooms are high abundances that of course happens on a linear scale because what comes into this is on a linear scale so these blooms then will actually dominate the average if i look at this strain and i look at the average it's going to be dominated by the bits when it's way up here so at any given time only a small number of islands will uh um will dominate okay and those crucial then the blooms because of course it's when it blooms that it can give a migrants into the other islands when it's down here it's not going to give much migration doesn't matter much but when it's up here it's important when it's up here it of course also when it has the biggest effects on the other strains okay um so the um um this crucial bit is going to be understanding some of what these blooms blooms are and how they get there and they get there in a very irregular way as you can see from the wiggles um uh wiggles in here okay so these blooms then as they dominate the um average the average of islands they also dominate the average over the um over time so they also dominate the average of new i alpha on the one island of t average over time and i'm going to use this angular brackets to mean average over um average over over time so they'll dominate this and then of course into all the islands are equivalent something that overall islands will give you should give you this island um um uh this island average okay okay so how do we um um understand this behavior so the the goal is to try to understand this um um uh this behavior okay well the nice thing here is that we can do a systematic um a theory of this and the systematic theory is strictly living the limit that the number of strains goes to infinity and the number of islands goes to infinity initially we'll also take the population size to infinity but we can um handle that um um afterwards and this is coming under the general approach is make things as simple as possible and then add features and of course one of the features we want to be able to add is local um uh local extinctions okay even with the deterministic dynamics so an n is infinite i can still have global extinctions i can have all the strains on all the islands one strain on all the islands just keeping coming down and um and die out so i can still have the global um um global extinctions okay so how do we do this well what we do is i mentioned in the last time is we focus on one type one strain on one um island so that's um um uh alternating between strains and types um so one strain strain i on one island and since the statistics are independent of the island i'm just going to drop the um alpha index and there and call it new i okay so we have this the dynamics of this has several um um has several parts okay so first it's got the sort of obvious things in its growth rate so here's growth rate it's got the s i it's got this interaction with itself if we take that into account but then it has um and then it's got the Lagrange multiplier which keeps the population on the island um constant okay and then it's got the migration coming from all the other islands um and the migration out from there okay but then the effects of all of the interactions with all the others are coming from these two pieces uh here and i've shown this piece with a double minus sign because gamma is negative so this overall sign will be um will be negative negative there okay so what are um what are these okay so the way we turn you could try to understand this is we add a type and we add one type um a new type and i'm going to call that um a type um a type zero i'm just to distinguish it from the others so we put that in and we ask what are the effects of the others okay so the effects of the others um all the other ones on same islands of the others what will they do well the effects of the others will give it an effect in its um growth rate um which is going to be it's going to be where the zeta naught is going to come from okay and that's going to be the sum over all the other strains of v naught j times nu j of t so that's going to be the the sum of the others and this is going to be something which is going to be approximately Gaussian it's the sum of a large number of things that these are independent so there's a large number of things this is going to be approximately um um this is going to be approximately Gaussian okay and it's going to be Gaussian and it's going to have some correlations so it's going to have mean zero so the average is going to be zero um equals zero and then it's going to have some covariance so it's going to have a covariance which we're going to call c of t and t prime okay which is going to be the average of the um zeta of um of t um zeta of t prime okay now that zeta of course because we've got the um the this is coming in for the zeros that's going to have a um zero in it um but the statistics of it is going to be the same for um uh for all of them okay but we'll have to um think carefully about what the effects of the um the particular type um are okay so that's the one type okay so that's this part I'm here how it's interacting okay but then there is a really crucial part and the really crucial part is the feedback of this on the other types the other strains okay where is that feedback going to um uh uh going to come from okay so I now imagine that this new new is now growing so it has some time dependence and we're going to want to look at its effects on the presence of the future so we've got new um the new naught um which is going to um have some um time dependence and I'm going to look at this say in the past so I'm going to call that um t prime and what does that do well what this will give rise to this will give rise to an extra force on the other islands so this will give rise to on each of the other islands it will give some delta j to j zeta j of um at that time okay and what's that going to be well what that's going to be that's going to be equal to then the just the sum sorry the vj naught is coming from this island times new naught of uh of t prime okay so that's going to be the um the zeta okay but what's that going to do well that's going to change the new j only of the other stream so this is going to then result in a change this is going to result in a change okay delta new j okay and we're interested in what that change can be at later times so this of course can be at later times there with the t bigger than t prime so that's its effect okay well what will its effect be well roughly speaking the effect of each one on each of the others is small because they're a very large number and there's a total number the effect of each one is small so we can approximate this effect here of this delta new j is it going to be the extra force which is the zeta j times delta the derivative of new j at t with respect to zeta j t prime okay so this is like the response this is the response of um of j to the um changing the force on it changing the zeta on it right because the other ones this is the force um the other ones are feeling um we've now got the other ones that are feeling they are those and so they'll get this extra force okay so this is the the effects of there but what does this do this is now change the new j so the changing of the new j then that'll give us a change back of the an extra sort of force on on new nought but now this force on new nought is going to be at this later time well well how's it going to do that well that's just got the vj nought we've now just got to sum this up here okay so we've now got this is v nought j so it's the feedback back coming on this and then we've got the sum overall um all j okay so that's our extra force that we're going to um that we're going to get that's the extra force back on the um um on new nought okay so what is that term this term is going to have some average value why is it going to have an average value because we've got these two things here which are correlated this one is correlated with that that is correlated exactly with this parameter um um uh gamma okay so what is this going to this is going to be in the limit of large numbers then this is going to be just someone j okay of now we're going to get this parameter gamma coming from that averaging times this average of new j with respect to zeta j t p prime okay and this is then going to be times and this is going to be then um our quantity which is just going to be r of t t prime okay times the new nought at um sorry the new yorkity file i've got i'm already there okay so this is the quantity here what this is here is this is this whole bit coming in here so this is this whole part all of that is going to come down and give us the r okay and sorry i've got the gamma in the r um um the r there um so i've got the gamma times the um um the r okay so what is the r r is then going to have to be equal to this okay so we now have a self-consistency condition so the crucial part here is we have the self-consistency conditions self-consistency for this approximation and that's that statistically all the the strains are equivalent so there's nothing special about this one that i called um zero so this self-consistency has to be that this r so here i applied the r into this equation i applied the r i applied the zeta the zeta had some correlations the r was coming from a response right so this r here is a response so that's the response okay and the c here is the correlations okay so those have to be determined um um um uh self-consistently and those are the zeta and of course then this i have to determine the self-consistently so i have to find the statistics of each of the new j statistics for each of those i have to compute the correlations and then the self-consistency is that c of t and t prime is going to be the sum on j of the um average of new j of t um new j of t prime okay and this is then the average over all of the noise so this is really the averaged over all of the effects of the um of the zeta j okay so that's that's the correlation function and then i have similar for the response function which i've already um um written out in terms of this which is the sum on j of the um d new t prime and then obviously have to integrate those effects over all previous um over all previous times okay so the effect in here this is the effects of all the others it's a Gaussian random variable with with correlations in it i've got this response from the feedback of i on the other types and back again on the on type i this is of course time lagged there can be general time lag there so it's integral all the way up to um t and the coefficient of that coming from the correlations in the v's of the effect of i on j and the effect j on back on i has this coefficient here minus gamma so this is a negative effect so it's a feedback effect that stops new getting large okay so this effect here this effect here really is this kill the winner um kill the winner effect okay the effect on the others is the when the new i is big so when new i gets large that has the effects on the others they then do well and they give the feedback on this okay so that brings it back down again and that's responsible for the this dynamics up here that's crucially responsible for this part here which is the turnaround and stops them getting too big okay if i do have the queue as well the queue also of course stops them i'm getting too big but for most of the time at least we're just gonna ignore this term um and um um it doesn't matter unless it's particularly large okay it has to be larger than the other effects to matter and i say that's corresponding to assuming niches which we specifically don't want to uh don't want to do okay so this is the basic structure then of the um of the dynamical um mean field theory um just one more second um this is the basic structure of the of the dynamical field mill dynamical mean field theory and now our task is a simple task seemingly simple task that one has to figure out then self-consistently these two um uh these two functions and so we have to get these once we figure out the new we assume something and then we solve okay then we've got an additional self-consistency we then with the migration so if i then plus adding the migration okay so how do i do that i assume some new i um bar general will be dependent on t so i assume some new i bar then i compute i get the actual new i is coming um uh coming out from from that and then this has to be equal to um that so i compute the new um and then i have to make itself consistent so the time average of the um of the new let's sorry that's actually the average of the islands um the one over i times the sum on the islands of of new i has to be equal to new bar okay of course i need to adjust that so i assume new bar i get a new i and then i have to adjust until i get this so what's gonna happen well there's gonna be some news which go extinct so sometimes we will get that this will go extinct um and so i'll have some fraction here where it's um extinct so some strains there's no solution which is bigger bigger those and those go globally extinct sorry then it can make a question so yes uh so the new i are broadly distributed you said that uh at any given time there is one uh few few of them that will dominate the sun yes yes well is this approach uh uh does is there any problem with this approach where you take uh where you assume essentially that uh things are self averaging yes so the condition that one needs is that the number which are large at any time is big and the number which is large at any time is basically going to be the total number divided by this factor because they're roughly uniform on that scale so the condition we actually need is not that k be much bigger than one but k be much bigger than this parameter here this log okay but this is this is you know modestly a modestly large parameter in practice the with you know in modus k that one can see this a phase so like i've got here in practice it often is only dominated by a few of them okay so this is not in the regime where the mean field theory is strictly valid and associated with that also we have these fluctuations because the number of islands is not very large okay so this is showing from modus numbers one could do it for larger numbers and there's ways of trying to get some convergence numerical so that's a good a good point is that strictly speaking i did have many large at the same time however it turns out they turn around fast enough that only having a small number large at the same time the behavior is essentially the same but that's one of those things that we can put in afterwards and understand understand that yeah thank you that's a good important question that's a raised hand my arm oh yeah um i guess could you remind me what you took to infinity um in terms of so the things i took to infinity is k to infinity so that gets around this problem that Mateo just just raised i mean i always have a large number of i'm affecting all the others and that's coming in here the fact that i've got a sum over large number of roughly independent things okay um and then the i also took the number of islands to um um uh to infinity um and the reason i can do that is then i can treat the um self consistently of the number of islands i can treat this as something which doesn't depend much on the islands these are going to be roughly independent of each other because of the chaos the uncorrelated chaos and so i'll get something which is well-behaved average there but again one can i'll make something you can go by number of islands so then the Lagrange multiplier then what at what level is it keeping um the okay so i also have to adjust the um the epsilon okay i need to adjust that um that as well that i can do as i go along with the um with the dynamics it's part of the same um same thing so i better put that also in um in here um i'm i'm finding a new event because this is i get this and i also get um i also have to get the um epsilon of t sorry for not having said that i'm that part thank you the epsilon on t um will be because again the the large number of types and the statistics being the same the epsilon t can be the same on each island after transit so i'm now going to simplify um simplify things so some strains have gone globally extinct and then the assumption is that the rest go to a statistical steady state okay the statistical steady state and that means that for example the correlation function will just be a function of t minus t prime okay the response function will also just be response function of t minus t prime okay the um epsilon will be approximately constant so i'll have a those um those simplifications and each new bar it'll still depend on i okay um this will also go to a constant but depends on i depends on the strain capable to lose its time dependence so now i've got a time translation variant problem and i can try to solve that okay now i would just like to make a side note for people who've seen dynamic theory and spin glass context or other contexts usually in people the situation people do you can take these um a correlation response function you can go back in and work it out and you can directly get a self-consistent equation for the correlation response function and once you've done that you no longer need to do the stochastic dynamics here we don't have that behavior here you have to do the full stochastic dynamics you have to assume as data you have to do the stochastic dynamics understand the statistics of the new get the average of it get the correlations of it these quantities here both have long tails in time this has a long tail in time this has long tails in time and you have to work through those self-consistent and that's what's hard okay so the real challenge here is sort of the applied math problem now of going and trying to understand this um uh self-consistently say that's done in detail in this um um um the pna s paper okay so we have to assume we get a steady state and then we work out all of these things off consistency okay so what i want to just do is i want to do a bit of the um uh a bit of the heuristics um to give a flavor of that and since the distributions are on abroad the natural thing is to look at the log variables to look on a log scale okay so i'm going to define li is going to be um the log of a new i okay so l actually got i mean it was negative because the new is bounded by um the new is bound by one and then i can write that i've got the log i um uh dot here so this is now just going um uh just going up and down so it has this part coming from the um the zeta i um of t minus the um epsilon which is roughly a constant and then it has this feedback um effect of the um of the r times the um up to t this is now two minus t prime of the n well what is the n the n is going to be e to the now l at an earlier time right and there you see the exponential waiting if you like in terms of the natural variables which are the um the l's okay then there's going to be part the other part well there's just going to be the minus the migration out but then the important thing is that the migration in the migration in has this average coming in um here which is now just a constant but now of course that's divided by n so that means it has an e to the minus l here okay has an e to the minus l of t okay so these are all functions of t has an e to the minus l of t so what does this do okay this one here cuts it off at the top end right when l gets large it cuts it off this one here gives you a floor that you're not likely to go down much below this okay so this term here this term here keeps l usually being um uh bigger um than um log one over m times this new i bar okay so what is that that's this exactly in this picture um this picture here that's exactly what this floor is so that's this floor which is set by um which is set by this we have to adjust that floor itself consistently if this is a not very good type then i'll go up and come down oh and i should i let me put in the um leave in the si um um in here okay so what do i want to do well i want to divide this into two things i want to look at this and then it's going to have some average value okay so in the steady state that's going to have some average value but that's going to depend on i so i have a quantity then which i'm going to call psi i okay and that's going to have several parts it's going to be si plus the average over time this is going to be the average over time um the um minus the epsilon okay and then of course i've also got a part of the zeta which isn't average over time so i've got an extra part which doesn't average um so i can write zeta is the average zeta plus some eta where the average of eta equals zero okay so i have some other part there and the eta has the correlations associated with the remainder part of the um of this okay so now this will depend upon the i and this i'm going to call the bias okay what is that well in connection of things that Stefano Alacina talked about well this is this is just the invasion eigenvalue from small numbers why is that well if i look back up at my um equation here if i look when nu is small they've been small in the past this is what's going to determine how it invades okay so this is exactly the invasion of the invasion eigenvalue okay so this is the invasion eigenvalue now the surprising thing is well the not surprising thing is if if minus psi i if it's strongly negative so it's strongly negative they'll get we go extinct okay but psi i can be less than zero but bigger than some critical um critical value which is negative it can be in that range it can still persist okay so a crucial things here is even things that are biased downwards on average even things that are biased downwards on average can persist and in fact they can be biased down quite strongly on um um um on average and the reason that they um um uh that they persist sorry um the reason that they persist here is associated with what this um um uh form is here of what comes okay so even though on average they're going downward they're being pushed down towards there they've got um this effect here which stops them but in order for that to work in order for that to work they have to burst upwards so they have to even though they hang out down here if the psi is quite strongly negative they hang out down here but occasionally they burst upwards those blooms are crucial without those blooms you don't survive okay so you need them to keep the new i bar up you have to um bloom and in order for this to happen they need to have occasional blooms okay and those blooms then is what dominates the new i um uh the new i bar okay so you can work after work out the statistics of those blooms that's uh that that's subtle and the reason that subtle is that these r and c's have long-range um correlations in time so the r and the c here and the correlation function both have long-range correlations in time they decay with a two-thirds exponent over some range they have two other different regimes as well um and it gets and life gets very complicated but the crucial thing is when that's to understand the statistics of these blooms okay and that's a rare event um um uh calculation and they say this is coming from this is going to be the average of e to the li right i'm so it's an exponentially weighted average it's dominated by when the l's are almost close to zero i'm not enormously well okay so this is now um the um uh this is now the problem that one has to um uh the one has to solve okay so i'm not going to go into more of that this gives you a sort of qualitative picture of the uh behavior um it turns out that most of them actually persist with most of the parameter range unless the migration gets large you actually get most of the persisting only a few of them um uh go extinct most of them persist you can work out what this is how it depends on the parameters at least roughly so this is all doing asymptotics there's almost no things you can write down exactly you can write down sort of bounds on things which really give you useful results but the crucial thing is understanding the statistics of these blooms the crucial thing is seeing how they give rise to long-time correlations in the response and correlation functions okay so that's that that's the basic um um uh basic behavior okay so i just want to think at a um at a page here um okay okay so what do i want to just ask about is the robustness of this phase okay so um uh Roy Felix Roy and collaborators have looked at this for gamma equals zero so independent interactions and um q being uh um positive but not um uh too large um well it's order k root k but not um not too large okay and they find similar behavior there's some um differences associated with what happens how they turn around when they get large so if something comes up and when it gets large it's turned around by the q um rather than by the response but rather than by the feedback but the behavior is qualitatively similar okay so it seems as if this actually applies over very large parts of the sort of phase diagram in the basic um basic model okay so that's one part but another part of looking for the robustness is what happens if i have finite number of islands okay so if it's finite n finite population of each island okay then if um if m n so the total migration into each island is large okay then it's still okay you so some extra ones um uh go extinct but most of the strains still um persist okay a few extras go extinct if the new bar falls below one over n okay if i hadn't the figure okay finite number of islands i'm finite number of islands as well then what you do is you get to this the survival time um the uh survival time in this population goes as e to the um i um uh a number of islands exponentially long divided by some characteristic scale which depends upon the bias of that type it depends on log m and it depends on log n and again figuring out what this um this is takes quite a bit of um quite a bit of work okay and one can check that numerically 10 islands and you know 100 types um i've forgotten what it is 80 or something like that have survived very long times and some of them will survive i say exponentially long time as the number of islands get large so even though this is asymptotics and in fact it's asymptotics in log variables in some sense it's asymptotics in log variables it turns out that it's very um robust and so works over a large large range we could add other features we could add in um some extra environmental stochasticity we could add slight differences from island to island um uh Felix and company have looked a bit at um that and again the behavior seems to be assist so we really seem to have this robust phase for an assembled community with correlations in the interactions which are not strongly competitive right gamma being positive it's gone to the more competitive interactions this certainly is going to persist for some positive gamma we may persist in some sense all the way up to gamma just below one um but that we don't know yet okay so we don't know how persistent this will be if i put in say interactions via chemicals in the um um in the environment so that's still a lot of questions associated with that okay so i want to um um to talk in the last bit about um um whether this community can um um can evolve but let me just pause here if there are questions at this um um at this stage um and i say i certainly don't expect you to understand this in detail but to understand the spirit and how one does the calculations and sort of the heuristics so let's now ask the absolutely crucial question we assume it was an assembled community we let things go extinct it stayed but we want to know is this phase stable to evolution okay and also can evolution give um uh give rise to this so how do i want to um want to evolve it i want to take just choose some strain i um okay so i'm we're gonna first look at this we're gonna look at slow evolution that's the hardest case as far as the thing's persisting so we're gonna add one mutant at a time one mutant at a time and we're going to take type i and it's going to mutate to some type um um i tilde so this is the mutant it has a given parent and these will be correlated in some way their s's and the v's will be correlated in some way with a strength with a strength rho so root one minus rho squared is essentially like the difference between them between the parent and the mutant okay now one has to do these correlations in the right way to keep the sort of cystics and so on um in there this bit we have some analytic understanding of it mostly um uh uh most numerical at this stage okay so this is now looking at the um what to what happens so i start with some number of types here i start with number of some number of strains initially here those um some fraction go extinct so there's the rapid extinction here on the ecological time scales then we we let it equilibrate and then we add one more okay so we've add this mutant we let it then equilibrate um that'll drive some extinctions um possibly some new extinctions and of course a crucial thing if we're adding one sometimes it'll drive the parent out but much of the time the parent and the offspring um coexist and the mutant um uh coexist they're slightly different from um uh from each other okay and then i pick another random parent i do the same thing again so this is looking what um what happens and if you start with a small number of types you tend to get that it goes extinct if you start with an intermediate number it sort of fluctuates around and then starts um going up and if you start with larger numbers it just goes up so this suggests that there's sort of roughly a threshold at which you need a threshold in the complexity that you need before it starts to take off okay now where this is it will depend upon the migration rate it depends on much more details about the model we don't understand this quantitatively one understand qualitatively what's associated with it so when you go in here the number of types ends up plummet if you start with small numbers you basically it's exponentially rare um in a small of large exponential factor in order to get up and get going okay so this behavior one won't see unless one sort of gets the community going in some way but one could do that in various um um various ways we haven't explored in detail okay there's a very subtle thing and this is what a detail amount of fun is working on as to how the invasions actually occur so how does a new type come in and that turns out to be rather hard and if it's too close to its parent it's even it's even harder um for this we haven't explored the details of that um yeah but say that we're working on now okay so what happens is this something which is persistent well here's now looking on longer time scales so this is of course sometimes the it doesn't invade at all I should say um sometimes the mutant doesn't come in um so we collaborate often the parent mutant mutant some that sometimes um or even often the mutant fails and doesn't invade and sometimes it does um if does sometimes it replaces the parent can replace the parent okay so what happens then well depending on what the correlation is so this is now putting in independent ones that's assembling the community more this is very highly correlated ones tiny differences between the parent and the offspring um there you don't really see much happening yet um mostly happens here is that the the mutants replace the the parent but as soon as you get you decrease the correlations by a tiny bit you started getting if you add you lose a half um and so you just go up at a steady um right actually the these ones so um you go even faster than that okay so this this tends to come up fast you get the community which gets richer and richer as it goes along however this here we assume there were no generalist mutations what does that mean you can do better in general by having all of your v's larger or having all the v's against you being less negative okay but the bigger the population the number of the community is the less likely you are to have that um um uh to have that happen okay so this has no generalist mutations what do i mean we mean that the s i's there's no s i's the s i's rule zero okay and s i would just mean that you'd better in general if your s i's bigger than your parent you do better your than your parent in general okay so what happens now if we put in generalist mutations okay so we're going to do this but we're going to make a variance of the sigmas um uh so the variance of the sigmas is going to be much less than um uh one that's the assumption initially that things are already pretty well adapted they're pretty well adapted but what then happens is you can still go out into the tail so if i look at the distribution of the um of the s's so i look at the distribution of the uh um s's here distribution of the the s's so that they'll say it's some Gaussian um a Gaussian distribution initially okay well even very early on if i look though the ones that um that survive i'll tend to not most of these ones down here won't survive so they'll go um they'll go extinct and as i go up i will start to get um as i as i evolve the start types i'll start to get that this distribution will tend to concentrate more and more towards the tail as i go up further it'll concentrate even more towards the tail and it'll keep creeping up towards the tail so this i go with successive um uh successive invasions um i go along and i start pushing it out towards the tail okay what happens then is one can see that um and you can look at the um uh the correlations here if you look at the mean of the the um s's that come in so this is the generalist mutations then you that that goes gradually upwards you push further and further out into the um further and further out into the tail okay the potus also gets slower and slower because if you're an s here if you find a smaller s most the time your mutant will have a smaller s right so the one once you get up in this regime here okay the most of mutants um um in fact even the successful mutants so the s of the mutant will be less than the s of the parent nevertheless if it has good v's there's good interactions it can um it can invade okay but generally it's more likely to invade if it's s is s is larger because it's got an overall average high average growth rate or higher bias so what happens in this case is it continues to diversify it continues to diversify just gets slower and slower it's harder and harder to invade it slows down but we have a um analytic understanding of this and for at least for the Gaussian tails here it should keep on growing um um indefinitely um but um just getting more gradually slower and slower okay so even if you allow the generalist mutations this phase can still exist it evolves more slowly it gets harder and harder to invade that's of course a general property the things will evolve for a same in constant conditions will tend to get harder for new things to come in interestingly if you didn't put specifically these generalist mutations in that doesn't real