15:03:22 Alright. Alright so anyhow, first of all thank you The organizers, it's really a delightful opportunity to be in Santa Barbara I've been here many times before my journey to biology started here in 2001.
15:03:35 And I never regret it at the age that I made the switch from physics to mostly biology back then that was actually a problem organized by theory and child time than others.
15:03:46 So, I'm also happy to talk on the third on the third week of the conference because it's kind of the number of niches which remain vacant is shrinking so fortunately nobody really clean the niche which I haven't going to talk about today.
15:04:01 And I'm going to talk about alternative stable states regime shifts and priority effects, which has not been emphasized much during the previous two weeks.
15:04:13 Also, nice to follow the talk by marine excellent talk by marine because she warmed up the audience to think about not only carbon but nitrogen, about how different organisms consume carbon and nitrogen there's certain ratios, and also has demonstrated
15:04:28 that sometimes it's good to use sort of economics, as a way of thinking about microbial ecosystems I will definitely follow on all those points in my talk.
15:04:39 So, again, I have been working on microbial ecosystems for maybe three or four years, and this is what motivates me to keep working on it. And again, this has been discussed forever, you're in the conference by the diversity, so I will not dwell on it
15:04:53 anymore.
15:04:54 This topic has not really been discussed so much and again just to repeat the topic here is how reproducible is a composition of an ecosystem if you assemble it.
15:05:08 In, in the same environment but independent. So what distinguishes some species which are always present from the species which are sometimes presence sometimes not.
15:05:17 and some really really rare.
15:05:20 And there is some evidence that many of the microbial ecosystems follow this U shaped distribution result of species being present.
15:05:30 Very rarely, and some species, isn't being present almost always 90 200% of the time and with the relative deficit of species in the middle.
15:05:41 Again, log scale log scale usually it looks best on the log scale and leader scaled is deep in the middle, can be some sometimes so deep that you feel like it goes to zero but again this is not what I am going to talk about even though this is a good
15:05:56 topic and we touched upon this in one of my papers with my group.
15:06:00 What I want to concentrate today is just the notion of multi stability which is different from reproducibility and for instance, here I am asking not just which President which pieces are present and absent but what are the combinations of species, what
15:06:15 are the sets of species which are being present and absent in an ecosystem for a given environment.
15:06:23 And sometimes quite a lot of times they actually you can have this situation in which the ecosystem for the same environmental parameters can be in one of several states, and they are very distinct and sharp based transition separating those states from
15:06:38 another from each other.
15:06:40 And this is exactly what I want to kind of flashing this slide.
15:06:46 Again, as every single biology, it's very complicated this is sort of textbook example of those regime shifts which is what physicists, call the first order phase transition, because it is characterized by historic behavior so you when you are changing
15:07:03 external variable, like a nutrient concentration phosphorus concentration in this case, you the ecosystem follows very different trajectories when the phosphorus is increased and when the phosphorus is decreased.
15:07:16 And people have been studying it in some real ecosystems This is ongoing study in some two lakes, Peter and Q's the lakes and Michigan. And as you can see, even though they are really next to each other they're very different.
15:07:30 One has a clear water word.
15:07:33 You can see all the way down to the bottom of the, of the lake. The other one is kind of murky water, and it's actually has been established that those transitions happened why this historic first order condition.
15:07:45 So again, this lake ecosystems that are caught too complicated to my days there are fishes their plans. I want to stick with something we should understand.
15:07:54 So is there any evidence that those historic transitions also happens with purely microbial ecosystems. This is a relatively recent paper with a bunch of authors out of whom I know personally very well, Chris Henry, and they studied to microbes, this
15:08:12 is kind of a lab experiment, it's not a human in vivo experiment.
15:08:17 And those two species, one of which is the strict an Arab and the other one is the faculty define Arab.
15:08:25 They are sought to be relevant for some disease we're in your small intestine which is sort of above the large intestine, you can have some of the prickly I'm Arabic species invading Reuters, and causing some problems.
15:08:39 So they put them in the chemo status and they married the two variables basically the glucose input rate, and oxygen flow rate, and they found some regions fairly broad region in which they had this is theoretic phase transitions and multi stability,
15:08:57 and you can see this is the beta which shows how when you change it in one direction it's different from another direction.
15:09:06 The other topic which I will kind of touch upon in my talk, this is sort of a motivating slide before I dive into models is the concept of priority effects, again a priority effects happened when you're starting colonizing an ecosystem from an empty state.
15:09:23 Here is one of such blogs from an experimental paper published in 2003, in which they were assembling a community of protests single cell you carriers.
15:09:41 And they started colonizing they had a 1236 types of protests six pieces of protest. They started colonizing the empty, empty ecosystem. And they tried all possible additions of products that already exist in college.
15:09:52 So in the first round, they were able to colonize the first four out of six added them to the empty, empty column here and there were some of the transitions were allowed, which were taking it from sort of a single species state to a double species state,
15:10:07 and sometimes they even had a reversals if they reached the three species state. Then in addition of a species, he for instance where to get back down to the single species state.
15:10:20 Again, nobody really understands what is going on here. What I want to show you is that those priority effects can be fairly complicated and characterized by this network which my personal association of with business game of letters and
15:10:34 so sometimes you go up, sometimes you slide down and, on average, you kind of maybe go up and at some point finish at the fully assembled ecosystem which you're like, whatever it means.
15:10:47 Now, the priority effects also happen in the gut. This time, it's the flight got another human got. This is a series of experiments done by William, London, and I like those experiments very much because they do something which we very rarely do in our
15:11:02 models, namely, instead of treating the invasions as just a single or perturbations where you just had a tiny bit of the population of an invasion species and see and looking how, how they grow or not.
15:11:16 Will in his experiments was pushing some species.
15:11:20 They were Adam those bacteria to the, to the food they were feeding flies, and they weren't adding it in different concentrations because I had no problem of Aden and the ratings pieces in a huge concentration.
15:11:32 They call it an Oculus those. And they were basically asking how high concentration Do I need so that the invading species will actually be be present in the ecosystem at a non negligible level.
15:11:48 And again, like highlighted this is sort of a pictorial abstract here. And they highlighted again that the priority effects definitely are at play. We kind of understand why because God is really a spatially segregated system with all of those Crips and
15:12:02 so on. And again, maybe it maybe not in the fly, but even the flights far from being a homogeneous environment.
15:12:10 And the if you, if some of this rains already to cop this spatially isolated relatively isolated area then it's harder to get it out of there by and completely replace it with another bacteria, unless you push very very hard.
15:12:27 And they even argued that whatever they absorb was very much consistent visa lottery type of phenomenon where the first colonizer was just a randomly selected strain from a bunch of strains introduced and then one, once this gets established it gets harder
15:12:41 and harder, is legit.
15:12:43 Again, this is a little kind of translation table or a college accepted to college in terms and physics terms so I've already told us that what the college is start calling and turn into stable states don't even attempt to abbreviated are actually what
15:12:58 we in physics call multi stability regime shift our first order phase transitions with histories is. And again priority factor is not a such a concept in physics, but he will argue that it's the same kind of type of phenomena as a regime shift because
15:13:13 the final state is not unique and it depends on water in which you, you are in the species to the ecosystem.
15:13:20 So, anyhow since I'm working primarily on consumer resource models.
15:13:26 I was always curious to see if, to what extent those alternative stable states and regime shifts are a feature of consumer resource models, and whether we can make a simple enough model which we can understand what is actually going on.
15:13:41 And I will tell you two stories so I'll kind of follow them. And that's example I'll try to tell you a story. Number one, and if I will have any time I'll tell you story number two.
15:13:52 So again the story number one is exactly the one where I'm happy to follow reverse speaker because it will be about cool utilization of two types of essential nutrients like carbon and nitrogen.
15:14:04 And each one of those nutrients can be represented by distinct.
15:14:09 You know multiple several metabolites so for carbon it could be different sugars for nitrogen could be an age for and or three or even and two if you have some nitrogen fixers in the community.
15:14:22 And the second story would be. I don't forget about nitrogen for a second story I will only concentrate on what happens when a bunch of material, our core utilizing different carbon sources but instead of query to lighten them in a pain of terrible fashion
15:14:39 where each bacteria is eating all sugars at the same time. My bacteria will be a strictly following a carbon utilization hierarchy, and using some preferred nutrient first and switching to the second nutrients and switching to the third nutrient and so
15:14:54 on. And this is a well known fact which is that for many bacteria is this is a preferred strategy for some of the nutrients, and those shifts from one nutrient and other nutrients are called dioxin shifts.
15:15:08 And finally, and this is where the economic center is my story.
15:15:12 I will try to convince us that a branch of game theory or economics if you want, which is known as a stable marriage or stable mentioned model is relevant for understanding both of those stories and actually gives you an answer.
15:15:32 handle on how to think about those communities. So let's start with story number one.
15:15:36 And I will first kind of refresh your memory about the first of all let me acknowledge the people who work with me on this and who did a lion's share of the work.
15:15:46 First of all let me acknowledge the people who work with me on this and who did a lion's share of the work. Veronica was a PhD student at UIC she is here in the audience right now.
15:15:58 First Name, who is the former postdoc at UIUC. And my sort of mathematical collaborator in Moscow and stuff scientist at courtyard of Institute. She actually developed the, you know, much more in depth mathematics for this than I ever expected to see.
15:16:09 And as you know you know beautiful mathematics usually ends up in supplementary material so if you're at all hooked on this on this talk, please take a look at the supplementary materials because there is a lot of neat, mathematics there.
15:16:24 So let me first just tell you what probably many of you already know and this is a website at work by David Tillman.
15:16:33 And this is sort of tells you how to think about what is a geometrical way of thinking about microbes, or any other species using essential nutrients and again this is has you saw an example of this in a model.
15:16:48 In a previous dog by marine.
15:16:50 So let's think about two nutrients so carbon and nitrogen So, so far, each one just as a single metabolite which is supply in carbon and simple single metabolite, which is supply, nitrogen, and those access show me the concentration of the carbon source
15:17:07 and the nitrogen source in the environment where microbial community will be will be growing.
15:17:15 Now imagine that I'm a red species, and a red species has this line which Pankaj talked about in his talk which is called a zero net growth is decline.
15:17:27 And the way to think about the slide is that everywhere here.
15:17:33 This material species is read bacterial species would exponentially grow everywhere below this line, it will exponentially die off and add the line, it will be in the status state.
15:17:45 So I always think about it in terms of chemo start, even though you can even think about it in the absence of chemo start thinking about some maintenance costs.
15:17:54 But for me, what is you know when I'm when I have a species in the chemo stat. This is this line tells you exactly where the species growth rate exactly matches up the dilution rate provided by, you know, said biochemist that.
15:18:10 Now it has this L shaped structure, which is a property of essential nutrients. And it's actually only characterized by two numbers, that's beauteous so what is it called his star and one is a gold star.
15:18:25 And what it was what this line basically tells us that the growth rate of the bug is given by the minimum of the growth rate on on nitrogen and growth rate on carbon, and the bug will not grow as soon as just one of those rates falls below this dilution
15:18:43 threshold. So, that is why it has this idealized L shaped so when you're down here, one of two nutrients is limiting the growth of this, of this bug and it starts to die off because it doesn't match it's lower than that they do.
15:18:59 Now, imagine now that I'm Adam this bug to emptiness that So initially, my chemo studies doesn't have any bacteria in it.
15:19:08 And my concentration of the nutrients in the chemistry is exactly what I provide in those flow tubes. So I'm providing quite a lot of carbon and nitrogen.
15:19:17 And I first just added this red bug. We start consuming the carbon and nitrogen. And as a result, this point describing the environment, starts flowing down towards lower values of CNN, and it will stop as soon as it intersects zero net grace Isaac line
15:19:36 line so you may think about it as a bug sort of depleting the concentration of nutrients in the environment, until it stops growing, and it stops growing and executives.
15:19:46 And at this point, you can see that this bug is becoming limited by one of the unless you are super super lucky any heat exactly this corner point, you will always end up being limited by ice or carbon, like in this case, or if this line would have ended
15:20:06 down, down there, it would have been limited by nitrogen.
15:20:06 Right. Now imagine what happens when you get a second bug to the ecosystem.
15:20:11 And I made in a blue bug, and the blue bag has a different year on that growth, Isaac line. And you can immediately see that if I'm trying to add the blue bug.
15:20:21 After I already added the red bug. The blue bag will not grow at all.
15:20:25 Why because the red bog depleted the carbon concentration in the chemo start down to this point and this point is located below zero net growth is a client of a blue bird, and it will simply not grow.
15:20:40 So basically at this point the blue bug is doomed. But it's still an interesting exercise to see what would have happened if I added the blue bag first.
15:20:48 And my blue bug has a different ratio at which you are, it is utilizing carbon, nitrogen, I forgot to mention one very important fact is that the slope of this line.
15:20:59 The reason why I'm drawing this line as a straight line is because I am a human that each of the microbes is using carbon and nitrogen in a well specified stoic your metric ratio.
15:21:11 They enter well doesn't have to be integer everybody has to be some ratio which is fixed, independent of the growth rate.
15:21:20 And this ratio is different from a red bug and a blue bug. So you see the red bug is using comparatively lots of carbon, and a little bit of nitrogen So the slope is has a long x axis and short y axis, whereas the blue bar is the opposite it uses lots
15:21:39 of nitrogen and relatively middle of carbon.
15:21:43 So that means that if if I was to air the blue bug First, it would have driven my environment to this point where the red bug would not be able to survive, simply because it would be it would not have enough nitrogen for the red bug to survive.
15:21:57 So you see this by stability very easy example of stability word dependent on which bug or a priority effect if you want, which bug I Ed said the environment, to a state, which is inhospitable to another to another species.
15:22:13 So, what about this point again when we are thinking about those here on that growth sized clients we usually think about the points where they intersect.
15:22:21 And the reason why we care about this intersection point is that if I know that those two bugs red and blue coexist with each other in the environment, the environment has no other place to be.
15:22:33 But this black point right here.
15:22:36 And this point unfortunately in this particular example is dynamically unstable. So if I carefully create the environment in which I had just the right amount of blue and red bug, and my nutrients would be just at the right concentrations.
15:22:49 It can keep for a while but then just like a ball at the top of a hill, it will either roll down to this fixed point or that point. So we have basically a bi stability between two states in this, in this system, and whenever you have to unstable points
15:23:05 in the simple model you always have one, kind of, sorry, whenever you have a bike stability between two stable steady state so you always have one unstable, they just stayed in between the two.
15:23:18 So a quick question about that. Do people play around with models where the sticky geometry isn't fixed but were you sort of pay a fitness penalty if you will for forgetting it subtly wrong, it doesn't change a huge deal.
15:23:31 Again, if you think about for instance MacArthur model you can think about it also in there.
15:23:36 In case where a tree is not fixed dependent on what how much carbon one and carbon to I heaven in medium, my state government free would change with this presence so it will be proportional to the carbon in the medium, and it can still have multi stability
15:23:52 is only problem is that it's a little bit harder to predict the pictorial it because those lines would be not a straight lines but a curved lines.
15:24:01 In fact, you can check that for MacArthur models they will be type parabolas, whereas here it will be just straight lines.
15:24:07 Right. And I was always kind of puzzled by images from human book Were you always do straight lines, even though he was occasionally talking about MacArthur model what he meant by straight line is that he was meaning the slope of this line years neural
15:24:24 net growth as a client but he MacArthur model it would be very different up here.
15:24:38 Sure. Oh yeah, absolutely. That is a good point. So even my bugs more the opposite so let me just by the way this is a good point because it draws me to the point I wanted to make and then it will answer your questions that in order for this by stability
15:24:50 to happen, you need a bug, to be a very competitive, because the lower is this line, the closer it is to the origin, the more competitive with a bug for the nutrient, it uses most, right, which is kind of a natural thing to have, if you think about it
15:25:07 but if it's the other way around. If the bug is less competitive for the nutrients that uses more, then there will be a stable fixed point here and it will have a certain region.
15:25:29 I think it's kind of related to the question of variable slopes, and I would go one step further, I would say,
15:25:36 least to me it seems that are concrete reasons to expect.
15:25:40 Why, as soon as the bark approaches. Let's say the red bug approaches the vertical line, it actually will start moving downwards.
15:25:51 And to mechanism. Yeah, so you would actually swipe towards the corner you considered that event. Yeah, I would, I would suggest this could actually be something that eventually will be reached because of either rapid evolution and or physiological adaptation
15:26:08 slash regulation and the reason I suggest that is when you reach that vertical point, you're limited by carbon if I read it correctly but yes, that means you have a syrup, and you can think about that box will probably have the ability to somewhat regulate
15:26:26 their end star undersea stars star perhaps at a trade off with one another by allocating more transporter enzyme, for example, more enzymes interested at all and so on.
15:26:35 So if you reach that vertical point you're limited by carbon and nitrogen that means you have a little bit of a buffer to allocate a little bit more towards the problem transporter yeah it's a good point that's a little bit of ability of the nitrogen
15:26:59 We heard about for instance a Redfield ratio in oceanic microbes which is roughly tells that all of them have close enough to the geometry in terms of CMP ratios again this is not a law, there are violations of it there are fluctuations but you don't
15:27:14 really need twice as much carbon, as you have, as, as in your state of state ratio because your biomass would not change that much. Again, it's a it's a deep deep rabbit hole which I don't want to enter.
15:27:27 But, again, in a, in an idealized model let's just live in a world where the story geometry, in which a bug is using CNN or later, maybe multiple nutrients would be completely fixed.
15:27:39 And then we can discuss about, again, what you're saying is quite possibly happens in in oceans because people are talking about core limitation and acquire limitation is exactly being at this sort of corner point we're not a single nutrient can increase
15:27:55 your abundance, you need to produce to nutrients.
15:27:59 Basically, when you allow the box to, I haven't seen a good model explaining this again I saw a recent high profile papers where they just were, you know, taking a boat trip around the world the ocean and studying, which nutrients are limited in different
15:28:15 points in ocean.
15:28:17 Haven't seen an explanation maybe what you're saying is right. But that's kind of a tangential story to my, my storyline so let's just move on. So what are the lessons here.
15:28:30 The lessons are that ask a technical point. Yeah, absolutely.
15:28:32 Use this thing about limiting most using the resource that you limit most, as far as I can tell it's only approved for two resources. Do you know any different.
15:28:41 Well, I will show you an example of three essential resources cnn p, which is again another classical work.
15:28:50 This is, again, nobody really knows anything so nobody really studied it as far as I know is true for infinite number species and to resources.
15:29:01 And in three you can already constructed. Well, actually, independently of how many resources, if the resource you are using the most in terms of study geometry.
15:29:11 You are also the most competitive for among other species, it will give you by stability stability quadrant stability and so on. That is effect that is effect that is always effect that if the resource you need the most you are most competitive for, it
15:29:26 will give you multiple to be independent of the shape of the.
15:29:41 Is that what is what attracts me to it is the simplicity of this functional forum, so I'd really don't need to care about anything except that I can compare speeches by how well they can, you know, I need basically to store in my brain only a rank order
15:29:55 of competitiveness of species on each of the essential nutrients, nothing else.
15:30:00 So as soon as I know those limit in concentration see star announced are, which are by the way completely independent of functional form of the growth curve as long as it's monotonic increasing.
15:30:11 I know everything I need to know about the who will win.
15:30:23 Between the species, if they are limited by the same huge.
15:30:23 The beauty of the problem of course is that usually in, well actually not usually always in those communities with multiple species, each one is limited by by different nutrients, right.
15:30:30 Otherwise, if two species were limited by the same nutrient one would always win over the other.
15:30:37 So, multi stability actually requires is difference in style geometry I would never get a multi stability. Without difference in stone geometry. This is a completely general question applies to my car SIR model Tillman model, we personally have a paper
15:30:50 with keep snapping about the same statement about Facebook your liquid systems, know the same story kilometer means normal stability in consumer resource models.
15:31:02 And finally, now the stability happens when species with high demand is also more competitive.
15:31:08 If the other way around, stable coexistence. Let me tell you another interesting story again which is not mine what I'm telling you right now is just the classical work.
15:31:17 This particular paper in the college in 2001 by Huisman wising. Yes. Can I ask that is a proof.
15:31:25 In terms of simulations mathematical proof I've just turned into almost like verbal proof you can verbal proof but but just just think about what happens, I mean, it can be converted into mathematical proof there's nobody bothers to convert it.
15:31:42 So, listen to this story, unless you don't know you know if you know it, then then then, this is a really neat story by who's minimizing the published a paper in 2000 I believe where they showed that there could be all kinds of chaos and cycles in, Mike.
15:31:59 world, in some model ecosystems with more than three or more nutrients. Let me show you. And again, you know, you look at it you see simulations the air chaos so it's fine, but they actually in this paper which is not a Wi Fi profile, less high profile
15:32:15 than their nature paper, they gave a very much in depth explanation verbal explanation of what is happening and they explained, for instance, if you have a three essential resources the carbon nitrogen and phosphorus.
15:32:29 You can have easily.
15:32:44 No, unavailable state at all, so you can have this endless cycles of the rock paper scissors type. And this is what you need to have a title so you need to have three species. Each one is has a highest demand for one of the three resources one has a high
15:32:49 high demand for nitrogen does or has a high demand for phosphorus a certain one high demand for carbon, but its competitiveness for this resource has to be not the highest but the second highest so each species is in the middle of their competitiveness
15:33:04 rank for resource it demands the most. Then imagine you are Adam species number one it depletes, and this thesis requires lots of phosphorus it depletes phosphorus.
15:33:16 But out now man my three speeches There is one other species which is more competitive than this one, or phosphorus so if you had this piece is number two to this ecosystem.
15:33:39 You can do it as a simulation or you can do it as a common sense argument what will happen is that it will outcompete species, number one, because it will drive down the phosphorus considerations below the survival limit for this species, but then it
15:33:40 will actually instead deplete the nitrogen.
15:33:45 And once it depletes the nitrogen because it needs a lot of nitrogen, then you can add another species, which is more competitive than number two on nitrogen, and then it will deplete the carbon and every single cycle will complete So, again, it's a little
15:33:58 bit of a mental exercise but you can see how it works and then this ecosystem will endlessly cycle between those three pieces so instead of adding them one by one you mix them all together, it will have those limit, the limit cycle, and there will be
15:34:12 no unavailable state. We are so familiar with the concept of unavailable state we don't want to think about what happens when our beloved unreadable state disappear.
15:34:22 This is a case where disappear. And of course if you have for more resources you can have killed.
15:34:28 I don't like chaos, I don't like cycle so I stay away from this mess right so this is the last time you hear me speak about cycles. I don't want cycles.
15:34:37 I want good clean status states. And I just wanted to see what happens to a human model, you know what, never made sense to me when I was reading the human book is that there is only only one source of carbon, or maybe universal source of carbon everybody
15:34:51 can eat all the carbon sources in the world and everybody can consume all nitrogen source to win the World. We know from our experience with bacteria, because human was primarily thinking about plants, is that bacteria are quite picky about what they
15:35:04 can consume and what they cannot conceive.
15:35:08 Like us the following question again this is this paper we can present and now so it's from this moment on, I'm starting to shift into my own work, what happens if you have a key carbon and nitrogen metabolites distinct ones.
15:35:22 And no cross freedom so no funny business, no violation of competitive exclusion principle, if I have k plus, em resources and Mickey Mouse that I can never have more than k plus EMS pieces.
15:35:34 In fact, I usually will have a little bit less.
15:35:37 Now, I'm trying to colonize the secret system from a pool of SPSS, and my species are specialists, again this is extremely bit away from Tillman towards real know specialist so each piece is can only use one specific carbon metabolite and one specific
15:35:53 nitrogen double.
15:35:55 So, in principle, you can have, you know, arbitrary number of species because species are defined by their parameters like growth rates and yields and the commentaries, but you can have k times m possible specialist right because there are key.
15:36:12 Carbon sources to match with em nitrogen sources, no more than k plus m species will ever exist at the same time.
15:36:19 but every one of those pieces will have some, some region where it will work it will survive, and that just the question of what kind of environment which each of the species will survive.
15:36:31 Okay.
15:36:33 Now, I will be talking about the states, which will quickly become an alternative stable state so my state is defined by which species from the pool are present in the state of state.
15:36:44 And I also want to know for each of the species what it is limited by. And again, so you may think about it that each species can be in one of three states, you can access our apps and completely present but limited by carbon and present limited by nitrogen,
15:36:59 that means that in principle I can have as many as three to the power s states to go over.
15:37:06 But of course, most of those states would not make any sense because they will have for instance two species limited by zc nutrient and so on. So, There are variety of rules which will be violated.
15:37:17 And so the question I'm asking is how many of those possible states, I will actually absorbing the ecosystem. And on want you to lose you at this point.
15:37:28 And I don't I'm not claiming that there will be a multi stability between all of those three to the power s state. When I'm talking about states I want to see what is in principle possible in this ecosystem so for some nutrients supply rates, maybe, imagine
15:37:43 just like human products to do imagine a huge Wide Field in one corner of your field you have lots of carbon in the other corner of the field you have lots of nitrogen.
15:37:53 If you sample your species in different local regions in this field you will get different species, never more than two because in his model you cannot get more than two because only two nutrients but out there you will have a species which are adapted
15:38:07 to low nitrogen out there, it will be low carbon and so on so I'm just asking the same question but in a multidimensional field in which there will be
15:38:20 stable unavailable. This is exactly where I'm doing here. And by the way, I still do not fully understand why I don't have a level of function nobody has to my knowledge has developed an app on a function for this model.
15:38:33 Well Pankaj is.
15:38:35 You don't have Okay good. Well actually not good but I would love you to have one but we ran those simulations Emily see we never saw anything but status states, be so unstable states with sustainable states we saw in available unavailable and so on.
15:38:51 So let me just show you a pictorial diagram how we are thinking about the states, this is my cartoon, he must add the illusion rate Delta, the flux is in for two carbons to nitrogen so I have three bacteria.
15:39:03 Bacteria are labeled by which carbon and which nitrogen sources they consume the first index is carbon The second is nitrogen. And then those little oval shaped things are actually indicated what this particular bug is limited by so for instance this
15:39:20 bug is limited by nitrogen sources bug is limited by carbon, and this bug is limited by car. So this is one state. One possible state.
15:39:28 It turns out that, again, we just roll the dice about some preferential ranking order. That was not particularly smart way to choose preferential rank in order we completely roll the dice.
15:39:41 And then we found that this particular ecosystem here can have seven unavailable states and 27 and available states and I'm counting even empty state as an invisible states altogether, you may say that I can imagine this ecosystem to be in a steady state
15:39:59 in 34 possible configurations of speeches and limitations. And 34 is still much less than 81 which would have been a theoretical maximum reduce power for is at one, then it turned out that out of those unavailable, of course, unavailable states i care
15:40:15 much more about because if I'm really colonizing the secret system repeatedly from a pool, sooner or later it will end up in one of those and unbeatable states and then it will stop changing unless I will mess up his nutrients so one of those states is
15:40:30 dynamically unstable.
15:40:32 The remaining six are stable.
15:40:35 And if you want to ask again I was showing you the results for two nutrients, but if I go for instance to nine nutrients and 81 species, the number of unavailable states goes to 80,000, the number of possible states goes to 10 to the power, your team.
15:40:53 And I can enumerate them all. Yes,
15:40:57 Sure.
15:41:00 What does it mean to be dynamically unstable but uninhabitable doesn't invasion automatically turn you. Well, we are we are sort of making this distinction even though I get your point so any invisible state.
15:41:13 You can think about it as dynamically unstable and stable with respect to introduction of and then read or consideration here I'm saying. Imagine a carefully prepared the state results for bugs, all present together, no other bugs out of 81 are allowed.
15:41:29 And then just perturbed to the tiny bit no invasion just the perturbation on their abundances and they will roll down one of those 60, so this is just like in this human diagram, there was this unstable state in the middle, which means that in the human
15:41:43 model if I put two bugs together they will either end up with one bug winning or the other bug. So it's the same story here except that we have four pieces instead of, instead of two.
15:41:59 Exactly.
15:42:01 Exactly, exactly. I'm things are stable. Exactly.
15:42:06 So, they know was just making a statement and I agreed with it so his statement was that there are seven fixed points. One of them is dynamically unstable six are dynamically stable, all of them do not accept any more species so if I tried to add any
15:42:33 the 81 minus minus whatever is a number of pieces present in the states species in my pool, it will simply not grow. Right. Exactly, exactly. And by the way, don't be fooled I always kind of for simplicity around this model where the number of species is
15:42:41 is equal to the number of nutrients squared, but it can be anything they can have a much bigger pool and have a much smaller pool it's completely you know it depends on how mature is my ecosystem with my ecosystem is just assembled maybe my pool is only
15:42:54 10 species and not at one.
15:42:58 So, any, any, any questions about it. So again, this is by the way, aligns, which I know how to count exactly all those states and this is exactly what is this stable marriage business centers.
15:43:13 I will not really overload you with how call exactly I calculated I can just tell us that this is sort of the agreement between the theory and simulations.
15:43:28 Right, so the squares are circles are from the model. It grows a little bit faster than the exponential In fact I. The formula is something like the right line also starts to curve up a little bit, in, in, in simulations, it doesn't curve up that much
15:43:42 but I believe that it's because we didn't sample and offer something, right, because remember that for each number of nutrients they have to sample over many models, and it's quite possible that some of them or give rise to a lot more states and others
15:43:56 others so exactly how the number of states is distributed between my particular choice of nutrients is a rich and fully understood question, but all I'm saying is that I'm not within the ballpark of the, of the observations, right now.
15:44:13 Oh this is by the way if you're curious what is the functional form of this black line this is the forum which I got by doing something I haven't done for over 10 years settled point integration of some functional which is too complicated to explain here
15:44:29 in in a, in a year now. Okay, it turns out that in order to predict all of those possible states again I'm not talking about I make them unavailable, that I can do, I cannot really predict which one of them will be dynamically unstable which one will
15:44:45 be stable, that will depend on something else. But just to tell you what states are in principle possible. All I need to know is the relative competitiveness of microbes for each of the nutrients, right.
15:44:58 So I pick one carbon source which is being consumed by a whole bunch of microbes, not, not all of them but we ever is using this as their sole carbon soil, we will have some star for this history for this nutrient so whichever has the smallest art will
15:45:14 be the most competitive second smallest second competitive so all I need to know is this ranking table and if I have this ranking table for all nutrients and all bugs they can predict every scene, in terms of this all possible feasible state in this community.
15:45:29 Okay.
15:45:36 We are in the middle carbon sources by this I mean that I have key types of carbon molecule carbon containing molecule ck sugars, or no no what no CrossFit no cross freedom is not allowed here just out of the fear of making it even more complicated, it's
15:45:54 trivial to add CrossFit in but it's not.
15:45:58 And also I don't allow for simplicity of some molecules which contain both carbon and nitrogen for instance like amino acids, which would be in hazard complication.
15:46:08 So, again, so I will have a whole separate section about the stable marriage so all I want to flash here is just how it actually maps.
15:46:19 By the way, the stable marriage problem is being used, maybe not all of you know it but when somebody graduates from medical school in this country, and he or she needs to go to the residency in the hospital around the stable marriage problem which is
15:46:35 no hope of hospital residents problem. The difference is that the number of, sort of, hospitals, and and and and applicants is not the same, like when you're having say marriages it's always one to one hospital can have multiple relevance.
15:46:51 Right. It turns out that basically the way to map this model to this hospital resonance problem is a two step process.
15:46:58 First, I want to say that first I will start dealing with the nitrogen limitation, and carbon will come late. Later, when I have in this particular model I have four carbons and three nitrogen.
15:47:09 So I need to have a for each nitrogen I need to assign the number of surviving consumers of this nitrogen. And let's get the alien numbers, all, all I take for them is to add up to four.
15:47:22 In this particular case I stay in 2111. I could have assigned 4000, which would mean that probably have a lot of this nitrogen in their environment and very little of and one and two.
15:47:33 So once I make this assignment I can actually run exactly this hospital residents problem and find all the, all the possible.
15:47:43 survivors here.
15:47:44 And then, there is a unique way to make this unavailable state so I had a few other speeches again. This is a confusing limitation each arrow here is a species consuming carbon and nitrogen and limited by nitrogen so when I have this red arrow going from
15:48:01 sea to 22. That means I have one species which consumes and see two and two and it's limited by and two when I have a blue arrow, that means I have a species consuming, and 23, and limited basis.
15:48:14 Again, just don't worry about it, it works, it has been proven to work, all I want to say is that I need to repeat it for any other random assignment of this 112.
15:48:24 And that's why there are exponentially many states. So for any particular assignment, the number of stable states is not terribly large it's kind of n log n are some limits and for some problems it's even affordable one, but because they have so many
15:48:40 other ways of distributing those numbers between the neutral nitrogen sources, I have a lot more diverse.
15:48:47 of distributing those numbers between the neutral nitrogen sources, I have a lot more diverse. Okay, so now, that's fine I can predict all the states but I cannot really predict any experimental data yet because in order to predict experimental data you need to
15:48:58 need to know thank you I'm a trace of species. And as Danielle correctly pointed out, if I just try to mess up the stroke of interest of two species by reversing them.
15:49:07 What used to be a perfectly stable coexistence points will turn into an unstable point and everything will flow elsewhere.
15:49:15 So this is indeed true. And again, there is no complete free lunch here so I have to say something about the economic free speech I in my simulations I draw from some relatively narrow log normal distribution.
15:49:30 What I want to tell you is that, again, we just. So here is a model with this two carbons two iterations where I made my selection of three geometry so think about it as a model of this state for one particular choice of parameters which unfortunately
15:49:47 is has more parameters in the eyes and model has.
15:49:51 So as I already explained in this choice of parameters that are six unavailable states I have to worry about, and the seventh one which is unstable so I don't need to worry about it.
15:50:00 Now let me show you what is going on in terms of transitions between states so remembers that I have carbon and nitrogen supply rates which I can modify it my own will.
15:50:11 There could be two states and the states, by the way, are nice sort of you know the the polygons in some multi dimensional space, the polygons can share a boundary.
15:50:24 And if they share a boundaries and if I very nicely carbon supply rate for some particular carbon source and I cross this boundary. Number Tirol abundance will go to zero and stay zero so it will be a second order phase transition Everything is fine.
15:50:38 So there will be no big jumps, but there could be also overlaps between those polygons and whenever they overlap within this overlapping region, I have a by stability.
15:50:56 but for larger number of nutrients that happens. And this is my kind of attempt to show you which states overlap, that would be represented by a red a red line here, and which are just sharing the boundary and do not overlap like one on five share a boundary,
15:51:09 if I am state one, and I am crossing those boundaries, there will be no no history says, If I'm in five and crosses boundaries, it will be a jump or one bacterial abundance will drop to zero and other one will, I will go to.
15:51:24 Oh, yeah, when I'm doing an opposite direction that will jump to a non zero value at a later point.
15:51:29 direction it will jump to a non zero value at a later point. Sorry, can you go back and tell us more about what determines the shape of the polygons and how they're related to each other. Well, again, they have to touch, like what happens if they're all non overlapping.
15:51:40 Is that OK, that's a very good point. So first of all, let's ask, could there be a point where there are no states about this.
15:51:49 In this model, no, but in this model with CNN p as I explained to you the three nutrients are could be a place where there is not a single state above this point here I'm guaranteed to have at least one stable state because I know that this model doesn't
15:52:04 Here I am guaranteed to have at least one stable state because I know that this model doesn't have any funny business so if I started with some microbial abundances will flow somewhere and it will be at least one stable state.
15:52:12 So every point in this multi dimensional space is covered by at least one point.
15:52:19 And sometimes it's covered by two sometimes it's covered by three. I'm kind of jumping ahead a little bit but this is the distribution of how many stable states are above a given fixed point.
15:52:29 It's kind of an exponential distribution or course on distribution whichever way you look at it. So, you are mostly having a single state for my choice of variability in Stoke your mid race, and then sometimes maybe about 10% of times a little bit less
15:52:45 5% of times they have two states, and very rarely it has five states above a point so that means that I can be one of the five places.
15:52:55 This is what I would do if I was to sample microbial community. We did a kind of a simulated managing omics experiment. We prepared a community in this point where there are five states above it.
15:53:08 And each state has a characteristic microbial species presence or absence so if you do principal component analysis which everybody does. You will see five blobs of different colors.
15:53:18 It's also interesting that you will see some gray areas between them which you will you will, which are dynamically unstable we checked, so basically whenever we created this great state.
15:53:30 It was stable but when we perturbed a little bit it was flowing elsewhere. So what you see is that for basically, there is some kind of a one dimensional manifold here and there are some transitions between the states on this manifold.
15:53:43 I don't know why. Because I don't know none of function, even if I knew level of function I would still not understand it but for every m dynamically stable states there is always n minus one dynamically unstable state, kind of, shown here five stable
15:54:00 states 12345 and four unstable states, 1234.
15:54:06 Why are they so I don't know, always works. We tried, even in places where we initially couldn't find we increase the density of our Monte Carlo agreed and found those unstable states.
15:54:19 It's kind of, if it's a level of function it's cozy one dimensional because only then I can map it to something called more severe it more Syrian apology.
15:54:28 One of the first courses I took at Stony Brook was noon or teaching me about moral theory.
15:54:35 Anyhow, this is by the way, what would happen in the community with six carbon six nitrogen and 36 pieces, you will have about 900 possible states with 8600 regime shifts so I'm only showing you a second first first order faced musicians here I forgot
15:54:51 about boundaries I just show you where I will have a discontinuous jumps.
15:54:56 You can see that they are kind of clustered so probably. Here I have a excess of one of the nutrients. Here I have maybe access have a couple of other nutrients.
15:55:06 So, if I am in a particular region and my nutrients supply rate I am kind of leaving in one neighborhood of this network and I may have, you know, two or three is continuous transitions, but if I start messing around with this multi dimensional space
15:55:19 a lot I will be jumping through those 9000 transitions.
15:55:25 Okay, I think I kind of, well, we also have a colonization order effects so for instance for coming back to my simple model relatively simple with six states.
15:55:49 Fluxus. I have a multi stability between states one and two. On my way to states one and two, I have those partially assembled states, the way to read this diagram is that I have four species here.
15:55:58 This is word none of the species are present. This is where is this species which is the one one is present and limited by carbon. This is word A B One, two is President limited by nitrogen.
15:56:11 Be to one present demon and by nitrogen and so on so I can in fact have some pretty funny, four step Rock Paper Scissors game where I will kind of replace those species in a cyclic fashion but only if I add them in a particular order.
15:56:25 If I add them out of order I will flow to one of those two states and stay there, so they are kind of fixed point trajectories Yes, this is a basic question I think I'm missing something, to get all to all those states.
15:56:39 So you use sample randomly from the set of possible microbial strategies of what they can consume and you end up somewhere in the sample again and you end up somewhere else, no not exactly so my microbes are fixed so again my strategies and all the parameters
15:56:53 of microbes are fixed those states are, if I change the fluxes supply rates are for nutrients right. So if I drive my system with supply in like high for instance here.
15:57:06 So everything which define depends, so the species are fixed but those four bars which show me, roughly how abundant resource for nutrients in the environment or how how quickly they flow into my community there is.
15:57:22 So here, for instance, they will have this by stability and to compete in alternative stable states here where I give you a lot of the nitrogen number two and very little of carbon one carbon, nitrogen one not zero because then everybody will die but
15:57:38 we are in middle I will have a unique statements we microbes, right.
15:57:42 So I will have Iser by stability or more stability and more complicated cases they can have price stability and so all right this is just in case you want to do some Cubism here.
15:57:55 This is Colorado state's look like and one thing I want to three quarter but but then if you if the environment is fixed so there is a flexible or fixed.
15:58:03 But then if you if the environment is fixed so there is a flux is are fixed. There's only one solution. Well, one or two in this particular model so for instance the equivalent, so they are, they have different species, right, for instance, this was just
15:58:17 a fixed environment fix Fluxus here I have two speeches, between one and B one two and both are nitrogen limited and alternatively for a very same environment it all depends on who was added first, you can have two other speeches, be one one and B to
15:58:31 two and both will be limited by carbon so it's a little bit like this. Human diagram where I can either have just pieces one or species two but I can never have both of them so except except here.
15:58:44 It's either this pair of species, or that Paris pieces, and this red and blue, red and blue is something which makes my job easier but experimentally it's impossible to measure that means that you have to really dig down into who is limited by what that
15:58:59 does make sense.
15:59:02 Yeah. Yeah, exactly, exactly, and also independent of if I now start messing around with.
15:59:09 We actually have a kind of appropriate not appropriate but a work almost in a pre printed stage where we are messing around with his supply rates and we show you how.
15:59:19 Sometimes it's hard to drive the system from this day to this day because he will. It's a historic transition so you use, you will have to raise nutrients well above the, the limit in order to make it switch from this day to this.
15:59:35 Okay. So anyhow, I think it's enough about this model because I want to have a half an hour to talk about the other model.
15:59:45 Any questions. Okay, here's a, here's a take home so I want to drive home just one point if you will just take one point home.
15:59:53 And is this is not a trivial point for you is that in consumer resource models of any type in order to have multi stability, you have to have a variation in.
16:00:05 In the yields of different microbes on the same on the same nutrients, it cannot be factories double because if it's kind of factories double then you can always include it into your redefined concentrations and everything will be the same.
16:00:18 So as soon as you have a differences, you open the door to multi stability we have shown it also works for ages and bacteria is a separate work with Kim snap and those of you who know Kim then he always asked what if you add features to it so when I told
16:00:32 him about this mouth stability story.
16:00:35 And when I kind of did a little bit of work here you obviously very quickly last last last my, my threat but after I explained him after several attempts he asked what if you wouldn't be in phases so here is the one nutrients a one carbon source to bacterial
16:00:52 spaces and faith which, in fact, in fact those two bacteria classical system, Bruce live in another study did.
16:01:11 We found that for some particular combination of those infection rates, those parameters, you can have a multi stability. But just like in this nutrient story in order to have it multi stability, you have to have a different state geometries and vice
16:01:17 Stoke geometry is a here. Here I mean, I want to follow the flux of carbon wrong carbon to bacterial biomass and from bacterial biomass to fish biomass so first here it's a yield.
16:01:29 And here it's a burst size. And if the multiplication of yield and undersized along this trajectory doesn't match the this trajectory, if I have a mismatch some of them, the biomass flows faster elsewhere it flows slower, you know, or conversion is more
16:01:46 complete in one direction than another, you can have a mountain.
16:01:51 Can I ask a question about that necessary again. Yes, I'm really bothers me because I'll just show you afterwards is a counter example, I know you're countering dynamics, and it's not necessary.
16:02:02 So I think, sorry, sorry, I never said necessary maybe I said necessary I didn't mean necessarily sufficient.
16:02:10 Good, good. I know your example indeed you can have, and we are now seeing multi stability in our other dioxin shifting systems without this.
16:02:20 It's much more exotic though, again my experience I cannot put a number on it but my likelihood of hit upon a multi stable point.
16:02:29 In this way, it's much more likely than doing another way but I agree I should I should not say nessa. And can we think about both of these cases is just being self limitation.
16:02:41 By self limitation you mean what, maybe it just parts of the little bit for me. Maybe a limitation in that.
16:02:50 You know like if you are the moat you're the most impacted by the set of resources that you have the most impact on.
16:03:00 It's, it's kinda I think that's what I explained to you is just more complicated version of what I showed you in the first slide, again, it's still an interesting story.
16:03:10 I, you know, because it's probably released in, but you saw in this classical human by stability case what happens is exactly is this.
16:03:23 The first species I add will deplete the carbon down to some very low level that the second piece is cannot survive.
16:03:24 And that means that it will reach this environmental state which makes it hostile to the other one, but if I added the second species first it will reach another environmental state which will be friendly to the.
16:03:37 I forgot which one it was blue species but hostile currents pieces, and you need to have this strong enough in order for for this to be really exclusion otherwise, you know maybe inverse self limitation.
16:03:54 I'm just I'm trying to.
16:03:54 I am right. I think it is exactly, exactly.
16:04:00 Yep. Yeah, and again in this face story we also told an interesting anecdote about it. Now what we are currently thinking is how to generalize the story to something which has been discussed at this conference lot and Redux state so imagine the Tower
16:04:27 Redux peers and so on. You can also, I don't really care if my carbon and nitrogen go into biomass, or are being used to generate energy. They are still being consumed, they are being converted into a byproduct which at this present, you know, version of the model I don't
16:04:35 of the model I don't have any byproducts, but imagine a grab a particular electron donor and a particular expert acceptor I combine them again energy from it.
16:04:45 The only thing which is again I can construct a model and we are constructed model but it will not be multi stable because the conversion is always one electron to one electron.
16:04:57 So there is this is dictated by physics or by consideration of electrons if you want. So there will be. You may argue that well maybe the seat to M ratio is not variable enough to give you a lot of stability in the real world, but at least it could be
16:05:11 variable here I know it cannot.
16:05:15 Okay, so now shifting gears, and those are a principles, this is the Marxism I subscribe to so Pankaj I know that you, you are fond of some other type, this is, this is my Marxism.
16:05:29 Okay.
16:05:31 Anyhow, so now the story again Veronica was in both stories. She's unfortunately a graduating student looking for a postdoc I'm sad, she's happy.
16:05:42 Actually it is already a long since left.
16:05:46 About, he's not left the collaboration but no no not not really at USC anymore. Anyhow, so this is a story we kind of predates the first one.
16:05:56 And it's so forget about nitrogen now Now let's come back to the carbon only. And let's consider what happens when I mix a bunch of microbes which use carbon dioxide manner, again, I think we are done with this kind of general introductions.
16:06:13 This is what we always discuss MacArthur model. This is what I told you about in the previous part of the talk now consider what happens when you have a bunch of microbes which are shifting by dioxin clear.
16:06:24 Switching between different carbon sources and that means that each of the resources disappears from your tyranny diluted environment at different times.
16:06:35 Let's say if you had a just one species which likes a lot this carbon source number one, it will be the first one to disappear. And then there will be some lag time when achieved through one to two then it will start growing on a secondary source and
16:06:52 second will disappear and third will disappear and so on. There is a story which I will not tell you about but maybe, actually, in his talk we'll touch upon it I don't know his plans.
16:06:59 This is a fresh, a work, which we submitted just this year, about exactly how to think about those tyranny diluted diversity shift in communities and how to apply some of the standard consumer resource techniques, except everything needs to be applauded
16:07:15 not in a carbon coordinates, but in the time until the resource disappears, born again, I probably not making too much sense here but just whatever diagrams you saw, or you plot a carbon one carbon to carbon three here I will say, tier one tier two tier
16:07:34 three and that's the time from the, from the start of the dilution cycle until a particular nutrient resource disappears from the environment, then if you think about it this way, everything becomes much more geometrical.
16:07:48 Now, what I want to tell you about is much more stylized model, and the stylized model is consider this situation which by the way maybe it's something I don't want really to be fixed on dioxin shifting microbes maybe it's more general than this.
16:08:03 So let's just assume that you are a generalist microbe which uses multiple nutrients substitute for nutrients and they have a certain preferences, means that you will rather use nutrient one the nutrients to.
16:08:17 But if you don't have nicer one or two you will use nutrients three, and this is another bug the yellow box, which has a different ideas about which nutrients are good which are bad.
16:08:27 It really loves nutrient three second nutrient to and third nutrient second different one and certain nutrients.
16:08:35 But that's not enough. It's one thing to love nutrient. It's another thing to really competition with other bug schools to love this nutrient.
16:08:44 So here is another table. Notice that there are no parameters. So in some sense it's a zero parameter model because everything is a rank order tables.
16:08:52 I still need to assume something about how those rank order tables are being generated but there are no PDFs no lamb there's no signals no office just ranked on tables.
16:09:05 The other table is those bugs are having a different competitive abilities on different nutrients so if you match yellow bug and red bug on nutrient one yellow will will read will win in a battle for nutrient two and yellow will be win the battle for
16:09:23 nutrients, three. So that's all I need to know for this model. And I want to know what will happen if I will mix those bikes together, and they will start kind of picking their nutrients having their battles and settling down onto some steady state.
16:09:37 So what would be low status states, again this is kind of a stylized model.
16:09:43 And we kind of it was inspired by another stylized model, which was proposed I already mentioned Gail and cheaply who published this paper college admissions stability of the marriage.
16:09:54 Then they got a Nobel Prize actually not both of them unfortunately David Gail passed away in 2008 and normal price was working 2012.
16:10:02 So he Lloyd shapely got it together is another person who worked a lot on those stable mentioned problems. This is economics game theory I personally learned about it when I was taking the games he recorded during my PhD at Stony Brook know a lot of people
16:10:18 of people will know like and respect worked on in Donald Knuth of tech and latex fame published this book really seminal book in 1976, and this is I call it my little green book I have it on my table by to MIT professor of mathematics about a stable marriage
16:10:36 problem.
16:10:37 So let me just explain what is a stable marriage so again I'm using the Jane Austen Pride and Prejudice and example for men for women. Each man ranks.
16:10:53 Each man ranks women in order of attractiveness to him every women ranks men in order of attractiveness to her rankings do not agree with each other, and then everybody gets married like in those novels, always everybody gets married.
16:11:02 You've got four couples. These are the stable set of marriages that are forming marriages.
16:11:08 So, it would be unstable. So, it will be stable if I cannot find at least one block in pair again block in theory the terminology of this marriage game theory, I wouldn't call it blocking pair but I would call it a runaway lovers right so, Elizabeth likes
16:11:25 Darcy more than her husband.
16:11:28 Darcy likes Elizabeth wars and his wife, so they would be happy to run away with each other, they say, to hell with everybody else they may be super unhappy if you run away but we would be happy.
16:11:40 We would be more happy. And this is by the way this is my kind of economics response to, to the
16:11:50 what,
16:11:53 exactly.
16:11:54 Exactly.
16:11:56 Anyhow, so are there stable marriages,
16:12:02 Not just the other ones, they can be super unhappy they may not even get married, they may go and kill themselves they don't care. Those guys are selfish, they just all they care is they are maximizing their own utility and that's why there is a certain
16:12:17 em gate here is that in order for something to happen the transaction has to be favorable to both parties, just like an in your system between razor the implants.
16:12:26 So, ours are still marriages how to find them all.
16:12:30 So let me tell you some anecdotes and again I have about 15 minutes that's good.
16:12:34 So, Galen shapely actually proposed the constructive algorithm to find at least one stable marriage. Usually, I will not go into details of the algorithm, even though it sounds a lot like this proposal game we all play when we are in the marriage market
16:12:49 or on the job market for that. For that sake. So, if you have one side of this to you know to to to part is, is being active being proposed in starting from the top of their list and the other one is being passive.
16:13:06 What you will end up is a stable match stable marriage, which will be the best for all active elements here so men are proposing, they get the best deal they can ever hope to get stable marriage every single one of them.
16:13:20 Every woman gets the worst deal they can ever have in a stable matching This is a mind blowing fact which I still don't cannot wrap my head around because I would expect that well some men are a little bit better some men are a little bit worse.
16:13:34 The see where it says, Every man is the best partner, you can have in a stable are each and every woman is the worst one. If you run this algorithm from the other side you get the one which is the best for all women, and the worst world man.
16:13:48 They will be in general two different states and the rule via quite a few of states in between, something like analog and states for em on average for random rankings you expect to find analog and states.
16:14:01 So, one interesting factoid is that there is a strict hierarchy of stable mentioned.
16:14:23 And each time you go down this line. Some men get worse off some women get better off and so on.
16:14:27 So, you may argue well are they all kind of similar in terms of quote unquote unhappiness so let's define unhappiness and happiness with the rank of your partner, if you get your top choice.
16:14:40 Your unhappiness is one. If you get your bottom choice your unhappiness is m.
16:14:46 So, it turns out that there is this relationship issue as well yeah no but it's easier, the math is easier when you I know, it should be easier I agree with you but just be.
16:14:59 So, Zach was my old time collaborator Mattel mercy Leah, learn from me about the stable marriage model back in 96, and they quickly published the paper which I still like to reread, they have shown that there is a interest in formulas that every can happen
16:15:19 If you run this algorithm, which is best for women should be for women. The men are super unhappy and login. Women are super happy login on before large and of course the difference would be huge number out there is there is an optimal solution where
16:15:22 as a whole, man, times every time happens so for women, is equal to em.
16:15:37 men and women are equal and happy and that would be square root of them.
16:15:42 So that's the best the society can get while you're still being stable.
16:15:46 And now this is bizarre and Peri Zm their paper about replica and optimization they consider it a version of this story in which instead of having requirements the abilities are required in Neverland dictator, they didn't write it this way and translated
16:16:01 you imagine I'm in charge of who gets married in the little community so I told Exactly. And I don't care about. I only care about maximize minimize and unhappiness of this side here, you can bring it down to chi square over 12 will have this bicycle
16:16:17 We're the fam this pie square or 12 was the replica symmetry, which is about 18%, less, you know, 18% better than the best stable solution so you may call it this 18% is the price you pay for, you know, in other words, since since I'm stable if you, if
16:16:45 dictator dies or something like this it will break and then it will go to one of those stable status or offense so this is my sort of anecdote about happy and unhappy marriages. Now back to microbes. Yes.
16:16:49 Yeah, yeah, yeah, there is no constructive algorithm how to do it but but you, in principle, there is an algorithm of how to get all the stable states, a little bit cumbersome you first which one which is best for men or women, and then you start initiating
16:17:04 divorces right.
16:17:06 And the worst follows particular directories somebody leaves somebody randomly is an abandoned spouse starts going down his or her list and so on and then you end up in another state and under certain conditions it will be stable so you can find all the
16:17:21 stable solutions, this is what actually the end, and actually it was actually assimilate and when he was looking for, stable states and our micro build.
16:17:41 So, as I already said. You only need to know to rank tables, but those of you who were actually paying attention, you will say well look nutrients are not choosing anyone nutrients are being chosen.
16:17:51 So how does it map to a stable marriage, it turns out it doesn't matter.
16:17:55 It's like, you still may, you know, so there are two tables one is preferences so microbes are clearly an active site here so they are having certain preferences, Doc nutrients nutrients are nutrient dense on competitiveness on a given nutrient can be
16:18:12 can be thought as a nutrients choice so it's like, you know, the nutrient will choose the most competitive micro pauses for this nutrient, and again, even though the nutrient is completely passive here.
16:18:24 Mathematically, it will be completely equivalent. So if I know those two tables preferences and competitiveness, I know everything about the system, interesting stories that you can trigger some transitions or regime shifts commit back to the title of
16:18:39 my book by either add or remove them extra microbes are nutrients.
16:18:47 Now this is kind of a maybe too long of a story for our little brain but it's instructive story. Nevertheless, consider a state with two microbes purple and dark green, and they're happy, they get their top choice they're super happy is their bliss level
16:19:01 and bliss, until this light green bug invades light green bug likes the same nutrient as the dark green microbe their boss prefers the nutrient number two.
16:19:09 Moreover, as you can see for nutrient number two, the light green bar is more competitive than the dark green, it's number one versus number three. The light green one starts to out compete the dark green one green one is frustrated like hell I had a
16:19:31 complete bliss. Top nutrient, but it switches, because it has this visibility of switching switches to its second choice.
16:19:39 When the box which is a second choice, it turns out that this green bar is more competitive than the purple box for a second nutrient it starts out competing the second nutrient.
16:19:51 The purple box purple bog also is frustrated like hell he had the best nutrient here it switches back and the purple bug out compete the light, green light green body disappears from the environment.
16:20:03 The entire community is worse off so you started with two happy microbes and you ended up with two unhappy microbes. And it's all because of an invasion of microbes, which is not even didn't survive the invasion, it was out competed out of the ecosystem.
16:20:20 So, the, the lesson from the story to whatever extent this has anything to do with the human gut and that's a big, big, if you have to be careful what you're thinking as a probiotic which is sort of life species because it can in principle trigger a transition
16:20:35 from one state to another. And even if you're no longer visible in any of the meta genomics data you will still end up in the community in a different state.
16:20:46 Okay. How well does this intuition scale up to higher dimensions.
16:20:51 I didn't try my brain doesn't cannot handle more than two, even to it barely can handle so uh but actually we simulation wide we of course right I will show you in a moment.
16:20:58 What happens when you have a seven.
16:21:07 OK, so the way.
16:21:12 Okay, you can bring back the microbes to the state of bliss. If you are instead of instead of extra microbes yet extra nutrients. When you add extra nutrients you can start with two microbes in a bad state, that should be a small circle small circle means
16:21:31 micro Buzzell happier and turn it back into a good state, and again there is some philosophy here. Adam a species that this transiently increases competition for resources competition for resources.
16:21:45 Make sure that the species would be worse off because they will be harder press and they will have to settle for worse, of nutrients, but adding a nutrient relaxes the configuration which brings it back to this blissful state where every microbe was getting
16:22:00 somewhere near near the top resource. This is how it feels to higher dimensions.
16:22:06 Seven microbes seven nutrients. Macy tables.
16:22:10 We found that in this particular example we have seven states this is a state which is the best for microbes. This is a state which is best for nutrients, those are all possible transitions between states, as I already told you.
16:22:26 They are kind of hierarchical, but still to trigger a transition from one state to another by removing one nutrient you have to be very specific. So, this transition can be triggered by removing different five this production can be triggered by removing
16:22:40 nutrient one. Notice that I am not adding nutrients but removing here just because we're seven nutrients. We are too lazy to recalculate everything it's the same story.
16:22:50 You can also remove microbes and then it will be going up this hierarchy again, following the same philosophical rules that in order to make life better for microbes, you need to transiently decrease the competition in order to make life harder for bankruptcy,
16:23:07 you need to. Presently, increasing the competition. Okay. Perfect diamonds so okay USA. Let's be practical so this is all fine but it's completely irrelevant to anything which, which happens in any microbial ecosystems, but maybe you will be right, but
16:23:25 people start to think about right now about utilization of nutrients of political rights by but through the species of which we have tons of species in our human gut.
16:23:36 And it turns out this is a level of Eric Martin's at University of Michigan and they did some experiments, three years or four years ago, where is a peak species but growing is available and victory to set a year and all my crown.
16:23:48 completed but they tested their preferences on those 123456 Polish soccer right and they found that they have a kind of distinct preferences, so don't ask me how those fingers work but they figure only pay attention to this number, according to their
16:24:05 study this Aveda likes those nutrients in this order 123456 a range them in order. A way to like them. If you now go to say the Euro micron, it still likes a CS which is con writings sulfate or something.
16:24:23 But then it suddenly like the second it likes something here, then it has a kind of indistinguishable like in between all three. And finally, whatever used to be second choice for this bug is only the six choice for for the ad on all my Chrome, so maybe,
16:24:42 just maybe, what is really happening in our gut, is this a bunch of bugs which are divided the turf between themselves so one each bug would like one particular nutrient best, and they will be kind of mutually complimentary game unfortunately this story
16:25:00 doesn't corroborated because they like a nutrient but maybe it's not present in the diet or something.
16:25:05 So again we ran a little toy model we don't know unfortunately those preferences. All we know is that for this seven bugs, we have a study in which the kind of analyzed what they can eat, and some of them eat more some of them eat less, and then we try
16:25:24 to see what will happen if all of them like the same thing so they are like competing for the same top make then everybody will end up rather I'm happy and our way of expressing that there I'm happy to give them a lower abundance.
16:25:39 If you just do a random selection, you will be kind of in the middle of the happiness scale and middle of abundance scale. If on the other hand you will just initiate them with the kind of mutually complimentary conditions whatever for each nutrients,
16:26:08 is a unique bug which has it as a top choice. They will all be super happy and interesting story again where you may hear from actually it is that when we ran this dioxin assembled communities and are much more realistic still idealized but much more
16:26:11 realistic serial dilution scenarios, what emerges out of this is that the bugs which their wife will be perfectly complimentary under option that concert choice doesn't matter but the top choice in.
16:26:21 In the dioxin hierarchy will be absolutely didn't.
16:26:26 Okay. Alright so let's see, so I'm almost on time four minutes, know my deadline take home messages, stable stable marriage problem is really really cool and useful stylized conceptual way to think about stable states and transitions and ecosystems.
16:26:45 And what makes me perfect personally attract attracted to it is that I don't need to come up with anything which I don't know and I still have to come up results tables but tables are discrete entities, they just need to rain quarter of a bunch of things,
16:27:03 and the remnants through my models and tells me We will survive, we will not. And this is contrasted with others era model again it's not a criticism I'm working on CRM model, myself, where you need to lots of quantitative parameter quantitative parameters
16:27:12 which I think all of us agree we will never in life of us we will be able to measure in really cases will never be able to tell exactly the update create of this nutrient under this condition for millions of conditions.
16:27:25 Maybe we will have a better luck just having some rank order list and wherever it's appropriate in predict something.
16:27:33 Now final slide is sort of a bit of an advertiser you can see that. I like the models was a few parameters as possible. We all heard about a top down, bottom up models.
16:27:44 I propose a new type of model the bottom down model where you take something which is already relatively simple and strip it down to the pier more I mean, so, as much as you can without losing the meaning, and then you are.
16:27:58 This is your membership criteria or in grade if you on the Euro primary plus one parameter a Bible more parameters fu are not the bottom down board or so if you have a candidate, again this is a joint effort by me and my colleague Kim snap and who is
16:28:14 as much in love with simple model says I am so this is my conclusions. Thank you.
16:28:27 One minute guys make one one comment to stay next upon God to hate space as well as chaos.
16:28:34 If you add on some, a bit of spatial structure here but everything is uniform. So then you can have these multiple different stable communities in different places, yes then you put on a little bit mixing between them.
16:28:45 And then what will happen is in each one. You will also get some of the others there yes engines. Yes, but if you turn off the mixing they would they would go back extinct again.
16:28:54 Now if you cranked up and mix them together too much of course then you decrease the number of back down. Yeah, yeah. So it's certainly a mechanism to have much much more there.
16:29:05 Were you still even if you sample locally, it says enough mixing you'll see many, but the other ones are sort of their, you know their more transiently.
16:29:13 If you turned off the if you turn off the mixing. I'm completely agree in fact this is when we are, I hate space as much as Pankaj does because it's an extra complication but I'm forced to consider that this probably had this with the remark which I forgot
16:29:29 to mention under is that, you know, we have this two things which are in the middle is death in Texas and in microbial communities space and features right without space and features your models are far from reality.
16:29:41 And I completely take your point in fact i think that the only realistic way of modeling something like this, Reeboks towers like Winogradsky column or something like this, is to reintroduce as much as I hate it, just somebody has to hold my hands and
16:29:56 introduce those spatial gradients. And then, initially air gradient, that's the thing right you can have everything can be the same, except you have the different stable communities in different places and some mixing between them right but then you still
16:30:08 But then you still have to somehow self organize this flux of nutrients is by the way was my kind of thoughts when I was listening to your talk, they know that it would be nice to have this self.
16:30:18 That's why I even asked you do you have those self organized Fluxus of something between your islands, or the.
16:30:34 Well if pages yeah pages can be this but if we are on the consumer resource model level then we want to I know that Matt has a model about it and this is a good model I like your model is gradients and spatial self organization of communities we need
16:30:41 to have those gradients we self organized, solve this diversity paradox at a cost but sold them and intervene and rescue color will figure in sample just one little tiny layer.
16:30:53 I would have kind of mostly the community which is optimized to this level with some bleed over from species from nearby layers. And this is a human also was advocating it with this idea about a big field with different nutrient considerations and this
16:31:08 and that and and you will have a lot of diversity, but it will be only if you collect from the entire field.
16:31:16 This is actually a powerful idea, it's, it's unfortunately and by the way, Mikhail also had this plot, you had a one plot where you keep all your environment parameters fixed.
16:31:27 And then once you start lead them very, the exponential increase.
16:31:34 You know, re emerges, and I have this exponential increase in my model if of course a model it in this tiny neighborhood of fixed flux is a will have a much lower diverse, but if I have a huge, which by the way, there is one interesting spot which might
16:31:49 be worthwhile to show here so you may argue that well maybe a really losers in terms of thesis will be will disappear from the system, you have this plot, which we rank, all of our God six microbes so we have six carbon six nitrogen here we have 36 microbes.
16:32:09 And this is the average competitiveness rank, what it means is, one, if, if this is a species which is best one, its carbon and nitrogen, both man all as our species using this carbon and nitrogen.
16:32:25 And this one is among the worst, it's five and a half, meaning that it's at the bottom of the list for its carbon and second from the bottom one is nitrogen.
16:32:46 And I guess what I want to point out is that even this complete loser is still present in one out of thousand samples. So if you buy us your Fluxus strongly enough, you will give this species an opportunity to shape it kind of relaxed.
16:32:53 So it kind of relaxed. You know competition if you want if you think about it globally just like as Daniel was saying,
16:33:04 is the end and then a comment on that is the set of non invisible I've heard of stable states in your models.
16:33:16 Can it be mapped 1212 alternative concentrations of vectors for your nutrients that is that this is true in a sandbox.
16:33:28 an exact statement one is that if you have a maximum diverse state if number of species matches the number of nutrients, you always your nutrient considerations are always fixed in any model MacArthur Tillman because you have those Isaac lines which intersect
16:33:52 you have six size clients we have a unique point that's the only place you can be.
16:33:57 Once your diversity is lower than some of the nutrients, or some combinations of nutrients are not fixed. So some nutrients are completely fixed, others are variable.
16:34:07 So if you are to go and sample in those ecosystems, you will find this nutrient is the same the same the same but this nutrient is variable.
16:34:16 And the number of variable nutrients is equal to the number of nutrients minus the number of species so survivor mean variable.
16:34:30 The different concentrations sustained the same community. Yeah, Yeah, exactly. So the biomass this will be variable because biomass is we will be dependent on the Fluxus, it's almost like one to one it's linear combination of a give Fluxus.
16:34:41 And I think the state I know the biomass nutrients are almost fixed the nutrients which are limiting the growth of at least one species are fixed simply they're fixed at the sea star or m star in the more in this model again it's the stylized model but
16:34:56 it just means that this nutrient is just at this point where it can. It is being controlled, if you want this nutrient is being controlled by this pieces it limits.
16:35:07 Okay, so it looks like.
16:35:11 For a given diversity, you would have a
16:35:18 one to one mapping to a subset of the, of the notions, all the nutrients exactly what that lead to testable hypotheses that you could then test in good point.
16:35:28 Good point. Whenever you find different speeds configuration that should also map. Yeah, good point, I almost like this is you know I have never thought about writing this essay but what you're saying is that there is a massive database of some community
16:35:44 where I have the reasons to believe that as being colonized by the same pool of species. Once I find that this particular nutrient is roughly constant.
16:36:07 Yeah.
16:36:21 You're not the one explaining that variability in speech competition. Yeah, yeah. Again, everything will be approximate because this L shaped curve in reality is like a parable so there will be a little bit over, but again this concept of the limiting
16:36:23 nutrient consideration our stars his star is a well known one and people like Tillman tested it in say like Michigan some, some for the synthetic bacteria or algae I forgot what it was.