 Okay, so here we are for number three and I applaud you all for surviving two weeks. So you're almost at the end of this. Today I'm going to continue to talk about issues of control in biology but whereas yesterday I talked mostly about pattern and positional information today I'm going to talk about growth, proliferation, and organ and tissue size. And I thought I'd begin by acknowledging the fact that you know among people who take quantitative approaches to to growth and proliferation a lot of the time people are focusing on you know the growth of populations. E. coli in a flask or populations out in an environment where you know you have a situation of material comes in, wastes go out, and the net result is you have cells or organisms that expand according to a simple exponential law. But then that's usually modified by the fact that organisms are depleting their environment of something and so you get some sort of modification to that law that reaches a steady state on its own. This is a very typical version, a logistic rule where you know the environment has some fixed capacity and when you exhaust that capacity you stop growing. So there's a lot of beautiful quantitative biology and mathematics and so on associated with this kind of growth. But that's not what I'm going to talk about today. What I'm going to talk about today is the growth of organs and tissues. Okay so within a body how things grow and reach the appropriate sizes. And the difference there is of course you still have a situation of things that are proliferating according to an essentially exponential process and they're still limited by nutrients and raw materials. But the issue of the depletion of those nutrients and raw materials is pretty much not important. The reason being that from the perspective of any one organ or tissue in your body it's not a particularly big drain on the raw materials. And so if it wants to grow bigger it can. And of course the sad fact of that is you get a cancer it will simply rob all the rest of your organs of nutrients in order to grow because it's really not limited in any way. So this is not the limitation in organ and tissue growth. There is cell death. There are changes in cell shape and size and these are important in particular organs but by and large they're a kind of minor correction to the major player here which is the regulation of this exponential proliferative process. Okay so that basically controlling the size of things is primarily about controlling this exponential proliferative process. Now yesterday when Boris talked he mentioned this very important work from Darcy Thompson now a hundred years ago on growth and form which was really kind of sparked the interest of many people in approaching the issue of size and form and shape and so on in a very mathematical way trying to find fundamental laws. He pointed out that very complex shapes like this could be derived from simple growth laws and as you heard both from Boris and Marianne that different shapes can also be related to each other by very simple you know kind of scaling and shearing principles which suggests that there are kind of fundamental rules behind morphogenesis. So the relationship between growth and morphogenesis is actually a fairly complicated one and I'm not going to get to it until the very end of my talk. I'm going to start by talking about a very even simpler issue which is the regulation between growth and simply size just how much you make and the reason why to talk about that is because observation suggests to us that growth is very tightly regulated to produce specific sizes and of course you know Leonardo noticed a long time ago the beautiful symmetry in the body and also the beautiful proportions and how reliable they are from individual to individual. If you think about the fact that your arms match it's a quite remarkable thing given that these cells stopped talking to these cells some time before gastrulation right so there's really been almost no direct communication between the two almost perfectly matched sides right and we know this is a genetically controlled thing because if you look at individuals who are monozygotic twins right you see tremendous amount of kind of reproducibility of some of even the finest finest features in terms of sizes of things so all of these things suggest that size is strongly under control and so we can approach that same question the way we have when we talked about pattern or other phenomenon biology that if something's under control we should be asking questions about what are the control objectives right what has evolution selected for in terms of the producing that control and so remember these are the objectives that I talked about in both of my previous lectures and we're going to go through some of these in the context now of tissue and organ size and as before a good way to begin is to start with the observations what do the data that are out there tell us about which things are under control and how well controlled they are so the nice thing about studying tissue and organ sizes there's some really really beautiful data many of which stretch back almost a hundred years some of which are much more recent but they really provide remarkable evidence of tremendous control so one thing is that if you look across genetically identical individuals and you just look at what's the coefficient of variation of size for many organs you'll find is quite small so for example if you dissect out the brains of you know mice that are genetically you know inbred and you just get the coefficient of variation on brain weight brain dimensions or even brain cell number it's on the order of 3% so that's pretty good if you go out in the wild and you collect snakes of the same species so you know snakes have this enormous number of vertebrae this particular snake species has 300 almost 300 vertebrae and the coefficient of variation from snake to snake is 2% and what's remarkable about that is the way that those vertebrae are formed and you heard a little bit about somitogenesis right earlier in this in this course so here's the somitogenesis going on in a snake embryo and it's just forming this coil that's going around and around to getting longer and longer and it stops at 296 plus or minus 4 somites so that's quite a remarkable thing right for this repeating process that seems to stop right at the right time another thing we know this quite remarkable is how independent organ and tissue sizes are from the initial conditions and so here's an example of a mouse embryo in which the investigator has sucked out one or two of the cells to do something with them and so you get back an embryo that's missing maybe 10 or 20% of its cells does it make a mouse that's 10 or 20% smaller no it makes a perfectly normal mouse in fact you can take four embryos smush all the cells together so you have a mouse with four times as many starting cells it still makes a normal sized mouse and this is actually clinically very important because people do pre-implantation genetic diagnosis now on human embryos and that requires throwing away some cells and if we were worried that people would come out short as a result of that we wouldn't do that right so but we're confident that the size the final size is independent of the initial size we also know that organs and tissues often have a remarkable ability to reject disturbances in other words you could slow down growth or take away cells early on during the growth process and the tissue will catch up and go back to its original size or original sort of intended or desired size and classically one sees this in the clinic when you have children who suffered some kind of a disease long-term disease or cancer chemotherapy or something like that during early childhood when they're cured they will very frequently go faster in growth until they catch up to their original growth curve and end up at the size that was predicted for them at birth and so that's called catch-up growth another thing we know about size and tissue control is that it's independent of the rate of the cell cycle and again that's a bit counterintuitive right you'd think of cells cycle faster you make more stuff right but some beautiful experiments in fruit flies where the wing disc remember the wing disc has posterior and anterior halves that you can genetically manipulate separately so if you separately manipulate the posterior half of the wing disc so that the cells grow 35 percent faster or 35 percent slower than in the anterior what happens is the two half discs grow until they reach their right size the faster one stops and the slower one continues going until it catches up and then everything comes out giving you a perfectly normal wing so the rate of the cell cycle isn't important and remarkably the sizes of the cells don't seem to be that important either because there are mutations you can introduce into the posterior half they give you cells that are extra big or extra small so if you make the cells twice as big it simply goes through half as many cell cycles in in order to produce a tissue of the same size and so again you get a normal sized wing now sometimes we know a little bit about the mechanisms underlying size control and sometimes we know they're kind of long range across the whole animal and we know that from these kinds of experiments which are called parabiosis these two rats have been sewn together right there their skin was cut and connected and so the two rats are now connected to each other and what happens is their bloodstreams begin to connect to each other and as a result everything circulating one rat is now communicating with the other rat and if you go into that and you resect or remove part of the liver from one rat the liver starts growing in the other rat and basically it continues growing until the net liver in the two rats is equal to two rats worth of liver even though now one is bigger than the other so somehow you're getting the right size across the rat so that's an example of kind of long range or global communication most often that's not what you see most experiments suggest that size control is quite local and some of the oldest experiments go back to the nineteen thirties in which you have two species of salamander one would make a big salamander one's one like a small salamander and you can graft the the buds that are going to make the limbs from one salamander onto the other right so you make a chimera and the result is that the limb that comes out is specific for the donor right the one that came from and not the one that was grafted onto as though each limb autonomously knows what size it's supposed to produce now another thing we notice is if we look at the rates the dynamics of size control uh... we see that growth often occurs at nearly an exponential rate so this is a drosophila wing disc these are some recent data looking at uh... here on a logarithmic axis the number of cells uh... in the wing disc and it's going up as a straight line on a log uh... plot so that's exponential growth up until about the end of the third larval instar and then just stops you know rather abruptly it stops another example of that phenomenon is uh... in the liver which is a very good regenerator you can take out three fourths of a mammal's liver and the mammal will survive usually and will regrow all the cells of that liver in precisely two cell cycles so basically all the remaining cells go back into the cell cycle they divide once twice so now you're back up a factor of four and then they stop dividing and then a few minor cell types appear somewhat later okay so this kind of process where you go as fast as possible to the goal and then you stop remember has a name in engineering it's called bang bang behavior bang bang control and actually there's some recent work this doesn't have to do with tissues and organs but even with cell sizes suggesting that's the growth of individual cells is bang bang so some nice studies from uh... mark kirchner's lab that very carefully plotted the growth rate of cells up until the point that they undergo cell division and you see the same thing it's an exponential growth rate and then stops and you know if you reach the final size you haven't divided yet you simply stop growing right and then you eventually divide so somehow there's bang bang behavior so when you add all this up tissue and organ size control is is a tremendous example of fantastic engineering right you have great accuracy you have bang bang kinetics you have uh... parametric robustness right we can vary the cell cycle we can vary the initial conditions can vary the size of the cells it doesn't seem to matter you can reject disturbances right you have this uh... catch up growth stable right when it's unstable we call it cancer but most of the time well not just cancer there are other hyper proliferative disorders but they're rare most of the time tissues and organs stop and that's it right either they maintain a steady state or they just stopped for good and it's also adaptable right we can adjust the sizes of things some of the organs for example your spleen your liver these will grow if the needs require it your lymph nodes you know various things can also adapt so of course with this much control there has to be feedback right you can't do all this kind of stuff open loop so there has to be feedback somewhere and the idea that feedback plays a role in the control of tissues and organs is an old idea uh... it was uh... kind of most uh... clearly articulated back in the nineteen sixties by this guy bullock who coined what he called the k-loan hypothesis so k-loan is a word that comes from chaos and hormone the idea is that it's you know some kind of circulating factor except instead of doing good things which is what people thought hormones did in those days you know promote growth it's kind of destroying growth and the idea is it was some kind of a negative feedback on growth cells would produce k-loans and then as they accumulated to a certain level that would stop cell growth so that's basic negative feedback control but the idea was this is this was the way biologists put it in the nineteen sixties uh... that spawned a lot of work in tissue culture and various other fields to go search for these mysterious k-loans nobody found any by the nineteen late seventies early eighties everyone pretty much gave up started working on molecular biology and other things sometimes things take time the first real k-loan did not emerge until the late nineteen nineties and that was mainly through studies of knockouts in mice when it was discovered that there is a molecule called gdf-8 or myostatin i think is the official name now is myostatin it's produced by muscle and when you delete it from a mouse you get a double-muscled mouse you get a mouse with at least twice as much muscle as normal it's produced by muscle and you can show because it's easy to culture muscle you can show that it acts to prevent muscle pre-genitors from proliferating so it stops the production of muscle and it's made by muscle so it fits the characteristics of the k-loan uh... incidentally there are naturally occurring breeds of animals that uh... are gdf-8 mutants because they have twice as much steak uh... this is the belgian blue i think you can get that here in europe in the butcher shop uh... i've never had it i'm told to taste good but in any case there's a lot more muscle per cow here uh... so it's since the late nineteen nineties it's become clear there are many molecules which are secreted proteins many of them fall into one particular family of proteins that tgf-beta superfamily you can see a lot of them here some of them don't but there are many that have the characteristic that they're produced by tissues and organs and when you throw them on those tissues and organs they stop or uh... or restrict growth in some way so at some very simple qualitative level you can say okay the problem solved we know the basis for control it's you know these secreted things but in fact of course it's not so simple because you have to do the math right so let's do the math very briefly uh... here's an exponential process right the number of cells goes up proportional to the number of cells we can draw that here as a cell with a little curved arrow showing that it's self-renewing and it has some rate v at which that's happening if you'd like to think in terms of cell cycle times v is just one the log two over the cell cycle time okay so now if we want to uh... to consider x uh... negative feedback we can do what we've been doing before just put in a term that uh... goes down with the number of cells and we can give it an exponent here so we can make it very steep feedback or very shallow feedback depending upon what we choose here and we have a parameter here for how strong the feedback is and so you can write this equation and so will this equation give you good control of growth well you know just simulate it what do you see well what you see is instead of a tissue that grows exponentially if n is one it grows linearly if n is two it grows sub-linearly if n is three it grows very sub-linearly but what it doesn't do is ever stop growing okay so you can achieve stability in an exponential circuit with proportional negative feedback so that alone is not enough to explain even why a tissue stops something's missing now of course we can say why our model is not right because not only the cells proliferate but they also die at some rate so we can add in a death term to our model and indeed now this model does reach a steady state okay but if you look at that steady state here you see it's very non robust to its parameters so for example if you look at the value of the steady state it's pretty much proportional to the rate of the cell cycle it's also pretty much proportional to the rate at which cells die but I already told you that real tissues they don't care about the rate of the cell cycle they come to stop at the right size so that's telling you this model is probably not right there's another reason why this model clearly can't be good enough and that's that many of our tissues aren't actually steady states at all so some of our tissues like the intestinal epithelium, the epidermis most of the epithelia that line your organs those are tissues that turn over all the time they're constantly having cells die and constantly having cells replace them okay those are steady state tissues but all of the rest of our tissues don't fall into that category they're created by developmental proliferation that effectively ceases or if it doesn't cease it goes to such a low rate that you may consider it essentially as having stopped so our bones our brain the corny of the eye, the retina of the eye and in invertebrates like Drosophila and C. elegans virtually every tissue falls into this category except the germline okay you grow and you stop so it's not really a steady state right because there's nothing dynamic at that point you simply run out of any cells that are proliferating and so we'll call that final state tissues right so obviously if there's no death you can still stop and so again the simple model of just growth balanced by death with negative feedback that can't be the right model okay incidentally this is an example of homeoresis not homeostasis right the fact that you reach the right trajectory the right end point but it's not a not something you're maintaining as a steady state as in homeostasis so how do you do that right how do you actually get this exponential system to robustly stop in a way that's independent of a lot of the parameters and so as you might expect you have to have integral feedback control okay and if you remember there was one type of integral feedback control that I already mentioned and Boris mentioned it as well that uh... uh... affects tissue growth which is mechanical uh... negative feedback through essentially pressure or compression of cells if you have cells that are growing faster than the cells around them then that will cause pressure on those cells both the cells themselves and their neighbors and so if growth is affected negatively by excess pressure right then you essentially have a negative feedback loop and since the pressure will continue to rise right as long as there's any excess growth at all right you essentially have integral negative feedback as the pressure forced to decrease growth is going up and up with time okay and this is just a reminder that we do now know of transcription factors that are sensitive to the amount of stretch or compression on cells and so that gives us mechanisms by which this kind of thing can be implemented although to be honest there's a lot of the details of how this works that we really still don't know about so i mentioned that last time as one mechanism by which you can achieve integral negative feedback control but today i'm gonna focus on another man another mechanism which i didn't mention on wednesday which is a mechanism that involves the use of cell lineages and with feedback between different elements in a cell lineage okay so you've already heard about cell lineages you heard from alone about cell lineages that you know cells go through different stages on the way to become for example hemopoietic cells but this is really the rule in all your tissues that they're organized in terms of a proliferative cascade in which cells uh... are going through different stages on the way to making terminal cells cells that don't divide okay and by and large you hear a lot about these stages as being kind of the unfolding of a genetic program involved in the functions of these terminal cells right you need to go through these stages to you know turn on the right genes to be an erythrocyte or be uh... uh... you know a megakaryocyte or something like that but today i'm going to suggest that there's another reason why cells go through lineages and that has to do with control and it has to do with the fact that when cells do go through lineage now the process of growth depends on two parameters not one there isn't just a speed of the cell cycle that matters obviously that's important but there's this other parameter which is really a probability which we call the renewal probability renewal parameter it's between zero and one and for every cell it describes the fraction of the proliferative output of that cell that retains the character of that cell as opposed to progressing on to the next stage so p equals one means a cell is just renewing itself all the time p equals zero means every division gives rise only to daughter cells that are different p equals one half means every division on average gives rise to half of one progression and half of one renewal it doesn't necessarily mean that that's true for every cell this is just a probability on average p equals half means half goes this way and half returns so i'm gonna argue that the critical thing for calons is feeding back on the self-renewal parameter okay so what's so good about feeding back on this self-renewal parameter which describes the division between these two activities and so in order to see that we're gonna write this the dynamics of this process in terms of two uh... remarkably simple differential equations so these are just differential equations for the stem cell and the terminal cell all right now this is your basic uh... exponential growth equation the stem cell c zero grows a proportional to c zero with the rate constant equal to the cell cycle speed v so that much is the same and over here this minus one plus two p is just another way of saying with probability p the cell divides so you make two but you lose the parent so you minus one and likewise for the next cell with probability one minus p you make two of them same speed and then they die at some rate so very very simple equation to talk about negative feedback on p all we have to do is replace big p with some function that declines according to c one so we use our same function the hill function where we have some constant which is probably going to be close to one but it really doesn't matter and then we're going to have something into the denominator that's proportional to c one in this case we won't will just have a hill coefficient of one but we can use whatever we like there now if we take that simple differential equation and just plot it how it behaves starting from initial conditions of very low numbers of c zero and nothing else you get behaviors like this where red marks the c zero and blue marks the c ones and you'll see what happens the systems generate a lot of stem cells this first cell type and they were new and they were new and they were new and then at a certain point they had a kind of inflection point and they level off and they start going down and now the terminal cells start going up and they reach a steady state and the stem cells reach a steady state as well now if you play with the parameters different initial conditions different cell death rate different speed of the cell cycle you change the dynamics but the steady state is exactly the same okay so we're on to something good here right we have something that's robust to all of its parameters which certainly suggests that difference between concentration of what of the cell the same this is the actual numbers of these cells okay you can have number or concentration it doesn't matter because these equations are kind of blinds to what you're measuring okay yes uh... no in terms of terminal cells that's all we care about not the stem cells the stem cells are not robust but the terminal cells that's what actually can see the majority of the tissue consists of terminal cells and they're the ones that do the job of the tissue okay so everything's going to be looked at from the vantage point of the terminal cells all right so if you change for example the terminal cell turnover of this rate of the cell cycle what happens is the stem cell numbers adjust uh... so that the terminal numbers are exactly the same okay in terms of the initial conditions actually both end up exactly the same okay you'll see why in a second okay so one of the good things about this type of feedback is right away you see you get stability all right and furthermore you get stability without the stem cell needing to know anything in advance uh... that's fine they don't need to be fine-tuned in any way this actually uh... solves a long-standing problem in stem cell biology right is that most people realize if you have a stem cell that makes the terminal cell and you want the system to be stable right what does the effective p-value have to be in order for this to be stable where you're making these guys at a constant rate and then they're dying with a constant probability anyone want to guess has to be one half exactly right has to be exactly one half other because if it's not one half say it's more than one half then these numbers are going to go up exponentially and if it's less than one half these numbers are going to decline exponentially the only way for stem cells to stay constant is if half of their offspring progress and half of their offspring renew right people have known that for a long time but no one's quite known how that comes about right why is it the case that it's exactly one half and until recently the most popular idea was that stem cells were obliged to make asymmetric divisions where they were just because of the mechanism by which division occurs they were forced to have one of their offspring become differentiated and one of their offspring stay at stem cell and there's about ten thousand papers in the literature about you know how all this is done but again this is an example of open-loop fine-tuned behavior and an engineer would look at that and say that's crazy right and more recently it's become very clear that from the standpoint of each cell this is only true in a probabilistic sense right some cells make two of these some cells make two of these some cells make one of each and somehow the whole population achieves exactly one half so that's been a mystery for a long time but really it shouldn't be a mystery at all because it's exactly what feedback does for you as a result of feedback the net p will go to exactly one half uh... and so you don't really need to have any fine-tuning of the stem cell you also get remarkable speed if you look at this behavior the terminal cells are appearing almost exponentially until the last moment right so it's not gradually approaching the final size it's getting there quickly and as i mentioned essentially perfectly robust it doesn't matter what the parameters v or d or the initial conditions are and you can see why it's perfectly robust simply by solving in the steady state because if you solve this equation in the steady state right what do you get on the assumption that c zero is non-zero and v is non-zero the only steady state available as p equals one half but if this is the equation for p the only way that can be one half is if c one the terminal cells have this value and that value is independent of all the parameters except p bar and gamma yes exactly right so this model is saying somehow right the feedback is tracking the total number of cells and we'll come back uh... in the second part of the talk conditions under which that becomes difficult okay but for now remember the cells are producing some secreted molecule right the more cells the more secreted molecule you have so at least at some level the cells ought to be able to track how many of those cells there are by the amount of k loan they're producing as you suggest there there will be problems that arise from that yes so this it's it this is the steady state but if you do stability analysis you can see it's a it's a it's an attractive or st or st asymptotically stable steady state and just to kind of drive this point home a little more you might think you should right influence the rate of the cell cycle with your feedback right why not slow the thing down when you make too many of these cells and you can model that too by putting in v as a function of c but if you do that first of all it doesn't solve the stability problem you still need to have p be completely fine-tuned if you have no feedback on it but the other thing is you can calculate the parametric robustness remember using sensitivity coefficients again this is one of those cases where you can't solve it in the general case for all and but you can carry you can generate the sensitivity coefficients for all and uh... this term cap here combines all the parameters and basically this is is what the number of terminal cells would be if you didn't have feedback so uh... if you do have feedback that means the number of cells is smaller than that and so therefore this is a small number so basically this just goes to one plus one over and one plus one over and and this becomes and log the size of the population over one plus and and so that's pretty terrible robustness you have to have really big ends really steep feedback in order for that to be very good contrary to intuition if you want to control a population negatively feeding back on the rate of the cell cycle is pretty useless but feeding back on self-renewal is really useful you had a question there's no stability that comes from this feedback the only way it can be stable is by setting that parameter to exactly one half okay that's because it's an exponential process and you're trying to stop it with a proportional feedback right it's kind of like that case i alluded to with the uh... asthma regulation in yeast right you can you can get over the process by having your perturbation climb in time right here you need a essentially have a process that's climbing in time the only way to catch up with it is to have an integral feedback that itself is growing in time no it doesn't become any more useful now there can be used when you combine the two when you have feedback on both of the renewal probability and the rate of the cell cycle you can help kind of fine-tune your uh... your behavior that way as as the sole strategy feeding back on the cell cycle is pretty useless alright so um... this is a little bit surprising if you take these equations you remember i said not every tissue is formed by a terminal cell that turns over and is balanced by a continual renewal a steady state so you can take the same equations and just delete that turnover term and say how does this behave right so it turns out uh... if you try to solve for a steady state here what's the problem if you try to solve just by setting the rates to zero so the problem is your model here is that you make lots of terminal cells and all the stem cells disappear which means that time equals infinity c zero is zero so if you try to solve for the steady state you just get zero equals zero so you can't get anything out of it right so you have to figure out where the final state of this this set of equations is going to be by taking an integral right and letting that integral go to infinity so you have to so this is one way to do it you define a new variable which is the sum of the two so essentially it's the size of the whole system and so consequently the rate of growth of the whole system is just the sum of the rates of growth of these two but because this plus that equals one when you add these two equations up you just get the rate of growth of the whole system is just v times the number of stem cells well that that's also pretty intuitive the stem cells are the only things dividing so of course the rate of the growth of the whole system is v times the number of stem cells so now if you integrate that right you get that the the size at any time in this case will pick infinity is just the initial size plus v times the integral of that okay but fortunately from the second equation you can represent that integral in terms of c one and its own derivative right we just solve this equation for c zero stick it in there we notice the v's cancel and then the trick is you realize that you can rewrite this integral in terms of limits of integration based on c one so essentially you can absorb the derivative of c one and the differential with respect to time into each other and now you have the same thing expressed as an integral of with respect to c one going from its initial value to its final value and the only thing you have in here is the form of the feedback function and its argument okay so of course the final value of c one is just the total system size at infinity because all the stem cells go away and so now you get this equation which has the final size and the final size in two different places here so of course that's a little tricky to solve but if you happen to know the form of the feedback function for example if you use that simple hill equation in this case with an exponent of one then you can evaluate that integral and you get this slightly ugly formula here but what's nice to see about this formula is the initial conditions enter here and there and so right away you can see if the initial conditions are small if you start from a relatively small number of cells compared to the final size then essentially they're negligible right you can ignore this ignore that ignore that and you get an equation for the final size which again is not explicit because you have s infinity on both sides and so you try to solve it you involve weird things like product logs but the net result is the only parameters of any consequence in here are p and gamma okay so essentially you get that same kind of almost perfect robustness even though it's not a steady-state system and so when you plot the behavior of these equations it looks almost like the last graph with the slight difference that now the stem cells go to zero and these are not exactly identical they're slightly different from one another but again you get the same behavior you can vary the initial conditions you can vary the rate of the cell cycle nothing much happens you arrive at the same final state so it's integral feedback but in a homeoretic rather than a homeostatic system funny you should ask that question because that's exactly what i'm going to show you so you would predict if this is the way a real tissue works like the brain that if you measured the effective p-value over time it would fall according to a curve like this that would look just like the curve for the production of final cells and that's actually been done for the mouse brain this is for the cerebral cortex and it shows you exactly that behavior if you measure the renewal behavior of each of the cells it falls according to a very fast trajectory that matches the exponential production of the final cells it effectively goes to zero i mean there are a few places in the brain where you keep some around even then they're very very slow so from the mathematical standpoint they're pretty negligible you can take the same equations and you know uh... play with them have a period of growth depression and you see they catch up and if you plot these results on a log scale you'll notice it goes up almost perfectly exponentially until the final point at which it turns around and stops and so it exhibits that bang bang behavior so just this one very very simple feedback gives you all of those things and so not surprisingly when calones have been looked at uh... in every case where anyone bothered to check for it they're doing exactly this they're inhibiting self-renewal it's true for gdf8 and my or myostatin in the muscle factory epithelium is another really nice situation where there's two calones that have been well described in a neural tissue they do exactly the same thing and although in the hemopoietic system there's a lot we don't know about what the actual factors are there's some lovely work that's been done in vitro that suggests and again whatever the substances are that are being shared between cells they're primarily working by impacting these p-values or self-renewal so the nice thing about having this based on such simple math right is it's very easy to demonstrate how this is working and why this is integral feedback control because remember i said for integral feedback control you need something that acts as an integrator and here it's very tricky to kind of see what's the integrator because you have these two coupled equations and so let me show you a quick way to sort of see why this works it uses that same trick of defining a new variable that represents the sum of the two variables and therefore it's the total size of the system so again the rate of change of that is just the sum of the two rates of chains and when you sum them up you get this right so why is that useful because these facts allow us to reinterpret all the c variables now in terms of s and its derivatives so for example c is just s prime normalized to v from this equation and c1 is just s minus s prime from this equation and that equation and c0 prime is just s double prime from this equation and so now you can take the first equation and rewrite it just in terms of s where p now instead of being a function of c1 is a function of s minus s prime okay so for example if you use a hill function for the form of that feedback you get an equation like this okay so why is that helpful to have that equation so the first thing is that equation is an equation for the growth acceleration the rate of change of the rate of change of size and so that means it's a control equation for the growth rate this is the thing being controlled the growth rate and this is the rate of change of the growth rate so it's a control equation for the growth rate and what's in the denominator of this control equation for the growth rate the integral of the growth rate, which is the size. That's why it's integral feedback control, because the integral is in the denominator. But you can see a few more things. Because of this minus one over here, it tells you that as s goes to infinity, s double prime has to go below zero and therefore you have guaranteed stability. The system has to growth decelerate at some point and eventually will go to a growth rate of zero. Because it must go negative with a big enough s. And then the other thing you notice is this funny little term here. Minus s prime in the denominator. So something in the denominator, right, is kind of like a negative feedback, but you have a minus in front of it, so it's really a positive feedback. The growth rate is feeding back positively on the growth rate. And if you take that term out, you still get a system that's stable, that has integral feedback, but it doesn't display that bang bang behavior now. It gradually approaches the final state instead of very abruptly approaching the final state. So the bang bang quality is dependent upon this extra little term of positive feedback of the growth rate on the growth rate. And there's nothing I know of an engineering that's based on this type of circuit, but there is something in economics. It's called the Ponzi scheme. So this is Carlo Ponzi, and this is Bernie Madoff. He's the greatest Ponzi schemeer of all time, but Ponzi, of course, is the one who is named after. And he had this scheme in the early, I think the 1920s, right, when he would borrow money from one set of investors, promising to pay them at a wonderful rate of return. And the way he paid them was by borrowing more money from an even larger set of investors, paid with that, you know, and borrowed even more money from an even larger set. And as long as he can keep finding more investors to borrow from, he could keep paying off people at fabulous rates, right? And so people were getting rich and he had to keep finding more and more and more investors, you know, and of course eventually, right? That's not gonna work over time. And this is a graph of actually Ponzi's assets during his scheme, which ran in the 1920s. And you can see his assets are growing more or less exponentially. And then suddenly the whole scheme collapses, right? When he can't do this anymore. That's usually what happens with the Ponzi scheme. And, you know, if you're a physicist, you're probably gonna look for like some fancy phase transition here, right? You know, something where there's a critical behavior and so on. But really it's much simpler than that. What's going on is there's a lot of negative feedback in this system. As his assets are growing, right? There's newspaper articles and lawsuits and everybody's calling this guy a crook. But it's not stopping, why? Because there's positive feedback of the growth rate on the growth rate, right? The faster his assets are growing, the more people wanna get in on the scheme. And that's balancing the fact that the bigger the assets are, the more negative feedback there is to tell people not to get in on the scheme. And so you end up with that same bang-bang behavior where it grows exponentially until the point at which it's slightly unsustainable and now suddenly the whole thing crashes, right? And you see that. Anyway, so that's same basic equation. All right, so let's see, where are we now? Ooh, we got, all right, got a fair bit to get through. But let me stop and take any, you have any questions at this point? This is kind of a good break point. Yeah, very good question. So you can of course have an even more realistic lineage, right, where you have something like this, where you have stem cells and terminal cells and what some people call progenitor cells or transit amplifying cells and so on, right? And you can get all kinds of very interesting behavior if you have multiple loops and so on. But the bottom line is, no matter how you configure this, as long as you have one loop that goes back to the initial cell in the lineage, this will be stable and the concentration or number of these cells will be set point controlled by the gain of this or the strength of this loop. Doesn't matter what's in the middle, you can have a hundred stages, right? But the end result is this one loop controls everything, okay? Usually you don't have a hundred stages, you usually have like one or two. So often during embryonic development you have growth and proliferation if you don't have necessarily a sense of a discrete stem cell population giving rise to a discrete final differentiation, think so. How do you have to shift things around a little bit? Okay. You don't pay even like an imaginable bit. Yeah, so okay, okay. So yeah, there are cases in which you have things that reach final states where there doesn't appear to be a stem cell, right? And so one possibility is that you're achieving integral feedback control by different mechanism. For example, in the disk it's been suggested that mechanics plays a bigger role. But the other thing is you don't really need to have a distinct stem fate and terminal fate. You simply need to have cells that are different from each other in terms of one cell sending a negative signal and one cell's not sending the negative signal. That's really all you need. And they can be inter-convertible. Yes, they don't have to be a permanent state. Yeah, so you can make it work without having a fixed state or a state that's not inter-convertible with the initial state. You can make it work. And if you look at things like the wing disk, in fact, there are changes in cell state, for example, expression of vestigial and things like that that are changing across the disk as time goes on. So cells clearly, where some cells are vestigial positive and some are vestigial negative. So that may well be a way that you do it in those kinds of tissues. Okay, good. Let me go on. And the remainder of my talk is gonna focus on this issue of what is the price that you pay for strategies like this? Because remember, I tried to emphasize every strategy comes at a price, right? There's always trade-offs, right? You build a fast fuel-efficient car. It's bad in a crash. You build a crash-tolerant car. It's not fuel-efficient, right? You animals show the same sort of thing. So consequently, you would expect that these integral feedback control strategies themselves are subject to trade-offs and that what you see in the animal is going to be influenced by having to overcome some of these trade-offs. So for each of these strategies, we can divide the different kinds of trade-offs into three sort of domains. The temporal, the spatial, and the stochastic. Okay, that's just for convenience. So for example, on the temporal side, I already pointed this out on Wednesday that pressure, when you put pressure on cells, it's absorbed in some way. It relaxes over time. The cells get smaller or die or whatever. And so consequently, the integral feedback associated with this process is leaky. And therefore it's not really that great over long time scales, but it's spectacular on short time scales. So that's a problem potentially with that that suggests for certain types of long time scale growth control, that may not be the optimal thing to do. Now there's also a temporal set of trade-offs associated with this sort of renewal feedback, and that's something that engineers call non-minimum phase dynamics, which I'll explain to you because just having the name doesn't really mean anything. Let me give you an example. If we take this very, very simple system, so here it is written again where I've gone ahead and put in the hill function with a hill coefficient of one for our feedback term. And you just simulate that under conditions whether the D parameter is large compared to the V parameter. Then you get that kind of beautiful bang-bang behavior for the terminal cell that is what I showed you earlier, right? It zips up to the final state and stops. That's great. But of course, I cheated. I picked the parameters to make this work well. If I happen to make D equal to V, then the system's a little less well-behaved. It undergoes a damped oscillation on the way to the final state. Now what does D equal V mean? It means that the length of the cell cycle is the same as the lifetime of the cell. Now that's not very realistic, right? Cells usually live longer than one cell cycle, don't they? I mean, there are cases where they don't, but very commonly they do, and so you would normally have D be much less than V, and then you'd have things like this, right? Where you have these long oscillations that go on for many, many cell cycles before the system settles down. That's not good, right? So this kind of system is a terrible way to build the tissue from scratch unless you're killing cells at a very, very fast rate. But it might not be a bad way to keep the tissue. You build the tissue by some other means and then you kind of keep the tissue at a homeostasis. With this, then maybe those oscillations wouldn't be such a problem. Unfortunately, you'd still have a problem with rejecting disturbances. So this tissue, the olfactory epithelium, like many epithelia, the death rate of the terminal cells is environmentally driven. This is a great example because basically whenever you get a cold, you kill a lot of those neurons and then they have to come back. So there's a very, very variable death rate. And we can simulate that by taking our equations and just introducing, add a sine wave on top of this and just see how the system responds. And what you can see is as long as that sine wave is a slow one, that you're slowly varying the death rate, the stronger and stronger the gain of the feedback, the better and better you dampen those oscillations. So the feedback's pretty good at rejecting disturbances if they're slowly varying disturbances. But if you have fast varying disturbances, notice this time scale is 10 times faster than that one. Now look what happens. The feedback actually makes the disturbances worse. It's actually worse than if you had no feedback. And so remember I said yesterday there's a thing called a Bode plot in which you plot the sensitivity to the disturbance as a function of the disturbance frequency. And this is the Bode plot for that system for different values of the feedback steepness or gain. And what you see is for slow disturbances you reject the disturbances but for fast disturbances you actually amplify them. And if you hit the frequency just right you can actually go into permanent oscillations. And another way to see that is to take a tissue that's happily at steady state just take half the cells away abruptly. So that's an extremely high frequency perturbation and look what these high gain systems do. They just go for a very, very long time. So why does that happen? Okay, so this is called non-minimum phase dynamics because it refers to a dynamical system that has a negative feedback loop which restores a system to its state after a perturbation but first before it restores it initially sends it in the wrong direction. And the reason I have this picture is this is a picture of someone trying to drive in reverse. If you've ever tried to drive backwards, right you notice how hard it is to drive backwards especially if you try to do it fast. Has anyone ever tried to drive in reverse very fast? It's crazy, it's really, really difficult. If you drive really slowly in reverse it's okay. But if you drive fast it's hard and the reason is let's say you want to move in this direction and here you are and here's the front wheels which is where the steering is and so you want to go this way and so you turn your wheels this way, right? So that your car will go that way. But what's the first thing that happens once your car starts moving and you've turned your wheels this way is that you're drawn that way, right? So if I followed you your trajectory would be like that. You would first move this way and then you'd move that way because of the way you turned your wheels. And so consequently your brain is going wait a minute I'm going in the wrong direction and you turn the wheels the other way and then of course that's wrong too and so now you turn the wheels the other way and you set up exactly the same type of oscillation, okay? So that's called non-minimum phase dynamics and what's going on in these systems is that let's say you have two many terminal cells, right? What's the first thing that happens is it reduces P and when you reduce P what's the first thing that happens you make more terminal cells. Now then later you have fewer stem cells and so you make fewer terminal cells and you eventually correct. But in the first cell cycle you get it completely wrong and that's why these things oscillate and just as with driving backwards the faster you try to do it the worse the problem is and you actually make the error worse, okay? So this is by way of saying this is a structural problem with the system. This is not a parameter adjustment problem. There's nothing you can do about it without changing the system. So you have to change the system, right? Okay, so what can we do? Well here's one thing you can do, branch the lineage. Allow that cell to produce more than one type of cell. And then instead of simply having a negative feedback on P you can allow feedback to promote the branching. Essentially you now have an extra degree of freedom here, right, you have another parameter here which is the fraction of the output that isn't renewal that goes this way instead of that way and you can promote that so that now you can if you have too many of these you can start making those instead without forcing yourself to make some of these first, okay? And in fact that works really well. You can take the very same system using a branch lineage now you can have extremely high gains and you just damp out all oscillations at all frequencies, okay? Because now the Bodhi plot never goes above one. Well you sort of do have the negative feedback because the problem is all these three probabilities here, here, here have to add up to one. And so if you promote any one of them the others have to fall. And so as long as P falls to any extent then you still have negative feedback on P. That's a good point, okay? It's also true that now you can start a system from scratch even with a low D and you don't have that much in the way of damped oscillations. You still have exactly the same bang bang kinetics because you still have that same basic kind of Ponzi scheme kinetics. You're still insensitive to the cell cycle speed, the initial, so you get to keep all the good stuff but you've added one degree of freedom and that enables you to get around these bad kinetics. Okay, now if you're thinking yes, I'm glad you asked that question because this is exactly what this slide is about, right? You know, suppose you want to control this guy. Doesn't this guy now have the same problem that you got rid of for this guy? And the answer is of course it does but of course there's a simple solution. Put another branch in the lineage and another branch in the lineage. You can keep doing this and have every single one of these terminal cells have perfect robustness, no non-minimum phase dynamics until you get down to the last one, right? And then what can you do with the left one? You can throw it away so you can just have those cells die or you can have it a cell that maintains its number by some other reason, for example, it can simply match its numbers to the numbers of these by sort of proliferating until it matches one of those. Now this may seem like kind of fanciful, right? But in fact, this is exactly what several tissues do and the best examples of this are in the nervous system particularly the retina is a great example. In the retina there's a single stem cell that makes seven different types of neurons but it doesn't make seven of them all at once, it makes them in overlapping waves, okay? And it only starts making the later ones when it's getting close to being finished making the earlier ones and guess what's controlling some of those transitions and at least two stages we know it's calones that are doing the job. GDF-11, a close homologue of GDF-8 and TGF-beta-2 are essentially produced by the terminal cells at one stage and pushing the progenitor down to the next stage. And so this kind of layered lineage is in fact something that you observe a lot. Okay, second set of trade offs, spatial and this relates to the question of the cells somehow have to be able to measure how many terminal cells they have in order to implement the feedback. And the problem is that whether you have mechanical signals or signals from diffusible molecules they're going to decay over space and that's gonna impair your ability to measure. So for example, remember we said for anything that diffuses with constant uptake you can define a characteristic length scale, decay length which is the ratio of the diffusivity to the uptake rate constant, right? And so if you have a bunch of cells that are gonna become a tissue and they're growing and they're growing suppose the red cells here are producing red substance and the red substance is gonna accumulate in a gradient around that. And as long as that tissue isn't too big, right? If that red substance say is coming from the C1 cells as long as the tissue isn't too big that red substance will continue to increase with tissue size but eventually at a certain point, right? It's not gonna get any redder in between these cells, right? Because when the tissue gets much bigger than the decay length, right? The concentration of anything the cells are secreting is gonna be stable. It's not gonna change anymore, right? You can see that in a lot of ways. One is that the decay length simply characterizes how far a molecule travels before it's captured. So of course if the tissue is really big molecules made in the center are never gonna make it to the edge. So there's no way that the edge cells can possibly know the size of the tissue anymore because they can't measure anything coming from the middle. Okay, so for anything that has a spatial length scale the bigger and bigger you get the harder and harder it is to measure how much tissue you have. And so what that means is if you looked at if P were dependent on some substance produced by C1 then the sensitivity of P to the size of the tissue was just going down and down and down the bigger the tissue gets until eventually, right? The P is insensitive to the size of the tissue. At that point P could only tell you the proportion of cells that are red. It would still reflect that but it wouldn't reflect the size of the tissue anymore. So in engineering that's called a declining gain feedback problem, right? You have feedback but the strength of that feedback is getting worse and worse the bigger and bigger the tissue gets. So if you want to maintain a really large steady state it becomes impossible to do so. And in fact you can show that categorically, right? Because when you're in a regime where you're sensitive to the number of cells, right? This will stabilize that. But if you're so big that you can't sense the number of cells you can only sense the proportion that's C1. If you plug this into that you can see that it's categorically unstable. It will never stabilize. So that's kind of bad news because the factors we've been talking about things like GDF11 and GDF8 and active in and TGF betas. Their ranges in vivo are probably on the order of 100 microns for their decay lengths, 200 microns. So it suggests it would be very hard to maintain a steady state tissue at bigger than a few hundred microns. Now interestingly, most of your steady state tissues are based on epithelia that are no bigger than a few hundred microns, right? The length of the tissue, like the length of your intestine is much bigger, right? But that's not really being maintained at a steady state. What's being maintained at a steady state is the thickness of the epithelium. The length of the intestine was determined pretty much during development. So actually it's okay for thin things but you know we're interested in big things like the brain and the arms and so on. And this doesn't look like it's gonna cut it, right? But of course, those aren't steady states. Those are final states. All your big things like your bones, like your brains, those are final state tissues. So you gotta use that equation, the one without the death term here. And it turns out if you put this in to here, it is stable as long as there's a high enough feedback strength, okay? So it is possible to reach a steady state even when you can no longer sense the size of the tissue as long as you have enough feedback strength. So that's good news. Yes, that's right. No, so the problem is it becomes singular as you send D to zero. And actually what the solution is when you send D to zero, it's the maximum amplitude of the oscillation you get. So we can talk about it later. But yeah, you can't just set D to zero or take the limit as D goes to zero to figure out how these things behave. So a final state system is truly different. Okay, so anyway, it is stable, but it's definitely not robust. If you have a system that's solely being regulated by the proportion of cells of one type or another, you can get it to reach a stable state, but if you change the initial conditions by a factor of two, you change the size of the tissue by a factor of two. And if you change the feedback gain even slightly, you dramatically change the size of the tissue. In fact, you can show that for a big final state, it's gonna vary with the 17th power of the feedback gain, which is, of course, ridiculous, right? Nothing in biology can cope with that kind of sensitivity. But of course, that's probably not how things really happen in development. Because in development, you always start small. You start with a very small number of cells and you let them expand to a large number of cells, which means your system has the opportunity to be sensitive to its own size early, and then only late is not sensitive to its own size. And so you have to consider the whole trajectory. Maybe what's happening is the system achieves robustness before it leaves this regime and then achieves stability when it's in that regime. And it turns out that actually works. If you model that, of course, you have to model this now as an actual spatial system. So you have to put some boundary conditions on this and you have to write this as differential equations where you deal with both the advection and the diffusion. But if you look at how that behaves over time, so this is kind of like a chimograph where we're looking over time at the size of a tissue and the color is telling you which fraction is stem versus terminal cells. And you can see here's a case where the initial conditions have been doubled, but there's almost no effect on the final size. And if you want to see that in a more two-dimensional format, you can see these movies from three different initial conditions over a factor of four. They all end at pretty much the same final size. And the interesting thing is that the final size is about 10 times bigger than the decay length of the information that's coming from these molecules. So you can make something 10 times bigger than you ought to be able to because you're taking advantage of this movement from one regime into another regime. So that's good, but if you explore the parameter space, you see that you, yeah? Sorry? Yes, because the stem cells are gone. Ah, will they, yeah, will they return? That depends on when you disturb them. Nothing will happen. Right, if you disturb at the end, they will not recover. And the same is true with catch-up growth observed in tissues. If a child has a disease too close to adolescence, right, they will not recover their size. If they have that disease when they're two or three years old, they will recover their size. So it's also what you observe in biology. Anyway, if you explore the space of parameters, you find that there's just so big you can get using this strategy. So this is just kind of exploring all the different parameters plotted against one particular lump parameter that gives you a general sense of the feedback strength. And you can see if you want the system to be robust, say, to its initial size, you really can't get much more than, say, 30 times the decay length. And actually, if you put noise in this, you really can't even get more than 10 times the decay length. And all the things that approach that all have a particular value of the feedback strength. And that's because if you drop it any lower than that, you go into a regime where the system becomes unstable. So unlike the ODE version of this, where we ignored space and it was categorically stable for all values of parameters, when you consider the spatial version, it becomes unstable if the feedback is too weak. Because essentially, for every increase in cell number, you can't get enough feedback to dampen that back down again. Okay, so basically, there's just so big you can get. And if you remember that the K-loans typically have decay lengths on the order of 100 microns, that means you can build a tissue of a couple of millimeters. But this is a meter in length, right? How do we gain three orders of magnitude, right? So this is a very interesting problem, right? This is a kind of problem physicists like. How do you gain three orders of magnitude? And it turns out biology, as is often the case, divides the problem up into parts, right? So one way is you don't actually build an arm, you build multiple segments. And each segment is independently sized controlled. And so the segments altogether are a good deal smaller. And in fact, the segments are made out of two parts called growth plates, one on either end. And so you independently control each of those. So the net result is you don't really have to control a meter, you have to control more like 100 millimeters. Exactly, and that was yesterday's lecture, that's pattern formation. Okay, so pattern formation and growth control are very closely tied together. One of the reasons to create segmental pattern is because then you can create independent subunits that can growth control. There's some beautiful work from James Briscoe, I don't know if you talked about it in the spinal cord that actually points to exactly that, that the segments that are generated, they're not segments, but that the subdomains that are generated each independently growth controls. Okay, and so that's probably what's going on here too, right? Each growth plate is growth controlled and you have about eight to 10 of them as you go all the way out to your fingers. So now the problem is a bit easier. And then you also generate cells called chondrocytes that as a result of the planar cell polarity pathway, they all elongate in one direction by about a factor of 10. So now you get the problem down to about 10 millimeters. Okay, so really it's not such a bad problem, but you're still not there yet, right? You still gotta get from one millimeter to 10 millimeters. You gotta get another factor of 10 somewhere. And so the question is, are there any other ways of overcoming the declining gain control problem? And so again, it's really helpful to turn to engineering because engineers face declining gain control problems, right? In fact, when you send a probe out to Pluto, right? You have a declining gain control problem because the farther the probe gets, the harder it is to communicate with it because you have these huge time delays, right? And so you really can't finally control anything the probe is doing as it's reaching Pluto, right? It just takes too long to do it. So what do you do? So you run a little model in the lab of what's going on and you take the control cues from the model rather than from the actual observations and communications with the satellite. And it turns out that not only do engineers do that, but the human body does that in the nervous system. And this is one of the best examples of overcoming declining gain control is when you have to catch a ball that's moving really, really fast, right? The way you control your muscles typically is you get feedback from receptors in your bones and muscles, your joints and muscles that tell you where they are in space. And you combine that with information from your vision to send signals to the muscles to put them in different places in space. This works just fine as long as you're doing slow activities. But when you're trying to catch a baseball moving really, really fast, you have a big problem which is the baseball gets close to you, right? There's too big a delay in the sensory information coming from the muscles because they have to travel down axons and those are relatively slow. So halfway between when the baseball is thrown and when it gets to you, you forget it. You can't use any actual feedback from the muscle. It's useless. So what the nervous system does instead is it generates in the spinal cord on the fly, it generates a little model, a little internal model of the entire scene, the baseball, you, the arm, and it runs that model forward in time and predicts where everything is going to be and it uses that to drive the feedback control. And so that's called the internal model principle. And the question is, can you use the same thing, right? So you have a case here where you have terminal cells that are produced by stem cells and as this thing gets bigger and bigger, this gets weaker and weaker, right? That's what we've been talking about. But what if these cells produce another cell type in very small numbers? So the proportion between here and here is large to small and these stay around, one of the ways you can get them to stay around is by having these guys inhibit that transition, right? And then these feedback, and since these don't actually move very far away, in theory they should be able to feedback for a much longer amount of time. And it turns out that works really well. You can, again, write the same equations. You can see that now you can get up to many times lambda and still be robust to initial conditions. And in fact, if you look into the growth plate of the bone, you can see hints that this is indeed what's going on. So it turns out we know that in a way growth plates of bones are controlled. There's a proliferating zone and a differentiated zone and the differentiated zone feeds back negatively on P. All of this is known. And then these cells are pushed farther and farther away, which means you're gonna have declining gain feedback very, very quickly. But there is another cell type in this system which is called a resting chondrocyte stem cell. Now they call it resting because it doesn't divide very often. And if you read the literature, they'll say, well it's there to act as kind of a support for the stem cells in case they run out. Now that's the silliest thing in the world because those cells are proliferating exponentially. There's no chance they're gonna run out for no good reason, right? So it turns out that developmentally, these cells are produced by those cells. They're set aside by those cells and they accumulate in numbers and then as time goes on they decrease in numbers or absorbed back into this pool and eventually you make only hypertrophic chondrocytes and evidence suggests that these guys actually feedback negatively on P as well. So this would fit that model that I just showed you. Now once you begin to think about internal models, actually there's many, many ways you can imagine internal models. For example, you can imagine one dimension acting as an internal model for other dimensions. So the retina is essentially a disc that grows. It gets wider and wider over time. We can sort of view that as something like this. We want this to grow to this size and stop, right? But the problem is this size is too big for size control. That's the radial dimension. But the retina also grows in the apical basal dimension, the other dimension. And that's a short distance. That's a couple hundred microns. And what we know is at the beginning in development differentiation starts in the very center of the retina where the apical basal dimension has reached a certain size. You start differentiating. So that could be the same negative feedback mechanism that we've been talking about. But because the two dimensions are coupled, once these guys start differentiating they're gonna make the guys around them differentiate. They're gonna make them differentiate. And you're gonna get this moving wave, this trigger wave of differentiation that overtakes the entire tissue and when it reaches the edge causes the tissue to stop. So again, this is not fanciful. This is what actually happens in the retina. There is a trigger wave of differentiation. It does proceed at a more or less linear rate once it starts going, which is what modeling predicts. And if you build a model of that kind of a trigger wave phenomenon, you can show that, see you start a little bit of differentiation here and it essentially quickly overtakes the entire tissue and you get the tissue to stop easily at a hundred times. The decay length, it obeys the kinetics that look a lot like the way the retina goes. You have a trigger wave that moves out more or less linearly. And as long as you start from a reasonable number of cells in your initial condition it's very insensitive to the initial conditions. So you have that same property of robustness. Okay, running a little bit low on time. Let me go very quickly through the last two items. Everything I've been talking about has been a deterministic view of cell populations which assumes we're dealing with large numbers. Of course, there's a whole stochastic side to this and I don't have time to get into that much of it but I will mention that either of these strategies has the problem that it rewards cheaters. Okay, so you imagine if you have a bunch of cells, stem cells that produce terminal cells that feed back on P if one cell steps out of line and becomes less sensitive to feedback it will now grow exponentially until there are enough of these to keep it under control. But the only way that can happen is if there are so many of these that all of the normal stem cells now decline exponentially and go away. Okay, so this is what sometimes developmental biologists call cell competition where one cell that's even slightly less sensitive to a negative signal than all the others quickly takes over all the cells and both of these strategies, any one of these integral feedback strategies is gonna be sensitive to that. Something that's a little less sensitive to feedback will quickly take over something that's totally insensitive to feedback will blow up, right? The tissue will expand to infinity. Unfortunately, we can do the experiment in the olfactory epithelium of taking out both of the negative feedback factors there and we see the surprising result that although the stem cell numbers go up the terminal cell numbers don't change. So, or they change very, very slightly, right? So how is it that the tissue manages to achieve control even though we've totally taken away the integral negative feedback scheme? But remember there's more than one integral negative feedback scheme going on the same time. Now, if you have two integral negative feedback schemes working on the same parameter, that's no good because neither one of them can really be integral negative feedback at that point. But if you have them working sort of orthogonally on very different aspects of growth, so for example, the, if this mechanism is primarily just controlling proliferation, divisions of cells, right? It has nothing to do with lineage progression, then these two schemes can operate orthogonal to each other. So basically, even though you knock this one out, you still have that one in place. And that has an interesting impact if you do stochastic simulations because you can actually see this kind of phenomenon where if you take a cell that make it completely insensitive to feedback from K-loans, it will start to overtake but then all the neighbors around it because they're being deprived of the K-loan feedback from it because it's not making any terminal cells, they'll start to overgrow and you'll actually get an arms race which will very frequently extinguish the original cell that's stepped out. And if you look at the probability of extinguishing the mutant clone, or in this case, the probability that we'll fix, that you won't extinguish it, is actually highest if the mutant clone doesn't completely lose responsiveness to feedback. In other words, it predicts that from the standpoint of developing a cancer, a cancer that immediately jumps to doing nothing but proliferating is actually very bad for the cancer. The much smarter thing to do is for it to be somewhat less sensitive to feedback but still keep the feedback control a little bit so that essentially sort of stealthfully overtakes its neighbors without being wiped out by them. Okay, so let me just end up by saying that not only do all these feedback mechanisms that I've mentioned give you a way to control size, but now that you know about them, you can begin to sort of reinterpret or interpret some of these issues about the relationship between growth and form in the light of the fact that these feedback mechanisms are all there and therefore they're all available to be used as ways of creating structures and forms. And one of the things that makes those mechanisms particularly able to generate form and structure is the fact that the negative feedback that I've been telling you about is often mixed with positive feedback. So the very same tissues that have calones almost always also produce factors that increase P in the same tissue by the same cells often. And initially we thought this was crazy. They would just cancel each other out but of course they don't cancel each other out unless their feedback strengths are identical. If they saturated exactly the same point. If they don't and if in fact the positive feedback is stronger than the negative feedback, then you very easily get what's called growth by stability. You get systems which have two stable growth states. One in which the stem cells are generating lots of terminal cells and one of which is stem cells are generating fewer terminal cells. And you can easily switch back and forth between these growth states simply by adjusting one of these parameters or even adding or subtracting external amounts of those feedbacks. This feels very right to developmental biologists because one of the things it predicts is that when you weaken processes and development you don't get smaller organs you just get organs that vanish. This is actually something you see called agenesis. Lots of mutations lead to organs that start growing and then pfft Peter out and never make it. But if they do get going then they usually get to the right size. So this is predicted by that kind of behavior. So you, ooh, wow, we crashed. Let's get that back. Sorry about that. Let's see if that works. Okay, so what it means is not only can you have inherent by stability but you can use transient levels of things to create form, right? So a little bit of the factor that does the positive feedback here will cause a structure to grow into a very sharply demarcated bud. You can take things that are growing and turn them off with transient signals. So this thing's growing on its own and then we have a little bit of this factor and we could turn it off. And you can even get things to self-organize, right? So these are, it's a very simple system in which we just have slightly noisy initial conditions. And here the noise is just in the shape. We didn't even have to put any noise in the parameters. And what happens is this will spontaneously grow into fingers and buds because of the mixture of positive and negative feedback that you have here. And we know that this is self-organization because you can look at the spatial frequencies and the noise that you start with and the spatial frequencies and the noise that you end, in the pattern you end with and they have nothing to do with each other. Okay, and interestingly, even though these look like repeated patterns, they're not Turing patterns. Turing patterns are based on things that diffuse with different ranges. But here, we can do this with things that diffuse with exactly the same ranges and they'll still produce this. So this is based on kind of a bistability of growth rather than the patterning substances themselves being spatially unstable. So what does this kind of patterning mean? Well, you know, you do see it in nature. This is what human skin looks like and this is an epithelium. And it looks a whole lot like that sort of thing. Now, that doesn't mean that this has anything to do with that, although actually there's very good reason to believe that skin operates based on a mixture of positive and negative feedback as well. But this is just kind of a hint to suggest that this kind of interplay between feedback that's used for size regulation and positive feedback can kind of give you a toolkit out of which to build various kinds of morphogenetic structures. Okay. Well, the feedback in both cases is proportional to the amount of one or the other cell types, okay? It's just that when you implement that feedback and the consequence of the lineage, when it's negative feedback, it ends up being an integral. Okay, that was when I showed the equations. So the positive feedback is mediated just the same way as the negative feedback, but I don't know if you can really talk about integral positive feedback. Perhaps it's derivative feedback. I'm not really sure, right? Okay, it's a good question. Okay, so in summary, right? I hope to have shown you that this is one of the most remarkable feats of engineering in all of biology, that effective strategies rely on lineage progression. So you may still hear lots about lineages from the standpoint of the terminal fates and how you control them and so on, but there's this whole other aspect of lineages that they're the substrate upon which you do feedback so that you can achieve control. We talked about integral feedback, bang-bang kinetics, growth by stability, branching as a way of improving controllability and the use of internal models and fail-save control, all showing up within the context of this very, very simple thing. And then finally at the end I mentioned that the coupling between cells that's created by control also gives you a toolkit to generate pattern in form. And then as Boris said, biologists always put their acknowledgements at the end, so these are some of the people in my lab who've worked on growth control and these are some of my collaborators, in particular some wonderful mathematicians and engineers and experimental biologists. So thanks, take your questions.