 Who is going to talk about understanding exponential trajectory, balance growth, and cell size? Is it audible in the back? Audible? Thanks very much to the organizers for inviting me to this stimulating meeting and giving me this opportunity to present some work. I'll be talking about some work that has been done in Delhi with PhD students, Parth Pratim Pandey, Pooja Sharma, Harshad Singh, and Nagpal Sethi, Shagun, who is in the meeting, who has a poster up. I'd like to talk about some theoretical questions related to bacterial physiology and cells in general. So first is the origin of exponential growth and how come a non-linear system exhibits something which is characteristic of a linear system, exponential growth. How do cells, as they grow, all the diverse chemicals which are all whose dynamics is decentralized, how do they all manage to double together? How do they all manage to grow, exhibit what's called balanced growth? And so this would be some general sort of, I'll talk about some structural properties of models which have these features and how these properties might arise. And then I will briefly turn to some specific things, a class of models that I'll be describing. I'll show how bacterial growth laws arise from an optimization principle. And if time permits, I will say something about stochastic dynamics, about a timekeeper molecule, the stochasticity of a timekeeper molecule whose threshold commits the cell to division, the adder property, and intracellular molecular fluctuation patterns, and so on. So this is the early result that suggests that single cells grow exponentially from birth to division in exponential cultures. So the length of the cell on a logarithmic scale is plotted on the y-axis versus time from division on the x-axis. And these straight lines means that cells are growing exponentially at a certain rate. These things have been measured in more precision in recent times. Of course, since the time that the cell takes to divide only allows it a two-fold growth, there could be other fits. Nevertheless, an exponential fit seems to be quite good. And at least it's a working hypothesis that certain cells, certain bacterial cells, grow exponentially in time from birth to division. Now, so what kind of a description of cellular dynamics can address questions like this? So a cell consists of lots of molecules of various types, metabolites, amino acids, nucleotides, messenger RNA molecules, proteins, ribosomes, of various kinds. And one description of the state of the cell at time t is just the set of populations of all the molecular species. So there are, let's say, capital N distinct molecular species of various kinds inside the cell. And we specify the number, how many molecules of each type. And again, this is a crude description of what's going on inside the cell, but it's a description, which is very useful. And of course, these variables xi of t, they are integers. They are populations. So they are non-negative integers. And they change stochastically. But it is often convenient to write down deterministic dynamics for the change of their populations. In the form dxi by dt, that is xi dot, is some function of the populations, of the populations at that time. And these functions fi contain all the complexity of the cell, all the reactions and interactions and processes that are going on inside the cell. And these are coming from reaction kinetics. They are highly non-linear functions of the variables in question. So exponential growth is a characteristic of linear systems. And here is the simplest possible system that one can talk about. A one-dimensional variable, a single variable system x, dx by dt is r, a positive constant, times x to the alpha. Where if alpha is one, so that's a linear system, x dot is equal to r of x, that's the central picture. And x grows exponentially, r is positive. And that's a hallmark of a linear system. But if you depart from linearity, let's say alpha becomes less than 1, then the growth pattern, let's say alpha is half, then the growth pattern is not exponential. It at large times is given by a power of time. And if alpha is greater than 1, then it grows faster than exponential. And there is a finite time singularity that you can have in such things. So departure from linearity means also departure from exponential trajectories. That's a standard understanding. And what we are trying to understand is why do cells grow exponentially, assuming that they do, assuming that that approximation is correct, why do they grow exponentially if they're described by nonlinear systems? So here is a higher-dimensional linear system. And again, because of linearity, you see exponential growth. So here, you have n variables, but the functions fi are linear functions, aij xj. So a is a matrix of real numbers, let's say. And that describes the dynamics. And now, in this case, so one can represent it x as a column vector and write down the dynamics in terms of a column vector and matrix. And the eigenvectors and eigenvalues of a play an important role in this dynamics. And in particular, asymptotically, as time becomes large, the dynamics x of t, the vector x of t, is attracted to the eigenvector of a corresponding to its largest eigenvalue. So x lambda are the eigenvectors of a. Lambda are the corresponding eigenvalues. And the largest of these, assuming it's real, is the one that determines the asymptotic behavior of the trajectories. And in that asymptotic regime, you get exponential behavior again. Transients are mixtures of exponentials of different kinds. But once the system has, once the transients are over, then there is exponential growth again. So in that sense, you see exponential growth in this linear system. And this is also an example of balanced growth because the largest eigenvector, it has some components which are fixed because a is a fixed matrix. So x is a fixed vector. x lambda 1 is a fixed vector. And therefore, it's a fixed vector up to normalization. And therefore, the ratios of its components are constant. And so once you have gotten onto this attractor, once you've gotten to this eigenvector, all the populations, all the variables x are increasing with the same rate. They are increasing in proportion to each other. They're all growing exponentially with the same rate. It's an example of balanced growth. So for a cell, just to take a nonlinear example, just to consider a concrete nonlinear example, we'll consider a particular coarse-grained model of a cell described by three variables. So x is a triple. And the populations are p of t, denoted p of t, capital T of t, and capital R of t. So p is supposed to be the pool of precursors of macromolecules, that is amino acids. Let's take p to be just the amino acid pool. T of t is the pool of all transporter and other metabolic enzymes, which bring in the food and convert it to amino acids. That's what t does. And R is the pool of ribosomes, it's the number of ribosomes, which catalyze the production of t as well as themselves from p, from the amino acids. So that's a coarse-grained version of the model of the set of variables. And here is a possible set of equations that one might use to describe certain aspects of the cell. So the box brackets, of course, denote concentrations of these pools. And without box brackets, it's populations. And I will sometimes use box brackets for concentrations and sometimes use lower case letters for concentrations and capital letters for populations. So the amino acids are produced by the transporter and metabolic enzymes. And Kp is a proportionality constant that represents many things. It represents the efficiencies of these enzymes, of this enzyme pool. It represents the nature of the medium, et cetera. So Kp is doing many duties there. That's the production term for amino acids. And amino acids are used up to make proteins and ribosomes. So in proportion to their concentrations, p-concentration and r-concentration with some rate constant K. And that results, of course, in the production of T and r. And the positive terms are, of course, of the same type, proportional to the r and p-concentration. And the rate constants are proportional, in some sense, fractions of. So Ft and Fr are the fraction of ribosomes that are making, respectively, the transporter and enzymes. That's Ft. And the ribosomes, that's Fr. These are, at the moment, parameters of the model. And mt and mr are the number of amino acids used up in each transporter molecule or enzyme molecule that's mt. And mr is the number of amino acids used up to make a ribosome. So that explains those kinds of equations. I just want to put this up here in part because I wanted to just say that this is a nonlinear model because the right-hand side has quadratic terms. And it's more complicated than linear. And in models of this kind, which are nonlinear, you do not expect. I mean, even for a single variable, you don't expect exponential growth. Models of this kind, which have many variables that are nonlinear, you don't expect exponential growth. Now, we are in an expanding cell. So therefore, we must add the dilution terms to all these three equations. And these dilution terms are minus v dot by v into the respective concentration. So those are the equations on the right-hand side. And that's typically the kind of starting point that we have in this core-screen modeling of cells. Now, it's important to note that this dynamic, supposing I know all the constants, kp, kft, mt, fr, et cetera, everything, supposing I know those, and I start with some initial concentration, this dynamics does not tell me how to do it because it's not completed yet because v dot by v has not been specified. So v dot by v is appearing in this dynamics, but we don't know what it is. And v dot by v is the growth rate, which I will be denoting by mu of the cell. And if the cell is expanding, then mu is a function of time. And unless one specifies that as a function of time, we have not specified the dynamics and we can't really integrate it and can't really find out the trajectories of things. So we need to specify this. But we don't want to do it exogenously by just assuming that v dot by v is some constant mu or lambda, or that will be then determined later. We want the dynamics to tell us what the growth rate will be of the cell. And therefore, this v dot by v has to be determined in some endogenous way. And what I will do in the next slide, where the model will be completely described, is that this v dot by v, this mu, it will turn out to be in the model a function of the concentrations themselves. So all these p, t, and r with box brackets, mu will turn out to be a function. And that function has to be substituted here. And then that specifies the dynamics completely. That function will be explicitly specified. And that then specifies the dynamics, which you can then use to integrate these equations. The system will be completely defined. So in order to do that, I first want to think in terms of populations instead of concentrations. So the populations xi are just the concentrations times the volume. And therefore, the d by dt of xi has two terms in which you differentiate the concentration and one in which you differentiate the volume with respect to time. And for the first term, you can use the equations that we had on the previous slide. And so when you do that, you find that the v dot term, so there is a v dot term which existed in that equation on the previous slide. That was the v dot by v terms. These v dot by v terms cancel the new v dot term that is there on the right-hand side of this equation. And you get even simpler looking equations for the populations. So those are the equations that you get for all the populations. But notice that even this is not well defined because it contains v. And we've not specified what v is. So that's something that we would like to do. And this is where I will make a key assumption that we will take v to be, in this case, a linear function of the populations themselves. So v is an extensive variable. And it can be expressed in terms of populations but not in terms of concentrations directly. So that's why we have gone to the population level. We express v in terms of the populations as a linear function. In general, I will be considering v to be a homogeneous degree one function of the populations. And this is going to be an ingredient of the model. And these constants vp, vt, vr are therefore also parameters of the model. The model is not specified without that. So and these constants, it will turn out, affect everything that the cell does. So let's take a particular case in which all the v's, little v's are equal. And so I can just take it out as a capital V is some little v, some constant, the parameter of the model, times p plus t plus r as a particular case. And now you can see that with this choice, both the models, both the versions of the model, the population as well as the concentration is completely defined, are completely defined. Because if you were to now calculate, so the population version is clearly defined because you've specified v. But as far as the concentration version is concerned, note that v dot by v, since capital V is little v into p plus t plus r, so you have v dot, capital V dot by v, is given by that expression, little v by capital V, into p dot plus t dot plus r dot. And we know what p dot, t dot, and r dot are from those equations, from those equations. So you can substitute that. And so the p dot term, there is a kp t term. And when divided by the v, which is sitting in this expression, it will become kp into box t. And then there are the quadratic terms, pr by v terms in each of those p dot, t dot, and r dot. And they will all combine. And notice that there was a volume factor that was sitting there before. That volume factor will be augmented by another volume factor that is coming in the definition of mu in the denominator. And so you will get concentration of p, concentration of r, multiplied by the constants. So you have expressed the growth rate in terms of concentrations. This particular choice, this linear, this choice of v being linear in the populations, is crucial to be able to express mu as a function of the concentrations. So if it was not, then it would involve populations themselves and not just concentrations. So now, once we have this mu, you can substitute this back in this equation, in these three equations. And now the concentration dynamics is defined and self-contained. So you have dynamics which has been defined at both levels, at the level of the populations, and at the level of the concentrations. And they are both consistent with each other. So that's what we've done. And now you can, of course, start this dynamics off with any concentration, initial concentration, of the three chemicals. And you can ask what the concentrations will do. So notice, I mean, so this is a cell that is growing, maybe whose volume is changing. But the concentrations themselves have a dynamics of their own, given by this. But notice that the dynamics has been crucially modified by what we've done on the next slide. It's not as straightforward as it appears here. These v dot by v terms are themselves concentration dependent. And in that particular case, v dot by v had a linear term and a quadratic term, which makes this dynamics even more nonlinear. So that's the structure that we have of this dynamics. Now, imagine that you have some concentration dynamics, some nonlinear concentration dynamics, and we can analyze it by the usual tools of dynamical systems. And we will be looking for fixed points, we'll be looking for stable fixed points, we might be looking for unstable fixed points, we might be looking for limit cycles, various kinds of attractors of this concentration dynamics. And let us say that you have, you found some fixed point of this dynamics, which is a stable fixed point. Let's call that stable fixed point p star, t star, r star. That's our stable fixed point of this dynamics. The moment you have a stable fixed point of this dynamics, you have the following situation. So what would happen to the system after it has reached that fixed point? So once the system has reached that fixed point, p star, r star, p star, t star, r star, then after that, of course, being a fixed point, being a stable fixed point, these would not change with time. And therefore, mu has become a constant in time. So once the system reaches that fixed point, mu becomes independent of time, which means that v dot by v, which is mu is independent of time, and therefore, v increases exponentially with time. So v will have the behavior, something like v naught e to the mu star. So mu star is the value of mu at the fixed point. So we will have some kind of a behavior like that, where the zero of time, let's say, is the time where once it has reached the fixed point. So that's from there you start, and we will then, from there on, increase exponentially. And it is clear that all the other variables must also increase exponentially. They must increase exponentially, because v is, after all, a linear function of p, t, and r. And if v is increasing exponentially, and p by v, t by v, r by v, they are all constants, then it must be that p, t, and r are also increasing exponentially at the same rate. So we have found a situation using this approach, where the very existence of a stable fixed point of the concentration dynamics that we have constructed means that we have constructed an exponentially growing trajectory of the cell in time. And although we have a nonlinear system here, we have been able to construct exponentially growing trajectories. And the crucial ingredient here, of course, was this key assumption here. It is with that that we are able to have nonlinear dynamical systems, which still obey exponential trajectories. And these trajectories, if the fixed point is an attractive fixed point, then these trajectories would be attractive trajectories for the population dynamics as well. So one can actually do this for this PTR model, and if you just simulate it at some choice of parameters with different initial conditions. So the top two plots are different, starting p, t, and r with different initial conditions. There is some transient, and then you can see that on a semi log plot, they are growing linearly with the same slope. So they're all growing exponentially at the same rate, yes. Yes, so I'll show you a simulation later in which you can choose the volume to be, let's say, two thirds the power of the surface area of the cell. And you can say that the surface area is proportional to let's say t, okay? And you can say, let's say, supposing you were to say that v being instead, yeah, maybe I should, yeah, I'll show you those similarly. And if you do that, if you depart from linearity or in general, homogeneous degree one character of the function v, then you find that you don't have you know, things reaching constant ratios asymptotically. They don't get attracted to being constant ratios. You can see in the lower two curves here, the ratios t by r and p by r doesn't matter what initial condition you start from. This is a model in which the concentration dynamics has one stable fixed point. This is a simple example in which it has one stable fixed point and which is a global attractor of the entire phase space. And so doesn't matter what initial condition you start with. Once the parameters are fixed, you tend to that those particular ratios, the concentrations are fixed, okay? So that's what you end up with, which you do not if the volume does not satisfy this property. I'd like to demystify this as to why we are getting exponential trajectories in a non-linear system, okay? And the reason is that this particular choice of v has rendered our dynamics closely related to linear systems. So these are non-linear, but homogeneous degree one function. So our dynamics, let's write down our population dynamics, which are the ones which are growing exponentially. Of course, concentrations are going to a fixed point. So it's the populations that are growing exponentially. So let's write down xi dot in terms of these functions fi, okay? And look at these functions f one, f two, f three for p dot, t dot, r dot respectively that we have, okay? These functions have the following property that if you scale all the variables, all the three variables by the same parameter beta, okay? Then these functions also scale by the same single power of beta, okay? That's the feature that they have. So that's why they're referring to them as homogeneous degree one functions. And we'll see in a later slide that this is a consequence, of course, as you, I mean, you can see that it is a consequence of v being linear because v is linear, it scales linearly with beta. Is that the same as saying that reaction rates actually depend on concentrations? Is that the same statement? It is. It is the same statement. I will just elaborate on that, yes, indeed. That's what it is. Cool. Okay. So, okay, so this is what we have this character. And whenever you have this character of a set of nonlinear differential equations, which is defined by homogeneous degree one functions, then you always have, you can always find exponentially growing solutions. You can substitute the non-zarts. Xi of t is equal to Xi of zero e to the mu t, where mu is the same for all the i's, okay? Same mu for all the i's, all growing exponentially at the same rate. And you can take this anzarts and plug it into that equation, the differential equation. The left-hand side is mu times Xi of zero e to the mu t, trivially by differentiating this anzarts. And the right-hand side, you have to substitute the anzarts. And when you substitute the anzarts, it has n arguments, but each one of them has this factor e to the mu t. And this e to the mu t, is like our beta, right? So, since it is multiplying all the arguments, and since our function has this homogeneous degree one property, we can use it to scale it out of the function. That's the last step of the second-last line, okay? So it scales out. And therefore, you can see that e to the mu t cancels out from both left and right-hand sides. And you are left with a condition on the assumed mu in the anzarts, on the growth rate, and the initial conditions from which you've started. So, this means that if you start with certain initial conditions, okay? There is a set of initial conditions which you can discover by solving these equations. These will typically be algebraic equations. The last equation, mu Xi of zero, is equal to fi x1 of zero, x2 of zero, xn of zero. It will be typically for mass action, kinetics, and so on. A set of algebraic equations, no time. And you can, you know... So, how many unknowns are there? There are actually... It might seem that there are n plus one unknowns, mu and x1 to xn. But actually, there are only n unknowns here that you can determine from these because you can only determine the ratios, x1 of zero to all the others because of this property of f. So, I'm calling this the generalized eigenvalue equation because for the special case where these functions are linear, okay? So, let's say fi was what we had earlier. Fi of x is aij xj. Then what does that equation look like? That equation looks like mu Xi of zero is equal to aij xj zero, summed over j. That's what that equation is because I simply have to substitute it in the function. The last equation on the left-hand side is mu Xi of zero, and the right-hand side is fi at the initial point, okay? So if we do that, that equation is nothing but the eigenvalue equation of the A matrix, okay? And for a linear system, that naturally comes out, of course, and we know that every eigenvector of A, assuming it's a real eigenvector, defines a direction, and if you start from that direction, then you will keep going exponentially at a rate lambda i in that direction for a linear system. We know that, okay? So something like that is what is happening for this nonlinear system. This equation, the right-hand side is nonlinear. It's not linear in the x's, in the variables x1, 0, x2, 0, xn, 0. It's not linear. So it's some kind of a nonlinear generalization of the eigenvalue equation of a matrix, and I don't know whether such structures, I would be curious to know in case someone knows whether such structures have been studied by mathematicians in some context, or somebody else in some context. But they seem to very naturally arise in the context of bacterial physiology in this way of thinking, okay? So anyway, right. So here I'm just pointing out that if you define the ratios of the xi's, psi i is given by xi by xn. You've divided everything by the last xn, okay? Then that same equation is, you know, you can write it down in terms of... So now there are explicitly n-1 ratios and 1 mu, and you can see that you have n equations for these n variables. And generically, you could find a solution. Of course, you have to find a feasible solution in which these x's are non-negative, if they're populations. And if you want a growing solution, you want mu to be real and positive and so on, things like that. But in general, you have n equations and n unknowns, and there would be generically solutions for that. And those solutions are exactly, it turns out, the fixed points of the ratios, I mean, as far as the ratios are concerned, at those solutions, you can find out xi by v, and that's the concentration. So those are precisely the fixed point, the concentration. So this set of x's which solves this equation, it defines a line which passes through the origin of this n-dimensional space. So we have this phase space which is n-dimensional, and you have a solution of this that defines a line, and what this means is that if your initial conditions lie on that line, you will have exponential trajectories, all the x's will grow exponentially with the same growth rate mu. So it's a very special curve in your n-dimensional space, which is this line of balanced growth, if you like. But the point is that is an attractor of the dynamics. So you can start from anywhere, if your fixed point is a stable fixed point, and that is an attractor of the dynamics, and you can start from anywhere and you will converge to this line. That's what's happening. So here I will address precisely the point as to how this class one property is arising. And what I want to say is that it is simply a consequence of two things, that the rate of change of concentration is a function of concentrations, point number one that you mentioned, and the second point is that v is a homogeneous degree one function of the x's. So we started with some particular functions g1, g2, g3 in the PTR model, but now let's be general, some nonlinear function of the concentrations. So little xi denote concentrations now, and capital xi will denote the populations. So little xi dot, this is mass action kinetics, you can include Michaelis-Menten, whatever one is interested in, as long as it's a function of the concentration. In an expanding volume, you will add the minus v dot by v xi term, which we did, and then in the capital, in the population language, when you write down the equations, the equations will, the v dot by v term will cancel out, and you will have equations xi dot is equal to v times g i of little x. Capital xi is given by that. And now, let's assume that the volume v is some function of the populations. So then, we can write down capital xi as a function of capital xi dot, as a function of the capital x's, just to complete that dynamical system. We can do that by plugging in v as a function of x into the above equation, and that's what I'm calling this function capital, little fi of x. That's the other ones that we wrote down earlier for the PTR model. So these are the functions fi, and we need to ask whether these functions have this homogeneous degree one property or not. And the claim is that they will always, doesn't matter what the form of g is, it's not that we've taken some very special dynamics and we're getting this homogeneous degree one property, it's generically going to arise always. It will, provided v itself is a homogeneous degree one function of its arguments. So that's the last line of the proof. So basically, this property is, you can see where this property is coming from and it's very generic. So the key assumption here again, I want to emphasize is this linearity or homogeneous degree one character of v, which is giving us exponential trajectories which is giving us this character. So just to note a couple of points that the solution to that generalized Eigenvalue equation, any feasible solution of that one gives us a quote unquote line of balanced truth, already mentioned. A special set of initial conditions. And I've already mentioned that to every stable point, there corresponds a line of balanced truth to which the population are attracted. And notice here that we've not assumed any regulatory mechanism. So things are growing, they're growing in a coordinated way, they're all growing with the same ratios. It is being assumed here that all the components of this vector, x10, x20, x30, they're all non-zero so that all the populations are growing and that actually requires a further assumption of a kind of a auto-catalytic nature of the whole system such that every chemical species is being created somewhere in that system. Then you will have all the components non-zero and all of them will then be growing exponentially at that same rate. And so one point is that no regulatory mechanism is needed, so it's a very generic mechanism. And there's no fine tuning involved here. It's as long as v has this property, it'll work. And notice that the constants in v, they do affect things. So just to remind you that in this equation for mu, we had this little v sitting out here. But if we had vp, vt and vr, those constants would also have been sitting here and they would have affected the growth rate, they would have affected the eigenvector components and so on. They are observables in that sense, they are affecting our observables. So that's an important thing. This bringing in this linear answers for v is not harmless in that sense, it is actually introducing some new parameters that matter to the dynamics. So you can write down a much more complicated model. We wrote down one which had about 30 odd cellular variables with sort of 40 odd interactions and so on. And we wrote down a set of equations and just put it on the computer. And because it was class 1, because it was homogenous degree 1, we found that, again, for this complicated model, it's a very generic thing that's going to happen. That you had these exponential trajectories and balanced growth. On the other hand, in this example, this is the one that I mentioned that you change the homogenous degree 1 character of v. v is now t to the 2 thirds instead of a linear function. And what you find is that there is no, it's not going to do any constant ratios asymptotically. So this is the generalities about these homogenous degree 1 systems and how they might help us to understand the phenomenon of balanced growth and exponential trajectories in a generic fashion. I will now briefly, since time is short, I will mention that this PTR model that I introduced in this talk as an illustration of this homogenous degree 1 is actually a model that can teach us a few things. I mean, teach us in the sense that at least it allows us a very explicit working out of things. So it's an exactly solvable model. You can actually work out explicit analytical expressions for the growth rate and all the ratios. So in spite of the non-linearity, you can just work those out. And because you can work out the ratios, you can also talk about the ribosomal fraction, which is phi r is m r r by m t t plus m r r. And if you fix all the parameters of the model, which include the ones that were listed earlier, including the fr and the ft, which add up to 1, and the v's and all that stuff, if you do that, then you don't see any growth loss. And you just get something. So you, for example, get phi r is equal to fr plus some other terms that depend upon. So in these formulae, I've introduced death rates of the t and the r species, d t and d r. So one can solve it analytically with those. And you get something, which if you keep all the other parameters fixed, I mean, you can keep changing the other parameters, like, for example, k or k p. k p was the medium, the medium and the efficiency of the enzymes. So you can change them, but nothing happens to, in this case, where d t and d r is 0, phi r is just fixed. So there's no growth loss seen. And you see growth loss, however. Here, if you employ an optimization principle, if you say that given that this parameter f, fr, ft plus fr is 1, but if this parameter fr is the one that the cell is actually regulating in some fashion or the other, and you say that you assume that the cell is always tuning this parameter in such a way that when all the other parameters are fixed, this parameter is chosen so that you get the maximal growth rate. If you make that assumption, then you find that you can analytically derive the growth rate. So you can get these kinds of things from this. I just wanted to mention this. And of course, you can make the model more complicated by adding the q-sector. That's not much of a complication. You can do various other things with it. And okay, so that's some of those things Shagun has on her poster that she is trying to get around. And yeah, notice that this model, you know, if you look at the model originally, yeah, so here the equations are the same as the PTR model, although this is, the volume is defined differently in this case, but just to have the equation on the board, the the peptide elongation rate is actually not K, but it's K times P by V. So it's also a dynamical variable in the model. It's not a fixed constant in the model. And one of the results that Shagun has observed is that as you increase the growth rate, this has qualitatively an increase in saturating character. But the amount of increase is not by a factor of 2. It's much less. So anyway, this is a very simple model and so it's not expected to reproduce all the nice, all the features that we would like. So at any rate, I will then briefly mention some other things that we've done in terms of extending the model that we added a timekeeper molecule which we called Z. So there is this X sector which is the PTR sector and it can contain other things, whatever. Which is homogeneous Degree 1. But then there is this other molecule which is a molecule whose I'll wrap up in two or three minutes. Yeah, so this timekeeper molecule is yeah, so anyway, so there are these, so it has you know, it's also produced, some protein let's say, and it is produced you know, in the same way as let's say T is same kind of dynamics. Homogen is Degree 1 and this molecule however has the role in the cell that when it reaches a critical value Zc the cell divides or commits itself to division. And when it divides then all the XIs are halved and Z might also be halved but it might also be reset to some other value which we are calling Zb. So you can do this, this is motivated by DNAA type dynamics. And anyway, so you can again analytically solve this with the PTR model when the X sector is the PTR model and and now of course, we are talking of stochastic dynamics because Z the dynamics of Z, we are interested in cell-to-cell variability here and so you can do stochastic dynamics and Z in order to reach its threshold will take a variable amount of time and so you will have an inter-division time variability and birth volume and division volume variability. So you can work out those things and we compared this with the data of I have not mentioned that Tahiri Aragi et al Sakchun June's group and what we found was that it matches the data best when this value Zc is between is of the order of 20 to 60 that's the range in which it matches the data best. It reproduces, I mean it matches to a fair extent the other distributions but it doesn't reproduce the growth rate distribution very well. So there's something else happening in the growth rate distribution which it reproduces the adder property and also this mechanism you can use to address this question which was raised by the work of Tahiri et al where they measured protein number fluctuations and they found this crossover from intrinsic noise to something that they called extrinsic noise. And this behaviour is actually reproduced by this model putting another protein in the dynamics and study its number fluctuations and of course at small numbers it has the universal fluctuations but it also has this crossover because at some point its variability is set by this timekeeper Z molecule so that is what determines its what is considered extrinsic noise is actually something which is part of the cell itself which is this generated noise generated by the cell. So I will conclude by summarizing that this mechanism for exponential trajectories and balanced growth is probably also relevant for proto-cells because proto-cells also near the origin of life needed to have identity and so they needed to have maintain their composition across generations so because it is very generic this kind of dynamics maybe it was used by proto-cells okay and I would like to emphasise that it is perhaps relevant to work out empirical consequences of the new parameters, the VIs that have been introduced and simple models of this type are possibly useful for bacterial physiology and they are at least very explicit and clean at least in the sense that the assumptions are all on the table and they are explicit okay thank you very much