 Yeah, you have to turn it on. Great, so next speaker is Lead Nur and he's gonna talk about optimal enzyme profiles, principle and simple solutions. Okay, is it working? Yeah? Maybe I'll put it higher. Clicker, okay. There's no one? No, okay. Somebody talk to me. That one is for the... Who? That sounds really great. Okay, never mind. I'll manage. Okay, yeah. Thanks for staying so long and also thanks for the organizers for giving the questionable privilege of being last on the week. I hope I won't exhaust you completely after all the pizzas I had. I'm also a little bit digesting, but I hope it's not... I tried to make it light, so please also stop me in the middle. I mean, you already know that you can do that, but stop me in the middle if there are any questions. So, yeah, so the idea of this talk is to talk about optimality again, the cursed word, about the optimality of pathways, but I think I'm not... Just want to say in advance we're not taking optimalities too seriously in this talk, okay? We're just trying to see what happens when we assume optimality and I'll start with a quote of Terry, like everybody else does. Yeah, we should always tweak files before assuming it. The problem is I made this presentation before the quote was made, so I couldn't really change it in the last minute, but I still want to try to justify why we do this. So, first of all, when you do it and it works, it's very, very satisfying. So that's why we keep doing it, even though it almost never works. But, yeah, that's one reason. Another reason is that usually, optimality doesn't work across all experiments and all conditions and all species, whatever, but it might help us understand why if something's up with them and what's the constraint of the system or maybe it doesn't care, but that's another option. But it's still interesting to know that, basically. And this is a bit closer to my heart specifically, is that it could have synthetic biology applications. Okay, so it's not, we're not just thinking about understanding current biology, but also how can we engineer biology maybe in the future? So that's another reason to think about optimality. And so, yeah, so, I wanted to start with genome scale models. Actually, today's tutorial is about FBA, so maybe it's a bit premature. But in general, these kind of very large metabolic models, they assume everything is linear or they need it for the calculations relatively fast, which means that you have to assume something about the reaction rates and what's binding them. And the most typical assumption is that nothing is binding them, which is not very realistic. Sometimes they're just bounded by some upper bound. So this is usually used in FBA for flux bound sciences for uptake rates. They're just bounded, and that creates a little bit more reasonable result. But the more advanced models just give an upper bound that's depending on enzyme levels. So if you measure enzyme levels, you know more or less what's the upper bound. But they assume the maximal efficiency can be achieved. And the last generation, I guess, of these models, they try to make some effective upper bound. So it's not really the Kcat of the enzyme, but something more like a K apparent, we call it. But it still doesn't change really between conditions. And that's a little bit of a problem, because as you know probably, some of you, that the flux of a reaction is determined by the enzyme level, but also some kind of nonlinear function that depends on parameters and some kind of metabolite concentration. So a brief history of what these functions look like. So I like this because it kind of makes an order of what a lot of people just use the word because, went in for everything. And actually I think you almost should never use their names, maybe. So Henri is the first one who actually came up with a kind of phenomenological reaction, which already looks like this rate law that we know. Mika Esmenten just a bit simplified, but also justified it based on some kind of time separation model, basically, that the equilibrium of the binding is very fast. But actually the people who I usually like to cite are Briggs and Halden, and Halden later, basically came up with this reversible rate law, which we call reversible Mika Esmenten, but actually they never did this work. Okay. So we are going to use Halden's rate law. I just write it a little bit differently here. So I hope it's clear enough. The parameters of the rate law are the enzyme, which we use epsilon, a little bit non-standard notation. Sorry about that. K-cats are the forward and backward rates. The maximal rates, forward and backward. And then you have S, which is the substrate level, and KS, which is the Mika Esmenten constant of the substrate, and the same for P. Okay. So this is basically a rate law that depends both on substituted product, and it can go both ways also, depending on the equilibrium state. And Halden also found that there is a connection between all these parameters and the equilibrium constant of the reaction. So enzymes cannot really change that. So that's another constraint on the system. The four kinetic parameters are connected in a way that enzyme cannot change. And then about 10 years ago, we tried to reformulate this law. So the formula at the bottom is exactly the same as the one at the top. It's just using different parameters. It's the same exactly. We just did it to basically separate it to three terms, which we call efficiency terms. So the maximal efficiency is the Vmax, like always. The middle term is like something that tells us something about thermodynamics. Basically, there is stability of the reaction. And it only depends on the driving force, or the delta G. We'll go into more, if you come to my tutorial on Monday, I'll go into more details about this. But it's also written in this reference. Sorry, it's not here, the reference. Oh, no, yeah, it is. And the last term is just the saturation term, which you already should know. Maybe there's a slight difference here that you also have the product in the denominator. But that could be ignored if the binding is very inefficient. Okay. So that was the introduction. Now the goal, I was talking about automality, so the automality problem we're trying to solve, and Bob yesterday already mentioned it, is how to minimize the allocation of enzymes in a pathway. And a pathway can also be a wholesale. We'll talk about it in the end. So it's very easy to do it when everything is linear, but when things are very nonlinear, it starts to be a complicated problem. So now we have all the parameters of the system are the enzymes, the metabolites, and whatever, the steady state if it exists. So this rate law case, basically, to solve this, we said let's assume we have a pathway. Like it's a simple example where we know all the fluxes. Let's say you can say they're all the same. And we just basically say the demand for every enzyme in this pathway is the inverse of all the efficiency terms. So the inverse of the Kcat, the inverse of the efficiency of the thermodynamic term and the inverse of the saturation term. And this is how much enzyme we need per flux for every enzyme. So summing all these up give us the total demand or the total cost in some way. And I think also Bob mentioned this, we proved that this is convex. So it's solvable numerically, at least. And there's only one optimal solution. I'm not sure cells care about this, but us as scientists, it's very useful because we can find it at least pretty easily and then check if it actually works or not. And we did it and we found some correlations. It's not perfect. There are a lot of things that are not optimal, of course, according to what we know now. Okay. So just to show a little bit, example of how this works. So if we assume it, actually throughout the whole talk, I'm only talking about linear pathways with no branching and only series of steps. This is a very simple assumption, I know, but this is what we need for really doing some kind of deep analytics. So let's assume this kind of series of three reactions from X to Y. And there are three enzymes here. Okay, so three demands basically. And what I'm plotting in these three plots are the sum of the demands. So all the three demands as a function of, sorry, this is only the first demand, the second and the third. The second and the third. And the last one is the sum of all three. And the X and Y axes are just the metabolite levels. Okay, so we have two or three variables, A and B metabolite levels in steady state and three dependent variables, which are the enzyme levels. Okay, and then if you plot them, you see that each one of them is kind of a convex function and the sum is also a convex function. The gray areas are the infeasible regions. So you cannot even run the reaction in the right direction when you have let's say too much of B or too much of A depending on the reaction. Okay, so that's why we get like this kind of triangle where things are feasible and within this triangle, the function is convex. I think you saw something similar in Bob's talk also. Okay, any questions so far? It's not extremely necessary for the second part, but it's an introduction. Okay, so we did this already a few years ago, but like lately we said we thought to ourselves, can we actually do this solution analytically? Can we, instead of the saying it's convex, saying what actually is the optimum point? Okay, as a function of all the parameters. And in general, okay, I'll tell you in the second part. It's generally not, but there are cases where it is. And why do we want it to be analytical? First, it's nice to have an analytical solution because numerical solutions sometimes tend to give you, not precise, or have numerical problems, basically, where you're not really sure that it's the optimum. If you have a formula, you can sort of see the patterns, you know, just from the math. You sometimes see some kind of connections that you wouldn't see, or you could see with a numerical solution, but you have to look for them, right? You have to plot them in order to see them. And from the math, you can maybe just find it just by looking at the equation. What's more interesting for me is that we might find some design principles. So why things are geared in some way because there's a constraint, basically, based on the algebra. And the last part is like maybe there's a mechanism behind all this, which is probably the main point here. So the problem is, as I said, is that there is no analytical solution for the general problem. So what I showed you before, that's convex, with a held-in rate law and everything. It's not solvable as far as we know. But we do... I mean, I'll show you in a minute that we can do it for simple cases where everything is linear and there are no branching points. And one other constraint I didn't say so far is that we cannot use the full rate law. We have to simplify it somehow. And you'll see that in a minute. So the model we're looking at, as I said, is a linear pathway. The number of reactions here is three, but it could be any number. We put a constraint on the total amount of enzymes. So we're trying to allocate the enzyme in the most efficient way to give us a certain flux. And in one of the cases, we also have to have another constraint on the total amount of metabolites in the pathway. But that's a side note. I'll get to it later. Of course, we can also do the optimization if we say that the cost of the enzyme is not proportional to their weight, to their concentration, but not necessarily in the same weights. So we can do it with weighted apertures. I won't talk about this today. Okay, so the problem, as I said, is that even the linear pathway, you cannot solve this rate law. It's just too complicated, as far as I know. So we did three different approximations and solved it for each one of them. The first one I'll start from the right, actually, has already been done many decades ago. And basically, it's assuming that the saturation basically is very low. Like, there's no saturation, there's no linear regime of the rate law. It doesn't look like that in this equation, but I'll explain in a minute how it works. The second approximation is close to equilibrium approximation, basically, or something that you can also say happens when you're completely saturated. So only delta G can actually change the rate, because the substrate saturation is complete. And the third example, or the more interesting one today is the Mikas-Menten approximation, which is going back in time to the one where there's no thermodynamic stir. Basically, the reaction is completely irreversible. So we solved all three of them. So I'll start with a mass-action rate law. So first of all, why is it called mass-action? That's because a chemical reaction that works typically, like, the only thing that matters for the rate is the binding of, or unbinding of the substrates. And that basically means that everything is linear and then ends up never saturates. So if you have more substrate, the rate will continue to increase indefinitely. So the rate, going back to the original formulation of Haldane, so this means that you're below saturation for both substrate and product, which means that this denominator is one, and then you get only the numerator, which is this kind of mass-action description where you have the forward rate, which is something times s, and then the backward rate, which is something times p. So this, it's derived from Haldane, but we call it mass-action because it looks like that. And if we look at the chain and say the flux in this pathway is, like, it's a steady state, right? So all the fluxes are the same. I don't need to actually reach my hand. All right, so the enzyme level times this right will be equal to j, and then we invert it and we solve it. And basically this has been done already in 64 by Whaley, and then you can see that you get the optimal flux when you distribute basically enzyme in a certain way, and the optimal flux will be a function of the first substrate minus the last substrate times divided by k-equilibrium. So this is kind of the equilibrium, this equilibrium of the whole system, in a way, divided by a very annoying term, but it depends only on kinetic parameters. So there's nothing else there. It's a bit hard to even understand, but it's like a product of a lot of the kinetic parameters. So this is a very, very nice result. It's been used actually today, even it was mentioned in Matthias' talk. So the Costa paper that tried to use it to justify why you don't want to have long pathways and you want to split, and they actually use this result in some way. So it's been known for a while. Okay, so now to the second one. Sorry, so again, I'm not sure if it's clear. I'm not showing you the derivation, okay? It's a bit long to show here, but you can read the paper. The second option is to say, okay, we're still kind of constrained by the... So we're still reversible, we still have forward and backward flux, but we don't care about the kinetic term, right? So we're saying it's one. And the dynamic is the only thing that matters. Okay, so all the reactions are, let's say when you're very close to a equilibrium, this is a pretty realistic. And this is our solution, but it's a bit, again, a bit too long to show here, but we couldn't actually solve it completely. So we have a closed-form formula, but it's not solvable. It's not invertible, basically, the function, but we found a very good approximation that works for any parameter as we tried, and it's almost with no error. So I'm showing only the approximation. And what's nice about it is if you compare it to the individual rate law, it looks kind of similar, right? So the total enzyme times some expression that looks a little bit like the rate law itself, right? One minus e to the date, delta g, but this is the total amount of Gibbs free energy in the system. So from the beginning to the end, ignoring everything in the middle. So we only care about the total disequilibrium. Okay, so this is, again, the second result. The last rate law we're considering is the Micaela cement one, which is unique in the sense that it's not reversible. So, basically, all fluxes go only in one direction. Okay, there's no reverse flux. And therefore, you don't see the product in the rate law. Okay, only the substrate. To solve it, actually, first we had to realize that you cannot solve it with the assumptions we made so far, because, like, basically to maximize the flux, you just increase the substrate indefinitely. You won't lose anything, right? So you just more and more saturated, you always increase the flux. So if you try to solve it like this, you get infinite solutions for the substrates, for the metabolites. So you have to add some constraint on the total amount of metabolites. So we added, like, the sum of them. All s i's is s dot. And with this, actually, you can solve it using just, like, grand multipliers, basically, inverting this function, deriving it. It works. And you find that the total flux, the optimal total flux is, again, proportional to the total amount of enzyme times some kind of term which is, depending on K-cats, the Km over K-cats, and this s dot. If s dot is very, very large, you lose this term, and then you just get the sum of one over K-cats, which makes sense, basically, because you can always be saturated in all the steps, right? So then you don't care about the Km. Good. So these are the three solutions I'll discuss today. And, yeah, okay, so, actually, we wrote this first as a kind of a chapter for the question. Perfect. Yeah, perfect timing. Yeah. This is nice. And, of course, now I see that it probably depends on s dot, but in this optimal solution with s dot, how the substrate considerations internal substrate considerations are relative to K-i's. They can be larger, smaller. Well, I guess they can be anything. More interesting when we... You're asking, like, how do they change with, what, with K-i? Yeah, and how do they... Okay, I'll get back to it, but basically, the system is not stable because it's irreversible. So if you... If you want to, like, a J, which is bigger than one of the K-i's, then you won't find the solution. Right? Sorry. Bigger than the total enzyme times one of the K-i's. Like, you won't be able to find a fast enough enzyme. And also, it's not stable in the sense that any... Like, it's... You cannot, like, there's... If you don't find the steady state, then it won't reach a new steady state. Right? If you just allocate the enzyme wrongly, you won't reach a steady state. It will just collapse. But the S-i's, basically, are very easy to solve. If you know the enzyme allocations, right, there's only one solution for the S-i's. So they're kind of... And as you'll see... Okay, as you see, everything is kind of flat. Anyway, you'll see it. Okay, so we were already pretty happy just with the mathematical results without any implications in biology. But then we thought, okay, maybe we can connect it a little bit also to some kind of results we know, you know, from... We discussed all week these results, so I don't want to repeat them. And the basic, I think, one is something that we know well now is the monocurve. Basically, why does... We describe this increase in rate law based on the initial substrate concentration. And... Yeah, and we... Actually, the question we had is, which one of these assumptions would work best to fit this model? I mean, I think you can already guess, but you'll see in a moment. So we considered, again, a very basic cell model with three steps. So one representing a transporter, one representing all the metabolism, and one representing ribosomes. And for metabolites, so the glucose concentration or the sugar, the intermediate imported sugar, like, let's say, the phosphorylated version of it, the precursors, which you can imagine are the amino acids or the tRNA charge amino acid, and the biomass, which is the proteome, mostly. Okay. And of course, there is dilution only of the last only of the last step. The other phenomenon we wanted to describe is this linear allocation of enzyme. Here I'm actually showing other ribosomes, because we already saw them many times. But the allocation of the lack operon, right? So when you increase growth rate, the lack operon actually increases. So basically when you're limited, you need to, carbon limited, you need to express more of the transporter to get more carbon in. And now I'm just laying down the assumptions that I've said. So again, the model has three steps. Yeah, three steps and then growth. The parameter we're going to change, basically see how we respond to it with the sugar concentration. The total enzyme is constant. The growth rate is maximized and the rate laws, so we're going to use the same rate law for all three reactions. Either mass action, thermodynamic, or mucous menten. Something we didn't include and we would like to, but it's a bit hard to do at the moment is the dilution of the metabolites. So the enzymes, of course, are diluted. We have to regenerate them. They are not diluted in the model, which is not realistic, because at least amino acids dilution could have a strong effect. We're not sure. Okay, and the extra constraint only for the mucous menten case, I told you. So the solutions are already written before, but I'll just, I rewrote them in the notation of the model, so there are three reactions. There's K, yeah, so there's Km of the transporter, the Kcat of the transporter. Again, r is ribosome and m is metabolism. These are the equilibrium constants. So for the mass action case, these are the three gammas. So basically what's nice is that we know how much to allocate. So sorry, I didn't say this. The allocation is the square root of this gamma. In this case. So we know how much to allocate to transporter, to metabolism and to the ribosome if we want to be optimal. And it's the square root of this function. Okay, and this doesn't change. So what's not realistic in this model is that it doesn't change. You see, there's no effect of the sugar concentration on any of these. And that's actually why you get this kind of monochrome, which I don't think is very realistic, which is a linear increase with the sugar, right? So allocation doesn't change. You increase sugar, all the efficiencies increase linearly and everything increases linearly and that's what you get. Okay, any questions? Okay, the equilibrium line is, of course, you cannot grow when s0 equals sn over k equilibrium, right? There's no driving force. So that's where you hit 0. Okay. Yeah, so as I said, the allocation of enzyme is constant. Again, not the very good solution, I think. The second one we tried is the thermodynamic rate law. I showed you the approximate solution, which we'll just use as is because it's very good. I just changed the delta G to the explicit description of it just to have the parameter we want, the sugar concentration, so we can see it here. So this basically, this function looks like this. A bit more realistic. But again, I have to say that there's two things that are wrong. First of all, it's not what we call a Micaelos-Menten curve. So it doesn't have the k50 in the same way. So the curvature is not the same. It could still fit the data, we don't know. But the main problem is that there's an equilibrium point here again. So you're not hitting 0 when you lower the sugar. There's some kind of limiting sugar concentration, which is based on k50. And if you make it very, very far from equilibrium, you will just get a very different shape. So these two things are connected. The other sad thing about this model is that the allocation changes not linearly, but also not in a nice way. It's even hard to, there's no formula for them. But they're basically changing in a very unexpected way and it doesn't fit any data. And the Micaelos-Menten kinetics is required a bit more work because we have four metabolites, but we want the first one to not be included in this constraint because it's supposed to be an external metabolite now. The sugar concentration is not part of this metabolic constraint on metabolites. So if you do that, you get this expression which looks very nice because it's again already you can see that it's going to work for the monoker. And you get the analytical expression for the maximal rates. Again, this maximal rate depends only on kinetics parameters. And this constraint as well. And we already called it the mono coefficient, which is the same denominator with a different numerator. So this is how it looks. What's nice I'm not comparing it to data yet, so all these numbers are just completely arbitrary. But you can already see that very easily if you have a measurement for the monoker, you can kind of get an idea of how it relates to these kinetics parameters. Because it's an analytical formula, we can also if it works, it can predict maybe how things change if you change one of these parameters. Hopefully. Of course, I should have said it before, every point on this curve assumes that the cell optimizes everything at every sugar concentration from scratch. So it doesn't matter what happened last night. It will always optimize everything again at this point. Otherwise the growth rate will be lowered. I'm always thinking about it. So the last thing I probably want to show you is the enzyme. So in this case this is the solution. The enzyme allocations are also expressions more or less on the same parameters and it also looks very, very complicated. But what's really, really nice is that you can rewrite them as a function of mu. Which is already a complex function of all these other things. But when you do that you actually see that it's a linear function. So the transporter actually has a negative sign so it's the total minus a linear function of mu and the metabolites and ribosomes go up with mu. And this is what it looks like. Of course the numbers, again, don't look at the numbers. I just used all the parameters are one I think in this model. Something else to know about this. So the lines go, yeah the sugar concentration goes from zero to infinity as you go with growth rate. But the line stops at some point. They don't go all the way down because there's the maximum growth rate you can reach is here. So they just have to stop somewhere. Okay, so I'll just summarize. So we tried three options. I would say one of them works and the other two basically is very simple terms. And we tried to figure out why the caspenton case works. And I think we can generalize it a bit. So we don't have to assume everything is a caspenton. What really matters is that the transporter is a caspenton. So that this reaction basically is saturable and not reversible. Then you always will get something like this. So the allocation of enzymes will shift between the transporter and everything else. So the transporter will go down with mu and the rest will go up with mu. But inside the cell, things will stay always proportionally the same because if you solve it once, you solve it for every, all the time because there's, again, the transporter is irreversible. So there's no there's nothing changing inside the cell except the flux. So everything just scales up proportionally all the time. In terms of enzymes, not metabolites. So metabolites don't change at all, actually. So that's actually a reason to get, that's where we get this curve. It makes sense a lot in retrospect. And we also tried it with a general enzyme cost minimization when we made the transporter more reversible, it looks exactly the same. Like in all the range of the model. Okay, so I'll summarize the, I have five minutes. Oh, okay. I started early, I think. Okay, so the mass action assumption gives us a mass action, a growth law. The terminamic assumption gives us something that approximately looks like a thermodynamic growth law. And then the cementer assumption gives us a monochrome which is also a cement type growth law. So I think, just forget about cell models I think, just if you have a few reactions that have the same kind of kinetics, approximate kinetics you can lump them together to one as long as of course everything is optimized you can lump them to one. And maybe this is the kind of mechanism that could be used by cells. Especially the last one is nice because of this linearity. So you can lump them to one operon and just like scale up the operon together. And that will give you almost the optimal solution all the time. Okay, so that's basically all I have to say so I repeat the mantra that the enzyme customization is a convex problem. We have a few analytical solutions I showed you today that are a bit more informative than just saying it's convex I hope. And we applied them to coarse grain models we find that only one of the three approximation gives us more or less reasonable growth laws. Of course the reality is more complicated but I think it will necessarily have to have this at least part of it will have to look like this kind of irreversible because it cannot be linear all the way of course. But I do want to say one last note that I think that and that's something maybe interesting to look at for you Terry is that I think the thermodynamic growth law could be relevant for some organisms that are really like the methanogens that are very close to equilibrium people have shown that actually like you can give them a methane and they made hydrogen so they're definitely all everything is reversible. So maybe their growth laws look very different right there's no irreversible step. So it could look more like the other one. Yeah and that's all I have these are the list of references and if you want to read anything then this QR code is the link to the presentation. You can just get all the references. Yeah, thank you very much. Perfectly on time so we can ask some questions so we are. So the question is what is the definition of the concentration you assumed that the epsilon total is a constant right? Yeah. So it's the fraction which is a constant one right? But the whole thing can change. I know in your information everything is kind of copy numbers but here we only have all the parameters that you saw in concentration. They should say that from now on. Can I repeat my question now because I still did not quite get it because it's sort of not my brain is tired after full day of talks but in particular this S total if I want to think of I know that of course biology does not optimize we learn the mantra so very thank you but if S total is something then it's some sort of constraint on you know you cannot pack too much metabolites inside the cell and easier model you told that if you don't impose this constraint everything blows up and I understand why because there is no limit how high you can take each one so it's tragedy of the commons easier model operates far away from the tragedy of the commons because in biology it probably is not super you know, a similarity unless it's caused by pH is not a huge problem so how the internal considerations S compare to S total and how they compare to capital case? Yeah so I think there are two questions one is if we assume that some of all metabolites is this actually S total is it really close to a physical constraint and this I think it's hard to answer but but I think I mean in E coli it's probably a bit low but there are organisms where like it's many almost a molar I guess so I think the total of all the metabolites so I think it could be a constraint for sure and the other question is like how is it compared to the saturation of the enzymes and I think Jaffer Binowitz has a very nice paper comparing KNs to actual measurements and it's actually they know like the variation is huge but basically the average is more or less like the same place so most the average enzyme will have the substrate around the KN which means that they don't want they could almost double the rate if they this wasn't a constraint and what emerges from your optimal model with Mekaitis-Menton part? what's the optimum yeah what is the OPS I can't answer that because we didn't use any real parameters yet so we just I just use one for all the parameters so it will be just some monstrous expression which you cannot I mean isn't it like could you flash back the formula again for the Mekaitis-Menton yeah something like something like this something like this right if I just completely stupidly think that the capital K over lowercase K is the same for all then this the whatever you have in square will be just basically 4 times capital K divided by lowercase K and basically then you will have comparable to 3 over K in the left-hand side except that there will be K over S total so I'm just I'm trying to digest this in some simplest I think the important question is how much S tells you how much you need to care about the left side versus the right side if it's very very low then you will be very far from being saturated and you really need to care about these optimizing everything goes to the KM if the S total is very very high and you have no constraint basically you can just saturate everything like nothing matters we used to call the pathway specific activity it's just like summing all the specific activities and you get like solution for saturation conditions so the relevant parameter is capital K whatever it is that's a dimensionless parameter right? capital K has units of considerations and S total has units of considerations the ratio of these two things so just to answer your question so for E. coli with these curves that I showed we can actually determine each of these one over KK of this and that and this works very well but roughly just ignoring the last term so empirically we kind of know because we have no handle on KM right? we don't know what S total means okay what works is this mono matches up you can estimate the one over KK for each of these and so it works out very well if you ignore the last term which means last term is small and in the limit where KT is smallest one then of course we get a trivial result that mono is just a transporter mono constant but the thing is this is all effective so when you do the proteomics you see the C-Sector changing whatever that's an effective KK for all of the kind of value that's expressed of which only like a few percent is used to transport the carbon you're actually using so you cannot use the actual KK the entire lack of operon that I was trying to it's not just the transporter but the transporter is again it's lumping of all things from the external metabolites to where it joins the rest of the metabolites so it's a really weird thing because optimization works beautifully I mean we know this number except that when you look at the actual enzyme if it's optimizing you want to say use all of the transporter you want to give it to LATOS it is not doing that the framework it's using the optimization framework it's almost like you call I know it's the framework it's stealing it I agree that's still a very interesting any other question ok if not thanks a lot we now have the tutorial on FBA