 times log Pi. Some up over all possible energy states. Of course, this is subjected to some restrictions, including how you define the probability, which is just the number of molecules that are in a given state normalized by the total number of molecules, and also with some experimental data that makes it condition-specific, phenotype-specific, like, for example, the average energy that the amount of molecules have. So the last equation is just the average or expected value of the energy level. And with that, using an analytical solution, he derived the Boltzmann distribution, which is an exponential distribution, showing that most molecules have low energy levels, and few of them have high energy levels, and between them, there is an exponential decay. The same equation was derived independently by Claude Shannon, and I'm going to exemplify how this works using a game. So I never saw this TV show, but they were supposed to be a game where a set of people was living in a corner, and you were living aside with headphones, and one of these persons had to write a word. Let's assume that Florence, in this case. And then you need to probe this person asking just, just, or no question. You can just get bits of information, one question at a time, and you win if you ask the least amount of questions than your competitor. So for the sake of simplicity, let's assume that you are already familiar with Florence, and you know that Florence thinks mainly in three categories, which is half of the time thinking on cars, one for the time thinking on watches, and the rest of the time thinking on books. So with that information, with that distribution, you can play the game in an optimal manner. So knowing this distribution, you should always start asking if the war is a car. Because half of the time, you will win the game with just one question. The other times, so this is stochastic, Florence can be from time to time thinking in the other categories, you should need to ask two questions. So in average, the number of questions that you need to ask, so using log two, applied to the probabilities, you get the number of questions per item, and in average, you can just get the same equation as before, so there are two extreme cases. Let's assume that you also play the game with Aldo, and Aldo is always thinking only about cars, 100% of the time. If you apply the formula, you get that H is equal to zero. You don't need to ask any question to Aldo. In the forehand, you know that it's going to be a car, the answer. So you win the game with zero question. The other extreme case is where, for example, here is the one that has to write the war, and here is using a uniform distribution. It's equally likely to be thinking in cars, watches, or books. That's the worst case scenario because there is no structure in the probability distribution that allows you to take a shortcut. So that's the case where you, in average, need to ask the most number of questions. So hopefully, when you play the game, it's her and not, sorry, Aldo, not here, who is writing the war. So using this expression, if you don't know anything about the person, for example, maybe you are playing with fun, you should assume that he's using a uniform assignation of words because that implies that you are most ignorant about what this person has in his mind. So you are the least biased about it. Okay, so with these two pillars, so you have a system that can be several configurations according to Boltzmann. The one that maximizes the entropy is the one most likely to happen because it can occur in the greatest numbers of ways. And according to Chanon, it's the one that you are most ignorant about it. Therefore, you are the least biased. You are not assuming anything that is not warranted by prior information. So James said these are just two sides of the same coin, and we can use it as a statistical inference procedure. So what we want to do is, from the problem that I defined before, the polytope, so this is the space of all possible fluxums, we want to select the one that has the highest entropy. So for this, we need to define what is H in the context of the fluxum space. And later on, we also want to incorporate some phenotype specific data that restricts the solution to a particular condition. And I'm going to argue that gene expression data is a good candidate to do that. So this is at the cost of doing some assumptions. So we're going to apply this, assuming that the cell has a limited amount of enzymes and need to decide to which reaction is going to assign these enzymes. Let's assume that capital E is a total number of enzymes. What is the most unbiased, most likely way to assign these enzymes? That's the problem that we're going to solve. But since we're interested in the fluxes B, we're going to use this expression. So the flux through the reaction I is equal to the amount of enzymes assigned to that reaction times this rule of engagement. So this could be, for example, amicuris menten expression, or it could be anything that you can have in mind in addition by product, substrate or something more complicated. We don't know in advance. So since we don't know, we don't know either the parameters of this rule of engagement, we are going to assume there are values in the steady state that just uniformly distributed, which is not true. This is a simplifying assumption because with that, we can go from a probability distribution that is a function of the enzymes to another that is a function just of the fluxes. So with this simplifying assumption, we can factor out the F values from the numerator and the denominator. So doing that simplification, we get this expression for the entropy. So therefore, if you have polytop, you should select the one that maximizes that expression. Yeah, we're assuming that are all the same. So to see if this simplification is just too abrupt, we compare it with the base reference that we have, which is carbon labeling to estimate some fluxes of glycolysis and the TCA and also compare it to alternative methods to compute the flux. So one popular method is just flux sampling. So you take a uniform sampling from the flux and from that you can then compute average or the whole distribution of each flux. So now I'm going to simplify it with a metabolic network that has a loop. So again, there is 10 units of green molecule going in and just by mass balances, there are 10 molecules of purple molecule going out of the cell. But in between, you have this where one path will convert this molecule into this one and there is another that does the opposite. So this is a thermodynamically infeasible cycle, which is a problem that play the networks that are typically used in this case. And this is particularly important for flux sampling because if you have that type of loop, B3 can be anything. It can go to infinity if you are not bounding that flux, which is not realistic. And flux sampling is just with sample everything that is given within your bounds. So in this case, the polytope is actually just that blue line. Flux sampling in principle is going to take any point along that line. One way to avoid that problem is to run the FBA and after that select the B that has the minimum amount of total flux. So this is under the assumption that the cell tried to use the minimum amount of enzyme to produce a given phenotype. And this can be done computing the amount of flux by the square, by the Euclidean metric. And without you effectively get rid of this loop. Another way to do it is taking the absolute value. That's what the geometric method does. And it's called geometric because when you use the absolute value, you get a linear optimization that doesn't have a unique solution. And from the space of possible solutions, you select the one that is in the geometric center that is represented there. And finally, using the principle of maximum entropy, we select the point that has the maximum end. So let's see how Maxen compared to these three popular methods that are available in literature. Yes. I missed this, but how do you evaluate the maximum entropy on this polytope or on this topology? My main question is then, how do you actually sample from green to purple? How many paths go and you evaluate each of those? I know, it's a strictly complex optimization problem. So you just run the optimization problem and you get 1B. That's it. So that is the optimization problem and H is a strictly complex function. So you just run the optimization and get your B. Okay, thanks. So the flux sampling, the min flux, and the max end, I can conceptually imagine. But can you just describe, where does the geometric approach, what does that conceptually mean? The way you described it is perfect, but I'm just having a hard time going from the way that you described it to where that leaves us in the metabolism. So can you just talk a little bit more about what the geometric solution is? Yeah, so I'm going to repeat what I said before. Just let me know how it can improve it. So in this case, that whole line is the space of alternative solutions. So the extremes of this space are given by the upper bounds that you are assigning to your fluxes. Geometric, select the point that is at the center of these bounds. Okay, thanks. Could you repeat how substrate concentration affects a VI or a pathway? Like even probably you said it like there is V1 and there is a certain quantity of substrate. How does that affect the V entropy of a pathway? If I understood correctly how the flux of the intake affects the entropy. So in this case, we just take the probability distribution. So the fluxes are normalized. So if I go back to this equation, I should have said that P times a lot of P is not equal to that expression. You need to use the exponential of the sum of fluxes times that expression. So that takes into account the total flux of the network. But since in general we don't have the magnitude, we're most interested in the relative weights of the fluxes, we just use that expression. So you can consider that you get the distribution of the metabolic fluxes. Substrate VI enzymes are working on important. Does it appear? Does it not appear? It affects, but we consider not the concentration. We consider the flux that intakes that substrate. And you can use something like Michaelis Menten kind of expression to derive that flux. But we are not considering metabolic concentrations in this model. So this is a limitation of this type of distribution. Yeah. If we can see the slide when you compare the different methods. So Maxen will give you a distribution. And I have a similar question that was asked before. Are you able to infer it only as a multiplication problem because you are only solving the mode of this distribution or you happen to have the whole distribution at the end of the process? No, I only have one bit. Okay. Yeah. And I'm not sure if the distribution is symmetric that would be equivalent to the mean value or the mode. But I'm not sure. Okay. Maybe it's asymmetric and then it's not equivalent. So you have some short cut that you don't need to infer the whole distribution only at this point that you like. Yeah, that's a benefit and of course, because sometimes you want to study the heterogeneity in the system, you cannot do that with this. Okay. You showed me the details. So one of the nicest properties of maximizing entropy is that you fix one flux like the intake of glucose and everything trying to go to that same level without reaching the bounds that you are arbitrarily for technical processing in the system, unlike flux sampling. So maybe you can explain it with exactly this example. So again, this is a toy network and purpose I put a thermodynamically infeasible cycle. So for example, B Phi can be reverted back C towards A and B Phi can be anything and still respect the mass balances. But if we fix the input flux, for example, 10 and you maximize the entropy, since the preferred distribution maximizes entropy is the one that tends to be the most uniform one. Automatically it's going to resort to produce a distribution of fluxes that is similar to 10 without reaching the upper bounds. So in this case, it's actually a uniform distribution. Every reaction has a flow of 10. If you can do flux sampling, then you can have anything or B Phi and then most likely you're going to reach the upper bounds. I don't know if I answered the question. Supposing the upper bound of one of those reactions was less than 10, then you would not arrive at that solution. In that case, it will match the upper bound. In general, your max end solution does depend upon what the upper bounds are. Yeah, although we try to put the upper bounds that are big enough to never reach them, unless we have a very good inkling of what would be the value of that flux. So all fluxes will become basically equal because your PIs are just the flux values themselves. All fluxes will become equal. Is that the solution for the max end? It will depend on the topology of the network. In this case, all of them will become equal. But for power law distributions of the metabolites, it will not. It will behave like an exponential decay. What about the previous example that you showed us, right? V1, V2, V3, and V mu? So to be split equally. No, there was a loop there. This is the second example, the previous, yeah, this example. What would be the answer in this case? I think it's going to be, V3 is going to be around 5. So 15 goes from green to purple and 5 going back. So in this case, here's an example where they're not going to be all equal in the max end solution. Yeah, in general, they are not going to be equal. But it depends on the topology of the network. Just a will intuition, I put a thermodynamically invisible cycle there. And as you can see, a max end solution is not free of this artifact. It still predict flux going back from C to A. This problem is solved by mean flux and the geometric methods because they effectively kill all flux loops. But they do it in different ways. If you use the Euclidean matrix or the absolute value, you could use different results. If the flux just makes zero, this one. Whereas the geometric makes zero, also this path. So then you have the problem of which one is the correct implementation of this idea that you are minimizing the total flow through the network. But we're going to argue that although max end is susceptible to this problem, it's not as sensitive as flux sampling. So I presented this loop also because there are some reactions forming loops that are thermodynamically feasible because they are tapping in another reaction to provide the energy to do the opposite conversion. One of them is the glyoxylate chunk, which has been chunked to carry flux. So this is a formal loop, but it's thermodynamically feasible. And if you use the alternative methods, mean flux and geometric, for these flux in particular. So the dots are the experimental values reported to 0.1. This is panel B and 0.2, a specific growth rate of E. coli. You get systematically that these two methods predict zero flux through ICL. On the other hand, since flux sampling over represents the thermodynamically feasible fluxes, one of them being sukoas, which is not shown here, but it partakes in a thermodynamically feasible loop, so flux sampling predicts an extremely large value, reaching the upper bound that is assigned to the reactions. And you have max-n in between. So max-n is not as sensitive to flux sampling, and it doesn't produce these artifacts where you can still have thermodynamically feasible cycles. So going back to the question of the uniform distribution of the fluxes, one of the reviewers asked us actually the same. So this is very boring because you get a uniform distribution, but when we apply it to the E. coli metabolic network, we actually get an exponential decay. So these are again for 0.1 and 0.2. The same conditions are presented before. And this match a proxy of what would be the flux, which is the messenger RNA, which we took from that reference, which also decays exponentially. So it seems to match the literature in that sense. So then we asked if with these alternative methods, you can still get a solution that produce a high entropy. And this is what is shown in panels A and C for 0.1 and 0.2 row rates for E. coli. And flux sampling, we took like 10,000 samples, and in average it produces an entropy that is way lower than what we get with a maximum in both cases. And from the other two alternative methods, the clear and distant tend to produce a most uniform distribution, but not as high to reach the same entropy value as maximum. And since each also publish the carbon isotope label derived fluxes, we also compute the difference between the predicted values of glycolysis and TCA and the reported values by each. And the results show that if you take the average of flux sampling, the distance tend to be artificially large because you are taking into consideration reaction like sukoas, which belongs to a thermodynamically infeasible cycle. The other three methods, Maxent, is on par with the alternative methods. So produce without adding these extra assumptions of minimizing the total number of flux, produces a quality that is similar. Okay, so then I'm going to jump now to how we can incorporate gene expression data to make your predictions phenotype specific. So one convenient way to make it specific to a particular condition is to take RNA-seq data. So RNA-seq has been becoming cheaper and cheaper to perform, and now there are databases where you can, for free, download the gene expression data of your favorite microorganism. So you can take it for several conditions, and for each one of them you can somehow plug it into the Maxent framework to get your B estimation. So these have been done before, and one of the most successful methods is called SPOT, where the idea of this alternative method is that you maximize the correlation between the G vector. The G is a vector that contains the gene expression data for each enzyme that partakes in the metabolic network, and B is just a normalized version of the metabolic fluxes. So by maximizing the correlation, you should get something that makes B similar to the distribution of J. Again, from the same poly-dot you select that one. So we compare our method against SPOT, because in that paper they compare SPOT against i, mat, and other type of methods that also take into account gene expression data. Okay, so how do we add gene expression data into the entropy function? Again, we start with this equation, that the flux of reaction i is equal to the amount of enzyme times this rule of engagement. And in this case, we use the gene expression data as a proxy of the amount of enzymes that is attached to each reaction, which, again, is a simplifying assumption. We know that the amount of the messenger RNA is not exactly the same as the protein concentration of the enzyme that is catalyzing the reaction. So this is another way in which the problem is simplified. So we took this simplification, changing the E for the gene, the gene expression of reaction i, and then we compute the distribution of these terms, the flux divided by the gene, which can be interpreted as the flux per enzyme. So that's defined the probability, then we apply again the entropy function, and now we need to run it for every reaction, from reaction 1 to capital R, for each reaction for each one of the enzymes, from enzyme j equal to 1 to G sub i. So assuming that the flux per enzyme is the same for each copy of the enzyme, which, again, is not necessarily true, we can factor out this term from this inner summation, and we get this term here, which later on cancel out. So that term over there is the flux per enzyme. So now we're applying the distribution for each reaction and for each enzyme. I mean, your PI is just a relative fi. It's no longer a flux, it's just a sort of engagement, right? Yeah. The distribution of engagements now, another distribution of fluxes. So I don't understand why they get now multiplied. Yeah, maybe the notation is tricky, but the idea is that we have our reactions, and now we know the number of enzymes that are partaking in each reaction. And what we want to maximize the entropy of is the flux through each one of those enzymes. So that's, we annotated there as bi over gi, called it maybe alpha. We're assuming that the flux through each one of these enzymes is the same. So if you think this as flow traffic, the number of enzymes is the number of lanes on a street. So we're assuming that the number of cars per lane is the same. Yeah, I didn't have a problem with that, but I thought now you're going to maximize the distribution of these P's, which is essentially the distribution of F's. Yes. Right? But then I don't understand why you have to multiply by gi, which is effectively what you're doing there in defining the entropy. So you don't agree with the inner summation, I guess. Well, it's just going to multiply by gi, right? I don't. So the inner summation taking into account that you are considering the flux through each one of the enzymes that is assigned to each one of the reactions. Different reactions can have different numbers of enzymes assigned, and we're taking that as data. So those are the gi there in the inner summation. Maybe we can discuss it later. So the last part, it can be easily shown that it's equivalent to the Kullback-Liebler divergence between the vector g and the vector b. So maximizing the entropy is the same as minimizing the Kullback-Liebler divergence between b and g. So this defined a new strictly convex optimization problem where now entropy is conditioned on the g data. So to will intuition, now we have this simple network and we have now information about the relative abundance of the messenger RNA assigned to each reaction. We can now make an h function that depends on this transcript. So in this case, it's going to be preferred the flow that goes through reaction 2 because you have more messenger RNA. Whereas for transcript on b, you have the opposite case. So again, we will a toy network to see how spot and feed flux, we can let phenotype the specific summation of fluxes depart from each other. So in the toy network, we have again a cycle there, you have the fluxes in black and blue, you have the gene expression assignation to each reaction. And then we have three cases, b, c, and d. The first case is a uniform assignation of gene expression data, all of them get a unit of one. C is when you have the double, double the amount of gene expression here compared to that one. And the third one is where you have one order of magnitude higher gene expression assigned to that reaction. So when you have a uniform distribution of gene expression data, feed flux and spot, they don't look too different. They're very similar, one another. But when you start having this type of differences, then the kind of predictions became evidently different. So in this case, with one order of magnitude difference between these two, feed flux still predicts that it's an intake of metabolites from the medium, whereas spot predicts that all the flux is just running in circles here with no intake and no output of fluxes. So another way to try to validate this is compared to reported values of fluxes. So we use as a benchmark a carbon labeling, which typically just report fluxes for glycolysis, TCA, and some ramifications from there. And we collected a database for several microorganisms that has both fluxomic data and transcriptomic data, either microRNA or RNA-seq. So here are the results. So there are five micro-organisms. B-subtilis and E. coli have several conditions, depending on the type of carbon source that is being fed into the system. And in average, feed flux produce better results compared to spot. But we also compare it to FBA because FBA doesn't take into account any gene expression data. We compare it to the mean flux version of FBA, where after you run FBA, you minimize the total sum of fluxes. And surprisingly, it's on par with feed flux. This is surprising because you don't have this extra layer of information and FBA still predict in average fluxes are on par with feed flux. And this is something that was reported by the paper of the spot as well. That FBA, without any gene expression data, is better than a spot. So what could be the reason of that? We argue that it's because we are sampling just a small set of reactions. Basically glycolysis and TCA. But what happens if we take into consideration all these other fluxes? Would the performance of FBA be as good as with this small set? Unfortunately, these alternative pathways and ramifications cannot be measured experimentally. So we have to resort to computational simulation. So basically we sampled the polytope space and we created a gene expression vector matching the same V vector. So that could correspond to a perfect correlation between G and B. That is not realistic, so we took several scenarios where only a fraction of the genes have a gene expression that is perfectly correlated with the fluxes. So here are the results for FBA panels A and C. And the X axis is where we destroy the signal between gene expression and fluxes. We totally shuffled one vector with respect to the other. So this is equivalent to MaxM when we don't have any genomic information. The other case is where fluxes are perfectly correlated with the gene expression data, which is the ideal case for free flux, although not we don't expect to be the case. The question in between must be the case. So we have several cases in between. The results for FBA shows that FBA actually have a very good performance when you only focus on these 20 fluxes, Lycolysis and TCN. But performance drops when you consider the whole metabolic network. It goes close to signal. On the other hand, the predictions of free flux when you only consider this small set of reactions is all over the place, but above 0.5. And the situation gets better as this assumption becomes more and more ideal. So having very good predictions where the fluxes are perfectly correlated with gene expression data. But when we look at the whole set of fluxes, the whole genome scale network, we see a consistent increase in performance. So after 60% of the reactions are perfectly correlated with gene expression data, the performance is above 0.5, measure as a correlation and gets better and better from there. So then we also look at if this method can recapitulate some known metabolic traits, some phenomena like cancer. In this case, we took from the cancer genome atlas, gene expression data, RNA-seq libraries from patients that have samples from the tumor mass and the adjacent normal tissue. And we focus on three cancer types that have the most number of points. So it's around 1,000 RNA-seq libraries for the cancer type and around 500 points for the normal adjacent tissue. So each point represents the prediction done with free flux of this ratio, the production of lactate normalized by the consumption of glucose. We should expect this one in average to be greater in cancer cells. And that's what we found for breast, kidney and bronchus lung cancer types. But as you can observe, there is a big overlap. So this seems to suggest that not all cancer cells overproduce lactate compared to their normal counterparts. It's just that in average, they produce more. So again, one of the reviewers said, but I can know this just doing a simple bi-informatic test. Can you show me something new? So then we created this figure where we compute the enrichment of each pathway. So we took the information from Keck, which report that interaction belongs to glycolysis, oxidative phosphorylation and since we can compute the fluxes for each one of them, we took the average of the flux through the whole metabolic network, normalized by the total flux and we call that enrichment. So I'm going to focus just in the extremes. So here you have glycolysis where it's dominated by tumor cells. So tumor cells have more weight on glycolysis compared to normal cells, which matches the accepted view and on the other hand oxidative phosphorylation is more prevalent on normal cells. So in this case these two road pathways are what we would expect. But things like TCA flip sides. So for example in kidney the TCA is more prevalent among normal cells but changes size for breast and bronchus lung. So this is just a speculation because we need experimental validation but seems to indicate that there is a potential therapeutic value using this type of methods. So this is a non-linear, it's a strictly complex obtuse problem but it's not linear, so it's going to take longer than FBA but still we could run it in a very reasonable time. So for micro organisms, E.coli, Saccharomyces herbicide runs in a matter of seconds, seven seconds and for Recon 3D for Homo sapiens which has around 10,000 reactions it runs in about three minutes. So it still can be done in your laptop. So as a conclusion at least with the data set that we can compile free flux seems to have better prediction of metabolic fluxes compared to the set of alternative models that we choose to study. It can be used to produce a phenotype specific dissolution of fluxes like in the Warbore effect and based on that it can inform therapeutic treatments and also it can be run in your laptop in a very reasonable amount of time. So room for improvement this kind of modeling is still susceptible to thermodynamically infeasible cycles although not as sensible as fluke sampling. And one of the simplifying assumptions that we did that mRNA is equivalent to the number of proteins catalyzing reactions can be proved if we have protein information instead of messenger RNA. So some work in progress as few was mentioning that is when you submit a cell like saccharomycerobeside to scarcity of resources and then you feedback again with carbon to this one you can have an oscillation. So this is the G's metabolic cycle where you have these periodic oscillations which are dominated by the rates of oxygen consumption. So in a paper of one at all they measure the gene expression data at each of these time points. So using the profiles of the dissolved oxygen we can estimate the consumption rate of oxygen and with that we can feed it into FIFLOX with both RNA-seq data and the intake of a metabolite from the medium. What Barbara was interested in was the relationship between the metabolic state and the epigenetic state. As you know the DNA is not freely floating within the nucleus or the cytoplasm is wrapped around these proteins in the nucleosomes which can be reversibly modified. For example you can have a methyl group or an acetyl group which can have several outcomes some of them facilitate gene expression some of them prevent them and in this database Quangarol also measure sero-histon marks a 3K4-3 methylation and H3K9 acetylation among them. So with the RNA-seq data and the fluxes of oxygen Barbara predicted how the precursor of the acetylation and the methylation of histones changes over time. So the blue line is the production of the precursor and the methylation of the histones and the green one is for the acetylation of the histones and they show a complementary pattern when one peaks, the other doesn't and vice versa. So then Barbara looked at the genes whose level of acetylation and methylation of the histones in the promoter region was correlated with these patterns and then submitted these genes to David to do a genotology analysis and if there is no signal these peaks should be all over the place they shouldn't be enriched in any function but we found that the ones that are correlated with acetylation are mainly associated with metabolic processes whereas the ones that are enriched with methylation are associated for example with chromosome production and workings of the genome suggesting that the alternative production of these two metabolites can have some play in the regulation of the expression of the genome but this is work in progress. Another salient angle of this is how we can extrapolate this to explore the diversity of a clonal population for example E. coli not all cells in the population are growing at the same rate there is some distribution so there is a method that Andrea and Daniel the Martino produced in 2018 using also the principle of maximum entropy but they were modeling the growth rate so if we use again this example they were interested in before how before is distributed in the population so they have a distribution like this which is not dependent on the internal distribution our method on the other hand just predict one distribution that one over there and capable of to modulate the different growth rates in the population and the idea is to mix both types of entropies to have a sampling that produce diversity between growth rates and for each growth rate across these and inner so thank you for your attention and happy to answer any questions thank you very interesting I have three questions comments so one is just a comment so I'm not surprised that the method from my background the method with the gene expression works better than the other one from resource allocation models you often get the prediction that optimal flux distribution should be EFMs or sums of few EFMs which is kind of the opposite of your first method because that would mean that many fluxes are zero while the entropy would exactly try to to avoid many zero fluxes and I think the fee flux method solves this because you can have zero gene expression and then you can account for EFMs I was just saying I like second is more a question so I was wondering in the fee flux method if you normalize the fluxes not just by the gene expression but by the gene expression times the k-cat value in cases where k-cat values are known or can be guessed I imagine that this may even improve the predictions did you think about this? Do you think it makes sense? Yeah, it makes sense I don't know how to estimate the k-values though Is there a methodology to estimate them in a genome way? Well, there are machine learning methods that try to do this and there's for some enzymes there are at least some So that definitely would improve and the third is it's more like a fundamental critique so you treat fluxes as variables that basically probabilities so variables that are always positive but fluxes can be negative so I'm wondering if there's maybe some some tinkering going on or something that you need to fix to avoid Yeah, we consider reversible reactions as the sum of two reactions that are going in opposite ways So you split them in both? Yes, but if you split them I think then there may be the other problem that your method will try to make them relatively equal so that will create huge cycle fluxes So some of the reactions has to be unidirectional and that solves the problem Let's assume that you have two metabolites The one in between is going both ways But since the ones that are adjacent are going in one predominant way that is going to oblige the system to have one of these in between to go predominantly in one way Okay So you split depending on what you know about the reaction and then you accept the huge cycle flux? Yeah, we just split the reversible ones Okay, cool, thanks Eric, last question Yeah, just for my clarification So before you started including the expression data which I agree is very nice the Maxent approach so I remember reading a paper from Daniela de Martino Gaspercacci's lab at some point, so is that similar or is this fundamentally different I was trying to understand Yeah, is it different? It's not that different but so they try to produce this distribution So they are concerned only on before in this case, or biomass growth rate So they have the metabolic network actually you don't need the metabolic network if you can have a method that produce the biomass growth rate and they maximize the entropy of the distribution of the biomass growth rate not necessarily looking into the distribution of fluxes inside the cell So basically it's that distribution over there So if you would apply it to this you would get that line uniformly distributed So a point here on here is indeterminable in the Daniela de Martino framework We'll talk about the questions Alright, let's thank the speaker again