 Right good afternoon. I know everybody's tired, but we are almost done. So please I mean just a bit of effort All right Let's welcome Warfram. He's one of the organizers of the school. Thank you You hear me. All right Okay, good. So before I start I just wanted to repeat Some of the advertisements I made last week for those who haven't heard them before but very very quickly So we collect teaching materials for systems biology and related fields. So whoever of you who Who teaches and has some slides exercises videos, whatever to share that are not yet Publicly available. You're very welcome to send them to us or tell us a lot and me about it so that we can enrich and increase and This this body of yeah knowledge or materials for the community Then short repetition of the other announcements so for the forum As almost all of you know, it's once a month zoom meeting Zoom seminar on economic principles. You're all very much invited to join us. It's alternates between time that is good for us and Europe 1 p.m. In Europe 530 in Europe and a time that is good for Europe and Asia 1 p.m. In Europe and if you You would like to give a talk there. You can also Always approach us and we'll try to figure out. What is the interesting general question the context of the talk and make a session So be very nice also to increase a bit this community The young scholars group I think you Probably all of you who are concerned by this participated in the session last week and For the book project. I just wanted to say briefly what what happened now in last week and this week so we have a few people that Consider writing chapters and worked on this a bit and we'll continue working on this. So we have Sanjay and Some others who discussed Origin of life questions then Sergio will probably write a chapter on scaling laws for microbes for prokaryotes and eukaryotes and do this with a student then I hope that Bob Planke will also write something and There was also a group discussing a bit microbial communities I With a lot so I don't know how this ended whether this ended if any one of you Particularly those that were in the group Would like to continue this You're also very much invited so we can continue this with zoom meetings and then see where it goes If for you it was just a nice experience now last week That's also fine. Yeah, so and everyone in general is is always very much invited to to join us for writing for reading for thinking yeah, and Now I'm going to start with my talk I think It's a bit sci-fi I would say so I hope that some of it will be proper science and part of it I think is fiction or Very speculative and it's also a bit chaotic So I invite your questions many questions because then I can pretend afterwards that the case was not because of the Slides but because of the dynamics of the session Okay, let's get started. I Know I took some inspiration from Andrea Who basically gave a summary of his talk in the first slide? So I'm going to do the same These are the take-home messages already now And a bit the the line of argument that I will make in general the idea is to Have a new perspective on optimality problems in metabolism, which is not based on amounts and Transformation of mass but based on an idea of value or prices and in practice This can be defined by a Lagrange multipliers or shadow prices and I think this is in a perspective that could potentially become very interesting and and Useful mostly for maybe didactical or intuitive reasons I don't know how interesting it will be for actual calculations, but I want to just tell about this and So I start with a problem. The problem is that It's really difficult to Model and predict optimal metabolic states because you actually need to think of fluxes metabolite concentrations and enzyme concentrations at the same time Of course, you can also just talk about fluxes that but then you're missing an important part of the picture And you need to fix this by adding more and more I talk assumptions So if you want to go for the whole problem, it's going to be difficult because it's all interconnected between these types of variables Then also in the network and in principle you could know of course you can you can try to optimize a single pathway But since everything is also connected throughout the network in theory for an optimal solution You either need to optimize the entire cell or you need to assume for your System of interest that it interacts with other systems that are also optimized so it's a bit of a vicious circle and So in principle the optimal solution for one part of the network May depend on parameters everywhere in the cell like any any change anywhere in the cell could potentially Have an impact on what you're looking at in one place in terms of optimality So what I'm proposing here is to Do something different where one can get actually local laws where you can think of a subsystem It's optimality and isolated from the rest of the network by defining values that Live first on the boundary of the subsystem and that tell you about the effects of this system on the rest of the system and the fitness effects that that emerge from this and From these values on the boundary of your system You can also then go to values inside the system and eventually there's a each variable each concentration flux and so on Will be associated with a second variable, which is the value of this physical variable and This the values are not just anything so they can be to have defined as Lagrange multipliers. I'll say this later They do not take any numerical values, but there are laws that connect them between neighboring elements So if if you know something about the value of one metabolite, it can also tell you something about values in the in the immediate surrounding Okay, I said that already one way to get to these values are Lagrange multipliers or which are also called shadow prices if you evaluate them in a in an optimal state and for kinetic metabolic models you can also alternatively and equivalently define them by metabolic control coefficients and If you apply this thinking to different kinds of models for example an FBA model And a kinetic model or even a cell model You can always get to the same laws Because all these models share some common Constraints the mass balance constraint for fluxes exists in all these models and that gives you laws That emerge from the mass balance constraint and always have the same form You will not necessarily get the same numbers in different models, but you can arrange You can adjust the model so that also the numbers will be will be the same and The the form of these laws is actually it looks like thermodynamics so the values are very similar to chemical potentials and In the same way as the chemical potentials in metabolism determine flux directions These economic values also determine flux directions But they come they don't come from the same place as thermodynamics So they're really it's a complimentary a set of constraints that just takes the same mathematical form and then the two can be considered at the same time together and The laws can also be Interpreted as conservation laws. So if you think of a single reaction You can think of value flowing in from the substrate and from the enzyme and value flowing out through the product and potentially also co-factors and so on and the value flow is conserved within the reaction So you can you can think of value and this holds only for optimal states. It this Derives from the from the assumption that one describes an optimal state So this is also a way to say that values are connected through the network because values are flowing and there's a there's a conservation law for value Then for non-optimal states in the formulas you get extra terms that Describe the deviation from optimality, but also locally and so a positive deviation in a reaction could tell you that the cell should have an incentive to increase the enzyme in this specific reaction or Negative to decrease it in the specific reaction. So I compare it to stresses in in biomechanics was for example in in the growth of bones where local stresses are taken as Signals that tell Cells in the bone to either reinforce the bone or take away material in the bone There's a feedback that translates stresses into a rule for adapting the shape and so for me, this is an analogy to a Mechanism that is is probably not it cannot exist in reality. It's a hypothetical mechanism so if the cell were able to sense these Stresses of non-optimality it could know which enzymes should be up or down regulated in order to get to an optimal state and I claim that The formalism because I'm borrowing words like prices costs investments I claim that the formalism also gives kind of an interpretation to the language that we're sometimes using for describing cells like this metaphor Okay, so I think you probably already see that this is a bit of sci-fi Okay, well try to to get to the to the thing Okay, so the basic question one one basic starting point could be How can we make sense of the proteome of a cell? Why does the cell invest a lot of enzyme in certain? Pathways or single reactions why so much in ribosome so on and to Think about this It's the plug To think about this it would greatly help to know the fluxes already So if we know the fluxes in the metabolic system In each reaction, then we can already guess a bit Where much enzyme needs to be one needs to be invested so what else do we need to know aside from the fluxes? What what further information would help us get more precise estimates of the enzyme? Any ideas the kinetic parameters, how would that help? How would you use them? Mm-hmm Yeah, exactly. So if you know the rate law, you know the parameters, you know the basically the enzyme efficiency So it's and if you multiply the enzyme with the efficiency you get the flux So if you know the flux you can solve for the enzyme so it's will be The flux divided by the efficiency and if the efficiency is for example proportional to k-cut then a high k-cut Will tell you it's very efficient so you need less of that enzyme high k-cut is good. What else what what else is good? Aside from a high k-cut low Km Yeah So you're you're far in the in the rate or you're far on the on the right Yes, exactly in this picture if a pathway is a region if if the region contains fewer Single enzymes and it does the same thing then the whole region is smaller Yeah, and some of dynamics also plays a role if a reaction is very much forward driven then Also, the enzyme is efficient. So you need less of a reaction is close to equilibrium then you need more and So if the flux is unknown We can figure out the enzymes if we have a lot of Extra knowledge and so this is something that I think a lot showed more or less the same in his talk you can You can split Reversible rate loss in this way so the flux can be written as the enzyme level times the forward k-cut Times a thermodynamic term that basically starts at zero with zero driving force And it goes to one with an infinite driving force. So good high driving force is good and here's an extra term that Is related to to a saturation so the Km value would would come in in this term if you Now you say that the total enzyme cost is a weighted sum of the enzyme levels You will solve for the enzyme level here Insert then you see that the enzyme cost Is basically it's like the inverse so High k-cut is good. It makes the enzyme cost small High like thermodynamic efficiency is good and so on so in this way you you could Figure out the enzyme levels if you have all this information and this picture just shows The where the different effects are coming from so if you have a fully saturated enzyme like only substrate and no product then you basically get a formula that only contains this term and not these terms if You have a lower driving force, but are still fully saturated Then you need to account for the reversibility and you also get this term and in the general case that you have non-saturation You know you get the same formula and so you go from if you have more and more of these terms You go from higher to lower rates or in the reverse At the given flux you go from lower to higher enzyme demands. Okay, so We could be happy, but we don't know the metabolite levels We don't know the kinetic constants and we don't know the metabolite levels now in from now And we assume that the kinetic constants could be known, but still what are the metabolite levels that will change? we either need to measure them or we need to guess them somehow and One idea since we're already talking about optimality is that maybe the cell adjusts the metabolite levels in an optimal way such that this whole enzyme demand is is minimized if you do that Yeah, you get to You get to the method that a lot described and some cost minimization Well for a given set of fluxes you run in a convex optimality problem that gives you the enzyme levels and the metabolite levels So basically then you have a solution but With this method you need to know the fluxes and Okay, in order to get the fluxes you probably need to run FBA But FBA like a serious Variants of FBA need to know about enzyme efficiency and for enzyme efficiency you need the concentration so you need ECM and then you're in the situation where in order to run FBA you need ECM in order to and in the end You you are again at the point where you need to optimize all all variables at the same time and This the same problem would also occur if you split a network into several parts You want to optimize something in one part actually for us for a good optimization here You would need the boundary conditions that come from another part. So you need the optimal solution It's always Always a problem and So now how can we actually Optimize all the variables at the same time. So this little picture shows just how how the different variables are Connected for just one single reaction Show this so in a in one reaction. We assume we have three variables the substrate level the flux and the enzyme level Let's assume the product level is known fixed and given Then we have the rate law and the rate law Defines a relationship between these three between these three variables so given Metabolite level and the flux you can solve for the enzyme level so you can plot the enzyme level as a function of the two and then you get this surface and Any feasible state is a point on that surface now you can try to make sense of that surface so you know the flood enzymes Increase positive Linearly with the flux so you know the surface has straight lines here And if you look at these curves you look at from from the top at this given enzyme level you see the rate law You see the Michaelis Enten curve and so on but if you now think of this for a whole network you have three times or like Roughly three times the number of metabolites or reactions as a dimensionality. So this surface becomes really really complicated so it's important to to understand its structure and one thing that is striking here is That the surface if you project it down It only covers a part of the of this metabolite flux space So some parts Can lead to feasible states and some can't do you have an idea how what's what could be the reason? Certain concentration and flux combinations don't work This is thermodynamics. Yeah, so here in this example, there are certain metabolite levels where Actually, actually this arrow should go in that direction. I think I think it's wrong I know it's right. It's correct So there are certain certain metabolite levels high metabolite levels that only allow for positive flux because of thermodynamics so you are in this region and Certain metabolite levels only allow for negative flux. So you're in that region and here the concentrations would not be compatible with the flux direction so just by looking at the at the thermodynamics between concentrations and flux directions you already know the pattern on to which the the Surfaces projected and thermodynamic flux analysis does exactly this it it operates in the space in this in this Projected space which distinguishes between feasible and unfeasible regions and you can you can optimize and sample fluxes and metabolites at the same time Just okay But now how can we navigate? How can we optimize in this high-dimensional space? so in theory one possibility would be to Think about this projection first so here we are in Flux space only But we already we already Analyze the system and we know which patterns of flux directions are feasible at all can be feasible at all This is I'm I will not say how this is done This is just assume that we we can distinguish them So we have some authors in flux space that are feasible and others are not now for each feasible author We know constraints on the fluxes in metabolite levels So we get a feasible polytop in flux space and a feasible polytop in metabolite space and now here we are free to To choose so any point that we choose here will be compatible and any point we choose here will be compatible and Knowing the two we can also solve for the enzyme profile using our enzyme demand formula So this is in principle a scheme for navigating in the space of all possible metabolic states in Like a secret in an iterano like a layered manner first thinking about the flux directions and then navigating sampling Optimizing in these two subs subspaces and the the problems here are relatively simple once For example, once you chose a flux distribution Optimizing in metabolite space is a convex problem once you chose a metabolite point Optimizing in flux space is a linear problem. So you break this complicated problem down into more more simple simple problems and with a lot and Make a water Frank Röchmann we Proposed one way of actually finding the optimum So in this case we have a layered Optimization we first think of a search in flux space So this is this is here. This is a depiction of flux space So we pick a point in flux space and then for this point in flux space We do an optimization in metabolite space An optimization for minimal cost for minimal enzyme cost and that's This part from here to here is something that a lot showed in his in his talk But there was a triangle with blue colors inside. Maybe you remember. So this is exactly his blue triangle Once you have the optimal metabolite profile it comes with with an enzyme cost and the enzyme cost now you can Think of it as them the the minimal cost the minimal enzyme effort that comes with this flux distribution And in principle you could do this for any point on that on that triangle So any point would have its own enzyme cost that comes from this underlying optimality problem now if you plot this as a function and You absolutely welcome, but which network do you use how you define everything or this is for a toy model those uses for a coli and In okay, so in the paper we used a small model of central metabolism with about 30 reactions The thing that I will in a moment. You will see why not bigger. Yeah so We could or it was already more or less known that this function that you get from this procedure as a concave function Meaning it's it's curved like this and you want to minimize So where are the optimal points? Not in the middle, but potentially on the on the boundaries and that means that Even if we don't know the function in detail, we know that the optimal point can only be one of the corners and The corners with some further simplifying assumptions are elementary flux modes so basically the The whole procedure is like this We take our model We need to know the kinetic constants We enumerate all the elementary flux modes For each of the flux modes. We do this calculation Get the enzyme cost and then we pick the one that has the lowest cost and we predict this as the global optimum Perhaps I'm wrong, but the elementary flux modes are exponentially large. I think that's why we cannot we cannot handle whole network For for our model with 30 reactions. We had about a thousand flux modes, but yeah So that limits the the applicability of this Okay So all this Was more or less like an introduction just to make the point that one can do such optimizations but it's complicated and It won't work for a large system And if we know that to understand a small system we in principle, we need to understand the large system There's there's a problem one more question for myself. Yeah how Credible is the fact that you have all the kinetic parameters for a real problem In what sense in the sense that I Actually don't know but the biologists can give you which are the kinetic parameters of each reaction in the metabolic network Of a few of them or it's reasonable to assume that they're all the same So they are they are not known some of them are known and we use Special special tricks to find a parameter set that we believe makes sense and So the I think one test is if we know the flux distribution Experimentally and we run this procedure with the kinetic constants the model with the kinetic constants can we make good predictions of metabolite and enzyme levels or not and we did that in the in the previous paper on Enthusiasm cost minimization and we got okay predictions. So they were not like in physics but they were they were kind of okay and each single K cat value that is wrong will give you a wrong prediction and Yeah So I think what what we did is not is not completely off but Yeah, since the kinetic constants have huge error margins Especially K cat values that you don't know that you have to invent This will all reflect in the enzyme cost predictions and this will also reflect in the EFM That you choose in the end. So it could be that let's say one K cat value in your model is extremely small Then the corresponding enzyme level will be extremely large and then all the EFM's that use this enzyme will be This this predicted. Yeah, okay another good reason Another good reason to say that a global modeling is it can have its problems Okay, no Summary of this first of all, it's very Difficult to to get the predictions second. It's also a bit unintuitive. You have this huge black box This model then it spits out some solution Can we understand why it's this and not another solution? I'm not clear and Also the network may not even be known in fully maybe we only know part of the of the network and we've missed some parts maybe if we added them then Whatever, maybe maybe the solution becomes different and now the question is can we can we do something local and The idea behind it is very different from from this the idea is really You think of an enzyme as an investment So there are already the enzyme levels and you ask if the cell invested a bit more in this enzyme What would it gain from it? How much would it cost? How much would the cell gain? And the idea that is that in an optimal state what you what you invest and what you gain marginally so the difference Should be should equal out to zero if you gained more than you invest And you should you would probably you would probably invest more so the state that you were looking at was not optimal yet You could you could improve it same if you gained less than you Invest then also the initial state is not optimal. So the logic is in an optimal state The cost and benefit of small further changes should always be Yeah cancel out and This is now the logic Seeing enzyme and seeing the proteome not necessarily as amounts which there are amounts of course, but also as investments in terms of A large chunk of the proteome will have come with a high cost for the cell and It indicates that probably these enzymes also provide a large benefit There's lots of things that can be argued about this and I think that question if it comes up now for you Maybe we postpone it to the end because that's really Tricky and the aim of this is to go basically in a circle of Thinking so we first started with molecule properties kinetic constants of enzymes Then in order to see what they are doing we arrange them in networks Then we know about the data we set up our Optimality problems in flux space and then here we are in a point where we don't understand things anymore It's really unintuitive and now the the idea is really to go back to this network To this local network picture where we can look at things one by one try to figure them out by ourselves and we do this by Placing values on the different variables on the enzymes on the metabolites and so on and Yeah, and then we are we have basically again a local description of molecules But in terms of values that encapsulate knowledge about about these solutions okay, and first of all an existing Not theory, but an insight that is has very much the same spirit So first of all In kinetic modeling there's a notion of control coefficients control coefficient tells you if you increase the Locally one reaction rate or you increase a parameter that has the Has an increasing effect on on that reaction rate You wait for a long time. You look at the changes in steady state and you ask How strongly the different fluxes and concentrations in the system change so you perturb here And you wait for a while and then you look at you look at something else and the flux control coefficient tells you this for Looking at fluxes you perturb this reaction. You look at the flux Naively one could say okay if I perturb a reaction and the reaction will just be higher but this is not the whole The whole truth because if the reaction rate goes up Then there will be an accumulation of product and depletion of substrate and the whole system will counteract this effect So the actual change in flux will be different from the Initial change that you applied and the flux control coefficient it describes exactly this and There's a result has been found already like in the 70s. I think for an example, but it's been generally shown by 200 Heinrich and edda clip that in an optimal state The enzyme levels are proportional to the flux control coefficients of the enzymes themselves so even in an optimal state of Theoretically if an enzyme has high flux control, then it should also have a high abundance and the opposite and This is this kind of practical because if you know the Control coefficients, you can you already know the the amount if you and We with a lot we recently generalized this to a to a different setting so in the original setting the underlying optimality problem was Optimizing maximizing the flux in the system at with a cap on the enzyme levels. I get the fixed total enzyme then yeah Max maximizing the flux we we generalized this To systems where there's a cap on a weighted sum of the enzyme levels and metabolite levels so it's a general Density constraint that takes into account both of them and then we got a formula that contains flux control coefficients and also concentration control coefficients yeah, and now How can you make sense of this so the way I would make sense of this is really just saying it's a cost benefit balance the enzyme level Tells you something about the investment of the cell how much the cell invests in this reaction and the flux control coefficient in a way tells you how much the cell gets from in Increasing this reaction. It's a bit simplified the explanation, but in fact, this is an Is an example of a cost benefit a relation as you will see them later And how can this be shown? So this is this is yeah gain of the flux How do you you'd like to put these costs into I'll show how it is derived and maybe this answers the question. Yeah, so Let's and we'll see also how First Lagrange multiplier comes in so how is this derived we assume a Metabolic pathway it can also be done for a metabolic network, but pathways simpler to to explain so we assume this pathway with enzyme levels and We say that if some of the enzyme levels is fixed and given so this is a constraint now given the enzyme levels and the external a fixed external concentrations We can solve for the flux and now if we play with the enzyme levels We can make the flux higher or lower and we want to find The enzyme vector a level vector that maximizes the flux Okay, so In order to do that Constraint optimization we define a Lagrangian which is the flux itself plus or minus minus is Convenient here a Lagrange multiplier times the the constraint now we Take derivatives of the Lagrangian with respect to the individual enzyme levels and we get the Derivative between the steady-state flux and the enzyme level Minus the Lagrange multiplier because each of the enzyme levels appears only once in that some and that Derivative is What is called the metabolic response coefficient in MCA and it can be written as the metabolic control coefficient times The rate of the reaction divided by the enzyme level something Trust me on that. That's just a fact And yeah, it's an approximation or the fact an approximation for the derivative. No, it's So can you basically you can see it as a definition of C here. It's not an approximation It's the definition of side. It's exact and there's also formula for C, but this is exact See it's a complicated. It's a it's a global sensitivity. It's not a simple No, it does not It depends. Yes, it depends on the flux So this will it's a function of the fluxes in a given in a given state where you are already These C's will be defined And you can you can compute them, but if you if you are in different states The C's will have different values. Yeah in that sense they depend on the flux, but they are not here we don't We don't have to Differentiate by the flux again, so it's not important that they are a function of the flux because we already differentiated This is not a constant it's something that it's not a constant if you if you look at It depends on the fluxes. Yes, you know, you're just interested in the values of this in the state that you're looking at so and Okay, now You know that There's only one lambda for all the react so there's this holds for every single reaction But lambda is always the same value so if this is equal to zero and The you'd also know that the reaction rates are always the same as the flux in the state that you're looking at So V is also constant you already see that the two must be a proportional. So if you if you solve this for C over Ensemble level you get a constant and you know that the two are proportional along the chain Did that answer your question yes, yes Yes, you can do the same thing. It's you It's very easy. You you put different cost You knew the cost. No, I was just wondering how you're gonna know but I guess I still have also a little bit his problem because now you're gonna say at whatever solution you're gonna arrive at If that solution is a locally optimal one Right so that we only talk about locally optimal ones. Yes that the derivatives with respect to the enzyme concentrations vanish of the flux Then the enzyme concentrations must be equal to to this coefficient going to happen It's in this in this kind of model. It's not going to happen that an enzyme has no No effect on the flux it would happen In in models where you have irreversible Reversible reactions you can have all kinds of crazy things and then this in fact breaks down. So you're you're touching on phrase my question What have you learned because it really just seems to me that you've defined this control coefficient to basically just satisfy this equation No, not at all. The control coefficients are defined are defined They are defined as the as this derivatives and then scaled by by V and EL so you can define them as D lock V over D lock Yeah, that's the definition this and it has nothing to do with this optimality problem It's just a it's just the sensitivities of the fluxes with respect to changes in the in the enzyme levels Just a dynamical property of the system Yeah, a linear No, it's not a linear relation between fluxes and enzyme levels. This is a derivative. Yeah Okay, I I would try to it's does this is you can say it's a trick in MCA Let's say you have an enzyme level that has an effect on the flux on this reaction rate and Then you measure the flux here Let's call it J for distinguishing it from the from the V as a as a function of Concentrations and E Now there are two kinds of sensitivities. There's There are the so-called elasticity's that describe direct relationships so between The enzyme and the rate you have an elasticity that is defined as DV over DE and Because this rate law is so the enzyme Is a pre-factor in this rate law? You can for this elasticity, you know that it's a it's a ratio. It's because enzymes are rights and and enzymes are assumed to be proportional But this is very simple. It's just be like a something between these two What you actually want to know is the relationship between e and the flux after a rearrange complete rearrangement of the of the system and This you can define. It's called the flux response coefficient You would define it as DJ over D e Let's say for the L's enzyme and now J is really the solution of a problem that finds you the stationary state in this in this yeah, and this Because you know that the enzyme level only affects This reaction and nothing else it is convenient to Split this into a part That looks like that and a part so this TV and A part that looks like looks like that. So here that describes how the enzyme level changes V This and you can see this as a matrix where all enzyme levels and always appear and then a part this part That shows how a change in V will eventually Change the stationary fluxes and this is called the control coefficient. There are formulas for this so you can Yeah, you you can write this down So if this is an identity matrix another elasticity matrix or geometric matrix, so So this formula if you know your system Can can tell you about about the control coefficients, but we're not going there Here we just want to say that in an optimal state whatever the control coefficients are They will always be proportional to the enzyme levels Which is very beautiful. You don't you don't have to know them You know, whatever they are they are proportional to the enzyme levels because you're in an optimal state Yeah, it seems to me much more intuitive to say if I'm in a local optimum of the flux Under the constraint that the sum of the enzyme levels is the same Yes, any kind of vector that changes by a tiny bit the enzyme levels, but such that there some Remains the same. Yes in order for such change to not change the total flux The derivative of each flux with respect to enzyme level must be the same exactly That's what you get. Yes the same statement. No one that you it's exactly what you get So if you try the same, no, I just think yours You're writing this constant as if something is a constant, which is not a constant. No, let so what from your logic What you get is that all the response coefficients must be the same these guys so they are all the same constant and this const tells you that these two guys Must be inversely proportional and this is exactly v over e. I Understand that yeah, I I understand that but But these the thing that you call these coefficients Right, it's not me. It's they have been invented in the 70s field Coefficient it's the 50 50s anniversary of the of these coefficients this year 100 year anniversary of the p-value and so but But the problem is that if it was true that this is sort of an intrinsic property that is always the same No matter the state of the system. Yes, then I and then I understand that it's a powerful result But right now we have no idea how this how these constants may change. Yes as the fluxes in enzyme level may change Well, you can say this in any engineering problem You could say take any system with its Alice with its sensitivities If if the sensitivities were always the same then it would be a powerful theory if they change. I'm not so interested So that's that's what with M times a is is helpful because M is a constant is not changing in every condition Yeah, I don't know for me, it's the interesting M to be and there's a there's a whole field that that builds on it, but yeah So there's more things that you can do with them there are there are beautiful summation rules that that tell you more general things about these this Sensitivities how they are related exactly with the network structure what so summation rules that connect them only to the stoichiometric matrix and Other rules summation rules that connect them to the elasticity is only and that even if you don't know the control coefficients You can know something about their sums and you can for example the sum of the flux control coefficients must be one that means Your you know that at least one of them always must must be positive then you have other control coefficients for concentrations and Here the summation theorem gives you a sum of zero, which means that whenever one Ensign has a positive control Exactly, you could you get that from the theorem of homogenious functions. Yeah, I mean It's if it's obvious That's already a good thing, but to make it into something that helps you compute stuff analyze stuff say maybe which What is a good drug target or not in a given system? I think that's that's another That's a worthwhile endeavor for me. Yeah Okay, so here this was just to say From a simple optimality problem. We can get this cost-benefit relationship And then okay, this is a bit is a bit detail So a lot and I also had a closer look at this we use the second theorem that I just mentioned That control coefficients I In a specific relationship with the elasticity. No, it's another theorem. I didn't mention it. Sorry another theorem that connects control coefficients and elicities and that tells you for each metabolite a relationship between the control of the previous and the following enzyme on the one hand and the elasticity of that between that metabolite and the reactions on the other hand and From that Using this and also using the result for enzyme levels and control coefficients what we get is a relationship between the Investment in this enzyme and the investment in this enzyme the ratio between this investment and this investment Must be the inverse ratio between the effect of this metabolite on that reaction and the effect of the metabolite on that reaction That's again, it's very general if you use if you accept this optimality problem again, the problem is The elasticities are not fixed numbers. So you could say if I don't know them in general. I'm not interested The elasticities change depending on the optimal state But once you know them you directly get the ratio of the enzyme levels and with the two rules together This and the summation theorem You you can basically Yeah With relatively little knowledge Only of the elasticities in the state between metabolites and fluxes You can know the entire Profile of Optimal enzyme levels directly Yeah, without solving any any optimality problem anymore explicitly because it's a very general rule. Okay Now this was the Ten minutes. Okay, so I think I'm maybe a quarter Which is great, which is which is fine, so I think I'll go a bit More quickly just give some some Comments here and there the next after this we We think of cost and benefit of enzymes we can also think of cost and benefit of individual metabolites can also be defined through like branch multipliers or control coefficients and then we get an kind of analogy to Labor value theory, it's in like an old very old-fashioned Way of looking at economics marks was very like it was very important for marks and There are people still think that there's a value in in looking at this Also for empirical as the understanding things empirically, but it's very old and the idea in labor value theory is that The value of something that you produce is basically the condensed labor that was necessary to make this thing so you start with Materials that you get for free Somehow that's already a problem, but let's assume you can collect things for free And then somebody works on this material for an hour and then what is produced is worth one hour of labor another one works for two hours and then and then This accumulates and the end product is Yeah Contains a value that is basically the embodied the embodied labor in this in this object and the theory that I Propose is a bit similar for metabolic systems Here I define values for the different metabolites and they don't like fall from the sky, but they can be defined Precisely from optimality problems and I get a balance Relation between the val in each reaction between the value of the substrate the value in the product The flux and the enzyme investment so the enzyme investment could either be the enzyme level or if as you said we We weigh the different enzymes differently It would be the weighted the weighted enzyme level in this reaction And So the relationship is very simple. So this would be the enzyme level This would be the weight of the enzyme level. So together is the weighted enzyme level. It must be the flux in the reaction times the difference in value and So this will always be positive because every enzyme costs and can only have positive expression levels So that also means that the delta In the value must have the same sign as the flux. So the values always increase along the flux direction and This only holds for expressed enzyme. So in the case that an enzyme is not expressed. He is zero Then this doesn't necessarily hold and so you can see the enzyme investment as something like the invested labor You can Yeah, see the the flux as a as a speed of like how much items are produced per time and the values are then Yeah, basically they embodied the equal to the embodied labor How can we define this? I think since I don't have a lot of time I will go very quickly So it can be defined based on different optimality problems. This is just one of them You have a metabolic system now. You don't have a cap on the enzyme levels, but an actual cost function So you have a cost that is a function of the enzyme levels You have a benefit function that is a function of the fluxes and metabolite levels and The fitness is the the difference of the two benefit minus cost now you can Look at the fitness as a function of one of the enzyme levels as you the other enzyme levels are fixed You'll screen the enzyme level then you would get an increasing cost and an increasing benefit But the benefit would level off because if you have more and more of this enzyme it would be Less and less important to have even more of it because then other reactions become more limiting and in that case the difference between the two The fitness would have some optimal point and this optimal point is exactly where the two slopes are the same Because if the slopes are the same you take the difference you get zero and that's the that's the condition for local optimum and You can do the same thing Also in for the log The log enzyme levels and you get a different picture Okay, so the optimality problem is this maximize the fitness as a function of E where the benefit is some of Positive the function of the fluxes which are a function of E Minus some cost function for the concentrations as a function of E minus the actual cost function for the enzymes you can see this as a metabolic objective minus the enzyme cost and So now you can either Set the two derivatives like you The derivative of this must be zero so the derivative of this must be the same as derivative of this so you can either Equate them directly and then you have a An equation between This derivative which has to do with concentration with control coefficients If you think of it and you apply the chain rule and this derivative, which is just the pre-factor like the different enzyme costs Or you can take logarithmic derivatives, which basically means you multiply with E and Then you get something that looks a bit more like like the result by clip and Heinrich Well if all enzymes cost the same across the same then this would always be the same So you have E on the one hand and then you have something on the on the other hand that is complicated and composed of control coefficients Now the interesting thing is This expression is not really is it's really complicated How you can make sense of this? expression and I claim that it can be Written okay, so I jump over this I Claim that this can be written in a local way So for a local expression You need to define the values of metabolites and you do this again by sensitivities So you consider the metabolic system And you assume that the objective metabolic objective scores production of the end product Now you ask what will happen to the system if you Give the cell for free a certain influx of one metabolites It's just a hypothetical influx. That's basically means you break the mass balance condition in this point You ask if this influx existed and everything was rearranged how much would that change your your benefit the production of value and So you can just this express this as a again as a control coefficient between this variable and this and Now I want I won't show how but it's this whole expression if You think of it in terms of control coefficients. You apply the theorems that you know it can be rewritten exactly as The difference between the values of the two subsequent metabolites Times the flux between the two metabolites And so you have this you have this local relationship that looks pretty much like in labor value theory Okay, I think in the interest of time I will probably jump over many things So this can you can show it with Lagrange multipliers and there are general rules that look a bit like Kirchhoff's rules in electricity You have this analogy to chemical potentials Which can also be derived from an optimality principle and therefore satisfies a very similar relationship. So Yeah, on the left you see thermodynamics where fluxes go from high chemical potential to low chemical potential on the right You see the same thing where fluxes go from Low value or economic potential to high economic potential And then you can use this in flux analysis and FBA as an as an additional constraint so you can have type of a for example flux sampling with three kinds of laws the first is mass balance like in normal FBA the second is thermodynamic constraints like in energy balance analysis or other kinds of thermodynamic FBA and the last would be a new constraint that comes from Yeah from this analysis based on kinetic underlying kinetic models and optimality problems and that says that For a flux distribution to exist in an optimal state. There must be a pattern of values That is compatible with exactly this flux distribution and this this condition you can use to discard certain certain cycles like you can discard cycles in thermodynamics because They would not dissipate any Any Gibbs free energy in the same way you can discard other cycles and in some cases also the same cycles based on this economic principle Because you cannot the cycle would mean that you go from in a cycle from a low value to a high value and Then you are you're at the start point and it doesn't it doesn't fit. Yeah So this is this okay, I jump over this Here this is just what I said before If you look at this balance equation closely, you can interpret the different terms as value flowing in and flowing out So in that sense because you know the the investment in the enzyme must be exactly the same as the the difference in value that you get Between the two metabolites times the flux so you can interpret this as one term one value term coming in here one value term in Coming in here and another one going out here and they need to balance so in that sense you can see a metabolic pathway as As a machine that that takes value from the substrate and Value being a derivative of the objective function, but if you think that consumption of substrate is costly While you flowing in and then value flowing in from the enzymes adding up adding up adding up and then you have the out the the value flowing into the Into the end product that determines the the benefit Sorry, so it can be it can be shown with the formulas, but I don't have the time and Yeah, so this is just the analogy to that I mentioned before to to biomechanics The the structures of bones the outer structures and inner structures have been explained by a feedback mechanism whereby Cells in the bone can sense stresses and can either add or remove material and This is also is a good way to to optimize the shape of in engineering shape of parts to make them stress resistant like here you analyze where there are high stresses and you add material you get an optimal shape and so I take this as an analogy to an hypothetical process where one starts in a non-optimal state of a metabolic system one looks at the balance Relations in in each reaction and one checks how much the balance reactions are Violated at the moment in the given state and then in order to improve the state one would Have a feedback that says if there's a like too much Too much on the investment side and too little on the value production side in a given moment then investment should be reduced or The other way around and the hope would be to then get at least to a locally optimal state So not global optima are not not considered here as a as a question It's hypothetical. Yeah. No, it's just fiction. It's just too No, it's just a different way to think about the system without a practical application in mind It's what what a cell ideally would have to know to take the best decisions And then you maybe you can compare this to information that the second actually have and you see the problems the missing information at a second actually have Yes, I don't believe a second ever manage a measure this tension. It's really like in like in Utopian Solution to which actual solutions could be compared. Yeah Okay, um to Comments I so I followed this on a very Sort of course level because I was not ready for them But I think the first that could be a mechanism where this cell senses this and this is not physiology But evolution right you can like make different Genotypes that I don't know rays and lower enzyme levels sort of like adding material here to this building part And then you let them compete So I think evolution could be a mechanism that actually finds a similar solution and then this entire loop of sort of adding Things and then testing and reminds me of a statistical method. That's called expectation maximization Then sometimes used to estimate Parameters of statistics statistical distribution when you don't have the full data Yeah, so then you use your current best estimate to generate the missing data fit The parameters using likelihood to generate new data and then you get into the similar loop. Yeah, and in these algorithm They converge empirically when people use it And I think a big question for this is also what it converge If you would apply it it will Think it will converge because enzymes as the level goes up. They they're control decreases. There's they yeah, there's some Diminishing returns thing that you get automatically in in these systems I think it would work and for the first thing that you say I think that the tensions could probably be interpreted as selection pressures Yeah, there's a connection basically You could think of it as a simple simple idea of of a selection pressure and you can even make it very explicit So, I mean we can talk about it more. Yeah. Yeah, I think That's it This is the repetition of the first slide these are some papers and some preprints well, I wrote down some of the stuff that I Told you about Yeah