Optimization of metabolic states

Optimization of metabolic states Andreas Kremling1, Wolfram Liebermeister2, Elad Noor3and Meike T. Wortel4 1Systems Biotechnology, School of Engineering and Design, Technical University of Munich, Germany; 2Université Paris-Saclay, INRAE, MaIAGE, 78350 Jouy-en-Josas, France; 3Department of Plant and Environmental Sciences, Weizmann Institute of Science, 76100 Rehovot, Israel; 4Swammerdam Institute for Life Sciences, University of Amsterdam, Amsterdam, the Netherlands Abstract Cells in a well-mixed and nutrient-rich environment can be expected to experience selection on their growth rate. Such cells optimize their metabolic state to achieve a high growth rate. Metabolic states that lead to a high growth rate are states that realize a maximal biomass production rate at a minimal enzyme cost. Metabolic states that optimize a specific flux at minimal enzyme cost are called enzyme-efficient metabolic states and in this chapter we refer to them as optimal metabolic states. The calculation of optimal metabolic states is facilitated by the result that, in models without further constraints, the flux distributions in enzyme-efficient states are Elementary Flux Modes (EFMs). This result allows for an algorithm to find enzyme-efficient states by: 1) Enumerating the EFMs, 2) calculating the minimal enzyme cost per EFM, and 3) choosing the one with the lowest enzyme investment. This algorithm finds optimal metabolic states for larger models which cannot be optimized by ’brute force’, but are still small enough to enumerate the EFMs. Finding optimal metabolic states uncovered the effect of changing external nutrient conditions: As growth conditions are changing, the optimal flux profile either changes continuously (and metabolite and enzyme concentrations change continuously as well) when the same EFM remains optimal, or fluxes change discontinuously together with metabolite and enzyme concentrations when a different EFM becomes optimal. Contributions: This chapter was drafted and written by Andreas Kremling, Woflram Liebermeister, Elad Noor, and Meike Wortel. It was initially discussed with Jürgen Zanghellini and later reviewed by David S. Tourigny and Hugo Dourado and discussed with Stefan Müller. We thank Coen Berns for comments. Keywords: elementary flux mode (EFM) - enzyme cost minimization - enzyme-efficient state To cite this chapter: A. Kremling, W. Liebermeister, E. Noor, M.T. Wortel. Optimization of metabolic states (Version October 2025). doi: 10.5281/zenodo.8164468. Chapter from: The Economic Cell Collective (2025). Economic Principles in Cell Biology. No commercial publisher | Online open access book | doi: 10.5281/zenodo.8156386 The authors are listed in alphabetical order. This is a chapter from the open textbook “Economic Principles in Cell Biology”. Free download from principlescellphysiology.org/book-economic-principles/. Lecture slides for this chapter are available on the website. c 2025 The Economic Cell Collective. Licensed under Creative Commons License CC-BY-SA 4.0. An online open access book. No publisher has been paid. doi: 10.5281/zenodo.8156386 1 Chapter overview ◦Optimal metabolic states in this chapter refer to enzyme-efficient states, which are metabolic states that realize a given objective flux at a minimal enzyme cost. ◦In models without further flux constraints, flux distributions of enzyme-efficient states are Elementary Flux Modes (EFMs). ◦Elementary Flux Modes can be used to find enzyme-efficient states in networks that would be too large to optimize metabolic states "by brute force". ◦Biomass per enzyme efficiency can be converted to cell growth rate by simple approximate formulae. ◦The Elementary Flux Mode that is realized in an enzyme-efficient state depends on the external conditions. ◦As growth conditions change, either the flux profile changes continuously (together with metabolite and enzyme concentrations), or fluxes change discontinuously, implying jumps also in metabolite and enzyme concentrations. 7.1. Introduction In a simple economic picture of cells, we assume that cells adjust their metabolic state in each environment to obtain a maximal fitness advantage. This may be impossible in reality, but it remains an interesting question what this best metabolic state would look like, according to our knowledge of cells. So what is the best metabolic state overall (comprising metabolic fluxes, metabolite concentrations and enzyme levels)? What pathways should a cell use, which enzymes should be induced or repressed, and how should this change in a new environment? To answer these questions, we need to remember that all metabolic variables (fluxes, metabolite levels, enzyme levels, and enzyme efficiencies) depend on each other. Physically, fluxes depend on metabolite concentrations through kinetics and enzyme regulation (e.g. competitive inhibition) and metabolites are produced and consumed by the fluxes until a steady state is reached. Hence, if we think in terms of cellular economics (treating enzymes as control variables), then all metabolic variables must be optimized together. In the previous chapters we saw some ways to predict optimal metabolic fluxes, metabolite concentrations and enzyme levels separately: in Flux Balance Analysis (FBA, Chapter 5 in [1]), we optimized fluxes by maximizing an objective function (typically biomass) while in Enzyme Cost Minimization [2,3] (Chapter 6 in [1]) metabolite concentrations were optimized by minimizing cost (or, equivalently, maximizing the enzyme efficiencies). Each of these methods is based on a strong assumption: FBA requires measured flux ranges and/or apparent catalytic rates and assumes enzyme saturation effects can be neglected, while enzyme cost minimization requires a given flux distribution. But what if we don’t know any of the variables in advance? How can we predict all of them from first principles? Before thinking about this, let us briefly step back: what do we actually mean by an “optimal state”? What quantity should be maximized in metabolism? There could be very different aims (e.g. production in biotechnology, versus number of offspring and survival in a wild-type cell). However, in both cases an important aim is cell growth – or at least, avoiding strong growth deficits. Below we will see that cell growth depends, to a good approximation, on biomass/enzyme efficiency, that is, biomass production per total enzyme invested. Hence, whenever fast growth is important, cells should maximize this efficiency. In conclusion, we will consider the following optimality problem: maximize biomass/enzyme efficiency, defined as the production flux per invested enzyme with respect to all metabolic variables (metabolites, enzymes and fluxes) and under all constraints (steady state, enzyme kinetics, etc.). Solutions to this problem are considered optimal states. 2 7.2. Enzyme-efficient metabolic states use elementary flux modes The optimization problem in this chapter is to reach maximal objective flux with minimal enzyme investment. The biological interpretation is that this would lead to the highest growth rate, because it optimizes the ratio between gains (fluxes) and costs (enzymes). When we solve this optimization problem with mathematical tools, it is convenient to either find the minimal enzyme investment for a certain flux, or the maximum flux for a fixed enzyme investment. Although one could think of different biological explanations for those two ways to state the optimization problem, mathematically they are equivalent. For the outline of the proof that optimal states are elementary flux modes, it is convenient to fix the objective flux to an arbitrary value (we choose 1) and then minimize the enzyme investment. This leads to the following optimization problem over the fluxes (v), enzymes levels (e) and internal metabolite concentrations (s): minimize v,e,s r X i=1 hiei(7.1) subject to: N·v=0steady state ∀i:vi=eiκi(s)enzyme kinetics e,s≥0positive concentrations vr= 1 fixed objective flux s≤smax metabolite bounds where hiare the weights, Nis the stoichiometry matrix, iis the index of the reactions (ranging from 1 to r), with the last reaction (with index r) representing the objective. This optimization problem states that by adjusting the fluxes (v), metabolite concentrations (s) and enzyme concentrations (e), the total cost (sum of costs – hiei– for every reaction) is minimized, while keeping the objective flux constant (any arbitrary constant can be chosen, here we chose 1). The weights (hi) can be thought of as the size or production costs of the enzymes (measured, for example, in molecular weight or gene length) We require certain constraints: (i) the metabolic network needs to be in steady state to avoid built-up of intermediates, (ii) enzyme kinetics – the flux of each reaction (vi) has to be equal to the enzyme concentration (ei) times a metabolite dependent (e.g., saturation) term (κi(s)), (iii) all enzyme and metabolite concentrations have to be positive, (iv) the objective flux is equal to 1, and (v) the metabolite concentrations are within their given bounds. The latter constraint is optional and is mostly necessary when dealing with irreversible kinetics. Reversible kinetics usually lead to bounded metabolite levels because very high concentrations of products inhibit the reaction that forms the products. In this section, we will explain why the optimal state is reached at an Elementary Flux Mode (EFM). One important starting point is that, as we have seen before in Chapter 4 in [1], convex optimization problems with only positivity or equality constraints (no other inequalities) lead to an optimal solution at an extreme point of the feasible solution space, and those extreme points are Elementary Flux Modes. However, the optimization problem (7.1) is not convex, mainly due to the hyperbolic dependence of reaction rates on the concentrations of metabolites (κi(s)is usually not linear in the internal metabolite concentrations). There are several ways to prove that the solution of this optimization problem is an EFM, of which some are outlined in the papers by Wortel et al. [4] and Müller et al. [5]. Here we will outline a proof by contradiction: assuming a solution to the optimization problem that is not an EFM and showing that this leads to a contradiction. Theorem 1. The flux distribution that maximizes an objective flux over the total enzyme cost in a metabolic network without additional constraints is an Elementary Flux Mode. 3 v2 v3 v1 E2 E3 E1 flux space enzyme space E2 E3 E1 enzyme space E2 E3 E1 enzyme space optimum optimum optimum metabolite set I metabolite set II metabolite set III increasing enzyme cost increasing enzyme cost increasing enzyme cost Figure 7.1: Translation from flux to enzyme space retains EFMs as extreme rays – The top left panel shows the feasible flux space with the steady state constraints, all fluxes positive (using splitting of fluxes, as explained in the text, if necessary) and a fixed objective flux. The extreme points here are points where one flux becomes 0 and are elementary flux modes (see Chapter 5 in [1]). Here we show that when we have assumed metabolite concentrations, such as when we keep them at an optimal solution, we get a linear transformation and the extreme rays are maintained. Different metabolite levels, for example solutions to different environmental conditions, can lead to different transformations and therefore different optima (minimal total enzyme), but those are always located at an EFM. Proof. Assume we have some optimal state where the flux distribution is not an EFM. Any optimal solution is associated with a set of fluxes, enzyme concentrations and metabolite concentrations. Now we set the metabolite concentrations to the concentrations of the assumed optimal state. Then all metabolite-dependent terms (κi(s)) become constants, and we return to a convex problem. As explained in Chapter 10 in [1] and Figure 7.1, the optimum of this problem (now in terms of enzyme concentrations and fluxes) is a flux distribution that is an EFM. But this contradicts our initial assumption that the optimal state from which we took the set of metabolite concentrations was not an EFM. The proof by contradiction shows that the optimal state must be an EFM. 7.3. Enzyme-efficient states in an example network To illustrate the proof, we study a simple network representing growth on glucose and pyruvate that we have seen previously in Chapter 5 in [1] (Figure 7.2). We use Gand Pfor glucose and pyruvate in the equations, we use the subscript ex when a metabolite is extracellular and square brackets to denote a concentration. For the use in this 4 Box 7.A Kinetics of the example network The detailed kinetic equations for the example model (Figure 7.2) using the factorized rate law (see Equation (7.2) and Chapters 3 in [1] and 6 in [1]) are: v0=e0·k+ cat,0· [Gex]/KGex 1 + [G]/KG+ [Gex]/KGex ·1−e∆rG00/RT  v1=e1·k+ cat,1· ([G]/KG)([ADP]/KADP) 1 + ([P]/KP)([P]/KP)([ATP]/KATP) + ([G]/KG)([ADP]/KADP) ·1−e∆rG01/RT  v2=e2·k+ cat,2· [P]/KP 1 + [Pex]/KPex + [P]/Kp ·1−e∆rG02/RT  v3=e3·k+ cat,3· ([P]/KP)([ADP]/KADP)([O2]/KO2) 1 + ([CO2]/KCO2)([ATP]/KATP) + ([P]/KP)([ADP]/KADP)([O2]/KO2) ·1−e∆rG03/RT  v4=e4·k+ cat,4· [Pex]/KPex 1 + [Pex]/KPex + [P]/KP ·1−e∆rG04/RT  vBM =eBM ·k+ cat,BM · ([P]/KP)([ATP]/KATP) 1 + ([BM]/KBM)([ADP]/KADP) + ([P]/KP)(ATP/KATP) ·1−e∆rG05/RT  (7.3) Note that Pis a product twice in v1, as v1produces 2P. Note that v2and v4are the same reaction, but defined in the opposite direction. The standard set of parameters we used for the toy model is all k+ cat,i = 10 s−1except k+ cat,3= 0.1s−1, all ∆rG0◦ i/RT =−440 and all KM= 1 mM. For the external metabolites [Pex]=1mM, [Gex] = 0.05 mM, [O2] = 0.1mM, [BM] = 1 mM and [CO2] = 10 mM unless mentioned otherwise. chapter, we add enzyme kinetics to this network. We will use the factorized rate law as in Chapter 6 in [1], but then generalized for nssubstrates and npproducts (also compare Eq. (3.10 in [1]) in Chapter 3 in [1]): v=e·k+ cat ·Qj=1 nssj/KS,j 1 + Qk=1 nppk/KP,k +Qj=1 nssj/KS,j ·1−e∆rG0/RT (7.2) See Box 7.A for all detailed rate laws of the example network. We can simplify this equation by combining the forward catalytic constant, the thermodynamic efficiency factor, the saturation efficiency factor, and the regulation efficiency factor (if that exists) in a function κ(s), which only depends on the metabolites, and not on the enzyme concentrations. We will below write κfor κ(s). vi=ei·κi(7.4) Now we take vBM = 1 and optimize all fluxes, enzyme concentrations and metabolite concentrations to minimize the enzyme costs (etot =Piei), while satisfying the constraints posed in Equations (7.1), for different levels of external glucose and standard levels of the other external metabolites. We see that for different concentrations of external glucose, lead to different optimal fluxes, enzyme levels and metabolite levels (Table 7.1). [Gex]etot v0v1v2v3v4vBM e0e1e2e3e4eBM [G] [P] [ATP] [ADP] 0.01 156.2 5 5 0 9 0 1 54.4 4.4 0 94.4 0 2.9 0.08 15.14 0.05 20.09 0.1 91.3 50 50 99 0 0 1 61.3 11.3 14.2 0 0 4.4 0.13 4.55 0.11 20.09 1 36.2 50 50 99 0 0 1 13.0 8.0 12.5 0 0 2.7 0.60 7.65 0.11 20.09 Table 7.1: Outcomes of the optimization of the example network with standard kinetics, parameter values and external concentrations (see Box 7.A) for varying levels of [Gex]. 5 The table shows that the total enzyme needed for a biomass flux of one decreases with increasing glucose levels, as we expect. In addition, the optimal level of internal glucose increases with increasing external glucose. This is because a higher external glucose allows for a higher internal glucose while still maintaining a steady glucose influx, and a higher internal glucose allows fewer enzymes to drive further metabolism. Moreover, the fluxes of the solutions follow an EFM (see Figure 7.2b). We can now reformulate the problem for only the flux and enzyme levels while keeping the metabolite levels as they are in the table. With the metabolite levels in the first row of the table, we can linearly relate the enzyme and flux levels (with the factors κithat have become constants now we have set the internal metabolite concentrations), and thus the extreme rays of the enzyme and flux space will be equal and EFMs, as pointed out above (see also Chapter 5 in [1] and Figure 7.1). Optimization in this space will lead to the optimal flux distributions following an EFM (see Box 7.B for the detailed calculations). As fixing part of the optimal solution should lead to the same optimal solution, this required the flux distribution of the optimization over all variables to follow an EFM, as was indeed the case. We point out two important aspects, using the network (Figure 7.2) as an example. First, it is convenient to split reversible reactions such that fluxes are always positive. In this case, that means that the reversible reaction from P to Pex is split into the forward reaction v2and the reverse reaction v4, both of which can have only positive flux. This splitting makes sure that EFMs are the extreme rays of the flux space (see Chapter 5 in [1]). This splitting is purely a mathematical convenience; we still assume this to be one reaction in the biological sense, and therefore the kinetic equations of both the forward and the backward reactions will be exactly the same. Depending on in which direction the flux goes, either one of the reactions will be positive and the other zero. Any solution with both reactions positive is infeasible, but minimizing enzyme levels will never lead to such a solution; therefore we do not need to set additional constrains to avoid this. Second, the feasibility of EFMs can depend on external concentrations. In this network, the biomass reaction (vBM) is the objective flux and there are three EFMs leading to the production of biomass: EFM1 consisting of v0,v1,v2and vBM, EFM2 consisting of v0,v1,v3and vBM and EFM3 consisting of v4,v3and vBM. However, if Pex is absent in the environment, the uptake flux v4will always be 0 and therefore EFM3 will not be feasible. 7.4. Calculation of optimal states We can now use the result that states of maximal enzyme efficiency are reached at an elementary flux mode to calculate optimal states in a metabolic network using the following steps: 1. Enumerate the elementary flux modes that include the objective flux 2. Calculate the minimal enzyme for each EFM scaled to an objective flux of 1 3. Compare the EFMs and select the one with minimal enzyme demands Step 1 is possible for relatively large networks, although usually not for genome scale metabolic networks. Step 2 is a convex optimization problem as we have seen in Chapter 6 in [1] and Step 3 is straightforward. These three steps together are called Enzyme Flux Cost Minimization, because it is similar to Enzyme Cost Minimization, but while that is focused on fixed fluxes, Enzyme Flux Cost Minimization simultaneously finds the optimal fluxes, enzyme and metabolite levels. In this section we will show the method on the example network of Figure 7.2. First, we describe the network with the stoichiometric matrix (N) and the concentration vector (s): N=       1−1 0 0 0 0 0 2 −1−1 1 −1 0 2 0 10 0 −100 0−2 0 −10 0 100       ,s≡       [G] [P] [ATP] [ADP]       (7.6) 6 Box 7.B Optimal metabolic states in the example network We minimize the enzyme investment for vBM = 1 with [Pex] = 0 (and therefore v4= 0 and EFM3 is not feasible) for the network in Figure 7.2 (the optimization problem in Equation (7.1)). Assuming all hi= 1, the objective function Pr i=1 hiei=e0+e1+e2+e3+eBM. The constraints vBM = 1 and e,s≥0in Eq. (7.1) are straightforward. The steady state of all internal metabolites (G, P, ADP and ATP) leads to the following equalities (the steady states of ADP and ATP lead to the same equality): Steady state ATP =⇒100 vBM = 2 v1+ 10 v3 Steady state P =⇒2v1+v4=v2+v3+vBM Steady state G =⇒v0=v1 Substituting vBM = 1 and v4= 0 and solving this set of linear equations, we can write all fluxes as functions of v2:v0=v1= 5 + 5 11 v2and v3= 9 −1 11 v2(there is only one independent flux in this system). This means we can draw the feasible flux space on the v2line and we can express the objective function in terms of v2: r X i=1 hiei=e0+e1+e2+e3+eBM =v0/κ0+v1/κ1+v2/κ2+v3/κ3+vBM/κBM = (5 + 5/11v2)/κ0+(5+5/11v2)/κ1+v2/κ2+ (9 −1/11v2)/κ3+ 1/κBM = (5/κ0+ 5/κ1+ 9/κ3+ 1/κBM) | {z } α + (5/(11κ0)+5/(11κ1)+1/κ2−1/(11κ3)) | {z } β v2 =α+βv2 (7.5) The kinetic functions (κi) depend on several parameters (external metabolite levels [Gex],O2,[CO2]and [Pex], catalytic constants, Michaelis constants and Gibbs free energies) and the variables [G],[P],[ATP] and [ADP]. That means that once we have a set of internal metabolite concentrations s, the enzyme levels in the objective function can be written as a constant times the flux: ei=vi/κi, with κia constant. For a set of parameters, αand βare positive or negative depending on the choice of s. It is clear that when we minimize this objective function by adjusting v2, we will always have an optimum at v2= 0 (when βis positive) or v2= 99 (when βis negative). v2= 99 is the maximum of v2because then v3= 9 −1 11 v2= 0, and higher values of v2would lead to negative values for v3. In conclusion, the optimum cannot be at a value of 0< v2<99. If there would be an optimum with 0< v2<99, we can determine sand calculate whether β > 0to find a lower objective value at v2= 0 or v2= 99, contradicting that we started with an optimum. Only if β= 0 there is a range of optima, but this requires very precise parameter values. v2= 0 and v2= 99 correspond to EFMs of this network (Figure 7.2). And with the stoichiometric matrix we can describe the steady state constraints: d dts=N v =       1−1 0 0 0 0 0 2 −1−1 1 −1 0 2 0 10 0 −100 0−2 0 −10 0 100                  v0 v1 v2 v3 v4 vBM            =       0 0 0 0       (7.7) 7 (A) (B) biomass glucose O2 pyruvate pyruvateex CO2,ex 2 ADP 2 ATP 100 ATP 100 ADP 10 ATP 10 ADP v0 v3 v1 vBM v4 v2 glucoseex O2,ex CO2 EFMs glucose respiration glucose fermentation pyruvate respiration (C) (D) 10−210−1100101 0 200 400 600 800 1,000 default value [Gex] etot glc. resp. glc. ferm. pyr. resp. 10−210−1100101 10−3 10−2 10−1 100 101 [Gex] Specific flux v0 v2 v3 v4 vBM Figure 7.2: States of maximal efficiency in an example model – (A) Example network from Chapter 5 in [1] with added stoichiometry. (B) Three elementary flux modes of this network. (C) Calculated enzyme investment needed for a biomass flux of 1. At a very low concentration of extracellular glucose ([Gex]), EFM3 has the lowest cost. But as we move along the x-axis, at around [Gex] = 0.02 there is a switch to EFM1 and later, at around [Gex] = 0.07, EFM2 becomes the one with the lowest cost. (D) Specific fluxes (flux divided by total enzyme) associated with the optimal EFM for different levels of Gex. Note that v1is not shown as it is always equal to v0. The rates show a discontinuity when there is a switch from one optimal EFM to another. Now we find the EFMs (for example with EFMtool [6]). It can easily be checked that the following EFMs (denoted by vectors f(i)) are in the nullspace of the stoichiometric matrix: f(1) =            5 5 0 9 0 1            ,f(2) =            50 50 99 0 0 1            ,f(3) =            0 0 0 10 11 1            (7.8) 8 (A) (B) Fixed biomass fraction Metabolic fraction Ribosomal fraction Growth rate Fraction of biomass Biomass rate per enzyme demand Growth rate Figure 7.3: Translation of enzyme-specific biomass rate to growth rate – (A) Both from experimental data and a cell-optimization point of view, the ribosomal fraction of the proteome increases with the growth rate, while the metabolic fraction decreases. (B) This leads to a hyperbolic dependency of the growth rate on the biomass production rate per amount of enzymes. The next step is to perform the convex optimization over the metabolite levels for each one of the three EFMs. Therefore, we express the enzyme levels as a ratio of the flux and the function f(s), using Equation 7.4. Summing over all enzymes, we get a function for the total enzyme cost (level) as a function of fluxes, metabolite concentrations and parameters: etot =X i ei=X i vi κi(s).(7.9) We use the standard parameters (Box 7.A) and replace viby the values given by each EFM. We are then left with a convex optimization over the metabolite levels, an Enzyme Cost Minimization problem as in Chapter 6 in [1]. For [Gex] = 0.05 we obtain a total enzyme of 111.1 for EFM1, of 146.3 for EFM2 and 136.5 for EFM3. That means that for these conditions we will conclude that EFM1 is optimal. From the optimization we obtain the metabolite concentrations: [G] = 0.08,[P] = 3.93,[ATP] = 0.11 and [ADP] = 20.09 (that the internal glucose concentration is higher than the external is because we described the transport with regular enzyme kinetics instead of transporter enzyme kinetics, which would have been more realistic). We can next use the rate equations to calculate the enzyme levels from the fluxes and metabolite levels, using the values for the parameters and external concentrations. We can repeat this procedure for different levels of external concentrations and see that the optimal EFM can change depending on the external concentration (Figure 7.2c). When the optimum shifts to using a different EFM, there is a discontinuity in the fluxes at the external metabolite concentration (Figure 7.2d). Many cells show shifts in metabolic strategies depending on the external conditions and Enzyme Flux Cost Minimization is one way of explaining those shifts. Above, Enzyme Cost Flux Minimization was used to find the metabolic state with the maximum enzyme efficiency. Although in our calculation we obtain the enzyme concentrations last, it is by enzyme concentrations that cells actually control metabolism. If cells produce enzymes at the concentrations we calculated and reach a steady state, this state will realize the fluxes and metabolite levels that lead to our optimal state. 7.5. Translating enzyme efficiency into cell growth rate In the section above, we learned how to optimize metabolic states for a maximal overall enzyme efficiency. Why is this quantity relevant? One reason is that overall enzyme efficiency, according to some simple reasoning, determines the cell’s growth rate. If microbes compete by growing fast, their fitness is largely determined by their momentary growth rate in their respective environment. In such environments, the biomass/enzyme efficiency will be under selection, which makes it one of the important objective functions in this book. If higher enzyme efficiency means higher growth rate, and if we have a conversion formula for this, we can plot the growth rate of the different EFMs instead of overall enzyme efficiency. 15 Metabolite name Biomass stoichiometric coefficient AcCoA -41 ADP 547 2-oxoglutarate -14 ATP -547 H2O -547 Pi547 CO22 CoA 41 DHAP -5 G6P -4 NAD+178 NADH -178 NH3-139 2-oxoglutarateAcetate -24 PEP -32 Pyruvate -38 E4P -5 R5P -13 Table 7.2: Stoichiometry of biomass reaction – R70 Reaction ID EC number Reaction name Formula R1 2.7.1.69 pts Glucose + PEP G6P + Pyruvate R2r 5.3.1.9 pgi G6P F6P R3 2.7.1.11 pfk F6P + ATP FBP + ADP R4 3.1.3.11 fbp FBP + H2OF6P + Pi R5r 4.1.2.13 ald FBP DHAP + G3P R6r 5.3.1.1 tim G3P DHAP R7ra 1.2.1.12 gap G3P + NAD++ PiBPG + NADH R7rb 2.7.2.3 pgk BPG + ADP 3PG + ATP R7rc 5.4.2.11 / 5.4.2.12 pgm 3PG 2PG R8r 4.2.1.11 pgh 2PG PEP R9 2.7.1.40 pyk PEP + ADP Pyruvate + ATP RR9 2.7.9.2 pps Pyruvate + 2 ATP PEP + 2 ADP + Pi Table 7.3: Glycolysis Reaction ID EC number Reaction name Formula R10a 1.1.1.49 zwf G6P + NAD+6PGL + NADH R10b 3.1.1.31 glh 6PGL 6PGC R10c 1.1.1.44 pgd 6PGC + NAD+NADH + CO2+ Ru5P R11r 5.1.3.1 rpe Ru5P X5P R12r 5.3.1.6 rpi Ru5P R5P R13r 2.2.1.1 txt1 R5P + X5P S7P + G3P R14r 2.2.1.2 tal G3P + S7P E4P + F6P R15r 2.2.1.1 txt2 E4P + X5P G3P + F6P R60 4.2.1.12 edd 6PGC KDPG R61r 4.1.2.14 eda KDPG G3P + Pyruvate Table 7.4: Pentose Phosphate Pathway Reaction ID EC number Reaction name Formula R20 2.3.1.54 pfl Pyruvate + CoA AcCoA + Formate R21 1.2.4.1 / 2.3.1.12 pdh Pyruvate + NAD++ CoA AcCoA + CO2+ NADH R22 2.3.3.1 csn 2-oxoglutarateacetate + AcCoA Citrate + CoA R23r 4.2.1.3 acn Citrate iso-Citrate R24 1.1.1.41 icd iso-Citrate + NAD+2-oxoglutarate + NADH + CO2 R25 1.2.4.2 kgd 2-oxoglutarate + NAD++ CoA NADH + Succinateinyl-CoA + CO2 R26r 6.2.1.5 scs Succinateinyl-CoA + ADP + PiSuccinateinate + ATP + CoA R27 1.3.5.1 sdh Succinateinate + ADP + O2[e] + PiFumarate + ATP R27b 1.3.5.4 frd Fumarate + NADH Succinateinate + NAD+ R28r 4.2.1.2 fum Fumarate Malate R29r 1.1.1.37 mdh Malate + NAD+2-oxoglutarateacetate + NADH Table 7.5: TCA Cycle 16 Reaction ID EC number Reaction name Formula R40 4.1.1.31 ppc PEP + CO22-oxoglutarateacetate + Pi R41 1.1.1.38 me Malate + NAD+Pyruvate + NADH + CO2 R42 4.1.1.49 ppck 2-oxoglutarateacetate + ATP PEP + ADP + CO2 Table 7.6: Anaplerotic Reactions Reaction ID EC number Reaction name Formula R53r 1.1.1.27 ldh Pyruvate + NADH Lactate + NAD+ R54ra 1.2.1.10 ada AcCoA + NADH Acetaldehyde + NAD++ CoA R54rb 1.1.1.1 adh Acetaldehyde + NADH ETOH + NAD+ R55a 2.3.1.8 pta AcCoA + PiAcetyl-P + CoA R55b 2.7.2.1 ack Acetyl-P + ADP Acetate + ATP Table 7.7: Redox-associated reactions Reaction ID Reaction name Formula R80 oxphos NADH + 2 ADP + 0.5 O2[e] + 2 PiNAD++ 2 ATP + 3 H2O R82 atpmain ATP + H2OADP + Pi+ ATPmain Table 7.8: Oxidative phosphorylation Reaction ID Reaction name Formula R90 exetoh ETOH ETOH[e] R91 exace Acetate Acetate[e] R93 exNH3NH3[e] NH3 R94 exlac Lactate Lactate[e] R95 exsuc Succinateinate Succinateinate[e] R96 exfor Formate Formate[e] R97r exCO2CO2CO2[e] Table 7.9: Membrane Transport Reactions 17 Reaction ID k+ cat [1/s] Keq [unitless] Enzyme molecular weight [Da] R1 100 N/A 2.6·105 R10a 240 N/A 5.6·104 R10b 410 N/A 3.6·104 R10c 110 N/A 1.0·105 R11r 130 2.3 2.5·104 R12r 1400 2.3 1.9·104 R13r 46 3.7 7.3·104 R14r 17 0.9 3.5·104 R15r 75 38 7.3·104 R20 4800 N/A 8.5·104 R21 38 N/A 2.8·105 R22 360 N/A 9.6·104 R23r 33 0.074 9.6·104 R24 110 N/A 4.6·104 R25 150 N/A 1.2·106 R26r 89 0.52 7.1·104 R27 78 N/A 7.9·105 R27b 180 N/A 1.8·105 R28r 280 4.7 6.0·104 R29r 210 6.1·10−53.2·104 R2r 320 0.51 6.2·104 R3 110 N/A 1.4·105 R4 25 N/A 3.7·104 R40 120 N/A 2.0·105 R41 76 N/A 6.3·104 R42 51 N/A 6.0·104 R53r 140 2.1·1043.7·104 R54ra 0.35 2.3·10−39.6·104 R54rb 320 2.8·1039.6·104 R55a 91 N/A 7.7·104 R55b 59 N/A 4.3·104 R5r 8.0 3.0·10−43.9·104 R60 250 N/A 6.5·104 R61r 80 9.6·10−32.2·104 R6r 7800 11 5.4·104 R70 99 N/A 6.0·104 R7ra 230 0.088 3.6·104 R7rb 390 730 4.1·104 R7rc 53 0.16 2.9·104 R80 4.0·106N/A 9.1·105 R82 180 N/A 6.0·104 R8r 210 3.5 4.6·104 R9 510 N/A 5.0·104 R90 100 N/A N/A R91 100 N/A 5.9·104 R93 100 N/A 4.5·104 R94 100 N/A 5.9·104 R95 100 N/A 4.5·104 R96 100 N/A 3.1·104 R97r 100 N/A N/A RR9 13 N/A 8.7·104 Table 7.10: Kinetic parameters associated with reactions 18 Reaction ID Metabolite name KM[mM] R1 G6P 0.102 R1 Glucose 0.116 R1 PEP 0.0983 R1 Pyruvate 0.102 R10a G6P 0.314 R10a 6PGL 0.129 R10a NAD+0.863 R10a NADH 0.129 R10b 6PGL 0.168 R10b 6PGC 0.0594 R10c CO20.0626 R10c 6PGC 0.101 R10c Ru5P 0.0626 R10c NAD+0.0591 R10c NADH 0.0626 R11r Ru5P 0.0878 R11r X5P 0.114 R12r R5P 1.25 R12r Ru5P 0.558 R13r G3P 1.23 R13r R5P 0.972 R13r S7P 2.11 R13r X5P 0.157 R14r E4P 0.175 R14r F6P 0.888 R14r G3P 0.578 R14r S7P 0.206 R15r E4P 0.0934 R15r F6P 0.737 R15r G3P 1.27 R15r X5P 0.152 R20 AcCoA 0.0352 R20 CoA 0.0168 R20 Formate 6.35 R20 Pyruvate 2.18 R21 AcCoA 0.159 R21 CO20.159 R21 CoA 0.0629 R21 Pyruvate 0.291 R21 NAD+0.0629 R21 NADH 0.159 R22 AcCoA 0.0867 R22 Citrate 0.0756 R22 CoA 0.0756 R22 2-oxoglutarate 0.0287 R23r Citrate 3.49 R23r iso-Citrate 2.42 R24 2-oxoglutarate 0.483 R24 CO22.02 R24 iso-Citrate 0.0227 Reaction ID Metabolite name KM[mM] R24 NAD+1.06 R24 NADH 0.0119 R25 2-oxoglutarate 0.0670 R25 CO20.108 R25 CoA 0.0927 R25 Succinyl-CoA 0.108 R25 NAD+0.0927 R25 NADH 0.108 R26r CoA 0.00731 R26r Succinate 0.237 R26r Succinyl-CoA 0.0105 R26r ADP 0.0560 R26r ATP 0.0812 R27 Fumarate 0.0812 R27 O2[e] 0.371 R27 Succinate 0.0756 R27 ADP 0.371 R27 ATP 0.0270 R27b Fumarate 0.0201 R27b Succinate 0.205 R27b NAD+0.0431 R27b NADH 0.232 R28r Fumarate 0.314 R28r Malate 0.615 R29r Malate 3.19 R29r 2-oxoglutarate 0.0283 R29r NAD+0.460 R29r NADH 0.0321 R2r F6P 0.162 R2r G6P 0.273 R3 F6P 0.116 R3 FBP 0.113 R3 ADP 0.113 R3 ATP 0.141 R4 F6P 0.171 R4 FBP 0.0161 R40 CO20.115 R40 2-oxoglutarate 0.0426 R40 PEP 0.364 R41 CO20.0885 R41 Malate 0.361 R41 Pyruvate 0.0885 R41 NAD+0.0691 R41 NADH 0.0885 R42 CO25.21 R42 2-oxoglutarate 0.571 R42 PEP 0.0643 R42 ADP 0.0484 R42 ATP 0.0750 R53r LACTATE 0.517 Michaelis constants – part I 19 Reaction ID Metabolite name KM[mM] R53r Pyruvate 0.0193 R53r NAD+0.517 R53r NADH 0.0193 R54ra AcCoA 0.0242 R54ra Acetaldehyde 1.80 R54ra CoA 0.00786 R54ra NAD+0.0415 R54ra NADH 0.113 R54rb Acetaldehyde 0.0593 R54rb ETOH 5.49 R54rb NAD+0.169 R54rb NADH 0.0593 R55a AcCoA 0.0424 R55a Acetyl-P 0.313 R55a CoA 0.0860 R55b Acetate 3.44 R55b Acetyl-P 0.154 R55b ADP 0.402 R55b ATP 0.0714 R5r DHAP 0.0782 R5r FBP 0.204 R5r G3P 0.0782 R60 6PGC 0.0434 R60 KDPG 0.150 R61r G3P 0.00146 R61r KDPG 0.561 R61r Pyruvate 0.00146 R6r DHAP 0.0750 R6r G3P 0.745 R70 AcCoA 0.462 R70 2-oxoglutarate 0.352 R70 BIOMASS 0.0998 R70 CO20.0996 R70 CoA 0.891 R70 E4P 0.0144 R70 G6P 4.31 R70 NH30.0151 R70 2-oxoglutarate 0.00672 R70 PEP 0.169 R70 Pyruvate 0.319 R70 R5P 0.881 R70 ADP 0.0293 R70 ATP 0.342 R70 NAD+1.43 Reaction ID Metabolite name KM[mM] R70 NADH 0.0913 R7ra DPG 0.0576 R7ra G3P 0.687 R7ra NAD+0.0558 R7ra NADH 0.0576 R7rb DPG 0.0426 R7rb 3PG 0.235 R7rb ADP 0.0426 R7rb ATP 0.235 R7rc 3PG 0.132 R7rc 2PG 0.0755 R80 O2[e] 0.116 R80 ADP 0.136 R80 ATP 0.0737 R80 NAD+0.0859 R80 NADH 0.116 R82 ATPmain 0.130 R82 ADP 0.130 R82 ATP 0.0769 R8r PEP 0.131 R8r 2PG 0.108 R9 PEP 0.291 R9 Pyruvate 0.0476 R9 ADP 0.218 R9 ATP 8.45 R90 ETOH 0.100 R90 ETOH[e] 0.100 R91 Acetate 0.100 R91 Acetate[e] 0.100 R93 NH30.0999 R93 NH3[e] 0.100 R94 Lactate 0.100 R94 Lactate[e] 0.100 R95 Succinate 0.100 R95 Succinate[e] 0.100 R96 Formate 0.0999 R96 Formate[e] 0.100 R97r CO20.0999 R97r CO2[e] 0.100 RR9 PEP 0.0934 RR9 Pyruvate 0.0864 RR9 ADP 0.0873 RR9 ATP 0.0350 Michaelis constants – part II 20 (A) (B) Figure 7.9: Metabolic strategies in the E. coli model, depending on external glucose and oxygen concentrations, In the EFM pahse diagram. Each region represents the winning EFM as explained in Figure 7.8. Here, the colors represent the flux in one specific reaction based on the winning EFM in that region. (A) The lactate secretion flux is strikingly equal to 0in most regions. The only conditions where lactate is secreted is at low oxygen and medium/high glucose concentrations. (B) The biomass yield is, in general, high if and only if lactate is not secreted. This makes sense because the carbon coming from the glucose is often the limiting nutrient for growth, and there is a trade-off between using it for biomass versus fermentation products such as lactate. Interestingly, the region with high glucose and high oxygen levels (upper right quadrant) is occupied by an EFM that doesn’t achieve the highest possible yield (i.e. max-gr). In low glucose and high oxygen, or in medium oxygen levels, the winning EFMs are the ones with relatively higher biomass yields. 7.9. More results for the E. coli central metabolism model This section contains additional results for the E. coli central metabolism model from Chapter 7 in [1], in particular, fluxes plotted in the EFM phase diagram and in flux space, as well as ideal and real enzyme costs for all EFMs. Each point in the Monod landscape in Figure 7.8 (A) corresponds to a state of the model, and the calculations that lead to the growth rate and the "winning EFM" shown yield a full description of this state, including all fluxes, metabolite concentrations and enzyme levels. These data can be explored and visualized in many ways. For illustration, Figure 7.10 shows a variant of Figure 7.8 (A) in the horizontal axes do not describe the external concentrations of glucose and oxygen, but their uptake rates, and the vertical axes shows the biomass production rate. Since uptake rates are a function of external concentrations, and the growth rate directly depends on the biomass production rate, we might have expected that this yields the same picture, just a bit stretched along each of the axes. However, the picture looks very different: instead of forming a continuous surface, the points now fall on disconnected rays, apparently with one ray for each colored region of the surface. In fact, when looking at the picture closely, we can see that each of the winning EFMs gives rise to exactly one ray. But this, after all, is logical. In the new plot, all axes refer to reaction rates, and for each EFM all rates come in fixed ratios, giving rise to a ray. So, if our solutions are EFMs, this picture cannot be continuous – in line with the fact that, in the original Monod landscape 7.8 (A), when moving from one region to the other one, one would notice a discrete jump of the reaction rates. But why is the new picture not a continuos surface, if uptake rates depends smoothly on external metabolite concentrations? In fact, they do not only depend on these concentrations, but also on resource allocation to the transporter. If this allocation shows a discrete jump (again, when moving from one region to another one), then also the rate shows a jump. The comparison between the two plots shows us what we gain by considering enzyme kinetics as compared to a pure stoichiometric model. With the biomass rate as a proxy for cell growth, each EFM defines fixed ratios between this growth rate and each of the metabolic fluxes, including the uptake rates. When continuously scaling an EFM, the glucose uptake, 21 Figure 7.10: Proportional scaling of fluxes within each EFM – In the diagram with glucose uptake, oxygen uptake, and biomass production rate on the axes, each colored line corresponds to one EFM, and shows the possible combinations of fluxes obtained from the model behind Figure 7.8 (which also shares the EFM colors with this figure). Importantly, here the xand yaxes represent uptake rates and not substrate concentrations. Therefore, as expected, each EFM yields a straight line (because of the proportional scaling of different fluxes for each EFM). Since – according to our reasoning – optimal flux distributions must be EFMs, only these combinations of fluxes are actually possible. When glucose and oxygen concentrations are varied smoothly in Figure 7.8, the corresponding movement in this plot would be along the lines and sometimes, jumps between different lines (when the system moves from one region to another one in Figure 7.8). oxygen uptake, and biomass production will scale proportionally. So the rays in Figure 7.10 reflect what we can know about possible metabolic fluxes based on network structure alone; but to get to the Monod landscape, as a function of concentrations, we had to use kinetic information and a extra principle of economical enzyme usage. The phase diagram of “winning EFMs” can also be used to visualize other (optimized) quantities as functions of glucose and oxygen concentrations. Figure 7.9 shows as example the (biomass-specific) lactate secretion (showing that also a number of other winning EFMs, apart from ana-lac, secrete lactate) and the biomass yield on glucose. Finally, a statistics over all EFMs shows that the range of possible enzyme demands per biomass production rate is quite large: as shown in Figure 7.11, they vary over more than two orders of magnitude, making some EFMs a hundred-fold more enzyme-expensive than others. The same plot also shows how enzyme costs depend on the fact that enzymes do not operate at their full capacity (reaching their kcat value), but at best at the enzyme efficiencies predicted by enzyme cost minimization. For our E. coli model and the aerobic glucose conditions studied, if all enzymes could operate at their kcat values, this would decrease to overall enzyme demand by a factor of at least 1.4, or maximally 4.7, depending on the EFM in question. But still, in this case, for determining enzyme costs the choice of the right EFM (even assuming "ideal" enzymes) is much more important than considering the actual, "non-ideal" way in which enzymes operate. But this may not always hold: under low-oxygen conditions, the enzyme demands of some EFMs may increase much more drastically. Solutions to problems Problem 7.1 (Effect of oxygen concentration) The oxygen concentration affects only the rate v3, and an increase in oxygen increases this rate for the same enzyme concentration. Since EFM2 does not contain v3, the enzyme cost of this EFM will not change. EFM1 and EFM3 22 10−210−1100101 10−2 10−1 100 101 y = 4.7 x y = 1.4 x min. ideal cost = 0.039 [h−1] min. real cost = 0.083 [h−1] ideal cost [gr enz / gr dw h−1] real cost [gr enz / gr dw h−1] Figure 7.11: Ideal and real enzyme costs of elementary flux modes – For each EFM (shown as a cyan dot), the ideal enzyme cost per biomass production rate (i.e. assuming that all the enzymes are saturated) is compared to the actual cost (calculated using Enzyme Cost Minimization, assuming standard aerobic glucose conditions). The costs span a wide range from the most enzyme-efficient EFMs on the lower left to the least enzyme-efficient ones on the upper right. For different EFMs, the ratio of actual and ideal costs varies between 1.4 and 4.7. Here, the EFM with the minimal actual cost is among the top 5 in terms of ideal cost. benefit from an increase of the oxygen concentration, because they will have to invest less enzyme in v3to obtain the same rate. Therefore, those EFMs can become more beneficial and the optimal EFM could shift to one of those EFMs. Problem 7.2 (Effect of external metabolites) Increasing [Pex]will benefit EFM3 and decrease the benefit of EFM2 (because for EFM2 [Pex]will inhibit reaction v2and therefore more enzyme is needed for reaction v2and the enzyme cost of EFM2 will increase. Qualitatively, with increasing [Pex], EFM2 might become more expensive, and either EFM1 or EFM3 will become beneficial, or both at different Pex concentrations, depending on the kinetics of the reactions. Problem 7.3 (States of maximal growth rate) (a) v1=e1(k+ 1s1−k− 1X),v2=e2(k+ 2s2−k− 2x)and v3=e3(k+ 3x−k− 3p) (b) etot =v1 (2s1−x)+v2 (30−x)+v3 x (c) When e1= 0,v1is also 0 and to achieve steady state v2=v3and using the rate equations and filling in the parameters we get etot =v3 (30−x)+v3 x. We now set etot = 1 to obtain v3= (1−v3 30−x)xwhich we can rewrite to v3=x 1+ x 30−x. We can take the derivative to xand set it equal to 0 to find the optimum, which leads to x= 15 and v3/etot =15 2. Note that in this specific case we did not need to set etot = 1 and could have maximized v3/etot directly, but in general this does not always work. In the rest of the answers we assume we set etot = 1 and therefore v3=v3/etot. (d) e2= 0 implies v2= 0 and filling in s1= 10 leads to v3= (1−v3 20−x)x, which can be rewritten to v3=x 1+ x 20−x. This is optimal when x= 10 and v3/etot = 5. (e) e1=e2implies v1 20−x=v2 30−x. From the steady state we know that v2=v3−v1. Filling this in and solving for v1leads to v1=v3x−20 x−50 . From the total enzyme and by replacing e1by e2we get e3= 1 −2v1 20−x. Putting this in the equation for v3and using the previous equality to replace v1leads to: v3= (1 + v3 x−25 )x. Solving this for v3gives v3=x−x2 25 , which is optimal for x= 12.5with v3/etot =25 4. (f) It was optimal to invest all enzyme in e2and none in e1, because that lead to the highest specific flux v3(namely v3= 7.5). (g) Since v1= 0,S1is not involved in any reaction and the solution is the same as above, x= 15 and v3/etot =15 2. 23 (h) Similar calculations as above but now with s1= 50 lead to v3=x 1+ x 100−x, which is optimal when x= 50 and gives v3/etot = 25. (i) Similar calculations as above but now with s1= 50 lead to v3=x−x2 65 .v3/etot is maximal at x= 32.5and takes the value 16.25. (j) Now s1increased the optimal strategy would be to invest all enzymes in e1, and have e2= 0, because that leads to the highest specific flux of v3, namely v3/etot = 25. (k) The two EFMs in this pathway that produce Pare v1=v3with v2= 0 and v2=v3with v1= 0. In the problem we saw when we optimize the specific flux, we always obtained one of those EFMs as the best solution, from the options that we tested. Therefore, these results are in agreement with the proof outlined in this chapter. Bibliography [1] The Economic Cell Collective, editor. Economic Principles in Cell Biology. Free online book, 2023. doi: 10.5281/ zenodo.8156386. [2] Avi Flamholz, Elad Noor, Arren Bar-Even, Wolfram Liebermeister, and Ron Milo. Glycolytic strategy as a tradeoff between energy yield and protein cost. Proc. Natl. Acad. Sci. U. S. A., 110(24):10039–10044, June 2013. doi: 10.1073/pnas.1215283110. [3] Elad Noor, Avi Flamholz, Arren Bar-Even, Dan Davidi, Ron Milo, and Wolfram Liebermeister. 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