E. coli, calculating essential genes and comparing with experimental flux data.Constraint-based optimization, such as flux balance analysis (FBA), has become a standard systems-biology computational method to study cellular metabolisms that are assumed to be in a steady state of optimal growth. The methods are based on optimization while assuming (i) equilibrium of a linear system of ordinary differential equations, and (ii) deterministic data. However, the steady-state assumption is biologically imperfect, and several key stoichiometric coefficients are experimentally inferred from situations of inherent variation. We propose an approach that explicitly acknowledges heterogeneity and conducts a robust analysis of metabolic pathways (RAMP). The basic assumption of steady state is relaxed, and we model the innate heterogeneity of cells probabilistically. Our mathematical study of the stochastic problem shows that FBA is a limiting case of our RAMP method. Moreover, RAMP has the properties that: A) metabolic states are (Lipschitz) continuous with regards to the probabilistic modeling parameters, B) convergent metabolic states are solutions to the deterministic FBA paradigm as the stochastic elements dissipate, and C) RAMP can identify biologically tolerable diversity of a metabolic network in an optimized culture. We benchmark RAMP against traditional FBA on genome-scale metabolic reconstructed models of  The foundation of constraint-based analysis is a representation of a cell’s metabolism as a linear system of differential equations in the (unknown) fluxes of the metabolic reactions. The additional assumption that the metabolic network has reached steady state results in a homogeneous system. It is customary to optimize an appropriate objective over the fluxes that achieve steady state to model how a cellular metabolism reacts as it is held in a constant environment5. Different objectives have been proposed and studied7, with the most prevalent being the maximization of the rate at which biomass is created.Constraint-based analysis is an approach to study metabolic networks that has become a primary computational tool with a well established literature8, but the use of constraint-based methods has expanded to encompass genome-scale metabolic models and a diverse range of applications and computational formulations9. However, the basic premises of optimality and steady state have remained intact, although these assumptions are well known to be inexact from a biochemistry perspective. That said, there are two somewhat evident reasons to leave these premises unchecked. First, the resulting optimization problems are most often linear and less often quadratic (convex), and in both cases the problems are easily solved with standard methods and software. Second, traditional constraint-based models are useful even with questionable premises. For example, with a high level of accuracy, FBA models can identify essential genes10, predict metabolic responses as pathways are interrupted11, determine central metabolic pathways14, develop synthetic biology engineering strategies18, as well as identify genes showing promise as inhibitors of cancer migration19. Hence, a justified retort by the constraint-based analysis community to those who might question the two defining premises is that the modeling approach yields meaningful and computationally tractable science even though the cornerstone premises are imperfect.The most commonly applied version of constraint-based analysis is known as flux balance analysis (FBA). The original study of FBA investigated small portions of the central metabolism4. Second, experimental flux measurements constitute averages over such heterogeneous populations. Moreover, these averages provide the only possible experimental prototype of cellular flux since single cell measurements are currently impossible. Third, heterogeneity of metabolic rates is innate within a culture because cells vary within, e.g., their cell cycles and replication states. The traditional paradigm that a constraint-based model represents a single, optimized cell is certainly challenged by such experimental realities upon which models are vetted.The traditional constraint-based paradigm assumes a model of an individual cell that has reached an ideal, limiting setting of steady state. Models are benchmarked with experimental outcomes under the assumption that experimental measurements are conducted on a population of identical cells that have evolved to optimal performance. However, experimental reality differs from this assumption. First, even well optimized cellular populations of identical cells exhibit prominent heterogeneity in uptake, secretion, and growth rates20 counterpart to FBA, called Robust Analysis of Metabolic Pathways (RAMP). The robust optimization implementation allows us to model a system stochastically where, instead of imposing the rigid and simplistic condition of deterministic coefficients and steady state, we allow these assumptions to be relaxed. In RAMP we control departures from steady state by limiting their likelihood of deviation. Since the stochastic flexibility of RAMP permits us to postulate metabolic models for cultures that deviate from steady state, this opens the possibility of computationally studying functional states of cellular metabolisms as they transition toward a steady state. Such an investigation begs the questions of (1) whether or not RAMP solutions are continuous in their stochastic elements and (2) whether or not convergent trajectories of RAMP solutions yield deterministic FBA solutions as the probabilistic assumptions abate. We mathematically answer both questions in the affirmative, and hence, FBA is justly thought of as the limit of evolutionary adaptations of a cellular metabolic network that achieves steady state.Here, we challenge and relax the foundational assumption of a system being in a steady state, and we instead introduce a robust optimization23. Our mathematical analysis identifies the probabilistic situations that coincide with our modeling framework once the limiting deterministic setting is reached. Consequently, if we assume that RAMP’s modeling paradigm aligns with the stochastic nature of a culture, then we can mathematically characterize the ways in which a culture may harbor variation.Additionally, the development and stochastic formulation of RAMP explicitly allows us to systematically address heterogeneity in metabolic phenotypes that exists in isogenic cellular populationsE. coli models25 demonstrate that RAMP rivals FBA in its ability to identify essential genes. We further show that RAMP’s efficacy in identifying essential genes is predominantly stable as individual coefficients of the biomass equation are assumed to be uncertain. The ranges over which individual coefficients can vary are disparate, having a surprising spread of many orders of magnitude. All coefficients could accommodate uncertainty of at least 0.42%, with most extending beyond 100% uncertainty. We also compare FBA’s and RAMP’s consistency with experimentally determined fluxes27, finding that RAMP significantly outperforms FBA for both aerobic and anaerobic conditions.Possible concerns with RAMP are that it could lose the predictive successes of FBA, and that it could be less computationally tractable. We have designed our computational experiments to compare RAMP with FBA in such a way that both models can be solved with the default simplex algorithms of several linear solvers, and our results on two 29. Such robust models were developed to overcome problems with over-optimizing designs, meaning that a design could be optimal with respect to the estimated data on which it was built but (much) less so as the data varied over realistic possibilities30. A similar concern about over-optimization has been expressed in the FBA literature7, and RAMP directly addresses this issue by expressing variability within the model itself.With regards to computational tractability, RAMP is a robust linear program, which is a second-order cone program (SOCP) that is known to be solvable in polynomial time31. This method differs significantly from RAMP since it uses a robust least squares method in a bi-objective model. Such a model can illustrate the trade-off between the optimal growth rate and the deviation from steady state, whereas RAMP uses robust linear programming. Furthermore, the numerical work in ref. We note that an alternative robust model has been suggestedOur conclusion is that RAMP opens a new paradigm for constraint-based optimization and analysis of genome-scale metabolic models that is also directly applicable to a large number of the recent constraint-based extensions to FBA. Our computational implementation and benchmarking of a specific version of RAMP using both linear and non-nonlinear objective functions, show that it is a computationally tractable alternative and complement to traditional FBA.c is defined by a system of n reactions, with the j-th reaction beingi-th metabolite in cell c. The cellular superscript c is absent in FBA’s standard development due to the assumption that an average cell is being studied. Thus, the ordinary differential equation for the i-th metabolic concentration in cell c is,i in reaction j, the flux of reaction j is c denotes the cell. Consequently, genetically identical individuals of a culture share the same collection of metabolic reactions. Hence, the preponderance of stoichiometric coefficients of any particular genome-scale metabolic reconstruction are static across the individuals of a culture. However, a complete metabolic model includes several inferred and less certain reactions to model processes, such as those involved in the creation of biomass. In many of the recent extensions of FBA9, a combination of alternative objective functions, global constraints or upper/lower bounds on sets of reaction fluxes are added to the system. Consequently, many extensions, such as FBAwMC32, GIMME33, MOMENT34, and GX-FBA35, are implemented by adding rows and/or columns to the FBA stoichiometric matrix, and these new stoichiometric coefficients do not generally consist of integer values. Instead, depending on the method of interest, these added stoichiometric coefficients may consist of a combination of data from large-scale’ omics sources, biochemistry knowledge, or theoretical modelling assumptions. Thus, they are carriers of inherent uncertainty.We begin by re-visiting the foundational assumptions of FBA, where the metabolism of an individual cell E. coli metabolic model iJR90436. These less certain coefficients would be stochastic, time varying parameters per individual cell. Thus, instead of the growth reaction in Table In particular, the experimentally determined coefficients of a biomass equation are noticeably different from their integral counterparts in the shared metabolism, see Table There exists also other sources of uncertainty for the performance of a genome-scale metabolic reconstruction, such as the value of the phosphate/oxygen (P/O) ratio. Changes in how the network performs the ATP production and drain has a significant potential to affect model-predicted growth performance. In the following discussion, however, we will base our presentation and discussion on uncertainty in the biomass growth coefficients.t → ∞. Furthermore, it has been suggested that cellular metabolic performance evolves toward an optimal growth state as a culture’s environment is left unchanged and as populations are repeatedly sampled and re-grown10. FBA is predicated on the assumption that cells have been tuned through such an evolutionary processes to an “optimal” use of their resources, where “optimal” is measured by a function of the fluxes, g(v). Hence, FBA studies metabolic processes through optimization problems, and adaptations thereof, of the formg(v) is typically chosen to estimate the rate at which biomass is created, and for convenience, we assume throughout that g(v) is the sole flux of the growth reaction, i.e. g(v) = vGrowth. While several alternative objectives have been proposed and investigated in different biological settings7, the production of biomass is the most prevalent choice and is the default in the COBRA37 toolbox. The vectors of lower bounds, j is reversible.Suppose i the collection of c of the culture is a random sample of the stochastic rate of change of the concentration. Let i) over the sample, that isWe assume that a culture has gone through a long-enough time of growth and re-sampling to be largely optimized to the resources of its (invariant) environment. Heterogeneity within the culture is modeled as the random experiment that is the growth of the culture, and we consider the variational states of the genetically identical cells of the culture as independent and identically distributed outcomes from that growth experiment. Hence, the known heterogeneity within an optimized culture is the result of a large sample of possible outcomes. Consequently, for each metabolite μi and σi be the unknown mean and standard deviation of this normal distribution. This approximation is expected to be trustworthy since cultures regularly contain hundreds of millions of cells.The central limit theorem mandates that Mi| is a bound on the feasible fluxes that define the allowed deviation from steady state. Unlike the FBA model formulation that collapses onto an ideal cell and ignores cultural variations, these stochastic constraints account for the innate variations that are known to exist. Since δε and δε1− are the ε and 1 − ε percentiles respectively, i.e. δε1− = −δε, we findWe replace the traditional static assumption of steady state with the probability constraintsSi to be the i-th row of the random stoichiometric matrix in which the uncertain elements are random variables, e.g. for the creation of biomass, then the implicit FBA assumption of an ideal cell results in the following stochastic differential equation for each metabolite,The stochastic constraints in Eq.  are exprv we haveConsequently, for any flux vector μi = 0. The first and second equalities then constrain the expectation of the sample mean, E(Si)v = 0, which shows that the stoichiometric coefficients of an FBA model are reasonably interpreted as averages, i.e. that An apt interpretation of FBA’s projection of a culture’s heterogeneity onto an average is somewhat laid bare by these equalities: FBA assumes that e of Eq. , and thementclass2pt{minimE(Si)v = μi shows how the vector of fluxes v defines the expected value of the rate of change of the concentration (of metabolite i), and the equation begs for detailed random models of the uncertain stoichiometric coefficients. Such detail is challenged by experimental limitations. For example, it is impossible to study individual cells or even groups of cells that are in identical metabolic or proteomic states. Hence, inferring accurate distributional characteristics on parameters such as the growth coefficients is currently impossible. Beyond averages, our ability to express Eq.  stoichiometric coefficients for the i-th metabolite in scenario k and let pik be the probability of scenario k, for k = 1, 2, 3, …, q. Allowing pi to be the (positive) vector of probabilities indexed by k, Pi to be the (positive definite) diagonal matrix formed by pi, and k-th row is Sik, the mean and variance of Siv aree is a vector of ones. Defining Assuming that each The expected value and standard deviation are related through constraint (4), from which we haveAlthough Eq.  is more with Eq. , we haveLc = L and Uc = U for all c. This default is somewhat for notational convenience since stochastic variation in L or U could be remodeled as variation in the coefficient of the associated flux, and the resulting constraint would then be part of the stoichiometric matrix20.The RAMP model defaults to a commonality of the upper and lower flux bounds among the cells so that A feasible RAMP solution is a collection of fluxes under which the mean and variance of the normal distribution of The left-hand side SOCP constraint can be similarly re-written as,Si are drawn are called uncertainty sets in the robust optimization literature, and these sets define the collection of stoichiometric possibility. A feasible flux must satisfy being within |Mi| of steady state for all possible stoichiometric possibilities, meaning that Si can deviate from its average uTRi as long as The sets from which s1 = 1 and s2 = −1, corresponding to the standard static values of a stoichiometric matrix. Assume we have q = 3 scenarios, andThe re-expression of the SOCP constraints in terms of their uncertainty sets provides a geometric description. As an illustrative example, consider the two variable systemδε1− = 3 so that ε ≈ 0.0015, we have thatThe expected value of the constraint is simplyM1 = 0.2, then Eqs M1 = 0.2This infinite collection of constraints is neither more nor less restrictive than the FBA constraint. One interpretation is that RAMP first relaxes the average equilibrium constraint of A depiction of the geometry for the above example is shown in Fig. In the following we develop three important mathematical qualities of the RAMP model by establishing three Theorems, proofs of which are contained in the Methods section. Our first result demonstrates that feasible RAMP flux vectors satisfy a Lipschitz continuity property in their probabilistic elements, provided that bounding constraints can be marginally relaxed. The second result demonstrates that convergent RAMP solutions are indeed FBA solutions as stochastic uncertainty diminishes. The third theorem characterizes the types of uncertainty in the stoichiometric matrix that can be tolerated by an optimal FBA solution.pi and L ≤ v ≤ U prevent an immediate application of Corollary 1 because realistic genome-scale metabolic models commonly enforce some fluxes with implied equalities, i.e. Li = Ui for some i. An example is the often called ATP maintenance flux, which is the rate at which ATP is lost due to non-growth-associated processes. These fixed fluxes are included to ensure that FBA models account for cellular demands that are not explicitly part of the genome-scale metabolic reconstructed model39. The continuity results depend on scaling, and Lemma 1 and Corollary 1 (see Methods) do not provide the Lipschitz continuity for RAMP since scaled fixed fluxes do not remain feasible under the requirement that vi = Li = Ui.Corollary 1 (see Methods) suggests that RAMP’s SOCP constraints might imbue continuity of the feasible flux vectors with regard to the stochastic modeling elements Sv = 0, these constraints cannot be guaranteed to be of the form required by Lemma 1. For instance, if vi is the flux for the reaction that removes ATP for non-growth-associated demands, then the equality vi = Li would naturally transition to P(Li − Mi ≤ vi ≤ Li + Mi) ≥ 1 − 2ε, which would associate with an SOCP constraint of the form Li − Mi and Li + Mi would agree for small values of Mi, which would be required to remain realistic, the requirement of a positive right-hand side in Lemma 1 can not be ensured after writing both inequalities in the correct form.The equalities imposed by the bounding constraints could be re-cast probabilistically. However, unlike the steady state equations of Mi for the variable bounds so that we satisfy them with probability 1.Nevertheless, if we adjust the variable bounds dependent on the perturbed stochastic elements instead of prescribing them independent of perturbation, then we can relax the bounds to ensure feasibility. Probabilistically this means that we can set the values of For a collection of probability vectors piand scenario matricesand for the lower and upper bounds U and L, letbe the nonempty set of feasible fluxes satisfying the constraints of the RAMP model in Eq.  model.We note that a recent discussion in the literature, see refs Li = vi = Ui is replaced with Li − η  ≤  vi ≤ Ui + η for any arbitrarily small η > 0. The proof of Theorem 2 (see Methods) is straightforward and follows from the fact that if the primal and dual solutions converge as the stochastic elements dissipate, then the resulting necessary and sufficient conditions are that of FBA. However, the proof importantly identifies that not all random elements need to disappear, a point that prompts Theorem 3.A reasonable question is if the deterministic FBA model is a limiting case of RAMP as the stochastic elements diminish? Theorems 2 and 3 show that convergent RAMP solutions are indeed FBA solutions, and hence, they answer this question in the affirmative. They also show that interpreting FBA as a limiting RAMP model characterizes the possible random flux states among cells in an optimal growth, steady state culture. The convergence result of Theorem 2 assumes an interiority condition, which is tacit as long as the equalities imposed by the bounding constraints are relaxed to allow arbitrarily small adjustments, i.e. as long as Letbe sequences such that for each t the corresponding RAMP model satisfies Slater’s interiority condition. Let vtbe an optimal solution of the RAMP model corresponding toandand assume vt → v as t → ∞. Assume likewise that a corresponding dual sequence of optimal solutions converges. Further assume that as t → ∞ we haveandThen v is a solution to the FBA modelwhereis the matrix whose i-th row isRi satisfy Riv = 0 and Ri can have low rank, e.g. the rank of R1 in Eq.  to appropriately set the values of Mi so that RAMP’s growth rate exactly matches that of FBA’s. The problem ensures that the sum of permissible variations from steady state is as small as possible.The forthcoming computational experiments introduces various stochastic assumptions by querying either different percentiles The goal of our first computational experiment is to gauge the relationship between RAMP and FBA for individual uncertainties in the growth reaction. This is motivated by the observation that stoichiometric coefficients of the growth reaction are empirically determined non-integers, with values spanning many orders of magnitude . As expected, RAMP and FBA coincide for the case with no randomness in the growth stoichiometric coefficients, i.e. with σ = 0. However, as σ increases it is possible that RAMP will find a different set of essential genes than FBA. Consequently, we determine the maximal value of σ for which RAMP and FBA return identical sets of essential genes.We tackle this challenge by individually increasing the level of randomness in a single growth coefficient, multiplying it with a factor containing ±Assuming that the genome-scale metabolic reconstruction is a high-quality rendering of the biochemical processes associated with the individual biomass components, we make the following observations: If RAMP returns identical essential gene sets as FBA (i.e. stable predictive ability) for large percent variations of a single coefficient, then the cells of the culture could possibly be disparate in how they incorporate the associated metabolite in biomass. If the predictive ability instead degrades with only slight deviations, then the growth coefficient is more likely exhibiting little variation across the culture.σ before RAMP departs from FBA’s gene-knockout predictions, for each of the 72 growth coefficients in iJO136625. We find that for 18 of the coefficients  the value of the mean  the standard deviation . Note that, for all of the 54 coefficients, the numerically determined RAMP solutions are more permissive than the FBA solutions, meaning that at the determined maximal multiplier values, RAMP identified as essential a subset of the FBA-determined essential genes. In total, we found that 23 of the 54 coefficients supported maximal permissible multipliers σ > 1. This is quite surprising, as it suggests that both the sign and the magnitude of the corresponding stoichiometric coefficient may vary while the growth rate and RAMP’s predictive power are maintained.We find for the remaining 54 biomass coefficients that the maximally allowed multiplier spans a surprising 4 orders of magnitude. Upon investigating a possible relationship between these maximal multiplier values and the corresponding stoichiometric coefficients, we find that they are uncorrelated , are tabulated in Table 24 and iJO136625 in response to the different RAMP implementations. While the majority of biochemical reactions are the same in these two models, the RAMP analysis emphasizes that small details of the reconstructed model affect the identification of essential genes. Here, RAMP may be an additional approach in vetting the quality of model reconstructions.The tallies in Table 1), which only permits uncertainty past the stated accuracy of FBA’s growth equation, precisely replicates the essential gene predictions of FBA for both genome-scale metabolic model reconstructions. This fact computationally substantiates Theorem 2 and numerically demonstrates further that minor probabilistic variations mimic FBA.Table 2) is designed to magnify the perturbations of Model 1 by a multiplicative factor ρ and ‘spread’ the scenarios in the last few digits. From Table ρ decreased the true positives by 9 and increased the true negatives by 1, which means that RAMP2 had 8 fewer correct predictions than FBA with increased scenario variability as magnified by ρ. In contrast, the iAF1260 model agreed with FBA for all values of ρ. RAMP’s predictive power compared with FBA for RAMP2 decreased from 91.14% to 90.56% with the iJO1366 model and matched FBA’s 91.32% with the iAF1260 model. The similarity between FBA and RAMP for RAMP1 and RAMP2 shows that probabilistic deviations in the growth coefficients beyond what is reported by the genome-scale metabolic reconstructions only lead to minor adjustments in predictive power.Model 2  has a slightly more varied effect than multiplying the default variations by ρ. Results with σ ≥ 0.4 had little value since these models relaxed the FBA constraints to the point at which few, if any, essential genes were identified, and for this reason we only report the results for σ ≤ 0.3. The trend as σ increases is that RAMP3 reduces the number of predicted essential genes, which lowers the number of true and false positives but increases the number of true and false negatives. Five true positives move to false negatives in the iJO1366 model for RAMPa3 and RAMPb3, which reduces RAMP’s predictive ability to 90.78%. However, for larger σ, RAMPc3 increases its number of true negatives and reduces the number of false positives, which increased predictive ability to 90.92%. The trend for the iAF1260 model is similar.Scaling the scenarios proportionately (RAMPε (RAMP4) had no effect on RAMP’s ability to predict essential genes, and for all tested values of ε, RAMP agreed with FBA. This clearly suggests that it is more important to tune RAMP with regard to the scenarios than it is with the probabilistic guarantee of satisfying the near equilibrium constraints in Eq.  objective function for RAMP that minimizes the distance between a set of experimentally determined flux data and the corresponding reaction fluxes in a genome-scale metabolic model. We directly compare the results from RAMPn in Eqs  and 23)3 with σ lly Fig.  and anaelly Fig. , as well1/h Fig.  and 0.4/1/h Fig. ).Figure N = 28. The results reported in Fig. A striking visual feature of Fig. The expanding suite of systems-biology computational methods described as constraint-based analyses has significantly improved our ability to probe the properties of genome-scale metabolic reconstructions. However, the basic premises of these methods, the assumptions of steady state and no heterogeneity in a cellular population, has stayed unchanged. In this article, we have developed the conceptual framework for a new paradigm of constraint-based methods by explicitly relaxing the basic assumption of steady-state while allowing for heterogeneity. We have presented the mathematical foundation proving that, as levels of uncertainty are reduced, the RAMP method will become identical to standard FBA.32, GIMME33, MOMENT34, and GX-FBA35. However, also methods that add gene-regulatory effects, such as rFBA43, and thermodynamic considerations44 may be directly implemented with RAMP, since many of these methods generate modified metabolic networks, or temporal sequences of such. Note that, since it is known that uncertainty in variable bounds, e.g. restrictions such as upper bounds on reaction fluxes, mathematically is not different from uncertainty in the stoichiometric coefficients20, the RAMP formulation is also capable of modelling uncertainty in flux constraints.Building on the RAMP foundation, it will be possible to develop robust versions of many of the contemporary constraint-based methods. For some of the new extensions to FBA, the inclusion of RAMP formalism is straightforward, e.g. FBAwMCL2 RAMP flux-minimization problem  methodincrease variation beyond the growth coefficients to replicate stochastic environmental bounds.Theoretically, we are able to formulate a wide host of linear and nonlinear RAMP models as SOCPs. However, the nonlinear ones have not been uniformly translatable into computational reality. While we have been able to solve the (see Eq. ) and shoWhile we are able to solve these problems occasionally, we were unable to identify a choice of solver settings and convergence criteria to achieve consistent and stable success for the numerical implementation with current SOCP software for any of these extensions. Establishing a stable computational platform for the general RAMP paradigm is an important goal for future work.The SOCP constraints of RAMP appropriately inject statistical variation into the FBA paradigm, but the conclusion of whether or not RAMP’s stochastic perspective is more or less restrictive than FBA’s deterministic setting remains unclear. From a mathematical point of view, RAMP is neither a direct relaxation of FBA nor a direct restriction of FBA. However, our computational results demonstrate that RAMP is, in practice, a relaxation of FBA. When investigating uncertainty in individual growth coefficients, we uniformly show that discrepancies in gene essentiality are due to RAMP being less restrictive, as evidenced by RAMP having predicted at least one essential gene less than FBA, i.e. a subset of the FBA results. Similarly with all growth coefficients assumed uncertain, RAMP’s stochastic adaptation results in a model whose trend is to predict fewer essential genes, especially as uncertainty increases.These computational outcomes favorably agrees with the biological sentiment that cellular metabolic networks are more robust in their wild-type setting and are increasingly fragile as they evolve toward an optimal, and specialized, steady-state. Hence, the number of essential genes should reduce as statistical variation increases, which is predicted by RAMP. Moreover, our mathematical analysis and computational outcomes show that RAMP mimics FBA if the stochastic elements are sufficiently small. In particular, if variation is not allowed beyond the assumed accuracy of an FBA model, then RAMP behaves like FBA for predictions of essential genes.46, and hence, learning to calculate FBA solutions that can witness as much variability as possible is a promising direction of future research.Theorem 3 provides a new approach to discriminate FBA’s optimal flux states: For example, some FBA solutions may have no biologically possible scenarios while others may have many. This suggests that the former is less stable with regards to random variation, whereas the latter is optimal under a wider range of stochastic possibilities. One may intuit that actual flux states should be robust against random variationFinally, our choice of letting the objective function coincide with the growth reaction was purely a choice based on making RAMP easily comparable to traditional implementations of FBA, and RAMP is valid beyond this choice.In this section we provide the mathematical foundation and proofs of the theorems presented in the main text. We also discuss aspects of the computational stability and implementation of RAMP.Lemma 1The following lemma establishes a general property from which our result follows.Letbe an n-element row vector, R be a q × n matrix, and b be a positive scalar. LetThen, for anythere are scalars z and λ satisfying 0 < z ≤ 1 and λ ≥ 0 such thatwhereand λ is independent ofandv satisfy z = 1 and λ = 0.Should τ such that 0 < τ ≤ 1 andSuppose instead that R′ = R + (1 − τ)ΔR. Also select z such thatTo ease notation, let Then,We conclude that Let λ is independent of Hence,b > 0, can be scaled to remain feasible, and moreover, that the magnitude of the adjustment to remain feasible is uniformly bounded by the magnitude of the perturbations. Importantly, this result immediately extends to finite collections of SOCP constraints of the same form, a fact we formalize in Corollary 1.Lemma 1 shows that vectors satisfying SOCP constraints of the form For i = 1, …, m, letbe an n-element row vector, Ribe a q × n matrix, and bibe a positive scalar. LetThen, for anythere are scalars z and λ satisfying 0 < z ≤ 1 and λ ≥ 0 such thatwhereand λ is independent of all  andzi and λI be the scalars for each I and setting z = mini{zi} and λ = maxi{λi}.□The proof follows directly from Lemma 1 and its proof upon letting α = 0. Notice that v1 = v2 = t solves the system for any t if α = 0, but that the only solution for α ≠ 0 is v1 = v2 = 0. Hence the solution v1 = v2 = 1 at α = 0 is not arbitrarily close to a solution with α ≠ 0.Unfortunately, systems of linear equalities like those of FBA do not satisfy similar continuity properties, and changing the stoichiometric matrix in FBA can lead to discontinuities. As a simple example, the system of homogeneous equationsProof for Theorem 1Let where μik between pik and pik + Δpik such thatFrom the Mean Value Theorem there is a Hence,lity Eq.  that16\dEach of RAMP’s SOCP constraints may be re-written asv′ satisfying these perturbed SOCP constraints and a From Corollary 1 and inequalities Eqs  and 16)16) we knv′ must also satisfySince Proof for Theorem 2t let f be a vector so that fTv = vGrowth. The dual can be assumed to satisfy Slater’s interiority condition since the necessary and sufficient primal-dual conditions in this case would be satisfied by the convergent sequences:For each rd through 6th constraints, from which we can see that the dual is always strictly feasible, and hence, the dual satisfies Slater’s interiority condition. Select any y and w variables such that ρt − σt can attain any vector, these two variables can be chosen so that ρt > 0, σt > 0 and thatDual feasibility is defined by the 3Hence the system is indeed necessary and sufficient for optimality.t → ∞, we have by assumption that Allowing v is an optimal solution to the FBA model.Since these are the necessary and sufficient conditions for FBA, Proof for Theorem 3Let pi is a probability vector, q − 1 for the eigenvalue 0, and one of dimension 1 for the eigenvalue 1. All eigenvectors for the eigenvalue of 1 are scalar multiples of the all ones vector e. Hence, I satisfy αI ≠ 0.The last equality shows that Mi ↓ 0 areRAMP’s SOCP constraints as Mi ↓ 0 is the linear programHence, the limiting RAMP model as f is the vector so that fTv = vGrowth.The necessary and sufficient conditions areI that S′iv′ = αie for some αI ≠ 0. Then Riv = 0 for all i. Moreover, since wi = 0. Alternatively, if for some I we have S′iv′ ≠ αie for all nonzero αI, then Suppose for each lves Eq. , the strlves Eq.  with \doe in Eq. . Hence, consistent achievement of optimality for the general RAMP problem with a wide variety of state-of-the-art algorithms to solve SOCPs. The computational challenge is likely caused by two characteristics specific to genome-scale metabolic reconstructed networks. First, a substantial number of variables are unsigned and, essentially, unbounded. Second, the FBA problems arising from genome-scale reconstructed metabolic networks are highly degenerate (a detailed discussion of this point is provided in ref. A straightforward implementation of RAMP (see Eq. ) as a coi-th constraint is the sole random variable. Then for some q-vector s we haveMotivated by biological considerations, we therefore implemented a limited version of RAMP as a computational model. If the probabilistic variability of each constraint is restricted to a single coefficient, then the associated SOCP is linear. The linear constraint is constructed by, for example, assuming that the last coefficient of the This calculation shows that if we restrict probabilistic variability to, for example, the coefficients of the growth reaction, then each of the SOCP constraints of the form49.We choose to use the growth coefficients for two reasons. First, we have already noted that this is an empirically derived reaction that is based on the aggregate properties of a cell culture, and thus, the stoichiometric values are inherently associated with uncertainty. Furthermore, it is well known that biomass composition  of a cellular culture is affected by nutrient conditions and the culture’s growth phase©, both of which worked well with the COBRA toolbox37.The linear re-formulation in Eq.  allows RE. coli genome-scale metabolic models iAF126024 and iJO136625 in a minimal nutrient environment consisting of (i) the models’ presets to emulate M9 medium, (ii) a maximal glucose uptake rate of 10 mmol/gDw/h, and (iii) unlimited oxygen uptake.We present two computational experiments to assist in assessing how stochastic variation affects RAMP’s ability to predict gene essentiality. The first experiment iteratively induces individual randomness in each of the growth coefficients, holding the others at their nominal value as stated in the model. The second experiment assumes simultaneous randomness in all growth coefficients. In all of the computational experiments, we used the Here, we increase the level of uncertainty in a single growth coefficient in an iterative fashion, and we systematically go through all of the coefficients. The goal of this experiment is to gauge predictive ability against individual uncertainties in the growth equation, and we measure the predictive ability simply by comparing the number of essential genes with that predicted by FBA. If the predictive ability is stable for large percent variations of a single coefficient, then the cells of the culture could possibly be disparate in how they use the associated metabolite. If instead the predictive ability degrades with slight deviations, then the growth coefficient is more likely conserved across the culture.q = 5 scenarios for each coefficient. The two most extreme scenarios multiply the coefficient by ±σ with probability P(3/2 ≤ z) = 0.0351, where z is standard normal variable. Two other scenarios multiply the coefficient by ±σ/2 with probability P(1/2 ≤ z ≤ 3/2) = 0.2389, and the other scenario leaves the coefficient unchanged with probability P(−1/2 ≤ z ≤ 1/2) = 0.4520. The value of δε1− was 3, which meant that the probabilistic constraints were guaranteed with probability 0.9997.The experiment uses σ was calculated within an accuracy of 0.0001 and with a final value that allowed RAMP’s predictive accuracy to match that of FBA’s. Thus, starting from an initial value of σ = 0.0001, we (i) increased σ by a factor 2 until RAMP’s gene knockout predictions no longer matched that of FBA, and (ii) at this point, we initiated a binary search to identify the maximal value of the multiplier σ, which resulted in RAMP and FBA gene knockout predictions matching.The largest value of In this experiment we assumed simultaneous randomness in all growth coefficients. Four probabilistic models were used to assess RAMP’s overall sensitivity to stochastic variation, and each was compared against FBA’s gene knockout predictions as calculated by the COBRA toolbox (see Table Model 1(default) Our default RAMP model assumes that the means of the growth coefficients are the values stated in the FBA model and that probabilistic variation is restricted to the first unspecified significant digit. Assuming i-th growth coefficient, we further assume that each growth coefficient has the 5 scenarios in which it is perturbed by η is one of 0, 1, or 2. The probability of the scenario with no perturbation is P(−1/2 ≤ z ≤ 1/2) = 0.4520, with perturbation P(1/2 ≤ z ≤ 3/2) = 0.2389, and with perturbation P(3/2 ≤ z) = 0.0351, where z is a standard normal variable. Two examples of these scenarios are illustrated in Table δε1− = 3, so that the probabilistic constraints have a 99.97% guarantee of satisfaction.Model 2 The second probabilistic model multiplies η in model 1 by the scalar ρ to assess how RAMP solutions adjust as the scenarios deviate from those of FBA. Eight tests with ρ = 2, 3,…, 9 were considered. Integers beyond 9 resulted in sign changes and were not considered. The probability scenarios from Model 1 were used, as well as δε1− similarly being set to 3.Model 3 Since Model 2’s scaling by ρ disproportionately effects small growth coefficients, we compensate by replacing the scenarios of Model 1 with percentages of the growth coefficient itself. We choose different ranges for σ, using the scenarios η is one of 0, ±1/2, or ±1. The probabilities are unchanged from those in Model 1, and δε1− is 3.Model 4 To assess how RAMP reacts to changes in the certainty of satisfying the probabilistic constraints, model 4 changes δε1−. All other model parameters are inherited from the default model.In our testing of the predictive ability of these four computational RAMP models  is an ivEXP is the set of experimentally determined fluxes. The Mi values were calculated to achieve the FBA-optimal growth rate as noted in the previous section. The scenarios were those of Model 3 (RAMP3) with a 20% scaling of the growth equation, i.e. σ = 0.2. The (FBA) optimal growth rate γ* was scaled down by θ = 0.9 because the experimental fluxes were not guaranteed to coincide with optimized cultures. Note that Eq. (We tested the ability of FBA and RAMP to agree with experimentally determined fluxes. For RAMP we solved,Likewise, the ability of FBA to realize the experimental fluxes was tested by solving the following adaptation of Eq. ,23\documθ was set to 0.9, and the solution was again as close as possible to vEXP.As with RAMP, Supplementary information


Escherichia coli and fatty acyl-ACP in the cyanobacterium Synechocystis sp. PCC 6803 by the UP Finder showed high degree of agreement with the reported key genes in the literatures.Overexpression of key genes is a basic strategy for overproducing target products via metabolic engineering. Traditionally, identifying those key genes/pathways largely relies on the knowledge of biochemistry and bioinformatics. In this study, a modeling tool named UP Finder was developed to facilitate the rapid identification of gene overexpression strategies. It was based on the COBRA toolbox under MATLAB environment. All the key gene/pathway targets are identified in one click after simply loading a Systems Biology Markup Language model and specifying a metabolite as the targeted product. The outputs are also quantitatively ranked to show the preference for determining overexpression strategies in pathway design. Analysis examples for overproducing lycopene precursor in   •A modeling interface is developed for identifying gene overexpression targets.•It evaluates the yield contribution for each overexpression candidate.•The outputs are ranked to show the preference for determining strategies. In the successful examples, overexpressing key genes of metabolic pathways is a widely used strategy for achieving overproduction . The purin silico analysis of metabolic fluxes  is defined as the ratio of fluxopt to fluxwt of a reaction (Ratio = fluxopt / fluxwt). And the Yield is simulated product yield of the targeted product by using fluxopt of a reaction as the constraint, in which maximizing cell growth is the objective function.The parameter 2.3(1)Identification of up-regulated fluxes. The main concept is to compare the flux distributions between the wild-type and overproducing metabolic networks by calculating theoretical maximum yield of a targeted product. Thus, up-regulated fluxes and their associated pathways (termed as up-regulated pathways in this study) can be identified through this comparison.(2)fluxopt) to achieve theoretical maximum yield of the product. The simulated product yields (Yield) constrained by each fluxopt for the wild-type network were used to evaluate the best contribution of each up-regulated pathway toward overproduction. Pathways with Yield > 0 are considered as the key targets that directly lead to the overproduction.Re-verification of identified pathways. Since not all the identified pathways from Step (1) are directly related to the overproduction, a re-verification is necessary to filter the low-relevant targets. For these identified pathways, their fluxes under overproducing networks were considered as the optimum fluxes ((3)Ratio, which is the ratio of each fluxopt over their associated wild-type fluxes (fluxwt) was used for ranking the output from high to low. Because Ratio reflects the up-regulated level for each reaction, the one with the highest Ratio value suggests the highest preference when considering gene overexpression in engineering of the targeted overproduction.Rank of the output. The output of the UP Finder is the abbreviated reaction names of the selected key pathways in Step (2). A termed parameter, UP Finder is an interfacial modeling tool based on the COBRA toolbox in MATLAB, which is developed by the MATLAB Graphical User Interface Development Environment (GUIDE). It is used to identify all the key gene targets for overexpression that directly related to the overproduction of a metabolite in a microorganism. The working procedure of the UP Finder is composed of following steps :(1)Ident2.4Organism item, all metabolites included in this model will be shown in the Targeted product item. Users can simply specify one metabolite as the target for overproduction. By choosing UPA in the Method option and running the program, a list of reaction names presented with their associated genes, reaction formulas, Yield and Ratio values will be returned as the output. All computation is based on the COBRA toolbox and MATLAB, and all optimization uses FBA for the solutions .Through the interface of the UP Finder, after loading a Systems Biology Markup Language (SBML) model in the olutions . Initialnditions . The UP 33.1E. coli, overproducing farnesyl pyrophosphate (FPP) is critical for increasing the product yield of lycopene because FPP is the native precursor in lycopene biosynthesis (c]) as the Targeted product for overproduction in the UP Finder , which represented the higher relevance to the related overproduction. Compared with the reported key genes for FPP/lycopene overproduction in the literatures  was chosen as the Target product in the UP Finder. accABCD, fabD, fabF, fabG, fabZ, fabI) were identified as the key pathways/genes for palmitoyl-ACP overproduction. 28 out of the 30 reactions are catalyzed by fatty acid synthases in Synechocystis sp. PCC 6803, in which 4 key genes  are involved. The reactions were ordered by their associated Ratio values, and the different Ratio values of the 30 identified up-regulated pathways represent different up-regulation levels to achieve the same Yield . Thus, the higher Ratio values indicated the higher demands of metabolic flux for up-regulation toward palmitoyl-ACP overproduction. On the other hand, the completed fatty acid synthesis pathways of the constraint-based model enabled the detailing of fluxes for each single reaction in multiple-repeated fatty acid biosynthesis. It was found that the reactions with same Ratio values reflected the similar level of metabolic flux for going through, which might be used as a quantitative standard for identifying metabolic modules in complex metabolic pathways.Directly converting COe energy . Fatty ae energy . Herein,Similar to the finding of Example 1, it was also found that the higher preferred gene targets presented closer metabolic distance to palmitoyl-ACP. Compared with reported key genes regarding to fatty acyl-ACP overproduction in the literatures , results4fluxopt, which evaluated their contribution toward theoretical maximum yield and enabled to pinpoint the results as yield/overproduction related. Although some key upstream pathways, such as glycolysis play an important role in improving target product yields when combining with downstream pathway enhancements, the sole overexpression of these genes might not directly contribute to the overproduction.In this study, a modeling software package named UP Finder was developed based on the COBRA toolbox in MATLAB. It facilitated the rapid identification of gene overexpression strategies for the metabolic engineering of targeted overproduction. Gene overexpression targets can be rationally determined by a quantitative evaluation procedure. Development of this interfacial software package was designed to provide the “one-click” convenience, and to facilitate the access for potential users without specific biochemistry and programming backgrounds. By taking advantage of standardized format of SBML models, UP Finder provided broad access for analyzing various targeted products in different microorganisms. Unlike OptForce and FSEOF, UP Finder specifically identified gene targets that were highly related to overproduction rather than all potential important targets. The UPA method used in the UP Finder investigated product yields for each single potential up-regulated pathway by constraining their fluxes with Ratio and Yield parameters improved the relevance of identified pathways for directly leading to the overproduction. In the UP Finder, gene overexpression targets were first identified as up-regulated pathways (the reaction with Ratio > 0) through a flux comparison between the theoretical maximum yield model and the wild-type model. By using fluxopt as a constraint, the contribution of each single identified up-regulated pathway on targeted overproduction was quantitatively evaluated based on the yield simulation. The restriction of Yield > 0 was used to exclude the reactions that are indirectly related to the overproduction, such as the pathways in central carbon metabolism. Thus, the UP Finder provided a quantitative procedure for identifying the key pathways toward overproduction. In addition, it was found in the examples that pathways with higher Ratio values showed closer metabolic distance to the targeted product. Since the greater Ratio value (up-regulation level) indicates the higher demands of metabolic flux for up-regulation to approach the theoretical maximum yield, overexpression of the particular gene would contribute more to the overproduction of targeted products. Therefore, results from the UP Finder not only presented all the key pathway/gene targets related to the overproduction, but the ranking of the outputs with their associated Ratio values also reflected the preference for considering overexpression strategies in pathway design.The quantitative evaluation that combining Ratio parameter can be used for identifying functional metabolic modules. In the analysis examples, it was found that pathways with similar level of Ratio values presented the adjacent locations with regard to metabolic functions. For example, in Example 1 , the isoprenoid unit synthesis module , and the FPP synthesis module . In Example 2 , and Synechocystis sp. PCC 6803 (iJN678.xml) have been tested with valid outputs in this UP Finder. Analysis examples demonstrated that the UP Finder is feasible to analyze gene overexpression targets for overproducing secondary metabolites and complex metabolic pathways, such as fatty acid biosynthesis. Given the decreasing cost of DNA synthesis, fast strain development for overproducing targeted products is becoming possible based on the large-scale DNA synthesis. Therefore, a user-friendly interfacial modeling tool that provides rapid pathway design would play an important role in the era of synthetic biology (In this version of the UP Finder, not all metabolites listed in the reconstructed model will have valid results. It is usually good for analyzing terminal metabolites, such as secondary metabolites. For metabolites with considerable degradation pathways, it may not have valid results because the up-regulated flux would be further consumed by the degradation without accumulation. Since the evaluation process only works for single pathway, the UP Finder does not provide the best combination of overexpression yet in this version. To achieve accurate prediction and high identity with experimental verification, a high quality of metabolic network reconstruction is necessary. Some current SBML models such as  biology .5In this study, a modeling tool named UP Finder was developed based on the COBRA toolbox. It facilitated the rapid identification of gene overexpression strategies to assist pathway design in metabolic engineering of targeted overproduction. Gene targets with highly related to overproduction were determined by a quantitative evaluation procedure. The graphical user interface of the UP Finder provided easier access for analyzing various targeted products in different microorganisms. Analysis examples for overproducing lycopene precursor and fatty acyl-ACP by the UP Finder showed high degree of agreement with the reported key genes in the literatures.


NADP+ is especially important in the metabolic engineering of yeasts for xylose fermentation, since NADPH is required by most yeasts in the initial step of xylose utilisation, including the fast-growing Kluyveromyces marxianus. In this simulation study of yeast metabolism, the complex interplay between these cofactors was investigated; in particular, how they may affect the possible roles of fructose-1,6-bisphosphatase, the pentose phosphate pathway, glycerol production and the pyruvate dehydrogenase bypass. Using flux balance analysis, it was found that the potential role of fructose-1,6-bisphosphatase was highly dependent on the cofactor specificity of the oxidative pentose phosphate pathway and on the carbon source. Additionally, the excessive production of ATP under certain conditions might be involved in some of the phenomena observed, which may have been overlooked to date. Based on these findings, a strategy is proposed for the metabolic engineering of a future xylose-fermenting yeast for biofuel production.The cofactor balances in metabolism is of paramount importance in the design of a metabolic engineering strategy and understanding the regulation of metabolism in general. ATP, NAD Kluyveromyces marxianus was performed with glucose or xylose as the carbon source under aerobic conditions [Saccharomyces cerevisiae in the absence of glucose, where the response is due to carbon source responsive transcription factors such as Adr1 and Mig1. Although the overall pattern of regulation in K. marxianus grown with D-xylose as carbon source instead of glucose resembled that of glucose derepression in S. cerevisiae, including the strong up-regulation of peroxisomal metabolism [S. cerevisiae have been dubbed “gluconeogenic” genes. Of particular interest is the fructose-1,6-bisphosphatase (FBP) reaction, catalysed by the Fbp1 protein and encoded by the FBP1 gene, which is under the control of the transcription factor Mig1 and glucose repression [K. marxianus [Recently, differential RNA-seq transcriptomics of nditions , 2. It ipression . What diarxianus . Hence, S. cerevisiae strain fermenting xylose has been postulated as a mechanism to increase NADPH production by a type of cyclic pentose phosphate pathway (PPP) [K. marxianus, in which xylose reductase utilises NADPH for reducing power [The up-regulation of the FBP1 gene in a recombinant ay (PPP) . NADPH ing power , 1. Howeng power , 8. In ong power . Heat ge1F0 ATPase. Thermogenin (UCP1) and similar proteins in mammals are the primary uncoupling proteins that allow the passage of protons [Yarrowia lipolytica is due to the promiscuous activity of an anion carrier protein [Another type of non-shivering thermogenesis is present in mammals, in which brown fat tissue catabolises acetyl CoA from lipids to generate heat. To avoid excessive ATP production, or equivalently, to avoid a limitation in ADP, uncoupling mechanisms exist. Hence, ATP can be excessive and metabolism is not always geared towards producing the maximal yield of ATP from a substrate. In fact, the uncoupling proteins found in mammals are an elaborate mechanism to avoid excessive ATP production. These allow the pumping of protons across the mitochondrial membrane, generating heat in the process and avoiding the need for protons to pass through the F protons . However protein , 12.a) Could an experimental condition or a genetic manipulation exist that would cause an ATP imbalance by excessive production of ATP (and thus an ADP limitation) in a yeast such as K. marxianus, or is there always a sufficient demand for ATP by processes such as biomass formation? (b) If such a situation indeed existed and considering that dedicated uncoupling proteins apparently are absent in yeasts, would substrate cycles like those induced by the FBP and PFK reactions be good candidate replacement mechanisms for uncoupling proteins under such conditions? (c) Further, since the FBP reaction is such a central reaction, with links to both ATP as well as to NADPH (as it is associated with gluconeogenesis), what is the true purpose of up-regulation of FBP in K. marxianus in a xylose medium when there is no apparent need for the FBP reaction? The transaldolase and transketolase reactions of the non-oxidative PPP, which catalyse the route of carbon entry from xylose, indeed produce fructose-6-phosphate which is upstream of FBP in the glycolytic pathway. (d) Finally, might there be as yet unidentified genetic manipulations which could theoretically enable anaerobic xylose fermentation for bioethanol production, given the complexities imposed by cofactor balances and flux constraints? These complex questions call for a rigid mathematical modelling framework such as Flux Balance Analysis (FBA). In this report, the complex interplay between cofactor balances, major metabolic pathways and redox cofactor specificity was investigated using FBA as a predictive simulation framework.A number of questions arise from the above observations: . The upAnalysis , implemeK. marxianus grown on glucose and xylose as respective carbon sources are presented below. For clarity, a visual guide to the reversibility of reactions and the names of reactions is provided as The results of the flux balance analyses of -1 g-1, specific to cell dry weight. Although the experimental xylose uptake and growth rates were approximately 50% of those for glucose, the power of FBA lies in calculating fluxes relative to a reference flux; in this case, the sugar uptake flux. Also, the measured production rates of ethanol and acetate were not included here as hard constraints, to facilitate exploring the theoretical limits of the model. The biomass formation rate in these simulations have arbitrary units and should be treated in a comparative manner among conditions. Throughout, fluxes and exchange rates were interpreted in a comparative sense and the units of mmol h-1 g-1 were omitted.Simulations of aerobic metabolism in glucose and xylose media were initially performed. Figs in silico carbon source, this was not the case for the PFK/FBP step and the FBP reaction did not become active when switching to the xylose in silico medium, in contrast to suggestions from literature that the FBP reaction was required to produce additional NADPH when xylose was the carbon source [Note that the reversible glucose-6-phosphate isomerase reaction flux switches direction between the conditions of glucose and xylose utilisation under simulated aerobic conditions Figs  and 2. Tn source .By optimising for excessive unbalanced NADPH production from xylose, instead of the growth rate, it was found that, in principle, two modes of cyclic PPP flux were possible Figs  and 4. Ain vivo as a limitation in NADP+ regeneration from NADPH. The NADPH imbalance would be further increased in a scenario of a ‘gluconeogenic’ net flux via aldolase and FBP towards glucose-6-phosphate, which produces three-fold more NADPH per xylose molecule  allowed steady state only when NADPH was allowed to accumulate in the model  concomitshown in . Thus, iAn interesting observation was that intermediates such as glucose-6-phosphate would accumulate if the NADPH balance was closed and if both PFK and FBP were inactivated . The gro+ limitation, in one of several ways. Transdehydrogenases that oxidise NADPH while reducing NAD+ would be one such mechanism, but transdehydrogenases are absent in yeasts in general, except for Pichia angusta [+ or NAD+ would affect the NADPH balance and, therefore, the potential role of FBP. The cofactor preference of oxidative PPP enzymes is known to vary among yeast species. The effect of cofactor preference in the oxidative PPP and the possibility of trans-oxidation-reduction reaction cycles in other parts of metabolism are explored in a later section. However, tightly linked to the redox cofactor balances is the ATP balance. These are linked not only via the oxidative metabolism in mitochondria, but also via the flux constraints in central metabolism, in which a given pathway may produce or consume ATP, NADH, NADPH and their partners in various ratios. Given that the total uptake rate of the carbon flux is constant in these simulations, alternative catabolic pathways with different cofactor stoichiometry in effect could cause a negative correlation between the production rates of different cofactors. Focussing still on upper central metabolism and the central role of FBP, the role of the ATP balance in flux routing is explored next.A cell may respond the constraint of NADPH overproduction, or NADP angusta . Conversin silico medium compared to the glucose in silico medium . This effect is however induced as an ATP avoidance strategy. Dedicated mechanisms in the electron transport chain for uncoupling ATP synthesis in yeasts have not been reported to date. Therefore, other mechanisms may be required to deal with the higher relative ATP production from xylose. The FBP reaction may perform exactly such a function by inducing an ATP hydrolysing substrate cycle together with PFK. Like the other glycolytic enzymes, the PFK1 and PFK2 genes were down-regulated in the xylose medium . NotablyS. cerevisiae strain to xylose under anaerobic conditions was recently determined, which also showed a four-fold up-regulation of FBP1 in the presence of xylose [Activating both the FBP and PFK activities and balancing ATP restored the rapid growth rate and avoided glycerol formation . The PFKf xylose . Details+, but can also use NAD+. In an attempt to obtain an increased cyclic PPP flux in the model, the cofactor specificity of glucose-6-phosphate dehydrogenase was changed in the model from NADP+ to NAD+ while allowing all biomass precursors and ATP to accumulate. As there was a deficiency of NADPH production in this model, cyclic PPP flux through FBP was observed as well as a higher growth rate  malic enzyme activity was required to be active in order to obtain a steady state in these simulations. Together with pyruvate carboxylase and malate dehydrogenase of the TCA cycle, these enzymes formed a substrate cycle in the simulation that effectively oxidised the excess NADH, originating from xylulose reductase activity, to NAD+ and supplied additional NADPH for xylose reductase. At the same time, excess ATP was consumed by the cycle, which resulted in the balancing of ATP production and utilisation. These reactions are also shown in in silico also manifested as xylitol accumulation, a phenomenon often observed in yeasts growing on xylose.The FBA model, which is independent from the RNA-seq data, also provides a means to explore the theoretical potential of future engineered yeast strains. From this perspective, metabolism is only limited by the flux constraints, the activity bounds assumed for the uptake rate of the carbon source, and thermodynamic feasibility of the individual reactions, and assumes that gene expression levels can be suitably set by the metabolic engineer. In contrast to the aerobic scenario, under simulated anaerobic conditions  ATP prodK. marxianus under anaerobic or oxygen-limited conditions has not been described thus far, which might be the relevant condition for the proposed malic enzyme cycle. It is also important to note that no flux was predicted in the oxidative PPP flux under the anaerobic condition utilising xylose. Sufficient NADPH for growth was produced by the oxaloacetate-decarboxylating malic enzyme. Ribose-5-phosphate was derived from xylulose-5-phosphate via ribulose-5-phosphate. PDB, which reportedly is utilised under anaerobic conditions [in silico knock-out of the PDB enzyme only resulted in a minor decrease in growth rate. From a flux perspective, its role is thus not clear for anaerobic conditions.The gene for malic enzyme, MAE1, was found in our annotation against UniProt, and was constitutively expressed under the aerobic conditions tested in glucose and xylose media using RNA-seq . To our nditions , only coS. cerevisiae strain to xylose under anaerobic conditions was recently determined [S. cerevisiae did not evolve for fermenting xylose, the latter observation cannot be extrapolated to K. marxianus. There was, however, a four-fold up-regulation of FBP1 in the recombinant S. cerevisiae strain when fermenting xylose.Responses of an engineered termined , showing+, NADP+ and ATP, combined with a variable and uncertain cofactor specificity for redox cofactors, render central metabolism complex to understand. FBA as a framework was demonstrated here to provide insight into the implications of alternative cofactor specificity of oxidative PPP enzymes. The potential role of the FBP reaction in cofactor balance was explored and, at the same time, its possible involvement in the ATP balance by forming a substrate cycle with PFK. It was shown that FBP could indeed contribute to additional NADPH production during xylose utilisation by causing a cyclic PPP flux, but only in a scenario where the oxidative PPP enzymes were not exclusively specific for NADP+ and only under aerobic conditions. Additionally, it was shown that in an in silico xylose medium, the excess ATP production was substantially greater than on glucose. This effect was increased further when the cofactor specificity of oxidative PPP enzymes for NAD+ increased. Thus, FBP may have a dual function under aerobic conditions—both to increase NADPH production in yeasts with one NAD+ -specific oxidative PPP enzyme and to hydrolyse excessive ATP.Cofactor balance is an important consideration in developing a metabolic engineering strategy. However, even without taking enzyme kinetics into account, the multiple reactions that involve NADS. cerevisiae that the glycolytic flux is strongly coupled to the yield of ATP during catabolism [S. cerevisiae, the ethanol yield on sugar was increased while the biomass yield was decreased [+ in another pathway, which might be the oxidative PPP, putting further flux constraints on the system. Notably, it was shown in this report that a potential malic enzyme cycle involving oxaloacetate-decarboxylating, NADP+-dependent malic enzyme could theoretically fulfil all three roles, enabling xylose fermentation, which currently is not theoretically possible in K. marxianus and other natural yeasts that do not possess a xylose isomerase gene. The cycle would oxidise the excessive NADH that cannot be oxidised by alcohol dehydrogenase due to flux constraints. In doing so, it also provides additional NADPH for the xylose reductase step for xylose utilisation. Finally, it forms an additional ATP sink, which may further pull glycolytic flux forward to ethanol production. Thermodynamic feasibility of this cycle involving malic enzyme should to be calculated based on the concentrations of ATP, ADP, Pi, NADH, NAD+, NADPH and NADP+ for this species, which is currently unknown. In addition, the PFK/FBP cycle may constitute an additional ATP sink, if it was found that ATP was inhibiting glycolytic flux. This cycle might, in fact, already be active in xylose-utilising K. marxianus, since our RNA-seq data showed that the FBP1 gene was derepressed in the xylose medium [Considering metabolic engineering of a future xylose-fermenting, bioethanol-producing yeast, at least three main aspects would be critical for success. Firstly, anaerobic growth requires an additional fermentative route for oxidation of the additional NADH that is produced by xylitol reductase and which cannot stoichiometrically be oxidised by alcohol dehydrogenases, as there is a shortage of electron acceptors. Secondly, it has been shown in tabolism . For insecreased . Since ue medium .+, PDB may also improve growth since it produces additional NADPH. However, the potential contribution of PDB to these effects was low, as this flux was constrained by the flux in lower glycolysis. In a scenario of a defined mineral medium where the organism has to synthesise all biomass components from a sugar as carbon and energy source, as was simulated here, the potential flux through the PDB is further decreased as anaplerotic reactions from lower glycolytic intermediates subtract from the potential flux to PDB to replenish oxaloacetate for amino acid synthesis.Other unanticipated findings were brought to light. Aerobic glycerol production as a physiological strategy to avoid excessive ATP production was also proposed here. Glycerol production is often observed in yeast fermentations and has primarily been associated with oxygen-limited growth conditions, but here the simulation demonstrated a different mechanism. The role of the PDB was also investigated. It was shown that allowance for a PDB flux increased the simulated growth rate under aerobic conditions when ATP was not allowed to be over-produced, as it hydrolysed ATP in the ATP over-producing scenario of aerobic xylose utilisation. In the case when oxidative PPP enzymes were more specific for NADK. marxianus UFS-Y2791 annotation, whereas PEP carboxykinase was constitutively expressed at low levels, as was pyruvate carboxylase [+-ATPase of the cell membrane [S. cerevisiae [+-ATPase exports protons that originated from the proton symport of nutrients, including NH4+. The ATP-hydrolysing capacity of the efflux pumps, however, results in the pumping of protons and NH4+, and hence cannot act as condition-independent replacements for uncoupling proteins.Two additional potential ATP sinks exist in yeasts. The first is another cycle in anaplerosis in which pyruvate and phosphoenolpyruvate are interconverted and dissipate free energy by effectively hydrolysing ATP , 25. PEPmembrane , 26, 27 revisiae . H+-ATPaK. marxianus strains, since the FBP1 gene was up-regulated in the glucose-free xylose medium, as observed in RNA-seq data. In addition, it was proposed here that a highly active malic enzyme cycle, which effectively exchanges redox equivalents from NADH to NADP+, would not only solve a redox imbalance under anaerobic, xylose fermenting conditions, but would draw the glycolytic flux forward due to forming an ATP sink. Finally, it is evident that ATP should not be thought of as always in demand. Depending on the condition, it might become excessive, especially in the case of pentose utilisation. The FBP/PFK substrate cycle and its regulation probably plays a key role in energy homeostasis, even in yeasts under selected conditions, which could determine phenotype switching. Alteration of the activity of the FBP/PFK substrate cycle and its effects on cofactor balances of ATP, NADH and NADPH may also be a mechanistic explanation for the recent discovery of the strong causative link between mutations affecting the FBP1 gene and renal cell carcinoma in humans.This work shows that cofactor balances should not be interpreted separately. Furthermore, it highlighted the importance of experimentally determining the cofactor specificity of oxidative PPP enzymes, as the prediction or calculation of fluxes will change, depending on these parameters. The ATP balance is highly relevant to metabolic engineering, since ATP not only has a negative feedback on glycolysis, but also a high ATP concentration might lead to excessive biomass formation, reducing the yield of a primary fermentation product such as ethanol. Thus, from a practical perspective, the proposed substrate cycle induced by FBP may be useful in decreasing the effective ATP yield on glucose, which may lead to an increase in ethanol production. This cycle might already be employed by natural S1 Figs(PPTX)Click here for additional data file.


The development of synthetic biology and metabolic engineering has painted a great future for the bio-based economy, including fuels, chemicals, and drugs produced from renewable feedstocks. With the rapid advance of genome-scale modeling, pathway assembling and genome engineering/editing, our ability to design and generate microbial cell factories with various phenotype becomes almost limitless. However, our lack of ability to measure and exert precise control over metabolite concentration related phenotypes becomes a bottleneck in metabolic engineering. Genetically encoded small molecule biosensors, which provide the means to couple metabolite concentration to measurable or actionable outputs, are highly promising solutions to the bottleneck. Here we review recent advances in the design, optimization and application of small molecule biosensor in metabolic engineering, with particular focus on optimization strategies for transcription factor (TF) based biosensors. Figure 1). And there are two major field of application for metabolite biosensors: (1) monitoring and screening the production of target products or important intermediates in engineered cells .Metabolic engineering has enabled the production of many chemicals currently produced from non-renewable resources using inexpensive and readily available raw materials . In ordeed cells . (2) ested cells  , with particular focus on optimization strategies for TF based biosensors.In the following sections, we review recent advances in the design, optimization and application of small molecule biosensor in metabolic engineering  resonance energy transfer (FRET) based biosensors typically have a pair of donor and acceptor fluorophores , which ain vivo microscopy. The FRET-based sensors were generated via fusion of the fluorescent proteins eCFP (enhanced cyan fluorescent protein) and Venus (yellow fluorescent protein) with trehalose repressor TreR from Escherichia coli. In addition to being directly involved in the metabolism of trehalose, T6P also regulates glucose metabolism and physiological processes as a signaling molecule. In order to test the utility of these biosensors in vivo, they examined the accumulation of T6P in E. coli cells under low- and high-osmolarity pressure. In low-osmolarity conditions, the addition of trehalose resulted in a FRET signal increase, indicating accumulation of T6P in cells. On the contrary, the signal was greatly reduced under high osmotic conditions, indicating that bypassing T6P, trehalose is directly converted to glucose. In the future, these biosensors can potentially be used for real-time monitoring of trehalose pathway regulations in any cell types  plays a key role in sugar metabolism in bacteria, yeasts and plants. In order to directly visualize the concentration of intracellular T6P, ll types .Amino acid industry, with annual production of more than 4 million tons, is a major pillar for industrial biotechnology. Lysine, one of the most demanding amino acids, has a global annual demand of about 1.5 million tones . RecentlE. coli with the fluorescent proteins mTFP and Venus. The application of this biosensor was performed in neurons, in which mitochondrial pyruvate consumption was discovered to go up by 300% within seconds by a strong calcium transient, while glucose levels remained unchanged . In order to test the performance of the biosensor in E. coli, the addition of exogenous butyrolactam, valerolactam, and caprolactam were carried out, respectively. Valerolactam and caprolactam induced 2.38 and 2.08 fold fluorescence increase compared to the uninduced baseline, while a moderate 1.47 fold of induction was detected for butyrolactam. The TF-based lactam biosensor has sufficient specificity to differentiate lactams with a wide dynamic range, which would be potentially helpful for the high-throughput screening of high-titer lactam producing strains  biosensor in S. cerevisiae by the fusion of MetJ repressor of E. coli and transcriptional activator domain B42. The resulting hybrid activator will bind SAM, which then can interact with the operator (metO) inserted upstream of the TATA box of CYC1 promoter, leading to over-expression of downstream reporter gene. They used fluorescent protein Venus as the reporter for identifying unknown genes that will affect intracellular SAM concentration. As a result, GAL11 was discovered as a novel multicopy enhancer of SAM level. Overexpression of GAL11 resulted in an approximately 3.3-fold improvement in the concentration of SAM  mutants by fluorescence activated cell sorting (FACS). The activity of the enzyme mutant toward the substrate 4-methyl-2-pentanone was improved by 36% with an 8-fold increase in the Km value.A large number of metabolite-responsive TFs exist in the cell, providing a rich source for engineering TF-based biosensors. And they have become the most widely used high-throughput screening tool in metabolic engineering. However, there are some key bottlenecks in construction and application of TF-based biosensor. It is still very challenging to transplant heterologous TFs into new hosts to generate functional biosensors. And detecting extracellular target metabolite concentrations seems out of the scope of TF-based biosensors.2+ cations) (Riboswitches are regulatory regions of mRNA molecules with conserved ligand-binding (sensor) domains and a variable sequence that can regulate downstream gene translation . Comparecations)  and fluocations) .ykkC motif RNA, which is the longest riboswitch candidate, exists in much of the bacterial domain of life. ykkC motif RNA commonly controls the expression of proteins may function as guanidine carboxylase and guanidine transporter, which were previously incorrectly annotated as urea carboxylases and multidrug efflux pumps  and Bacillus subtilis (BSRS) to regulate the expression of citrate synthase (gltA) gene and TCA cycle activity in Corynebacterium glutamicum LP917, respectively. The lysine production of the engineered strain ECRS-gltA and BSRS-gltA was 63% and 38% higher than their parent strain . A 89% increase in lysine yield was achieved comparing to that of the C. glutamicum LPECRS strain with deregulated aspartokinase  platform, L-Phe producing strain with a yield of 9.28 g/L.Although biosensor devices for small molecule detection are gaining increasing attention, very limited TFs have been successfully used in metabolic engineering. One of the majors challenges is their poor sensitivity and dynamic range . In a reE. coli, In order to increase the dynamic range of a FadR-based biosensor for sensing fatty acid in E. coli ara operon to sense D-arabinose  and the synthetic promoter was modified by incorporating six upstream operators. Therefore, the PhlF can repress downstream gene expression by steric hindrance. By introducing the DAPG biosensor, the new system will respond to extracellular DAPG with 50-fold induction ratio and 0.9 μM DAPG detection limit, which is comparable to the classic Dox-induced TetR system  leading to GFP expression. The fluoresce intensity of E. coli cells carrying the biosensor would change in accordance with benzoate concentrations, but were completely unresponsive to the substrate benzamide. They used this benzoate biosensor to screen benzamidases from metagenomic library in E. coli and successfully obtained 11 different amidases from more than 96,000 clones.Environmental microbes is an excellent source of industrial enzymes . Howeveretc. Directed evolution is a powerful strategy for artificially improving enzyme properties in a short period of time. However, how to obtain a mutant enzyme with desired property from a mutant library usually with a significant size is a key challenge in directed evolution. A small molecule biosensor is a perfect solution for setting up a screening system in directed evolution experiments. For instance, Caldicellulosiruptor saccharolyticus for improved lactulose production. The selected mutant C2E enzyme was discovered to have ∼32-fold higher expression levels, which demonstrates that it is possible to use lactulose biosensor for screening lactulose high-producing strain in the future (E. coli TF AraC to induce gene expression by responding to intracellular triacetate lactone (TAL). Via high throughput screening, they obtained a highly active 2-pyrone synthase, with 20% increased activity.Under industrial settings, wild-type enzymes often have limitations in different aspects such as activity and thermostability e future . In anotE. coli. The authors changed the ligand specificity of AraC by changing the hydrophobic region of AraC, and obtained mutant AraC which can respond to ectoine. As a result of following ectonine biosensor based screening, the yield of ectoine was significantly improved by about 3.3-fold comparing to the wild-type ectoine production strain.One of the main goals of metabolic engineering is to obtain a strain that can produce desired product with high yield. Many of the TF-based sensors targeting final products are used for high-throughput screening of high-yield microorganism. Ectoine is a compatible solute widely produced by bacteria for adapting to changes in osmotic stress . In the E. coli using the TF BmoR from Pseudomonas butanovora and its induced promoter PBMO. By combining the expression of tetA and gfp gene, the concentration of 1-butanol in the cell was coupled with the cell growth and fluorescence intensity. After a single round of positive selection, the 1-butanol production phenotype was improved by120-fold.Biofuel is another major class of target chemicals that is actively pursued in metabolic engineering efforts. Many different biosensors were developed to sense biofuel molecules. L-threonine is an important industrial amino acid, mainly used in animal feed industry, with annual production of 300,000 tons in 2014. And its market growth was expected to be over 20% in the next 5–10 years  (N-acetylneuraminic acid (NeuAc) biosensor was constructed based on a NeuAc-specific aptamer and a sfGFP fluorescence reporter in order to increase the yield of NeuAc, which is popular for its promising edible and pharmaceutical applications. With the designed biosensor, the NeuAc biosynthesis pathway was optimized by RBS library screening and key enzyme evolution, which improved the titer by 34% and 23%, respectively. Finally, a mutant with 8.31 g/L NeuAc production was obtained after two-stage fermentation in minimal medium with the glucose as sole carbon source  , Xiu et n source .2. Improving yield and production efficiency via metabolic engineering is a key step for their real industrial applications. Small molecule responsive biosensors have become and will continuously be one of the most powerful tools in real-time monitoring and systematic optimization of biosynthetic pathways in the metabolic engineering works.With the continuous advancement of biotechnology, more and more petroleum-based chemicals, fuels as well as pharmaceuticals can be produced through microorganisms from biomass even directly from COIn this review, we have examined the recent developments in metabolite-responsive biosensors and their various applications. Although there are many successful examples of metabolite-responsive biosensors from all three major categories, which were used in either the “proof-of-concept” experiments or achieved real impact in the industrial scale applications, there are some challenges need to be solved in a systematic way. First is how to streamline the discovery, design and optimization process of novel biosensors. It is very likely that every single natural metabolite has its own corresponding biosensor somewhere in the nature. However, how to quickly find them through enormous accumulated bioinformatics data and properly incorporated into a desired host via rational design is a major challenge. Second, despite that metabolite-responsive biosensors are extremely useful as a simple reporter/screening system, they will increasingly be incorporated into cellular control networks as a key layer that bridges small molecule and macromolecule. And how to design and optimize multiple metabolite-responsive biosensors that are compatible with each other and other genetic parts/devices is another grand challenge.All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.