Metabolic reactions of single-cell organisms are routinely observed to become dispensable or even incapable of carrying activity under certain circumstances. Yet, the mechanisms as well as the range of conditions and phenotypes associated with this behavior remain very poorly understood. Here we predict computationally and analytically that any organism evolving to maximize growth rate, ATP production, or any other linear function of metabolic fluxes tends to significantly reduce the number of active metabolic reactions compared to typical nonoptimal states. The reduced number appears to be constant across the microbial species studied and just slightly larger than the minimum number required for the organism to grow at all. We show that this massive spontaneous reaction silencing is triggered by the irreversibility of a large fraction of the metabolic reactions and propagates through the network as a cascade of inactivity. Our results help explain existing experimental data on intracellular flux measurements and the usage of latent pathways, shedding new light on microbial evolution, robustness, and versatility for the execution of specific biochemical tasks. In particular, the identification of optimal reaction activity provides rigorous ground for an intriguing knockout-based method recently proposed for the synthetic recovery of metabolic function.
A fundamental problem in systems biology is to understand how living cells adjust the usage pattern of their components to respond and adapt to specific genetic, epigenetic, and environmental conditions. In complex metabolic networks of single-cell organisms, there is mounting evidence in the experimental [1]–[6] and modeling [7]–[14] literature that a surprisingly small part of the network can carry all metabolic functions required for growth in a given environment, whereas the remaining part is potentially necessary only under alternative conditions [15]. The mechanisms governing this behavior are clearly important for understanding systemic properties of cellular metabolism, such as mutational robustness, but have not received full attention. This is partly because current modeling approaches are mainly focused on predicting whole-cell phenotypic characteristics without resolving the underlying biochemical activity. These approaches are typically based on optimization principles, and hence, by their nature, do not capture processes involving non-optimal states, such as the temporary activation of latent pathways during adaptive evolution towards an optimal state [16], [17]. To provide mechanistic insight into such behaviors, here we study the metabolic system of single-cell organisms under optimal and non-optimal conditions in terms of the number of active reactions (those that are actually used). We implement our study within a flux balance-based framework [18]–[23]. Figure 1 illustrates key aspects of our analysis using the example of Escherichia coli. For any typical non-optimal state (Figure 1B), all the reactions in the metabolic network are active, except for those that are necessarily inactive due either to mass balance constraints or environmental conditions (e. g. , nutrient limitation). In contrast, a large number of additional reactions are predicted to become inactive for any metabolic flux distribution maximizing the growth rate (Figure 1A). This spontaneous reaction silencing effect, in which optimization causes massive reaction inactivation, is observed in all four organisms analyzed in this study, H. pylori, S. aureus, E. coli, and S. cerevisiae, which have genomes and metabolic networks of increasing size and complexity (Materials and Methods). Our analysis reveals two mechanisms responsible for this effect: (1) irreversibility of a large number of reactions, which under intracellular physiological conditions [14] is shared by more than 62% of all metabolic reactions in the organisms we analyze (Table 1 and Note 1); and (2) cascade of inactivity triggered by the irreversibility, which propagates through the metabolic network due to stoichiometric and flux balance constraints. We provide experimental evidence of this phenomenon and explore applications to data interpretation by analyzing intracellular flux and gene activity data available in the literature. The drastic difference between optimal and non-optimal behavior is a general phenomenon that we predict not only for the maximization of growth, but also for the optimization of any typical objective function that is linear in metabolic fluxes, such as the production rate of a metabolic compound. Interestingly, we find that the resulting number of active reactions in optimal states is fairly constant across the four organisms analyzed, despite the significant differences in their biochemistry and in the number of available reactions. In glucose media, this number is ∼300 and approaches the minimum required for growth, indicating that optimization tends to drive the metabolism surprisingly close to the onset of cellular growth. This reduced number of active reactions is approximately the same for any typical objective function under the same growth conditions. We suggest that these findings will have implications for the targeted improvement of cellular properties [24]. Recent work predicts that the knockout of specific enzyme-coding genes can enhance metabolic performance and even rescue otherwise nonviable strains [25]. The possibility of such knockouts bears on the issue of whether the inactivation of the corresponding enzyme-catalyzed reactions would bring the whole-cell metabolic state close to the target objective. Thus, our identification of a cascading mechanism for inducing optimal reaction activity for arbitrary objective functions provides a natural set of candidate genetic interventions for the knockout-based enhancement of metabolic function [25].
We model cellular metabolism as a network of metabolites connected through reaction and transport fluxes. The state of the system is represented by the vector v = (v1, …, vN) T of these fluxes, including the fluxes of n internal and transport reactions, as well as nex exchange fluxes for modeling media conditions. Under the constraints imposed by stoichiometry, reaction irreversibility, substrate availability, and the assumption of steady-state conditions, the state of the system is restricted to a feasible solution space (Materials and Methods). Within this framework, we first consider the number of active reactions in a typical non-optimal state v∈M. We can prove that, with the exception of the reactions that are inactive for all v∈M, all the metabolic reactions are active for almost all v∈M, i. e. , for any typical state chosen randomly from M (Text S1, Section 1). Accordingly, the number n+ (v) of active reactions in a typical non-optimal state is constant, i. e. , (1) The reactions that are inactive for all states are so either due to mass balance or environmental conditions, and can be identified numerically using flux coupling [26] and flux variability analysis [9]. We now turn to the maximization of growth rate, which is often hypothesized in flux balance-based approaches and found to be consistent with observation in adaptive evolution experiments [31]–[34]. Performing numerical optimization in glucose minimal media (Materials and Methods), we find that the number of active reactions in such optimal states is reduced by 21%–50% compared to typical non-optimal states, as indicated in the middle bars of Figure 2. Interestingly, the absolute number of active reactions under maximum growth is ∼300 and appears to be fairly independent of the organism and network size for the cases analyzed. We observe that the minimum number of reactions required merely to sustain positive growth [7], [8] is only a few reactions smaller than the number of reactions used in growth-maximizing states (bottom bars, Figure 2). This implies that surprisingly small metabolic adjustment or genetic modification is sufficient for an optimally growing organism to stop growing completely, which reveals a robust-yet-subtle tendency in cellular metabolism: while the large number of inactive reactions offers tremendous mutational and environmental robustness[52], the system is very sensitive if limited only to the set of reactions optimally active. The flip side of this prediction is that significant increase in growth can result from very few mutations, as observed recently in adaptive evolution experiments [35]. We now turn to mechanisms underlying the observed reaction silencing, which is spread over a wide range of metabolic subsystems (see Figure 1 for E. coli). The phenomenon is caused by growth maximization via reaction irreversibility and cascading of inactivity. Although we have focused so far on maximizing the biomass production rate, the true nature of the fitness function driving evolution is far from clear [44]–[47]. Organisms under different conditions may optimize different objective functions, such as ATP production or nutrient uptake, or not optimize a simple function at all. In particular, some metabolic behaviors, such as the selection between respiration and fermentation in yeast, cannot be explained by growth maximization [48]. Other behaviors may be systematically different in situations mimicking natural environments [49]. Moreover, various alternative target objectives can be conceived and used in metabolic engineering applications. We emphasize, however, that while specific numbers may differ in each case, all the arguments leading to Eqs. (2) – (4) are general and apply to any objective function that is linear in metabolic fluxes. To obtain further insights, we now study the number of active reactions in organisms optimizing a typical linear objective function by means of random uniform sampling. We first note the fact (proved in Text S1, Section 4) that with probability one under uniform sampling, the optimal solution set Mopt consists of a single point, which must be a “corner” of M, termed an extreme point in the linear programming literature. In this case, dopt = 0, and Eq. (2) becomes (5) With the additional requirement that the organism show positive growth, we uniformly sample these extreme points, which represent all distinct optimal states for typical linear objective functions. We find that the number of active reactions in typical optimal states is narrowly distributed around that in growth-maximizing states, as shown in Figure 4. This implies that various results on growth maximization extend to the optimization of typical objective functions. In particular, we see that a typical optimal state is surprisingly close to the onset of cellular growth (estimated and shown as dashed vertical lines in Figure 4). Despite the closeness, however, the organism maintains high versatility, which we define as the number of distinct functions that can be optimized under growth conditions. To demonstrate this, consider the H. pylori model, which has 392 reactions that can be active, among which at least 302 must be active to sustain growth (Table 3). While only a few more than 302 active reactions are sufficient to optimize any objective function, the number of combinations for choosing them can be quite large. For example, there are combinations for choosing, say, 5 extra reactions to be active. Moreover, this number increases quickly with the network size: S. cerevisiae, for example, has less than 2. 5 times more reactions than H. pylori, but about 500 times more combinations (). Our results help explain previous experimental observations. Analyzing the 22 intracellular fluxes determined by Schmidt et al. [50] for the central metabolism of E. coli in both aerobic and anaerobic conditions, we find that about 45% of the fluxes are smaller than 10% of the glucose uptake rate (Table 4). However, less than 19% of the reversible fluxes and more than 60% of the irreversible fluxes are found to be in this group (Fisher exact test, one-sided p<0. 008). For the 44 fluxes in the S. cerevisiae metabolism experimentally measured by Daran-Lapujade et al. [51], less than 8% of the reversible fluxes and more than 42% of the irreversible fluxes are zero (Table 5; Fisher exact test, one-sided p<10−7). This higher probability for reduced fluxes in irreversible reactions is consistent with our theory and simulation results (Table 6) combined with the assumption that the system operates close to an optimal state. For the E. coli data, this assumption is justified by the work of Burgard & Maranas [44], where a framework for inferring metabolic objective functions was used to show that the organisms are mainly (but not completely) driven by the maximization of biomass production. The S. cerevisiae data was also found to be consistent with the fluxes computed under the assumption of maximum growth [52]. Additional evidence for our results is derived from the inspection of 18 intracellular fluxes experimentally determined by Emmerling et al. [53] for both wild-type E. coli and a gene-deficient strain not exposed to adaptive evolution. It has been shown [21] that while the wild-type fluxes can be approximately described by the optimization of biomass production, the genetically perturbed strain operates sub-optimally. We consider the fluxes that are more than 10% (of the glucose uptake rate) larger in the gene-deficient mutant than in the wild-type strain. This group comprises less than 27% of the reversible fluxes but more than 45% of the irreversible fluxes (Table 7; Fisher exact test, one-sided p<0. 12). This correlation indicates that irreversible fluxes tend to be larger in non-optimal metabolic states, consistently with our predictions. Altogether, our results offer an explanation for the temporary activation of latent pathways observed in adaptive evolution experiments after environmental [16] or genetic perturbations [17]. These initially inactive pathways are transiently activated after a perturbation, but subsequently inactivated again after adaptive evolution. We hypothesize that transient suboptimal states are the leading cause of latent pathway activation. Since we predict that a large number of reactions are inactive in optimal states but active in typical non-optimal states, many reactions are expected to show temporary activation if we assume that the state following the perturbation is suboptimal and both the pre-perturbation and post-adaptation states are near optimality. To demonstrate this computationally for the E. coli model, we consider the idealized scenario where the perturbation to the growth-maximizing wild type is caused by a reaction knockout and the initial response of the metabolic network—before the perturbed strain evolves to a new growth-maximizing state—is well approximated by the method of minimization of metabolic adjustment (MOMA) [21]. MOMA assumes that the perturbed organisms minimize the square-sum deviation of its flux distribution from the wild-type distribution (under the constraints imposed by the perturbation). Figure 5A shows the distribution of the number of active reactions for single-reaction knockouts that alter the flux distribution but allow positive MOMA-predicted growth. While the distribution is spread around 400–500 for the suboptimal states in the initial response, it is sharply peaked around 300 for the optimal states at the endpoint of the evolution, which is consistent with our results on random sampling of the extreme points (Figure 4). We thus predict that the initial number of active reactions for the unperturbed wild-type strain (which is 297, as shown by a dashed vertical line in Figure 5A) typically increases to more than 400 following the perturbation, and then decays back to a number close to 300 after adaptive evolution in the given environment, as illustrated schematically in Figure 5B. A neat implication of our analysis is that the active reactions in the early post-perturbation state includes much more than just a superposition of the reactions that are active in the pre- and post-perturbation optimal states, thus explaining the pronounced burst in gene expression changes observed to accompany media changes and gene knockouts [16], [17]. For example, for E. coli in glucose minimal medium, temporary activation is predicted for the Entner-Doudoroff pathway after pgi knockout and for the glyoxylate bypass after tpi knockout, in agreement with recent flux measurements in adaptive evolution experiments [17]. Another potential application of our results is to explain previous experimental evidence that antagonistic pleiotropy is important in adaptive evolution [54], as they indicate that increasing fitness in a single environment requires inactivation of many reactions through regulation and mutation of associated genes, which is likely to cause a decrease of fitness in some other environments [15].
All the stoichiometric data for the in silico metabolic systems used in our study are available at http: //gcrg. ucsd. edu/In_Silico_Organisms. For H. pylori 26695 [77], we used a medium with unlimited amount of water and protons, and limited amount of oxygen (5 mmol/g DW-h), L-alanine, D-alanine, L-arginine, L-histidine, L-isoleucine, L-leucine, L-methionine, L-valine, glucose, Iron (II and III), phosphate, sulfate, pimelate, and thiamine (20 mmol/g DW-h). For S. aureus N315 [78], we used a medium with limited amount of glucose, L-arginine, cytosine, and nicotinate (100 mmol/g DW-h), and unlimited amount of iron (II), protons, water, oxygen, phospate, sulfate, and thiamin. For E. coli K12 MG1655 [75], we used a medium with limited amount of glucose (10 mmol/g DW-h) and oxygen (20 mmol/g DW-h), and unlimited amount of carbon dioxide, iron (II), protons, water, potassium, sodium, ammonia, phospate, and sulfate. For S. cerevisiae S288C [76], we used a medium with limited amount of glucose (10 mmol/g DW-h), oxygen (20 mmol/g DW-h), and ammonia (100 mmol/g DW-h), and unlimited amount of water, protons, phosphate, carbon dioxide, potassium, and sulfate. The flux through the ATP maintenance reaction was set to 7. 6 mmol/g DW-h for E. coli and 1 mmol/g DW-h for S. aureus and S. cerevisiae. Under steady-state conditions, a cellular metabolic state is represented by a solution of a homogeneous linear equation describing the mass balance condition, (6) where S is the m×N stoichiometric matrix and is the vector of metabolic fluxes. The components of v = (v1, …, vN) T include the fluxes of n internal and transport reactions as well as nex exchange fluxes, which model the transport of metabolites across the system boundary. Constraints of the form vi≤βi imposed on the exchange fluxes are used to define the maximum uptake rates of substrates in the medium. Additional constraints of the form vi≥0 arise for the reactions that are irreversible. Assuming that the cell' s operation is mainly limited by the availability of substrates in the medium, we impose no other constraints on the internal reaction fluxes, except for the ATP maintenance flux for S. aureus, E. coli, and S. cerevisiae (see Strains and media section above). The set of all flux vectors v satisfying the above constraints defines the feasible solution space, representing the capability of the metabolic network as a system. The flux balance analysis (FBA) [18]–[20], [22], [23] used in this study is based on the maximization of a metabolic objective function cTv within the feasible solution space M, which is formulated as a linear programming problem: (7) We set αi = −∞ if vi is unbounded below and βi = ∞ if vi is unbounded above. For a given objective function, we numerically determine an optimal flux distribution for this problem using an implementation of the simplex method [43]. In the particular case of growth maximization, the objective vector c is taken to be parallel to the biomass flux, which is modeled as an effective reaction that converts metabolites into biomass. To find a set of reactions from which none can be removed without forcing zero growth, we start with the set of all reactions and recursively reduce it until no further reduction is possible. At each recursive step, we first compute how much the maximum growth rate would be reduced if each reaction were removed from the set individually. Then, we choose a reaction that causes the least change in the maximum growth rate, and remove it from the set. We repeat this step until the maximum growth rate becomes zero. The set of reactions we have just before we remove the last reaction is a desired minimal reaction set. Note that, since the algorithm is not exhaustive, the number of reactions in this set is an upper bound and approximation for the minimum number of reactions required to sustain positive growth.
Cellular growth and other integrated metabolic functions are manifestations of the coordinated interconversion of a large number of chemical compounds. But what is the relation between such whole-cell behaviors and the activity pattern of the individual biochemical reactions? In this study, we have used flux balance-based methods and reconstructed networks of Helicobacter pylori, Staphylococcus aureus, Escherichia coli, and Saccharomyces cerevisiae to show that a cell seeking to optimize a metabolic objective, such as growth, has a tendency to spontaneously inactivate a significant number of its metabolic reactions, while all such reactions are recruited for use in typical suboptimal states. The mechanisms governing this behavior not only provide insights into why numerous genes can often be disabled without affecting optimal growth but also lay a foundation for the recently proposed synthetic rescue of metabolic function in which the performance of suboptimally operating cells can be enhanced by disabling specific metabolic reactions. Our findings also offer explanation for another experimentally observed behavior, in which some inactive reactions are temporarily activated following a genetic or environmental perturbation. The latter is of utmost importance given that nonoptimal and transient metabolic behaviors are arguably common in natural environments.
computational biology/metabolic networks
computational biology/systems biology
Spontaneous Reaction Silencing in Metabolic Optimization