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Proc Natl Acad Sci U S A. Feb 22, 2011; 108(8): 3124–3129.
Published online Feb 7, 2011. doi:  10.1073/pnas.1009772108
PMCID: PMC3044393
Applied Physical Sciences, Biophysics and Computational Biology

Dispensability of Escherichia coli’s latent pathways


Gene-knockout experiments on single-cell organisms have established that expression of a substantial fraction of genes is not needed for optimal growth. This problem acquired a new dimension with the recent discovery that environmental and genetic perturbations of the bacterium Escherichia coli are followed by the temporary activation of a large number of latent metabolic pathways, which suggests the hypothesis that temporarily activated reactions impact growth and hence facilitate adaptation in the presence of perturbations. Here, we test this hypothesis computationally and find, surprisingly, that the availability of latent pathways consistently offers no growth advantage and tends, in fact, to inhibit growth after genetic perturbations. This is shown to be true even for latent pathways with a known function in alternate conditions, thus extending the significance of this adverse effect beyond apparently nonessential genes. These findings raise the possibility that latent pathway activation is in fact derivative of another, potentially suboptimal, adaptive response.

Keywords: complex networks, flux balance analysis, metabolic networks, gene dispensability, synthetic rescues

Living cells are surprisingly robust against mutations and, in particular, against gene knockouts (15). The origin of mutational robustness—whether it is a directly evolved trait or a byproduct of evolutionary history—remains debatable (6). In either case, metabolic network analysis shows that the nonessentiality of enzymes and associated genes is largely due to the inactivity of the corresponding metabolic reactions under laboratory conditions (79). This leaves environmental robustness as the natural candidate to explain gene nonessentiality. Yet, apart from chemical stress-based assays (10), studies designed to test whether nonessential genes become essential under different conditions have failed to identify a phenotype for more than a small fraction of additional genes (11). A recent groundbreaking study has shown, however, that a large fraction of reactions not active under standard laboratory conditions become transiently active after a genetic or environmental perturbation (12, 13). Why? The prevailing interpretation has been that the transient activation of such latent pathways facilitates adaptation to new conditions, thereby attributing function to genes that have been classified as dispensable for the lack of phenotype in steady-state experiments. This is naturally formulated as the hypothesis that latent pathways have a positive impact on postperturbation growth (cellular reproduction), which is a measure of competitive advantage with a strong empirical basis (13, 10, 11, 13). Even for genes with known functions under different conditions, this hypothesis is appealing as it suggests the possibility of an alternate phenotype that would not be detected in traditional high-throughput screens of knockout mutants (14).

Here, we test this hypothesis using the most complete in silico reconstruction of the metabolic network of Escherichia coli K-12 MG1655 (15, 16) and perturbations caused by single-gene knockouts. The response of the metabolic network to knockout perturbations is modeled using both model-independent analysis and the two most accepted phenomenological approaches, minimization of metabolic adjustment (MOMA) (17) and regulatory on/off minimization (ROOM) (18) (Materials and Methods). Starting from a growth-maximizing state determined by flux balance analysis (FBA) (19), we compare the early postperturbation growth rate (Materials and Methods) of the original organism with that of a modified organism in which the latent reactions have been disabled. We consider glucose minimal medium and gene knockouts that necessarily change the original metabolic flux distribution but that nonetheless are compatible with nonzero growth according to FBA. There are 52 enzyme-coding genes associated with 97 metabolic reactions in the reconstructed network that satisfy this condition. We systematically predict the impact of latent pathway activation on growth rate following perturbations caused by the knockouts of each of these genes.


Phenomenological Analysis.

Fig. 1 illustrates the essence of our approach. In an optimal growth state, as observed for E. coli after adaptive evolution in fixed environmental conditions (20), many metabolic reactions are inactive (8, 12). Shortly after a perturbation, however, both the original and modified strain operate in a suboptimal growth state (17, 18), which we model using MOMA or ROOM (Fig. 1A). In silico (8) and laboratory (12) experiments show that this change is accompanied by a burst of reaction activity (Fig. 1B), reflecting regulatory changes that locally reroute fluxes in the short-term metabolic response to the perturbation (21, 22). If the perturbation is nonlethal, the perturbed organisms will undergo adaptive evolution—adopting beneficial mutations over longer timescales (23, 24) to achieve a new optimal growth state, which can be predicted by FBA (18, 25).

Fig. 1.
Hypothetical growth impact of latent pathways under a knockout perturbation. (A and B) The solid lines indicate the drop in growth rate (A) and the burst in latent reaction activation (B) that follow a gene knockout. The dashed lines indicate the possible ...

For the perturbations considered in this study, the average and standard deviation of the number of transiently active reactions is 291 ± 83 and 120 ± 59 for MOMA and ROOM, respectively. This difference is expected because ROOM, by design, favors a small number of significant flux changes, which reflects the fact that ROOM may model a later stage in the adaptive response pictured in Fig. 1B than MOMA (18). These numbers should be compared with the number of active reactions in the corresponding growth-maximizing states, which is 385 in the wild type and remains 385 on average in the knockout mutants, with an average of ≈98% overlap between the two sets for the simplex solutions we consider; these numbers are representative for other choices of optima (26, 27) within our models (Sensitivity to Alternate Optima section in SI Text). We emphasize that because the modified organism lacks only transiently active (latent) metabolic reactions, the optimal steady states are identical to those of the unmodified strain both before and after the perturbation. Our question is then whether the early postperturbation growth rate (before adaptive evolution) will be lower (red), remain equal (green), or become higher (blue) when these latent pathways are not present (Fig. 1A).

Our principal result is that the strains lacking latent pathways systematically show equal or better adaptation as determined by growth within our in silico model, regardless of the approach used to model the organisms’ response to perturbations. We assume that the organisms are in an optimal growth state both before and long after the perturbation, which accounts for cases that have received much attention in the literature (8, 12, 13), but we note that our conclusions remain equally valid when this assumption is relaxed (Effects of Nonoptimal Reference States section in SI Text). Table 1 summarizes the results for all 52 single-gene knockouts considered in our study. Relative to the unmodified strain in the suboptimal state following a gene knockout, the strain lacking the latent pathways exhibits equal or improved growth in 100% of the cases according to MOMA (Fig. 2A) and in 98% of the cases according to ROOM (Fig. 2B). Across all knockout perturbations, this corresponds to an average change of +8.5 and +1.2% of the optimal wild-type growth rate, respectively. With either approach, a large fraction of mutants (50% for MOMA, 77% for ROOM) show a negligible difference in growth rate (within ± 1% of the wild-type growth rate) when the latent pathways are disabled. If only cases exhibiting significant changes are considered, the removal of latent pathways consistently increases the early postperturbation growth rate for all mutants, by an average of +16.9% for MOMA (Fig. 2A) and +4.6% for ROOM (Fig. 2B). Thus, for almost all knockouts, the strain lacking latent pathways is predicted to suffer no competitive disadvantage compared to the latent pathway-enabled strain. On the contrary, we predict that it more often shows improved growth in the suboptimal regime shortly after the perturbation.

Fig. 2.
Predicted adaptive impact of latent pathways in E. coli under single-gene knockout perturbations. Each row indicates the difference in early postperturbation growth rate between the strains with and without latent pathways for the knockout perturbation ...
Table 1.
Summary of the predicted impact of latent pathways in E. coli K-12 MG1655 under single-gene knockout perturbations

The set of transiently active (latent) reactions depends on the perturbation. Even though we predict that, in general, the removal of one of these 52 sets increases growth under the corresponding knockout, the same removal may in principle have an adverse effect under a different knockout. To address this possibility, we first note that the sets of simultaneously nonessential latent reactions remain sizeable: an average of 258 ± 79 for MOMA and 109 ± 53 for ROOM. For a given knockout perturbation, this set is defined as the subset of the original latent reactions that are inactive in the optimal growth state we consider for each of the other 51 knockout mutants. These reactions are therefore dispensable for optimal growth, both in the wild type and all 52 single gene knockout strains we consider, but are nonetheless transiently activated in response to the given perturbation. We have tested the impact of disabling these reduced sets of latent reactions under the corresponding knockouts (Materials and Methods). As shown in Fig. 2 C and D, the presence of these simultaneously nonessential latent reactions has the same trend of inhibiting growth adaptation as found for the full sets of transiently activated reactions.

The possibility that latent pathway activation enhances cells’ viability following a perturbation is a compelling hypothesis, as it would reveal functions for genes that have thus far eluded high-throughput phenotype screens. We note, however, that our analysis also predicts the transient activation of pathways that do, in fact, have known phenotypes under different conditions. For example, activation of the glyoxylate shunt is known to mitigate growth defects of E. coli on glucose following phosphofructokinase mutations (28). Because we focus on single knockouts, the genes affected by such mutations, pfkA and pfkB, are not among the perturbations we consider. Nonetheless, out of the 52 unrelated knockout perturbations in our study, our models show the transient activation of the glyoxylate shunt in response to 25 of them according to MOMA and 7 according to ROOM. The same phenomenon can be observed for reactions that are essential under different environmental conditions but inactive in the aerobic glucose medium employed in our simulations. Pyruvate formate lyase is required for anaerobic growth in xylose medium according to experiments (29) and our models, but is transiently active for 2 (MOMA) and 18 (ROOM) of the 52 genetic perturbations in this study. This interesting effect—the nonspecific use of pathways under an array of perturbations quite different from the conditions under which they have an observed phenotype—indicates that the phenomenon of latent pathway activation extends beyond the set of apparently nonessential genes.

Model-Independent Analysis.

The analysis above shows that the availability of latent pathways inhibits growth in the short term after a genetic perturbation. But how sensitive are these conclusions to the models we used to simulate the response of the network? To provide model-independent evidence, we have determined how the volume of the space of feasible metabolic states (Materials and Methods) depends on the growth rate. As shown in Fig. 3 A and B (green lines) for the cyoA and lpd knockout mutants, the volume systematically decreases as a function of growth rate for the single-gene knockout mutants considered in our study, indicating that the number of metabolic states available to the unevolved mutant is much larger at lower growth rates. When the latent reactions are disabled, however, the relative volume, and hence the relative number of available metabolic states, increases for large growth rates (Fig. 3 A and B, blue lines). Therefore, the principal effect of removing latent pathways appears to be an increase in the relative frequency of high-growth states due to the preferential elimination of low-growth states. It should be noted, however, that a large number of high-growth states are also disabled in this process because of the “entanglement” between latent pathways and biomass-producing pathways that exist under the metabolic steady-state conditions of our models (Elementary Mode Analysis section in SI Text).

Fig. 3.
Properties of the space of feasible metabolic states. (A and B) Volume of the metabolic solution space as a function of the growth rate for the cyoA- (A) and the lpd- (B) knockout mutant. All values are normalized so that the area under each curve is ...

To appreciate the constraints imposed by this structure, imagine that the organism responds to perturbations by moving to a random metabolic state in the feasible space of fluxes. We simulated this hypothetical response using an implementation of the hit-and-run sampling algorithm (Materials and Methods). As shown in Fig. 3C, the postperturbation growth rate is nearly zero for all mutants with latent pathways and close to the theoretical maximum for all mutants without them. This random response is arguably a lower bound for the actual response of organisms that have evolved to cope with perturbations, but the conclusion is clear: unless we assume that organisms have evolved to respond to perturbations in a highly specific manner, which appears to be inconsistent with experiments (30), the availability of latent pathways does not facilitate growth, and this prediction is largely independent of the network response to perturbation. This holds true in particular for MOMA and ROOM, which incorporate (in different ways) the main flux rerouting features observed in the activation of latent pathways in E. coli (12).

Further mechanistic insight comes from the recently identified synthetic rescue interactions (31), in which the knockout of a gene inhibits growth but, counterintuitively, the targeted concurrent knockout of additional genes recovers the ability of the organisms to grow. The reactions catalyzed by such rescue genes are predicted to be active in typical suboptimal states and inactive in growth-maximizing states of the knockout mutant (8, 31). Now, given the observation above that the set of active reactions predicted by FBA is only slightly modified by a gene knockout, it follows that most rescue genes are in fact associated with latent pathways. This, in turn, explains why the latent pathway-disabled strains show improved growth. Note that this argument cannot be anticipated from intuition, because an enormous number of low-growth states (up to several orders of magnitude larger than for near-optimal growth) may exist even when latent pathways are disabled (Fig. 3 A and B). Furthermore, even in the extreme case when one disables all reactions that are inactive in the optimal state of the knockout mutant, metabolic states with a very low growth rate (≈10% or less of the wild-type optimum) exist in 47 out of the 52 mutants (Elementary Mode Analysis section in SI Text). This threshold is significant because all but five unmodified knockout mutants exceed this growth rate according to MOMA, and all but one according to ROOM. Thus, although our model-independent analysis suggests that latent pathway activation inhibits growth under a general choice of metabolic state after a perturbation, this is not directly imposed by the geometry of the solution space. Rather, the predicted growth benefit associated with latent pathway removal and synthetic rescues is partly a reflection of the cells’ adaptive response.

An extreme example of this rescue effect is provided by the cyoA-deficient mutant, which is predicted by MOMA to drop to < 10% of the optimal wild-type growth rate following the perturbation, but recovers to ≈60% if the latent pathways are also disabled. In addition, cases in which the single-gene knockout mutant operates near the theoretical optimum but growth nonetheless improves upon the removal of latent pathways, such as for the folD mutant, can be related to weaker forms of synthetic rescues (31).

This surprising relation to synthetic rescues is particularly interesting when we note that latent reactions define several pathways whose participation in E. coli’s metabolism has been controversial or elusive. The Entner–Doudoroff (ED) pathway, an alternative to glycolysis for glucose catabolism, is inactive in wild-type E. coli according to in vivo experiments in glucose (32) but becomes transiently active in mutants lacking phosphoglucose isomerase (12). This activation may serve to reduce NADPH accumulation accompanying increased flux through the pentose phosphate pathway (33). Both MOMA and ROOM predict a small, transient flux through the ED pathway in response to the knockout of pgi, the gene coding phosphoglucose isomerase. In triphosphate isomerase-deficient strains, our model predicts the activation of the normally inactive methylglyoxal bypass (34). Experimentally, this pathway is observed to channel excess dihydroxyacetone phosphate (DHAP) into pyruvate following glycolytic flux splitting into glyceraldehyde 3-phosphate and DHAP after the knockout of the associated gene, tpi (12). These findings emphasize the importance of probing multiple gene knockouts or perturbations—previously suggested in the context of synthetic lethality (35), synthetic rescues (31, 36), multitarget drug discovery (36, 37), and neutral mutations (38)—as a means to determine the puzzling role of transients.


Latent pathways and their associated genes are, by definition, dispensable for optimal growth under the given conditions both before and after a perturbation. Several explanations have been offered to justify the persistence of apparently nonessential metabolic genes within the genome. Environmental robustness is the most natural hypothesis, as it acknowledges the unpredictable, time-varying conditions that confront single-cell organisms in natural environments. Yet, in silico (7, 39) and experimental (11) studies of model organisms under various environmental conditions likely to occur in nature suggest that condition-dependent robustness is inadequate to fully explain the dispensability of metabolic genes. Part of the remaining redundancy may be attributed to the varying efficiencies of redundant pathways under different environments (40), or selective pressure for increased metabolic capacity across all conditions (39). An alternative to environmental robustness is genetic robustness, in which redundant pathways buffer against null mutations (41, 42). But regardless of which explanation might apply in the context of latent pathways, the question that follows from our analysis is not why these dispensable pathways are present in the genome, but rather what causes their transient activation in response to genetic perturbations. Our results indicate that this activation confers no advantage in fitness as measured by growth, and more often hinders growth in the short term following a perturbation.

The dispensability of latent pathway activation predicted in this study can be interpreted in three different ways, which are not necessarily mutually exclusive. First, it is possible that temporary reaction activation does not provide necessary intermediate states but is instead a byproduct of the network’s suboptimal response to perturbations. The fact that E. coli undergoes a period of suboptimal growth following genetic perturbations is well-established in both in silico and in vivo experiments (17, 18). This interpretation is thus supported by the recent observation that typical suboptimal metabolic states have a much larger number of active reactions (more than 2.5 times larger for the conditions considered here) than typical states that optimize growth (8).

A second interpretation is that cells activate a large number of reactions as a global nonspecific response to perturbations that nevertheless creates a library of possible metabolic states that can be subsequently fine-tuned by adaptive evolution. This scenario cannot be rejected by existing experimental results (30) and is appealing as it allows for an indirect, long-term benefit of temporary reaction activation even if, as predicted here, this activation inhibits growth in the short term. Although whole-cell regulation remains largely unexplained, there are known mechanisms that could subsequently lead to optimal growth (43, 44). This interpretation suggests that fine-tuning of whole-cell reaction activity is best achieved by down-regulating specific overexpressed reactions rather than by the coordinated up-regulation of entire underexpressed pathways, a principle that is widely appreciated in metabolic engineering but that is yet to be demonstrated for natural systems. The recent observation that E. coli’s response to perturbations is more stable from the viewpoint of metabolite concentration and protein or mRNA abundance (45) than reaction flux (12) may be relevant for the validation of this interpretation. The objection to the second interpretation is that it cannot explain (not even in the long term) the availability of pathways that are simultaneously dispensable for all known perturbations that have shaped the evolution of the organisms. This “paradox of latency” reinforces the need to study the mechanisms that govern latent pathway activation.

In experimental studies of gene function in microbial organisms, growth is the most often used indicator of fitness, owing to both its accessibility for measurement and its intrinsic importance in determining viability (13, 46). Although the results of this study suggest that the availability of latent pathways does not promote faster growth following a perturbation, the conclusion that this negatively impacts adaptation is reached by studying these pathways in the context of metabolism alone. Thus, a third interpretation is that, in general, the cellular objective invoked in the adaptive response is not growth. Instead, the activation of otherwise latent metabolic pathways may accompany other cellular processes, either inside or outside metabolism, that are initiated to ensure survival. For example, yeast adopts changes in cell shape and internal pressure in response to osmotic shock, a process that recruits the metabolism to accumulate specific metabolites (47). Moreover, enzymes and metabolic cofactors involved in transiently upregulated pathways may have regulatory or signaling roles in addition to their metabolic function (48). This third possibility allows for an advantageous, external function for pathways whose activation appears disadvantageous when only the metabolism is considered. In this way, latent pathway activation is incorporated into a larger, more sophisticated adaptive response. This explanation, however, seems inconsistent with our observation of the apparently highly nonspecific activation of pathways in response to different perturbations.

Regardless of which interpretation proves to be correct, this study leads to a clear, experimentally testable prediction; namely, that latent pathway activation does not enhance, and in fact often inhibits, early postperturbation growth. Although we expect this behavior to be observed experimentally under a wide range of conditions, regardless of the specific suboptimal response of the cell, deviations from this behavior would also be highly informative because they would uncover growth phenotypes not detected in previous steady-state experiments. Varying environments can also generate strong (45) and sometimes counterintuitive responses, such as the possibility of accelerating the evolution of unevolved strains (49). Therefore, it may well be the case that in time-varying environments, which are beyond current modeling capability, latent pathway activation will exhibit a different fitness effect.

Materials and Methods

Network and Perturbations.

We used the iAF1260 E. coli model (15), which is the most up-to-date reconstruction of the metabolic network of E. coli MG1655. The network consists of 2,082 unique reactions catalyzed by 1,260 genes and involves 1,369 metabolites, as well as 299 exchange fluxes and the biomass flux. We focused on the 52 single-gene knockout strains that are compatible with growth but for which the original growth-maximizing flux state becomes infeasible after the gene knockout. The volume calculation, which remains challenging for the full network, was performed using a reduced network consisting of 62 reactions, 101 genes, 49 metabolites, and 14 exchange fluxes representing the central metabolism of E. coli (50).

Medium and Constraints.

The simulated medium consisted of limited amounts of glucose (8 mmol/g DW-h) and oxygen (18.5 mmol/g DW-h), and unlimited amount of carbon dioxide, iron (II), protons, water, potassium, sodium, ammonia, phosphate, and sulfate. Irreversible reactions are constrained to have nonnegative fluxes. The flux through the ATP maintenance reaction was set to 8.39 mmol/g DW-h. A total of 1,432 reactions in the iAF1260 model, including 61 reactions in the subnetwork of the central metabolism, can be active under these medium conditions. An analysis of the effect of regulatory limitations (51) that may constrain this reaction activity in vivo is considered in the Regulatory Constraints section in SI Text. There, it is shown that our results remain valid under the additional constraints imposed by these limitations.

Feasible Metabolic States.

The state of the metabolic network is represented by a vector of all reaction fluxes ν = (νj). Because the time scale at which the network responds to perturbations is much shorter than the characteristic time for adaptive evolution, we focused on steady-state flux distributions both before and after perturbation. Steady-state fluxes are solutions of the mass-balance equation S·ν = 0, where S = (Sij) is the matrix of stoichiometric coefficients, under the constraint imposed by the medium, reaction irreversibility, ATP maintenance requirements, and possibly gene knockouts. A knockout of the enzyme-coding genes associated with reaction j is implemented through the additional constraint νj = 0. The solution space is the set of all such steady-state flux vectors and it has the form of a convex polytope. We refer to the individual solutions in this space as feasible metabolic states.

Objective Functions.

FBA (19) identifies a growth-maximizing state within the space of feasible metabolic states by maximizing the flux νb through a biomass reaction that drains biomass precursors. With respect to this original state, MOMA (17) and ROOM (18) identify feasible states that minimize the distance in the space of fluxes and the number of significant flux changes, respectively (the MOMA and ROOM sections in SI Text). Our implementations of FBA, MOMA, ROOM, and the hit-and-run algorithm used a commercial optimization package (ILOG CPLEX, version 11.0, www.ilog.com). For all FBA results, we have used the growth-maximizing states provided by the simplex algorithm. More information about the computational methods is provided in SI Text, where it is also shown that our results do not depend sensitively on the assumption of optimal growth in the reference states either before or after the perturbation (Effects of Nonoptimal Reference States), nor on particular choices for the growth-maximizing states used throughout the paper (Sensitivity to Alternate Optima).

Biomass Flux and Growth Rates.

The in silico model predicts the biomass flux, but for exponential growth the result can be expressed in terms of a normalized growth rate An external file that holds a picture, illustration, etc.
Object name is pnas.1009772108eq1.jpg. This follows from the observation that biomass production is governed by An external file that holds a picture, illustration, etc.
Object name is pnas.1009772108eq2.jpg, where κ is the growth rate, N measures the population size, and An external file that holds a picture, illustration, etc.
Object name is pnas.1009772108eq3.jpg is proportional to the biomass flux νb. Therefore, when normalized with respect to the wild-type or theoretical maximum, the normalized biomass flux νb/νb,0 equals the corresponding normalized growth rate κ/κ0 used throughout the paper.

Identification of Latent Pathways.

We define the latent reactions (pathways) associated with the knockout of gene A to be the set LA of all reactions predicted to be transiently active in the unevolved mutant shortly after the perturbation (according to MOMA, ROOM, or the hit-and-run algorithm), but inactive in both the optimal wild-type and mutant strains (according to FBA). This set is therefore nonessential for optimal growth under the knockout of A, although in principle some of these reactions may be necessary for growth under the knockout of a different gene, B. To account for this, we tested the impact of a set An external file that holds a picture, illustration, etc.
Object name is pnas.1009772108eq4.jpg of latent reactions that are simultaneously nonessential in the optimal growth states of the other 51 single-gene knockout mutants that we consider.

Volume Calculation.

The exact calculation of the volume of the (high-dimensional) solution space is computationally intractable and, because this set is very skewed (52), even approximate calculations are computationally demanding. To determine the volume of the solution space as a function of growth rate, we used an approximate inference algorithm based on belief propagation (53). In this approach, the convex polytope representing the constrained flux space is tiled with hypercubes of size ε. We then use a message-passing algorithm to approximate the probability distribution P(ν) over the discretized space. Using this, we define the associated entropy An external file that holds a picture, illustration, etc.
Object name is pnas.1009772108eq5.jpg, which counts the log of the number of ε-hypercubes that overlap the space of feasible states. The total volume covered by these cubes is then V = 10Sεn, where n is the dimension of the space. If m is the number of linearly independent mass balance constraints and f is the number of available fluxes after a given knockout, then the dimension of the unmodified metabolic space is n = f - m, and the solution space for the modified organism has dimension n = f - m - l, where l is the number of independent latent reactions removed. We used a granularity of An external file that holds a picture, illustration, etc.
Object name is pnas.1009772108eq6.jpg for all calculations presented in the paper.

Hit-and-Run Algorithm.

To randomly sample the metabolic solution space, we implemented a hit-and-run algorithm (54) with artificial centering (55), which is a quickly converging sampler for high-dimensional convex sets. The algorithm is based on selecting a randomly oriented line l passing through the current sample point and then selecting the point for the next iteration from a uniform distribution along l. For efficient sampling, we employed artificial centering (55), where the orientation of the line l is obtained from the direction defined by the current sample point and the center (mean) of a subset C of already-sampled points. The subset C was initially created by taking 10,000 warm-up points on the boundary of the space and was updated recursively by replacing a randomly selected point of C with the currently sampled point. In all calculations presented in the paper, we generated a set of 5 × 106 points to sample the solution space. All calculations were performed with the COBRA Toolbox (56).

Supplementary Material

Supporting Information:


This work was supported by the National Science Foundation under Grant DMS-0709212, the National Cancer Institute under Grant 1U54CA143869-01, and a Sloan Research Fellowship to A.E.M.


The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1009772108/-/DCSupplemental.


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