^{1}

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Conceived and designed the experiments: AEM. Performed the experiments: NG. Analyzed the data: TN NG AEM. Contributed reagents/materials/analysis tools: TN NG. Wrote the paper: TN AEM.

The authors have declared that no competing interests exist.

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

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

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

To provide mechanistic insight into such behaviors, here we study the metabolic system of single-cell organisms under optimal

The pie charts show the fractions of active and inactive reactions in the metabolic subsystems defined in the iJR904 database

Total number of reactions [ |
479 | 641 | 931 | 1149 |

Reversible | 165 | 220 | 245 | 430 |

Irreversible | 314 | 421 | 686 | 719 |

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

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 _{1},…,_{N}^{T}_{ex} 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

We can prove that, with the exception of the reactions that are inactive for all _{+}(

Part of the metabolic reactions are forced to be inactive solely due to mass balance, independently of the medium conditions. For example, glutathione oxidoreductase in the

Other reactions are constrained to be inactive due to the constraints arising from the environmental conditions imposed by the medium. For example, all reactions in the allantoin degradation pathway must be inactive for

The results for the typical activity of each organism in glucose minimal media (

For each organism, the bars correspond to a typical non-optimal state (top), a growth-maximizing state (middle), and a state with the minimum number of active reactions required for growth (bottom), which was estimated using the algorithm described in

Total number of reactions [ |
479 | 641 | 931 | 1149 |

Inactive reactions: | 87 | 222 | 322 | 570 |

Due to mass balance | 44 | 133 | 141 | 268 |

Due to environmental conditions |
43 | 89 | 181 | 302 |

Active reactions [ |
392 | 419 | 609 | 579 |

These reactions are inactive due to constraints arising from the availability of substrates in the media defined in

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

We now turn to mechanisms underlying the observed reaction silencing, which is spread over a wide range of metabolic subsystems (see

We identify three different scenarios in which reaction irreversibility causes reaction inactivity under maximum growth. The simplest case is when the reaction is part of a parallel pathway structure. While stoichiometrically equivalent pathways lead to alternate optima

(A) P1, P2, and P3 are alternative pathways for glucose transport and utilization. The most efficient pathway, P1, is active under maximum growth in glucose minimal medium. P2 and P3 are inactive, but if P1 is knocked out, P2 becomes active, and if both P1 and P2 are knocked out, P3 becomes active. In both knockout scenarios, the growth is predicted to be suboptimal. (B) Isocitrate lyase reaction in the pathway bypassing the tricarboxylic acid (TCA) cycle is predicted to be inactive under maximum growth due to its irreversibility. If it were to operate in the opposite direction, it would serve as a

A different silencing scenario is identified when no clear parallel pathway structure is recognizable. In this scenario there is a

A third scenario for the irreversibility mechanism arises when a transport reaction is irreversible because the corresponding substrate is absent in the medium. For example, since acetate, a possible carbon and energy source, is absent in the given medium, the corresponding transport reaction is irreversible; acetate can only go out of the cell (Note 3). For

We interpret these inactivation mechanisms involving reaction irreversibility as a consequence of the linear programming property that the set of optimal solutions _{opt} must be part of the boundary of _{opt} is characterized by a set of _{i}_{i}_{i}_{i}_{opt} and _{opt} is sensitive to changes in the constraints (changes in _{i}_{opt} would be an edge of _{opt} linearly independent constraints must be binding, where _{opt} are the dimensions of _{opt}, respectively. Since many metabolic reactions are subject to the irreversibility constraint (_{i}_{i}_{+}(

The remaining set of reactions that are inactive for all _{opt} is due to cascading of inactivity. On one hand, if all the reactions that produce a metabolite are inactive, any reaction that consumes this metabolite must be inactive. On the other hand, if all the reactions that consume a metabolite are inactive, any reaction that produces this metabolite must be inactive to avoid accumulation, as this would violate the steady-state assumption. Therefore, the inactivity caused by the irreversibility mechanism triggers a cascade of inactivity both in the forward and backward directions along the metabolic network. In general, there are many different sets of reactions that, when inactivated, can create the same cascading effect, thus providing flexibility in potential applications of this effect to the design of optimal strains

While the irreversibility and cascading mechanisms cause the inactivity of many reactions for all _{opt}, the inactivity of other reactions can depend on the specific growth-maximizing state, whose non-uniqueness in a given environment has been evidenced both theoretically _{opt} (_{opt}, leading to rigorous bounds for the number of active reactions _{+}(_{+}(_{opt}. This behavior is expected, however, under the concurrent optimization of additional metabolic objectives, which generally tend to drive the flux distribution toward the boundary of _{opt}.

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

We first note the fact (proved in _{opt} consists of a single point, which must be a “corner” of _{opt} = 0, and Eq. (2) becomes

We find that the number of active reactions in typical optimal states is narrowly distributed around that in growth-maximizing states, as shown in

The red solid lines indicate the corresponding number in the growth-maximizing state of

Active reactions under typical non-optimal states [ |
392 | 419 | 609 | 579 |

Active reactions under maximum growth |
308 | 282 | 297 | 289 |

Lower bound [ |
257 | 77 | 272 | 196 |

Upper bound [ |
351 | 414 | 355 | 426 |

Minimum number of active reactions for growth |
302 | 281 | 292 |
275 |

Inactive reactions under maximum growth |
171 | 359 | 634 | 860 |

Due to irreversibility | 29 | 3 | 147 | 72 |

Due to cascading | 12 | 2 | 107 | 81 |

Due to mass balance | 44 | 133 | 141 | 268 |

Due to environmental conditions | 43 | 89 | 181 | 302 |

Conditionally inactive |
43 | 132 | 58 | 137 |

With respect to the minimal media defined in

Based on a single optimal state found using an implementation of the simplex method

Estimated using the algorithm described in

Predicted to be inactive by the simplex method _{opt}, which tends to have more inactive reactions than a typical optimal solution.

The corresponding minimum number of active reactions for maximum growth is 293.

Our results help explain previous experimental observations. Analyzing the 22 intracellular fluxes determined by Schmidt ^{−7}). This higher probability for reduced fluxes in irreversible reactions is consistent with our theory and simulation results (

Aerobic | Anaerobic | |||

Reversible | Irreversible | Reversible | Irreversible | |

Number of fluxes | 8 | 14 | 8 | 14 |

Number of fluxes <10% of glucose uptake rate | 1 | 7 | 2 | 10 |

Glucose | Maltose | Ethanol | Acetate | |||||

Rev. | Irr. | Rev. | Irr. | Rev. | Irr. | Rev. | Irr. | |

Number of fluxes | 22 | 22 | 22 | 22 | 22 | 22 | 22 | 22 |

Number of zero fluxes | 2 | 8 | 2 | 7 | 1 | 11 | 2 | 11 |

Percentage of zero fluxes | 9.1% | 36.4% | 9.1% | 31.8% | 4.5% | 50.0% | 9.1% | 50.0% |

Reversible | Irreversible | Reversible | Irreversible | |

Number of reactions | 245 | 686 | 430 | 719 |

Number of inactive reactions | 139 | 495 | 301 | 559 |

Percentage of inactive reactions | 56.7% | 72.2% | 70.0% | 77.7% |

Same states considered in

Additional evidence for our results is derived from the inspection of 18 intracellular fluxes experimentally determined by Emmerling

Reversible | Irreversible | |

Number of fluxes | 30 | 24 |

Number of mutant fluxes that are larger |
8 | 11 |

Relative to the corresponding fluxes in the wild-type strain.

Altogether, our results offer an explanation for the temporary activation of latent pathways observed in adaptive evolution experiments after environmental

(A) The initial response is predicted by the minimization of metabolic adjustment (MOMA) and the endpoint of adaptive evolution by the maximization of the growth rate (FBA), using the medium defined in

Another potential application of our results is to explain previous experimental evidence that antagonistic pleiotropy is important in adaptive evolution

Combining computational and analytical means, we have uncovered the microscopic mechanisms giving rise to the phenomenon of spontaneous reaction silencing in single-cell organisms, in which optimization of a single metabolic objective, whether growth or any other, significantly reduces the number of active reactions to a number that appears to be quite insensitive to the size of the entire network. Two mechanisms have been identified for the large-scale metabolic inactivation: reaction irreversibility and cascade of inactivity. In particular, the reaction irreversibility inactivates a pathway when the objective function could be enhanced by hypothetically reversing the metabolic flow through that pathway. We have demonstrated that such pathways can be found among non-equivalent parallel pathways, transverse pathways connecting structures that lead to the synthesis of different biomass components, and pathways leading to metabolite excretion. Although the irreversibility and cascading mechanisms do not require explicit maximization of efficiency, massive reaction silencing is also expected for organisms optimizing biomass yield or other linear functions (of metabolic fluxes) normalized by uptake rates

Our study carries implications for both natural and engineered processes. In the rational design of microbial enhancement, for example, one seeks genetic modifications that can optimize the production of specific metabolic compounds, which is a special case of the optimization problem we consider here and akin to the problem of identifying optimal reaction activity

In particular, our results open a new avenue for addressing the origin of mutational robustness

A recent study on

We conclude by calling attention to a limitation and strength of our results, which have been obtained using steady-state analysis. Such analysis avoids complications introduced by unknown regulatory and kinetic parameters, but admittedly does not account for constraints that could be introduced by the latter. Nevertheless, we have been able to draw robust conclusions about dynamical behaviors, such as the impact of perturbation and adaptive evolution on reaction activity. Our methodology scales well for genome-wide studies and may prove instrumental for the identification of specific extreme pathways

In addition, under steady-state conditions in the media considered in this study, more than 77% of the reversible reactions become constrained to be irreversible, rendering a total of more than 92% of all reactions “effectively” irreversible.

This reaction is regarded in the biochemical literature as irreversible under physiological conditions in the cell, and is constrained as such in the modeling literature

Similar effective irreversibility is at work for any transport or internal reaction that is a unique producer of one or more chemical compounds in the cell.

For single-reaction knockouts, MOMA predicts that 662 out of the 931 deletion mutants grow at more than 99% of the wild-type growth rate. Among these 662 reactions, 95% are predicted to be inactive under maximum growth.

All the stoichiometric data for the

Under steady-state conditions, a cellular metabolic state is represented by a solution of a homogeneous linear equation describing the mass balance condition,_{1},…,_{N}^{T}_{ex} exchange fluxes, which model the transport of metabolites across the system boundary. Constraints of the form _{i}_{i}_{i}

The flux balance analysis (FBA) ^{T}_{i}_{i}_{i}_{i}

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.

Mathematical Results

(0.06 MB PDF)

The authors thank Linda J. Broadbelt for valuable discussions and for providing feedback on the manuscript. The authors also thank Jennifer L. Reed and Adam M. Feist for providing information on their