Pathways in Escherichia coli show large differences in the extent to which enzymes from the same pathway are expressed in a coordinated manner.Using dynamic optimization, we show that regulation of the initial and terminal reactions of a pathway is the minimum requirement for a precise control of flux.We find that in E. coli a regulation of initial and terminal reactions is predominantly used to control pathways with low costs of enzymes while a regulation of all enzymes occurs if protein costs are high.A trade-off between minimization of protein investment and minimization of response time can explain the preference for transcriptional regulation at key positions (leading to high protein costs, but low response time) or coordinated transcriptional regulation of all enzymes (leading to low protein costs, but high response time).

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To survive in fluctuating environmental conditions, microorganisms must be able to quickly react to environmental challenges by upregulating the expression of genes encoding metabolic pathways. Here we show that protein abundance and protein synthesis capacity are key factors that determine the optimal strategy for the activation of a metabolic pathway. If protein abundance relative to protein synthesis capacity increases, the strategies shift from the simultaneous activation of all enzymes to the sequential activation of groups of enzymes and finally to a sequential activation of individual enzymes along the pathway. In the case of pathways with large differences in protein abundance, even more complex pathway activation strategies with a delayed activation of low abundance enzymes and an accelerated activation of high abundance enzymes are optimal. We confirm the existence of these pathway activation strategies as well as their dependence on our proposed constraints for a large number of metabolic pathways in several hundred prokaryotes.

BackgroundAdjusting the capacity of metabolic pathways in response to rapidly changing environmental conditions is an important component of microbial adaptation strategies to stochastic environments. In this work, we use advanced dynamic optimization techniques combined with theoretical models to study which reactions in pathways are optimally targeted by regulatory interactions in order to minimize the regulatory effort that is required to adjust the flux through a complex metabolic network. Moreover, we analyze how constraints in the speed at which an organism can respond on a proteomic level influences these optimal targets of pathway control.ResultsWe find that limitations in protein biosynthetic rates have a strong influence. With increasing protein biosynthetic rates the regulatory effort targeting the initial enzyme in a pathway is reduced while the regulatory effort in the terminal enzyme is increased. Studying the impact of allosteric regulation for different pathway topologies, we find that the presence of feedback inhibition by products of metabolic pathways allows organisms to reduce the regulatory effort that is required to control a metabolic pathway in all cases. In a linear pathway this even leads to the case where the sole transcriptional regulatory control of the terminal enzyme is sufficient to control flux through the entire pathway. We confirm the utilization of these pathway regulation strategies through the large-scale analysis of transcriptional regulation in several hundred prokaryotes.ConclusionsThis work expands our knowledge about optimal programs of pathway control. Optimal targets of pathway control strongly depend on the speed at which proteins can be synthesized. Moreover, post-translational regulation such as allosteric regulation allows to strongly reduce the number of transcriptional regulatory interactions required to control a metabolic pathway across different pathway topologies.Electronic supplementary materialThe online version of this article (doi:10.1186/s12859-015-0587-z) contains supplementary material, which is available to authorized users.

Quasi-sequential methods are efficient and flexible strategies for the solution of dynamic optimization problems. At the heart of these strategies lies the time discretization and approximation of dynamic systems for nonlinear optimization problems. To address this question, we employ a time derivative analysis within the quasi-sequential approach and derive a finite element placement strategy. In addition, methods for direct error prediction are applied to this approach and extended with a proposed time derivative analysis. According to the information for current time derivatives, subintervals are introduced that improve accuracy of state profiles. Since this is only done in the simulation layer, the nonlinear programing solver need not be restarted. An efficient gradient computation is also derived for these subintervals; the resulting enhanced accuracy accelerates convergence performance and increases the robustness of the solution to initialization. A beer fermentation process case study is presented to demonstrate the effectiveness of the proposed approach.