Sequential Activation of Metabolic Pathways: a Dynamic Optimization Approach

Abstract

The regulation of cellular metabolism facilitates robust cellular operation in the face of changing external conditions. The cellular response to this varying environment may include the activation or inactivation of appropriate metabolic pathways. Experimental and numerical observations of sequential timing in pathway activation have been reported in the literature. It has been argued that such patterns can be rationalized by means of an underlying optimal metabolic design. In this paper we pose a dynamic optimization problem that accounts for time-resource minimization in pathway activation under constrained total enzyme abundance. The optimized variables are time-dependent enzyme concentrations that drive the pathway to a steady state characterized by a prescribed metabolic flux. The problem formulation addresses unbranched pathways with irreversible kinetics. Neither specific reaction kinetics nor fixed pathway length are assumed.In the optimal solution, each enzyme follows a switching profile between zero and maximum concentration, following a temporal sequence that matches the pathway topology. This result provides an analytic justification of the sequential activation previously described in the literature. In contrast with the existent numerical approaches, the activation sequence is proven to be optimal for a generic class of monomolecular kinetics. This class includes, but is not limited to, Mass Action, Michaelis-Menten, Hill, and some Power-law models. This suggests that sequential enzyme expression may be a common feature of metabolic regulation, as it is a robust property of optimal pathway activation.

Dynamic optimization identifies optimal programmes for pathway regulation in prokaryotes

“…For this scenario, we observe a sequential activation of enzymes according to their order within the pathway. This type of activation strategy is similar to the so called 'just-in-time-activation' strategy 19 , also reported in other studies [13][14][15][16] . In the case in which each individual enzyme synthesis rate is smaller than the free protein synthesis capacity but their sum is larger than the free protein synthesis capacity, we observe an intermediary behaviour in which parts of the metabolic pathway are sequentially activated (Fig.…”

“…We find that the interplay between the protein production capacity of the cell and the amount in which a particular enzyme needs to be produced (that is, its abundance) can explain the optimality of a wide variety of pathway activation strategies. In particular, we find that the previously reported sequential activation strategy of enzymes along a pathway [13][14][15][16][17][18][19] is only optimal if large amounts of proteins need to be produced, whereas the simultaneous activation of all enzymes within a pathway is optimal in the case where only small amounts of protein need to be produced. Thus, we show that, depending on protein abundances, an operonic organization of a metabolic pathway is optimal to reduce activation time, whereas previous work postulated activation time-independent effects to explain the operonic organization of metabolic pathways 18,23 .…”

“…Being able to quickly adjust fluxes through metabolic pathways is of critical importance to reduce lag times upon depletion of essential biomass components and during major growth transitions 6,7,11,12 . Previous work has established that, assuming a limited total abundance of proteins as well as the minimization of the invested protein, a sequential induction of enzymes along a pathway is optimal for a rapid activation [13][14][15][16][17][18][19] . That metabolic pathways show a pattern of sequential activation, also known as just-in-time-activation 19 , has been demonstrated experimentally in Escherichia coli for a selected number of amino-acid biosynthetic pathways 18 and on a global scale in Saccharomyces cerevisiae 20 .…”

Synthetic biology and regulatory networks: where metabolic systems biology meets control engineering

“…A previously reported sequential activation strategy was only optimal if protein abundance relative to protein synthesis capacity was high; as protein abundance decreased, the strategy shifted to the simultaneous activation of all enzymes. These numerical approaches can be formulated into a more rigorous theoretical framework using optimal control theory [133,134] and by formulating the following optimization problem: The objective function represents either a minimization of the transition time to reach a given state of the system (or enzyme cost) or a maximization of biomass limited by for example the synthesis of the product of the pathway. U(t) is the vector of independent control variables representing enzyme concentrations E(t); m(t) is the growth rate profile.…”

Optimal regulatory programs for the control of metabolic pathways: The case of feedback inhibition

“…This mathematical framework has a significant number of applications and variations ( [3], [4], [5]). While classical MCA have focused on steady states, several authors have used a dynamic optimization approach to study the behavior of metabolic networks ( [6], [7], [8], [9], [10]), including the study of optimal strategies to shift between different steady states in a dynamic environment [11].…”