Optimality in cellular storage via the Pontryagin Maximum Principle

Abstract

Abstract:We study an optimal control problem arising from a resource allocation problem in cellular metabolism. A minimalistic model that describes the production of enzymatic vs. non-enzymatic biomass components from a single nutrient source is introduced. The basic growth modes with this model are linear growth, where only the non-enzymatic component is produced, and exponential growth with only enzymatic components being produced. Using Pontryagin's maximum principle, we derive the optimal growth trajectory depending on the model's parameters. It turns out that depending on the parameters, either a single growth mode is optimal, or otherwise the optimal solution is a concatenation of exponential growth with linear growth. Importantly, on the short time scale, the choice of growth mode depends only on catalytic rate constants and biomass weights of the two component types, whereas on longer time scales, where the nutrient amount becomes limiting, also the yield coefficients play a role.

Modeling metabolic networks including gene expression and uncertainties

“…with initial values x0 and fixed end-time T . We have already shown analytically in [14] that this system can experience basically three different growth modes depending on the chosen parameters and end-time T . Growth in this sense means the change of the biomass over time b T ẋ.…”

“…We present a simple model, which we already analyzed very detailed in [14], to give the reader an idea how changing the catalytic constant k cat or other system parameters, e.g. the end-time T , can impact the quality of results.…”

“…This change in the reproductive capabilities of E is enough to make purely linear growth the most effective solution. Following [14], we can even pinpoint the exact conditions necessary for optimal linear growth…”

Dynamic metabolic resource allocation based on the maximum entropy principle

“…The observed accumulation of intracellular carbohydrates in response to nutrient limitation is not intuitively rationalised based on the instantaneous maximisation of growth rate alone, because investing resources in any process not contributing directly to growth would be considered a sub-optimal control policy. For this reason, among others, authors have considered alternative metabolic objectives to maximisation of total catalytic biomass, such as maximising total carbon uptake, or have explicitly included intracellular reserves as an integral component of biomass [44,45,46]. However, here it is demonstrated that no further assumption beyond φ(X) = x is necessary for explaining the accumulation of intracellular storage compounds in response to nutrient limitation.…”

A Protocol for Generating and Exchanging (Genome-Scale) Metabolic Resource Allocation Models

“…However, this has to be done with caution because this constraint will then prohibit the model from using up storage molecules in a situation where it would actually be needed. Also, for most cases, we recommend not to include the weight of storage in the biomass objective function (see Equations ( 2 ) and ( 3 )), because inclusion of storage may lead to unrealistic growth modes in some situations as discussed in [ 56 ]. Nevertheless, depending on the usage of storage in the model, one can of course add the storage molecules to the objective function if needed.…”