Optimal feedback strategies for bacterial growth with degradation, recycling, and effect of temperature

Abstract

For both fundamental biology and engineering applications, it is relevant to investigate how microorganisms adapt to changing environmental conditions. In this work, we consider a continuous-time dynamic problem of resource allocation between metabolic and gene expression machineries for a self-replicating prokaryotic cell population. In compliance with evolutionary principles, the criterion is to maximize the accumulated structural biomass. In the model, we include both degradation of proteins into amino acids and recycling of the latter (i. e., using as precursors again). Based on the analytical investigation of our problem by Pontryagin's maximum principle, we develop a numerical algorithm for approximating the switching curve of the optimal feedback control strategy. The obtained field of extremal state trajectories consists of chattering arcs and one steadystate singular arc. The constructed feedback control law can serve as a benchmark for comparing actual bacterial strategies of resource allocation. We also study the influence of temperature, whose increase intensifies protein degradation. While the growth rate suddenly decreases with the increase of temperature in a certain range, the optimal control synthesis appears to be essentially less sensitive.1 BIOCORE team, INRIA Sophia-Antipolis Méditerranée (as a part

Minimal time crisis versus minimum time to reach a viability kernel: A case study in the prey‐predator model

“…† In general, this terminology is used when an optimal control has an infinite number of switching points over a finite horizon (in presence of a singular arc of order 2); see, for instance, the Fuller's example. 21,22 Using the previous proposition, we can apply the hybrid principle on the time crisis problem, which allows to state the following necessary optimality conditions (see the works of Bayen and Rapaport 7,8 and Haberkorn and Trélat 11 ).…”

“…(i) If z 0 ∈  , optimal solutions for Problems (13) and (22) coincide. (i) If z 0 ∈  ⧵  , then one has…”

“…As u ⋆ ∈  is admissible for (22), we have (z 0 ) ≤ J(u ⋆ , z 0 ). If we have (z 0 ) = J(u ⋆ , z 0 ), then u ⋆ is necessarily an optimal control for (22). The switching points in K(x) of the associated trajectory occur on the line {y = r} only from Proposition 7.…”

“…Intuitively, this result says that for initial conditions in  , the time spent in K c by an optimal trajectory of the minimum time problem (13) is greater than the one spent in K c by an optimal trajectory of the time crisis problem (22). To conclude this study, we provide properties of the minimal time crisis problem when the viability kernel Viab(x) is empty, that is, when the condition m +ū < x is fulfilled (see Proposition 2).…”

“…Notice that in this case, the value of ∈ [0, 1] is a parameter (as K(x) is nonsmooth at (x, r − ), there exist infinitely many extremal trajectories arising from (x, r − )). Numerical determination of extremal trajectories for the minimum time crisis problem (22). The initialization of (A1) is the same as for Problem (13).…”