Optimal control of bioproduction in the presence of population heterogeneity

Abstract

Cell-to-cell variability, born of stochastic chemical kinetics, persists even in large isogenic populations. In the study of single-cell dynamics this is typically accounted for. However, on the population level this source of heterogeneity is often sidelined to avoid the inevitable complexity it introduces. The homogeneous models used instead are more tractable but risk disagreeing with their heterogeneous counterparts and may thus lead to severely suboptimal control of bioproduction. In this work, we introduce a comprehensive mathematical framework for solving bioproduction optimal control problems in the presence of heterogeneity. We study population-level models in which such heterogeneity is retained, and propose order-reduction approximation techniques. The reduced-order models take forms typical of homogeneous bioproduction models, making them a useful benchmark by which to study the importance of heterogeneity. Moreover, the derivation from the heterogeneous setting sheds light on parameter selection in ways a direct homogeneous outlook cannot, and reveals the source of approximation error. With view to optimally controlling bioproduction in microbial communities, we ask the question: when does optimising the reduced-order models produce strategies that work well in the presence of population heterogeneity? We show that, in some cases, homogeneous approximations provide remarkably accurate surrogate models. Nevertheless, we also demonstrate that this is not uniformly true: overlooking the heterogeneity can lead to significantly suboptimal control strategies. In these cases, the heterogeneous tools and perspective are crucial to optimise bioproduction.

Revisiting moment-closure methods with heterogeneous multiscale population models

“…We denote the population density of cells of class k over the continuum quantities x at time t by p k (x, t). We are interested in models governing the evolution of the collection of population densities in all states, {p k (x, t)} k , of the form [31] ∂ ∂t p k (x, t) = Population growth + Population dilution + Discrete dynamics + Continuum dynamics, (1a)…”

“…Considering the full distribution of populations over these stochastic growth events for any real-world system is computationally intractable. Studying only the expected population produces the model we pose here, which strikes a balance between retaining the important population heterogeneity [31] while retaining a more manageable state space size [12,30].…”

“…In this section we study a model of a microbial consortium (from Ref. [31]), where two strains are maintained within a single bioreactor with the aim of producing a protein of interest (denoted by the state y). We consider a construct in which a photoreceptive transcription factor (TF) is constitutively produced.…”

“…Despite a thoroughly mechanistic understanding of the behaviour of individual cells and communities [50] many population models, particularly for the purposes of control, neglect this heterogeneity due to the complexity it introduces [16,17]. When this nonuniformity is ignored the resulting dynamics and optimal controls can deviate significantly from the ideal performance [31].…”

“…We refer to the resulting finite system of moment equations as the momentclosure approximation. In contrast to previous work, where the same class of PDE models are studied alongside moment-closure approximations up to first order [31], we study moment closures of arbitrarily high order. As we will demonstrate, retaining orders above the first allows us to preserve heterogeneous features not accounted for in the lower-order (homogeneous) models.…”