Microbial metabolism and growth under conditions of starvation modelled as the sliding mode of a differential inclusion

Abstract

We consider a model of bacterial growth with variable internal stores, extended with adaptive resource allocation and investigate the behaviour of this model under conditions of starvation, i.e. severe nutrient shortage, treating the behaviour under the starvation regime in terms of a differential inclusion and derive Filippov solutions. This Filippov sliding mode representation appears to be a simple but sound qualitative description of metabolic 'shut down' in response to starvation. We discuss a natural connection between biologically motivated modelling approaches to metabolic 'shut down' and numerical regularisation techniques to approximate Filippov solutions.

Optimal management of nutrient reserves in microorganisms under time-varying environmental conditions

“…The assumption of acceptor-driven kinetics, which is implicit in the equations of Section 2, breaks down under metabolic shutdown conditions, which means that under such conditions the macro-chemical model could be extended with explicit donor-controlled rate multipliers, or equivalently, as we have shown elsewhere (Nev and van den Berg, 2017a), by postulating a sliding mode for the dynamics. The model is based on n + 1 feedback loops, one between each reserve density and the allocation of molecular building blocks towards the machinery dedicated to the assimilation of that nutrient, in addition to a basic growth-control loop that is based on homeostasis of the density of synthetic (zero-type, i.e.…”

Mathematical Models of Microbial Growth and Metabolism: A Whole-Organism Perspective

“…We believe that approaching the subject as an allocation problem is a natural and elegant way of fulfilling all of these desiderata 7,[19][20][21] . At the heart lies a simple principle, expressed by the following equation: where α i is the fraction of molecular building blocks devoted to the synthesis of machinery of type i, r i is a regulatory law (or r-function, for short), and r Σ = i r i is a normalising constant which assures that α i ∈ [0, 1] and i α i = 1.…”

“…We enunciated three desiderata: the possibility of connecting with transcriptomics/proteomics, the possibility of connecting with classic models, and accordance with conservation principles and suchlike. The first two have been accounted for; as regards the latter, it turns out that the recipe "physico-chemical principles plus r-functions" suffices in many cases to specify a model completely 19,20 . In the next section we will review various aspects of microbial physiology that justify and motivate the dynamic allocation theory.…”

“…The formalism proposed here extends effortlessly to any number of reserves, and to the inclusion of maintenance costs and how the dynamics change when these costs can no longer be covered by the available reserves or external supplies 19,20 . However, the reserves have to correspond to essential (non-substitutable) nutrients, and there are many known cases where alternative (substitutable) nutrients converge on a common reserve polymer.…”