Elementary growth modes/vectors and minimal autocatalytic sets for kinetic/constraint-based models of cellular growth

Abstract

Elementary vectors are fundamental objects in polyhedral geometry. In metabolic pathway analysis, elementary vectors range from elementary flux modes (of the flux cone) and elementary flux vectors (of a flux polyhedron) via elementary conversion modes (of the conversion cone) to minimal cut sets (of a dual polyhedron) in computational strain design. To better understand cellular phenotypes with optimal (or suboptimal) growth rate, we introduce and analyze classes of elementary vectors for models of cellular growth. Growth modes (GMs) only depend on stoichiometry, but not on growth rate or concentrations; they are elements of the growth cone. Elementary growth modes (EGMs) are conformally non-decomposable GMs; unlike elementary flux modes, they are not support-minimal, in general. Most importantly, every GM can be written as a conformal sum of EGMs. Growth vectors (GVs) and elementary growth vectors (EGVs) also depend on growth rate, concentrations, and linear constraints; they are elements of a growth polyhedron. Again, every GV can be written as a conformal sum of EGVs. To relate the new concepts to other branches of theory, we define autocatalytic GMs and the corresponding (minimal) autocatalytic sets of reactions. As a case study, we consider whole cell models (simple kinetic models of selffabrication). First, we use EGMs to derive an upper bound for growth rate that only depends on enzyme kinetics. Next, we study growth rate maximization (via control parameters for ribosome kinetics). In particular, we analyze growth states (GSs) and elementary growth states (EGSs) as introduced in [de Groot et al, 2020]. Unlike EGMs, EGSs depend on (metabolite) concentrations and growth rate. Most importantly, (i) we show that EGSs are support-minimal, (ii) we give a simple proof for the fact that maximum growth rate is attained at an EGS, and (iii) we show that, at every optimal EGS, the ribosome capacity constraint is active. Finally, we determine the dependence of EGSs on growth rate, and we study the relation between EGSs and minimal autocatalytic sets, EGMs, and elementary flux modes. Along the way, we point out (and resolve) mathematical issues in [de Groot et al, 2020].

Elementary vectors and autocatalytic sets for computational models of cellular growth

“…Comparison with (semi-)kinetic models. Recently, de Groot et al [9] analyzed (semi-)kinetic models of cellular growth, and one of the authors of this work started a study of constraint-based and (semi-)kinetic models; for a preprint, see [48]. Here, we compare the theoretical concepts introduced in [9] and in this work by means of an example, namely, the minimal growth model shown in Fig 9(a).…”

Elementary vectors and autocatalytic sets for resource allocation in next-generation models of cellular growth

“…Recently, de Groot et al [12] analyzed (semi-)kinetic models of cellular growth, and one of the authors of this work started a study of constraint-based and (semi-)kinetic models; for a preprint, see [52]. Here, we compare the theoretical concepts introduced in [12] and in this work by means of an example, namely, the minimal growth model shown in Fig 9a . The following discussion is based on a complete mathematical analysis given in Section D in S1 Text.…”

Combining Kinetic and Constraint-Based Modelling to Better Understand Metabolism Dynamics

“…The latter is limited to asymptotics, as only equilibrium fluxes are in the cone. Other decompositions, such as Elementary Growth Modes proposed by Müller [27], may be used to generalise this approach and go toward a decomposition of a whole trajectory.…”

“…• Minimal Constraint Flux Modes (MCFMs) by Morterol et al [25] to take into account Boolean constraints on the reactions, • Elementary Flux Vectors (EFV) by Klamt et al [26] to take into account linear constraint on the reaction rates' values, • Elementary Growth Modes (EGM) by Müller [27] to bring ODEs and CBM closer by taking into account the dilution term in the definition of CBM.…”