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

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 nondecomposable 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 self-fabrication). 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].