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

Traditional (genome-scale) metabolic models of cellular growth involve an approximate biomass “reaction”, which specifies biomass composition in terms of precursor metabolites (such as amino acids and nucleotides). On the one hand, biomass composition is often not known exactly and may vary drastically between conditions and strains. On the other hand, the predictions of computational models crucially depend on biomass. Also elementary flux modes (EFMs), which generate the flux cone, depend on the biomass reaction. To better understand cellular phenotypes across growth conditions, we introduce and analyze new classes of elementary vectors for comprehensive (next-generation) metabolic models, involving explicit synthesis reactions for all macromolecules. Elementary growth modes (EGMs) are given by stoichiometry and generate the growth cone. Unlike EFMs, they are not support-minimal, in general, but cannot be decomposed “without cancellations”. In models with additional (capacity) constraints, elementary growth vectors (EGVs) generate a growth polyhedron and depend also on growth rate. However, EGMs/EGVs do not depend on the biomass composition. In fact, they cover all possible biomass compositions and can be seen as unbiased versions of elementary flux modes/vectors (EFMs/EFVs) used in traditional models. To relate the new concepts to other branches of theory, we consider autocatalytic sets of reactions. Further, we illustrate our results in a small model of a self-fabricating cell, involving glucose and ammonium uptake, amino acid and lipid synthesis, and the expression of all enzymes and the ribosome itself. In particular, we study the variation of biomass composition as a function of growth rate. In agreement with experimental data, low nitrogen uptake correlates with high carbon (lipid) storage.


Mol
set of molecular species Rxn set of chemical reactions N ∈ R Mol×Rxn stoichiometric matrix (unit: 1) ω ∈ R Mol > molar masses (unit: g mol −1 ) X ∈ R Mol ≥ amounts of substance (unit: mol) R(·) ∈ R Rxn reaction rates (extensive) (unit: mol h −1 ) The chemical reactions induce the dynamical system We define mass, M = i ω i X i = ω T X, (unit: g) (2a) the intensive quantities and growth rate Thereby, we use mass instead of volume to define the "concentrations" x, the (intensive) reaction rates v, and growth rate µ. In practice, cellular composition is often given in the unit mol g −1 (dry weight).
Finally, we recall the chain rule (of differentiation), d dt January 24, 2022 1/10 Equations (1), (2), and the chain rule yield the dynamic model of cellular growth: Thereby, we assume given cell density. Recall that reaction rates depend on (volumetric) concentrations X/V , with volume V (unit: L) and cell density ρ = M V (unit: For constant cell density ρ, v = v(x) only depends on concentrations.
For alternative derivations, see e.g. [2] or [1]. By multiplying the mass balance equation with a vector c ∈ R Mol , we obtain We highlight two observations that hold for any model of cellular growth.
Fact (conservation laws). In a model of cellular growth, there cannot be any conservation laws. In mathematical terms, ker N T ∩ R Mol ≥ = {0}. To see this, assume c T N = 0 with 0 = c ≥ 0, for example, assume c 1 = c 2 = 1 and Fact (dependent concentrations). In a model of cellular growth, there can be dependent concentrations. In mathematical terms, ker N T = {0}.

B Example: membrane constraints
For the small model of a self-fabricating cell studied in the main text, we derive the membrane constraints (7c) and (7d).
The cell membrane area A is formed by lipids L and importers IG and IN, where A L and A I denote the areas of lipids and importers, respectively, and #X denotes the number of molecule X. After division by Avogadro's number N A , we have where s X = #X N A denotes the amount of substance. Further, after division by cell mass m, we have where x X = s X m denotes the (mass-specific) concentration. Finally, using cell volume V , the surface-to-volume ratio r = A V , and cell density ρ = m V , we obtain A m = A V V m = r ρ and hence Additionally, we require that a minimum fraction α of the surface area is formed by lipids, where we use concentrations instead of numbers of molecules. Equivalently,

C Example: figures and tables
Name (In)equality

D Minimal growth model with alternative pathways
Consider the following minimal model of cellular growth with two alternative pathways: The cell takes up external substrates and forms amino acids (AA) via two "reactions", catalyzed by the "enzymes" E1 and E2, respectively. Amino acids are then used by the ribosome (R) to synthesize the enzymes and the ribosome itself, The set of molecular species is Mol = {AA, E1, E2, R}, and the set of reactions is Rxn = Rmet ∪ Rsyn with metabolic reactions Rmet = {r 1 , r 2 } and synthesis reactions Rsyn = {s 1 , s 2 , s R }.
Every EGM "produces" exactly one molecular species, as indicated by its name, thereby using either pathway 1 or 2. (For every EGM, there is exactly one species with nonzero associated concentration.) Due to the factor µ/ω, all EGMs have associated growth rate µ.
GMs need not be AC, in particular, no EGM is AC. In fact, every (nonzero) GM is BC, since all reactions are catalytic, however, a GM need not be CC. MAC subsets of reactions are the supports of AC GMs. There are two MAC sets, namely M 1 = {r 1 , s 1 , s R } and M 2 = {r 2 , s 2 , s R }, corresponding to the two alternative pathways. AC GMs with support M 1 are generated by the EGMs e AA,1 , e E1,1 , e R,1 . (Analogously for the MAC set M 2 .) For a GM v ∈ C g , the associated growth rate amounts to determined by the "exchange" fluxes v 1 and v 2 . For fixed growth rate µ, the growth cone becomes a growth polytope, Further, for scaled fluxesv =ω/µ · v, the polytope becomes independent of µ, In particular, its projection to the synthesis fluxesŵ ∈ R Rsyn ≥ is the "growth simplex" • In a constraint-based model, one considers inequality enzyme capacity constraints, • in a (semi-)kinetic model, one considers equality constraints arising from enzyme and ribosome kinetics, for i ∈ {1, 2} and j ∈ {1, 2, R}. Thereby, κ i , τ j are functions of the amino acid concentration x AA , and α j are control parameters (ribosome fractions) for studying growth rate maximization, cf. [1,3].
Moreover, one often considers ribosome capacity constraints: w 1 + w 2 + w R ≤ k tl x R in constraint-based models and j∈{1,2,R} α j ≤ 1 in (semi-)kinetic models. However, the (inequality) ribosome capacity constraint is treated separately in the (semi-)kinetic model, and for reasons of comparison, we just require x R > 0 in the constraint-based model. Additional constraints involve concentrations. For a GM v ∈ C g , the associated concentrations x are given by In particular, x E1 = w 1 /µ and x E2 = w 2 /µ.
In the following, we consider the growth polyhedra and EGVs arising from the additional constraints.