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

['Stefan Müller', 'Faculty Of Mathematics', 'University Of Vienna', 'Diana Széliová', 'Department Of Analytical Chemistry', 'Jürgen Zanghellini', 'Austrian Centre Of Industrial Biotechnology', 'Vienna']

Date: 2022-02 

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.

Next-generation, genome-scale metabolic models allow to study the reallocation of cellular resources upon changing environmental conditions, by not only modeling flux distributions, but also expression profiles of the catalyzing proteome. In particular, they do no longer assume a fixed biomass composition. Methods to identify optimal solutions in such comprehensive models exist, however, an unbiased understanding of all feasible allocations is missing so far. Here we develop new concepts, called elementary growth modes and vectors, that provide a generalized definition of minimal pathways, thereby extending classical elementary flux modes (used in traditional models with a fixed biomass composition). The new concepts provide an understanding of all possible flux distributions and of all possible biomass compositions. In other words, elementary growth modes and vectors are the unique functional units in any comprehensive model of cellular growth. As an example, we show that lipid accumulation upon nitrogen starvation is a consequence of resource allocation and does not require active regulation. Our work puts current approaches on a theoretical basis and allows to seamlessly transfer existing workflows (e.g. for the design of cell factories) to next-generation metabolic models.

Funding: This study was funded by the Austrian Science Fund (P33218) to SM and by the Austrian Centre of Industrial Biotechnology (acib - Next Generation Bioproduction) to DS and JZ. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Data Availability: All relevant data are within the manuscript and its Supporting information files.

Copyright: © 2022 Müller et al. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Finally, we relate EGMs and EGVs to other branches of theory. Obviously, purely stoichiometric models do not reflect implications from autocatalysis (namely, that all catalysts of active reactions need to be synthesized) and kinetics (namely, that all species involved in active reactions need to be present with nonzero concentrations). We observe that additional (capacity) constraints often ensure that EGVs are autocatalytic.

In this work, we introduce elementary growth modes and vectors (EGMs and EGVs) that are given by stoichiometry and irreversibility (EGMs) as well as by additional constraints and growth rate (EGVs). In fact, we develop a mathematical theory that enables an unbiased characterization of all feasible flux distributions in constraint-based models of cellular growth. In analogy to EFMs and EFVs, EGMs and EGVs can be interpreted as unique functional units of self-fabrication. Even more importantly, they provide an unbiased understanding of all possible biomass compositions. In a small example of a self-fabricating cell, we highlight that the experimentally observed lipid accumulation upon nitrogen starvation is a general feature of a comprehensive, next-generation metabolic model and not the result of active regulation.

In traditional, first-generation models, elementary flux modes and vectors (EFMs [ 4 – 6 ] and EFVs [ 7 , 8 ]) allow an unbiased study of metabolic pathways, in particular, an interpretation of any feasible flux distribution in terms of unique functional units. On the other hand, flux balance analysis (FBA) enables the computationally efficient identification of optimal fluxes [ 9 ]. In this respect, RBA can be viewed as a generalization of FBA that also accounts for enzyme costs. It represents the corresponding tool to identify optimal fluxes (and corresponding optimal allocations) in comprehensive, next-generation models. However, the analogues of EFMs and EFVs have not been identified, yet. For kinetic models with one enzyme constraint, optimal solutions have been characterized as EFMs [ 10 , 11 ], and “elementary growth states” have been introduced for kinetic models of cellular self-fabrication [ 12 ], but for constraint-based models, the corresponding theoretical concepts are still missing.

Arguably, some of the most successful approaches to study (microbial) growth processes are rooted in constraint-based modeling [ 1 ]. At their heart sits a (genome-scale) metabolic model, which captures all possible (cell-specific) biochemical transformations in an annotated and mathematically structured form. The resulting stoichiometric matrix, coupled with environmental and physico-chemical constraints, is already sufficient to study (fundamental aspects of) growth. Various biased and unbiased methods have been developed to predict (steady-state) metabolic phenotypes [ 2 ]. However, first-generation genome-scale metabolic models do not involve the expression of proteins and hence do not account for the individual enzyme costs of metabolic fluxes. These models rather use a fixed biomass composition that represents the average costs of growth. Recent efforts focus on the development of next-generation genome-scale metabolic models that also account for the enzyme demands of individual reactions. As one prominent example, we mention resource balance analysis (RBA) [ 3 ]. Although specific approaches differ in their degree of mechanistic detail and mathematical formulation, the central concept in the analysis of next-generation models is the (optimal) allocation of resources. However, we currently lack a theoretical understanding of all feasible allocations in next-generation metabolic models.

A major characteristics of life is self-fabrication, involving self-maintenance and self-replication. Cellular self-fabrication requires the acquisition and transformation of nutrients, not only to maintain the cell, but also to replicate, that is, to grow. During one cycle, a cell needs to duplicate all its building blocks and to cover the related costs. In constant environments, this process is balanced and leads to exponential growth. Indeed, exponential growth means that all cellular components are synthesized in proportion to their abundance. Clearly, metabolic activity depends on growth rate since higher growth rates imply higher synthesis rates which in turn require more ribosomes that produce the additional enzymes (and the ribosomes themselves). Thus, growth can be seen as a process that allocates cellular resources, limited by environmental and physico-chemical constraints.

2 Results

In this section, we introduce and analyze elementary growth modes and vectors for computational models of cellular growth. Further, we relate our approach to the theory of autocatalytic sets. In order to motivate the new concepts and to illustrate our theoretical results, we use a running example (a small model of a self-fabricating cell).

In Section 3 (Methods), we provide the relevant mathematical background (elementary vectors in polyhedral geometry), and in Section 4 (Discussion), we compare our theory with the constraint-based study of traditional models and the analysis of (semi-)kinetic models [12].

To begin with, we summarize the mathematical notation used throughout this work.

Notation. We denote the positive real numbers by and the nonnegative real numbers by . Let I be an index set; often I = {1, …, n}, and stands for . For , we write x > 0 if , x ≥ 0 if , and we denote its support by supp(x) = {i ∈ I∣x i ≠ 0}. Recall that a nonzero vector is support-minimal (in X) if, for all nonzero x′ ∈ X, supp(x′) ⊆ supp(x) implies supp(x′) = supp(x). For , we define its sign vector sign(x) ∈ {−, 0, +}I by applying the sign function component-wise, that is, sign(x) i = sign(x i ) for i ∈ I. The relations 0 < − and 0 < + on {−, 0, +} induce a partial order on {−, 0, +}I: for X, Y ∈ {−, 0, +}I, we write X ≤ Y if the inequality holds component-wise. For , we denote the component-wise product by , that is, (x ∘ y) i = x i y i . For and subindex set , we write for the corresponding subvector.

2.3 Elementary growth modes At steady state, Eq (1a) implies (4) Further, in a given setting, some reactions may have a given direction, as determined by thermodynamics. That is, (5) where denotes the set of irreversible reactions. The inequalities Nv ≥ 0 and (for the fluxes v) specify a polyhedral cone which suggests the following definition. Definition 1. Growth modes (GMs) for the dynamic growth model (1) are elements of the growth cone A GM v ∈ C g has an associated growth rate μ(v) = ωT Nv ≥ 0 and, if μ(v) > 0, an associated concentration vector . Elementary growth modes (EGMs) are conformally non-decomposable GMs. At this point, we refer the reader to Subsection 3.1 for an introduction to elementary vectors in polyhedral geometry (in particular, for the definitions of polyhedral cones and conformal non-decomposability). Still, we also try to provide an intuitive understanding of Definition 1: GMs are fluxes given by stoichiometry and irreversibility; in particular, they do not depend on concentrations or growth rate. Still, GMs have an associated growth rate and an associated concentration vector, as given by Eqs (2) and (4).

A growth cone C g is a general polyhedral cone. In contrast, a flux cone

is a general polyhedral cone. In contrast, a flux cone Whereas the elementary vectors of a flux cone are support-minimal, the elementary vectors of a growth cone are conformally non-decomposable. Thereby, a conformally non-decomposable vector cannot be written as a sum of other vectors without cancellations [14].

In applications, the growth cone (as the flux cone) is often pointed (due to irreversibility). GMs are scalable, but the associated concentrations are scale invariant. Proposition 2. For a GM v ∈ C g with associated concentration x(v) and λ > 0, it holds that x(λv) = x(v). Proof. Further, we can characterize the case of zero growth rate. Proposition 3. For a GM v ∈ C g , μ(v) = 0 is equivalent to Nv = 0. Proof. Obviously, Nv = 0 implies μ(v) = ωT Nv = 0. Conversely, μ(v) = ωT Nv = 0 with and implies Nv = 0. GMs v with μ(v) = 0 are flux modes (FMs), that is, elements of the flux cone EGMs v with μ(v) = 0 are elementary flux modes (EFMs), that is, support-minimal elements of C f . They are not support-minimal elements of C g , in general. (Again recall that N is the stoichiometric matrix of a comprehensive model of cellular growth, and hence C f and its EFMs are different from the corresponding objects in traditional models). Finally, we apply the general theory of elementary vectors and state the main result of this subsection. Theorem 4. Every nonzero GM is a conformal sum of EGMs. Proof. By Theorem 11 in Subsection 3.1. The growth cone C g is a general polyhedral cone, and its elementary vectors are the conformally non-decomposable vectors, that is, the EGMs. In fact, EGMs with nonzero growth rate can be scaled to have the same growth rate as the given GM. Corollary 5. Let v be a nonzero GM with associated growth rate μ(v) ≕ μ. Then, there exist (possibly empty) finite sets E 0 and E μ of EGMs with associated growth rates 0 and μ > 0, respectively, such that λ e ≥ 0, and . Moreover, if μ > 0, then . Proof. By Theorem 4, there exist (possibly empty) finite sets E 0 and E > of EGMs (with associated growth rates 0 and >0, respectively) such that In particular, . If μ > 0, that is, E > ≠ ∅, then where with μ(e′) = μ and with . Finally, if μ > 0, then . In Corollary 5, we actually fix growth rate which turns the growth cone into a polyhedron. Hence, the result is also an instance of Theorem 12 in Subsection 3.1. Example (EGMs). Recall the small model of a self-fabricating cell given in Fig 1 and note that all reactions are assumed to be irreversible, . As it turns out, there are 11 EGMs (up to scaling), corresponding to the 11 molecular species, that is, with i ∈ Mol = {G, N, AA, LD, L, IG, IN, EAA, ELD, EL, R}. Explicitly, Thereby, we introduced the factors μ/ω i to obtain correct units (mol g−1 h−1) and equal growth rate μ for all EGMs. We note the following points: Every EGM “produces” exactly one molecular species, as indicated by its name. (For every EGM, there is exactly one molecular species with nonzero associated concentration.) Formally, Ne i = (μ/ω i )u i , where u i is the ith unit vector in G produces G (glucose), e N produces N (ammonium), …, and e R produces R (ribosome).

This special situation arises from the fact that the stoichiometric matrix N is square. In general, there may be more than one EGM for the exclusive production of some molecular species or, conversely, no EGM for exclusive production (just joint production with another species).

= (μ/ω )u , where u is the ith unit vector in produces (glucose), e produces (ammonium), …, and e produces (ribosome). This special situation arises from the fact that the stoichiometric matrix N is square. In general, there may be more than one EGM for the exclusive production of some molecular species or, conversely, no EGM for exclusive production (just joint production with another species). EGMs e G and e N are support-minimal, but all other EGMs are not.

and e are support-minimal, but all other EGMs are not. For the associated growth rates, we obtain μ(e i ) = ω T Ne i = μ/ω i ⋅ ω T u i = μ/ω i ⋅ ω i = μ. (This is just the consequence of introducing the factors above.) For the associated concentrations, we obtain x(e i ) = Ne i /μ(e i ) = (μ/ω i )u i /μ = u i /ω i .

) = ω Ne = μ/ω ⋅ ω u = μ/ω ⋅ ω = μ. (This is just the consequence of introducing the factors above.) For the associated concentrations, we obtain x(e ) = Ne /μ(e ) = (μ/ω )u /μ = u /ω . There are no EGMs with zero growth rate, that is, there are no EFMs. To give an example of a conformal sum, we consider the GM with support in the import/enzymatic reactions r IG , r IN , r EAA and in the synthesis reactions s IG , s IN . In particular, v′ is a GM without lipid synthesis. Now, let μ(v′) = μ. By Theorem 4 (and Corollary 5), v′ is a conformal (and convex) sum of EGVs, with λ G , λ N , λ AA ≥ 0, λ IG , λ IN > 0 and λ G + λ N + λ AA + λ IG + λ IN = 1. Note that v′ produces IG, IN (since λ IG , λ IN > 0), however, it produces G, N, AA if and only if also λ G , λ N , λ AA > 0. Since GMs are given by stoichiometry (and irreversibility), they do not reflect constraints implied by autocatalysis and kinetics. Example (autocatalysis and kinetics). Obviously, the GM v′ above involves catalytic reactions. In particular, it involves r EAA (amino acid synthesis) as well as s IG , s IN (the expression of the importers). However, it is not catalytically closed in the sense that reactions r EAA and s IG , s IN carry fluxes, but the corresponding catalysts, the enzyme EAA and the ribosome R, are not expressed. Now, EGV sEAA involves e EAA (the expression of EAA), and analogously eR involves s R . Hence, we form a convex sum with λ EAA , λ R > 0, to obtain a GM that is catalytically closed. Further, the EGM eG is kinetically consistent in the sense that it involves the reaction r IG and hence the species G and it has a corresponding nonzero associated concentration x G (eG) > 0, as implied by kinetics. By the discussion above, the GM v′ is kinetically consistent if and only if λ G , λ N , λ AA > 0. In this case, all species involved in active reactions have nonzero associated concentrations. In the next subsection, we elaborate on the concepts of catalytic closure and kinetic consistency.

2.4 Implications from autocatalysis and kinetics Cellular growth is autocatalytic in the sense that the cell fabricates itself (thereby exchanging substrates/products with the environment). One needs to distinguish this notion of “network autocatalysis” from “autocatalytic subnetworks” [15, 16]. Consider the overall reaction corresponding to (the flux through) a subnetwork. If a molecular species appears on both the educt and product sides, in particular, with a larger stoichiometric coefficient on the product side than on the educt side, then it is formally autocatalytic (cf. [17]). In fact, there are several competing notions of autocatalytic species and subnetworks (cf. [15–17]). In this work, we consider network autocatalysis. Before we state possible definitions, we distinguish two modeling approaches. Detailed models (without individual catalytic reactions):

In this approach, catalysis occurs on the level of (small) subnetworks. In particular, individual reactions are not catalytic. For example, a simple catalytic mechanism (involving enzyme E , substrate S , and product P ) is given by E + S ↔ ES ↔ EP ↔ E + P .

(without individual catalytic reactions): In this approach, catalysis occurs on the level of (small) subnetworks. In particular, individual reactions are not catalytic. For example, a simple catalytic mechanism (involving enzyme , substrate , and product ) is given by + ↔ ↔ ↔ + . Coarse-grained models (with individual catalytic reactions):

In this approach, catalysis occurs on the level of individual reactions. For example, the catalytic mechanism above is written as E + S ↔ E + P or For detailed models, one may call a growth mode autocatalytic if it contains an autocatalytic species or subnetwork and it is catalytically closed. Formal definitions and their comparison are beyond the scope of this work. For coarse-grained models (like the small model of a self-fabricating given in Fig 1), we give a formal definition of network autocatalysis. Definition 6. For a coarse-grained model, let Cat ⊆ Rxn be the set of catalytic reactions. A GM v ∈ C g is basically catalytic (BC) if there is a catalytic reaction r ∈ supp (v) ∩ Cat. Further, a GM v ∈ C g is catalytically closed (CC) if, for every catalytic reaction r ∈ supp (v) ∩ Cat, it holds that (Nv) s > 0 for the corresponding catalyst s ∈ Mol. Finally, a GM v ∈ C g is autocatalytic (AC) if it is BC and CC. A subset of reactions is autocatalytic (AC) if there exists an autocatalytic GM v ∈ C g with S = supp(v). A nonempty subset of reactions is minimally autocatalytic (MAC) if it is AC and inclusion-minimal. In the literature, a closure condition is also crucial in the definitions of “reflexive autocatalysis” [18–20] and “chemical organizations” [21–23]. We note the following points: AC is implied by two conditions: BC guarantees that there is at least one active catalytic reaction, and CC ensures that all active catalysts are produced.

For an illustration, recall the motivating paragraph “Example (autocatalysis and kinetics)” just before this subsection. In the running example, all reactions are catalytic, and hence all GMs are BC. Whereas the GM v′ is not CC (the active enzyme EAA and the ribosome R are not produced), the GM v″ is CC and hence AC. Its support supp(v″) = {r IG , r IN , r EAA ; s IG , s IN , s EAA , s R } is an AC subset of reactions; in fact, it is the only MAC subset of reactions.

For an illustration, recall the motivating paragraph “Example (autocatalysis and kinetics)” just before this subsection. In the running example, all reactions are catalytic, and hence all GMs are BC. Whereas the GM v′ is not CC (the active enzyme and the ribosome are not produced), the GM v″ is CC and hence AC. Its support supp(v″) = {r , r , r ; s , s , s , s } is an AC subset of reactions; in fact, it is the only MAC subset of reactions. Catalytic closure is defined for fluxes, but it refers to concentrations via x(v) s = (Nv) s /μ(v). In particular, (Nv) s > 0 implies x(v) s > 0. In addition to autocatalysis, also kinetics implies constraints on growth modes. In particular, a growth mode is kinetically consistent if all species involved in active reactions (not necessarily all species in the model) have nonzero associated concentrations. Definition 7. A GM v ∈ C g is kinetically consistent if, for every r ∈ supp (v) ⊆ Rxn and s ∈ Mol, N sr ≠ 0 implies (Nv) s > 0. We note the following points: Like catalytic closure, kinetic consistency is given by stoichiometry.

Interestingly, kinetic consistency implies formal autocatalysis for all (not necessarily catalytic) species involved. In the next subsection, we consider additional (capacity) constraints which often ensure catalytic closure.

2.5 Constraint-based models For many systems, kinetic models are not yet available, and constraint-based models are used. Steady-state reaction rates (fluxes) v are considered as independent variables, that is, the non-linear dependence of the kinetics on the concentrations x is neglected. Most importantly, catalytic processes imply linear capacity constraints for x and v. Additional constraints can be formulated for processes that are not catalytic (in the given model), e.g. lower bounds for concentrations or fluxes. In compact form, linear constraints can be can be written as with , , and . Altogether, constraint-based growth models involve steady-state, irreversibility, (dry) mass, and additional linear constraints, (6) Example (additional constraints). Recall the small model of a self-fabricating cell given in Fig 1. In addition to steady-state, irreversibility, and (dry) mass constraints, we consider capacity constraints for all catalysts (importers, enzymes, and the ribosome) and membrane constraints. Let k cat be the turnover number of the importers IG, IN and enzymes EAA, ELD, EL and k el be the elongation rate of the ribosome R. The resulting capacity constraints are given by (7a) for I = IG, IN and E = EAA, ELD, EL and (7b) The cell membrane area is formed by lipids L and importers IG and IN, leading to the (equality) constraint (7c) where A L and A I denote the areas of lipids and importers, respectively, r denotes the surface-to-volume ratio, ρ denotes cell density, and N A is Avogadro’s number. Additionally, we require that a minimum fraction α of the surface area is formed by lipids, leading to the (inequality) constraint (7d) For the derivation of the membrane constraints, see Section B in S1 Text. As stated above, the additional constraints (7) can be summarized as Ax + Bv ≥ b. All constraints are based on realistic data. The parameter values for the small model of a self-fabricating cell are given in Table 2. In the final subsection, we introduce the basic “building blocks” of any possible flux distribution in a constraint-based model of cellular growth.

[END]

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