Notation

Vectors are denoted by boldface, e.g., v. The vector u_j denotes the j-th elementary vector, filled with zeros except for the j-th element, which is 1. The dot product between vectors v and w is denoted by v ⋅ w, and |v| denotes ∑_i |v_i|, which in this paper always simplifies to ∑_i v_i because the vectors in question have positive entries. Time derivatives are denoted by overdots, e.g., . The inequality v ≥ 0 should be interpreted as v_j ≥ 0 for all j. Modelling assumptions are labelled as A1, A2, etc. An overview of all variables and parameters used in this paper can be found in S1 Table.

Linking metabolic activity to the growth rate

The following is closely related, and in part identical, to the exposition in [24].

Let n(t) = (n₁(t), …, n_K(t)) be the vector of copy numbers (in moles) of K compounds in a population of cells with total volume V(t) at time t. The concentration of compound k is defined as c_k(t) ≔ n_k(t)/V(t). The modelling assumption is that changes in copy numbers of the compounds are due to chemical reactions which take place in a well-stirred cell [25], (1)

Here N denotes the stoichiometry matrix, and v_j(c) is the j-th reaction rate, as a function of concentrations c. Since n_k = Vc_k, we have (2)

In balanced growth, all concentrations are constant over time [26], , which means that (3)

To relate the growth rate of the volume of a cell to all the metabolic reactions making new cell material, we have to relate cell volume to the molecule copy number. We consider experimental conditions such that the volume is a function of all the copy numbers only, V(t) = V(n(t)). In particular, we assume (see S1 Text for justification and detail) that V only depends linearly on copy numbers, (4) which is equivalent to (5)

The (instantaneous) growth rate is now defined as (6)

It directly follows (see S1 Text) that at balanced growth (7)

(This derivation differs only from [24] in the exact form of the “constant density” assumption (A2). In our case, each type of molecule contributes its own specific volume, instead of assuming an overall constant biomass concentration. As explained in the S1 Text, (A2) is not directly based on data, but on the requirement that the steady state Eq (3) have solutions at all).

Since ρ_l equals the molar volume of molecule l, relation (7) simply states that the growth rate of a cell equals the total volume synthesis rate of all its reactions per unit volume. This total volume synthesis rate is calculated by summing the volumes of all synthesized metabolites in all reactions per unit time.

Eq (2) is equivalent to (8) which is the familiar form showing reactions and dilution [3, 24]. If we substitute the balanced growth rate (7) into (8) at steady state conditions, we obtain (9)

This equation shows that at steady-state, the net production of molecule k (first term) should balance the dilution of molecule k by cellular growth, growth that is due to the volume contribution of all produced molecules in the cell (second term). We could not analyse these equations further to understand the structure of steady state solutions, without additional mathematical assumptions.

Introducing a whole-cell model of self-fabricating cells

To obtain a better understanding of how the molecular state of a cell, its ‘phenotype’, relates to its growth rate, we have to add more biological information to the general model.

We distinguish concentrations of metabolites x, enzymes e and ribosomes r. We ignore several biological entities, especially mRNA and genes. Genes are implicitly defined by the reaction stoichiometries occurring in the stoichiometric matrix; they are not synthesized nor degraded. DNA itself may be synthesized as a macromolecule in our description. Messenger RNA is not synthesized in our model, but it does play a role as will become clear later. We note that the model can be extended with mRNA, without qualitative changes. We left it out for simplicity and because, in terms of volume, it is a minor cellular component, in contrast to rRNA, which can be considered as a regular macromolecule.

We note that in all population perspectives on cell growth, including ours, the division and birth process is generally not incorporated as a molecular process. All cells are growing and dividing asynchronously, but in the model description they are considered in the same state of producing cell components. At each moment in time, a fixed fraction of cells is dividing in a population of cells at balanced growth [30]; considering their activities would require a different modelling formalism than the one pursued in this paper. We have chosen this formalism because it is the overarching description of all genome-scale modelling formalisms [4, 22] that are currently in use in systems biology.

The dynamical system for all concentrations c = (x, e, r) is still given by (8). We subdivide the stoichiometric matrix N into parts, corresponding to the two levels of “metabolism” and “enzymes and ribosome synthesis”, as follows, (10)

Here P is an m × n-matrix (the usual stoichiometric matrix in a metabolic pathway), with entries generally of different signs. The (stoichiometric) number of metabolites needed for the synthesis of each enzyme is recorded in M, an m × (n + 1)-matrix with mostly positive (or zero) entries. The only negative entries in M correspond to metabolites that are produced when enzymes are synthesized, such as ADP. The (n + 1) × (n + 1) identity matrix, I, denotes that each enzyme is made by one enzyme synthesis reaction. The vector of reaction rates is also split, into metabolic rates v = (v₁, …, v_n), and synthesis reactions w, consisting of n enzyme synthesis rates (w₁, …w_n) and a ribosome synthesis rate w_n+1.

The assumptions underlying (10) are that each individual metabolic reaction v_j(c) has a unique enzyme associated to it. Since enzyme j catalyses metabolic reaction j, we assume, in agreement with basic enzyme kinetics [31], that (11)

The ribosome catalyses synthesis of all enzymes and itself, and amino acids (which form part of the metabolites x) are consumed in the process. We therefore choose (12)

The rate laws f_j(x) are assumed to be known nonlinear functions, and contain the catalytic rate parameter k_cat,j for the enzyme e_j in this notation; moreover, they incorporate the thermodynamic constraints imposed by the substrates and products of each enzyme. Also the g_j(x) are assumed to be known and contain k_cat,r. The linear dependence of the metabolic rates v_j on enzyme concentration e_j follows generally from quasi-steady-state type derivations of rate laws [31], and are a good approximation when enzymes are at concentrations that are much lower than the substrate or product concentrations. The linear dependence of enzyme and ribosome synthesis rates on ribosome level in (12) is assumed for the same reason. Both linear dependencies are crucial for the development of the theory.

The coefficient α_j is the fraction of the total ribosome pool that is allocated to produce enzyme j, and are the result of gene expression. A possible mechanism is that the different mRNA molecules passively compete to be translated by the ribosome. The fraction of the ribosome allocated to produce enzyme j is then assumed to be proportional to the fraction of mRNA_j over total mRNA (but not necessarily equal to it, since a certain fraction of the ribosome may not be allocated to anything, in principle). Our theory does not depend on the specific mechanism that is used in cells, but is based solely on the assumption that the α_j-factors can be influenced by gene expression. We have (13) where λ signifies the fraction of all the ribosomes that are actively involved in synthesis of new protein and ribosomes. We emphasise that the α_j’s are the main control parameters of the cell in the perspective taken in this paper. By changing them, the cell changes phenotype.

Let us lastly introduce the molar volume parameters for the compounds. We denote these by ρ₁, …, ρ_m for the m metabolites, σ₁, …, σ_n for the n enzymes, and σ_n+1 for the ribosome. Assumption (A2) now reads, analogously to (5), (14)

In summary, assumptions (A1)–(A6) give the following dynamical system for a whole-cell model, (15)

External environmental concentrations, such as nutrients, are incorporated as parameters in the relevant f_j(x). An overview of the main ingredients of the model is given in Fig 1.

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Fig 1. Illustration of our formalism describing self-fabrication, with three layers of nonlinearity.

The top figure shows how the synthesis of cellular building blocks leads to cell growth, with rate μ. In a balanced growth state, all currently present cellular building blocks (denoted by the coloured shapes on the left side) should be synthesized in the right proportions (depicted on the right side). Our theory is as general as possible: it identifies all cellular make-ups that could give rise to a balanced growth state, the specific coloured shapes are for illustration purposes only. The bottom part shows three layers of nonlinearity that have to be incorporated in a self-fabrication model. 1) The rate of a reaction depends nonlinearly on the concentrations of substrates, products, and often on different compounds via (allosteric) regulation. 2) The dilution by growth of each compound is proportional to its concentration and to the growth rate. Since the growth rate depends on the synthesis rate of volume, this again depends on the concentrations of compounds that contribute to volume synthesis, such as the ribosome. Thus, the dilution of such compounds depends nonlinearly on their own concentrations. 3) Self-fabrication brings an inherent nonlinearity: the cellular composition should be tuned to produce the exact demand for precursors, but the demand for precursors is again determined by the cellular composition.

https://doi.org/10.1371/journal.pcbi.1007559.g001

The balanced growth equations

To make notation more concise, set so that μ in (15) simplifies to (16)

We now derive a set of identities that need to hold at balanced growth, i.e., when . The aim is that they are written in such a way that the variables under control of the cell, the ribosomal allocation parameters α_j, stand out.

Since , μe_j = rα_j g_j(x), and the definition of μ in (16) may be rewritten as the following equation for μ, (17) which allows us to compute a steady state value for r in terms of x, α₁, …, α_n+1 and μ, (18)

Demanding for k = 1, …, m in Eq (15) gives another m identities for μ, (19) With Eqs (17) and (19), we now have two different expressions of that should be equal whenever x_k ≠ 0. This yields (20) for k = 1, …, m, which are m equations for x and μ only.

Lastly, we still need to require , which yields (21)

It is tempting to use this relation straight away in (20), but it is much better not to submit to this temptation as it would destroy the geometric structure of the solution space in the remaining fractions α₁, …, α_n. This is a vital ingredient later in the construction of Elementary Growth States and Modes.

For future reference, we collect the pertinent Eqs (20) and (21), together with the assumptions on α in (13), and collectively call these the Balanced Growth equations,

Note that when a specific ribosome allocation is chosen (by fixing α), we get m + 1 equations that should be met. In principle, this is enough to fix the m metabolite concentrations and the growth rate, μ. Subsequently, the enzyme and ribosome concentrations, e, r, can also be calculated. The α_j-factors can thus be seen as controlling all other variable quantities in the model. However, not all choices for α will yield a solvable system of equations, and not all ribosome allocations will thus lead to balanced growth.

Definition of Elementary Growth States (EGSs)

The balanced growth Eq (22) may be succinctly written as (26) where (27) and for k ∈ {1, …, m} and j ∈ {1, …, n} we have (28)

This is a system of linear equations in α. Because the Eq (23) is also linear in α_n+1, we can incorporate it in the system by introducing a new matrix, (29)

With these definitions, the balanced growth Eqs (22)–(25) read (30) where u_m+1 is the unit vector: [0, …, 0, 1]^T.

Remark. The rank of the matrix B(x, μ) may in principle depend on the choice of x and μ. Certainly, rows corresponding to a metabolite with zero concentration, x_k = 0, are no valid constraints. If these rows are left out, it is clear, considering the definition of the elements of B(x, μ) in (28), that the set of x and μ for which B(x, μ) does not have full rank has measure zero, i.e., represents negligible exceptions—even if the original stoichiometry matrices may have lower rank. We therefore from now on assume, that the rank does not depend on the choice of x and μ. Most arguments that follow are for fixed x, and so we will require that the rank does not depend on μ.

To introduce Elementary Growth States, we fix x and μ, and ignore the ribosome allocation inequality |α| ≤ λ for now. Denote the row vectors of B = B(x, μ) by , k = 1, …, m + 1. The set formed by all positive solutions to the first m equations, is a pointed polyhedral cone. The cone becomes a convex polytope if we restrict to those vectors α in such that α_n+1 g_n+1(x) = μ. Fig 2a gives the reader a sketch of this construction. Let us define (31)

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Fig 2. Construction of Elementary Growth States and Elementary Growth Modes.

a) For fixed metabolite concentrations x and growth rate μ₁, the balanced growth Eqs (22) and (24) define a cone . Intersecting this cone with (23) defines a polytope containing all balanced growth solutions. Its vertices are the EGSs at this growth rate and metabolite concentrations. b) Changing the growth rate from μ₁ to μ₂ changes the cone to and the polytope to , because (22) depends on μ. (A similar change would be seen if x had been changed.) c) The EGSs change continuously with μ (and also with x, not shown), but their support does not. The green lines, connecting the vertices in the different polytopes, together form the EGMs.

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Definition 1. For a given set of metabolite concentrations x and growth rate μ, let be the corresponding convex polytope defined by (31). Then the Elementary Growth States with metabolite concentrations x and growth rate μ are the vertices of this polytope.

Each EGS is thus a vertex of and therefore a basic feasible solution of the Linear Program (30). It has a corresponding feasible basis D such that B_D(x, μ)α_D = μu_m+1 and α_j = 0 for all j ∉ D. Moreover, for each D, B_D(x, μ) is square (after restricting to a set of linearly independent rows) and nonsingular. The support supp α for an EGS satisfies supp α ⊆ D.

EGSs are quintessentially objects that depend on the chosen metabolite concentration vector x and growth rate μ. For each choice, the polytope will be different, with different vertices. This is because the constraint equations that define this polytope include the stoichiometry, the kinetics of enzymatic reactions, and the metabolite concentrations of the cell.

With the definition of EGSs, we immediately have the following fundamental result.

Theorem 1. Any balanced growth solution with growth rate μ is a convex combination of EGSs with the same growth rate and the same metabolite concentrations.

Proof. For fixed x and growth rate μ, any balanced growth solution satisfies B(x, μ)α = μu_m+1 and α ≥ 0. The vectors that satisfy these relations form precisely the polytope . This polytope is the convex hull of its vertices, the EGSs.

A convex combination of EGSs with different growth rates does not automatically satisfy the balanced growth equations, i.e., if α₁,α₂ are two EGSs with growth rates μ₁ and μ₂, then the convex combination tα₁+(1 − t)α₂ does not necessarily satisfy

The main reason is that B(x, μ) changes with μ. For fixed x, each μ defines a different polytope , see Fig 2b.

We may assume without loss of generality that B(x, μ) has maximal rank m + 1. If not, we can, without consequence, delete rows from B that are linearly dependent. This would mean that in balanced growth, not all metabolic concentrations are independent: there are ‘moieties’, but not exactly in the classical stoichiometric fashion. There is always dilution by growth. When selecting a feasible basis D, the last column in B should always be kept, since it enforces α_n+1 g_n+1(x) = μ, and ensures that the ribosome is made; without ribosomes, there are no enzymes.

Theorem 2. If a vector α is an EGS, then its support supp α corresponds to a subnetwork that has a number of metabolic reactions that is less or equal to the number of independent metabolites. Each set D that is a feasible basis for an EGS, and for which D is also its support, has exactly as many metabolic reactions as independent metabolites.

Proof. We fix x and μ throughout, and thereby also the polytope . The EGSs are the vertices of , and equivalently the basic feasible solutions of the LP. (A standard LP has an objective function, but that does not concern us here. For now, we only need to focus on the basic feasible solutions).

Each basic feasible solution has a corresponding feasible basis D, so that B_D(x, μ) is square and invertible. Note that |D| must therefore be equal to the number of independent rows of B. Independent rows are given by the Balanced Growth equations of independent metabolites with non-zero concentrations. The vector has length |D|. For all j ∉ D we have α_j = 0, but even for j ∈ D we can have α_j = 0, so that we have at most D non-zero α_j’s. Certainly n + 1 ∈ D, because it corresponds to the ribosome, without which enzymes cannot be synthesised. That leaves |D| − 1 elements j ∈ D\{n + 1}, each of which corresponds to a unique α_j, and thus to a unique enzyme. The number of active reactions, corresponding to non-zero α_j’s, is thus less or equal to the number of independent metabolites (in the sense described above) with non-zero concentrations.

If D is also the support of α, then the number of non-zero α_j’s is equal to |D|, so that the number of active reactions is exactly equal to the number of independent metabolites with non-zero concentration.

Elementary Growth Modes are equivalence classes of Elementary Growth States

The support of an EGS, defined as the set of nonzero α_j (see S1 Text), induces a natural equivalence class of EGSs which all share the same support and hence constitute steady state networks with the same enzymes. This equivalence class does not depend on the μ and x for which the individual EGSs were calculated.

Definition 2. Two EGSs, α₁(x₁, μ₁) and α₂(x₂, μ₂), are said to be equivalent if they have the same support. The equivalence class to which α₁ belongs is denoted by α₁(x, μ)], and is called an Elementary Growth Mode.

The set of EGMs, contains all essentially different minimal networks that can sustain balanced growth, each with its unique set of participating reactions. In mathematical terms, the set of all EGMs, , is the quotient set of all EGSs under identification of EGSs with the same support.

Note that for each x and μ individually, a representative EGS for each EGM equivalence class can be calculated using standard algorithms. It is just a matter of finding the vertices of the polytope defined in (31), a task for which computational algorithms are already available. These representatives will change with x and μ, but their support (and thus the EGM that they belong to) will not. It is however possible that an EGM has a feasible representative at some set of metabolite concentrations and growth rate, but not for another set.

We here prove that an EGS can be continuously extended both in an open neighbourhood around μ and in a neighbourhood around x. This shows that if the whole-cell model allows a steady state at some fixed x and μ, then steady states also exist with the same participating reactions for metabolite concentrations and growth rates close to the fixed values, see Fig 2c. In other words, given a representative of an EGM, this EGM also has a feasible representative in a small neighbourhood.

Theorem 3. For a given growth rate μ₀ and set of metabolite concentrations x₀, there exists an open neighbourhood U such that for all (x, μ)∈U, each EGS with support equal to its feasible basis, α(x₀, μ₀), can be continuously extended to a vector α(x, μ) that solves the balanced growth equations and belongs to the same EGM.

Proof. Choose an arbitrary EGS: . We know that it solves

The EGS has a feasible basis D, and without loss of generality we restrict B(x₀, μ₀) to the columns indexed in D. As discussed before, we can select a set of rows corresponding to independent metabolites with non-zero concentrations such that the resulting matrix is square and invertible. We may therefore choose n = m and the dimension of B to be (m + 1) × (m + 1).

We would like to apply the Implicit Function Theorem to see that there are continuously differentiable functions , j = 1, …, m + 1, such that in an open environment of (x₀, μ₀), the Balanced Growth Equations are still met: . Since the are continuous, and since no α_j(x₀, μ₀) is equal to zero by the assumption in the Theorem, we can then also choose a neighbourhood in which . This would thus indeed be a continuous extension of the EGS that belongs to the same EGM, since its support does not change.

Let s be the number of components in x. For the Implicit Function Theorem we need a function that is zero at (α₀, x₀, μ₀). For this function, we can use the Balanced Growth equations as components: the first m are given by (32) where k = 1, …, m. The last component is given by (33)

Let us check the conditions for the Implicit Function Theorem:

• The function with components given by (32) and (33) is continuously differentiable in α, x, and μ in a neighbourhood of (α₀, x₀, μ₀).
• By assumption, (α₀, x₀, μ₀) is a solution of the balanced growth equations: F(α₀, x₀, μ₀) = 0.
• The entries of the Jacobian of F at (α₀, x₀, μ₀) with respect to α are, for k = 1…, m, given by and the last row of the Jacobian is zero except for the last entry, which is g_m+1(x). Note that this Jacobian is exactly the matrix B(x₀, μ₀), which was invertible by construction.

The IFT may therefore be invoked, which shows that for each EGS at (x₀, μ₀), with support equal to its feasible basis, there is an open neighbourhood around (x₀, μ₀) such that the EGS can be continuously extended. By taking the intersection of these neighbourhoods we can indeed find a neighbourhood in which all of the EGSs can be extended.

The previous theorem shows that the set of EGMs is largely conserved when the growth rate or the metabolite concentrations are changed. Indeed, the functions that were found in the previous proof in principle give rise to balanced growth vectors at a wide range of growth rates μ and concentrations x. However, these vectors are not necessarily all EGSs, because the positivity constraint α ≥ 0 may be broken. If one of the components becomes smaller than zero, then the corresponding EGM ceases to exist. We will use this fact in Theorem 4.

Maximal growth rate is attained in EGMs

We now prove that maximal growth rate is attained in an EGS, and hence in an EGM. Until now we have ignored the upper bound on the ribosome allocation, expressed in

From now on, we will consider this bound, and study the optimisation problem of finding the maximal balanced growth rate for the whole-cell model (15), phrased here as

Since μ is part of the construction of the solution polytope , we formulate a different optimisation problem, in which the metabolite concentrations are fixed, and we do not maximise the growth rate but we minimise the total fraction of the ribosome necessary to attain a given growth rate,

Problem (P2) should be considered for growth rates μ that are relevant for (P1); for instance, for the growth rate μ_max that maximises (P1). Also note that (P2) may be reformulated as

Fig 3 gives some intuition behind the proof of the following theorem.

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Fig 3. Intuition behind the proof of Theorem 4.

As μ is increased, the polytope cuts through the ribosome allocation constraint , with |α| = λ shown in blue. Exactly at the maximal growth rate, the intersection of the polytope with |α| = λ is reduced to one vertex (black dot), i.e., one EGS.

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Theorem 4. For each choice of λ, (P1) is maximised in an Elementary Growth State.

Proof. Let us consider the maximisation problem for an arbitrary fixed set of metabolite concentrations x (inspired by the proofs from [17, 19]). For this fixed x we will prove that the problem is always maximised in an EGS. This is enough to prove the theorem because this means that the growth rate is also maximised in an EGS if we would have picked the optimal metabolite concentrations x.

We only need to consider those x for which (P1) has a nonempty set over which to maximise. The guiding insight for this proof is that, for each fixed value of μ such that the polytope is nonempty, the minimum of (P2) is attained in a vertex of this polytope, i.e., in an EGS with growth rate μ. (This is true for any linear objective function of α).

Let μ_max be the value of the growth rate that maximises (P1), with maximiser α^max, and |α^max| ≕ λ^max ≤ λ. It is clear that . For μ > μ_max, is either empty, or it is not, so we distinguish two cases.

Case 1: If is not empty for some μ > μ_max, then |α| = λ for all ; if not, those vectors would satisfy all the constraints of (P1), so μ_max could not have been maximal. Moreover, vertices of change continuously as a function of μ in a neighbourhood of μ_max (Theorem 3), and since α^max is a convex combination of those vertices, α^max itself also changes continuously. Finally, because |α| is a continuous function of α, the change in |α^max| will also be continuous. Adding this to the observation that |α| > λ for all , μ > μ^max, we can conclude that |α| ≥ λ for all .

Since also |α^max| ≤ λ, we must have |α^max| = λ, and α^max is thus a minimiser of (P2) for μ = μ_max. The minimum of (P2) is attained in a vertex of the polytope, so we can choose α^max such that it is an EGS.

Case 2: If is empty for all μ > μ_max, then all α that solve B(x, μ)α = μu_m+1 for μ > μ_max, must have at least one index j such that α_j < 0. However, we also know that the polytope was nonempty for μ = μ_max, and a nonempty polytope must have at least one vertex. We can thus choose α^max to be such a vertex, and the maximal growth rate is thus indeed maximised in an EGS.

Note that in both cases there could, in principle, be more than one choice for α^max: in Case 1 there could be an entire edge of the polytope that minimises (P2) for μ = μ_max, and in Case 2 the polytope at μ = μ_max could have more than one vertex. This however does not contradict the theorem since the theorem does not state that (P1) is only maximised in an EGS, but rather that it is maximised in some EGS. This is of course a standard situation in Linear Programming theory.

The proof shows that there are two situations in which the growth rate is maximised. Either the constraint |α| ≤ λ is hit, or one of the α_j = 0 and becomes negative at larger growth rates, causing the network to lose an enzymatic reaction and thereby ‘disintegrate’ (it cannot support balanced growth anymore).

We noted that the maximiser of (P1) does not have to be unique. It is however very unlikely to be non-unique if enough information about the enzyme saturation functions, f_j(x), and the volumetric parameters, ρ_k, σ_j, is provided. Having a non-unique maximiser means that there are at least two EGMs, α₁, α₂ that have exactly the same maximal growth rate, thus these EGMs either both have |α₁| = |α₂| = λ or both become infeasible, at μ = μ_max. However, the two EGMs must by definition have a different support, such that there is at least one reaction j that is only used in the first EGM, and a reaction j′ that is only used in the second. The Balanced Growth equations for an EGM, and thus the maximal growth rate for that EGM, depend on the parameters of all participating reactions. The different parameters (e.g., different stoichiometry, volumetric constants, and enzymatic parameters) for the reactions j and j′, will thus influence the maximal growth rate, making it very unlikely that the maximal growth rates for the two EGMs are exactly the same.

We also remark that (P1) is a problem for all metabolite concentrations and all ribosome allocations, whereas (P2) is a problem only for the ribosome allocation. We can also consider an optimisation problem for the metabolite concentrations alone: if we pick an EGM we can maximise the growth rate in that EGM by varying x. In this stage we do not know whether this optimum is unique, or what the convexity properties are of this problem.

The proof of Theorem 4 also shows that increasing λ must coincide with increased maximal growth rate.

Corollary. Consider (P1) for two values of λ, 0 < λ₁ ≤ λ₂, and denote the maximal growth rate attained with λ = λ_i, i = 1, 2. Then .

The converse of this corollary does not necessarily hold: if μ₁ ≤ μ₂ and for i = 1, 2, it does not follow that |α₁| ≤ |α₂|.

The relation between EFMs and EGMs

Elementary Flux Modes (EFMs) are the fundamental building blocks of genome-scale stoichiometric models [6]. EFMs are minimal in the sense that all the reactions in an EFM are essential for a steady state flux. The EFMs can be found as the extreme rays of the flux cone (see Fig 4), if reversible reactions are split in two separate irreversible reactions. The set of all feasible flux vectors, given the reaction stoichiometry, can then be described as the set of all convex combinations of EFMs.

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Fig 4. Comparison between EFMs and EGSs/EGMs for a simple metabolic network (a) with one metabolite M, and three reactions, and the corresponding self-replicating network (b) with two enzymes E₁ and E₂ and the ribosome R.

In case (a) the cone of possible steady state flux solutions is a plane through the origin, bounded by v₁ ≥ 0 and v₂ ≥ 0. Restricting to solutions with those with a prescribed biomass flux v_BM (orange horizontal plane) gives a line segment bordered by two EFM vertices. In EFM₁, reaction v₁ participates; in EFM₂, reaction v₂. In (b) for given metabolite concentration x and growth rate μ₁, the balanced growth solutions for the corresponding self-replicating network also form a line segment of a cone, again spanned by two vertices, the EGSs at these fixed values of x and μ. Only this line segment contains balanced growth solutions, the rest of the cone does not. In EGS₁ only enzyme 1 and the ribosome are expressed, while in EGS₂, enzyme 2 and the ribosome are expressed.

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To relate EFM theory to EGM theory, we start with stoichiometric matrices P and M of the whole-cell model (15). Choosing an EGS requires first choosing metabolite concentrations x and a growth rate μ, and then choosing a feasible basis D which gives rise to a square invertible matrix B_D, corresponding to a vertex of the polytope . We may therefore also restrict the stoichiometry matrices to P_D and M_D, accordingly.

In genome-scale metabolic networks that only model the metabolic part and not the enzymatic and ribosomal part of self-fabrication, a virtual biomass reaction is added. This reaction consumes all components necessary for cell synthesis in (approximately) the right proportions. The growth rate is assumed to be proportional to the rate of the biomass reaction. In the following theorem we find, for each EGS, a suitable biomass reaction, such that the metabolic part of the network (P_D) appended with this virtual reaction has one EFM. The flux values of this EFM approximate the flux values of the EGS. The correction is of order μ, which is usually several orders of magnitude smaller than the flux values.

Theorem 6. Let α be an EGS for metabolite concentrations x and growth rate μ, with feasible basis D. Let be the corresponding flux vector and the corresponding enzyme/ribosome synthesis flux vector. Let ϕ = M w, where M is the enzyme synthesis stoichiometry matrix, and assume that [P_D − ϕ] has full rank. Then we may write (35) where is the unique appropriately scaled EFM of the stoichiometric matrix [P_D − ϕ].

Proof. Let P = P_D. At balanced growth, the flux vectors v and w satisfy where

These vectors satisfy the steady state equations , i.e.,

Setting ϕ = M w and P_ϕ = [P − ϕ], we can see that v is given by the linear set of equations, (36)

We analyse this problem for fixed x and ϕ, which must have solutions, since v is already one such solution. We call a solution to P_ϕ u = μx. Since we have assumed that P_ϕ has full rank, we know that there must be a one-dimensional solution space. We further exploit that P_ϕ has full rank by using row-reduction to write . Row-reduction does not change the solution space, so, if we let u_n denote the first n entries of u, solves if and only if it also solves . The latter is solved by , so the first must be too, such that this gives us a particular solution. The one-dimensional solution space is then found by adding a scalar multiple of a vector, u*, from the nullspace of P_ϕ. So,

The vector u* is the EFM of the matrix P_ϕ. Let t* be chosen such that . (This is always possible, since we already have a solution v). Setting , we deduce

The theorem states that [P_D − M w] should have maximal rank. This requires at least that P_D ≥ m − 1. This does not follow directly from the fact that B_D(x, μ) has full rank; one can easily make a counterexample. We do expect that P_D has full rank in many cases.

The approximating EFM is constant if the precursor consumption rates change proportionally.

The relevant EFM for an EGS is thus the vector v spanning the null space of the matrix P_ϕ, excluding the last element V_n+1. The total precursor consumption rate, ϕ = M w, is part of the EGS solution and is therefore not known a priori. In this section, however, we assume that the relative precursor consumption rates are constant, making ϕ a fixed vector, up to a proportionality constant. We show that under this assumption, the EGS is approximated by the same EFM for all metabolite concentrations and growth rates.

Theorem 7. Let α⁰ be an EGS for metabolite concentrations x⁰ and growth rate μ⁰, with feasible basis D. Let be the corresponding flux vector and the corresponding enzyme synthesis flux vector. Let ϕ⁰ = M w, where M is the enzyme synthesis stoichiometry matrix, and assume that [P_D − ϕ⁰] has full rank. Let V_n be the EFM that approximates the flux values of α⁰, according to Theorem 6. Moreover, assume that ϕ(x, μ) = h(x, μ)ϕ⁰ with h some scalar function. Then, for all μ and x such that D is a feasible basis, the flux values v(x, μ) are approximated by the same EFM: (37)

Proof. Following the proof of Theorem 6 we know that the flux values of an EGS are approximated by the first n coordinates of a vector that is in the nullspace of P_ϕ = [P −ϕ]. Since , we can set V_{ϕ(x, μ)} ≔ (h(x, μ)V₁, …, h(x, μ)V_n, V_n+1)^T. We have

Note that the first n entries of V are all multiplied by the same scalar. So, the ratios of all metabolic reaction rates (not including the virtual biomass reaction) are constant.

In Fig 5 we provide evidence that the amino acid composition of cells growing in different media are practically equal. With the above theorem, this indicates that flux values of EFMs likely closely resemble those of EGSs, provided that these EFMs were calculated for an accurately estimated biomass reaction.

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Fig 5. Side-by-side comparisons of relative amino acid frequencies in growing cells, in minimal and rich media.

Bar plots give relative amino acid frequencies in minimal (left column) and rich medium (middle column). Right column illustrates the cumulative frequency distributions used in the Kolmogorov-Smirnov (KS) test. KS statistic values (which measure the maximal difference between these two cumulative distributions) for the three studies are 3.5 ⋅ 10⁻⁴, 3.6 ⋅ 10⁻³ and 9.3 ⋅ 10⁻³, indicating that these distributions can hardly be distinguished (as is evident from the plots). Data (available at proteomaps.net) from [45] (top row; E. coli) [46] (middle row; S. cerevisae) and [47] (bottom row; E. coli). See [48] for another experimental example from Lactobacillus plantarum.

https://doi.org/10.1371/journal.pcbi.1007559.g005

A biochemical theory of balanced, unicellular self-fabrication

Self-fabrication, self-repair and phenotypic adaption to new environments are defining characteristics of autonomously living organisms. Understanding them in terms of the underlying biochemistry is a key challenge in cell biology. In this work, we focused on the average cell in a population that is growing balanced (arguably the only well-defined state in microbiology [35]) and in a static environment. We defined the phenotype of such an average cell as the complete set of concentrations of all biochemical components, and all rates of chemical reactions. An expression for cellular growth was derived in terms of the production rates of cellular components. We then asked which phenotypes could give rise to steady state balanced growth.

To get a mechanistic understanding of how these different phenotypes can be sustained by a cell, we have derived a theory in which the quantities that are directly controlled by gene expression—the allocation fractions of the ribosome—are the free variables (inspired by [3]). We ignored the precise mechanism how gene expression gives rise to these allocation fractions, but this does not influence the main findings of our theory. In currently used modelling approaches, the free variables are often only indirectly controllable, e.g., the reaction rates or the enzyme concentrations. By taking this new perspective, we can get a full description of the problem that needs to be solved by cells by means of gene expression, instead of only getting a description of the possible solutions of this problem.

EGMs: The minimal phenotypes that lead to balanced growth

The key modelling assumption that turns the inaccessible balanced growth system (9) for general chemical systems into one with sufficient structure (22)–(25) for biological systems is the fact that enzymes and ribosomes catalyse reactions in a linear fashion. Doubling the concentration of these catalysts leads to a doubling in reaction rates. This introduces just enough linearity into the balanced growth equations to make linear programming techniques accessible.

To describe all balanced growth phenotypes, we identified EGMs: the minimal modes of gene expression that lead to balanced growth; all possible phenotypes are convex combinations of these minimal modes. The EGMs can thus be seen as the regulatory degrees of freedom of the cell. In contrast, one could suggest to view the expression of single enzymes as the most important degrees of freedom, but an enzyme alone can never lead to a self-fabricating system. Even an entire pathway that carries a steady-state flux does not lead to a self-fabricating system, unless it synthesises all cellular building blocks that are needed for the enzymes catalysing this pathway. Regulation should therefore not involve making numerous decisions about independent enzyme expression levels: the decision should rather be how and when to express one or a few EGMs.

In one specific environment, we can calculate all minimal growth-supporting enzyme expression states, which we called Elementary Growth States. However, when the environment changes the EGSs change with it, i.e., the specific allocation fractions of the ribosome change: changing nutrient levels cause changes in enzyme saturation and reaction rates, leading to new enzyme concentrations and a new growth rate. Since we wanted to find the regulatory degrees of freedom for the cell, the EGSs were not very useful: biological environments are noisy which means that these degrees of freedom would fluctuate constantly. That is why we defined the Elementary Growth Modes as equivalence classes of Elementary Growth States: two EGSs with the same set of expressed enzymes belong to the same EGM. Under small fluctuations in the environment, the EGSs change, but the set of EGMs is constant. The set of growth-supporting EGMs can change due to a larger change in the environment, e.g., due to the depletion or appearance of nutrients.

The EGMs are analogous to EFMs, objects defined in models where no explicit synthesis of enzymes and the ribosome is considered. The EFMs are defined as the minimal (i.e., non-decomposable) combinations of reaction rates that support a steady-state flux [9]. So, where the EGMs are the minimal building blocks of balanced growth, the EFMs are the building blocks of metabolism. While EGMs are defined as equivalence classes of EGSs, the EFMs are defined up to a normalization factor, and are therefore also equivalence classes of steady-state flux vectors. Moreover, we showed that, as long as dependent metabolites (for example due to so-called moieties) are removed, an EGM will have exactly as many active metabolic reactions as non-zero metabolites; the ribosome-catalyzed synthesis adds another reaction (Theorem 2). EFMs, also after removing dependent metabolites, have one active reaction more than the number of used metabolites. The additional reaction in the EFM is the (virtual) biomass reaction, thus playing a similar role as the ribosome-catalyzed reaction.

Growth rate is maximised by using a small number of active EGMs

Evolutionary theory suggests that in stable environments, those microorganisms are selected that express the phenotype that maximises the growth rate. One of our aims was therefore to describe, in any given condition, the phenotype with the maximal growth rate.

We proved in Theorem 4 that the highest possible growth rate in any fixed environment is attained in an EGM. This makes intuitive sense: by reducing the number of active reactions, the redundancy in the reaction network is reduced. When fewer different proteins are used, the remaining ones can be expressed to a higher level. To maximise the growth rate, the cell should thus express only the pathways that are most efficient in terms of resources. This ultimately leads to cellular states in which no redundant reactions are active, i.e., with only one active EGM, such that none of the reactions can be removed without stopping growth. Again, EGMs prove to be analogous to EFMs, since it was proven that the metabolic flux through a proteome-constrained metabolic network is maximised in an EFM [17, 18].

Because cells and their compartments have a limited size, and because each molecule takes up a certain volume, molecule concentrations in a cell are bounded. As a consequence, biochemical constraints arise for weighted sums of molecule concentrations, in particular on enzyme concentrations since most cellular biomass consist of protein. These constraints might change the optimal solution. For example, an EGM that is very efficient in terms of cytosolic volume might be very inefficient with respect to the limited membrane area. The first of these constraints was already incorporated when we assumed a constant average density (Eq (4)), but additional constraints may be active and can be imposed on the model. We proved (Theorem 5) that a small number of EGMs still constitute the optimal solution when additional constraints are added. However, maximally one EGM might be added for each constraint that is imposed. Given that the constant average density constraint was already active, the number of active EGMs is thus bounded by the number of constraints on enzyme expression. We hereby generalized yet another existing result [19] for EFMs to their self-fabricating counterparts.

Evolution of metabolism proceeds via mutations, either affecting the kinetics of enzymes, to increase the activity per unit enzyme, or affecting the regulation of protein expression. The first type of evolution will change the properties of the EGMs, while the second type will change the number or identity of EGMs that is expressed. The above theorems involve the second type of evolution; they predict that—no matter the environment, nor the specific properties of the growth-supporting EGMs—only a small number of EGMs should eventually be selected. How close microorganisms currently are to this optimal state is unclear, and awaiting experimental investigation, although indirect evidence is mounting that cells are very good indeed [19, 36, 37].

EGMs are the basic building blocks of cellular growth models

In order to fully describe the phenotypes that can sustain balanced growth, we needed to incorporate 1) a direct coupling between reaction rates and growth rate, 2) explicit synthesis of enzymes and the ribosome, and 3) nonlinear and metabolite-dependent enzyme kinetics. All three aspects added nonlinearity to our theory; the first makes it impossible to prescribe the growth rate and solve for all remaining variables in steady state, the second makes the solutions depend nonlinearly on the growth rate, and the third introduces a nonlinear dependence on metabolite concentrations. In the resulting theory, all phenotypes and their corresponding growth rates can in principle be calculated, but these computations are currently not feasible for a genome-scale model. We thus developed a quantitative framework that led to general theoretical results, but no quantitative predictions can be made for a specific organism at present.

Fortunately, our theory can be simplified in various ways, resulting in computationally feasible models (see [24] for a more extensive review of different modelling approaches). Elementary Growth Modes are still the minimal building blocks, and the above-mentioned maximiser theorems still hold. A commonly used simplification is to consider a small coarse-grained whole-cell model [3, 11, 13, 24]. These models become computationally feasible because of their small size, such that no further simplifications have to be made; EGM-theory is directly applicable to this type of models. Another popular modelling approach ignores the explicit synthesis of enzymes and the ribosome, replacing it by a constant biomass reaction as a sink for cell components. These approaches either consider medium-scale models with nonlinear enzyme kinetics [38, 39], or genome-scale models without metabolite concentrations and only a maximal catalytic rate (k_cat) as enzyme kinetics [40], or genome-scale models without enzyme kinetics (FBA [5]). In all these approaches, the solutions can be written as convex combinations of EFMs, which can be seen as the linear approximations of EGMs (Theorem 6). The recently developed Metabolism and Expression (ME) models currently make the least simplifying assumptions in modelling cellular growth [22, 41, 42]: enzyme and ribosome synthesis are explicitly modelled, but enzyme kinetics are replaced by a single catalytic rate. The minimal building blocks of these models are thus EGMs where the saturation function is replaced by a constant and the dilution of metabolites is ignored. Very recently, theory was also developed for kinetic models with explicit synthesis of one type of protein [43]. This approach replaces the constant density assumption by an upper bound on this density, and assumes that the ribosome (i.e., the one protein) is always fully occupied. This enabled the authors to further analyse the optimal growth state, which was indeed an EGM.

A constant amino acid composition greatly simplifies self-fabrication

Experimental data suggests that the amino acid composition of cells is constant across different growth conditions (Fig 5). This can only be maintained when amino acids are consumed at rates with constant ratios. Under that assumption, EGMs simplify considerably, and become much more “linear” objects: the flux values of the EGM can be approximated by an EFM. Under the additional assumption that the dilution of metabolites is negligible, and within one EGM at fixed metabolite concentrations, the fractions of the ribosome allocated to producing enzymes, the α_j’s, all have the same dependence on the growth rate (Theorem 8). In other words, the relative allocation of these enzymatic α_j’s remains equal. The ribosomal fraction allocated to producing ribosomes, α_n+1, does scale differently: this fraction changes linearly with the growth rate. The result of this simplification is therefore that to control enzyme/ribosome synthesis the cell should only tune two different variables: the overall fraction of the ribosome allocated to making enzymes, and the fraction of the ribosome allocated to ribosome production. The ppGpp-mechanism in E. coli, which acts on amino acid abundances and controls the balance between metabolism and ribosome synthesis, fits perfectly with this structure [33, 34].

Stresses and non-enzymatic reactions can be considered in terms of EGMs

Although the theory is presented for whole-cell models that are geared solely towards growth and self-fabrication, it does not preclude non-enzymatic (i.e., diffusive) reactions, stress responses, and maintenance or homeostasis activities. Processes that are not directly related to growth, such as heat shock responses or the removal of toxins, may all be incorporated. As long as the proteins involved in those processes act linearly on the reaction rates, growth rate maximisers will again be EGMs of the corresponding model. In Fig 6 (code in S1 Text) we provide an example of a situation where it is optimal to invest in a stress response, in this case the removal of a toxin. The effect of the toxin was modelled by adding an inhibitory effect of the toxin on all metabolic enzymes.

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Fig 6. Illustration of investment in toxin removal while maximising growth rate.

(A) The example network contains five metabolites; x₁ to x₄ are precursor molecules and x₃ and x₄ play the role of amino acids. An external toxin can diffuse over the membrane and, when inside the cell, inhibits the catalytic activity of all proteins. A fifth protein, in purple, destroys the toxin. (B) Without toxin present, investment is divided over the four proteins and the ribosome to maximise steady state growth rate. With toxin present, the toxin-degrading protein is also expressed, even though it does not directly contribute to growth. However, without expressing this stress protein, the maximal growth rate would have been lower. (C) Investment in proteins and ribosomes to attain maximal balanced growth rate as a function of toxin concentration. Note that in this example, a switch is observed, from an EGM with four metabolites to one with five. Full detail and code may be found in S1 Text.

https://doi.org/10.1371/journal.pcbi.1007559.g006

Open problems and outlook

Clearly, many open theoretical problems remain. For example, we have described the (optimal) phenotypes that lead to steady-state self-fabrication in static environments, but we do not know if the same phenotypes are evolutionary favourable in dynamic environments. Moreover, even if we know that the optimum is attained in one, or a combination of a few, EGMs, we do not know how a cell should find these optimal protein expression states with its regulatory circuitry. Further, it is unclear to us if more existing results for EFMs can be generalised to EGM theory. It is known, for example, that finding the metabolite concentrations that maximise the specific flux in an EFM is a strictly convex optimisation problem [37, 44], but we do not know if an analogous result for EGMs can be proved.

We believe that a sound fundamental theory of microbial growth could help to better understand the common denominators underlying qualitatively similar physiological behaviours of evolutionary distinct microorganisms. Systems biology is, however, remarkably short of experimentally testable theories, in contrast to other systems sciences, such as statistical physics and population genetics. This is somewhat surprising since it is apparent, even from the history of microbiology itself, that abstract theory can aid in the understanding of concrete phenomena. The understanding of single-cell physiology has profited greatly, for instance, from theories on stochastic fluctuations in molecular circuits, and the introduction of enzyme kinetics theories revolutionised enzyme biochemistry.

The EGM theory presented here should allow at least to improve our understanding of the biomass composition of cells, the relation between growth rate and reaction rates, better prediction of proteome constraints that limit growth rate, and prediction of transcription and hence also enzyme levels.

We hope that this paper contributes to a growing body of ‘biomathematical’ theories that eventually provide basic answers to the molecular basis of life—firstly of microorganisms. We are convinced that such a theory is within reach. The next frontier we foresee is understanding metabolic behaviour of microorganisms in terms of growth rate maximisation and constrained protein expression—a general biomathematical theory that merges enzyme biochemistry, metabolic network reconstructions and evolutionary theory.