Optimal metabolic strategies for microbial growth in stationary random environments

The standard theoretical view of bacterial growth posits that, in any growth medium, cells are capable of adjusting their metabolic degrees of freedom (i.e. the rates of metabolic reactions) within bounds dictated by thermodynamic and regulatory constraints (e.g. enzyme expression levels, reaction free energies, etc) so as to maximize their growth rate [1]. Besides evolutionary considerations, such a picture is supported by the fact that the expression levels of certain metabolic enzymes and of basic macromolecular machines like ribosomes actually appear to be tuned for growth-rate maximization in bacterial populations [2, 3]. On the other hand, the significant cell-to-cell variability in growth rates observed in experiments [48], together with the fact that constraints arising outside metabolism suffice to explain a large batch of empirical facts without assuming any growth-rate optimization [9, 10], appears to call for deeper insight into the notion of 'optimality' for bacterial growth.

Recent work has shown that the relationship between population growth and cell-to-cell variability is well described by a Maximum Entropy (MaxEnt) theory leading to a variable trade-off akin to the usual energy-entropy balance in statistical physics. More specifically, E. coli populations growing in carbon-limited media realize a close-to-optimal fitness-heterogeneity trade-off in rich media [11], while they seem to be less variable or slower-growing than optimal in poorer growth conditions [12]. Metabolic fluxes likewise appear to be better captured by accounting for such a trade-off than by a standard optimality assumption [13]. In each case, the balance between growth and variability is described by a finite (medium-dependent) 'temperature', where a zero-temperature (resp. infinite-temperature) limit corresponds to maximal growth (resp. maximal variability). Save for a few general ideas derived from broad-brush models [14], what determines the 'temperature' (i.e. the fitness-heterogeneity balance) of actual microbial systems is still unclear.

High variability can naturally arise from unavoidable inter-cellular differences in gene expression levels or regulatory programs (e.g. cell cycle) [15]. In models of metabolism, this would lead, at the simplest level, to cell-dependent changes in the constraints under which growth is optimized. In this respect, cell-to-cell heterogeneity might be interpreted as 'optimality plus noise', and the 'temperature' described above would quantify, in essence, the noise strength. Importantly, though, there might be an inherent advantage in maintaining a diverse population, especially in environments that fluctuate (due e.g. to natural variability) or when cells have imperfect information about their growth medium (due e.g. to limits in precision caused by the high costs cells face to maintain and operate a sensing apparatus). These factors are not usually included in standard models of metabolic networks, which therefore cannot address the fitness benefits of heterogeneity.

The problem of growth maximization clearly becomes more subtle under uncertainty about environmental conditions, as the straightforward optimization that can be carried out in a perfectly known medium is no longer an option. Recipes for selecting the optimal growth strategy in uncertain environments are, however, provided by information theory. Theoretical work aimed at understanding how efficiently populations can harvest, process and exploit information about variable or unpredictable media has indeed shown that bet-hedging (i.e. maintaining a fraction of slow-growing cells even in rich media or sustaining a lower short-term growth to ensure faster long-term growth) can yield significant fitness gains in a wide variety of situations [16]. Biological implications of these results have been explored against several backdrops [1719], albeit never specifically in the context of metabolism.

In this paper, inspired by the above studies as well as by [20, 21] (chapter 6), we look at a minimal, experiment-derived mathematical model to characterize optimal metabolic strategies for growth in uncertain environments, focusing for simplicity on static environments defined by the probability density of a single variable (the 'stress level'). Optimal strategies are parametrized by a 'temperature' that modulates the amount of information they encode about the growth medium or, loosely speaking, how precisely cells can match their phenotype to the external conditions in order to foster growth. We will show explicitly that, when the medium is sufficiently complex, optimal populations acquire a non-trivial phenotypic organization even when the metabolic strategy encodes the maximum possible amount of information about the external conditions. A broad spectrum of behaviours is uncovered upon varying the structure of the environment. Remarkably, however, the emerging scenarios are often robust to changes in the 'temperature'. This suggests that metabolic networks may yield outcomes close to globally optimal ones (at least in an information-theoretic sense) even when resources to probe the environment and adjust metabolic reactions are limited.

2.1. Model of metabolism and growth

We consider a coarse-grained model of microbial growth metabolism in which each cell's metabolic strategy is described by just two quantities, namely the specific uptake (or inverse growth yield) q, quantifying the nutrient intake required to grow per unit of growth rate, and the biosynthetic expenditure ε, quantifying the proteome mass fraction to be devoted to metabolic enzymes per unit of growth rate. (In more detailed models of metabolic networks, the former quantity relates to the rate at which the limiting nutrient is imported, while the latter is proportional to a weighted sum of the absolute values of the fluxes through metabolic reactions [20, 22].) For E. coli growing in carbon-limited media it has been argued that, for given q and ε, the growth rate µ is well described by the formula [20]

Equation (1)

where $s\geqslant 0$ represents the level of nutritional stress to which the organism is subject, while φ > 0 and w > 0 are constants representing respectively the fraction of proteome devoted to constitutively expressed proteins and the proteome share to be allocated to ribosome-affiliated proteins per unit of growth rate. (For glucose-limited E. coli growth, $\phi\simeq 0.48$ and $w\simeq 0.169$ h [9].) For our purposes, s can be assumed to be inversely proportional to the carbon level as argued in [22] (so $s\ll 1$ and $s\gg 1$ for carbon-rich and carbon-poor environments, respectively).

Let us assume that the stress level s in (1) is a homogeneous parameter whose value is controlled externally. If µ were to be maximized, the quantity $sq+\epsilon$ would have to be minimized. In carbon-limited E. coli growth, however, q and ε are subject to a trade-off such that high q implies low ε and vice versa [20]. Metabolic states with minimal biosynthetic expenditure are hence favoured in rich environments (small s), while states of minimal nutritional requirements prevail in poor media (large s). To make a concrete model, we draw inspiration from the fermentation/respiration duality that again characterizes E. coli growth in carbon-limited media and assume that the organism can regulate q and ε between two extreme strategies, denoted by indices F and R, respectively, distinguished by the fact that $q_F\gt q_R$ (i.e. the specific nutrient requirements of F are higher than those of R) and $\epsilon_R\gt\epsilon_F$ (i.e. the specific expenditure for R is higher than for F). (Estimated values of the specific nutrient intake and the specific proteome mass fraction devoted to metabolic enzymes required for E.coli to grow on lactate under fermentation are $q_F\simeq 8$ g$_}/$g$_}$ and $\epsilon_F\simeq 0.3$ h, respectively; the corresponding quantities under respiration are instead given by $q_R\simeq 5$ g$_}/$g$_}$ and $\epsilon_R\simeq 0.6$ h [20]. In this work, we choose the representative values $q_F = 10$, $\epsilon_F = 0.1$, $q_R = 1$, $\epsilon_R = 1$, omitting the units as the specifics of the carbon source are immaterial for us. Nevertheless, with these choices the growth rate µ can be interpreted to be measured in 1/h.) We then describe the trade-off between q and ε by assuming that both depend on a single variable x whose value ranges between 0 and 1, such that

Equation (2)Equation (3)

where ν > 1 is a constant. The growth rate of the organism will then be given by

Equation (4)

By taking the derivative of $\mu(x,s)$ over x at fixed s, one finds that, for any given q(x) and $\epsilon(x)$, µ is maximum when

Equation (5)

If q(x) and $\epsilon(x)$ are given by (2) and (3), the maximum is achieved for $x = \widehat(s)$, with

Equation (6)

This means that media with $s\ll s_c$ can be considered to be rich, while media with $s\gg s_c$ are effectively poor. (With our choices for the parameters, $s_c = 0.1$.) In turn, if µ is maximized, strategy F (i.e. x = 0) will be used in rich environments (where ε should be as small as possible) while strategy R (i.e. x = 1) will be used in poor ones (where q should be as small as possible), as shown in figure 1. As one modulates the stress level between these two extremes, growth is maximized by intermediate values of x (i.e. by strategies that use both F and R). Other choices generically lead to slower growth.

Figure 1. Representative scenarios for $q\left(x\right)$ vs $\epsilon\left(x\right)$ (equations (2) and (3), left plots in each panel) with the corresponding fitness landscape $\mu(x,s)$ (right plots), for ν = 1.01 (panel (a)), 2 (b), 4 (c) and 8 (d). The values of x that maximize µ for each s, i.e. $\widehat(s)$ (equation (6)), are represented by a magenta line. The constants characterizing the R and F strategies are set to $q_F = 10$, $q_R = 1$, $\epsilon_F = 0.1$ and $\epsilon_R = 1$.

Standard image High-resolution image

Notice that the $q-\epsilon$ trade-off gets stronger as ν approaches 1, when the corresponding optimal strategy is a step-like function. Conversely, it gets weaker and weaker as ν increases. For sakes of simplicity, in the following we shall always use the value $\nu = 3/2$, which qualitatively reproduces the trade-off reconstructed from empirical data [20]. A discussion of how results depend on ν (including the issue of why a specific value of ν may be evolutionarily preferred) is deferred to future work.

2.2. Optimizing growth in random environments: theoretical framework

Consider a microbe whose growth rate depends on x and s as in (4). For any fixed s, the cell can maximize µ by setting $x = \widehat(s)$ (see (6)). Suppose, however, that s is a random variable with a prescribed probability density p(s). In this case, cells face an uncertainty about the exact value of s they will encounter, although they have knowledge of the ensemble of environmental conditions in which they live (i.e. of p(s)). What is the optimal choice for x in this situation? The most sensible measure of performance is now arguably given by the mean growth rate

Equation (7)

whose value depends on the conditional distribution $p(x|s)$ that describes the stochastic rule used by the cell to select x for any s. The question of optimality therefore concerns the optimal choice of $p(x|s)$. It turns out that this choice depends on the amount of information about s that is encoded in x. In particular, if

Equation (8)

denotes the mutual information of x and s (in bits), and

Equation (9)

then the optimal $p(x|s)$ is given by the solution of

Equation (10)

i.e. (see appendix and [21])

Equation (11)

where

Equation (12)

while the 'inverse temperature' β is a Lagrange multiplier and $p^\star(x)$ denotes the probability density of x corresponding to the optimal choice. Equation (11) has a rather straightforward interpretation. When β → 0 ('infinite temperature'), the choice of x becomes independent of s, which implies I = 0. As β increases, I increases (i.e. cellular responses encode more and more information about the environment) and $p^\star(x|s)$ tends to get more and more sharply peaked around $\widehat(s)$. For $\beta\to\infty$ ('zero temperature'), in particular, one gets $p^\star(x|s)\simeq\delta[x-\widehat(s)]$, so that cells respond to each instance of s by exactly choosing the value of x that maximizes µ. Such an 'infinite-precision' response requires I to be maximal. In other terms, the higher I, the higher $\left\langle \mu \right\rangle$. (Conversely, as β becomes more and more negative, $p^\star(x|s)$ tends to get more and more sharply peaked around the value of x that minimizes growth for any s. For sake of clarity, we shall however limit the following analysis to the case β > 0.) β is therefore a parameter by which one interpolates between the case of optimal response to any environmental cue and that in which the cell's metabolic strategy is completely insensitive to s. Note that, at optimality, the mean growth rate and mutual information given by

Equation (13)Equation (14)

are both functions of β and are related by

Equation (15)

For different choices of p(s) and β one can solve equations (9), (11) and (12) numerically by iteration from an initial guess. Starting (iteration n = 0) from uniform guesses for $p\left(x\right)$ and $p\left(x | s\right)$, we iterated equations (12), (9) and (11) up to numerical convergence, which we assumed to be achieved when

Equation (16)Equation (17)

where the index n denotes the iteration step while σ a numerical precision threshold. (A Matlab code implementing the above procedure is available from https://github.com/anna-pa-m/OptMetStrategy.) Solutions will clarify how optimal metabolic strategy and population structure change with β, i.e. with the amount of information about the environment encoded in x, in any given environmental condition (described by the chosen p(s)).

Before moving on, we note that the above setting suggests how an 'optimal' value of β may arise in this scenario. It is indeed reasonable to think that $I^$ is directly related to the quantity of cellular resources devoted to probing the environment and tuning metabolic reactions, and, in turn, that higher costs for sensing and metabolism may negatively affect the fitness if resources are limited. If one assumes for simplicity that such a cost reduces the growth performance by a constant amount c per bit of information encoded, the fitness faced by the organism can be written as

Equation (18)

One easily sees that, contrary to $\left\langle \mu \right\rangle_\star$ and $I^$ (both of which increase steadily with β), $$ is maximum when $c = \partial\left\langle \mu \right\rangle_\star/\partial I^$, i.e. for

Equation (19)

In other terms, the costs associated with sensing and adjusting metabolism can lead to the existence of an optimal value of β, i.e. of an optimal level of trade-off between $\left\langle \mu \right\rangle$ and I, such that (expectedly) higher values of $$ are possible only if the cost of encoding information about the environment in metabolic strategies gets smaller. (A similar idea was discussed in a different context in [13].)

2.3. Tightly controlled stress levels

We start from the simple case where the stress level s is tightly regulated, as in a lab setting where the nutrient level is externally controlled with high precision. Specifically, we assume that p(s) is uniform and centered at a value s0, and that s can only vary by 5% with equal probability around s0 (figure 2(a), top panel). The ensuing landscape of $\mu(x,s)$ is shown in figure 2(a), bottom panel. One sees that, with our choice of s0 ($s_0\gt s_c$, see (6)), growth is maximized for $x\simeq 1$ (R strategy) for all values of s (magenta line in figure 2(a)). Figure 2(b) reports the optimal strategy $p^\star(x|s)$ computed numerically for five different values of β. As expected, the distribution concentrates around the growth-maximizing value of x for large enough β, while it becomes more and more uniform over the $[0,1]$ interval (with the mean x, green dashed line, getting closer and closer to

留言 (0)

沒有登入
gif