Optimal metabolic strategies for microbial growth in stationary random environments Anna Paola Muntoni and Andrea De Martino

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Optimal metabolic strategies for microbial growth

in stationary random environments

Anna Paola Muntoni and Andrea De Martino

Politecnico di Torino, Turin, Italy, and Italian Institute for Genomic Medicine, Turin,

Italy

E-mail: andrea.demartino@polito.it

Abstract. In order to grow in any given environment, bacteria need to collect

information about the medium composition and implement suitable growth strategies

by adjusting their regulatory and metabolic degrees of freedom. In the standard

sense, optimal strategy selection is achieved when bacteria grow at the fastest rate

possible in that medium. While this view of optimality is well suited for cells that

have perfect knowledge about their surroundings (e.g. nutrient levels), things are

more involved in uncertain or ﬂuctuating conditions, especially when changes occur

over timescales comparable to (or faster than) those required to organize a response.

Information theory however provides recipes for how cells can choose the optimal

growth strategy under uncertainty about the stress levels they will face. Here we

analyse the theoretically optimal scenarios for a coarse-grained, experiment-inspired

model of bacterial metabolism for growth in a medium described by the (static)

probability density of a single variable (the ‘stress level’). We show that heterogeneity

in growth rates consistently emerges as the optimal response when the environment is

suﬃciently complex and/or when perfect adjustment of metabolic degrees of freedom

is not possible (e.g. due to limited resources). In addition, outcomes close to those

achievable with unlimited resources are often attained eﬀectively with a modest amount

of ﬁne tuning. In other terms, heterogeneous population structures in complex media

may be rather robust with respect to the resources available to probe the environment

and adjust reaction rates.

1. Introduction

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 [4–8], together with the fact

arXiv:2210.11167v2 [physics.bio-ph] 21 Mar 2023

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

[17–19], albeit never speciﬁcally in the context of metabolism.

In this paper, inspired by the above studies as well as by [20] and [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 parametrised 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. Results

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 , 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 qand , the growth rate µis well described by

the formula [20]

µ'φ

w+sq +,(1)

where s≥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,

φ'0.48 and w'0.169 h [9].) For our purposes, scan be assumed to be inversely

proportional to the carbon level as argued in [22] (so s1 and s1 for carbon-rich

and carbon-poor environments, respectively).

Let us assume that the stress level sin (1) is a homogeneous parameter whose

value is controlled externally. If µwere to be maximized, the quantity sq +would

have to be minimized. In carbon-limited E. coli growth, however, qand are subject

to a trade-oﬀ such that high qimplies low 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 qand between two extreme strategies, denoted by indices F

and R, respectively, distinguished by the fact that qF> qR(i.e. the speciﬁc nutrient

requirements of Fare higher than those of R) and R> F(i.e. the speciﬁc expenditure

for Ris 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 qF'8 glac/gDW and F'0.3 h, respectively;

the corresponding quantities under respiration are instead given by qR'5 glac/gDW and

R'0.6 h [20]. In this work, we choose the representative values qF= 10, F= 0.1,

qR= 1, R= 1, omitting the units as the speciﬁcs of the carbon source are immaterial

for us. Nevertheless, with these choices the growth rate µcan be interpreted to be

measured in 1/h.) We then describe the trade-oﬀ between qand by assuming that

both depend on a single variable xwhose value ranges between 0 and 1, such that

q(x) = qR+ (qF−qR)(1 −x)ν,(2)

(x) = F+ (R−F)xν,(3)

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

µ(x, s) = φ

w+sq(x) + (x).(4)

By taking the derivative of µ(x, s) over xat ﬁxed s, one ﬁnds that, for any given q(x)

and (x), µis maximum when

s∂q

∂x +∂

∂x = 0 .(5)

If q(x) and (x) are given by (2) and (3), the maximum is achieved for x=bx(s), with

bx(s) = sn

sn+sn

, sc=R−F

qF−qR

, n =1

ν−1.(6)

This means that media with ssccan be considered to be rich, while media with

sscare eﬀectively poor. (With our choices for the parameters, sc= 0.1.) In turn,

if µis maximized, strategy F(i.e. x= 0) will be used in rich environments (where 

should be as small as possible) while strategy R(i.e. x= 1) will be used in poor ones

(where qshould 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 Fand R). Other choices generically lead to slower

growth.

Notice that the q−trade-oﬀ gets stronger as νapproaches 1, when the

corresponding optimal strategy is a step-like function. Conversely, it gets weaker and

weaker as νincreases. For sakes of simplicity, in the following we shall always use the

value ν= 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 νmay be evolutionarily preferred) is deferred to future work.

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OptimalmetabolicstrategiesformicrobialgrowthinstationaryrandomenvironmentsAnnaPaolaMuntoniandAndreaDeMartinoPolitecnicodiTorino,Turin,Italy,andItalianInstituteforGenomicMedicine,Turin,ItalyE-mail:andrea.demartino@polito.itAbstract.Inordertogrowinanygivenenvironment,bacterianeedtocollectinformationab...

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Optimal metabolic strategies for microbial growth in stationary random environments Anna Paola Muntoni and Andrea De Martino

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