Noisy metabolism can promote microbial crossfeeding
Abstract
Crossfeeding, the exchange of nutrients between organisms, is ubiquitous in microbial communities. Despite its importance in natural and engineered microbial systems, our understanding of how interspecies crossfeeding arises is incomplete, with existing theories limited to specific scenarios. Here, we introduce a novel theory for the emergence of such crossfeeding, which we term noiseaveraging cooperation (NAC). NAC is based on the idea that, due to their small size, bacteria are prone to noisy regulation of metabolism which limits their growth rate. To compensate, related bacteria can share metabolites with each other to ‘average out’ noise and improve their collective growth. According to the Black Queen Hypothesis, this metabolite sharing among kin, a form of ‘leakage’, then allows for the evolution of metabolic interdependencies among species including de novo speciation via gene deletions. We first characterize NAC in a simple ecological model of cell metabolism, showing that metabolite leakage can in principle substantially increase growth rate in a community context. Next, we develop a generalized framework for estimating the potential benefits of NAC among real bacteria. Using singlecell protein abundance data, we predict that bacteria suffer from substantial noisedriven growth inefficiencies, and may therefore benefit from NAC. We then discuss potential evolutionary pathways for the emergence of NAC. Finally, we review existing evidence for NAC and outline potential experimental approaches to detect NAC in microbial communities.
Editor's evaluation
In this article, authors propose a novel hypothesis that can help explain why microbes release metabolites. In their NAC (noiseaveraging cooperation) hypothesis, withinpopulation crossfeeding can arise due to noisy metabolism in microbes. The authors predict substantial noisedriven growth inefficiencies from singlecell protein abundance data, review evidence for NAC, and propose how to detect NAC in microbial populations.
https://doi.org/10.7554/eLife.70694.sa0Introduction
Microbial communities are found nearly everywhere in nature, inhabiting ecosystems ranging from hydrothermal vents (Ding et al., 2017) to mammalian guts (LloydPrice et al., 2016). One of the most striking properties of these communities is the ubiquity of cooperation: microbes frequently share resources, relying on other microbes for the acquisition of essential metabolites. This exchange of resources is broadly referred to as ‘crossfeeding’ and can involve a wide variety of compounds, ranging from organic acids to vitamins. Crossfeeding is widespread across both natural and engineered systems, with notable examples occurring in the human gut and in wastewater treatment systems (Smith et al., 2019b; Nelson et al., 2011). These metabolic interactions link organisms across the entire tree of life, occurring both within and between kingdoms (Sokolovskaya et al., 2020) and even between specialized microbes of the same species (Rosenthal et al., 2018).
Crossfeeding plays a major role in the structure and function of microbial communities. In natural settings, crossfeeding is known to be a significant driver of microbial diversity, allowing many species to coexist on a small number of primary resources (Goyal and Maslov, 2018; Goldford et al., 2018). This microbial diversity has been linked to a wide variety of community properties (Ptacnik et al., 2008; van Elsas et al., 2012), including influence on host fitness (Taur et al., 2014). Crossfeeding can even play a role in public health: it has been shown that metabolite exchange can allow pathogens to compensate for fitness losses associated with antibiotic resistance (Hammer et al., 2014). In engineered systems, crossfeeding can be necessary for efficient operation. For example, wastewater treatment reactors rely on crossfeeding to prevent the buildup of inhibitory waste products (Ciotola et al., 2014). Thus, a thorough accounting of the factors promoting crossfeeding is an important part of understanding both natural and engineered microbial communities.
As a result of crossfeeding’s key role in microbial communities, much work has been dedicated to unraveling its origin. The evolution of crossfeeding requires the emergence of two organisms: the organism secreting the metabolite and the organism consuming it. In some cases, the secreted metabolite could be a waste product that is costless or even beneficial to secrete (Pfeiffer and Bonhoeffer, 2004; Pacheco et al., 2019; Kreft et al., 2020). Thus, the secreting organism evolves as a consequence of metabolic optimization, while the resulting availability of the secreted metabolite provides an opportunity for the emergence of a consumer. However, many cases of crossfeeding involve the secretion of metabolites that are essential to both the secreting and consuming organism (Sokolovskaya et al., 2020). Why would an organism leak metabolites necessary for its own growth, and why would the consuming organism lose its capacity to produce an essential metabolite? The most popular theory for the evolution of this form of crossfeeding is the Black Queen Hypothesis (BQH), which focuses on the evolution of the consuming organism. The BQH posits that if there exists an organism that secretes a useful metabolite, there will be a selective advantage for some organisms to lose the ability to produce the metabolite and rely on leakage from the secreter (Morris et al., 2012; Smith et al., 2019a; Morris et al., 2014; Sachs and Hollowell, 2012). An alternative theory, the ‘economies of scale’ hypothesis, posits that crossfeeding is stabilized because organisms somehow gain an intrinstic fitness advantage from specializing their metabolite production and sharing with other such specialized organisms (Pande et al., 2014). Both of these hypotheses in principle provide theoretically sound mechanisms for consumer evolution and crossfeeding stabilization, but leave open the question of how leakage of essential metabolites first emerges. Thus, a complementary theory is needed that can explain the emergence of leakers.
It has been posited that some crossfed metabolites are naturally leaky (Morris et al., 2012), but this assumption is not wellsupported. For a limited set of functions, leakage is clearly unavoidable because key processes take place outside the cell, such as the hydrolysis of large polymers by extracellular enzymes (Gore et al., 2009). However, many crossfeeding relationships involve metabolites that are produced intracellularly, and it is generally not known how these metabolites exit the cell, much less that this leakage is inevitable. Polar or charged metabolites are known to have low membrane permeability, limiting the possibility of natural leakage through the cell membrane (Chakrabarti and Deamer, 1992). Indeed, even if a metabolite has a high membrane permeability, this does not necessarily indicate high absolute leakage rates. Cells could minimize leakage by maintaining only a small metabolite pool with rapid turnover or by storing the metabolite in an altered, less leakageprone form. Even if substantial quantities of a metabolite are observed to escape via a leaky membrane, it cannot be determined without further study whether this leakage is truly inevitable or, rather, advantageous in some way. If leakage isn’t inevitable, how might it emerge?
Here, we explore a novel cooperative behavior that promotes the emergence of leakiness and thus crossfeeding in microbial communities. The basis of this mechanism is that metabolic enzyme regulation has been shown to be noisy, particularly so in bacteria due to their small size (Taniguchi et al., 2010). The resulting imbalances in enzyme levels can lead to growth inefficiencies due to the under or overproduction of necessary metabolites. Rather than attempting to downregulate the activity of the excess enzymes, cells can in principle improve their communal growth rate by exchanging metabolites among their kin and effectively ‘averaging’ out intracellular noise. We term this mechanism noiseaveraging cooperation (NAC). The emergence of NAC among identical individuals creates the conditions for obligate consumers to evolve via gene deletions, as proposed by the BQH. Thus, the proposed mechanism provides a complementary theory in which metabolic leakiness is not assumed a priori, but rather arises from selective pressures. We first characterize NAC in an ecological model of a small population of cells, demonstrating that metabolite leakage can increase collective fitness. We find that in extreme cases, NAC can even prevent the death of cells whose poor enzyme regulation would otherwise lead to irreversible growth arrest. We then develop a generalized, experimentally accessible framework for estimating how NAC is influenced by community size and the complexity of metabolic pathways. Using this framework and singlemolecule data on Escherichia coli enzyme levels, we predict that typical bacteria suffer from significant growth inefficiencies due to imperfect regulation, and thus could benefit from metabolite exchange. We then show that the benefits of NAC can be privatized, a key requirement for the evolutionary accessibly of cooperative behaviors. Finally, we discuss potential experimental tests of the theory.
Results
Isolated cells
We begin by exploring the impact of enzyme noise on the growth of an isolated model cell. To focus on the role of enzyme level fluctuations, we consider a cell of fixed volume and track the numbers of internal enzyme and metabolite molecules. Cell growth rate is recorded, but does not explicitly lead to an increase in cell volume. Instead, to capture the effects of cell growth and the associated volume increase, the rates of enzyme production and enzyme loss by dilution are both taken to be proportional to the growth rate. Since metabolite production and consumption fluxes are generally large compared to the dilution of metabolite levels by growth, we neglect the small effect of dilution on the metabolite levels. In this simple model, we assume that cell growth requires two essential metabolites that have intracellular counts of $m}_{1}^{\text{int}$ and ${m}_{2}^{\text{int}}$. These metabolites are produced intracellularly by specialized enzymes with intracellular counts of E_{1} and E_{2}. Both metabolites are produced from the same precursor, which is imported such that a constant number of precursor molecules is maintained within the cell. Each enzyme produces its metabolite at rate $\kappa {E}_{i}$, with $\kappa $ encompassing both the precursor concentration and the enzyme rate constant. For simplicity, we neglect the role of posttranslational modification in regulating enzyme activity (see Discussion for details on the potential impact of such additional regulation). In accordance with Liebig’s law of the minimum, growth is proportional to the level of the least abundant of the two metabolites such that the growth rate is $g={g}^{\ast}{\text{Min}}_{i}({m}_{i}^{\text{int}})$, where ${g}^{*}$ is a constant relating the metabolite levels to the cell growth rate. Thus, the cell growth rate is maximized when the metabolites are produced and present in equal amounts. Metabolites within the cell are assumed to be consumed at a rate proportional to the growth rate. Metabolites are also exchanged with an extracellular space whose volume is ${r}_{\text{V}}$ times greater than a cell volume. This exchange occurs via membrane diffusion with permeability $P$ (we show in Appendix 1 that this form of exchange is mathematically equivalent to active transport in the linear regime). Metabolites both inside and outside the cell are passively degraded at a rate $\delta $. A schematic of this model is shown in Figure 1A. Formally, the intra and extracellular counts of the metabolites evolve according to the following timerescaled equations (see Appendix 1 for details):
We model these metabolite dynamics as deterministic, as there is generally a large number of each essential metabolite within cells (Bennett et al., 2009).
Enzyme production within the cell is regulated based on internal metabolite levels, with the cell exclusively producing the enzyme corresponding to the currently least abundant metabolite. A flowchart of this regulation scheme is shown in Figure 1B. To reflect the bursty nature of gene expression (Golding et al., 2005), we assume that enzymes are produced in Poisson distributed bursts with average size $\beta $. Cells produce enzymes at a rate proportional to their growth rate such that the rate of enzyme bursts is $(\gamma /\beta ){g}^{*}{\text{Min}}_{i}({m}_{i}^{\text{int}})$, where $\gamma $ is a constant controlling the steadystate abundance of enzymes (such that at steady state $\u27e8{E}_{i}\u27e9=\gamma /2$). Enzymes are diluted by growth at a rate proportional to their abundance ${E}_{i}{g}^{*}{\text{Min}}_{i}({m}_{i}^{\text{int}})$. The coupling of enzyme production and dilution to growth rate reflects the requirement that all cellular components must increase on average at the growth rate to prevent component imbalances. We model enzyme production as a stochastic process due to its intrinsically noisy nature, and model enzyme dilution as a deterministic process. The metabolite and enzyme equations are simulated numerically using a hybrid deterministicstochastic method (see Appendix 2 for details).
In Figure 1C and D, we show example timecourses of metabolite dynamics for different burst sizes in cells with an average level of each enzyme of $\u27e8{E}_{i}\u27e9=\gamma /2=25$. In Figure 1C, we show the metabolite levels of a cell with a small burst size $\beta =2$. The small bursts allow for precise regulation of metabolite production, with the cell maintaining nearly equal levels of the two metabolites. In contrast, Figure 1D shows a cell with a burst size of $\beta =20$. This cell’s poor enzyme regulation leads to imbalances in enzyme levels which in turn manifest as metabolite imbalances (see Appendix 9—figure 1 for the corresponding enzyme timecourses).
How does the growth of an isolated cell depend on membrane permeability? In Figure 1E, we show the mean growth rate of isolated cells for varying permeability $P$. As can be seen, growth rate decreases monotonically with permeability. This follows because permeability leads to a loss of metabolites to the extracellular space where they cannot be utilized by the cell, but can be degraded. The coefficient of variation (CV) of the intracellular metabolite levels does not substantially change with increasing permeability, decreasing only very slightly due to stored metabolites in the extracellular space buffering fluctuations within the cell (see Appendix 9—figure 2). Growth rate losses increase with growing extracellular volume, approaching the limit in which metabolites are permanently lost upon leakage from the cell (see Appendix 9—figure 3).
The growth of isolated cells is also strongly influenced by the enzyme burst size, with small burst sizes permitting faster growth. This is seen in Figure 1F, which shows that growth rate decreases monotonically with average burst size $\beta $. The decreasing trend reflects a type of ‘use it or lose it’ phenomenon in which cells grow poorly when they have large metabolite imbalances, as these result in metabolites being degraded instead of consumed for growth. The smaller the burst size, the lower the variance of the enzyme levels and the closer to equality metabolite production and levels can be maintained. Note that with sufficiently large burst sizes, the cell can experience irreversible metabolic arrest. This occurs because the cell must grow to produce additional enzymes, and if the cell experiences a sufficiently large metabolite imbalance, it may be unable to make another burst of enzyme before its existing metabolites are exhausted. In Figure 1E, this growth arrest is reflected in the $\beta =100$ curve that is near zero for all values of permeability. Similarly, in Figure 1F, there is a value of $\beta $ beyond which cells do not grow.
Multicell communities
We have characterized the behavior of an isolated model cell, but how do enzyme noise and metabolite leakage affect growth rates in a community? We now expand our model to a population of cells that share a common extracellular space with which they exchange metabolites, as depicted in Figure 2A. When cells leak, there is now a possibility that these leaked metabolites will be taken up by other cells. To explore how leakage influences the collective metabolism of a multicell community, we simulate a community of 10 cells growing under the same conditions as in Figure 1E. In Figure 2B, we show the intracellular metabolite CV of these cells as a function of permeability for a range of enzyme burst sizes. As in the singlecell case, metabolite CV generally increases with increasing burst size. Interestingly, however, the metabolite CV now decreases substantially with increasing permeability. This occurs because metabolite exchange allows the cells to ‘average out’ the noise arising from their individually poor enzyme regulation, a phenomenon we term noiseaveraging cooperation (NAC). With large burst sizes, cells are prone to overproducing one type of enzyme, and thus overproducing a single type of metabolite. An isolated cell has no avenue to remedy this imbalance, leading to degradation of the overabundant metabolite. In a sufficiently large community this changes: within the population of cells, it is likely that there exist cells with opposite imbalances, and by exchanging metabolites these cells can collectively balance their metabolism.
How does this decrease in metabolite noise impact the average growth rate of cells within the community? In Figure 2C, we plot the growth rates of the communities from Figure 2B. In the case of a large burst size ($\beta =100$), the improvement is extreme. With sufficiently high permeability, cells that were previously unable to grow at all due to their poor regulation can now grow at a substantial fraction of the optimal growth rate. We note that while metabolite CV decreases for all values of burst size, this does not always translate into a growth improvement. With very small burst sizes ($\beta =2$), the increased permeability has the opposite effect and slightly decreases the growth rate. Since these cells already have efficient enzyme regulation, the moderate decrease in metabolite CV is outweighed by the increased degradation of metabolites within the extracellular space. Thus, the benefits of NAC are greatest under two conditions: (1) when individual cells have poor enzyme regulation and (2) when cells exist in a crowded space with minimal free volume between cells (such as in a biofilm).
How does community size influence the noisereducing effect of NAC? To answer this question, we consider a simplified, linear version of our model in which a population of $n$ fully permeable cells are directly connected to each other. We track only a single type of metabolite and enzyme. We also remove the nonlinear feedback between growth and enzyme dynamics, assuming a constant growth rate such that the percell average rate of enzyme bursts is $\mathrm{\Gamma}/\beta $ and the rate of enzyme dilution is ${\mu}_{E}$. Metabolite consumption and degradation are aggregated into a single rate parameter ${\mu}_{m}$. For simplicity, we assume that all enzyme bursts are of size $\beta $, rather than being Poisson distributed. The Langevin equations for the total number of enzymes and total number of metabolites are therefore:
where the $\xi (t)$ are noise terms with $\u27e8\xi (t)\u27e9=0$ and noise strengths $\u27e8{\xi}_{E}^{2}(t)\delta t\u27e9=n\mathrm{\Gamma}\left(\beta +1\right)$ and $\u27e8{\xi}_{m}^{2}(t)\delta t\u27e9=2\kappa \mathrm{\Gamma}n/{\mu}_{E}$ (see Appendix 3 for further details). Since these equations are linear, we can exactly compute the expression for the CV of the total enzyme level (see Appendix 3):
Consistent with our simulations, the CV increases with burst size $\beta $. The dependence on population size can also be immediately seen from this expression, with the CV being proportional to $1/\sqrt{n}$. Thus, larger communities are expected to magnify the positive impact of metabolite exchange. We can also directly compute the metabolite CV:
As expected, the metabolite CV has a similar scaling with respect to $n$ and $\beta $ as the enzyme CV. In (Appendix 9—figure 4), we compare these predicted relationships to the full model and find good agreement.
These calculations, along with our simulations, characterize the potential benefits of NAC. The small size of bacteria make them inevitably noisy, possibly leading to growth losses. Metabolite leakage can act as a form of bacterial mutual aid, benefiting cells by allowing resource pooling.
Generalized NAC framework
We have demonstrated NAC in a simple model with two metabolites, but how would the benefits apply to more realistic metabolic networks? We now develop a general framework to determine the impact of enzyme fluctuations and metabolite sharing on the growth of bacteria with an arbitrary number of nonsubstitutable metabolites. We begin with an arbitrary probability distribution function (PDF) of the intracellular levels of an individual metabolite $f({m}_{i})$. For simplicity, we first consider the case of independent but otherwise identical metabolites. For a given number of nonsubstitutable metabolites, we can then apply Liebig’s law of the minimum to compute the distribution of growth rates ${q}_{n}(g)$ by determining the distribution of the lowest metabolite level among a set of $n$ metabolites. Note that while this minimum is technically only proportional to growth rate, for brevity we assume ${g}^{*}=1$ such that $g={\text{Min}}_{i}({m}_{i})$. To calculate ${q}_{n}(g)$, we use the cumulative distribution functions (CDFs) $F({m}_{i})$ and ${Q}_{n}(g)$:
Intuitively, the mean of the distribution of growth rates with $n>1$ nonsubstitutable metabolites will always be lower than the mean of m_{i}, as shown schematically in Figure 3A. Thus, as in our simple twometabolite model, the mean growth rate will depend not just on the means of the individual metabolite counts, but also their variability.
To examine this general model in more detail, we now use realistic metabolite distributions to quantitatively analyze the effect of various factors on the growthrate distribution. While the distribution of metabolite levels in single cells has not been directly measured, there have been extensive measurements of singlecell protein distributions showing that these levels are typically gamma distributed. As we expect that enzyme levels are the dominant source of metabolite noise, we therefore approximate the metabolite distributions as gamma distributions, that is we assume that the metabolite distributions inherit the shape and thus the CV of the underlying enzyme distribution. In particular, we use the median gammadistribution parameters measured for essential proteins in E. coli (Taniguchi et al., 2010) as a base. (Note that we consider the measured variability to be due to stochastic noise, but in principle some of this variability could emerge from deterministic dynamics, e.g. persistent oscillations. This distinction is not important for NAC, as fluctuations away from an optimal average arising from deterministic dynamics will have the same effect as stochastic noise.) For simplicity, these analyses assume that all metabolites have the same average concentration and are thus corate limiting. We show the corresponding growthrate distribution as a function of the number of nonsubstitutable metabolites in Figure 3B. One sees that as the number of metabolites increases, the mean of the growthrate distribution decreases. This occurs because the more metabolites the cell must manage, the more likely it is that at least one will be poorly regulated at a given moment and constrain growth. Thus, NAC is most beneficial to cells that require large numbers of nonsubstitutable metabolites. In Figure 3C, we explore how this decrease in growth rate depends on the CV of the metabolite distribution, with the CV of the E. coli essential protein distribution shown as the dashed curve. If the metabolite levels are poorly controlled resulting in a large CV, the addition of more metabolites drastically reduces growth. Conversely, if the cell has tight control of its metabolites, it can manage significant numbers of nonsubstitutable metabolites without too large a growth loss. Interestingly, the curve corresponding to the CV of essential E. coli proteins shows a significant loss in growth rate, suggesting that realworld bacteria may suffer substantially from poor enzyme regulation. It should be noted, however, that enzyme count noise may overestimate noise in the resulting metabolites, as the enzymes can be regulated posttranslationally, and metabolite fluxes may be buffered against enzyme fluctuations by network feedback effects (Hackett et al., 2016).
Thus far, we have assumed that the metabolites’ distributions are independent, but this likely does not hold in nature. Experimental studies have found that levels of different enzymes within the cell are correlated (Taniguchi et al., 2010), suggesting that metabolite levels are likely also correlated. Indeed, this phenomenon occurs even in the simple models we analyzed above (see Appendix 9—figure 1). To account for these correlations, we computed the mean growth rate for varying degrees of correlation between metabolites. As a baseline, we again use the median distribution of essential proteins in E. coli. The results can be seen in Figure 3D, and show that correlation between metabolites reduces the adverse effects of metabolite noise. This occurs because if the metabolite levels are correlated, it is less likely that there will be a single outlying low metabolite level constraining growth. Note that the growth benefit of correlated metabolite fluctuations is largest when metabolites are on average corate limiting: if there is one metabolite that is strongly limiting, it makes little difference whether the other metabolites are correlated with it or not. The correlation between certain proteins in E. coli has been measured (Taniguchi et al., 2010), and we show this value as a dashed curve. While this level of metabolite correlation does improve growth, the growth loss associated with realistic enzyme noise is still substantial.
What if the growth rate is not determined by Liebig’s law of the minimum? Real growth functions are unlikely to be quite so simple, and given the variation that exists in microbial metabolism, it is unlikely that there is a single universally applicable growth function. Despite this uncertainty, we can determine what classes of growth function will lead to noisedriven growth defects. Consider an arbitrary growth function $g(\mathbf{X})$ and vector of randomly varying metabolites $\mathbf{X}$. We can express the decrease in growth due to noise as $\mathbb{E}[g(\mathbf{X})]\le g(\mathbb{E}[\mathbf{X}])$. This statement is equivalent to the multivariate Jensen’s Inequality for concave functions, meaning that if $g(\mathbf{X})$ is concave, the introduction of metabolite noise will decrease the mean growth rate. Note that there are also functions which are not globally concave that also satisfy this inequality, see Appendix 5 for examples. Growth functions in which the benefit of increasing individual nonsubstitutable metabolite levels is saturating will generally be concave. Thus, most reasonable growth functions will lead to decreased mean growth in the presence of metabolite noise. To demonstrate this, in (Appendix 9—figure 6) we show a version of Figure 3C with an alternative growth function based on the rate of protein synthesis. Interestingly, the dependence of the noisedrive growth loss on the concavity of $g(\mathbf{X})$ implies that the magnitude of the loss may depend on the mean metabolite levels. If the metabolite levels are well above the saturation point of the growth process, such that the local growth function has low concavity, there will be minimal growth losses due to metabolic noise. A similar reduction in growth losses may occur if the metabolites are far below saturation.
With the above generalized framework, we were able to incorporate experimental measurements into our theory. Our preliminary analyses based on enzyme level measurements suggest that real bacteria may indeed suffer from substantial noisedriven growth defects. Combined with our analyses of simple metabolic models, this raises the possibility that bacteria can engage in NAC to improve their collective growth rate, particularly in tightly packed environments like biofilms.
Discussion
In this work, we develop a theory of noiseaveraging cooperation (NAC), a novel mechanism potentially underlying the emergence of both intraspecies cooperation and interspecies crossfeeding. NAC allows microbes with individually poor regulation to average out their metabolic noise and raise their collective growth rate by sharing metabolites. Since NAC is strongest in crowded environments, it suggests an additional benefit of the biofilm mode of growth. With respect to crossfeeding evolution, our mechanism can be viewed as complementary to the Black Queen Hypothesis: it provides a mechanistic explanation for the emergence of metabolite leakage, setting the stage for the evolution of metabolic interdependencies via gene deletions (McNally and Borenstein, 2018). However, while we have shown that NAC can be beneficial under plausible assumptions about metabolite noise and is consistent with some existing data, further study is needed to determine whether NAC is evolutionarily accessible and to what extent it may be a driver of crossfeeding in nature.
An important aspect of NAC to consider is its evolutionary stability. We have shown in this manuscript that NAC is a potential social optimum for densely packed bacteria. However, a state being socially optimal does not guarantee that the state is evolutionarily accessible or stable. In order for NAC to support the evolution of crossfeeding, cells engaging in NAC must be able to invade populations of cells not engaging in NAC and vice versa. The conditions for the maintenance of cooperative behaviors within a population have been studied extensively, and one of the key conditions is based on preferential interaction between ‘cooperators’. If cooperators primarily interact with other cooperators, growth losses due to ‘cheaters’ are minimized (Nowak, 2006; Nowak and May, 1992). We conjecture that within a biofilm NAC cells will segregate with other NAC cells simply due to proximity within a lineage, providing a form of preferential interaction and stabilizing NAC within the population. Consider a growing biofilm composed of cells not engaging in NAC. If a cell engaging in NAC invades, its growth will initially be lower than its neighbors, due to the fact that its neighbors are only uptaking metabolites. However, after a few cell divisions, the single NAC performing cell will grow into a small sector of cells performing NAC and will thus largely share metabolites with its kin and potentially gain a growth advantage over nonNAC cells. To estimate this sharing distance, we developed a simplified model of NAC in a biofilm and parameterized it with measurements from a Vibrio cholerae biofilm (see Appendix 7). We found that leaked metabolites travel only a few cell layers before being largely consumed, suggesting that NAC cells require relatively few divisions to establish a successful NAC sector. Now consider invasion of a nonNAC cell into a NAC biofilm: a cell not engaging in NAC can invade because it is only taking up metabolites and not leaking them. However, its growth will eventually lead to a sector of cells not engaging in NAC, thus losing the benefit that allowed the initial invasion. Studying the evolutionary stability of NAC is a promising direction of future study, requiring the development of a spatial model incorporating cell growth along with metabolic noise.
If NAC does occur in nature, how can we identify it? Our theory predicts, counterintuitively, that it may benefit organisms to secrete essential metabolites into the environment. Thus, deliberate leakage or export of essential metabolites, such as amino acids or vitamins, is a potential signature of NAC. The clearest signature of deliberate export would be the existence of dedicated transporters for essential metabolites. Deliberate export could also occur via membrane leakage, but this case is more ambiguous as it is difficult to determine whether such leakage is ‘deliberate’, that is allowed by the cells, or is an unavoidable consequence of membrane permeability. As a test case, we examined the export of amino acids, a key class of nonsubstitutable metabolites, in E. coli. How much of E. coli’s aminoacid production is lost to leakage? Using prior experimental measurements of membrane permeability, intracellular concentrations, and aminoacid production rates, we estimate that E. coli loses only a small fraction (<1%) of its aminoacid production to membrane leakage (see Appendix 9—figure 5). This suggests that membrane leakage is not a substantial avenue of amino acid export in E. coli. On the other hand, there exist multiple aminoacid exporters in E. coli (Pathania and Sardesai, 2015; Hori et al., 2011; Nandineni and Gowrishankar, 2004; Kutukova et al., 2005; Zakataeva et al., 1999), implying that E. coli does indeed engage in deliberate export of essential metabolites, consistent with NAC. Moreover, consistent with the idea that these exporters enable metabolic exchange, it has been shown that artificial auxotrophic E. coli strains can indeed crossfeed each other amino acids (Mee et al., 2014). Note, however, that there may be reasons other than NAC for deliberate export of amino acids, such as overflow metabolism or the use of amino acids as signaling molecules (Burkovski and Krämer, 2002; Park et al., 2003; Pathania and Sardesai, 2015). Indeed, the ‘economies of scale’ scenario, in which organisms are hypothesized directly benefit from specializing in production of fewer metabolites (Pande et al., 2014), might lead to similar amino acid exporting behavior.
Another way to probe the relevance of NAC to real bacteria would be to obtain more accurate estimates of intracellular metabolite distributions. Due to a lack of direct measurements of metabolite concentrations within single bacterial cells, we approximated the metabolite distribution using data from singlecell proteomics measurements. This approximation allowed us to estimate growth loss due to noise, but our conclusions depend on the assumption that the magnitude of metabolite noise roughly follows that of enzyme noise. In our simple metabolic model, this assumption is borne out, as the dominant source of metabolite noise is the burstiness of the enzyme dynamics. However, in real cells, there are likely additional regulatory feedbacks that suppress metabolite noise. Thus, our analysis may overestimate metabolite noise and thus the benefits of NAC. Future theoretical studies could yield a more accurate estimate of metabolite noise using realistic models of intracellular metabolite dynamics that incorporate complete pathways and phenomena such as posttranslational regulation. There is also the possibility that direct measurements of intracellular metabolite distributions will become available, as the technology for singlecell metabolite measurement is rapidly advancing (Evers et al., 2019; Yoshida et al., 2019).
In addition to more precisely quantifying intracellular metabolite noise, better predicting noisedriven growth losses will require understanding the relationship between individual metabolite levels and growth rate in real cells. We showed that in order for metabolic noise to decrease mean growth rate, the only requirement is that the growth function be concave. We expect most growth functions to meet this condition, as the benefit of increasing metabolite levels generally saturates. However, it is possible that cells mitigate the impact of noise by maintaining their metabolite levels in a region of the growth function with low concavity, for example in a linear regime or near saturation. Intriguingly, one experimental study examining the relationship between fluctuations in metabolic enzymes and growth rates found that enzyme fluctuations did translate into growth fluctuations (Kiviet et al., 2014). These findings suggest that metabolite noise driven by enzyme fluctuations can indeed influence growth, as is required for NAC to be beneficial. However, more definitive conclusions can only be drawn from direct correlation of metabolite levels and growth rates, an experiment that is difficult with current technology. Such experiments may become possible in the future, as the technology for realtime singlecell metabolite measurement is rapidly developing. For example, a fluorescent reporter for branchedchain amino acids has recently been demonstrated in eukaryotes (Yoshida et al., 2019).
Even if the growth loss due to metabolic noise is large, for NAC to be beneficial the reduction in noise must outweigh the cost of metabolite loss in the extracellular space. In the context of a biofilm, some of this loss will likely arise from diffusion away from the biofilm, and thus the loss rate can potentially be calculated from the geometry of the biofilm and the diffusion constant of the metabolite within the biofilm matrix. Estimating the impact of other, spontaneous or reactionbased, forms of metabolite loss will likely require experimental measurements. We explore the impact of low degradation rates on NAC in Appendix 8.
Analysis of intracellular metabolite dynamics and realistic growth functions may provide some support for NAC, but definitive testing of the mechanism will likely require dedicated experiments. E. coli would be a suitable organism for such experiments, as it is known to encode transporters for at least some essential metabolites, and has already been shown to engage in intercellular exchange of amino acids (Mee et al., 2014). To directly test the theory, experiments will require at least two conditions: one in which cells are isolated and another in which they exist at a relatively high density. One possibility is to compare planktonic and biofilm cells, while another would be to assemble varying densities of planktonic cells. With isolated and crowded conditions defined, there are two major predictions that could be tested: growth rate should increase when cells are in crowded environments, and metabolic noise should be reduced when cells are in crowded environments.
Testing of the growthrate prediction could be performed using populationlevel measurements of wellmixed cultures. The simplest way to test this would be measure the exponentialphase growth rates of bacterial cultures of different densities. An exponentialphase culture of E. coli could be resuspended at different densities in minimal media with saturating concentrations of nutrients, and growth rates measured (e.g. via OD). If NAC is occurring, the growth rate should be positively correlated with the culture density. Another possible experimental system is an E. coli chemostat fed with minimal media. The measured output in this system will be the steadystate biomass. NAC predicts that, compared to the case where growth rate is independent of density, the cell density will be higher than expected at low dilution rates (see Appendix 6 for details). For both of the above experiments, it will be important to determine whether the observed growth differences are due to metabolite exchange. To test this, one could employ mutants with different essential metabolite exporter genes deleted and measure whether the difference in growth rate still exists between isolated and dense conditions. Data from these mutants will need to be interpreted carefully, for example due to redundant/undiscovered transporters or unintended effects of the deletions. In these experiments, NAC should be distinguishable from the ‘economies of scale’ mechanism of crossfeeding promotion mentioned earlier. In theory, the ‘economies of scale’ benefit should only occur when some organisms are auxotrophs for the shared metabolites, while NAC is beneficial even if all organisms are prototrophs.
Testing whether metabolic noise decreases with cell density will require techniques with singlecell resolution. The difficulty in testing this prediction will stem from finding a method to measure singlecell metabolite concentrations. One possible method to estimate the metabolite distribution is to measure timeseries of metabolite levels using the fluorescent reporter approach discussed above. One could image these fluorescent reporters in twodimensional colonies and attempt to correlate the observed metabolic noise to local cell density. Similar to the earlier proposed experiments, one could employ export mutants to determine whether observed decreases in noise are due to metabolite exchange.
If NAC does exist in nature, how is it implemented and regulated by cells? It is unlikely that cells would exchange their entire metabolome with the external environment, but how would cells select which metabolites to exchange? If the metabolites within cells have different levels of noise, it might be optimal for cells to engage in NAC with the metabolites with the highest noise. This would imply that metabolites produced in small quantities are good candidates for NAC. The choice of which metabolites to exchange is also influenced by the architecture of the cell’s metabolic network. If there is a bottleneck in the network, for example if catabolism generates a small set of metabolites that are then used in a myriad of anabolic synthesis processes, it may be beneficial to exchange these bottleneck metabolites. In addition to selecting which metabolites to exchange, there is also the issue of deciding when to engage in NAC. It would be harmful for cells to engage in NAC at low cell density, as most of the secreted metabolites would be lost. Thus, regulation of NAC would likely be linked to quorum sensing. Note, however, that sensing of kin cell densities would not be sufficient for regulation of NAC. If the environment also contains a high density of nonkin, there will likely be a high effective loss rate of metabolites to these nonkin, making NAC disadvantageous. Thus, regulation of NAC should be dependent on multiple quorumsensing circuits, using both kin and nonkin autoinducers (Miller and Bassler, 2001).
Thus far, we have assumed that the level of noise within an isolated cell is immutable, but in nature metabolic noise is an evolvable trait. Thus, rather than engage in NAC, cells may have the option to increase growth by reducing intracellular noise. Substantial noise reduction is possible, but it is often costly with steeply diminishing returns. For example, one method of reducing noise originating from Poissontype processes is to increase the number of the molecules of interest, which will decrease CV by the square root of the the number of molecules. This is essentially what occurs in NAC, but with production split across multiple cells. Another option is for the cells to introduce feedbackbased regulation. This approach is even more costly, with information theoretic work showing that the CV of a molecule whose abundance is controlled by feedback only decreases with the fourth root of the number of signaling events (Lestas et al., 2010). These superlinear costs make it impossible for bacterial cells to be entirely noisefree given their size, but it may still be possible for a cell to eliminate most of the detrimental effects of noise. As a result, NAC must be considered as only one of multiple possible strategies for noise mitigation.
We have focused on bacteria in this manuscript, but it is also possible that NAC may be relevant in other domains of life. For example, NAC highlights a potential advantage of multicellularity: a multicellular tissue separated from its external environment is the optimal environment for NAC. NAC may also apply to macroecological systems if the outcome of foraging for resources is highly variable. Under such conditions, it may be beneficial for animals to engage in resource sharing, potentially supporting the development of social groups. Indeed, NAC is analogous to revenuesharing clubs that can be employed to damp income volatility (Tilman et al., 2018).
While much research is needed to determine the relevance of NAC to real bacteria, the theory highlights an interesting aspect of ecology: noise at even the smallest scales can have a dramatic impact on the entire ecosystem. In this manuscript, we have focused on the singlespecies case, but there is potential for more novel behaviors in the manyspecies context, as has been observed in other resourcecompetition models (Tikhonov and Monasson, 2017). Overall, we hope this work can serve as a foundation for further theoretical and experimental work on how noise in resource acquisition impacts ecology and evolution.
Methods
All code and data used in this manuscript can be found at https://github.com/jaimegelopez/NAC (Lopez, 2021, copy archived at swh:1:rev:d39206b0890340db8b4faad87ed544f345b09057). Details on individual analyses and derivations can be found in the relevant appendices.
Appendix 1
Timerescaling of the twometabolite model
To show how we obtained the timerescaled Equations 1; 2, we begin with a generalized set of equations in which a parameter $\alpha $ relates the minimum of the two metabolite levels to the consumption rate of the metabolites:
Here, ${m}_{i}^{\text{int}}$ is the intracellular count of metabolite $i$, ${t}^{\prime}$ is time, ${\kappa}^{\prime}$ relates enzyme count to metabolite production, ${E}_{i}$ is the count of enzyme $i$, $\alpha $ relates the minimum of metabolite levels to metabolite consumption, ${P}^{\prime}$ is permeability, ${m}_{i}^{\text{ext}}$ is the extracellular count of metabolite $i$, ${r}_{\text{V}}$ is the ratio of extracellular to intracellular volume, and ${\delta}^{\prime}$ is the metabolite degradation rate. We now show that by rescaling time, we can eliminate $\alpha $. We first introduce our dimensionless time variable ${t}^{\prime}=tu$, where $u$ is a parameter with units of time. Substituting into the above equations yields:
By defining $u=1/\alpha $, $\kappa ={\kappa}^{\prime}/\alpha $, $P={P}^{\prime}/\alpha $, and $\delta ={\delta}^{\prime}/\alpha $, we recover Equations 1; 2.
We present transport in our model as passive exchange with the environment, but these equations describe a broader range of transport phenomenon. In particular, our expressions for metabolite exchange are mathematically equivalent to active transport in the linear regime. To demonstrate this, suppose that import and export are mediated by two separate enzymes with different kinetics. In the linear regime, the transport expression will be ${\eta}_{1}{m}_{i}^{\text{ext}}/{r}_{\text{V}}{\eta}_{2}{m}_{i}^{\text{int}}$ where the $\eta $ are constants defining transporter performance (for MichaelisMenten kinetics, each $\eta $ would be the ratio of the maximum rate and the halfsaturation constant of the corresponding transporter). Rearranging terms, one obtains ${\eta}_{2}\left(\frac{{\eta}_{1}}{{\eta}_{2}{r}_{\text{V}}}{m}_{i}^{\text{ext}}{m}_{i}^{\text{int}}\right)$ such that there is an effective permeability ${\eta}_{2}$ and effective volume ratio $\frac{{\eta}_{2}{r}_{\text{V}}}{{\eta}_{1}}$.
Appendix 2
Hybrid numerical methods
In our twometabolite model, the enzymes are produced in intermittent bursts which must be modeled stochastically, while other reactions in the system (such as metabolite consumption) involve a large enough number of molecules or occur frequently enough that they can be treated deterministically. In order to efficiently simulate this system, we use a hybrid stochasticdeterministic algorithm that can be viewed as a generalization of the Gillespie algorithm. We briefly present the rationale for the algorithm here, with a full description available in (Griffith et al., 2006).
We begin with the traditional Gillespie algorithm. In this algorithm, the time between events is an exponential random variable with mean $1/{r}_{\text{tot}}$, where ${r}_{\text{tot}}$ is the sum of all reaction rates in the system. This formulation assumes that ${r}_{\text{tot}}$ is constant between stochastic reactions, something that will not be true in a system that also includes deterministic reactions. To account for timevarying reaction rates, we first recognize that the Gillespie event condition can be rewritten as:
where $\tau $ is the stochastic time before the next reaction. This rewritten condition makes intuitive sense: if one regards the integral on the lefthand side as a kind of cumulative probability, this states that for every one unit of cumulative probability, on average one event occurs. From Equation 12 we can rigorously construct a simulation algorithm for systems with timevarying reaction rates. We first sample a random number ${x}_{1}\sim \text{Exponential}(1)$. We then integrate the deterministic dynamics until we find a $\tau $ such that:
In our method, the integration is done by Euler’s method. We then sample another random number ${x}_{2}\sim \text{Uniform}(0,1)$ and use this to determine which reaction occurs, as in the traditional Gillespie method. The reaction event then occurs at time $t+\tau $, and we repeat the above procedure until the simulation reaches a termination condition. In our simulation, the only stochastic events are the enzyme production bursts, while all other reactions are modeled deterministically. For calculation of CVs and growth rates in Figures 1 and 2, simulations were run for $4\times {10}^{5}$ time units (which corresponds to 5000 time units when normalized to the maximum steadystate growth rate in Figure 1), with statistics computed from the last $3\times {10}^{5}$ time units. This time is more than sufficient for the system to forget its initial conditions, as can be seen in Appendix 9—figure 7. Simulations are run with initial conditions ${m}^{\text{int}}(t=0)={m}^{\text{ext}}(t=0)=\frac{\kappa \gamma}{2(1+\delta +P)}$ and $E(t=0)=\gamma /2$. This metabolite concentration is the mean steadystate intracellular metabolite concentration assuming an infinite extracellular volume, and the enzyme concentration is the mean steadystate enzyme concentration. For Figure 1E, F, 140 replicate simulations were run per point. For Figure 2B, C, 40 replicate simulations were run for most points, with 200 replicates run for certain simulations with large burst size and low permeability.
Appendix 3
Langevin description
To better understand the influence of community size on metabolite noise, we consider a simplified version of the twometabolite model. In this model, we only track the concentration of a single constitutively produced enzyme and its corresponding metabolite. The Langevin equations governing the dynamics are:
where the Langevin noise terms have zero noise $\u27e8{\xi}_{E}(t)\u27e9=\u27e8{\xi}_{m}(t)\u27e9=0$ and are deltacorrelated such that ${\xi}_{i}({t}_{1}),{\xi}_{j}({t}_{2})\u27e9=\mathrm{\Omega}{}_{ij}\delta ({t}_{1}{t}_{2})$, where $\delta (\cdot )$ is the Dirac delta function. These equations have mean steadystate fixed point at ${E}^{*}=\frac{n\mathrm{\Gamma}}{{\mu}_{E}}$ and ${m}^{*}=\frac{\kappa n\mathrm{\Gamma}}{{\mu}_{E}{\mu}_{m}}$. To analytically compute the CV of the enzyme and metabolite, we employ the method of Swain (Swain, 2004). This method relies on linearizing the system about steady state and assumes that fluctuations about the steady state are sufficiently small so as not to drive the system out of the linear regime around the fixed point. Since our system is linear to begin with, this method will actually provide us with exact expressions of the moments.
We begin by computing the elements of $\mathrm{\Omega}$. The squared deviation of ${\xi}_{E}(t)$ will obey:
Similarly, we can compute the value of ${\mathrm{\Omega}}_{mm}$:
The offdiagonal entries of $\mathrm{\Omega}$ will be zero, as the two equations share no common reactions.
Next, we represent our deterministic dynamics as a matrix equation by computing the Jacobian about the fixed point
This Jacobian has eigenvalues ${\lambda}_{1}={\mu}_{E}$ and ${\lambda}_{2}={\mu}_{m}$ with a matrix of column eigenvectors
Variances and covariances of the state variables can then be computed using the following expression:
where the ${X}_{i}$ are the state variables.
Substituting in the values of $B$, $\mathrm{\Omega}$, and $\lambda $ yields
From these expressions for the variance, we compute the CVs as
Appendix 4
Generalized NAC framework
To compute distributions of growth rates from metabolite distributions, we first compute the growth distribution’s CDF in accord with Equation 7 and then apply numerical differentiation to yield the PDF. To obtain the expected value, we numerically integrate according to the expression $\mathbb{E}[X]={\int}_{\mathrm{\infty}}^{\mathrm{\infty}}xp(x)dx$ where $p(x)$ is the probability density of value $x$. As a reference distribution, we used the median gamma distribution of E. coli essential proteins observed in (Taniguchi et al., 2010) ($k=6.4$ and $\theta =5.2$, taken from Table S3 of Taniguchi et al.). For distributions with different CVs, we maintain the same mean as the median distribution of E. coli essential proteins.
The case of correlated metabolite levels was analyzed by simulation. To generate correlated distributions, we first generate 50,000 samples for each metabolite from a multivariate normal distribution with the desired correlation structure. The MATLAB function mvnrnd is used for this purpose. We then use the method of copulas to generate samples from correlated gamma distributions (Nelsen, 2007). This method first involves transforming the normally distributed samples into uniformly distributed samples using the inverse CDF of the normal distribution. We then transform these uniformly distributed samples into gamma distributed samples using the appropriate gamma CDF. Note that this method is only guaranteed to exactly preserve the rank correlation, but we find that the linear correlation is also very well preserved.
Appendix 5
Detailed conditions for noisedriven growth reduction
Here, we present a more general set of conditions for metabolite noise to reduce growth. In the main text, we invoked Jensen’s inequality to argue that metabolite noise will reduce growth if a growth function is concave. This statement is true, but it is not a complete description: there are many nonconcave functions that can result in reduced growth in the presence of noise. In addition, as mentioned in the discussion, growth sensitivity to noise is not only a property of the growth function, but also of the metabolite distribution. The most general requirement is that, given a growth function $g(X)$ and a metabolite distribution $f(x)$, the ‘Jensen gap’ $\mathbb{E}[g(\mathbf{X})]g(\mathbb{E}[\mathbf{X}])$ must be negative.
To show that noisedependent growth reduction can occur with nonconcave functions, we consider two toy examples. First, consider a growth function that is largely concave, but is convex at low nutrient levels (this convexity could reflect a minimum nutrient level required for growth). We show such a function as the red curve of Appendix 9—figure 8. For metabolites uniformly distributed across the entire plotted growth function, the Jensen gap will still be negative despite the initial region of convexity in the growth function. Now consider a more extreme case: a growth function that is everywhere oscillatory, rapidly switching between concave and convex, but has a mean behavior matching a concave function. We show such a function as the black curve in Appendix 9—figure 8. As in the previous case, this function has a negative Jensen gap for uniformly distributed metabolites.
In order to determine whether noise will reduce growth rate for a particular function and distribution pair, one must estimate the Jensen gap. Analytically, this can be done using "sharpened" versions of Jensen’s inequality. These expressions do not provide exact bounds, but provide more accurate results than Jensen’s inequality. For example, Theorem 1 of (Liao and Berg, 2018) provides an simple inequality that more tightly bounds the Jensen gap.
Appendix 6
Chemostat NAC experiment theory
Here, we present the theory for an experiment in which the effects of NAC are measured in a chemostat. First, we begin with the equations governing the chemostat dynamics:
where $\rho $ is the biomass concentration, µ is the maximum growth rate, $S$ is the nutrient concentration, $K$ is the halfsaturation constant, $\delta $ is the dilution rate, and $\mathrm{\Gamma}$ is the inlet nutrient concentration. Note that we are measuring biomass and nutrients in the same units. The steadystate nutrient value will be:
The steadystate biomass will therefore be:
Thus, if we model NAC as an increase in $\mu $, a higher than expected dilution rate will be required to maintain a given high cell density. To test this, one can estimate $\mu $ and $K$ at low cell densities and compute the predicted density as a function of dilution rate using Equation 29. Then, the actual steadystate densities can be measured for a range of dilution rates. If NAC is occurring, the predictions will match the experimental data at high dilution rates, while there will be a significant discrepancy between predicted and observed densities at low dilution rates.
Appendix 7
NAC metabolites within a biofilm
In this section, we estimate how far leaked NAC metabolites can travel in a biofilm. Consider a biofilm with cylindrical symmetry and radial coordinate $\widehat{r}$. In the center of the biofilm is a sector of radius $\widehat{a}$ composed of cells engaging in NAC, while the outer shell of cells are cheaters that only uptake metabolite and do not engage in NAC. We model metabolite leakage in the NAC sector as a constant metabolite production rate $\xi$. In both sectors, metabolites are imported at a rate of $\mu m(\hat{r})$, where $\mu $ is the uptake rate and $m(\hat{r})$ is the local metabolite concentration. Note that for simplicity we are assuming that the metabolite concentrations are below the saturation point of the uptake transporters. Within the NAC sector ($\hat{r}<\hat{a}$), the steadystate concentration of $m$ will obey
where $D$ is the diffusion coefficient of the metabolites within the biofilm. We assume a noflux boundary at $\hat{r}=0$
and an arbitrary flux at $\widehat{r}=\widehat{a}$ such that
Outside of the NAC sector ($\hat{r}$), the steadystate concentration of $m$ obeys:
We assume a zero metabolite boundary condition at infinity
and the same flux boundary condition at $\hat{r}=\hat{a}$ as with Equation 30. Once we solve the boundary problems in the two sectors, we will solve for ${J}_{\text{out}}$ by requiring continuity of $m$ at .
We introduce a nondimensionalized radial coordinate $rv=\hat{r}$ where $\nu =\sqrt{\frac{D}{\mu}}$. With this nondimensionalization, we can rearrange Equation 33 into an inhomogeneous modified Bessel equation of zeroth order:
Similarly, Equation 30 reduces to a homogeneous modified Bessel equation of zeroth order:
With this nondimensionalization, the boundary condition at $\hat{r}=\hat{a}$ becomes:
where $a=\hat{a}\sqrt{\frac{\mu}{D}}$.
We begin by solving the boundary value problem outside of the NAC sector. The general solution of Equation 36 is a linear combination of modified Bessel functions
where I_{0} and K_{0} are the zeroth order modified Bessel functions of the first and second kind, respectively. Applying boundary conditions from Equations 37 and 34 yields:
This solution is consistent with intuition: modified Bessel functions of the second kind are monotonic decreasing and the solution reduces to $m(r)=0$ for ${J}_{\text{out}}=0$. Similarly, we can obtain the solution to the boundaryvalue problem within the NAC sector:
This solution is also consistent with intuition: modified Bessel functions of the first kind are monotonic increasing and the solution reduces to $m(r)=\xi /\mu $ if ${J}_{\text{out}}=0$. Now that we have the solutions for each sector, we can solve for the value of ${J}_{\text{out}}$ by requiring $m$ to be continuous between both solutions at $r=a$:
To parameterize this model, we use values measured from a Vibrio cholerae biofilm in which NAcetylglucosamine is a public good that is being produced and consumed (Drescher et al., 2014). In this system, $D=500\mu {\text{m}}^{2}/\text{s}$ and $\mu =3.4\times {10}^{3}{\text{s}}^{1}$. The value of $\xi $ cannot be easily estimated, but as long as metabolite uptake is linear, $\xi $ will only rescale the solution and not alter the boundary size. We take a high estimate of $\xi =2.5\times {10}^{6}\text{molecules}/\mu {\text{m}}^{3}\text{s}$, approximately equivalent to E. coli leaking its entire aspartic acid pool each second. We plot metabolite profiles for two different values of $\widehat{a}$ in Appendix 9—figure 9. As expected, the length of the boundary region in which the NAC metabolites are available to cheaters is approximately constant. This occurs because the decay length scale of the modified Bessel function of the second kind only depends on $\mu $ and $D$, not $\widehat{a}$. Therefore, for large NAC sectors the number of cells in the transition region is small compared to the number of cells in the NAC sector.
Appendix 8
Models without metabolite degradation
To explore the impact of metabolite degradation on our results, we now analyze a version of the metabolite model without metabolite degradation (i.e. with $\delta =0$). We first examine the behavior of individual cells by repeating the analyses of Figure 1EF. These results can be found in Appendix 9—figure 10A,B. How does permeability influence growth in a system where metabolites do not degrade? In Appendix 9—figure 10A, we plot the growth rate of single cells as a function of permeability at different values of burst size. Curiously, the cell’s growth is poor at low permeability, improving when permeability is moderate. For $\beta =10$ and $\beta =20$, growth decreases again at very high permeability. This is in stark contrast to the case of single cells with metabolite degradation, in which increased permeability simply lowers growth. We found that this nonmonotonic behavior arises due to a ‘Paradox of Plenty’ in which increased intracellular metabolite buildup can make a cell more likely to die via sudden metabolite ‘crashes’. We show the metabolite and enzyme timecourses of a cell undergoing this crash behavior in Appendix 9—figure 10C,D. At around $t=15$, the cell develops an excess of the blue metabolite with a deficit of the pink metabolite, limiting its growth. In the time it takes the organism to produce a new burst of the pink enzyme, it continually builds up its pool of the blue metabolite, as there is no degradation term to limit the pool’s size. Eventually, at around $t=35$, the cell produces a burst of the pink enzyme, generating more pink metabolite and allowing a resumption of steady growth. Since there is such a large buildup of the blue metabolite, it takes the cell a long time to deplete this metabolite pool. In this time, the cell produces many bursts of the pink enzyme, almost entirely replacing the cell’s enzyme pool and leaving the cell with almost no blue enzyme. At around $t=55$, the blue metabolite crashes, rapidly becoming the new limiting metabolite. At this point, the cell is left in a difficult position, suddenly having almost none of the now limiting blue metabolite and having an enzyme pool entirely dedicated to production of the pink metabolite. Growth rapidly ceases, as the remaining blue metabolite pool is insufficient to produce a corrective enzyme burst. Even if the cell had survived, it would once again be back to having a severe imbalance, eventually leading to another crash. Such crashes can be averted if the cell is able to transfer a portion of its metabolites to an extracellular space, reducing the magnitude of intracellular metabolite buildup. For this reason, increasing permeability initially promotes cell growth, as seen in Appendix 9—figure 10A. However, if permeability becomes too high, the intracellular and extracellular metabolites are always equilibrated and metabolites can flow rapidly back into the cell. In this case, the cell is less able to avert metabolite buildup, leading again to decreased growth for cells with moderate burst sizes.
In Appendix 9—figure 10B, we show the growth rate of isolated cells as a function of burst size. As expected, the growth rate decreases with burst size. The nonmonotonic relationship between growth and permeability also manifests in these results, with the $P=1$ case having the highest growth. Next, we explore the impact of growth on multicell communities. In Appendix 9—figure 10E, we show the growth rate of tencell communities as a function of permeability. For lower burst sizes, the effect of NAC persists, with growth rate increasing with permeability. The lack of degradation actually allows NAC to be beneficial for cells with $\beta =10$, where previously they did not benefit from NAC because of losses due to degradation. Interestingly, NAC is no longer sufficient to rescue growth for burst size $\beta =100$. This is due to the aforementioned Paradox of Plenty, which makes it even more difficult for these highburstsize cells to regulate their metabolism. In Appendix 9—figure 10F we show the corresponding CVs as a function of permeability. Notably, the trends are nonmonotonic, with increased permeability initially reducing noise, but eventually leading to increased noise. The initial decrease in noise arises from the NAC mechanism of noise reduction discussed in the main text. The eventual increase in noise with high permeability is due to the metabolite buildups discussed above. Once the cells are strongly connected and their metabolite pools are in sync, they are more prone to large metabolite buildups. However, while these buildups increase noise, the linking of cells allows for increased overall metabolite production, leading to an increase in mean metabolite levels that outweighs the increase in variance.
Appendix 9
Supplemental figures
In this section, we present supplemental figures that support the main text. Each figure’s caption contains all pertinent information.
Data availability
The current manuscript is a modeling study, and thus no data was generated for this manuscript. All modeling code is available at https://github.com/jaimegelopez/NAC, (copy archived at swh:1:rev:d39206b0890340db8b4faad87ed544f345b09057).
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Decision letter

Wenying ShouReviewing Editor; University College London, United Kingdom

Naama BarkaiSenior Editor; Weizmann Institute of Science, Israel
Our editorial process produces two outputs: i) public reviews designed to be posted alongside the preprint for the benefit of readers; ii) feedback on the manuscript for the authors, including requests for revisions, shown below. We also include an acceptance summary that explains what the editors found interesting or important about the work.
Decision letter after peer review:
Thank you for submitting your article "Noisy metabolism can drive the evolution of microbial crossfeeding" for consideration by eLife. Your article has been reviewed by 3 peer reviewers, including Wenying Shou as the Reviewing Editor and Reviewer #1, and the evaluation has been overseen by Naama Barkai as the Senior Editor.
The reviewers have discussed their reviews with one another, and the Reviewing Editor has drafted this to help you prepare a revised submission.
Essential revisions:
Reviewers overall liked this article. Reviewers want to see the evolution of noiseaveraging cooperation (NAC), and the evolutionary stability of NAC against cheaters. To help revision, all reviews are attached.
Reviewer #2 (Recommendations for the authors):
1. As authors already have discussed, they have shown the optimality of metabolite sharing, not the evolution of crossfeeding. I agree that the authors nicely discussed what scenario could be feasible to evolve crossfeeding from the metabolite sharing. But still, two steps are not explicitly shown: (1) evolutionary stability of sharing metabolites (2) evolution from metabolic sharing to crossfeeding by gene deletion. I do not insist that the authors should examine them, but the current title "Noisy metabolism can drive the evolution of microbial crossfeeding" is misleading. So, it should be revised.
Because of the mismatch between the title and the model, I was confused about what crossfeeding really means. I admit that crossfeeding can happen within a single species, but the model considers identical individuals. It makes me confused that the authors consider metabolite sharing between identical individuals as crossfeeding.
2. Also, the model strongly depends on the bursty behavior of enzyme production, which makes me difficult to find which noise (or stochasticity) is necessary for NAC (or what is a fundamental mechanism of it). It seems that deterministic dynamics with bursty behavior starting from different enzyme concentrations give similar results. If the enzyme production feedback is less frequent, the gap between metabolite concentrations will increase. That induces a lower growth rate. In this case, metabolite sharing will also be optimal.
In the end, I realized that the noise in metabolite concentrations should play a fundamental role not one in the dynamics. But still, I am not sure about the fundamental mechanism, "noise" or "imbalance" of metabolite concentrations. I think the imbalance induces a lower growth rate with a smaller population size leading to a larger CV. Hence, to me, imbalance seems to be the more fundamental reason why metabolite sharing becomes optimal.
3. As I am not an expert in the Black Queen Hypothesis (BQH), it was unclear which question the authors aim to answer in the introduction either the generalization of BQH or the origin of the leakage. At the end of the reading, I found that the authors have asked the latter one. I think this confusion arises because I have no enough background in BQH. Thus, adding an explanation of BQH would help the audience to understand what the main question is.
4. I found that the initial enzyme concentrations are missing in the main text. I guess that the same initial concentrations of all enzymes are used for Figure 1C and 1D, but the authors should mention the initial conditions in the main text. Also, I wonder whether the results are robust under different initial conditions such that the different enzymes have different concentrations.
5. In lines 237~238, the term "curse of dimensionality" is often used in computer science. When the volume of the space increases so fast as the dimensionality increases, the available data becomes sparse. Thus one cannot find the statistical significance. However, in the main text, the authors have used this terminology to emphasize the situation where a large number of metabolites suppress the growth rate. I think the "curse of dimensionality" is inappropriate to be used in this situation.
6. The authors said that the correlated enzyme levels suggest the correlated metabolite levels (lines 248251). However, in Appendix7 Figure1, only the correlation between enzyme concentrations is shown without the correlation for the metabolites. It would be nice to verify this assumption by showing the correlation between metabolite concentrations in simulations.
7. As authors explained in lines 253256, the correlation between metabolites could make their concentrations even but it is not always so. When each concentration has different average values, one can have an outlying low metabolite level even with a positive correlation. It is because the correlation does not tell the metabolite level itself. I think the results in Figure 3D are obtained at the same fixed average metabolite concentrations. If so, please clarify this.
8. In Equation (3) and (4), noise strength is not given in the main text.
9. Typo in line 53: that > that
Reviewer #3 (Recommendations for the authors):
I enjoyed reading the manuscript: the NAC framework is an interesting new take on the evolution of crossfeeding and the manuscript is well written and organized. However, I have some suggestions for improving the presentation of the model.
1. The relation between NAC and BQH should be rephrased in more neutral terms (see public review).
2. It would be important to also mention the economies of scale hypothesis on crossfeeding evolution (see public review). I suggest briefly mentioning it in the intro and discussing how it relates to NAC in more detail in the discussion. Much of this work has been done in the context of aminoacid crossfeeding (see work by Christian Kost) and this would, e.g., be an appropriate place to discuss this.
3. In the absence of an evolutionary model, I think it is essential to phrase any statements about evolutionary dynamics more carefully and make clear to the reader that additional work is needed to confirm these hypotheses (see public review).
4. As mentioned in the public review I think some of the model assumptions have debatable biological rationale. It would be important to justify why these assumptions were made and to discuss how they affect the main conclusions. Additional simulations to explore the sensitivity of the model to these assumptions could be helpful, but are not necessary, provided that the authors can give an intuitive understanding of how these assumptions affect their conclusions.
5. In lines 142144 and 185186 the authors talk about the role of extracellular volume/cell density; however, no explanation is given of how these results were derived. I think these statements need to be supported by either a SI figure or by an intuitive explanation of how r_{v} impacts model predictions.
6. In the analytical analysis presented in lines 187205 the authors make use of various additional simplifying assumptions. Without additional details it is very hard to judge to what extend they affect the presented results. I think some additional discussion on the rational/effect of these assumptions would be highly beneficial to the reader (alternatively one could add a SI figure to e.g. compare the predictions of Equation 5 with data from simulations).
https://doi.org/10.7554/eLife.70694.sa1Author response
Reviewer #2 (Recommendations for the authors):
1. As authors already have discussed, they have shown the optimality of metabolite sharing, not the evolution of crossfeeding. I agree that the authors nicely discussed what scenario could be feasible to evolve crossfeeding from the metabolite sharing. But still, two steps are not explicitly shown: (1) evolutionary stability of sharing metabolites (2) evolution from metabolic sharing to crossfeeding by gene deletion. I do not insist that the authors should examine them, but the current title "Noisy metabolism can drive the evolution of microbial crossfeeding" is misleading. So, it should be revised.
Because of the mismatch between the title and the model, I was confused about what crossfeeding really means. I admit that crossfeeding can happen within a single species, but the model considers identical individuals. It makes me confused that the authors consider metabolite sharing between identical individuals as crossfeeding.
We thank the reviewer for pointing out that the current title is misleading. We have now changed the title to “Noisy metabolism can promote microbial crossfeeding”.
We agree with the reviewer that our definition of crossfeeding was unclear. We do not consider sharing among identical individuals to be crossfeeding. Instead, we consider this to be a form of cooperation that can eventually lead to crossfeeding when one of the individuals loses its capacity to produce one of the metabolites (thus making the individuals nonidentical). We have now added clarifying statements to this effect.
2. Also, the model strongly depends on the bursty behavior of enzyme production, which makes me difficult to find which noise (or stochasticity) is necessary for NAC (or what is a fundamental mechanism of it). It seems that deterministic dynamics with bursty behavior starting from different enzyme concentrations give similar results. If the enzyme production feedback is less frequent, the gap between metabolite concentrations will increase. That induces a lower growth rate. In this case, metabolite sharing will also be optimal.
In the end, I realized that the noise in metabolite concentrations should play a fundamental role not one in the dynamics. But still, I am not sure about the fundamental mechanism, "noise" or "imbalance" of metabolite concentrations. I think the imbalance induces a lower growth rate with a smaller population size leading to a larger CV. Hence, to me, imbalance seems to be the more fundamental reason why metabolite sharing becomes optimal.
On the necessity of stochasticity: The reviewer is correct that stochasticity is not necessary for metabolite sharing to be optimal. For NAC to be potentially beneficial, there are two requirements: the metabolite levels must fluctuate from the mean, and the growth function must be concave as a function of metabolite levels. In principle, fluctuations originating from deterministic dynamics could also result in a drop in mean growth rate (for example, due to persistent oscillations around the mean). In our analysis of experimental data, we implicitly refer to all variation from the mean as “noise”, but the underlying mathematics are agnostic to the origin of the variation, i.e. the origin of the variation could be stochastic, deterministic, or some combination. We have now added sentences noting this to the relevant section.
On the necessity of imbalance: In our model, the fundamental mechanism underlying the optimality of metabolite sharing is variable metabolite concentrations coupled to a concave growth function. In general, there is no requirement for an “imbalance” per se of metabolite concentrations. Even with a single metabolite, NAC can still be beneficial. In the particular model we analyze, the requirement for imbalance arises from a feature of Liebig’s Law of the Minimum and is actually a proxy for the concavity requirement. Liebig’s Law of the Minimum is concave for only two or more metabolites. In the case of a single metabolite, it reduces to a linear function that is insensitive to noise because it is neither convex nor concave. To better explain the conditions necessary for noise to be detrimental to growth, we have now added an additional appendix “Conditions for noisedriven growth reduction”.
3. As I am not an expert in the Black Queen Hypothesis (BQH), it was unclear which question the authors aim to answer in the introduction either the generalization of BQH or the origin of the leakage. At the end of the reading, I found that the authors have asked the latter one. I think this confusion arises because I have no enough background in BQH. Thus, adding an explanation of BQH would help the audience to understand what the main question is.
We thank the reviewer for pointing out this opportunity to improve the clarity of our introduction. We have now restructured the introduction to better explain the manuscript’s goals and how these relate to the BQH. The evolution of crossfeeding requires the emergence of two types of organisms: the organism secreting the crossfed compound and the organism consuming the compound. The BQH provides a mechanism for the evolution of the consuming organism, but leaves open the question of how the leaking organism arises. Our goal is to develop a theory complementary to the BQH that explains the origins of the leaking organism, thus filling a gap in the overall theory of crossfeeding.
4. I found that the initial enzyme concentrations are missing in the main text. I guess that the same initial concentrations of all enzymes are used for Figure 1C and 1D, but the authors should mention the initial conditions in the main text. Also, I wonder whether the results are robust under different initial conditions such that the different enzymes have different concentrations.
We thank the reviewer for pointing out this omission. We now report the initial conditions used in our simulations in the “Hybrid numerical methods” appendix. We also now explore the impact of initial conditions on our simulations in a new SI figure (Appendix 9 – figure 7).
In this figure, we compute the mean correlation between enzyme levels and the simulation state at a later time. We find that the system forgets its initial conditions rapidly, on the timescale of ~5 time units (time units normalized to the inverse of the maximum steadystate growth). The simulations reported in the manuscript are run for 5000 of these time units and statistics are computed from only the final ¾ of the growth timecourse. Thus, we are confident initial conditions do not impact our results.
5. In lines 237~238, the term "curse of dimensionality" is often used in computer science. When the volume of the space increases so fast as the dimensionality increases, the available data becomes sparse. Thus one cannot find the statistical significance. However, in the main text, the authors have used this terminology to emphasize the situation where a large number of metabolites suppress the growth rate. I think the "curse of dimensionality" is inappropriate to be used in this situation.
We thank the reviewer for pointing out that our use of this term may be confusing. We have now removed mention of the term.
6. The authors said that the correlated enzyme levels suggest the correlated metabolite levels (lines 248251). However, in Appendix7 Figure1, only the correlation between enzyme concentrations is shown without the correlation for the metabolites. It would be nice to verify this assumption by showing the correlation between metabolite concentrations in simulations.
We now report the correlation coefficients between the metabolites in the relevant figure caption.
7. As authors explained in lines 253256, the correlation between metabolites could make their concentrations even but it is not always so. When each concentration has different average values, one can have an outlying low metabolite level even with a positive correlation. It is because the correlation does not tell the metabolite level itself. I think the results in Figure 3D are obtained at the same fixed average metabolite concentrations. If so, please clarify this.
We thank the reviewer for pointing out this lack of clarity. Figure 3D is indeed generated assuming all metabolites on average are colimiting. The reviewer is correct that the benefit of correlated metabolites is strongest when the metabolites are on average corate limiting. If one metabolite is on average more limiting, even perfect metabolite correlation will not lead to the optimal growth rate. We have now added additional text to the “Generalized model framework” section to clarify these issues.
8. In Equation (3) and (4), noise strength is not given in the main text.
We now report the noise strengths in the main text.
9. typo in line 53: that > that
We have now fixed this typo.
Reviewer #3 (Recommendations for the authors):
I enjoyed reading the manuscript: the NAC framework is an interesting new take on the evolution of crossfeeding and the manuscript is well written and organized. However, I have some suggestions for improving the presentation of the model.
1. The relation between NAC and BQH should be rephrased in more neutral terms (see public review).
We thank the reviewer for pointing out this shortcoming in our discussion of the BQH, as it has helped us better put our study in the appropriate context. We agree with the reviewer that the BQH and NAC should in fact be regarded as complementary. We have now revised our introduction to clarify that NAC and BQH are complementary, and we now address the issue of noise being an evolvable trait in the discussion. We also now distinguish the BQH from the assumption of “inevitable” leakiness, as indeed the BQH is agnostic to the source of the leakiness. Drawing from Reviewer #1’s comments, we also have noted another fundamental difference between NAC and the BQH that makes the two complementary: the BQH focuses on the emergence of consumers, while NAC focuses on the emergence of leakers.
2. It would be important to also mention the economies of scale hypothesis on crossfeeding evolution (see public review). I suggest briefly mentioning it in the intro and discussing how it relates to NAC in more detail in the discussion. Much of this work has been done in the context of aminoacid crossfeeding (see work by Christian Kost) and this would, e.g., be an appropriate place to discuss this.
We have now integrated the economies of scale hypothesis into the manuscript. It is now mentioned in the introduction and elaborated on further in the discussion. We found comparisons between the economies of scale hypothesis and NAC particularly useful in our paragraphs on identifying NAC in natural systems. These comparisons allowed us to further sharpen our experimental and observational predictions by outlining cases in which these two hypotheses would be distinguishable.
3. In the absence of an evolutionary model, I think it is essential to phrase any statements about evolutionary dynamics more carefully and make clear to the reader that additional work is needed to confirm these hypotheses (see public review).
We agree. Therefore, we have now rephrased our mentions of evolutionary dynamics to make it clear that the models we analyze are ecological in nature, and by themselves do not allow us to reach evolutionary conclusions. We have restructured the discussion to prioritize discussion of evolutionary dynamics, including appropriate references and connections to the existing literature on the evolution of cooperation.
4. As mentioned in the public review I think some of the model assumptions have debatable biological rationale. It would be important to justify why these assumptions were made and to discuss how they affect the main conclusions. Additional simulations to explore the sensitivity of the model to these assumptions could be helpful, but are not necessary, provided that the authors can give an intuitive understanding of how these assumptions affect their conclusions.
We thank the reviewer for pointing out that these assumptions merit additional discussion. We address each of these two assumptions below:
Growth feedback on enzyme production: This assumption was added to reflect the fact that, on average, all cellular components must increase at the same rate as growth (otherwise cellular component concentrations become progressively imbalanced over time). However, even in the absence of this feedback, the model can still exhibit noiseaveraging cooperation. This can be seen in our analysis of the simplified Langevin equation model. The primary difference between the Langevin model and the full metabolism model is the removal of the growth feedback. We have now added an SI figure (Appendix 9 – figure 4) comparing the two models. In large communities and at small burst sizes, the two models are nearly identical. The two models differ more in very small communities and when burst sizes are very large, with the full model predicting somewhat higher levels of metabolite noise. This occurs because the growth feedback makes it more difficult for a “struggling” cell to correct its metabolite imbalances. An imbalanced cell has a lower growth rate, which makes it less likely to produce a corrective enzyme burst, thus leading to overall more persistent imbalances and increasing noise.
Degradation of metabolites: This assumption was included in the model to prevent unrealistic buildups of metabolites within the cell. Without this assumption, a cell with poor regulation will build up massive, nonphysical pools of one metabolite which it can later use. This does not reflect the reality of bacteria, which have a finite osmotic capacity and will release excess osmolytes (see Buda 2016 in PNAS). In addition, there also exist solubility limits that constrain metabolite storage. The assumption of degradation in the extracellular space was added to reflect the fact that in natural environments there may be other organisms present that consume free metabolites, or metabolites may simply diffuse away in 3D.
However, we agree that it would be informative to examine a model without degradation. We therefore explored such a model and present the results in a new appendix called “Metabolism models without metabolite degradation”. Many core behaviors of the model, such as the negative effect of burst size and the benefit of resource sharing, remain. However, the lack of degradation also introduces a counterintuitive effect into the model: more metabolite storage can actually lead to an increased probability of death. This increased susceptibility to death is a consequence of large metabolite buildups leading to sudden metabolite “crashes” that can kill the cell. We show the metabolite and enzyme timecourses of a cell undergoing this crash behavior in Appendix 9 – figure 10CD. At around t = 15, the cell develops an excess of the blue metabolite with a deficit of the pink metabolite, limiting its growth. In the time it takes the organism to produce a new burst of the pink enzyme, it continually builds up its pool of the blue metabolite, as there is no degradation term to limit the pool’s size. Eventually, at around t = 35, the cell produces a burst of the pink enzyme, generating more pink metabolite and allowing a resumption of steady growth. Since there is such a large buildup of the blue metabolite, it takes the cell a long time to deplete this metabolite pool. In this time, the cell produces many bursts of the pink enzyme, almost entirely replacing the cell's enzyme pool and leaving the cell with almost no blue enzyme. At around t = 55, the blue metabolite crashes, rapidly becoming the new limiting metabolite. At this point, the cell is left in a difficult position, suddenly having almost none of the now limiting blue metabolite and having an enzyme pool entirely dedicated to production of the pink metabolite. Growth rapidly ceases, as the remaining blue metabolite pool is insufficient to produce a corrective enzyme burst. Thus, paradoxically, allowing microbes to store larger pools of metabolites by removing degradation can actually make them more likely to die.
5. In lines 142144 and 185186 the authors talk about the role of extracellular volume/cell density; however, no explanation is given of how these results were derived. I think these statements need to be supported by either a SI figure or by an intuitive explanation of how r_{v} impacts model predictions.
We have now added a new SI figure (Appendix 9 – Figure 3) with simulations showing the effect of extracellular volume on growth rates. We provide more detailed explanations of the behavior and also include a comparison of the simulations with the analytically predicted maximum growth rate at infinite extracellular volume, showing they are in agreement.
6. In the analytical analysis presented in lines 187205 the authors make use of various additional simplifying assumptions. Without additional details it is very hard to judge to what extend they affect the presented results. I think some additional discussion on the rational/effect of these assumptions would be highly beneficial to the reader (alternatively one could add a SI figure to e.g. compare the predictions of Equation 5 with data from simulations).
We have now added an SI figure (Appendix 9 – figure 4) comparing the predictions of the full model with the Langevin model. We have also modified the initial description of the Langevin model to make it clear that the primary modification is the removal of the nonlinear feedback between growth and enzyme dynamics.
https://doi.org/10.7554/eLife.70694.sa2Article and author information
Author details
Funding
National Science Foundation (DGE1656466)
 Jaime G Lopez
National Institutes of Health (R01 GM082938)
 Ned S Wingreen
The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.
Acknowledgements
We thank Matt Black for useful discussions on possible experimental tests of the theory. JGL was supported by an NSF GRFP (DGE1656466). This work was supported by the National Institutes of Health (R01 GM082938). This work was supported in part by the National Science Foundation, through the Center for the Physics of Biological Function (PHY1734030).
Senior Editor
 Naama Barkai, Weizmann Institute of Science, Israel
Reviewing Editor
 Wenying Shou, University College London, United Kingdom
Publication history
 Received: May 26, 2021
 Preprint posted: June 2, 2021 (view preprint)
 Accepted: February 21, 2022
 Version of Record published: April 5, 2022 (version 1)
Copyright
© 2022, Lopez and Wingreen
This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.
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