Particle foraging strategies promote microbial diversity in marine environments
Abstract
Microbial foraging in patchy environments, where resources are fragmented into particles or pockets embedded in a large matrix, plays a key role in natural environments. In the oceans and freshwater systems, particleassociated bacteria can interact with particle surfaces in different ways: some colonize only during short transients, while others form longlived, stable colonies. We do not yet understand the ecological mechanisms by which both short and longterm colonizers can coexist. Here, we address this problem with a mathematical model that explains how marine populations with different detachment rates from particles can stably coexist. In our model, populations grow only while on particles, but also face the increased risk of mortality by predation and sinking. Key to coexistence is the idea that detachment from particles modulates both net growth and mortality, but in opposite directions, creating a tradeoff between them. While slowdetaching populations show the highest growth return (i.e., produce more net offspring), they are more susceptible to suffer higher rates of mortality than fastdetaching populations. Surprisingly, fluctuating environments, manifesting as blooms of particles (favoring growth) and predators (favoring mortality) significantly expand the likelihood that populations with different detachment rates can coexist. Our study shows how the spatial ecology of microbes in the ocean can lead to a predictable diversification of foraging strategies and the coexistence of multiple taxa on a single growthlimiting resource.
Editor's evaluation
This manuscript tackles an underexplored area in understanding microbial coexistence in marine and aquatic environments. This manuscript adds to the recently renewed interest on applications of optimal foraging theory to the study of microbial growth on marine snow.
https://doi.org/10.7554/eLife.73948.sa0Introduction
Microbes in nature are remarkably diverse, with thousands of species coexisting in any few milliliters of seawater or grains of soils (Azam and Malfatti, 2007; Young and Crawford, 2004). This extreme diversity is puzzling since it conflicts with classic ecological predictions. This puzzle has classically been termed ‘the paradox of the plankton’, referring to the discrepancy between the measured diversity of planktons in the ocean, and the diversity expected based on the number of limiting nutrients (Ghilarov, 1984; Shoresh et al., 2008; Hutchinson, 1961; Goyal and Maslov, 2018). Decades of work have helped, in part, to provide solutions for this paradox in the context of freeliving (i.e., planktonic) microbes in the ocean. Many have suggested new sources of diversity, such as spatiotemporal variability, microbial interactions, and grazing (RodriguezValera et al., 2009; Muscarella et al., 2019; Saleem et al., 2013). However, in contrast with freeliving microbes, the diversity of particleassociated microbes — often an order of magnitude greater than freeliving ones — has been overlooked (Milici et al., 2017; Ganesh et al., 2014; Crespo et al., 2013). In contrast with planktonic bacteria, which float freely in the ocean and consume nutrients from dissolved organic matter, particleassociated microbes grow by attaching to and consuming small fragments of particulate organic matter (POM) (of the order of micrometers to millimeters). It is thus instructive to ask: what factors contribute to the observed diversity of particleassociated microbes, and how do these factors collectively influence the coexistence of particleassociated microbes?
The dispersal strategies of particleassociated microbes can be effectively condensed into one parameter: the rate at which they detach from particles. This rate, which is the inverse of the average time that microbes spend on a particle, is the key trait distinguishing particleassociated microbial populations from planktonic ones Yawata et al., 2020; Fernandez et al., 2019. The detachment rates of such particleassociated taxa can be quite variable (Grossart et al., 2003; Yawata et al., 2014). Bacteria with low detachment rates form biofilms on particles for efficient exploitation of the resources locally, while others with high detachment rates frequently attach and detach across many different particles to access new resources (Ebrahimi et al., 2019). Therefore, to understand how diversity is maintained in particleassociated bacteria we must be able to explain how bacteria with different dispersal rates can coexist. In this study, we address this question. Specifically, we ask how two populations with different dispersal strategies can coexist while competing for the same set of particles, under a range of conditions relevant for marine microbes.
We hypothesize that dispersal is key to the coexistence of particleassociated microbes and thus might explain their high diversity. The degree of species coexistence on particles depends on the balance between growth and mortality. On particles, net mortality rates can be higher than for planktonic cells because of the large congregation of cells on particles, which exposes them to the possibility of a large and sudden local population collapse. The collapse of a particleattached population can be induced by a variety of mechanisms, including particles sinking below a habitable zone (Boeuf et al., 2019), or predation of whole bacterial colonies by viruses or grazers. For instance, after a lytic phage bursts out of a few cells on a particle, virions can rapidly engulf the entire bacterial population, leading to its local demise (Ganesh et al., 2014; Dupont et al., 2015; LópezPérez et al., 2016). Such particlewide mortality may kill more than 30% of particleassociated populations in the ocean (Proctor and Fuhrman, 1991; Weinbauer et al., 2009). The longer a population stays on a particle, the higher the chance it will be wiped out. This tradeoff between growth and risk of mortality suggests that there could be an optimal residence time on particles. It is however unclear whether sucha tradeoff could enable the coexistence of populations with different dispersal strategies and, if so, under what conditions.
Here, we study this tradeoff using mathematical models and stochastic simulations. These models reveal that the tradeoff between growth and survival against predation can indeed lead to the stable coexistence of particleassociated microbial populations with different dispersal strategies (in our work, detachment rates). We also study how environmental parameters, such as the supply rate of new particles, determine the dominant dispersal strategy and the range of stable coexistence. Our results show that in bloom conditions, when the particle supply is high, fast dispersers that rapidly hop between particles are favored. In contrast, under oligotrophic conditions, when particles are rare, rarely detaching bacteria have a competitive advantage. Overall, our work shows that differences in dispersal strategies alone can enable the coexistence of particleassociated marine bacteria, in part explaining their impressive natural diversity.
Results
Overview of the model
To understand how differences in dispersal strategies affect bacterial coexistence, we developed a mathematical model that describes the population dynamics of bacteria colonizing a bath of particles with a chosen dispersal strategy. More specifically, in our model, bacterial cells attach to particles from a freeliving population in the bulk of the bath; they then grow and reproduce while attached. Detachment is stochastic with a fixed rate. After detachment, cells reenter the freeliving population and repeat the process. During the time spent attached to particles, all bacteria on a particle may die with a fixed probability per unit time, corresponding to their particlewide mortality rate (Figure 1A). Another important feature of the model is densitydependent growth, which means that per capita growth rates decrease with increasing population size. For this, we use the classic logistic growth equation, which contains a simple linear density dependence (Figure 1B; Methods). Freeliving subpopulations cannot grow, but die at a fixed mortality rate due to starvation. The probability of a bacterium encountering particles controls bacterial attachment, which we calculate using random walk theory as the hitting probability of two objects with defined sizes (Leventhal et al., 2019; Frazier and Alber, 2012; see Methods for details). We assume that the detachment rate is an intrinsic property of a bacterial population and comprises its dispersal strategy independent of the abiotic environment. In our simulations, it is the only trait that varies between different bacterial populations. Growing evidence has shown that bacterial detachment rates differ significantly across marine bacterial communities from solely planktonic cells to biofilmforming cells on particles (Yawata et al., 2014; Ebrahimi et al., 2019). Using this mathematical model, we asked how variation in detachment rate affects bacterial growth dynamics and the ability of multiple subpopulations to coexist on particles. For this, we simulated bacterial population dynamics on a bath of several particles and measured each population’s relative abundance at a steady state (example in Figure 1C).
Bacterial mortality determines optimal foraging strategies
Our model simulates growth, competition, and dispersal in a patchy landscape, similar to classical models of resource foraging, with the additional element of mortality, both within and outside patches (i.e., particles). We hypothesized that the inclusion of mortality could play an important role in affecting the success of a dispersal strategy (i.e., detachment rate), since it would alter the cost of staying on a particle. To investigate how mortality affects dispersal strategies, we studied its effect on the optimal strategy, which forms the focus of many classical models of foraging. According to optimal foraging theory (OFT), the optimal time spent on a particle is one that balances the time spent without food while searching for a new patch, with the diminishing returns from staying on a continuously depleting patch (Yawata et al., 2020; Charnov, 1976). In our model, particles are analogous to resource patches, and the detachment rate is simply the inverse of the time spent on a particle (residence time). We assumed that the optimal strategy maximizes the total biomass yield of the population.
As expected, OFT predicts the optimal detachment rate given a distribution of resources and search times, but only in the absence of mortality (Figure 2A). To test if our model agrees with the predictions of OFT, we calculated the optimal detachment rate (d_{opt}) using simulations of our model in the absence of mortality and compared it with OFT predictions (Methods). We found that the optimal detachment rate, which outcompetes all other detachment rates, was consistent with OFT predictions across a wide range of particle numbers in our system (Figure 2A). Strikingly, in the presence of mortality, the optimal detachment rate (d_{opt}) changed significantly, either increasing or decreasing depending on the type of mortality. When mortality was particlewide, the optimal detachment rate was much higher than predicted by OFT, often resulting in residence times that were many days shorter than the OFT prediction (Figure 2A). This is because it is more beneficial to detach faster when there is a higher risk of particlewide extinction. In contrast, when mortality was only present in freeliving populations (affecting individuals, not particles, at a constant per capita rate), the optimal detachment rate was much lower than predicted by OFT (Figure 2A). These results expand on our knowledge of OFT and explain that the source and strength of mortality – on individuals or on whole particles – can differently impact the optimal detachment rate.
A tradeoff between growth and mortality enables the coexistence of dispersal strategies
Having observed that mortality can greatly affect the success of a dispersal strategy, we next sought to understand whether it could enable the coexistence of bacterial populations with different strategies (detachment rates). Simulations where we competed a pair of bacterial populations with different detachment rates revealed that differences in detachment rates alone are sufficient to enable coexistence on particles (Figure 2B). We assessed coexistence by measuring the relative abundances of populations at equilibrium (Figure 2—figure supplement 1). Interestingly, such a nontrivial coexistence only emerged in the presence of particlewide mortality. In the absence of mortality on particles, we only observed trivial coexistence (coexisting populations had identical detachment rates, and for the purposes of the model, were one and the same; Figure 2—figure supplement 2). These results suggested that the presence of particlewide mortality, where the entire population on a particle suffers rapid death, was crucial for populations with different dispersal strategies to coexist.
To investigate the underlying mechanisms that may give rise to the coexistence of populations with different detachment rates, we quantified the growth return of particleassociated populations as well as their survival rate on particles (Figure 3A, B). We calculated the average growth return based on the average number of offspring produced per capita during one single attachment–detachment event. The survival rate on particles was obtained by subtracting the mortality rate per capita from the offspring production rate per capita (Figure 3B; see Methods). The results revealed that a tradeoff between bacterial growth return and survival rate emerged on particles, supporting the coexistence of populations with different detachment rates (Figure 3C, D). Populations that detach slowly from particles have higher growth returns but are also more susceptible to particleassociated mortality. In contrast, populations with low residence time on particles (high detachment rate) have low growth returns but they are less likely to die by predation or sink beyond the habitable zone. We next investigated whether such a tradeoff was necessary to enable coexistence in our model.
We developed a coarsegrained model to address the conditions under which we might observe coexistence between populations whose only intrinsic difference was their detachment rates in our system. Our simple model expands on classical literature which describes coexistence among various dispersal strategies in spatially structured habitats (Levin and Paine, 1974; Kneitel and Chase, 2004; Tilman, 1994; Amarasekare, 2003). We simplified many details in favor of analytical tractability. Chiefly, we assumed that the growth dynamics on each particle were much faster than the dispersal dynamics across particles. This allowed us to replace detailed growth dynamics on single particles with a single number quantifying the bacterial population, N, after growth on each particle. In the model, we considered two particleassociated populations that competed for a shared pool of particles. To keep track of populations, we quantified the number of particles they had successfully colonized as B_{1} and B_{2}, respectively. Individuals from both populations could detach from particles they had already colonized and migrate toa number E of yetunoccupied particles, with a rate proportional to their detachment rates, d_{1} and d_{2}, respectively. Once migrated, individuals rapidly grew on unoccupied particles to their fixed per particle growth returns, N_{1} and N_{2}. To model particlewide mortality, we assumed a fixed per particle mortality rate, $m}_{p$. The population dynamics for the system of particles could therefore be written as follows:
At equilibrium ($\frac{d{B}_{i}}{dt}=0\text{}\mathrm{\forall}\text{}i$), either population can survive in the system if and only if its net colonization and mortality rates are equal (${N}_{i}{d}_{i}E\approx {m}_{p}$). Consequently, the product of the growth return per particle and the detachment rate of either population should be equal (${N}_{1}{d}_{1}\approx {N}_{2}{d}_{2}$). By simplifying Equation 1 at equilibrium, this model predicts that for two competing populations to coexist, their growth returns and detachment rates on particles must follow the relation:
This relationship shows that coexistence demands a tradeoff between the growth return (N) of a bacterial population, and its detachment rate (d), that is, the inverse of an individual’s residence time on a particle. In other words, coexistence only emerges when the growth returns increase with the residence time on the particle ($\frac{{N}_{1}}{{N}_{2}}~\frac{{T}_{1}}{{T}_{2}}$). In agreement with this, simulations from our detailed model revealed that coexistence between two populations with different detachment rates only occurred in conditions where the two populations obeyed such a relationship, or tradeoff (Figure 3C, gray region). We obtain the same relationship in Equation 2 through an alternate calculation, where the relative abundances of both populations remain fixed, while the particle number varies.
While the tradeoff in Equation 2 allows coexistence and is necessary condition for it, it does not strictly hold across all parameter values, and hence prevents certain pairs of detachment rates to coexist (Figure 3C, white region). In particular, no detachment rate can coexist with the optimal detachment rate, thus rendering coexistence between any other set of detachment rates susceptible to invasion by this optimal strategy. Other strategies, when paired with the optimal strategy, disobey the condition in Equation 2, and thus cannot coexist with it. Therefore, if detachment rates were allowed to evolve, only one population would survive in the long run – the one with the optimal detachment rate (Figure 2—figure supplement 3). Motivated by this observation, we next asked whether environmental fluctuations would render coexistence evolutionarily stable, or whether they would further destabilize the coexistence of populations with nonoptimal dispersal strategies.
Environmental fluctuations stabilize and enhance the diversity of dispersal strategies
The existence of a unique optimal strategy, even in the presence of particlewide mortality (Figure 2A), suggests that the coexistence that we observed between populations with different detachment rates (Figure 2B) may not be evolutionarily stable. However, in the oceans, both the abundance of particles and the density of predators (such as phage) exhibit temporal and spatial fluctuations (Nilsson et al., 2019; GarinFernandez et al., 2018; Luo et al., 2017), in turn affecting the foraging dynamics of particleassociated bacterial populations. We used our model to study how the particlewide mortality rate affects the likelihood of two particleassociated bacterial populations to coexist (see Methods). Surprisingly, we found a negative correlation between the mortality rate and particle abundance that enhances the range of coexistence among different detachment rates (Figure 4A). At low mortality rates, slowdetaching populations outcompete faster ones, as it is more advantageous to stay longer on particles and grow, that is, these populations derive higher net growth returns. However, a higher mortality rate on particles allows fasterdetaching populations to instead gain an advantage over the slowdetaching populations, since they can better avoid particlewide mortality events.
We extended our model to ask how variation in the total number of particles (or particle abundance) affect population dynamics and the coexistence range of populations with different dispersal strategies. The results indicated that an intermediate number of particles maximize the likelihood of coexistence of two populations with different dispersal strategies (Figure 4A). Here, we simulated a range of particle abundances, between 1 and 80 particles L^{−1}, which corresponds to the commonly observed range of particle abundances in aquatic environments (mean ~25 particles L^{−1}; Figure 4—figure supplement 1). Low particle abundances (0–20 L^{−1}) promote the growth of slowdetaching populations while at high particle abundances, fastdetaching populations dominate. The reason for this is the following: at particle abundances less than 20 L^{−1}, the probability of freeliving cells finding and attaching to particles is less than 50% of the probability at high particle abundances (100 L^{−1} in Figure 1—figure supplement 1). This makes particle search times very high, thus explaining how slowdetaching strategies have an advantage. As the number of particles increases, the entire system can support more cells (has a higher carrying capacity). This drives a decrease in particle search times, and thus increasingly advantages faster detaching strategies.
Interestingly, our results indicate that the optimal detachment rate (d_{opt}) is affected by the particle abundance and increases with the number of particles in the system (Figure 4B). We thus hypothesized that fluctuations in particle abundance may also induce fluctuations in the optimal detachment rate, such that no specific detachment rate would be uniquely favored at all times. Thus, environmental stochasticity would continuously change the optimal detachment rate; low particle abundances would favor fastdetaching populations, while higher particle abundances would favor slowdetaching populations. Such a ‘fluctuating optimum’ may create temporal niches and promote higher bacterial diversity on marine particles. To test this hypothesis using our model, we simulated competition between 100 populations with different detachment rates under a periodically varying particle abundance (Figure 4C). The chosen frequencies of variation in particle abundance (F_{p}) were selected to be consistent with the observed frequencies in the ocean, with periods ranging between 10 and 100 hr (Figure 4—figure supplement 2; Lampitt et al., 1993). We quantified the range of detachment rates, a proxy for bacterial diversity, that could coexistat equilibrium (Figure 4D). The results revealed that the scenario with fastest fluctuations in particle numbers (F_{p} = 10 hr^{−1}) supported higher diversity among populations with different detachment rates (Figure 4D). Consistent with the fluctuation periods observed in the ocean, our simulations showed that fluctuation at the daily scale is sufficient to support the coexistence of different dispersal strategies. Overall, our model provides a framework to study how environmental fluctuations contribute to observed diversity in the dispersal strategies of particleassociated populations in marine environments.
Discussion
In this study, we have shown a mechanism by which diverse dispersal strategies can coexist among bacterial populations that colonize and degrade POM in marine environments using a mathematical model. In our model, coexistence among populations with different dispersal strategies emerges from a tradeoff between growth return and the probability of survival on particles. Such a tradeoff determines the net number of detaching cells from particles that disperse into the bulk environment and colonize new particles. While slowdetaching populations are able to increase their growth return on particles and produce a relatively high number of offspring, they also experience higher mortality on particles that reduces their ability to colonize new particles. In contrast, fasterdetaching populations can better avoid mortality by spending less time on particles, but this comes at the expense of lowering their growth return on a particle. Such populations can instead disperse and colonize a larger fraction of fresh yetunoccupied particles. Interestingly, our results indicated that in the absence of mortality on particles, no coexistence is expected and there is a single dispersal strategy that provides the highest fitness advantage over dispersing populations, indicating that mortality on a particle is a key factor for the emergence of diverse dispersal strategies. Such correlated mortality with dispersal is the direct result of spatial structures created by particleassociated lifestyle, unlike the planktonic phase where predation probability per capita is expected to be uniform among planktonic cells. This study expands on the existing knowledge that spatial structure plays a critical role in promoting bacterial diversity in nature (Eckburg et al., 2005; Zhang et al., 2014), by incorporating the idea of particlewide predation, which are events of correlated predation of an entire population on a particle. Such correlated predation could be an ecologically relevant mechanism that explains, in part, why we observe a higher diversity in particleassociated bacteria than planktonic bacteria in nature (Milici et al., 2017; Ganesh et al., 2014; Crespo et al., 2013). Our model assumes a general form of predation on particles that is insensitive to population type. However in the context of viral infection, field observations often show high strain specificity Holmfeldt et al., 2007; Roux et al., 2012; Suttle and Chan, 1994; Suttle, 2005 that is likely to contribute to higher diversity in particleassociated populations. Viral infection act as a driving force to create a continuous succession of bacterial populations on particles by replacing phage exposed populations with less susceptible ones.
Consistent with the literature on OFT (Fernandez et al., 2019; Taylor and Stocker, 2012; Vetter et al., 1998), our model predicts the existence of an optimal foraging strategy for bacterial population colonizing particles in marine environments. Building on previous studies (e.g., Yawata et al., 2020) that show the optimal detachment rate is a function of search time for new resources, our study suggests that optimal detachment rate could be significantly impacted by the predation rate on particles. Our results indicated that a high mortality rate on particles shifts the optimal foraging strategy to populations with fast detachment rates. This finding agrees with previous OFT models that considered mortality, showing that optimal foraging effort and residence time on patches decrease significantly as the density of predators increase (Newman, 1991; Abrams, 1993). Interestingly, we showed that the variability in optimal detachment rate, due to environmental fluctuations in particle number and predation rate, could lead to evolutionarily stable coexistence among diverse dispersal strategies. Our results indicate that in the absence of any environmental fluctuations, there is a unique optimal dispersal strategy. However, the optimal dispersal strategy depends on the abundance of particles, and thus fluctuations in their abundance at ecological timescales could sustain multiple dispersal strategies for long times. This finding is consistent with previous theoretical and empirical studies showing that environmental fluctuations such as light and temperature may lead to the stable coexistence of species (Litchman, 2003; Li and Chesson, 2016; Catorci et al., 2017; Sousa, 1979). Our model also predicts a loss of diversity when particle abundances significantly increase, consistent with field observations from algal blooms (Teeling et al., 2012; West et al., 2008; Wemheuer et al., 2014).
While we simplified bacterial colonization dynamics on particles by only considering competitive growth kinetics, variants of our model suggest that coexistence between different dispersal strategies is also expected under more complex microbial interactions observed on marine particles, including cooperative growth dynamics (Figure 2—figure supplement 4). Such simplifications allowed us to explore the role of dispersal in maintaining microbial diversity in natural systems, in addition to previously observed factors such as metabolic interaction, resource heterogeneities, and succession (Muscarella et al., 2019; Datta et al., 2016; Dal Bello et al., 2021). However, future studies, which can build on our model, could study how additional ecological factors contribute to bacterial marine diversity, such as complex trophic interactions leading to successional dynamics (Boeuf et al., 2019; Datta et al., 2016; Lauro et al., 2009; PascualGarcía et al., 2021). Additionally, while we assumed diffusional searching for simplicity, extensions of our work could include more realistic bacterial search strategies, such as active motility and chemotaxis, which can play a big role in foraging in aquatic microorganisms (Son et al., 2016; Stocker et al., 2008). Finally, though we assumed a fixed detachment rate for each population, dispersal strategies can be quite complex, depending on local conditions such as bacterial and nutrient density on particles; a more thorough exploration of the relative costs and benefits of such myriad of dispersal strategies remains another promising avenue for future work. Overall, our model provides a reliable framework to further study how diverse dispersal strategies and mortality could contribute to the emergence of complex community dynamics on marine particles and how environmental factors impact microbial processes in regulating POM turnover at the ecosystem level.
Materials and methods
In this study, a populationbased model is developed that represents the interactions between the bacterial cells with different detachment rates and particles in a chemostat system, where the total number of particles is kept constant. The following provides a detailed procedure of the modeling steps as represented schematically in Figure 1. We have made the simulation code available in the following GitHub repository: https://github.com/aliebmitedu/Bacterialdispersalmodel.
Modeling population dynamics on particles
Request a detailed protocolOur model simulates the dynamics of two competing particleassociated populations (${B}_{p}$) that colonize the same set of particles. Two populations (i and j) are assumed to be identical, except for their detachment rates, d, from a particle (${d}_{i}\ne {d}_{j}$). The dynamics of the particleassociated populations are determined by the rate at which cells attach to particles ($\alpha $) from the freeliving population (${B}_{F}$), the growth rate of attached cells ($\mu $) and detachment rate ($d$), as follows:
where $n$ represents the particle index and its associated population, i. Equation 3 can be formulated for any other population at the same particle. In a system with ${N}_{p}$ particles and M populations, we numerically solve a finite set of equations (${N}_{p}\times M$) at each time interval. The growth rate of population, i ($\mu}_{i$ is a function of total particleassociated cells ($B}_{p,n$)), as described later in Equation 6.
From number conservation, the freeliving bacterial pool ${B}_{Fi}$ of any population i results from particle detachment and attachment dynamics. The rate of change of all freeliving pools results from a combination of three factors: (1) the rate at which cells detach from the particles ${d}_{i}$, (2) the rate ${\alpha}_{i}$ at which cells attach to the particles, and (3) a mortality rate due to starvation ${m}_{Fi}$, as the following equation:
We run all dynamical simulations until an equilibrium is reached and there are no noticeable changes in the population size of particleassociated and freeliving cells, that is, $\frac{d{B}_{\mathrm{p}}}{dt}\approx 0$ and $\frac{d{B}_{\mathrm{F}}}{dt}\approx 0$.
Bacteriaparticle encounter rate
Request a detailed protocolWe assume that a bacterial cell can attach to the particle it encounters and stay attached for a period of time (‘residence time’). The encounter probability of a spherical cell with radius ${r}_{c}$ and a spherical particle with a radius of ${r}_{p}$ at a given time $t$ can be calculated using the hitting probability from random walk theory (Leventhal et al., 2019; Frazier and Alber, 2012):
where R is the total radius ($R={r}_{p}+{r}_{c}$), D is an effective diffusion coefficient ($D={D}_{c}+{D}_{p}$) for a bacterial cell (c) starting at a distance (${r}_{c,p}$), and erfc represents the standard complementary error function (1  erf). The diffusion coefficient can be calculated from an empirical model: $D={k}_{B}T/6\pi \mu r$, where $k}_{B}\approx 1.38\text{}\times {10}^{23}\text{}{\mathrm{J}\text{}\mathrm{K}}^{1$ is Boltzmann’s constant, T = 293 K is the ambient temperature, $\mu =1.003\text{}\mathrm{m}\mathrm{P}\mathrm{a}\text{}\mathrm{s}$ is the viscosity of water at the given ambient temperature. In aquatic environments, the size of marine snow (>100 µm) is often a lot larger than the cell size, we thus assume that the effective diffusion is generally controlled by cell diffusion coefficient ($D\approx {D}_{c}$). From Equation 5, we calculate the total number attaching cells to a particle at a given time (t) from freeliving cells of population i by multiplying the hitting probability to the total number of freeliving cells.
Growth and reproduction on particles
Request a detailed protocolWe assume that per capita access to particulate resources decreases in proportion to the total number of cells that colonize the surface. This leads us to model bacterial competition on a given particle, n, with a linear negative densitydependent growth function.
In this model, we assume that the bacterial growth on the particle is competitive in which the growth rate, μ_{i} is not constant but changes as a function of the total biomass on a particle. The negative densitydependent growth is modeled by assuming a linear function with the total particleassociated cells ($B}_{p,n}=\sum {B}_{p,n,i$) on particle, n,
where $B}_{p,n}=\sum {B}_{p,n,i$ represents the total number of particleattached cells, µ_{i} represents that growth rate of population i, ${\mu}_{max}$ indicates the maximal growth rate, in the absence of competition, and N_{t} represents the particlespecific carrying capacity. The net growth rate is assumed to be zero if more cells colonize a particle where bacteria have reached their carrying capacity; this occurs when bacteria have fully covered a particle’s surface, such that the death or detachment of any cell is quickly replaced by the growth of another cell. The model assumes that freeliving cells cannot grow. We performed a sensitivity analysis to competitive growth kinetic parameterizations (maximum growth rate μ_{max} and carrying capacity N_{t}) and showed that coexistence among bacterial detachment strategies is robust for a wide range of parameters (Figure 2—figure supplement 5).
Offspring production on the particle only occurs when particleassociated cells accumulate a total biomass that is larger than the biomass of a single cell ($m}_{d$). For simplicity, we only measured biomass based on the dry mass of the cells. The biomass accumulation rate on a particle for population i is proportional to the available biomass on the particle, n and its exponential growth rate ($\frac{d{B}_{p,n,i}}{dt}={B}_{p,n,i}{\mu}_{i}$). With this, the total number of offspring (${N}_{o,i}$) on a particle for a time interval of, $\u2206t$ can then be calculated as:
Particlewide bacterial mortality
Request a detailed protocolIn the model, a general form of mortality on particles is considered that accounts for mortality induced by predation or particle sinking, taking cells beyond their preferred habitat. A constant fraction of particles ($m}_{p$) is randomly selected at each time interval ($\mathrm{\Delta}t$) and their associated cells are removed from the particle. This fraction represents the particlescale mortality rate ($m}_{p$). To maintain particle number equilibrium, a fraction $m}_{p$ of uncolonized particles is introduced into the system and colonized by freeliving populations (${B}_{p,n,i}$ = 0).
Mortality of freeliving cells is assumed to be caused by loss of biomass over a prolonged period of starvations from the absence of substrate uptake in the freeliving phase. As described in Equation 4, freeliving cells ($B}_{F$) lose a constant fraction of their biomass (${m}_{Fi}$) every time step as the cell maintenance. Note that though detachment of cells from a particle appears similar to mortality on particles, in the former, detached cells move to the freeliving pool, while in the latter, cells die and do not add to either pool.
Particle degradation and turnover
Request a detailed protocolWe assume that a particle contains a finite amount of resources that is degraded by bacterial cells with a constant yield of converting the resources into biomass. From a previous study, we assume that the yield is about 5% and a significant fraction of particle degradation products are lost to the environment before being taken up by the cells (Ebrahimi et al., 2019).
Optimal residence time from OFT
Request a detailed protocolOFT describes the dispersal behavior of microbial populations in patchy environments assuming maximized growth return using the marginal value theorem. According to OFT, the growth return of particleassociated bacteria is maximized if a bacterial cell detaches from the particle when its timeaveraged uptake rate reaches its instantaneous uptake rate. We applied this assumption to obtain the optimal residence time on particles by tracking individual cells in our model and numerically calculating their instantaneous uptake rate ($u(t)$) on a particle from the attachment time (${t}_{a}$) to detachment using our populationbased model. The residence time (${t}_{r}$) is considered optimal when the following equation is satisfied (Yawata et al., 2020):
where ${\tau}_{s}$ is the search time and a function of the number of particles in the system. We calculated the search time from Equation 5 when the probability of the cell and particle encounter is above 95% (Figure 1—figure supplement 1).
Data availability
Ours is a modeling and theoretical study, and has no associated data. All associated computer code relevant for the study and for reproducing the results is available as a GitHub repository at the following link: https://github.com/aliebmitedu/Bacterialdispersalmodel, (copy archived at swh:1:rev:9629ae0b5214a8a7a1ea9b96cef5d91adfe4a6ca).
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Decision letter

Maureen L ColemanReviewing Editor; University of Chicago, United States

Aleksandra M WalczakSenior Editor; CNRS LPENS, France

James O’ DwyerReviewer
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 "Particle foraging strategies promote microbial diversity in marine environments" for consideration by eLife. Your article has been reviewed by 2 peer reviewers, and the evaluation has been overseen by a Reviewing Editor and Aleksandra Walczak as the Senior Editor. The following individual involved in review of your submission has agreed to reveal their identity: James O' Dwyer (Reviewer #1).
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:
1) Ensure modeling choices are clearly explained and documented (including making code available)
2) Discuss the generality and limitations of the model and results
3) Provide additional background context/discussion of prior work as suggested by the reviewers
Reviewer #1 (Recommendations for the authors):
 In framing the paper, I think the authors are right to focus on dispersal and detachment as underexplored mechanisms. But readers will benefit from reference to other work (even on particleassociated microbes) related to resource diversity, succession, and crossfeeding. That can only help put the current study in context with other mechanisms for the maintenance of microbial diversity.
To expand on this, I know the authors have worked also on how microbial interactions, crossfeeding, and succession can maintain diversity, or at least add to our understanding of it on marine particles. Is the situation here envisioned significantly different than those experiments? If not, i think it is fine to focus on this different trait axis, and just consider the particles to in effect be a single resource, without resource preferences, etc. But I wanted to make sure I was understanding that correctlymaybe there is something different about the situation envisaged here that would make it less likely to have all of those other interactions (which clearly can contribute to many species being maintained). Martina Dal Ballo and Jeff Gore's work also seems relevant to this (where many resources are produced endogenously in an experimental system) and also work on resource heterogeneity in DOM in natural systems (e.g. Muscarella, Boot, Broeckling , Lennon).
Again, I don't think any of this detracts from the motivation for the present study. Just might fill out a fuller picture.
 There is a population growth process when a cell settles on a new particle. This is assumed to be logistic growth, though in the end, it seems likely that the precise dynamics of the growth process don't matter so much as the final abundance (carrying capacity). However, this seemed subtle to me for three reasons.
(i) It seems to me that the detachment rate should directly affect this final abundanceas an additional source of "mortality" (in the sense of removing individuals from the particle) contributing to the net growth rate. Maybe that effect will always be small, but this could be made clearer for readers if so.
(ii) The authors' conceptual diagram shows that one possible end point of this process is that the microbial population in effect eat the whole particle (Figure 1). This sharpens the issue to me of what the true dynamics are likely to be on a particle. For example, should I think of the population growing to a capacity that is roughly the surface area of the particle, and then gradually changing thereafter as the particle's physical size is reduced? And what direction would this change be? Could I think of a thickening of the film of particles and population size continuing to grow (maybe linearly) in time as interior layers continue to eat the particle? Or should I think of outer individuals as being shed, and the carrying capacity in effect reducing as the particle size reduces?
(iii) The issue of what is actually going to happen once a population reaches carrying capacity is also at the center of my final point here. It seems from the thought experiment above that it is unlikely that growth as such will stop when cells fill out the surface of a particle, since there are still resources to take up. So I am interested to know whether zero growth rate means to the authors that cells stop reproducing, or are cells dying to balance growth, or are they being shed from the particle?
It's possible that none of this matters too much if all that's important is a final population size. However, it might help to clarify the process for readers if we have a conceptual picture of what this final population size represents (surface of particle being filled? or volume of particle entirely eaten up) and if there is a truer picture of the dynamics than logistic growth.
 The relationship between the tradeoff (between different detachment rates) derived in Equation 2 versus the optimal detachment rate (derived in the methods) is framed a little confusingly. If I understand correctly, the "tradeoff" actually comes from the condition that a population will have net nonnegative growth rate in the absence of other populations with different strategies. So it may be reasonable to frame this as a thresholda necessary condition rather than a sufficient condition for a given population to persist. The reason I say this is that it is a bit confusing to have a tradeoff that suggests a range of detachment rates can coexist so long as they differ in their carrying capacities, since it is then stated that the optimal detachment rate outcompetes all the others. Maybe I misunderstood something important being assumed about the carrying capacity for the optimal case, but a tradeoff that also has an optimum is an odd outcome.
In short, it was not clear to me whether to populations satisfying this tradeoff in Equation (2) would tend to coexist. Or would in general one population (say the one closer to the optimum strategy) tend to outcompete the other? If so it might help to define more clearly what this tradeoff means. If I understood this correctly, I would not say that this issue merely indicates that the tradeoff is not "evolutionarily stable".
 In the end, it seems critical that for multiple strategies to be maintained in the population that there is not only wholeparticle mortality (which in effect is highly correlated catastrophic dynamics for an individual microbial population), but that the inflow of resources itself fluctuates. Did I interpret that correctly? Readers may appreciate a slightly clearer description of how this environmental stochasticity differs from the previous possibility of wholecell mortality, and this also left me wondering how to quantity the kind of environmental stochasticity that will generally lead to multiple strategies coexisting.
So my understanding from this was that wholecell mortality was not on its own to avoid a single population outcompeting others. but I did not get such a clear picture of what environmental stochasticity WOULD allow for coexistence.
 One other reference that might be tangentially related, but I thought could be relevant: "The importance of being discrete: Life always wins on the surface" by Shnerb et al. This describes growth on particles in 2d or 3d, and shedding (which seems different from but not entirely different from detachment).
This could be an important reference because in this stochastic model, the effective growth rate is different from what you would have with the naive mean field model. So I am wondering if this might change any of the outcomes of Equations 1 and 2.
Reviewer #2 (Recommendations for the authors):
The authors investigated an interesting question related to the coexistence of bacterial species with different detachment strategies on particles. The results of this investigation are interesting and relevant for our understanding of microbial diversity in particleassociated communities, but the manuscript would benefit from a more in depth discussion of results from recent papers on microbial community dynamics of particles, many of which are cited in the current version but only in passing. There are also many possible extensions of this work, which the authors are probably aware of, which would be interesting to explore in future work. For example, the role of search strategies (e.g., random walks vs chemotaxis vs Levy walks) and detachment rates that depend nonlinearly on the concentration of bacteria on the particle.
The authors seem to adopt a somewhat strict definition of Optimal Foraging Theory (OFP) limited to the Marginal Value Theorem (MVT). There are examples of OFT studies that do consider mortality and predation risk, finding that predictions of the MVT do not hold in these settings, e.g.:
Abrams, PA. "Optimal traits when there are several costs: the interaction of mortality and energy costs in determining foraging behavior." Behavioral Ecology, vol. 4, no. 3, 1993, pp. 246259.
Newman, JA. " Patch use under predation hazard: foraging behavior in a simple stochastic environment." Oikos, vol. 61, 1991,
pp. 2944.
It would be important to know how this study relates to previous results of OFT that do include mortality and predation risk.
In its current form, there are a few places where better description of the numerical simulations performed would critically enhance the manuscript and the reproducibility of the results. Specifically:
 There is a deterministic particle mortality rate m_{p,i} in Equation 3, and an additional, stochastic particle mortality in the section "Bacterial mortality". Are the two implemented with the same rate, and what is the rationale for implementing both forms of mortality? The manuscript text only seems to describe the stochastic particle mortality, but is never too explicit about it. As described in lines 385394, it seems that the stochastic mortality rate depends on the numerical integration time step: because m_{p} is a rate, a fraction m_{p} * dt of particles, where dt is the integration time step, should be chosen at each time step to impose mortality, rather than a "fraction" m_{p} as suggested by the text at lines 387389.
 The most problematic section is "Bacteriaparticle encounter rate". The authors mention explicitly the encounter probability of spherical cells undergoing random walks, but they need a rate to incorporate in the equations via parameter α. An explicit expression for α is not provided, and I would have expected α to be the diffusive flux towards a spherical absorber (see, e.g., Berg's "Random walks in biology" page 27, Equation 2.20): I = 4 \pi D R C_{0}, where C_{0} is the concentration of detached bacteria, but this is not mentioned. Also, what is d in Equation 5? It should be D_{c,p}, and it can't be the detachment rate d. The estimate for the diffusion coefficient of cells via the EinsteinStokes relationship is too small if bacteria are motile, as suggested by Figure 1. For motile cells, cell diffusion can be orders of magnitude larger than the EinsteinStokes estimate (see, e.g., Berg's "Random walks in biology" page 93 – Movement of selfpropelled objects). The sentence "From Equation 5, we calculated the total number attaching cells to a particle at a given time (t) from free living cells of population i by multiplying the hitting probability to the total number of freeliving cells" is very hard for me to interpret: which choice of D_{c,p} was used in Equation 5? I would also mention explicitly that this entire section assumes instantaneous attachment of bacteria to particles with an infinite rate coefficient.
For a study that is mostly numerical such as this one, availability of the computer code for peer review would greatly enhance the reproducibility of the results and would have clarified some of the doubts expressed above. I would encourage the authors to post it on Github for peer review, or provide it as supplementary material with the submission.
It would be very informative to know under which conditions the analytical approximation described at lines 174216 breaks down. At low particle density, the search time may be much longer than the growth period on the particle, but at high particle densities this may not be true. Would the approximation work less well in those conditions?
https://doi.org/10.7554/eLife.73948.sa1Author response
Essential revisions:
1) Ensure modeling choices are clearly explained and documented (including making code available)
Following the suggestions of all reviewers, the revised manuscript contains new text detailing our model choices and explaining the rationale behind them. We have also made all the simulation code available in the following GitHub repository: https://github.com/aliebmitedu/Bacterialdispersalmodel
2) Discuss the generality and limitations of the model and results
In the revised manuscript, we now discuss our interpretation of the model, including how its assumptions can be generalized; we have also added text discussing its limitations.
3) Provide additional background context/discussion of prior work as suggested by the reviewers
We are grateful to the reviewers for offering suggestions on appropriately crediting prior work; the revised manuscript now contains new citations and references, which discuss how our work connects with past literature on e.g., optimal foraging theory.
Reviewer #1 (Recommendations for the authors):
 In framing the paper, I think the authors are right to focus on dispersal and detachment as underexplored mechanisms. But readers will benefit from reference to other work (even on particleassociated microbes) related to resource diversity, succession, and crossfeeding. That can only help put the current study in context with other mechanisms for the maintenance of microbial diversity.
To expand on this, I know the authors have worked also on how microbial interactions, crossfeeding, and succession can maintain diversity, or at least add to our understanding of it on marine particles. Is the situation here envisioned significantly different than those experiments? If not, i think it is fine to focus on this different trait axis, and just consider the particles to in effect be a single resource, without resource preferences, etc. But I wanted to make sure I was understanding that correctlymaybe there is something different about the situation envisaged here that would make it less likely to have all of those other interactions (which clearly can contribute to many species being maintained). Martina Dal Ballo and Jeff Gore's work also seems relevant to this (where many resources are produced endogenously in an experimental system) and also work on resource heterogeneity in DOM in natural systems (e.g. Muscarella, Boot, Broeckling , Lennon).
Again, I don't think any of this detracts from the motivation for the present study. Just might fill out a fuller picture.
Thank you for the constructive comment. We now expanded our discussion to include this point in the revised manuscript (lines 322336):
“While we simplified bacterial colonization dynamics on particles by only considering competitive growth kinetics, variants of our model suggest that coexistence between different dispersal strategies is also expected under more complex microbial interactions that are observed on marine particles, including cooperative growth dynamics (Figure 1 —figure supplement 1). Such simplifications allowed us to explore the role of dispersal in maintaining microbial diversity in natural systems, in addition to previously observed factors such as metabolic interaction, resource heterogeneities and succession (Datta et al. 2016; Dal Bello et al. 2021; Muscarella et al. 2019). Recent studies have observed the emergence of complex trophic interactions and successional dynamics across particleassociated communities that our model can provide a theoretical framework to evaluate contributions on such mechanisms on maintaining bacterial diversity (Lauro et al. 2009; Datta et al. 2016; PascualGarcía et al. 2021; Boeuf et al. 2019).”
 There is a population growth process when a cell settles on a new particle. This is assumed to be logistic growth, though in the end, it seems likely that the precise dynamics of the growth process don't matter so much as the final abundance (carrying capacity). However, this seemed subtle to me for three reasons.
(i) It seems to me that the detachment rate should directly affect this final abundanceas an additional source of "mortality" (in the sense of removing individuals from the particle) contributing to the net growth rate. Maybe that effect will always be small, but this could be made clearer for readers if so.
Yes, the reviewer is correct. In the revised manuscript, we have clarified the distinction between particlewide mortality and detachment in how affect bacterial numbers both on particles and in the total system (lines 424426).
The detachment rate can be seen as a source of mortality if the mortality is defined as removing cells from the particle. It should be noted that the cells detaching from the particle will be added to the freeliving population (BF) which can have a positive effect on the rate of attachment to the particle (a*BF). Therefore, the "mortality" induced by the detachment rate is different than the particlewide mortality (mp) which removes cells from the system and reduces the total number of freeliving and particleassociated cells (BF + BP). Since we performed our simulations for a wide range of detachment rates, the rate of cells removed by this effect could be comparable to or higher than the particlewide mortality in a small fraction of the simulations, but not large overall.
“Note that though detachment of cells from a particle appears similar to mortality on particles, in the former, detached cells move to the freeliving pool, while in the latter, cells die and do not add to either pool.”
(ii) The authors' conceptual diagram shows that one possible end point of this process is that the microbial population in effect eat the whole particle (Figure 1). This sharpens the issue to me of what the true dynamics are likely to be on a particle. For example, should I think of the population growing to a capacity that is roughly the surface area of the particle, and then gradually changing thereafter as the particle's physical size is reduced? And what direction would this change be? Could I think of a thickening of the film of particles and population size continuing to grow (maybe linearly) in time as interior layers continue to eat the particle? Or should I think of outer individuals as being shed, and the carrying capacity in effect reducing as the particle size reduces?
As the reviewer understands, we do not explicitly model the size and food content on a particle, only an implicit densitydependent net growth rate of a bacterial population on the particle. Nevertheless, we believe our model captures the process of a population growing on a particle’s surface and eventually covering its entirety, rather than a volumetric shrinking over time as the particle is consumed. This is because the typical time taken to completely consume a particle is much larger than the time it takes for the particle to sink below a certain ocean depth and lost to bacterial grazing.
In the revised manuscript, we have clarified our interpretation of bacterial growth in the model (lines 388395).
(iii) The issue of what is actually going to happen once a population reaches carrying capacity is also at the center of my final point here. It seems from the thought experiment above that it is unlikely that growth as such will stop when cells fill out the surface of a particle, since there are still resources to take up. So I am interested to know whether zero growth rate means to the authors that cells stop reproducing, or are cells dying to balance growth, or are they being shed from the particle?
In our model, a zero net growth rate of the population on a particle implies that cellular reproduction balances intrinsic cell death (not extrinsic wholeparticle mortality) and detachment. Thus, the number of cells on the particles reaches a steady state.
It's possible that none of this matters too much if all that's important is a final population size. However, it might help to clarify the process for readers if we have a conceptual picture of what this final population size represents (surface of particle being filled? or volume of particle entirely eaten up) and if there is a truer picture of the dynamics than logistic growth.
We thank the reviewer for detailing this concern. In the revised manuscript, we have added clarifying text which explicitly explains our interpretation of the population size on each individual particle in the model (lines 388395). Specifically, as a particle is colonized, the bacterial population on it grows in number and density, eventually covering the surface of the particle. At carrying capacity, there is no more space to grow – the population size reaches a steady state – and any spontaneously dying cells are replaced by newly growing ones.
“We assume that bacteria grow on each particle, covering its surface over time; as the surface gets covered, bacterial density increases, in turn slowing down growth. The net growth rate is assumed to be zero if more cells are colonizing the particle compared to its carrying capacity; this occurs when bacteria have fully covered a particle’s surface, such that the death or detachment of any cell is quickly replaced by the growth of another cell.”
 The relationship between the tradeoff (between different detachment rates) derived in Equation 2 versus the optimal detachment rate (derived in the methods) is framed a little confusingly. If I understand correctly, the "tradeoff" actually comes from the condition that a population will have net nonnegative growth rate in the absence of other populations with different strategies. So it may be reasonable to frame this as a thresholda necessary condition rather than a sufficient condition for a given population to persist. The reason I say this is that it is a bit confusing to have a tradeoff that suggests a range of detachment rates can coexist so long as they differ in their carrying capacities, since it is then stated that the optimal detachment rate outcompetes all the others. Maybe I misunderstood something important being assumed about the carrying capacity for the optimal case, but a tradeoff that also has an optimum is an odd outcome.
In short, it was not clear to me whether to populations satisfying this tradeoff in equation (2) would tend to coexist. Or would in general one population (say the one closer to the optimum strategy) tend to outcompete the other? If so it might help to define more clearly what this tradeoff means. If I understood this correctly, I would not say that this issue merely indicates that the tradeoff is not "evolutionarily stable".
We understand the source of confusion and are happy to clarify. In the revised manuscript, we have also clarified the text and explained the tradeoff more carefully.
For all populations, there is a tradeoff between detachment rate and growth return (population size per particle at steady state) due to the dynamics of the model; as one rises, the other falls. However, the mathematical relationship between the two quantities is complicated and parameterdependent; in some parameter region, growth return changes rapidly with detachment rate, while in another, it changes relatively slowly.
For two populations to coexist, their detachment rates must obey equation (2). Equation (2) is a necessary condition and demands a specific mathematical form of the tradeoff between detachment rate and growth return. As Figure 3C shows, not all pairs of populations satisfy this equation; those that do can indeed coexist. Typically, the growth return is a complex quantity that depends on various parameter choices regarding the onparticle dynamics, and cannot be obtained using our coarsegrained description.
When one of the two populations has a detachment rate equal to the optimal detachment rate, the second population always fails to satisfy equation (2), i.e., the ratio of growth returns of the pair lies in the white region of Figure 3C. Thus, no population can coexist with the optimal detachment rate. Lines 211217:
“While the tradeoff in Equation 2 allows coexistence and is necessary condition for it, it does not hold across all parameter values, and does not allow any pair of detachment rates to coexist (Figure 3C, white region). In particular, no detachment rate can coexist with the optimal detachment rate, thus rendering coexistence between any other set of detachment rates susceptible to invasion by this optimal strategy. Other strategies, when paired with the optimal strategy, disobey the condition in Equation 2, and thus cannot coexist with it.”
 In the end, it seems critical that for multiple strategies to be maintained in the population that there is not only wholeparticle mortality (which in effect is highly correlated catastrophic dynamics for an individual microbial population), but that the inflow of resources itself fluctuates. Did I interpret that correctly? Readers may appreciate a slightly clearer description of how this environmental stochasticity differs from the previous possibility of wholecell mortality, and this also left me wondering how to quantity the kind of environmental stochasticity that will generally lead to multiple strategies coexisting.
This is a good point. In the revised text, we have clarified the difference between wholeparticle mortality and environmental stochasticity, which brings in new particles. In general, any event that changes the number of empty particles should aid in the coexistence of multiple strategies. Particles can be emptied by high levels of wholeparticle mortality, or new empty particles can be brought in by events such as algal blooms (environmental stochasticity).
So my understanding from this was that wholecell mortality was not on its own to avoid a single population outcompeting others. but I did not get such a clear picture of what environmental stochasticity WOULD allow for coexistence.
We have now added new text providing intuition for how environmental stochasticity promotes coexistence despite an optimal detachment rate. The optimal detachment rate is not fixed, but depends on the total number of particles and the wholeparticle mortality rate. What environmental stochasticity, i.e., changing the total number of particles, does is that it changes the optimal detachment rate over time. Thus, two populations (fast and slow detaching) can coexist if the environment keeps oscillating between regimes where either population is advantaged. Lines: 254262:
“We thus hypothesized that fluctuations in particle abundance may also induce fluctuations in the optimal detachment rate, such that no specific detachment rate would be uniquely favored at all times. Thus, environmental stochasticity would constantly change the optimal detachment rate; low particle abundances would favor fastdetaching populations, while higher particle abundances would favor slowdetaching populations. Such a “fluctuating optimum” may create temporal niches and promote higher bacterial diversity on marine particles.”
 One other reference that might be tangentially related, but i thought could be relevant: "The importance of being discrete: Life always wins on the surface" by Shnerb et al. This describes growth on particles in 2d or 3d, and shedding (which seems different from but not entirely different from detachment).
This could be an important reference because in this stochastic model, the effective growth rate is different from what you would have with the naive mean field model. So I am wondering if this might change any of the outcomes of Equations 1 and 2.
Thank you very much for sharing this important reference with us. This is a valid point. The spatial distribution of cells on the particle could lead to a drastically different form of growth that may not be captured by meanfield models. In fact, in our previous study, we have shown that bacterial cells can spatially selforganize on the particles allowing them to cooperate and grow on recalcitrate particles. And without spatial organization, cells would go extinct.
We have performed sensitivity analyses for various growth dynamics on the particle (cooperative vs competitive) in Figure 2 —figure supplement 4. We have shown that our observations of the emergence of coexistence is robust to the form of growth on particles.
More studies are needed to explore the role that spatial distribution of cells could play in shaping the coexistence of cells on particles.
Reviewer #2 (Recommendations for the authors):
The authors investigated an interesting question related to the coexistence of bacterial species with different detachment strategies on particles. The results of this investigation are interesting and relevant for our understanding of microbial diversity in particleassociated communities, but the manuscript would benefit from a more in depth discussion of results from recent papers on microbial community dynamics of particles, many of which are cited in the current version but only in passing. There are also many possible extensions of this work, which the authors are probably aware of, which would be interesting to explore in future work. For example, the role of search strategies (e.g., random walks vs chemotaxis vs Levy walks) and detachment rates that depend nonlinearly on the concentration of bacteria on the particle.
Thanks for the constructive comments. In the revised manuscript, we have expanded our discussion to include assumptions of the model and opportunities for future works in the discussion (lines 322336):
“While we simplified bacterial colonization dynamics on particles by only considering competitive growth kinetics, variants of our model suggest that coexistence between different dispersal strategies is also expected under more complex microbial interactions that are observed on marine particles, including cooperative growth dynamics (Figure 2 —figure supplement 4). Such simplifications allowed us to explore the role of dispersal in maintaining microbial diversity in natural systems, in addition to previously observed factors such as metabolic interaction, resource heterogeneities and succession^{8,52,53}. Recent studies have observed the emergence of complex trophic interactions and successional dynamics across particleassociated communities that our model can provide a theoretical framework to evaluate contributions on such mechanisms on maintaining bacterial diversity ^{18,52,54,55}. In addition, the current work only assumes Brownian motion while bacterial active motility and chemotaxis are shown to play a big role in the foraging strategy of aquatic microorganisms^{56–58}. Overall, our model provides a reliable framework to further study how diverse dispersal strategies and mortality could contribute to the emergence of complex community dynamics on marine particles and how environmental factors impact microbial processes in regulating POM turnover at the ecosystem level. ”
The authors seem to adopt a somewhat strict definition of Optimal Foraging Theory (OFP) limited to the Marginal Value Theorem (MVT). There are examples of OFT studies that do consider mortality and predation risk, finding that predictions of the MVT do not hold in these settings, e.g.:
Abrams, PA. "Optimal traits when there are several costs: the interaction of mortality and energy costs in determining foraging behavior." Behavioral Ecology, vol. 4, no. 3, 1993, pp. 246259.
Newman, JA. " Patch use under predation hazard: foraging behavior in a simple stochastic environment." Oikos, vol. 61, 1991,
pp. 2944.
It would be important to know how this study relates to previous results of OFT that do include mortality and predation risk.
Thank you very much for sharing these important articles. We carefully read those articles that include mortality in the formulation of OFT. Interestingly, our work is consistent with the previous publications in showing that the presence of mortality/predation reduces residence time on the particle/patch.
We have now expanded our discussion to make this comparison (lines 309311).
“This finding agrees with previous OFT models that considered mortality, showing that optimal foraging effort and residence time on patches decrease significantly as the density of predators increase (Newman 1991; Abrams 1993).”
In its current form, there are a few places where better description of the numerical simulations performed would critically enhance the manuscript and the reproducibility of the results. Specifically:
 There is a deterministic particle mortality rate m_{p,i} in Equation 3, and an additional, stochastic particle mortality in the section "Bacterial mortality". Are the two implemented with the same rate, and what is the rationale for implementing both forms of mortality? The manuscript text only seems to describe the stochastic particle mortality, but is never too explicit about it. As described in lines 385394, it seems that the stochastic mortality rate depends on the numerical integration time step: because m_{p} is a rate, a fraction m_{p} * dt of particles, where dt is the integration time step, should be chosen at each time step to impose mortality, rather than a "fraction" m_{p} as suggested by the text at lines 387389.
Thanks for pointing out this. This is correct and we now modified the equation or the text to reflect the correct way of implementing the mortality on the particle. We removed mortality term in Equation 3 which was creating confusion. There is only one mortality that is considered and that is the one described in the second of “Bacterial mortality”. We now included time step, Δt and particle number in this section to reflect how the fraction of particles are determined for mortality (Lines: 411417).
“In the model, a general form of mortality on particles is considered that accounts for mortality induced by predation or particle sinking and becoming inaccessible at deeper layers of the water column in the ocean. A constant fraction of particles (N_{p}.m_{p}.Δt) that represent the mortality rate at particle scale (m_{p}) is uniform randomly selected at each time interval (Δt) and their associated cells are removed from the particle. Then, the uncolonized particle is again introduced into the system and colonized by freeliving populations ($B}_{p,n,i$=0). ”
 The most problematic section is "Bacteriaparticle encounter rate". The authors mention explicitly the encounter probability of spherical cells undergoing random walks, but they need a rate to incorporate in the equations via parameter α. An explicit expression for α is not provided, and I would have expected α to be the diffusive flux towards a spherical absorber (see, e.g., Berg's "Random walks in biology" page 27, Equation 2.20): I = 4 \pi D R C_{0}, where C_{0} is the concentration of detached bacteria, but this is not mentioned. Also, what is d in Equation 5? It should be D_{c,p}, and it can't be the detachment rate d. The estimate for the diffusion coefficient of cells via the EinsteinStokes relationship is too small if bacteria are motile, as suggested by Figure 1. For motile cells, cell diffusion can be orders of magnitude larger than the EinsteinStokes estimate (see, e.g., Berg's "Random walks in biology" page 93 – Movement of selfpropelled objects). The sentence "From Equation 5, we calculated the total number attaching cells to a particle at a given time (t) from free living cells of population i by multiplying the hitting probability to the total number of freeliving cells" is very hard for me to interpret: which choice of D_{c,p} was used in Equation 5? I would also mention explicitly that this entire section assumes instantaneous attachment of bacteria to particles with an infinite rate coefficient.
For a study that is mostly numerical such as this one, availability of the computer code for peer review would greatly enhance the reproducibility of the results and would have clarified some of the doubts expressed above. I would encourage the authors to post it on Github for peer review, or provide it as supplementary material with the submission.
The code is now available online at the following URL: https://github.com/aliebmitedu/Bacterialdispersalmodel
It would be very informative to know under which conditions the analytical approximation described at lines 174216 breaks down. At low particle density, the search time may be much longer than the growth period on the particle, but at high particle densities this may not be true. Would the approximation work less well in those conditions?
The reviewer is correct: as the density of particles increases, the time taken for bacteria to colonize new particles will decrease. Certainly, when particle densities increase to be very large (by more than an order of magnitude), our approximation will break down. However, for the numbers in our study, where particle densities vary by 23x, our numerics support our assumption that the approximation works reasonably well.
References:
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https://doi.org/10.7554/eLife.73948.sa2Article and author information
Author details
Funding
Gordon and Betty Moore Foundation (GBMF4513)
 Akshit Goyal
Simons Foundation (542395)
 Otto X Cordero
Swiss National Science Foundation (P2EZP2 175128)
 Ali Ebrahimi
Swiss National Science Foundation (P400PB_186751)
 Ali Ebrahimi
The funders had no role in study design, data collection, and interpretation, or the decision to submit the work for publication.
Acknowledgements
This work was supported by Simons Foundation: Principles of Microbial Ecosystems (PriME) award number 542,395. AE acknowledges funding from Swiss National Science Foundation: Grants P2EZP2 175,128 and P400PB_186751. AG acknowledges support from the Gordon and Betty Moore Foundation as a Physics of Living Systems Fellow through award number GBMF4513.
Senior Editor
 Aleksandra M Walczak, CNRS LPENS, France
Reviewing Editor
 Maureen L Coleman, University of Chicago, United States
Reviewer
 James O’ Dwyer
Version history
 Received: September 16, 2021
 Preprint posted: September 17, 2021 (view preprint)
 Accepted: March 1, 2022
 Accepted Manuscript published: March 15, 2022 (version 1)
 Version of Record published: March 25, 2022 (version 2)
Copyright
© 2022, Ebrahimi et al.
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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