The authors have declared that no competing interests exist.
Conceived and designed the experiments: M. Arnoldini, R. Mostowy, M. Ackermann. Performed the experiments: M. Arnoldini, R. Mostowy. Analyzed the data: M. Arnoldini, R. Mostowy, S. Bonhoeffer, M. Ackermann. Contributed reagents/materials/analysis tools: M. Arnoldini, R. Mostowy. Wrote the paper: M. Arnoldini, R. Mostowy, M. Ackermann.
Current address: Department of Infectious Disease Epidemiology, Imperial College London, London, United Kingdom
Most organisms live in ever-changing environments, and have to cope with a range of different conditions. Often, the set of biological traits that are needed to grow, reproduce, and survive varies between conditions. As a consequence, organisms have evolved sensory systems to detect environmental signals, and to modify the expression of biological traits in response. However, there are limits to the ability of such plastic responses to cope with changing environments. Sometimes, environmental shifts might occur suddenly, and without preceding signals, so that organisms might not have time to react. Other times, signals might be unreliable, causing organisms to prepare themselves for changes that then do not occur. Here, we focus on such unreliable signals that indicate the onset of adverse conditions. We use analytical and individual-based models to investigate the evolution of simple rules that organisms use to decide whether or not to switch to a protective state. We find evolutionary transitions towards organisms that use a combination of random switching and switching in response to the signal. We also observe that, in spatially heterogeneous environments, selection on the switching strategy depends on the composition of the population, and on population size. These results are in line with recent experiments that showed that many unicellular organisms can attain different phenotypic states in a probabilistic manner, and lead to testable predictions about how this could help organisms cope with unreliable signals.
Most organisms are occasionally exposed to adverse environmental conditions, and can express protective features that help them mitigate the harmful effects of environmental stresses, such as infections, exposure to UV light or chemicals, or sudden habitat changes. Interestingly, a number of recent experiments with unicellular microbes revealed marked variability in the responses to such stress between genetically identical individuals. Some individuals express protective features even in the absence of stress; others do not express these features even if stress reaches substantial levels. Why is stress response, which seems so important for organisms, not more tightly controlled? One possibility is that this variation can help organisms mediate between costs and benefits of protection. These protective features are usually expressed in response to environmental signals that indicate stress. However, most signals are not absolutely reliable. Sometimes stressful conditions will not be preceded by a signal; other times, a signal might not be followed by stress. We used analytical and individual-based models to investigate how a probabilistic expression of stress response can evolve in response to unreliable signals, and in how the ecological setting influences the evolutionary dynamics.
Most organisms, from bacteria to multicellular eukaryotes, have sensory systems that allow measuring environmental cues, and responding to these cues by adjusting gene expression and modifying patterns of development and growth
Here, we are interested in the evolution of stress responses under conditions where organisms are faced with unreliable environmental signals; a situation where episodes of stress are usually preceded by a cue to which organisms can react – but in some cases, signals are not followed by stress, or stress is not preceded by a signal. These assumptions are realistic in biological systems: examples for stress without signals could include infections by pathogens, exposure to solar radiation, or rapid translocation from one habitat to another. We assume signals to be low levels of any environmental condition that would, at higher levels, cause stress and impact organismal functioning if not countered by stress response. In such situations, a deterministic response to environmental cues might not be ideal. Organisms that always express protective features in response to signals, and never express them without signals, face two types of problems: they might suffer high metabolic costs by always responding to the signal, even if it is often not followed by stress; and if stress occurs without preceding signal, all individuals are in an unprotected state, and are thus vulnerable to the deleterious effects of stress.
A number of previous studies investigated evolutionary responses to uncertain environments
Both theoretical and experimental studies suggest that probabilistic expression of the phenotype can help organisms cope with uncertain environments, in two different ways
Here we consider phenotype switching in the absence of environmental signals, and phenotype switching in response to environmental signals as two different, evolvable traits, a situation that we think is realistic in a natural situation: consider, for example, a bacterium expressing a transcription factor at a certain base-line level. This expression might be due to leaky regulation of the gene encoding the transcription factor, and will vary slightly between individual cells in a population as a consequence of stochastic effects of gene expression
We are interested in two main questions. First, we investigate the simultaneous evolution of random and responsive switching. We are interested in the conditions that favor one over the other strategy, and we analyze how combining the two strategies can help organisms cope with environmental uncertainty. Second, we are interested in how the evolution of random and responsive switching depends on the ecological setting. Specifically, we address the question how selection on the response to unreliable signals can depend on the composition of the population.
We are using two theoretical approaches to address these issues. First, we use an analytical model to derive a mathematical expression of the long-term growth rate of a genotype, as a function of random and responsive switching, and of the properties of the environment. We use this approach to analyze the combinations of random and responsive phenotype switching values that maximize the long-term growth rate. We find that, when signals are only partially reliable, genotypes can evolve that use both strategies simultaneously. Second, we use an individual-based approach to assess the impact of the ecological setting on the evolutionary dynamics. We first consider unstructured environments, where the evolutionary outcome is simply dependent on how well different genotypes can match environmental fluctuations, and on how well they balance costs and benefits of entering a protected state. Then, we turn to environments that are divided into patches, and in which the population density is locally regulated in each patch. In these situations, the success of a genotype depends on the strategies of the other individuals in the population. In populations of risk prone individuals, risk averse types benefit, but this benefit vanishes once their numbers increase. This indicates that the evolutionary success of a given type depends on the composition of the population, and that the evolutionary dynamics of bet-hedging depends on the ecological setting.
In order to analyze the evolutionary dynamics of random and responsive phenotype switching, we use both analytical and individual-based modeling approaches. In both cases we assume that individuals carry a genotype consisting of two loci. The first locus encodes the probability of random switching (from the vegetative to the protected phenotype), and the second locus encodes the probability of responsive switching. In an environment without stress, the growth rate of the vegetative phenotype is
We first derive an analytical expression for the long-term growth rate of a population in a single habitat (single patch) and two habitats (two patches), given parameters of the model defined above:
In the case of a single patch, the long-term growth rate is the geometric mean of the growth rates in each of the four environmental states, weighted by the frequency of the four states:
In the case of two patches, we assume that the environmental state in the first patch is independent from the environmental state in the second patch. This results in sixteen combinations of
For the individual-based approach, we bin the two switching probabilities
If the number of individuals in a bin
To impose density regulation, we scale the number of individuals to a target population size (the target population size is defined below). This scaling is either done for the whole populations, or, in the case of local density regulation, within a patch. If the number of individuals before density regulation exceeds the target population size, the scaling represents density-dependent mortality of individuals competing for a finite resource. If the number of individuals before density regulation is smaller than the target population size, the scaling represents population growth. In the case of one patch, the target population size is equal to
In each generation, each individual changes its genotype category (mutates) with probability
A semi-deterministic version of this model has also been used. This was achieved by using the expected values as actual rates. In the case of binomial distributions, the expected numbers are a product of a sample size and a probability.
All simulations in this study were encoded in C++, and the data was plotted using R. The computer code can be found in
Our focus is on how organisms evolve to respond to environmental signals that indicate stressful conditions, and how the course of evolution depends on the reliability of the signals. We assume an environment that occurs in two distinct states, benign and stressful. We further assume discrete time steps. During each time step, the environment is in one of the two states; it can change the state during the transition to the next time step. There is a signal that tends to indicate stressful conditions. If there is a signal, it occurs at the beginning of a time step, and organisms can react to the signal during that time step. The signal is not necessarily reliable. Sometimes, signals are not followed by stress; other times, there is no signal, but there is stress.
The organisms can also exist in two states, vegetative (unprotected) and protected. The vegetative state confers a high fertility in time steps without stress, but a high mortality during time steps with stress. Individuals in the protected state have a lower fertility, but survive stress better. An individual's transition from the vegetative to the protected state is referred to as ‘phenotypic switch’. Responsive switching occurs in response to the signal, while random switching occurs without signal. Both traits are genetically encoded, and can thus evolve. Each individual has two loci to encode these two traits, and there are an infinite number of alleles at each locus, ranging from 0 (the organism never switches) to 1 (the organisms switches with probability one). Our goal here is to investigate how the evolution of these two traits depends on the environmental conditions. We employ two different modeling approaches: an analytical model, to calculate the long-term growth rate of a genotype, and an individual-based approach, to model the evolution of random and responsive switching in heterogenous environments. The results of the analytical model are then compared to the individual-based model, which gives us an idea of the impact of stochasticity as well as population effects on the evolution of phenotypic heterogeneity. (See
We first examine a situation where stress is always preceded by a signal, but where signals are not necessarily followed by stress. In other words, we assume that the probability of a signal,
In (
We then analyze the situation where every stress event is preceded by a signal, but there are more stress events than signals; formally, this corresponds to
We next consider a more general scenario where stress is not always preceded by signals and, as before, signals are not always followed by stress. Individuals can only protect themselves against stress that is not preceded by a signal if they sometimes switch randomly to a protected state, i.e., if their
We investigate how different signal reliabilities affect combinations of these two traits that maximize long-term growth rates. To vary signal reliability, we vary
Long-term term growth rate predicted by the analytical model in a single patch as a function of random and responsive switching values,
The consequences of the evolutionary dynamics of random and responsive switching are presented in
So far, we have assumed a simple ecological setting – a population that lives in a homogeneous environment, and where all individuals are always subject to the same conditions. Would the conclusions change substantially if we modified the ecological setting? To investigate this, we consider a situation where the population evolves in two spatially separated patches, and where the environmental conditions imposed in the two patches are independent from each other (see
We then again used an individual-based model to analyze the evolutionary dynamics with different types of density regulation, which are not captured in our analytical model. It is essential to include density regulation in our individual based model; without density regulation, the number of individuals will either decline to zero, or grow without limit. We thus assume that the environment has a constant carrying capacity,
Both local and global density regulation are relevant mechanisms in natural environments. An example for local density regulation is a bacterial infection: bacteria infect different hosts, and are exposed to selection and reproduce in those hosts, where population density is regulated locally. An example for global density regulation would be the following: individuals live in discrete patches that are spatially separated, but live off a resource that is freely diffusible. By consumption of this resource all individuals are equally affected, and their density is thus regulated globally.
With global density regulation, the phenotype that dominates the individual-based models after many generations is close to the combination of random and responsive switching that, according to the analytical model, maximizes the long-term growth rate (
To investigate this effect in more detail, we perform a pairwise invasibility analysis
Pairwise invasibility plots (PIPs) are shown for three different ecological settings: single patch (
The PIPs show that populations with small random switching values can be invaded by mutants with higher values, while populations with large random switching values can be invaded by mutants with lower values. There is an intermediate strategy (“singular strategy”) that cannot be invaded by any mutant. In other words, the singular strategy is convergence stable, and it is evolutionary stable
Interestingly, with the numerical resolution of our analysis, the singular strategy for one patch is indistinguishable to that for two patches with local density regulation. As discussed above, we expect two effects when increasing the number of patches from one to two. The variation in performance decreases, favoring risk prone types; and local density regulation promotes types that survive when most individuals in a patch die, favoring risk averse types. Our finding suggests that, at least for the conditions analyzed here, these two effects cancel each other, so that the singular strategy is the same for one or two patches. Increasing the number of patches further beyond two does not change the value of the singular strategy (not shown).
We have discussed above how local density regulation is expected to result in negative frequency dependent selection on the rate of random switching. This effect manifests in the PIPs: with local density regulation (but not with global density regulation, or with a single patch), there are combinations of
The two density regulation regimes we employ here have similarities to the concepts of ‘hard’ and ‘soft’ selection in ecology (
It is also interesting to note that the evolutionary dynamics of random and responsive switching does quantitatively depend on the population size: the individual-based model shows that, in small populations, both random and responsive switching evolve to slightly higher values than predicted by the analytical model (
Overall, our results point to the importance of probabilistic behavior in response to unreliable signals. We focused on environments where episodes of stress are usually preceded by a signal, but where this signal is not absolutely reliable. We find that such conditions promote the evolution of types whose phenotype expression is statistically associated with the signal, but also deviates from it in a significant way. In clonal populations of these types, not all individuals enter a protective state in response to the signal, and some individuals also enter this state when there is no signal. This probabilistic behavior balances the costs and benefits of stress protection. By limiting the number of individuals that respond to the signal, it decreases the average metabolic costs of protection. And by inducing the protective state in some individuals even in the absence of the signal, it increases the chance that the genotype survives rare events of stress that occur without warning. Interestingly, the costs and benefits of protection, and therefore the evolutionary dynamics of bet-hedging, depend on the ecological setting. Under conditions where the population is distributed across discrete patches, and where lone survivors of stress events in a patch benefit from reduced crowding, the benefit of surviving stress events increases. Consequently, such populations evolve towards a state where they are dominated by types that frequently enter the protective state even in the absence of a signal. These results emphasize the role of the ecological setting for bet-hedging. To describe the evolutionary dynamics of bet-hedging, it is not always sufficient to analyze the fit between the phenotypes expressed by a give genotype and the state of the environment. In some situations, the success of a bet-hedging strategy depends on the phenotypes expressed by others, and thus on the composition of the population.
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We thank David Johnson for critical reading of the manuscript, and Jan Engelstaedter and Frédéric Guillaume for helpful comments.