Feedback between environment and traits under selection in a seasonal environment: consequences for experimental evolution

Abstract

Batch cultures are frequently used in experimental evolution to study the dynamics of adaptation. Although they are generally considered to simply drive a growth rate increase, other fitness components can also be selected for. Indeed, recurrent batches form a seasonal environment where different phases repeat periodically and different traits can be under selection in the different seasons. Moreover, the system being closed, organisms may have a strong impact on the environment. Thus, the study of adaptation should take into account the environment and eco-evolutionary feedbacks. Using data from an experimental evolution on yeast Saccharomyces cerevisiae, we developed a mathematical model to understand which traits are under selection, and what is the impact of the environment for selection in a batch culture. We showed that two kinds of traits are under selection in seasonal environments: life-history traits, related to growth and mortality, but also transition traits, related to the ability to react to environmental changes. The impact of environmental conditions can be summarized by the length of the different seasons which weight selection on each trait: the longer a season is, the higher the selection on associated traits. Since phenotypes drive season length, eco-evolutionary feedbacks emerge. Our results show how evolution in successive batches can affect season lengths and strength of selection on different traits.

1. Introduction

Experimental evolution is a choice tool to study adaptation in controlled environmental conditions [1]. For microorganisms, the two main devices are chemostats and batch cultures [2]. In a chemostat, the population is kept under a constant growth rate while resources are added continuously. By contrast, for batch cultures, the resources are consumed and are not renewed. In the present paper, we focus on the consequences of environmental changes, as mimicked by successive batch cultures, on selective pressures and evolution of traits.

Batch culture often involves 24 h batches, with low resource concentrations, notably for yeast [3–11]. The cell population exhibits roughly an exponential growth followed by a stationary phase. However, population dynamics can be more complex. The depletion and production of resources and/or toxins can lead to a succession of different phases. When these phases are repeated periodically in successive batches, they can be seen as seasons. The make–accumulate–consume strategy of several yeast species [12] typically leads to a seasonal environment. In the presence of oxygen, several yeast species are able to ferment glucose if its concentration is sufficient. When glucose concentration reaches too low a level, cells enter a transition phase called the ‘diauxic shift’ during which they switch to respiration metabolism. In the respiration season, yeasts consume the ethanol they have previously produced and the population grows at a lower rate than during fermentation. Considering the ubiquity of metabolic plasticity in microorganisms [13,14], such examples of complex metabolic dynamics leading to seasonal structure might not be an exception but rather the norm, and should be accounted for in the interpretation of experimental evolution.

The consequences of seasonality have been studied experimentally in batch cultures [15] and theoretically using mathematical or computer models [16–18]. Different models (considering different levels of organization, from phenotypic to metabolic levels) have been developed and predict that coexistence may be observed when a trade-off exists between two traits selected in two different seasons [16,17,19]. This coexistence has been observed experimentally, notably in the long-term experimental evolution (LTEE) of Escherichia coli [20], where it has been shown that a few mutations distinguish the two coexisting strains, and affect resource consumption and the efficiency of the diauxic shift [17,21–23].

The analysis of fitness in seasonal environments still remains limited by our ability to take into account the interaction between genotype and environment. Usually, fitness is characterized and measured using the ratio of growth rates of two strains over the period of the whole culture; the strain having the higher net growth rate being the fittest [24–26]. However, this does not provide much insight into why one strain performs better than another in a given seasonal environment. One way to predict the population dynamics of competing strains is to decompose fitness into components [27,28]. Different decompositions have been proposed for seasonal environments [27–29]. Some only include growth rates [28] and others account for frequency dependence or trait dependence [29]. However, fitness also depends on the environment and its descriptors should be included in fitness [30].

In this paper, using a mathematical modelling approach, we propose a decomposition of the fitness function accounting for the strain-induced environmental changes. We show that this decomposition retrieves results on coexistence in seasonal environments [15–18]. It also provides a simple tool to understand which traits are under selection in a given environment and how the selection of these traits acts on the environment and feeds back on the weight of the different fitness components. This provides new insights into ecoevolutionary feedbacks and the traits potentially involved in historical contingency and intransitive fitness.

2. Models and methods

(a). The model

We modelled the population dynamics of n competing strains in one batch culture where cells produce biomass successively by fermentation and respiration. In this model, cells only interact indirectly via competition for resources and production of ethanol, which is both a resource and a toxic compound. It is described by a set of differential equations adapted from MacLean & Gudelj [31].

2.1

G(t), simplified to G, is the glucose concentration at time t, E(t), simplified to E, is the ethanol concentration at time t, and for all i in 1 to n, N_i(t), simplified to N_i, is the cell concentration of strain i at time t. When the glucose concentration is high, the cells consume the glucose by fermentation at a rate J_f,i.G/(K_f,i + G), where J_f,i (g h⁻¹ cell⁻¹) is the maximum glucose consumption rate of strain i and K_f,i is the glucose concentration at which the glucose consumption of strain i is half its maximal glucose consumption, called saturation constant for glucose. For fermentation, the ethanol/glucose yield is denoted p_i (in grams of ethanol produced per gram of glucose consumed) and the ‘cell’ yield is denoted y_f,i (in a number of cells produced per gram of glucose). The maximum population growth rate in fermentation is denoted r_f,i = J_f,i y_f,i (h⁻¹). When the glucose resource is depleted, yeast switch to respiration and consume the ethanol at a rate J_r,i.E/(K_r,i + E), where J_r,i (g h⁻¹ cell⁻¹) is the maximum ethanol consumption rate and K_r,i is the ethanol concentration at which the ethanol consumption is the half of the maximal ethanol consumption, called the saturation constant for ethanol. To model the Crabtree effect, we multiply the growth rate by the term K_c,i/(K_c,i + G) which simulates the inhibition of respiration by the glucose and where K_c,i (g ml⁻¹) is an inhibition constant. For respiration, the ‘cell’ yield is denoted y_r,i. The maximum population growth rate in respiration is denoted r_r,i = J_r,i.y_r,i (h⁻¹). Owing to the toxicity of ethanol, the yield y_f,i is decreased by a factor tox(E) = exp(−E/E_m,i) where E_m,i denotes the tolerance to ethanol (g ml⁻¹) [32]. The cells' mortality rate, denoted m_i (h⁻¹) is considered constant over the batch for the sake of simplicity. Indeed, the number of observations would not allow the inference of mortality variations along the batch.

For n strains in competition, our model is therefore composed of (2 + n) differential equations with a total set of 10n parameters (J_f,i, r_f,i, m_i, E_m,i, p_i, J_r,i, r_r,i, K_c,i, K_f,i, K_r,i) where G(t), E(t) and N_i(t) are, respectively, the glucose (g ml⁻¹), ethanol (g ml⁻¹) and cell concentration of strain i (cell ml⁻¹) at time t.

The system was implemented in C and solved by the Runge–Kutta method.

(b). Model calibration

(i). Experimental data

We fitted the ‘one-strain version’ of the model to data from 72 different strains obtained from a 6-month experimental evolution described in detail in [11] and in electronic supplementary material S1. To fit the one-strain model, for each strain, we used observed data of glucose concentration G(t), ethanol concentration E(t), cell density Z(t) and the survival rate s(t) monitored over a 96 h batch monoculture in a 15% glucose medium [11].

(ii). Parameters inference

For each strain, we decided to estimate seven parameters (J_f, r_f, m, E_m, p, J_r, r_r) of the model. The remaining parameters (K_f, K_r and K_c) were set at 5 × 10⁻⁴ g ml⁻¹. We checked that parameter estimates were not sensitive to (K_f, K_r and K_c) values. We also allowed the model calibration to involve the initial glucose concentration G₀ and the initial population concentration N_0. These initial conditions were estimated as they may vary from one batch to the other due to experimental variations.

To estimate the parameters θ = (J_f, r_f, m, Em, p, J_r, r_r, G₀, N₀), we used the ABC-SMC algorithm [33]. The distance d(X_obs,X_pred) between data and model predictions was calculated as the Euclidian distance between the corresponding values of G(t), Z(t), E(t) and N(t) = Z(t).s(t) at times 3, 6, 9, 12, 16, 24, 48, 72 and 96 h, normalized by the observed values and the number of observations (n_N, n_Z, n_E and n_G). Observed ethanol concentration at 16 h was always lower than at 12 and 24 h, presumably because an additional ethanol dilution had to be carried out to stay in the condition of proper linear quantification by the kit. That extra dilution may have impacted all dosages at this point. Therefore, we decided to discard these observations in the estimation procedure. In order to test the accuracy of the model, the distributions of standardized residuals were examined for each parameter. More details and the algorithm are provided in electronic supplementary material S1 and S2, as well as the detailed analysis of these results.

(c). In silico experiments of competitions

One thousand strains were generated in silico by sampling 1000 sets of parameters in the parameter prior distributions (calculated from the estimated parameters based on experimental data) using uniform latin-hypercube sampling [12] (electronic supplementary material, table S2). For each couple of strains, we simulated pairwise competitions over one batch between a resident strain (initial concentration = 10⁶ cells ml⁻¹) and a mutant strain (initial concentration = 10 cells ml⁻¹), leading to a total of 999 000 competitions.

For each competition, the effective length of seasons was calculated for each strain using Riemann formula with time step of 0.02 h and 4800 steps. The sensitivity of fitness and effective season lengths to the trait values was analysed using partial rank correlation coefficients (PRCC) using the package epiR [34].

We also quantified the frequency-dependence of fitness by simulating in silico competitions over one batch as described above with different initial frequencies of the mutant and resident strains. For each couple of strains, we simulated pairwise competitions with different initial frequencies of the mutant strain (10⁻⁶, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.999999 cell ml⁻¹). For each couple, we determined whether coexistence was possible by finding initial frequencies where the fitness reached 0.

Finally to address the long-term dynamics of competition between two strains, we simulated the competition over several successive batches. To go from one batch to the next, we reset the glucose concentration to 0.15 g ml⁻¹, the ethanol concentration to 0, the population concentration to 10⁶ cells ml⁻¹ while keeping frequency of each strain at its final frequency in the preceding batch.

3. Results

To explore traits under selection, we built a system of differential equations that simulates a population of S. cerevisiae and its effects on resource dynamics in respiro-fermentative batches. We considered three seasons according to resource availability: first a fermentation phase where glucose is consumed and ethanol is produced; then a respiration phase where ethanol is consumed; and finally a mortality phase where resources are exhausted and cells are quiescent or dead. The environment is described by glucose and ethanol concentrations. Ethanol is considered as both a resource and a poison inhibiting growth. The model has been built for multiple strains, interacting only through the environment. Each strain is characterized by its population dynamics, which depends only on resource concentration. To calibrate and validate the model, we fitted the ‘one strain version of the model’ to experimental data regarding 72 strains grown separately (figure 1; electronic supplementary material S1). These fits provided 72 sets of parameter values that were used to define their range in simulations.

Figure 1.

Example of predictions of the model for one strain evolved in a 1%–96 h environment (a) for the total density of cells, (b) for the logarithm of the density of living cells, (c) for the glucose concentration, (d) for the ethanol concentration. Grey curves are the 10 000 best simulations obtained from the ABC algorithm. Black points are experimental data.

The model was then used to analyse and simulate the dynamics of competing strains to study fitness components. We analysed the traits under selection during each season and how season lengths affect selection. We then highlighted the feedbacks between selection and the environment and explored their potential consequences on life-history traits adaptation.

(a). Fitness components

To assess which traits are selected during a batch culture, we used a decomposition of the fitness function. Using the Malthusian fitness function [14], the fitness in a batch of length T, W_1/2(T), can be defined as

3.1

If the two strains have the same traits, ln(N₁/N₂)(T) = ln(N₁/N₂)(0) leading to a null fitness. If W_1/2(T) > 0, strain 1 is fitter than strain 2.

Combining equations (2.1) and equation (3.1), one obtain:

3.2

To simplify notation in this expression, E(t) and G(t) were, respectively, denoted E and G. For each strain (i = 1,2), let us define

3.3

and

3.4

We called these quantities ‘the effective season length’ of fermentation and respiration . Considering the simplified model where E_m → ∞, K_r → 0, K_f → 0 and K_c → 0, T^ferm is the time to glucose exhaustion and T^resp the time to ethanol exhaustion (see electronic supplementary material S3). In this simple case where concentrations of resources vanish in a finite time, the ‘effective season length’ corresponds to the period of time where the resource is effectively used. In the general case, they depend on resources, the strains' parameters and how these strains react to abiotic conditions.

With (3.3) and (3.4), fitness can be written as

3.5

Alternatively, (3.5) can be written in terms of differences between strains as

3.6

For each parameter θ (in r_ferm, r_resp, m, T_ferm, T_resp), Δθ = θ₁ − θ₂. Therefore, the fitness is composed of a sum of terms, each one describing a different component of differences between two strains weighted by the length of a season.

The first three terms correspond to the direct advantages given by life-history traits: growth rates during fermentation (Δr_f) or respiration (Δr_r) and mortality (Δm), weighted by their respective season length for the reference strain 1. The last two terms depend on strains differences for their effective season length ΔT^ferm, and ΔT^resp, weighted by the corresponding seasonal growth rate of the competing strain 2. For each season, a positive ΔT term means that strain 1 is able to take advantage of the resources efficiently over a longer period than strain 2. Hence these terms, called transition terms in this article, measure how strains respond differently to environmental changes.

To interpret this equation, consider two strains differing by one single model parameter. Two cases can be considered. Case a: this parameter is a life-history trait (r_f, r_r or m): effective season lengths will be equal (equations (3.3) and (3.4)) so ΔT^ferm = 0 and ΔT^resp = 0 and or

3.7

depending on the considered parameter. Case b: this parameter is not a life-history trait but affects the effective season lengths T^ferm and T^resp of each strain. Then

3.8

or

3.9

Hence, any parameter θ for which a strain difference leads to a difference in effective season length is under selection.

The effective season length of one strain can depend on the characteristics of its competitor. For example, having strain 2 consume resources faster will lead to a shorter season for strain 1. As the effective season length weights the advantage conferred by a life-history trait (case a), any competitor trait that changes season length will change selection strength on life-history traits. Similarly, for traits for which strain differences impact ΔT (case b), selection will be higher if the growth rate of the corresponding season is high.

In summary, fitness is composed of five terms: three can be considered as life-history terms and two are transition terms, related to the ability to keep the same metabolic behaviour (fermentation or respiration) despite the environment changes. Life-history terms depend on differences in growth or mortality rates in each season weighted by its length. The transition terms are differences in response to environment changes weighted by growth rates. Hence, a strain has two ways to be fitter than its competitor: having beneficial life-history traits (i.e. having faster growth or lower mortality), or taking better advantage of the available resources.

(b). Impact of traits on effective season lengths

According to equation (3.6), traits subject to selection are either life-history traits (r_f, r_r or m) or ones changing ΔT^ferm or ΔT^resp. To explore how fitness may be affected by variations of traits, we therefore explored how these variations impact the terms , , ΔT^ferm and ΔT^resp.

We tackled this question using a set of in silico competitions of 1000 strains, computing the effective season lengths (, , , ). Each strain was characterized by a set of traits sampled in a prior distribution based on experimental data and the literature. In each simulated competition, strain 2 is the resident strain (R, high initial frequency) and strain 1 is a mutant strain (M) whose very low frequency is such that it does not impact the resource dynamics. Hence if a resident trait is correlated with the season length of the mutant, this trait impacts season length via resource dynamics. On the contrary if a mutant trait correlates with its own season length, this means that the mutant impacts season length via the strain's response to the environment.

The partial rank correlation coefficient (PRCC) between season length and the individual traits of the mutant and the resident were estimated. We first explored traits that are subject to selection and then traits that affect selection strength.

(i). Traits subject to selection

A trait was considered as a life-history trait when the difference for this trait between the mutant and the resident led to variations in one of the two transition terms ΔT^ferm or ΔT^resp. PRCC between the effective season length differences and the trait differences are shown in table 1. Only two metabolic traits were subject to selection: ethanol tolerance E_m and respiration saturation constant K_r. The difference in ethanol tolerance between both strains (ΔE_m) tended to increase both ΔT^ferm and ΔT^resp, indicating that a higher tolerance to ethanol will be selected for. In the same way, ΔT^resp slightly decreased when the difference between saturation constants increased (ΔK_r), indicating that a lower saturation constant will be selected for. Indeed, the lower the saturation constant, the longer a strain is effective in using that resource.

Table 1.

Link between difference of mutant and resident season length and traits. Partial rank correlations between the difference of length of fermentation and respiration season and traits of the two strains computed with 100 000 strain couples. The traits indexed R are the traits of the resident, the sign Δ indicates the difference between the traits of the mutant and the traits of the resident. Italicized correlations are correlations larger than 0.2 in absolute value.

	rf,R	rr,R	mR	Em,R	yf,R	yr,R	pR	Kf,R	Kr,R	Kc,R
ΔTferm	0.123	−0.004	−0.004	0.455	0.062	0.035	−0.327	−0.038	0.036	−0.023
ΔTresp	−0.001	0.115	0.002	0.274	0.150	−0.053	−0.374	−0.030	0.094	−0.051

	Δrf	Δrr	Δm	ΔEm	Δyf	Δyr	Δp	ΔKf	ΔKr	ΔKc
ΔTferm	0.014	−0.180	0.014	0.940	0.148	0.028	0.009	−0.029	0.006	0.001
ΔTresp	0.006	−0.010	0.019	0.837	0.009	0.021	0.001	−0.004	−0.230	0.033

To summarize, higher growth rates, lower mortality rates, higher tolerance to ethanol and, to a lesser extent, lower respiration saturation constants will be under selection.

(ii). Traits that change selection strength

We first investigated the effect of mutant and resident phenotype on the mutant season lengths and . PRCC between the mutant effective season lengths ( and ) and the traits are shown in table 2. A positive correlation is expected to enhance the efficiency of selection on growth rate, while a negative correlation is expected to decrease it.

Table 2.

Link between season length and traits. Partial rank correlations between the length of the fermentation and respiration season and traits of the two strains computed with 100 000 strain couples. The traits indexed by R are the traits of the resident, traits indexed by M are the one of the mutant. Italicized correlations are correlations larger than 0.2 in absolute value.

	rf,R	rr,R	mR	Em,R	yf,R	yr,R	pR	Kf,R	Kr,R	Kc,R
	−0.948	0.031	0.548	−0.272	0.788	0.057	−0.522	0.129	0.097	−0.044
	−0.174	−0.578	0.475	−0.442	−0.764	0.628	0.829	−0.013	0.116	−0.085

	rf,M	rr,M	mM	Em,M	yf,M	yr,M	pM	Kf,M	Kr,M	Kc,M
	0.004	0.007	0.002	0.751	0.020	−0.007	0.014	−0.019	0.007	0.007
	−0.005	−0.006	0.009	0.589	0.011	0.001	0.029	−0.007	−0.093	0.016

Tolerance to ethanol E_m,M is the only trait of the mutant which affects the effective season lengths of the mutant (PRCC = 0.75 for and 0.58 for ). Otherwise the mutant effective season lengths are affected by traits of the resident strains, via the resource dynamics which can be modified by different mechanisms. Increasing the cell concentration reduces resources more rapidly, reducing the corresponding season length. This is in line with the negative correlation of the season lengths and with the growth rates of the resident (r_f,R and r_r,R) and their positive correlation with the mortality rate of the resident m_R. In addition, the respiration season length is negatively impacted by the tolerance to ethanol of the resident E_m,R. and the fermentation yield of the resident y_f,R, as they both increase the cell concentration in this season. Similarly, traits that impact the efficiency of resource consumption (at constant growth rate) will spare the resources and consequently increase the corresponding season length. This is in line with the positive correlation of the fermentation (respectively respiration) season length (resp. ) with the fermentation of the resident (resp. respiration) yield y_f,R. Finally, traits that impact resource production such as the ethanol/glucose yield p_R can be positively correlated to the season length where this resource is consumed.

We also investigated the effect of mutant characteristics on the transition terms ΔT^ferm and ΔT^resp. As shown previously, the only trait under strong selection through the transition terms' fitness component is ethanol tolerance. If (ΔT) is correlated with the trait of the resident, this trait will therefore affect the efficiency of selection on ethanol tolerance. PRCC between the transition terms and the resident traits are shown in table 1. We found that an increase in ethanol tolerance increased both ΔT^ferm and ΔT^resp, while an increase in the ethanol/glucose yield (p_R) decreased both terms.

Altogether, our results showed that ethanol tolerance and saturation constants are under selection and the strength of selection on these traits is weighted by the ethanol/glucose yield of the resident strains. Besides, the strength of selection on growth rates is modulated by almost all resident characteristics, except for the saturation and inhibition constants, as well as by the mutant ethanol tolerance.

(c). Feedback loops between traits under selection

Just as trait values affect season lengths, season lengths drive the strength of selection on traits (equation (3.6)). As a consequence, a feedback loop between selected traits and the environment arise: selection affects trait values, the associated changes modify season lengths, and then these modified season lengths impact the trait values via selection. Such eco-evolutionary feedbacks have been described in different situations and are known to have consequences on the evolutionary trajectories by promoting diversification such as branching or coexistence [35]. In this section, we explore and illustrate these feedbacks.

(i). Description of feedback loops

Four traits are involved in feedback loops between selection and environment: the growth rates (r_f and r_r), the mortality rate (m) and the tolerance to ethanol (E_m). As seen in §3a, selective pressures tend to increase the fermentation and respiration growth rates (equation (3.7)), to decrease the mortality rate (equation (3.7)) and to increase the tolerance to ethanol (table 1). The evolution of these four traits has different consequences on the effective season lengths and thus on the strength of selection. All the results described here are a compilation of the results from the previous section, and are summarized in electronical supplementary information S4. We consider the effect of a change in the resident trait on the mutant's fitness, and therefore which mutants are selected.

An increase of the growth rate (resp. r_f and r_r) shortens the season (resp. T_f and T_r) (table 2) and thus decreases the weight of the difference of the growth rate in the fitness function (equation (3.6)). Growth rates also weight ΔT^ferm and ΔT^resp (table 1), and consequently increase selection on ethanol resistance.

A decrease of the mortality rate shortens both the fermentation and the respiration seasons (table 2). Therefore, selection on mortality rates tends to decrease selection strength on growth rates, and indirectly increases selection on mortality rate.

Finally, selection tends to increase the tolerance to ethanol, which in turn decreases the length of the fermentation and respiration seasons (table 2), but increases ΔT^ferm and ΔT^resp (table 1). Consequently, selection on tolerance to ethanol tends to decrease selection on growth rates and increase selection on itself.

The fermentation yield, respiration yield and ethanol/glucose yield are not selected in our model, and consequently they are not involved in any feedback loop. Indeed, if only the yield varies between two strains, every ΔT term and Δθ term will be null, and so will be fitness (equation (3.6)). However, yield can affect season lengths and therefore enhance or inhibit selection on one trait. For instance, a higher yield (resp. y_f and y_r) enhances selection for higher growth rate (resp. r_f and r_r) (table 2).

(ii). Consequences of eco-evolutionary feedbacks in batch cultures

As shown above, in batch cultures, because of the dynamics of nutrients and toxins, the strength of selection is not constant during the evolution and can sometimes depend strongly on the resident strain. One potential consequence is non-transitivity of fitness. For instance, C can invade B but not A, B can invade A but not C, A can invade C but not B (figure 2; electronic supplementary material S5). Another consequence is the possible coexistence between strains via frequency dependence. When the mutant is at very low frequency, its impact on the dynamics of the medium is negligible. However, as its frequency increases, its impact on the resources and toxin dynamics can become essential. As a result, the effective season lengths will be modified gradually as well as the mutant's fitness. Its fitness can decrease and even reach zero. Note that as W_1/2 = −W_2/1, coexistence implies that the two strains have a negative fitness when they are resident [36], and this is possible if at least two life-history traits are different between the strains. Indeed, if only one trait differs between two strains, the fitness is reduced to a single term (equations (3.7), (3.8) or (3.9)) and cannot vanish.

Figure 2.

Competition between three strains and non-transitivity. On each panel, the frequency of the strain (blue for strain A, red for strain B and green for strain C) at the end of each batch is represented. The strains are propagated through 100 batches of 90 h. The text indicates the traits which influence fitness the most and are responsible for the invasion in the first batches (on the left) and in the later batches (on the right). Tolerance to ethanol and mortality are selected together, but we only indicate the traits responsible for the positive fitness. Parameter values are given in electronic supplementary material S3.

To explore the relationship and the occurrence of coexistence between two strains, we have generated 100 strains and simulated 10 000 pairwise competitions (see Models and methods). In 44% of these competitions, fitness was negative frequency dependent, indicating a negative feedback. In 8% of the simulations fitness sign changed indicating possible coexistence. In half of these 8%, there was a stable coexistence equilibrium. Lastly 93% of the simulated strains coexisted with at least one other strain.

4. Discussion

There have been several studies in the literature regarding how to define and measure fitness [24–26,30,37–39]. For a seasonal environment, these definitions are often based on the average growth rate in competition but they do not incorporate explicitly the determinants of this growth rate nor the environment. Here, we developed a mechanistic mathematical model which explicitly includes life history and resource use, as well as toxicity traits and their impact on both the population dynamics and the environment. For that we followed previous authors who modelled environmental variations such as the duration of daylight [27] or the amount of different resources [20,31]. Because our purpose was to provide a definition of fitness that includes the effect of the environment on the strain dynamics, we defined the ‘effective season length’, which can be interpreted as the time needed for a specific strain to reach the same fitness if growth rates were constant and maximal. In a general case, effective season lengths do not only rely on the level of resources, they also depend on the strains interaction with environmental changes. This definition implies that season lengths are not exactly the same for every strain, even if they are put into the same initial environment. The fitness function can be interpreted as a sum of the advantages in each season, weighted by the season's length. This decomposition allows classifying traits and studying eco-evolutionary feedbacks.

Our work shows that traits can be grouped according to their response to selection and their impact on the environment. The first group is composed of life-history traits (growth and mortality rates). These traits are well known as fitness components [37] and are responsible for a ‘selective advantage’ in each season. The second group contains transition traits (saturation and inhibition constants related to resource consumption and metabolism switches) which are related to environmental changes: the strains which are less sensitive to changes in levels of resources will have a selective advantage. These traits are less frequently studied, because their effect on the usual function of fitness (i.e. ratio of growth rates [24,28]) arises solely via their impact on average growth rates. An example of selection on these two kinds of traits can be found in the LTEE on E. coli [20]. After 20 000 generations, evolved strains are able to grow faster, but also have a lower saturation constant and consume resources sooner than their ancestor when resources are renewed. Other works showed that the ability to switch faster from one resource to the other has been selected during the LTEE [15,17,19,40,41].

Selection pressure also depends on season length. The impact of season length on selected traits has been studied both theoretically and experimentally in batches by considering cheaters and cooperators [31], in chemostat cultures [27], and in organisms with more complex life cycle and thus having for instance more elaborated consumption strategies [42]. Several experimental evolution studies have also shown that season lengths drive traits selection in batch culture. This has been found for instance in a study by Pekonnen et al. [43] where bacteria (Serratia marcescens and Novosophingobium capsalatum) evolving in long batches of 7 days were mainly selected for a lower mortality rates rather than higher growth rates. On the contrary, in many other works [3–8,10,20,44,45] using one-day-long batches, the growth rate increased while the mortality rate remained unchanged. This is for example the case in the experimental work on which our model was based and in which we compared the evolution of life-history and metabolic traits in batches differing by their batch lengths [11]. In media with long batches, the growth rate seems to increase less than in media with short batches. These examples show how experimental evolution design can affect the outcome of selection.

Besides the effect of season lengths on selection outcomes, the dynamics of strains will affect season lengths. The variation of strain frequencies over successive batches may change season lengths, forming feedback loops that can favour or disfavour different traits. Fitness in batch systems is inevitably frequency dependent. The associated feedback loops are crucial to understand evolution and adaptation [36]. In an experimental evolution with successive batches, the emergence of different strains will typically alter season lengths, which will in turn change the strength of selection on different traits. For example, in case of cross-feeding [17,19,40,41], one strain produces a resource used by another one; the ability of the ‘producing strain’ to release resources will modulate the length of the season where this resource is consumed, and consequently selection on the consuming strain.

Here, we have highlighted several feedback loops where the strength of selection should decrease over the course of evolution. For example, fermentation growth rate increases due to selection, but the strength of selection on this trait will decrease with batch number. This phenomenon is likely to be quite general: as a trait responds to selection, the corresponding season where the selection is effective typically shortens, and consequently the selective pressure will decrease with evolutionary time.

Regarding experimental evolution, the feedback loops provide an explanation of some observed non-trivial patterns. For instance, there are examples of experimental evolution aiming at comparing evolutionary outcomes in stressed and non-stressed environments. One way to design a stressful environment is to add a poison [3–5], which can increase the mortality rate. However, according to our simulations, fermentation season length is positively correlated with mortality rate. Hence, in the ‘stressed’ environment, the fermentation season will be longer and, as a consequence, the stressed and non-stressed environments will also differ in the resource dynamics. Then, a higher selection on growth rate is expected in the ‘stressed’ environment than in the non-stressed environment. As the experimental measurement of the effective season length may be difficult, this effect will be hard to quantify and account for without the use of a mathematical model incorporating traits and their impact on the environment. However, taking the seasons or a proxy into account should allow us to predict the outcome of adaptation more reliably than looking solely at phenotypes.

Feedback loops in batch culture may lead to difficulties in practical fitness measurement due to frequency dependence. Fitness is often measured by performing competitions between the initial ancestor(s) and the evolved strains. In a single season environment, this measurement provides an accurate measurement of adaptation and changing the competing strain or its frequency does not affect the ranking of fitness values [39]. But in more complex cases, the environment and notably season lengths can evolve with competing strains, and consequently an evolved strain may be less fit than its ancestor due to non-transitivity of fitness. This non-transitivity of fitness, observed in several experimental evolutions [46], is described by the ‘rock–paper–scissors’ game [47] and has been studied in many scenarios within spatially structured environments or via clonal interference [48,49]. Our work highlights that modelling the population and environment dynamics for decomposing fitness over a season cycle can help understand such non-transitivity.

In the present work, we quantified the genericity of coexistence using in silico competitions, sampling the traits values independently. Adding some metabolic constraints between traits could either prevent or enhance the coexistence pattern [27,50] and could help to explain why in some cases no significant change of the fermentation growth rate over the course on the evolution is observed [51]. For instance, coexistence by cross-feeding is possible if a negative relation exists between the ability to produce the resources and the efficiency to use it [17,19,40,41]. In a future work, we are planning to study the interplay between trade-offs and eco-evolutionary feedbacks in seasonal environments.

Exploiting the data obtained in [11], the model captures the fermentation-respiration behaviour of most strains although two biases have been observed. First, for strains which begin respiration while the glucose concentration is still high, the model cannot fit concentrations of both resources. Second, the mortality tends to be overestimated. This trend could be due to the choice of a constant mortality. However, because of the low number of observations for mortality, it was not possible to consider more complex mortality functions. We cannot dismiss that mortality rate has increased along the batch due to a lack of resources or osmotic stress. To investigate the variation of mortality, one would need more detailed data throughout the batch. However, whatever the form of the mortality rate, it should appear in the decomposition of fitness as terms accounting for the trait differences between the two strains and terms accounting for the difference between efficient season lengths. Then, the fitness decomposition could be used to understand selection pressures on traits related to mortality. Hence, despite its simplicity and limits, this approach helps investigating how eco-evolutionary dynamics can shape evolutionary paths in experimental evolution in batch cultures of organisms exhibiting several metabolic phases.

To conclude, we proposed a decomposition of the fitness function which allows to better grasp selection pressure. In the perspective of setting up an experimental evolution experiment, these results can be useful at two levels. First, results on feedback loops can be used to propose alternative interpretations of evolutionary outcomes. Second, they can guide the scientist in the design of the experimental evolution conditions. The scientist could play on environmental factors to exploit positive feedbacks that will strengthen the selective pressure while avoiding negative feedbacks. In the context of technological applications such as strain selection for industrial fermentation, this could provide a way to better select for a targeted character and avoid collateral damage coming from indirectly selecting for the wrong trait. In practice, further work is needed to explore how to translate such theoretical results into a tool for adapting batch length or other environmental factors to real-time observations of evolutionary dynamics.

Data accessibility

From the Dryad Digital Repository: http://dx.doi.org/10.5061/dryad.5vn65dh.

Authors' contributions

D.C., T.N., D.S., J.L. conceived the study and developed the model; D.C., T.N., S.M., O.M., C.D., J.L. contributed to the mathematical development and statistical analysis; D.C., T.N., J.R., C.D., D.S., J.L. interpreted the results; D.C. performed the analysis, D.C., T.N., D.S. and J.L. wrote the manuscript which was edited by all authors. All authors read and approved the final manuscript.

Competing interests

We declare we have no competing interests.

Funding

This work is supported by the ‘IDI 2014’ project funded by the IDEX Paris-Saclay, ANR-11-IDEX-0003-02.