# Synthetic Symbiosis under Environmental Disturbances

## ABSTRACT

**IMPORTANCE**The power of synthetic biology is immense. Will it, however, be able to withstand the environmental pressures once released in the wild. As new technologies aim to do precisely the same, we use a much simpler model to test mathematically the effect of a changing environment on a synthetic biological system. We assume that the system is successful if it maintains proportions close to what we observe in the laboratory. Extreme deviations from the expected equilibrium are possible as the environment changes. Our study provides the conditions and the designer specifications which may need to be incorporated in the synthetic systems if we want such “ecoblocs” to survive in the wild.

## INTRODUCTION

*Symbiodinium*symbioses or plant-rhizobium interactions are well-known (4–6). Many such mutualisms have evolved over millions of years. But if mutualisms are fragile and susceptible to collapse, as hypothesized, then how do they survive for eons in continually changing environments?

*LYS*↑ and

*ADE*↑. The dynamics of this cross-feeding system are comparable to a simple but powerful theoretical model of self-organization—a hypercycle. This model has extensive applications from explaining how life may have originated to how complex communities could form (11, 18, 21–23). On the basis of this theoretical background, we develop a simple but powerful and easily extendable phenomenological model. We then use the S. cerevisiae yeast system to validate the model and finally explore beyond the synthetic system to understand the stability and cost of cooperative interactions in changing environments. We establish the baseline mutualistic properties of this system and then identify conditions that can potentially disrupt this natural state. The proportions of the mutualists are our property of interest, which favorably comes close to ≈50% here. We define a proportion range around the equilibrium, here 20% to 80% where both the mutualists can be observed at similar frequencies. Where the mutualists exist naturally at a different equilibrium, the proportion range would need to be defined accordingly. Ecology of an organism is made up of both biotic as well as abiotic factors. While it is clear that changes in biotic ecology (population densities) affect the interaction pattern, it is essential to take into account the effects of the abiotic part of the ecology (24, 25). With this aim in mind, we begin with our theoretical and experimental model.

## RESULTS

*LYS*↑ and

*ADE*↑, and the densities of the

*LYS*↑ and

*ADE*↑ strains are denoted by

*x*and

_{L}*x*, respectively. The

_{A}*LYS*↑ strain is deficient in adenine, while it overproduces lysine and vice versa for the

*ADE*↑ strain. Modifying the logistic growth equation gives the dynamical equations for the growth of the two strains:

*r*. The densities of the available metabolites are given by

_{i}*c*(adenine) and

_{A}*c*(lysine). The strains compete for a limited amount of space, given by

_{L}*K*. Similar to the study reported in reference 18, we neglect the death rates assuming that the birth rates sufficiently characterize the growth. Additionally, we show that death is negligible over the time course that we focus on (see Materials and Methods). Metabolite concentrations, together with Monod-type saturation kinetics, control the growth of the strains. Although deriving a mathematical model for such a pairwise system can exclude metabolites, the explicit inclusion of metabolites is crucial, as pairwise Lotka-Volterra models may not always provide a realistic qualitative picture of the dynamics (14). Metabolites, in this particular case, are a consumable resource. Furthermore, the level of abstraction provided in equation 1 is adequate to focus on the relative concentrations of the mutualists. In reference 26, the authors focus on the postlag steady-state growth rate. A detailed predictive mathematical model for growth rate would need to match with precise measurements of the birth and death rates and the metabolite release and consumption rates. Our goal is more straightforward, and hence, in this case, we forgo the use of a complicated system with multiple parameters.

*), the other strain uses them immediately (at a maximum rate of γ*

_{i}*). The metabolite density dynamics hence can be captured by:*

_{i}*by strains*

_{i}*x*also involves the formation of an intermediate, thus being subject to Michaelis-Menten kinetic parameters. The simple dynamics of such a system are depicted in Fig. 1.

_{i}*ADE*↑ and

*LYS*↑ growth rates for different amounts of supplementation are shown in Materials and Methods, and the raw data have been provided. Using changes in the growth rates, we estimate the relative values of

*r*and

_{i}*k*for the two strains. The production rate of the metabolites is assumed to an order of magnitude smaller than the growth rates, and the maximum rate of uptake is set to unity (γ

_{i}*= 1). The parameters used are*

_{i}*r*

_{1}= 1 and

*r*

_{2}= 2 and the Michaelis constants ${k}_{{c}_{A}}=2$ and ${k}_{{c}_{L}}=1$. The metabolites are produced by the strains at β

*= 0.1. We assume the Michaelis constant estimated for the uptake rate for adenine and lysine to be the same as the rate at which they degrade from the pool (γ*

_{i}*= 1). At various initial conditions of the*

_{i}*ADE*↑-to-

*LYS*↑ ratio, the final equilibrium values are consistently close to 0.6, corroborated by experiments shown in Fig. 2.

### Ecoevolutionary dynamics under environmental disturbance.

### Initial supplementation. (i) Theory.

### (ii) Experiments.

*LYS*↑ strain, resulting in a lower prevalence of the

*ADE*↑ strain compared to the unsupplemented condition. The mean

*ADE*↑-to-

*LYS*↑ strain ratios after 120-h growth of all starting ratios for 0.1, 1, or 10 μg/ml adenine supplementation are 0.53, 0.57, and 0.056, respectively. For both Fig. 2 and Fig. 3, we include ecoevolutionary dynamics. The effect seen is localized to evolutionary space rather than to ecological space, since the growth rates are always positive, but only relative growth rates are relevant. In the standard batch culture regime, the added metabolites will be used up, and then the culture will be dominated by the internal production levels of adenine and lysine. Without the addition of extra nutrients (or efflux of waste) as in a chemostat, the equilibrium of unsupplemented case is not recovered. These stringent ecological conditions also make the experimental conditions compared to the mathematical model.

*ADE*↑ strain at a given time point as we change the amount of initially supplied metabolites. For the unsupplemented case, we know the equilibrium density both from experiments and theory (Fig. 2). For the initial supplementation regime, we assume the system to have equilibrated once the values of the strain densities do not change more than 10

^{−4}between two consecutive numerical time steps. The equilibration takes typically 30 time steps, but we show the snapshot of the relative densities from time point 300 in Fig. 4 (left panel). Experimentally we tested the addition of only adenine for different initial fractions of the

*ADE*↑ strain, but theoretically, we explore the consequence of adding both adenine and lysine for the

*ADE*↑ strain starting at 0.5. The results, summarized in Fig. 4 (left panel), reveal that even a slight asymmetric increase in the amount of initial metabolite present in the environment is enough to destabilize the equilibrium. When the equilibrium values come close to one of the two edges of the system, one mutualist is overrepresented compared to the equilibrium. We define this overrepresentation, compared to the equilibrium, as being dominant in the population. For our system, since the equilibrium is close to equality, we symmetrically choose 0.2 as the minimum acceptable threshold fraction for mutualism to be fair in terms of the mutualist densities. As the effects of environmental perturbation are observed in evolutionary space, rather than ecological space, we use frequencies of the mutualists to define how far the system moved from the unsupplemented equilibrium. The asymmetry of the graph in Fig. 4 reflects the inherently different uptake rates of metabolites by the two strains in Fig. 2.

### Intermittent supplementation. (i) Theory.

*c*

_{A}_{cont}and

*c*

_{L}_{cont}(where cont stands for continuous supplementation) determine the amount of metabolite added to the culture. The addition takes place at the end of the time interval determined by Δ

*t*which defines the cycle length. This changes the model from a smooth dynamical system to a hybrid dynamical system (27). We start with the same initial conditions as for other supplementation regimes, where the initial condition of the metabolite matches the amount used for supplementation. Dynamics proceed as per equations in the main text. In total, they run for the same amount of time as the initial and continuous supplementation experiments before assuming equilibrium, but the time is split into numerous small cycles of length Δ

*t*. Each cycle runs for a short period. Thus, we have cycle length × number of cycles = total time until equilibrium. At the end of each cycle, we add the predetermined amount of essential metabolites and then allow the next cycle to continue.

**(ii) Experiments.**As the initially added supplementation is consumed, it is possible to add metabolites at fixed intervals. The model is now an example of a hybrid dynamic system (27) where concentrations of metabolites are adjusted at regular intervals (see Materials and Methods and reference 4). In contrast to initial supplementation, equilibrium, in this case, does not only shift but is also maintained. In a chemostat or a continuously fed batch culture, without perturbations, we would expect the system to bounce back to the unsupplemented equilibrium. However, as here, with an intermittently disturbed environment, we can maintain the system away from the naive equilibrium.

*ADE*↑ strain when supplemented with 1.0, 10, and 100 μg/ml of adenine every 12 or 24 h. Compared to initial supplementation (Fig. 3), the dynamics of intermittent supplementation result in a different equilibrium. For example, for 10 μg/ml under initial supplementation (Fig. 3, right column), the population fraction of the

*ADE*↑ strain is almost negligible; however, for intermittent supplementation, the unsupplemented equilibrium is maintained (Fig. 5, middle column). Delay in supplementation (12 h versus 24 h) changes the time required to attain equilibrium.

*t*= 0, the scenario is the same as that of initial supplementation (Fig. 4, middle panel would resemble the left panel). If the delay between successive supplementations is minimal, then the concept is similar to continuous supplementation (Fig. 4, middle panel would resemble the right panel). In Fig. 6, we show the effect of the timing of intermittent supplementation. If the equilibrium of the system results in an intermediate fraction of both the strains, then we can maintain it if the supplementation is delayed. When supplementation occurs early in the transient stage of the dynamics, there is the potential to change the eventual outcome. For a legitimate comparison with initial and continuous supplementation, we started the intermittent supplementation with a nonzero amount and supplemented further at regular intervals.

### Continuous supplementation.

*ADE*↑ strain for different amounts of continuously added adenine and lysine (relative to adenine concentration) (Fig. 4, right panel). Slight, but continuous addition of lysine immediately shifts the composition of the two strains as the

*ADE*↑ strain takes over the mixed culture. Compared to initial supplementation and compared at the same time point, continuous supplementation shows a drastic change in the strain composition.

### Intervention measures in the presence of an ultimate overproducer.

*ADE*↑

*LYS*↑ strains. While not participating in mutualism might be lethal, taking part in a mutualistic interaction does not need to be costly if it involves the overproduction of a single metabolite. Although specific, yeast can overproduce a fluorescent protein and only suffer a 1% reduction in cost per copy (30). Moreover, reliance on external sources for essential metabolites can have a considerable cost as well. Several auxotrophic yeast strains, such as those unable to produce their own lysine or adenine, have up to a 10% reduction in growth, even with environmental supplementation (31, 32).

*ADE*↑

*LYS*↑ strain that overproduces both lysine and adenine and does not require any supplementation. We assume that any cellular cost incurred will not matter unless it affects the growth rate. In the simplest case, the growth rate would be chiefly independent of the environment since the strain can satisfy its requirements.

*ADE*↑

*LYS*↑ strain can quickly become the dominant strain, despite supporting both the

*ADE*↑ and

*LYS*↑ strains. If we are interested in ensuring mutualism, then we must intervene, but the question is when and to what degree?

*ADE*↑ strain has a higher growth rate, we support it by providing lysine in the culture (environment). As the amount of supplementation is reduced, it must be provided earlier to maintain the

*ADE*↑ strain as the dominant strain. The more delayed the supplementation intervention, the more likely that predictions of the growth rate for unsupplemented populations hold. The same holds for scenarios in which the ultimate overproducer dominates. We need to provide a large amount of supplement early to offset the fitness benefit of the ultimate overproducer in Fig. 7. If supplementation occurs after the carrying capacity has already been reached, then it has no effect, as seen from Fig. 6.

*r*) is constant, whereas the growth rates of the

_{o}*ADE*↑ and

*LYS*↑ strains change over time as metabolite concentrations change. When we look for the dominant strain (the one with the highest frequency in the culture), it is not just the growth rates that need to be taken into account, but their dynamics over time in the presence of supplementation matter as shown in Fig. 7.

^{−1}lysine supplementation, every time point for

*r*= 0.1) to no supplementation. With no intervention, the dominant strain is the

_{o}*ADE*↑ strain. Intervening with a small amount of lysine (10

^{−1}) at later time points (in this case even time point 2 onward) does not destabilize the dominance. With higher doses, the time of intervention can be delayed (rows above point a). However, if we want to intervene to see a change in the dominance of a strain, then with minimal disturbance, we need to intervene early.

*r*= 0.1 under no supplementation, the growth rate of the overproducer is higher than those of the

_{o}*ADE*↑ and

*LYS*↑ strains. Metabolites start accumulating; the growth rates of

*ADE*↑ and

*LYS*↑ increase. The initial growth spurt of the overproducer ends with the dominance of the

*ADE*↑ strain. In this scenario, under minimal supplementation, we can get Fig. 7 (a). We can monitor the change in dominance as we increase the growth rate of the overproducer, as shown in panels a, b, and c in Fig. 7. The dominance shifts from the

*LYS*↑ strain to the

*ADE*↑ strain, and eventually for a high enough

*r*, to the ultimate overproducer.

_{o}## DISCUSSION

*LYS*↑ strain, the final ratio shifts in both the experimental system and mathematical model (Fig. 3). Confident in the exploratory power of our mathematical model, we determined the environmental conditions that maintain the equilibrium of the system close to one observed in unsupplemented conditions. The modeled supplementation regimes are typical in microbial culturing, but they also have clear ecological parallels. The nonsupplemented culture represents a niche baseline. Initial supplementation is akin to an isolated event like the sinking of a whale carcass to the floor of the ocean, resulting in extreme point source enrichment (34). Intermittent supplementation represents seasonal or periodic change, such as temperature fluctuations or the regular introduction of a nutrient to the gut of a host animal. Understanding the effect of fluctuating resource availability is extremely important regarding invasiveness (35). A temporary reduction in competition due to nutrient excess can make mutualisms vulnerable to invasion (36). Finally, continuous supplementation is a permanent change to an environment, as in a permanent temperature change or evaporation of a large body of water. In general, we observe that as one progresses from initial to intermittent supplementation and then to continuous supplementation regimes, one of the mutualists dominates the system (Fig. 4). Tracking the size of the zone is thus a good measure of the resilience of a mutualistic system under changing environments.

*ADE*↑

*LYS*↑ strain that allows us to explore both the cost of mutualism generally and to tease apart the influence of the timing and magnitude of environmental disturbances. Although even modest costs can be rapidly selected against, it is essential to recognize that mutualism need not be expensive from a cellular point of view. Cost-free or inexpensive metabolic cross-feeding could potentially give rise to mutualism (29). Moreover, metabolic dependency, requiring the cellular uptake of public goods, can also negatively affect fitness even in the presence of supplementation (38), although the reverse can also be true (39). Thus, it can be problematic to assume static fitness for a mutualist. Depending on the fitness of

*ADE*↑

*LYS*↑ individuals, they can persist in a population contributing to the public goods. These individuals are not cheaters, as they support the existing mutualistic interaction but could quickly become dominant members of an ecosystem if the cost of contribution is lower than the benefits gained from not relying on other members of the ecosystem (Fig. 7). Therefore, such a strain could eventually displace obligate mutualists. Even with low fitness, the

*ADE*↑

*LYS*↑ strain can persist in a stable equilibrium with the other strains, in part because they are not dependent upon cycles in abiotic factors. This strain could also potentially give rise to Black Queen dynamics, which would facilitate the evolution of mutualism via gene loss in

*ADE*↑ and

*LYS*↑ strains (40).

*ADE*↑ and

*LYS*↑ strains. In principle, targeted intervention in collapsing ecological niches that depend on mutualism could save these relationships or at least forestall their collapse. Further experimental tests along these lines will provide insight into the role of abiotic components in the resilience of mutualistic systems.

## MATERIALS AND METHODS

### Experimental materials. (i) Saccharomyces cerevisiae strains.

*ADE*↑ throughout this manuscript. It has the genotype

*MAT*

**a**

*ste3*::

*kanMX4 lys2*Δ

*0 ade4*::

*ADE4*(

*PUR6*) ADHp-DsRed.T4. The second, WS954, is a lysine-overproducing strain called

*LYS*↑ here. This strain has the genotype

*MAT*

**a**

*ste3*::

*kanMX4 ade8*Δ

*0 lys21*::

*LYS21*(fbr) ADHp-venus-YFP where fbr stands for feedback resistance.

### (ii) Media.

### Experimental procedures. (i) Culturing.

^{7}. These cultures were pelleted, washed, and resuspended as described above and were then diluted 1 in 20 with SC-aa. Thus, all experiments were batch cultures starting at approximately 1/20th of the carrying capacity. Without supplementation, experimental cultures failed to reach carrying capacity in the experimental time frames. Stock solutions of adenine or lysine were made using SC media. Supplements were added at 1 part in 500 parts. This dilution was well below detectable variation.

### (ii) Single-strain growth.

*LYS*↑ strain or lysine for the

*ADE*↑ strain. The cultures were sampled at 0 and 24 h.

### (iii) Coculturing and supplementation.

### (iv) Delayed intermittent supplementation.

*LYS*↑ to

*ADE*↑ volume ratio, 0.65 to 0.35, was used. Cultures were supplemented at either 12 or 24 h after establishment with 0, 0.1, 1, 10, or 100 μg/ml adenine. They were then supplemented with the same amount of adenine either every 12 or 24 h (depending on the initial delay). The cultures were also sampled every 24 h for CFU and strain ratios.

### (v) CFU and strain ratios.

### (vi) Estimating growth parameters.

*r*

_{1}and

*r*

_{2}) using the single-strain growth experimental data. Single-strain growth rates for the

*ADE*↑ and

*LYS*↑ strains were determined on growth media supplemented with various concentrations of the appropriate required metabolite. The two strains were grown in synthetic complete media either unsupplemented or supplemented with 0.1, 1, 10, or 100 μg/ml of the corresponding metabolite. Growth rates under increasing concentrations of supplementation provided us with the supplementation-dependent growth rate function (Fig. 9). This curve was used to parametrize the functional form of growth rate [

*r*/(

_{i}s*s*+

*k*] where

_{s}*s*is the supplementation and

*r*is derived from the maximum of the function. From the raw data provided in the GitHub repository, we can estimate the maximum of the growth function of the

_{i}*ADE*↑ strain to be twice as much as that of the

*LYS*↑ strain. The half-saturation constant of the

*ADE*↑ strain is estimated to be also twice as much as that of the

*LYS*↑ strain. Thus, we use the normalized values of

*r*

_{1}= 1,

*r*

_{2}= 2, ${k}_{{c}_{L}}=1$, and ${k}_{{c}_{A}}=2$ as derived from the data and summarized in Table 1. Over the time course of the experiment, death is shown to be minimal (Fig. 10).

^{a}

Parameter | Description | Value |
---|---|---|

r_{1} | Normalized maximum growth rate | ${V}_{\text{max}L}/({V}_{\text{max}L})=1$ |

r_{2} | Normalized maximum growth rate | ${V}_{\text{max}A}/({V}_{\text{max}L})=25.4/13\approx 2$ |

${k}_{{c}_{L}}$ | Normalized half-saturation constant | ${K}_{{c}_{L}}/{K}_{{c}_{L}}=1$ |

${k}_{{c}_{A}}$ | Normalized half-saturation constant | ${K}_{{c}_{A}}/{K}_{{c}_{L}}=32.9/15.3\approx 2$ |

^{a}

*V*

_{max}and half-saturation constants

*K*for the two strains, the

*ADE*↑ and

*LYS*↑ strains. For the theoretical model, we do not use these direct values, since our goal is not to quantitatively estimate the data. Using the relative quantities, we understand the differences between strain growth rates, and we present a qualitative model in the main text. The italic

*L*and

*A*subscripts in the variables stand for lysine and alanine, respectively.

### Availability of data and materials.

## ACKNOWLEDGMENTS

## Supplemental Material

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