If you take two bacterial species in a large population and allow them to compete for the same resources, which one wins? Naturally, we would assume that the species which reproduces faster (assuming each dies at the same rate) will end up taking over the population, ultimately leading to the other species dying out. We say that the faster-reproducing species is fitter and fixates the system.
When there is a difference in fitnesses between species, we say that the system selects for the fitter species, and specifically refer to the direction of selection as being towards the fitter species. The strength of selection depends on the difference in fitness; a larger fitness difference means a larger strength of selection and fixation happens faster.
In addition, if we track the number of individuals of a species through time, we would expect to see some noise around an average evolution. This occurs in finite populations, due to the births and deaths in the system happening stochastically. On average, they will follow some smooth evolution (and the evolution of large populations will follow this closely), but in reality, population growth is noisy. Importantly, a smaller population will experienece a stronger noise, and so we may have larger deviations from the average evolutionary path.
In the simple case above, it is clear that whichever species is selected for should fixate the system. But what happens if we allow the direction of selection and the population size to change? This is the question we seek to answer!
For the biologically minded, this system resembles a scenario that may occur in the evolution of antimicrobial resistance (AMR). Consider the stomach of a patient that has been prescribed antibiotics. In the stomach are two types of bacteria causing illness: one has a higher fitness but is affected by the antibiotics (sensitive), and one has a lower fitness that is resistant to the antibiotics (resistant). Left alone, clearly selection is in the direction of the sensitive bacteria, and they should fixate the system. However, we would like to treat the illness. So, the antibiotics are used, and now the sensitive bacteria cannot reproduce as well as before while the resistant bacteria remain unaffected – the direction of selection has changed. Over time, the antibiotics will be used up, and the direction of selection will revert. Similarly, the patient will periodically eat, increasing the level of nutrients available in the stomach so the bacterial population can grow in size. These will be consumed, dropping the nutrient level, and the population will shrink in size. This clearly fits our criteria for a system whose direction of selection and population size change in time.
Here, we consider a well-mixed population of resistant and sensitive bacteria competing for the same resources. The environment here changes in two ways: the toxin level and the nutrient level. Each of these switches stochastically, independently, and instantaneously between states of high toxin / nutrient and states of low toxin / nutrient. In the high toxin environment, the drug is present and resistant bacteria fare better, whereas in the low toxin environment, the drug is not present (or below the minimum inhibitory concentration) and sensitive bacteria fare better. Furthermore, in the high nutrient environment, there a plentiful nutrients available and the population size is large, whereas in the low nutrient environment, nutrients are scarce and the population size is smaller (it experiences a bottleneck).
Key findings:
1. Strong selection and fast toxin switching lead to long-lived coexistence
Through analysis of the mean-field equation of the composition, we find that by taking the fast switching limit of the toxin level, a stable coexistence point emerges, and the coexistence is stronger for larger selection strength. This is a coexistence that arises solely due to the environmental switching, and crucially is not present when the environment is static. Furthermore, we find that the composition predicted at the mean-field level matches quantitatively well to simulation results of the full system. It is interesting that we can accurately predict quantities of the full stochastic model with a relatively simple analysis.
2. Nutrient level switching can promote or jeopardise coexistence
By investigating the evolution of the population size, ignoring random births and deaths, we are able to obtain an accurate approximation for the quasi-stationary population size distribution. Crucially, this distribution changes dramatically by modifying the nutrient switching rate. From this distribution, we find that it is the modal value which controls the size of the coexistence region, where a larger value implies a larger coexistence region. We find that increasing the switching rate can either increase or decrease the modal value (and hence the size of the coexistence region) depending on which of the high or low nutrient states we are more likely to spend time in and the severity of the population bottleneck. Additionally, we find that increasing the strength of the populatio bottleneck more strongly jeopardises coexistence in the slow nutrient switching case than in the fast nutrient switching case.
3. Abundance of resistant bacteria depends non-linearly on toxin switching rate
In the mean-field limit, the population size and composition are decoupled. This suggests that correlations are weak, and therefore a composite quantity such as the number of resistant bacteria, is likely captured well by treating its components as independent. This allows us to predict the non-linear behaviour seen in simulations of the average number of resistant bacteria with toxin switching rate, and in fact matches accurately even at a quantitative level.
Fluctuating environments can change population evolution radically. We find that long-lived coexistence emerges for sufficient toxin level variation, while variability in the nutrient level can promote or oppose coexistence compared to the static case dependening its switching rate, preference for one state over another, and the strength of the population bottleneck. In considering twofold environmental variations, we show that these can have qualitative effects on the population evolution.
Paper link: New J. Phys. 25 123010
Authors: Matthew Asker, Lluís Hernández-Navarro, Alastair M. Rucklidge, Mauro Mobilia
Research Data Leeds Repository: https://doi.org/10.5518/1371 (Data)