What season suits you best? Seasonal light changes and cyanobacterial competition

Nearly all living organisms, including some bacterial species, exhibit biological processes with a period of about 24 h called circadian (from the Latin circa, about, dies, day) rhythms. These rhythms allow living organisms to anticipate the daily alternation of light and darkness. Experiments carried out in cyanobacteria have shown the adaptive value of circadian clocks. In these experiments a wild type cyanobacterial strain (with a 24 h circadian rhythm) and a mutant strain (with a longer or shorter period) grow in competition. In different experiments the external light dark cycle was changed in order to match the circadian period of the different strains, revealing that the strain whose circadian period matches the light-dark has a larger fitness. As a consequence the initial population of one strain grows while the other decays. These experiments were made under fixed light and dark intervals. However, in Nature this relationship changes according to the season. Therefore, seasonal changes in light could affect the results of the competition. Using a theoretical model we analyze how modulation of light can change the survival of the different cyanobacterial strains. Our results show that there is a clear shift in the competition due to the modulation of light, which could be verified experimentally.


INTRODUCTION
Circadian rhythms, oscillations with approximately 24 h period in many biological processes, are found in nearly all living organisms. Until the mid-1980s, it was thought that only eukaryotic organisms had a circadian clock, since it was assumed that an endogenous clock with a period of τ = 24 h would not be useful to organisms that often divide more rapidly [1]. However, in 1985, several research groups discovered that in cyanobacteria there was a daily rhythm of nitrogen fixation [2,3,4]. Huang and co-workers were the first to recognize that a strain of Synechococcus, a unicellular cyanobacterium, had circadian rhythms [5]. This transformed Synechococcus in one of the simplest models for studying the molecular basis of the circadian clock.
The ubiquity of circadian rhythms suggests that they confer an evolutionary advantage. The adaptive functions of biological clocks are divided into two hypotheses. The external advantage hypothesis supposes that circadian clocks allow living organisms to anticipate predictable daily changes, such as light/dark, so they can schedule their biological functions like feeding and reproduction at appropriate times. In contrast to this hypothesis, it has been suggested that circadian clocks confer adaptive bene-fit to organisms through temporal coordination of their internal physiology (intrinsic advantage) [6]. In this case, the circadian clock should be of adaptive value in constant conditions as well as in cyclic environments.
In order to study if circadian clocks provide evolutionary advantages Woelfle and co-workers tested the relative fitness under competition between various strains of cyanobacteria [7]. They carried out experiments where a wild-type strain (τ = 25 h) of cyanobacteria and mutant strains, with shorter (τ = 22 h) and longer (τ = 30 h) periods, were subjected to grow in competition with each other under light-dark (LD) cycles of different periodicity. They found that the strain which won the competition was the one whose free-running period matched closely the period of the LD cycle. This difference in fitness was observed despite the fact that the growth rates were not significantly different when each strain was grown with no competition. Also, mutant strains could outcompete wild-type strains under continuous light (LL) conditions, suggesting that endogenous rhythms are advantageous only in rhythmic environments [7]. This study provided one of the most convincing evidence so far in support of fitness advantages of synchronization between the endogenous period and the period of environmental cycles.
Ouyang et al. suggested an explanation for fitness differences: this could be due to competition for limiting resources or secretion of diffusible factors that inhibit the growth of other cyanobacterial strains [8]. Roussel et al. proposed mathematical models in order to test which of these hypotheses was more plausible [9]. They found that the model based on mutual inhibition was consistent with the experimental observations of [8]. In this model the mechanism of competition between cells involves the production of a growth inhibitor, which is produced only during the subjective day (sL) phase, and that growth occurs only in light phase.
Each of the experiments and computational simulations mentioned before had equal amounts of light and dark exposure. However, in Nature the relationship is not constant, and the duration of sunlight in a day changes according to the season and the latitude. The circadian system has to adapt to day length variation in order to have a functional role in optimizing seasonal timing and generating the capacity to survive at different latitudes [10].
In this work we will test how day length variation plays a role in the competition between different strains of cyanobacteria.

THE MODEL
For modelling the growth of each cyanobacterial strain we use the model introduced by Gonze et al. [11], that is based on a diffusible inhibitor with a light sensitive oscillator to represent the cellular circadian oscillator.
The evolution equations of cell population N i and the level of inhibitor I are: k in L and i in sL or I < I c 0 otherwise, In these equations, N i is the number of cells of strain i, k i is the growth rate of each strain, p measures the rate of inhibitor production, V max is the maximum rate and K M is the Michaelis constant characterizing the enzymatic degradation of the inhibitor.
Following the work of Gonze et al. we use a modified version of the Van der Pol oscillator to produce sustained circadian oscillations [11].
Also, we use the same parameters which were found to be in agreement with experimental data obtained by Woelfle et al. [7].
In Fig. 1 we present a schematic plot of the model, that shows how the growth of each population is coupled with the circadian oscillator. As can be seen in this figure, when we modify the length of the light (L) phase the overlaps 1 and 2 change, so the growth of each strain is altered affecting the competition.  We initiate competition between equal fractions of wild-type strain and long-period mutant and equal amounts of light and dark. We dilute the culture after 8 light-dark cycles by dividing by a factor of 100 the variables N 1 , N 2 and I. In this way we mimic experiments in cultures, that were diluted and sampled every 8 days [7].
First, we analysed the case in which there was a phase of coexistence.
For T = 28, the period of the LD cycle has an intermediate value between the free-running period (FRP) of the two strains and both strains can coexist for a long time. However, when we allowed the days to become longer and the nights shorter, after some days the coexistence was broken, as we show in Fig. 3. This is due to the external period T that starts to get closer to the FRP of the long-period mutant. The results of our numerical simulations could be tested experimentally. This would be very simple, since the cultures do not need to be diluted. It is only needed to sample the culture at regular intervals to determine the composition of the population and verify a difference of about ten percent in the two cultures after eight days.
Starting from this simple test, we looked for non trivial effects in a longer experiment. We found an interesting effect that can be observed in Fig. 5. In this simulation we added 12 minutes of light time each day. In the first days, the growth was the expected. The wild-type strain could out-compete the long period mutant strain, since the external cycle was LD12:12. But after eight days, when the day was longer than 13 hours, a crossover was observed. The mutant strain started to win the competition because its endogenous period was closer now to the external cycle. This effect could also be tested experimentally, however, in this case the dilution of the culture every 8 days would be needed.   Figure 5: Competition between long-period mutant and wild-type strains in a LD12:12 cycle for the same parameters as in Fig. 4, but adding 12 minutes per day the light time. A crossover is observed after 8 days.

CONCLUSIONS
The mechanisms underlying the enhancement of reproductive fitness remain still unknown. Despite numerous models have been tested, each has some evidence that supports it and none can be excluded at this time [12]. In this work we used a diffusible inhibitor model, so our predictions in the growth rates changes could be useful to test the validity of this mechanism.
Our study is motivated by fluctuations in the day length throughout the year which are reflected in organisms behaviour. We studied how these fluctuations affect the competition between different strains of cyanobacteria. We found non-trivial effects which could be tested experimentally.
In the first case we determinate the composition of two strains under com-petition after eight days when the light is modulated. The prediction of these numerical simulations can be tested in a simple experiment where no dilution is needed. We also propose a second experiment where dilution in the cultures is necessary, which allows for a non trivial effect such as a crossover to be observed.