Papers in Physics, vol. 7, art. 070005 (2015)

Received: 20 November 2014, Accepted: 29 March 2015
Edited by: C. A. Condat, G. J. Sibona
Licence: Creative Commons Attribution 3.0
DOI: http://dx.doi.org/10.4279/PIP.070005

www.papersinphysics.org

ISSN 1852-4249

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

G. Cascallares,1, 2∗ P. M. Gleiser1, 2†

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 and 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 chosen to match
the circadian period of the different strains, revealing that the strain whose circadian period
matches the light-dark cycle 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. In Nature, however, 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.

I. 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 en-
dogenous 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

∗Email: mgcascallares@gmail.com
†Email: gleiser@cab.cnea.gov.ar

1 Consejo Nacional de Investigaciones Cient́ıficas y
Técnicas.

2 Centro Atómico Bariloche, 8400 San Carlos de Bariloche,
Ŕıo Negro, Argentina.

a daily rhythm of nitrogen fixation [2–4]. Huang
and co-workers were the first to recognize that a
strain of Synechococcus, a unicellular cyanobac-
terium, had circadian rhythms [5]. This trans-
formed Synechococcus in one of the simplest mod-
els for studying the molecular basis of the circadian
clock.

The ubiquity of circadian rhythms suggests that
they confer an evolutionary advantage. The adap-
tive functions of biological clocks are divided into
two hypotheses. The external advantage hypothe-
sis supposes that circadian clocks allow living or-
ganisms to anticipate predictable daily changes,
such as light/dark, so they can schedule their bi-
ological functions like feeding and reproduction at
appropriate times. In contrast to this hypothesis,
it has been suggested that circadian clocks con-
fer adaptive benefit to organisms through temporal

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Papers in Physics, vol. 7, art. 070005 (2015) / G. Cascallares et al.

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 be-
tween 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) pe-
riods, 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 pe-
riod 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 com-
petition. Also, mutant strains could outcompete
wild-type strains under continuous light (LL) con-
ditions, suggesting that endogenous rhythms are
advantageous only in rhythmic environments [7].
This study provided one of the most convincing evi-
dence so far in support of fitness advantages of syn-
chronization between the endogenous period and
the period of environmental cycles.

Ouyang et al. [8] suggested an explanation for fit-
ness differences: this could be due to competition
for limiting resources or secretion of diffusible fac-
tors that inhibit the growth of other cyanobacterial
strains. Roussel et al. [9] proposed mathematical
models in order to test which of these hypotheses
was more plausible. 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 pro-
duced only during the subjective day (sL) phase,
and that growth occurs only in the light phase.

Each of the experiments and computational sim-
ulations mentioned before had equal amounts of
light and dark exposure. However, in Nature the re-
lationship is not constant, and the duration of sun-
light 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].
Using this as a motivation, we will test how day

length variation plays a role in the competition be-
tween different strains of cyanobacteria.

II. The model

For modeling 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 Ni and the level of inhibitor I are:

dNi
dt

= kiNi(1 −
n∑

j=1

Nj),

dI

dt
=

n∑
i=1

Ni(pi −
VmaxI

KM + I
) (1)

ki =

{
k in L and i in sL or I < Ic

0 otherwise,

pi =

{
p if i in sL

0 otherwise.
(2)

In these equations, Ni is the number of cells of
strain i, ki is the growth rate of each strain, p mea-
sures the rate of inhibitor production, Vmax is the
maximum rate and KM is the Michaelis constant
characterizing the enzymatic degradation of the in-
hibitor.

Following the work of Gonze et al., we use a mod-
ified version of the Van der Pol oscillator to produce
sustained circadian oscillations [11]. This simple
mathematical model can describe circadian oscilla-
tions.

dx

dt
= 24

π

12
(xc + 0.13(

x

3
+

4

3
x3 −

256

105
x7) + B(t)),

dxc
dt

= 24
π

12
(−x(

24

τx
)2 +

B(t)

3
xc − 0.15B(t)x) (3)

with

B(t) = (1 −
x

3
)0.39ρ0.23,

ρ =

{
5 in light phase (L),

0 in dark phase (D).
(4)

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Papers in Physics, vol. 7, art. 070005 (2015) / G. Cascallares et al.

Figure 1: Following the schematic explanation of
Gonze et al. [11], we show how the model works.
The inhibitor I is produced in 3, during the sL
phase, and it is degraded during the entire day.
Each strain grows in 1, if its sD phase overlaps with
L and I < Ic, and in 2, when its SL phase overlaps
with L.

These phenomenological equations produce os-
cillations in the circadian variable x with a period
close to τx. xc is a complementary variable and
B(t) represents the coupling between the external
LD cycle and the circadian oscillator, which de-
pends on the light intensity ρ. We use this sim-
ple and phenomenological model because the de-
tailed mechanisms, both at molecular and popula-
tion level, have remained unknown in Synechococ-
cus elegantus. For example, it is still unclear which
phospho-state of kaiC, one of the clock genes iden-
tified in this organism, is involved in activation
or suppression due to inconsistent reports. There
are different mathematical models proposed for the
molecular mechanisms of Synechococcus, but none
of them has been experimentally verified [14]. We
chose the model by Gonze et al [11] since the pa-
rameters on their equations are in agreement with
the experimental data obtained by Woelfle et al.
[7]. These values are Vmax = 1000, k = 1.8,
Ic = 0.01, p = 500 and KM = 0.05.

In Fig. 1, we present a schematic plot of the
model that shows how the growth of each popu-
lation is coupled with the circadian oscillator. As
it can be seen in this figure, when we modify the
length of the light (L) phase then the overlaps 1 and
2 change, so the growth of each strain is altered af-
fecting the competition. This effect is due to the
fact that each strain grows always when its subjec-
tive phase sL overlaps the external light phase (2),

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

21
th

Day of Each Month

4

6

8

10

12

14

16

18

D
a
y

le
n

g
th

 (
h

)

Quito, Ecuador

Jujuy, Argentina

Ushuaia, Argentina

Figure 2: Day length over the course of 2012 at
different latitudes in South America.

and when sD overlaps L only if the inhibitor is be-
low a threshold Ic (1). The inhibitor is secreted in
the sL phase and then it is degraded.

III. Results

In many organisms, a photoperiodic response is re-
flected in a physiological change. Photoperiodic
responses are common among organisms from the
equator to high latitudes and have been observed
in different types of organisms, from arthropods
to plants. Diapause (a suspension of development
done by insects), migration and gonadal matura-
tion are examples of these annual changes con-
trolled by photoperiod. These biological processes
are triggered as soon as the day length reaches cer-
tain duration, known as the critical photoperiod.
Even near the equator, where day length changes
are very small through the whole year, they are
used to synchronize reproductive activities with an-
nual events.

In Fig. 2, we show how day length varies de-
pending on the latitude [13]. The figure com-
pares the day length on the twenty-first day of each
month in three cities in South America. We show
Quito (Ecuador), which sits near the equator in
latitude 0◦15′, Jujuy (Argentina), which is located
near the tropic of Capricorn in latitude 24◦01′ and
Ushuaia, the southernmost city in Argentina which

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Papers in Physics, vol. 7, art. 070005 (2015) / G. Cascallares et al.

A

B

Figure 3: (A) The outcome of competition between
wild-type (continuous line) and long-period mutant
(dashed line) shows coexistence between the two
strains for T = 28 h. (B) Coexistence ends after
≈ 8 days of adding 3 minutes of light-time per day.
The long-period mutant can win competition as the
photoperiod now is closer to its free-running period.

lies in latitude 54◦48′. We can see that during the
equinoxes, all places receive 12 hours of daylight
[13]. Even when changes in photoperiod are not
constant over the year and depend on the season,
we decide to modify the amount of light per day
with fixed steps in order to simplify the equations.

We simulate the seasonal fluctuations in day
length by adding or subtracting minutes of light-
time every day to the external LD cycle. For ex-
ample, if we add 12 minutes of light per day in
a LD12:12 cycle, after five days the external cy-
cle of LD has 13 hours of light and 11 of darkness,
since the total amount of hours per day is constant.
We initiate competition between equal fractions of
wild-type strain (τx = 25h) and long-period mu-
tant (τx = 30h) and equal amounts of light and
darkness. In order to mimic experiments in cul-
tures that were diluted and sampled every 8 days
[7], we dilute the culture after 8 light-dark cycles
by dividing by a factor of 100 the variables N1, N2
and I.

First, we analyzed the case in which the exper-
iments of [7] showed a phase of coexistence. For
T = 28, the period of the LD cycle has an interme-
diate value between the free-running period (FRP)
of the two strains and both strains can coexist for

A

B

Figure 4: Effect of modulation in light time (30
minutes per day) on the outcome of competition
between strain 1 (wild-type, τx = 25 h) and strain 2
(long-period mutant, τx = 30 h) carried out in two
different conditions: (A) LD12:12 and (B) LD15:15.
Fraction of strain 1 (left panel) and strain 2 (right
panel) are shown as a function of time; red with
modulation, so the proportion of L and D changes
after each day, and blue with fixed cycles.

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 increase in the amount of light
hours that benefits the long-period mutant.

In Fig. 4, we show in the left panel the fraction of
cells belonging to the wild-type strain as a function
of time with fixed LD cycles (red continuous curve)
and in the case in which we added 30 minutes of
light time each day (blue dashed curve). The first
days both curves are similar, but from the second
day on, the long-period strain has a FRP closest to
the period of the LD cycle. The fraction of wild-
type strain starts to decrease and is out-competed.
In the right panel, we show the corresponding frac-
tion of long-period mutant cells in the same cases.

Our results show a new way to test competition
experimentally. Testing results would be very sim-
ple, since the cultures do not need to be diluted. In
fact, as shown in Fig. 4, after eight days of compe-
tition between the two strains, a difference of about
ten percent in the two populations would be needed

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Papers in Physics, vol. 7, art. 070005 (2015) / G. Cascallares et al.

0 10 20 30
Time (Days)

0

0,2

0,4

0,6

0,8

1

N
1
/(

N
1
+

N
2
)

Mutant

Wild Type

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 min-
utes per day the light time. A crossover is observed
after 8 days.

in order to verify the theoretical prediction.

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 as ex-
pected. The wild-type strain could outcompete 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 but in this case the dilution of the
culture every 8 days would be needed.

IV. Conclusions

The mechanisms underlying the enhancement of re-
productive fitness remain still unknown. Despite
the fact that 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 behavior. We studied how these fluc-
tuations affect the competition between different
strains of cyanobacteria. We found non-trivial ef-
fects which could be tested experimentally. In the
first case, we determined the composition of two
strains under competition after eight days when the
light is modulated. The prediction of these numer-
ical simulations can be tested in a simple experi-
ment 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.

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