Evolution of clonal populations approaching a fitness peak
Supplemental Information
Fisher Geometrical model
Here we use stochastic simulation to obtain predictions under Fisher’s geometrical
model (FGM) when assuming a static optimum and a set of parameters that is
different from those presented in the main text. In the static FGM, the distance to the
optimum depends on the value of W0. The optimum is located at the center of the
geometric space and the fitness at this point is 1. In the figures below we always
report the relative mean fitness of the population (W/W0).
Figures A1 and A2 show the effect of population size N and number of traits under
selection n (indicated in different colors), for two different values of 2, which leads
to different values for the mean effect of mutations.
Figure A3 shows that when the mean fitness effect of a new mutation, E(sd)w=1, is
very small, mean fitness increases very slowly, at a constant rate, and this process is
accompanied by a constant rate of mutation accumulation, at least during the 20000
generations period simulated. We note that for the cases where E(sd)w=1=-3x10-5, in
the initial genotype founder of the population (which has fitness 0.5 and E(sd)w=0.5=1.3x10-3) the fraction of effectively neutral mutations is 0.5%. This implies a rate of
accumulation of these mutations of about 0.003*0.5% =1.5x10-5. This is a much
lower rate than the one observed in the simulations, which is 0.0067, suggesting that
the majority of the mutations accumulated are beneficial.
Figure A1) Dynamics of fitness increase and rate of molecular evolution with
phenotypic complexity (n) for different effective population sizes N. Divergence
represents the number of mutations accumulated along time. U=0.001 which is the
order of the genomic mutation rate for DNA based microbes. 2=0.0001 leading to a
mean effect of deleterious mutations at the optimum of 0.001 or 0.005 depending on
n. 10 simulations were run for each parameter set.
Figure A2) The same as A1 but with  2=0.001 leading to a mean effect of deleterious
mutations at the optimum of 0.01 or 0.05 depending on n. 10 simulations were run for
each parameter set.
Figure A3) The effect of mutations in the dynamics of fitness and rate of mutation
accumulation. Parameters are N=105 U=0.003, n=30 and W0=0.5. Various mean
effects at the peak (various 2), where Sd= E(sd)w=1 =2/n are indicated in the figure.
For all parameters values, deleterious mutations are never effective neutral
(N*E(sd)>1). 10 simulations were run for each parameter set.
Fisher geometrical model with a non-constant optimum
Here we examine predictions of Fisher’s geometrical model when assuming a shaking
optimum for different sets of parameters than those presented in the main text. We
assume that the environment is not static because it encompasses the organisms
themselves. If this is the case then non-transitive fitness interactions maybe expected
to occur.
Recent theoretical models have also considered the case of non-static fitness
landscapes. For example, Collins et al. [1] studied the dynamics of adaptive walks
assuming different rates of environmental change. They considered a population of
RNA molecules characterized by two traits (folding stability and secondary structure),
which are correlated, and contribute to fitness in a pre-defined way. Their model does
not consider clonal interference, and assumes a constant direction for environmental
change. These are different assumptions from those in the model studied here and so
can lead to different outcomes for the dynamics and statistical laws of adaptation. In
the special case of a sudden environmental change and adaptation towards a static
optimum they find large jumps in the first steps, just as we see here in static optimum.
Kopp and Hermisson [2] have also studied the distribution of adaptive substitutions in
a model where a population adapts to a gradually changing environment. In their
model, the optimum moves deterministically at a constant speed in a given direction,
instead of randomly as we assume here, and the distribution of beneficial mutations
fixed depends on the speed of the optimum movement relative to mutation supply.
Figure A4) Dynamics of increase in fitness and number of mutations under a model
with a shaking peak. N=105 2=0.001, n=20, U=0.001, W0=0.5 and the value of v is
indicated in the figure. 10 simulations were run for each parameter set.
Figure A4B) The effect of varying the distance to the peak (W0). N=105 2=0.001,
n=10, U=0.001, under FGM (v=0) and the shaking peak (v=0.00001). 10 simulations
were run for each parameter set.
Emergence of mutators
Invasion of mutators is observed in several bacteria evolution experiments [3,4,5]. In
the 12 lines evolved by Lenski and collaborators, 3 mutators of intermediate strength
emerged around 2400 and 8500 generations and a fourth mutator was detected later,
after generation 20000. Below we study by simulation the invasion of a modifier of
the genomic mutation rate (mutator) in the shaking peak model. The mutator allele
introduced confers no intrinsic fitness advantage. Invasion of mutator alleles, that
increase globally the mutation rate and therefore produce a higher frequency of both
beneficial and deleterious mutations, has been studied under different theoretical
models [6,7,8].
In the context of FGM with a shaking peak, we show that in conditions of the
classical FGM, the probability of fixation of mutators is large in the beginning of an
adaptive walk, but very low when the populations are already well adapted (Figure 2
A in the main text and Figure A5, for the cases where v/2 is very small). This makes
intuitive sense since, at the beginning of the walk the fraction of beneficial mutations
available is high, whereas when the peak is reached they get depleted and mutator
genotypes experience a higher load due to deleterious mutations. Under the shaking
peak (larger values of v/2), beneficial mutations are plentiful and mutators can
hitchhike even at the later stages of the walk, when the fitness has already increased
substantially. Figure A5 also shows that in larger populations the advantage of
mutator alleles is higher than in smaller populations, all else being equal.
Fig A5) Advantage of mutator in large populations. The parameters values are the
same as in Figure 3) of the main text except that N=106. Typically 105 simulations
were done for each parameter set.
Distribution of pairwise epistasis between beneficial mutations
Using stochastic simulations, we have obtained the amount of pairwise epistasis of the
first beneficial mutations that escape stochastic loss and can therefore contribute to
adaptation, under FGM. For that we consider the single step mutation neighbors (W1)
of an initial clone (with fitness W0) and sampled these according to their probability
of escaping stochastic loss pfix=(1-e-2s), where s=W1/W0-1. For each generated
random mutant, corresponding to the first beneficial mutation, we sampled a second
mutation to fall on this background following the same procedure (i.e. sampled with
probability pfix, with s now being W12/W1-1), obtaining therefore a clone with 2
beneficial mutations (W12). We then calculate pairwise epistasis (e) according to
e=W0*W12-W1*W2. Sign epistasis was counted when the second mutation, which is
beneficial in the first genetic background, was deleterious in the ancestral
background. In Figure A6 we show the distribution of e, for parameter values W0=0.5,
n=20, 2=0.001, which corresponds to a mean effect of arising mutations compatible
with estimates of mutation accumulation in Escherichia coli [9,10] and with strong
increments in fitness observed in adaptation experiments with this species [11,12,13].
Figure A6) Distribution of pairwise epistasis (e) during the first adaptive steps. The
fraction of pairs exhibiting negative epistasis was 60%, and rare cases of sign
epistasis were detected (0.7%). The line corresponds to a Normal Distribution of
mean -0.00068 and variance 5.9x10-6, estimated from the mean and variance of the
simulated data. 1000 simulations were run.
Analysis of ability to detect a shaking peak
In Table S1 we explore a potential way of assaying the occurrence of a shaking peak
through detecting fluctuations in the mean fitness of the population. We measure
mean population fitness, after 10000 generations of adaptation, at periodic time
intervals (100 generations) during a further 2000 generations. We calculate the
amplitude of fitness fluctuations and assume that fitness differences below 1% are
indistinguishable from experimental error. We run several independent populations
under the same parameters and compute the fraction of the simulations where the
amplitude of the fitness fluctuations is above experimental error. The mutation rate
and population size considered are within the range of typical microbial evolution
experiments [11,12,13]. The mean effect of arising mutations E(s)=-n2 is in the
order of that estimated from mutation accumulation experiments in bacteria [9,10].
Table S1. The power to detect a shaking peak
Mean
Proportion of simulations
W/W0
Amplitude
above experimental error
v/2
0.0001
1.98
0.002
0%
0.0005
1.97
0.004
0%
0.001
1.96
0.008
8%
0.005
1.89
0.04
98%
0.01
1.86
0.04
100%
0.02
1.78
0.07
100%
7
The parameter values are as follows: N=10 U=0.001 n=20 2=0.001 and W0=0.5.
100 simulations were run for each set of parameters.
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