GENETICS | INVESTIGATION Evolution of microbial growth traits under serial dilution Jie Lin *, , Michael Manhart †, ‡ and Ariel Amir *,1 * John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, US, † Department of Chemistry and Chemical Biology, Harvard University, Cambridge, MA 02138, USA, ‡ Institute of Integrative Biology, ETH Zurich, 8092 Zurich, Switzerland ABSTRACT Selection of mutants in a microbial population depends on multiple cellular traits. In serial-dilution evolution experiments, three key traits are the lag time when transitioning from starvation to growth, the exponential growth rate, and the yield (number of cells per unit resource). Here we investigate how these traits evolve in laboratory evolution experiments using a minimal model of population dynamics, where the only interaction between cells is competition for a single limiting resource. We find that the fixation probability of a beneficial mutation depends on a linear combination of its growth rate and lag time relative to its immediate ancestor, even under clonal interference. The relative selective pressure on growth rate and lag time is set by the dilution factor; a larger dilution factor favors the adaptation of growth rate over the adaptation of lag time. The model shows that yield, however, is under no direct selection. We also show how the adaptation speeds of growth and lag depend on experimental parameters and the underlying supply of mutations. Finally, we investigate the evolution of covariation between these traits across populations, which reveals that the population growth rate and lag time can evolve a nonzero correlation even if mutations have uncorrelated effects on the two traits. Altogether these results provide useful guidance to future experiments on microbial evolution. 1 2 3 4 5 6 7 8 9 10 11 12 KEYWORDS Microbial evolution; fixation probability; adaptation rate 13 L aboratory evolution experiments in microbes have provided 1 insight into many aspects of evolution Elena and Lenski 2 (2003); Desai (2013); Barrick and Lenski (2013), such as the 3 speed of adaptation Wiser et al. (2013), the nature of epista- 4 sis Kryazhimskiy et al. (2014), the distribution of selection coeffi- 5 cients from spontaneous mutations Levy et al. (2015), mutation 6 rates Wielgoss et al. (2011), the spectrum of adaptive genomic 7 variants Barrick et al. (2009), and the preponderance of clonal 8 interference Lang et al. (2013). Despite this progress, links be- 9 tween the selection of mutations and their effects on specific 10 cellular traits have remained poorly characterized. Growth traits 11 — such as the lag time when transitioning from starvation to 12 growth, the exponential growth rate, and the yield (resource 13 efficiency) — are ideal candidates for investigating this ques- 14 tion. Their association with growth means they have relatively 15 direct connections to selection and population dynamics. Fur- 16 thermore, high-throughput techniques can measure these traits 17 for hundreds of genotypes and environments Levin-Reisman 18 doi: 10.1534/genetics.XXX.XXXXXX Manuscript compiled: Friday 1 st May, 2020 1 Corresponding author: 29 Oxford St., Harvard University, Cambridge, MA, 02138. Email: [email protected]et al. (2010); Warringer et al. (2011); Zackrisson et al. (2016); Ziv 19 et al. (2017). Numerous experiments have shown that single 20 mutations can be pleiotropic, affecting multiple growth traits si- 21 multaneously Adkar et al. (2017); Fitzsimmons et al. (2010). More 22 recent experiments have even measured these traits at the single- 23 cell level, revealing substantial non-genetic heterogeneity Levin- 24 Reisman et al. (2010); Ziv et al. (2013, 2017). Several evolution 25 experiments have found widespread evidence of adaptation in 26 these traits Vasi et al. (1994); Novak et al. (2006); Reding-Roman 27 et al. (2017); Li et al. (2018). This data altogether indicates that 28 covariation in these traits is pervasive in microbial populations. 29 There have been a few previous attempts to develop quanti- 30 tative models to describe evolution of these traits. For example, 31 Vasi et al. (1994) considered data after 2000 generations of evo- 32 lution in Escherichia coli to estimate how much adaptation was 33 attributable to different growth traits. Smith (2011) developed a 34 mathematical model to study how different traits would allow 35 strains to either fix, go extinct, or coexist. Wahl and Zhu (2015) 36 studied the fixation probability of mutations affecting different 37 growth traits separately (non-pleiotropic), especially to identify 38 which traits were most likely to acquire fixed mutations and 39 Genetics 1 Genetics: Early Online, published on May 4, 2020 as 10.1534/genetics.120.303149 Copyright 2020.
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GENETICS | INVESTIGATION
Evolution of microbial growth traits under serialdilution
Jie Lin∗,, Michael Manhart†, ‡ and Ariel Amir∗,1∗John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, US, †Department of Chemistry and Chemical
Biology, Harvard University, Cambridge, MA 02138, USA, ‡Institute of Integrative Biology, ETH Zurich, 8092 Zurich, Switzerland
ABSTRACT Selection of mutants in a microbial population depends on multiple cellular traits. In serial-dilution evolutionexperiments, three key traits are the lag time when transitioning from starvation to growth, the exponential growth rate, and theyield (number of cells per unit resource). Here we investigate how these traits evolve in laboratory evolution experiments usinga minimal model of population dynamics, where the only interaction between cells is competition for a single limiting resource.We find that the fixation probability of a beneficial mutation depends on a linear combination of its growth rate and lag timerelative to its immediate ancestor, even under clonal interference. The relative selective pressure on growth rate and lag time isset by the dilution factor; a larger dilution factor favors the adaptation of growth rate over the adaptation of lag time. The modelshows that yield, however, is under no direct selection. We also show how the adaptation speeds of growth and lag depend onexperimental parameters and the underlying supply of mutations. Finally, we investigate the evolution of covariation betweenthese traits across populations, which reveals that the population growth rate and lag time can evolve a nonzero correlation evenif mutations have uncorrelated effects on the two traits. Altogether these results provide useful guidance to future experimentson microbial evolution.
Reisman et al. (2010); Ziv et al. (2013, 2017). Several evolution 25
experiments have found widespread evidence of adaptation in 26
these traits Vasi et al. (1994); Novak et al. (2006); Reding-Roman 27
et al. (2017); Li et al. (2018). This data altogether indicates that 28
covariation in these traits is pervasive in microbial populations. 29
There have been a few previous attempts to develop quanti- 30
tative models to describe evolution of these traits. For example, 31
Vasi et al. (1994) considered data after 2000 generations of evo- 32
lution in Escherichia coli to estimate how much adaptation was 33
attributable to different growth traits. Smith (2011) developed a 34
mathematical model to study how different traits would allow 35
strains to either fix, go extinct, or coexist. Wahl and Zhu (2015) 36
studied the fixation probability of mutations affecting different 37
growth traits separately (non-pleiotropic), especially to identify 38
which traits were most likely to acquire fixed mutations and 39
Genetics 1
Genetics: Early Online, published on May 4, 2020 as 10.1534/genetics.120.303149
Copyright 2020.
the importance of mutation occurrence time and dilution fac-1
tor. However, simple quantitative results that can be used to2
interpret experimental data have remained lacking. More recent3
work Manhart et al. (2018); Manhart and Shakhnovich (2018)4
derived a quantitative relation between growth traits and se-5
lection, showing that selection consists of additive components6
on the lag and growth phases. However, this did not address7
the consequences of this selection for evolution, especially the8
adaptation of trait covariation.9
In this work we investigate a minimal model of evolutionary10
dynamics in which cells interact only by competition for a sin-11
gle limiting resource. We find that the fixation probability of a12
mutation is accurately determined by a linear combination of13
its change in growth rate and change in lag time relative to its14
immediate ancestor, rather than depending on the precise com-15
bination of traits; the relative weight of these two components16
is determined by the dilution factor. Yield, on the other hand,17
is under no direct selection. This is true even in the presence of18
substantial clonal interference, where the mutant’s immediate19
ancestor may have large a fitness difference with the popula-20
tion mean. We provide quantitative predictions for the speed of21
adaptation of growth rate and lag time as well as their evolved22
covariation. Specifically, we find that even in the absence of an23
intrinsic correlation between growth and lag due to mutations,24
these traits can evolve a nonzero correlation due to selection and25
variation in number of fixed mutations.26
Materials and Methods27
Model of population dynamics28
We consider a model of asexual microbial cells in a well-mixed29
batch culture, where the only interaction between different30
strains is competition for a single limiting resource Manhart31
et al. (2018); Manhart and Shakhnovich (2018). Each strain k is32
characterized by a lag time Lk, growth rate rk, and yield Yk (see33
Fig. 1a for a two-strain example). Here the yield is the number34
of cells per unit resource Vasi et al. (1994), so that Nk(t)/Yk is35
the amount of resources consumed by time t by strain k, where36
Nk(t) is the number of cells of strain k at time t. We define R to37
be the initial amount of the limiting resource and assume differ-38
ent strains interact only by competing for the limiting resource;39
their growth traits are the same as when they grow indepen-40
dently. When the population has consumed all of the initial41
resource, the population reaches stationary phase with constant42
size. The saturation time tc at which this occurs is determined43
by ∑strain k Nk(tc)/Yk = R, which we can write in terms of the44
growth traits as45
∑strain k
N0xkerk(tc−Lk)
Yk= R, (1)
where N0 is the total population size and xk is the frequency46
of each strain k at the beginning of the growth cycle. In Eq. 147
we assume the time tc is longer than each strain’s lag time Lk.48
Note that some of our notation differs from related models in49
previous work, some of which used g for growth rate and λ for50
lag time Manhart et al. (2018), while others used λ for growth51
rate Lin and Amir (2017). Although it is possible to extend the52
model to account for additional growth traits such as a death53
rate or lag and growth on secondary resources, here we focus on54
the minimal set of traits most often measured in microbial phe-55
notyping experiments Novak et al. (2006); Warringer et al. (2011);56
Jasmin and Zeyl (2012); Fitzsimmons et al. (2010); Adkar et al.57
Time (t)
Dilution
a
b
(Relative growth rate change)
(Re
lative
la
g tim
e c
ha
ng
e)
0
0 WT
Benefical
mutants
Deleterious
mutants
Figure 1 Model of selection on multiple microbial growthtraits. (a) Simplified model of microbial population growthcharacterized by three traits: lag time L, growth rate r, andyield Y. The total initial population size is N0 and the initialfrequency of the mutant (strain 2) is x. After the whole pop-ulation reaches stationary phase (time tc), the population isdiluted by a factor D into fresh media, and the cycle startsagain. (b) Phase diagram of selection on mutants in the spaceof their growth rate γ = r2/r1− 1 and lag time ω = (L2− L1)r1relative to a wild-type. The slope of the diagonal line is ln D.
(2017); Levin-Reisman et al. (2010); Ziv et al. (2013); Zackrisson 58
et al. (2016). 59
We define the selection coefficient between each pair of strains 60
as the change in their log-ratio over the complete growth cy- 61
cle Chevin (2011); Good et al. (2017): 62
sij = ln
(Nfinal
iNfinal
j
)− ln
(Ninitial
iNinitial
j
)= ri(tc − Li)− rj(tc − Lj),
(2)
where Ninitiali is the population size of strain i at the beginning 63
of the growth cycle and Nfinali is the population size of strain 64
i at the end. After the population reaches stationary phase, it 65
is diluted by a factor of D into a fresh medium with amount 66
R of the resource, and the cycle repeats (Fig. 1a). We assume 67
the population remains in the stationary phase for a sufficiently 68
short time such that we can ignore death and other dynamics 69
during this phase Finkel (2006); Avrani et al. (2017). 70
Over many cycles of growth, as would occur in a labora- 71
tory evolution experiment Lenski et al. (1991); Elena and Lenski 72
2 FirstAuthorLastname et al.
(2003); Good et al. (2017), the population dynamics of this sys-1
tem are characterized by the set of frequencies xk for all strains2
as well as the matrix of selection coefficients sij and the total3
population size N0 at the beginning of each cycle. In Supple-4
mentary Methods (Secs. I, II, III) we derive explicit equations5
for the deterministic dynamics of these quantities over multiple6
cycles of growth for an arbitrary number of strains. In the case7
of two strains, such as a mutant and a wild-type, the selection8
coefficient is approximately9
s ≈ γ ln D−ω, (3)
where γ = (r2 − r1)/r1 is the growth rate of the mutant relative10
to the wild-type and ω = (L2 − L1)r1 is the relative lag time.11
The approximation is valid as long as the growth rate difference12
between the mutant and the wile-type is small (Supplementary13
Methods Sec. IV), which is true for most single mutations Levy14
et al. (2015); Chevereau et al. (2015). This equation shows that the15
growth phase and the lag phase make distinct additive contribu-16
tions to the total selection coefficient, with the dilution factor D17
controlling their relative magnitudes (Fig. 1b). This is because a18
larger dilution factor will increase the amount of time the popu-19
lation grows exponentially, hence increasing selection on growth20
rate. Neutral coexistence between multiple strains is therefore21
possible if these two selection components balance (s = 0), al-22
though it requires an exact tuning of the growth traits with the23
dilution factor (Supplementary Methods Sec. III) Manhart et al.24
(2018); Manhart and Shakhnovich (2018). With a fixed dilution25
factor D, the population size N0 at the beginning of each growth26
cycle changes according to (Supplementary Methods Sec. I):27
N0 =RYD
, (4)
where Y = (∑strain k xk/Yk)−1 is the effective yield of the whole28
population in the current growth cycle. In this manner the ratio29
R/D sets the bottleneck size of the population, which for serial30
dilution is approximately the effective population size Lenski31
et al. (1991), and therefore determines the strength of genetic32
drift.33
Model of evolutionary dynamics34
We now consider the evolution of a population as new mu-35
tations arise that alter growth traits. We start with a wild-36
type population having lag time L0 = 100 and growth rate37
r0 = (ln 2)/60 ≈ 0.012, which are roughly consistent with E.38
coli parameters where time is measured in minutes Lenski et al.39
(1991); Vasi et al. (1994); we set the wild-type yield to be Y0 = 140
without loss of generality. As in experiments, we vary the di-41
lution factor D and the amount of resources R, which control42
the relative selection on growth versus lag (set by D, Eq. 3) and43
the effective population size (set by R/D, Eq. 4). We also set the44
initial population size of the first cycle to N0 = RY0/D.45
The population grows according to the dynamics in Fig. 1a.46
Each cell division can generate a new mutation with probability47
µ = 10−6; note this rate is only for mutations altering growth48
traits, and therefore it is lower than the rate of mutations any-49
where in the genome. We generate a random waiting time τk50
for each strain k until the next mutation with instantaneous rate51
µrk Nk(t). When a mutation occurs, the growth traits for the52
mutant are drawn from a distribution pmut(r2, L2, Y2|r1, L1, Y1),53
where r1, L1, Y1 are the growth traits for the background strain54
on which the new mutation occurs and r2, L2, Y2 are the traits for55
the new mutant. Note that since mutations only arise during the56
exponential growth phase, beneficial or deleterious effects on lag 57
time are not realized until the next growth cycle Li et al. (2018). 58
After the growth cycle ceases (once the resource is exhausted 59
according to Eq. 1), we randomly choose cells, each with proba- 60
bility 1/D, to form the population for the next growth cycle. 61
We will assume mutational effects are not epistatic and 62
scale with the trait values of the background strain, so that 63
tary Methods Sec. V). Since our primary goal is to scan the space 66
of possible mutations, we focus on uniform distributions of mu- 67
tational effects where −0.02 < γ < 0.02, −0.05 < ω < 0.05, and 68
−0.02 < δ < 0.02. In the Supplementary Methods we extend 69
our main results to the case of Gaussian distributions (Sec. V) as 70
well as an empirical distribution of mutational effects based on 71
single-gene deletions in E. coli (Sec. VI) Campos et al. (2018). 72
Data Availability 73
Data and codes are available upon request. File S1 contains the 74
Supplementary Methods. File S2 contains data of growth traits 75
presented in Figure S3. 76
Results 77
Fixation of mutations 78
We first consider the fixation statistics of new mutations in our 79
model. In Fig. 2a we show the relative growth rates γ and the 80
relative lag times ω of fixed mutations against their background 81
strains, along with contours of constant selection coefficient s 82
from Eq. 3. As expected, fixed mutations either increase growth 83
rate (γ > 0), decrease lag time (ω < 0), or both. In contrast, the 84
yield of fixed mutations is the same as the ancestor on average 85
(Fig. 2b); indeed, the selection coefficient in Eq. 3 does not de- 86
pend on the yields. If a mutation arises with significantly higher 87
or lower yield than the rest of the population, the bottleneck 88
population size N0 immediately adjusts to keep the overall fold- 89
change of the population during the growth cycle fixed to the 90
dilution factor D (Eq. 4). Therefore mutations that significantly 91
change yield have no effect on the overall population dynamics. 92
Figure 2a also suggests that the density of fixed mutations 93
in the growth-lag trait space depends solely on their selection 94
coefficients, rather than the precise combination of traits, as 95
long as other parameters such as the dilution factor D, the total 96
amount of resource R, and the distribution of mutational ef- 97
fects are held fixed. Mathematically, this means that the fixation 98
probability φ(γ, ω) of a mutation with growth effect γ and lag 99
effect ω can be expressed as φ(γ, ω) = φ(γ ln D − ω) ≡ φ(s). 100
To test this, we discretize the scatter plot of Fig. 2a and compute 101
the fixation probabilities of mutations as functions of γ and ω 102
(Supplementary Methods Sec. VII). We then plot the resulting 103
fixation probabilities of mutations as functions of their selection 104
coefficients calculated by Eq. 3 (Fig. 2c,d,e,f). We test the depen- 105
dence of the fixation probability on the selection coefficient over 106
a range of population dynamics regimes by varying the dilution 107
factor D and the amount of resources R. 108
For small populations, mutations generally arise and either 109
fix or go extinct one at a time, a regime known as “strong- 110
selection weak-mutation” (SSWM) Gillespie (1984). In this case, 111
we expect the fixation probability of a beneficial mutation with 112
selection coefficient s > 0 to be Wahl and Gerrish (2001); Wahl 113
and Zhu (2015); Guo et al. (2019) 114
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Figure 2 Selection coefficient determines fixation probability. (a) The relative growth rates γ and the relative lag times ω of fixedmutations against their background strain. Dashed lines mark contours of constant selection coefficient with interval ∆s = 0.015while the solid line marks s = 0. (d) Same as (a) but for relative growth rate γ and the relative yield δ. The red dots mark therelative yield of fixed mutations averaged over binned values of the relative growth rate γ. In (a) and (d), D = 102 and R = 107.(b,c,e,f) Fixation probability of mutations against their selection coefficients for different amounts of resource R and dilution factorsD as indicated in the titles. The red dashed line shows the fixation probability predicted in the SSWM regime (Eq. 5), while theblack line shows a numerical fit of the data points to Eq. 6 with parameters A = 0.1145 and B = 0.0801 in (c), A = 0.0017 andB = 0.0421 in (e), and A = 0.2121 and B = 0.2192 in (f). In all panels mutations randomly arise from a uniform distribution pmutwith −0.02 < γ < 0.02, −0.05 < ω < 0.05, and −0.02 < δ < 0.02.
4 FirstAuthorLastname et al.
φSSWM(s) =2 ln DD− 1
s. (5)
This is similar to the standard Wright-Fisher fixation probability1
of 2s Crow and Kimura (1970), but with a different prefactor2
due to averaging over the different times in the exponential3
growth phase at which the mutation can arise (Supplementary4
Methods Sec. VIII). Indeed, we see this predicted dependence5
matches the simulation results for the small population size of6
N0 ∼ R/D = 103 (Fig. 2c).7
For larger populations, multiple beneficial mutations will be8
simultaneously present in the population and interfere with each9
other, an effect known as clonal interference Gerrish and Lenski10
(1998); Desai and Fisher (2007); Schiffels et al. (2011); Good et al.11
(2012); Fisher (2013); Good and Desai (2014). Our simulations12
show that, as for the SSWM case, the fixation probability de-13
pends only on the selection coefficient (Eq. 3) relative to the14
mutation’s immediate ancestor and not on the individual com-15
bination of mutant traits (Fig. 2d,e,f), with all other population16
parameters held constant. Previous work has determined the de-17
pendence of the fixation probability on the selection coefficient18
under clonal interference using various approximations Gerrish19
and Lenski (1998); Schiffels et al. (2011); Good et al. (2012); Fisher20
(2013). Here, we focus on an empirical relation based on Gerrish21
and Lenski (1998):22
φCI(s) = Ase−B/s, (6)
where A and B are two constants that depend on other parame-23
ters of the population (D, R, and the distribution of mutational24
effects); we treat these as empirical parameters to fit to the sim-25
ulation results, although Gerrish and Lenski (1998) predicted26
A = 2 ln D/(D− 1), i.e., the same constant as in the SSWM case27
(Eq. 5). The e−B/s factor in Eq. 6 comes from the probability28
that no superior beneficial mutations appears before the current29
mutation fixes. Since the time to fixation scales as 1/s, we expect30
the average number of superior mutations to be proportional31
to 1/s (for small s). This approximation holds only for selection32
coefficients that are not too small and therefore are expected to33
fix without additional beneficial mutations on the same back-34
ground; Eq. 6 breaks down for weaker beneficial mutations that35
typically fix by hitchhiking on stronger mutations Schiffels et al.36
for a wide range of selection coefficients achieved in our simula-38
tions and larger population sizes N0 ∼ R/D > 104 (Fig. 2d,e,f).39
Furthermore, the constant A we fit to the simulation data is in-40
deed close to the predicted value of 2 ln D/(D − 1), except in41
the most extreme case of N0 ∼ R/D = 106 (Fig. 2f).42
Altogether Fig. 2 shows that mutations with different effects43
on cell growth — for example, a mutant that increases the growth44
rate and a mutant that decreases the lag time — can neverthe-45
less have approximately the same fixation probability as long46
as their overall effects on selection are the same according to Eq.47
3. To test the robustness of this result, we verify it for several48
additional distributions of mutational effects pmut(γ, ω, δ) in the49
Supplementary Methods: a Gaussian distribution of mutational50
effects, including the presence of correlated mutational effects51
(Fig. S1); a wider distribution of mutational effects with large52
selection coefficients (Fig. S2); and an empirical distribution of53
mutational effects estimated from single-gene deletions in E. coli54
(Fig. S3). In Fig. S4a we further test robustness by using the55
neutral phenotype (orthogonal to the selection coefficient) to56
quantify the range of γ and ω trait combinations that neverthe- 57
less have the same selection coefficient and fixation probability, 58
and in Fig. S4b we show that the selection coefficient on growth 59
alone is insufficient to determine fixation probability. 60
While the dependence of fixation probability on the selec- 61
tion coefficient is a classic result of population genetics Hartl 62
et al. (1997), the existence of a simple relationship here is non- 63
trivial since, strictly speaking, selection in this model is not 64
only frequency-dependent Manhart et al. (2018) (i.e., selection 65
between two strains depends on their frequencies) but also in- 66
cludes higher-order effects Manhart and Shakhnovich (2018) (i.e., 67
selection between strain 1 and strain 2 is affected by the presence 68
of strain 3). Therefore in principle, the fixation probability of a 69
mutant may depend on the specific state of the population in 70
which it is present, while the selection coefficient in Eq. 3 only 71
describes selection on the mutant in competition with its imme- 72
diate ancestor. However, we see that, at least for the parameters 73
considered in our simulations, these effects are negligible in 74
determining the eventual fate of a mutation. 75
Adaptation of growth traits 76
As Fig. 3a shows, many mutations arise and fix over the 77
timescale of our simulations, which lead to predictable trends 78
in the quantitative traits of the population. We first determine 79
the relative fitness of the evolved population at each time point 80
against the ancestral strain by simulating competition between 81
an equal number of evolved and ancestral cells for one cycle, 82
analogous to common experimental measurements Lenski et al. 83
(1991); Elena and Lenski (2003). The resulting fitness trajectories 84
are shown in Fig. 3b. To see how different traits contribute to the 85
fitness increase, we also calculate the average population traits at 86
the beginning of each cycle; for instance, the average population 87
growth rate at growth cycle n is rpop(n) = ∑strain k rkxk(n). As 88
expected from Eq. 3, the average growth rate increases (Fig. 3c) 89
and the average lag time decreases (Fig. 3d) for all simulations. 90
In contrast, the average yield evolves without apparent trend 91
(Fig. 3e), since Eq. 3 indicates no direct selection on yield. We 92
note that, while the cells do not evolve toward lower or higher 93
resource efficiency on average, they do evolve to consume re- 94
sources more quickly, since the rate of resource consumption 95
(rk/Yk for each cell of strain k) depends on both the yield as well 96
as the growth rate. Therefore the saturation time of each growth 97
cycle evolves to be shorter, consistent with recent work from 98
Baake et al. (2019). 99
Figure 3 suggests relatively constant speeds of adaptation for 100
the relative fitness, the average growth rate, and the average 101
lag time. For example, we can calculate the adaptation speed of 102
the average growth rate as the averaged change in the average 103
growth rate per cycle: 104
Wgrowth = 〈rpop(n + 1)− rpop(n)〉, (7)
where the bracket denotes an average over replicate populations 105
and cycle number. In the Supplementary Methods (Secs. IX 106
and X) we calculate the adaptation speeds of these traits in the 107
SSWM regime to be 108
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Figure 3 Dynamics of evolving populations. (a) Frequenciesof new mutations as functions of the number n of growth cy-cles. Example trajectories of (b) the fitness of the evolved pop-ulation relative to the ancestral population, (c) the evolvedaverage growth rate, (d) the evolved average lag time, and(e) the evolved average yield. In all panels the dilution factoris D = 102, the amount of resource at the beginning of each cy-cle is R = 107, and mutations randomly arise from a uniformdistribution pmut with −0.02 < γ < 0.02, −0.05 < ω < 0.05,and −0.02 < δ < 0.02.
Wgrowth = σ2γr0(ln D)
(µRY0 ln D
D− 1
),
Wlag = −σ2ω
r0
(µRY0 ln D
D− 1
),
Wfitness =Wgrowth
r0ln D−Wlagr0,
(8)
where σγ and σω are the standard deviations of the underlying 1
distributions of γ and ω for single mutations (pmut(γ, ω, δ)), r0 2
is the ancestral growth rate and Y0 the ancestral yield (we as- 3
sume the yield does not change on average according to Fig. 3e). 4
Furthermore, the ratio of the growth adaptation rate and the lag 5
adaptation rate is independent of the amount of resource and 6
mutation rate in the SSWM regime: 7
Wgrowth
Wlag= −r2
0σ2
γ
σ2ω
ln D. (9)
Equation 8 predicts that the adaptation speeds of the average 8
growth rate, the average lag time, and the relative fitness should 9
all increase with the amount of resources R and decrease with 10
the dilution factor D (for large D); even though this prediction as- 11
sumes the SSWM regime (relatively small N0 ∼ R/D), it never- 12
theless holds across a wide range of R and D values (Fig. 4a,b,c), 13
except for R = 108 where the speed of fitness increase is non- 14
monotonic with D (Fig. 4c). The predicted adaptation speeds in 15
Eq. 8 also quantitatively match the simulated trajectories in the 16
SSWM case (Fig. 4d,e,f); even outside of the SSWM regime, the 17
relative rate in Eq. 9 remains a good prediction at early times 18
(Fig. S5). 19
Evolved covariation between growth traits 20
We now turn to investigating how the covariation between traits 21
evolves. We have generally assumed that individual mutations 22
have uncorrelated effects on different traits. Campos et al. (2018) 23
recently systematically measured the growth curves of the single- 24
gene deletions in E. coli. We compute the relative growth rate, 25
lag time, and yield changes for the single-gene deletions com- 26
pared with the wild-type and find that the resulting empirical 27
distribution of relative growth traits changes shows very small 28
correlations between these traits (Fig. S3b,c), consistent with our 29
assumptions. We note that these measurements, however, are 30
subject to significant noise (Supplementary Methods Sec. VI), 31
and therefore any conclusions ultimately require verification by 32
further experiments. 33
Even in the absence of mutational correlations, selection may 34
induce a correlation between these traits in evolved populations. 35
In Fig. 5a we schematically depict how the raw variation of traits 36
from mutations is distorted by selection and fixation of multiple 37
mutations. Specifically, for a single fixed mutation, selection 38
induces a positive (i.e., antagonistic) correlation between the 39
relative growth rate change and the relative lag time change. 40
Figure 2a shows this for single fixed mutations, while Fig. 5b,c 41
shows this positive correlation between the average growth rate 42
and the average lag time across populations that have accumu- 43
lated the same number of fixed mutations. For populations in 44
the SSWM regime with the same number of fixed mutations, the 45
Pearson correlation coefficient between the average growth rate 46
and the average lag time across populations is approximately 47
equal to the covariation of the relative growth rate change γ and 48
the relative lag time change ω for a single fixed mutation: 49
6 FirstAuthorLastname et al.
102
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Figure 4 Speed of adaptation. The average per-cycle adaptation speed of (a) the average growth rate, (b) the average lag time, and(c) the fitness relative to the ancestral population as functions of the dilution factor D and total amount of resources R. The adap-tation speeds are averaged over growth cycles and independent populations. (d) The average growth rate, (e) the average lagtime, and (f) the fitness relative to the ancestral population as functions of the number n of growth cycles. The dilution factor isD = 104 and the total resource is R = 107, so the population is in the SSWM regime. The blue solid lines are simulation results,while the dashed lines show the mathematical predictions in Eq. 8. All panels show averages over 500 independent simulated pop-ulations, with mutations randomly arising from a uniform distribution pmut with −0.02 < γ < 0.02, −0.05 < ω < 0.05, and−0.02 < δ < 0.02.
GENETICS Journal Template on Overleaf 7
0.012 0.014 0.016 0.018 0.02 0.022 0.024 0.026
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fixed mutations
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a
Figure 5 Evolved patterns of covariation among growth traits. (a) Schematic of how selection and fixation of multiple mutationsshape the observed distribution of traits. The sign of the Pearson correlation coefficient between the average growth rate and lagtime depends on whether we consider an ensemble of populations with the same number of fixed mutations or the same number oftotal mutation events. (b) Distribution of average growth rate and lag time for 1000 independent populations with the same num-ber of fixed mutations. Each color corresponds to a different number of fixed mutations (n f ) indicated in the legend. (c) Pearsoncorrelation coefficient of growth rate and lag time for distributions in panel (b) as a function of the number of fixed mutations. Thedashed line is the prediction from Eq. 10. (d) Same as (b) except each color corresponds to a set of populations at a snapshot in timewith the same number of total mutation events. Each color corresponds to a different number of total mutations events (nt) indi-cated in the legend. (e) Same as (c) but for the set of populations shown in (d). The dashed line is the prediction from Eq. 11. In (c)and (e) the error-bars represent 95% confidence intervals. In (b–e) we simulate the SSWM regime by introducing random mutationsone-by-one and determining their fixation from Eq. 5 with D = 103.