# Analytical description of mutational effects in competing asexual populations.

**ABSTRACT** The adaptation of a population to a new environment is a result of selection operating on a suite of stochastically occurring mutations. This article presents an analytical approach to understanding the population dynamics during adaptation, specifically addressing a system in which periods of growth are separated by selection in bottlenecks. The analysis derives simple expressions for the average properties of the evolving population, including a quantitative description of progressive narrowing of the range of selection coefficients of the predominant mutant cells and of the proportion of mutant cells as a function of time. A complete statistical description of the bottlenecks is also presented, leading to a description of the stochastic behavior of the population in terms of effective mutation times. The effective mutation times are related to the actual mutation times by calculable probability distributions, similar to the selection coefficients being highly restricted in their probable values. This analytical approach is used to model recently published experimental data from a bacterial coculture experiment, and the results are compared to those of a numerical model published in conjunction with the data. Finally, experimental designs that may improve measurements of fitness distributions are suggested.

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**ABSTRACT:**Recent advances in the mathematical analysis of models describing evolutionary dynamics are rapidly increasing our ability to make precise quantitative predictions. These advances have created a growing need for corresponding improvements in our ability to observe evolutionary dynamics in laboratory evolution experiments. High-throughput experimental methods are particularly crucial, in order to maintain many replicate populations and measure statistical differences in evolutionary outcomes at both phenotypic and genomic levels. In this paper, I describe recent technical developments which have greatly increased the throughput of laboratory evolution experiments, and outline a few promising directions for further improvements. I then highlight a few ways in which these new experimental methods can help to answer simple statistical questions about evolutionary dynamics, and potentially guide future theoretical work.Journal of Statistical Mechanics Theory and Experiment 01/2013; 2013(01):P01003. · 2.06 Impact Factor

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Copyright ? 2007 by the Genetics Society of America

DOI: 10.1534/genetics.107.075697

Analytical Description of Mutational Effects in Competing

Asexual Populations

Daniel Pinkel1

Comprehensive Cancer Center and Department of Laboratory Medicine, University of California,

San Francisco, California 94143

Manuscript received May 7, 2007

Accepted for publication October 1, 2007

ABSTRACT

The adaptation of a population to a new environment is a result of selection operating on a suite of

stochastically occurring mutations. This article presents an analytical approach to understanding the

population dynamics during adaptation, specifically addressing a system in which periods of growth are

separated by selection in bottlenecks. The analysis derives simple expressions for the average properties of

the evolving population, including a quantitative description of progressive narrowing of therange of selec-

tioncoefficientsofthepredominantmutantcellsandoftheproportionofmutantcellsasafunctionoftime.

A complete statistical description of the bottlenecks is also presented, leading to a description of the sto-

chastic behavior of the population in terms of effective mutation times. The effective mutation times are

related to the actual mutation times by calculable probability distributions, similar to the selection coef-

ficients being highly restricted in their probable values. This analytical approach is used to model recently

publishedexperimentaldatafromabacterialcocultureexperiment,andtheresultsarecomparedtothoseof

a numerical model published in conjunction with the data. Finally, experimental designs that may improve

measurements of fitness distributions are suggested.

T

on the suite of stochastically occurring mutations, each

of which may confer a different selective advantage.

The mutations occur throughout time, so that multiple

clones of mutant cells are present simultaneously.

Mathematical modeling of the population dynamics

typically employs numerical simulation to calculate a

particular instance of the system, sampling probability

distributions to include stochastic effects. The calcu-

lations may involve detailed tracking of each mutant

clone through the history of the population. Running

the model many times allows determination of its char-

acteristic behavior as a function of the parameters de-

scribing mutation and selection. Estimates of the values

of these parameters in a living system are obtained by

comparisons of the statistical properties of the model

with those of experimental data.

By contrast, this article presents an analytical de-

scription of the population dynamics. The analysis be-

gins by establishing the identity of the average behavior

of sequential finite cultures separated by bottlenecks

with the growth of an exponentially expanding effec-

tively infinite culture. Consideration of the infinite sys-

tem provides analytical expressions for characteristic

properties of the finite populations. The results include

HE adaptation of asexual populations to a new

environment is the result of selection operating

quantitative descriptions of growth of the proportion of

mutant cells with time and of the accompanying nar-

rowing of the frequency distribution of their selection

coefficients. Next, the stochastic behavior of finite sys-

tems is considered, resulting in a comprehensive and

convenient description of the selection in the bottle-

necks. The stochastic description is then used to de-

velop a model of coculture experiments such as the one

recently published by Hegreness and Shoresh (HS)

(Hegreness et al. 2006). Application of the results ob-

tained for the average behavior is used to simplify the

model and facilitate its comparison with experimental

data. Finally, the results are discussed and alternate

experimental designs that may allow better measure-

ment of the fitness distribution of the mutant cells are

suggested. Significant additional information concern-

ingtheanalysisispresentedinthesupplementalmateri-

alsathttp:/ /www.genetics.org/supplemental/,including

an Excel workbook for calculation of the statistical

distributions that are developed in this article.

THE EXPERIMENTAL SYSTEM

Consideration of the experimental system of HS

(Figure 1a) provides concrete motivation for the anal-

ysis presented in the remainder of this article. Approx-

imately23105cellswere seeded intoaculture,one-half

labeled with yellow fluorescent protein (YFP) and one-

half with cyan fluorescent protein (CFP). Otherwise the

1Address for correspondence: University of California, Box 0808, San

Francisco, CA 94143. E-mail: pinkel@cc.ucsf.edu

Genetics 177: 2135–2149 (December 2007)

Page 2

ancestral cells were identical. Mutations with selection

coefficients s occurred stochastically in both popula-

tions of ancestral cells with a probability distribution

r(s) and an overall frequency m per generation per cell.

The cells grew exponentially for 24 hr, and the culture

wasthensampledtoseedthenextpassagewith?23105

cells. After this bottleneck, exponential growth re-

sumed, followed by sampling to seed the next daughter

culture after another 24 hr, etc. The full experiment

lasted ?40 days or ?450 doublings for the starting

(ancestral) cells. Figure 1a illustrates the growth of the

CFP and YFP populations just before and after the kth

bottleneck for a time early in the series before the mu-

tant populations have become significant. Each growth

phase lasts a time t ¼ 11.7 population doublings and

resuts in an ert? 3300-fold increase in cell number,

where r ¼ ln(2). At the bottleneck, a sampling of e?r t?

1/3300 of the mutant and ancestral cells proceeded to

the subsequent daughter culture. The rest of the cells

were discarded.

YFP/CFP fluorescence ratios in each of 72 series of

such cultures were measured at each bottleneck, pro-

viding a measure of the relative numbers of the two cell

types as a function of time. Because of the stochastic

nature of the mutational process and the passage of

mutants through the bottlenecks, one expects that

under some conditions the ratio may change with time,

differing in each series of cultures. Figure 1b shows

examples of four of the possible time courses for the

ratio, using green and red to distinguish the popula-

tions. The ancestral cells in both populations are repre-

sented as light colors, and the mutants (of any s) as dark

colors. This distinction is made for illustrative purposes

only since ancestral and mutant cells cannot be distin-

guished by their fluorescence. While the colors are

illustrated as separated, in the actual experiment the

two populations are thoroughly mixed.

In the first example, mutant cells are assumed to

cometoprominenceinitiallyinthe‘‘green’’population.

This time is indicated by a green T1, defined in our

analysisasthetimewhentheproportionsofmutantand

ancestral cells in the green population are equal. The

overall growth rate of the green population detectably

increases, and green cells begin to overgrow the ‘‘red’’

ones. Mutant cells with approximately the same s as

those in the green population are assumed to come to

prominence in the red population at a later time,

indicated by the red T1. After this time, the green and

red populations tend toward the same growth rate, and

thus their ratio stabilizes. Mutant cells completely dom-

inate both populations with time, as indicated by the

darkening colors. This growth pattern corresponds to

Figure 1.—Schematic of the YFP/CFP cocul-

ture experiment. (a) The exponential expansion

of the YFP and CFP populations (distinguished

by green and red colors) during the growth peri-

ods before and after the kth bottleneck at a time

early in the experiment when mutant cells are

present at low abundance. The cultures begin

with N0of each type of cell and exponentially ex-

pand for time t so that there are N0ertof each. At

the bottleneck N0of each type of cell is removed

and transferred to the next culture. The remain-

der is discarded. In the actual experiment, t ¼

11.7 doublings, and the expansion factor is ert?

3300. (b) Four representative time courses for

the behavior of the color ratios. Ancestral cells

are indicated by light colors and mutants (of

any selection coefficient) by dark colors. The col-

ors are drawn separately for illustrative purposes,

but in the actual system the cells are uniformly

mixed. Schematic growth curves for the mutant

and ancestral populations during three of the

growth periods for the first series of cultures

are shown in boxes above that series. The behav-

ior of each culture series is described in the text.

The red and green T1’s indicate the times when

mutant and ancestral cells are of equal abun-

dance in the respective populations.

2136D. Pinkel

Page 3

curves2and3inFigure6,aandb.Graphsaboveseries1

schematically show the growth of the mutant and an-

cestral populations during three cultures of this series.

In the second example, red mutants completely over-

take the culture before green mutants of sufficiently

large s come to prominence, although green mutations

with low s will have occurred. This corresponds to curve

1 in Figure 6,a and b. Inthethird example, red mutants

are assumed to come to prominence first and the

fluorescence ratio begins to change in favor of red. At

a later time, mutant cells with larger s come to prom-

inence in the green population, and the ratio changes

infavorofgreen.Stilllater,mutantsthathaveaselection

coefficient equivalent to the green population come to

prominence in the red population, and the ratio sta-

bilizes. The initial part of such behavior is shown by

curve 5 in Figure 6b. Subsequent variations in the ratio

may occur as mutants with ever larger s come to prom-

inence in the two populations. But if r(s) has a suf-

ficiently defined maximum s, then the ratio will finally

stabilize when both populations are dominated by

mutant cells with this maximal selection coefficient un-

less one population completely displaces the other as

shown in example 2. In the final example, roughly

equivalent mutants arise in both populations at about

the same times, so that the ratio remains constant as

both populations become dominated by mutant cells

with selection coefficients tending toward the maxi-

mum possible. This is the behavior that always occurs if

the size of the cultures is sufficiently large so that many

mutations occur.

THE ANALYTICAL FRAMEWORK

Imagine an arbitrarily large ensemble of initial cul-

tures, each starting with N0cells (Figure 2a). Only one

cell population is shown for clarity. If additional popu-

lations are present as in the coculture experiment, their

behaviors will be statistically independent. Each initial

culture is sampled after growth time t, but instead of

seeding only one next passage culture as in the actual

experiment, all of the material from each initial culture

Figure 2.—The conceptual experi-

ment. (a) An arbitrarily large number

of initial cultures, each containing N0

cells, are started at t ¼ 0. Two such cul-

tures, h and h 1 1, are shown. After time

t,thetimeofthefirstbottleneck,thecells

ineachinitialculturearedividedamong

ertdaughter cultures so that each re-

ceives N0 cells. Thus no cells are dis-

carded. After each subsequent time

interval t, each daughter seeds ertnext-

generation daughters.

growth phase mutants received from

the previous culture expand and de novo

mutants randomly occur. At each bottle-

neck there is statistical variation in the

number of mutant andancestral cells re-

ceived by the daughters. This process is

indicated by the inset in the oval where

m(s)anddm(s)indicatetheaveragenum-

berandthefluctuationfromtheaverage

of the mutants transmitted to the next

daughter, and N and dN indicate the

same for the ancestral cells. The time

(in population doublings) and the num-

ber of the bottleneck is indicated along

thebottom,assuming11.7doublingsbe-

tween bottlenecks. (b) The shape of

W(s, t), normalized to its value at s ¼

0.1, is shown as a function of s for repre-

sentative values of t. As t increases the

weight of this function rapidly moves to

larger s. The distribution of mutant cells

in the pooled culture of a is given by

r(s)W(s,t).Foreachofthepossiblepaths

through a series of daughters, the selec-

tioncoefficientsofthemutantswilldiffer

due to the stochastic variation, but the

Duringeach

powerful shape of W(s, t) ensures that those that become prominent will be confined to a small range of s almost regardless of

theshapeofr(s).Ifr(s)isaconstantouttosomesmax,thenW(s,t)directlygivestheexpecteddistributionofthemutants.Thevalues

of s, termed s10, above which 90% of the mutant cells occur are indicated on the graphs for this case (see Equation 8).

Analysis of Asexual Competition2137

Page 4

is used to seed ertdaughter cultures (3300 in the case of

the specific experiment in question). The details of this

exhaustive sampling are illustrated in the oval inset in

Figure 2a for the kth bottleneck. Each of the (k 11)st

cultures receive ?N 1 dN ancestral and m(s) 1 dm(s)

mutant cells, where the d’s indicate stochastic fluctua-

tions that affect the individual cultures. The actual

experiment is equivalent to selecting 72 of the initial

cultures, and for each of these selecting a sequence of

daughters, thereby producing 72 series with 40 sequen-

tial cultures in each. Two such series are indicated in

Figure 2a by the shaded boxes. Examination of Figure

2ashowsthatthetimedependenceofthecharacteristics

of the mutant cells averaged over a large number of

series of daughters would be identical to the behavior of

the total mutant population if all of the daughters were

pooled. This pooled culture is just a population in un-

bounded exponential growth, which can be analyzed

with straightforward approaches. Since the structure of

this conceptual experiment preserves cell lineages, it

also provides the basis for the subsequent calculation of

the stochastic properties of the system.

This conceptual experiment is, of course, impossible

to implement. Using the parameters of the actual ex-

periment, by the 13th of the 40 days, the volume of cul-

ture medium required for the progeny of a single initial

culture would fill a ball with a radius about 22 times that

ofour solar system out to Pluto, and the radius would be

increasing faster than the speed of light. Coincidentally,

this isabout the characteristic time, T1? 150 doublings,

that mutants became equal in abundance to the ances-

tral cells in the actual HS experiment (see below).

DESCRIPTION OF THE AVERAGE CHARACTERISTICS

Basic relationships: Consider one of the cell popula-

tions in the infinite pooled culture of Figure 2a. Its

development is described by standard equations for

exponential growth. Let N‘(t) and M‘(s, t) be the

number of ancestral and mutant cells at time t. Then

_N‘ðtÞ ¼ ð1 ? mÞrN‘ðtÞ ffi rN‘ðtÞ

ð1aÞ

˙ M‘ðs;tÞ ¼ mrðsÞN‘ðtÞ1rð11sÞM‘ðtÞ;

where the dots indicate the derivative with respect to

time, m is the overall mutation rate per generation for

theancestralcells,r(s)istheprobability distributionfor

theselectioncoefficientssofthemutants,andr¼ln(2).

Equation 1a describes the increase of ancestral cells due

to division and the decrease due to conversion to

mutants. Since m > 1, it is neglected in what follows.

The coefficient r scales time so that it is measured in

units of the doubling time for the ancestral cells.

Equation 1b describes the increase in the number of

ð1bÞ

mutant cells with selection coefficient s through de novo

mutationandthedivisionofexistingmutantswitharate

1 1 s times that of the ancestral cells. In the real world,

singly mutant cells are susceptible to additional muta-

tions. The possibility of multiple mutations raises com-

plex modeling issues that are discussed in section 2 of

the supplemental materials at http:/ /www.genetics.org/

supplemental/. Multiple mutants are neglected in what

follows.

This continuous growth model (with bottlenecks

introduced below) does not include one important

component oftheactual experiment. Inthe real experi-

menttheculturesenterastationaryphasewheregrowth

stops prior toseedingthe daughtercultures. Asmutants

become prominent in the population and the overall

growth rate increases somewhat, this stationary phase

will be reached earlier during each passage. By contrast,

the model allows expansion to continue during this

stationary period. The error introduced by this simpli-

fication is small for the HS experiment. As is shown

below, the maximum selection coefficient for the mu-

tants in the experiment is ?0.1. Thus, after cultures

become dominated by mutants, in the time equivalent

to 11.7 doublings of the ancestral cells the model allows

about one additional doubling per passage (rst ? 1).

Therefore, during the latephasesof the experimentthe

timescale in the model may be somewhat accelerated

compared to that of the actual cultures. However, in-

clusion of this small effect in the analysis is not war-

ranted given the noise in the experimental data with

which it will be compared. The presence of the sta-

tionary phase raises additional interpretive issues, since

asindicatedinthediscussiontheselectiveadvantageof

mutants may be expressed by changes in their behavior

as they cease and resume proliferation.

ThesolutionofEquation1aisN‘(t) ¼N‘ert,whereN‘

is the number of ancestral cells at t ¼ 0. Inserting N‘(t)

into Equation 1b allows solving for M‘(s, t). It is con-

venient for the subsequent discussion to calculate

Rm(s, t), the ratio of mutant cells with selection coeffi-

cientstothetotalnumberofancestralcells,becausethis

describes how significant the mutant population has

become:

Rmðs;tÞ ¼ M‘ðs;tÞ=N‘ðtÞ

or

M‘ðs;tÞ ¼ N‘ðtÞRmðs;tÞ:

Inserting this into Equation 1b, one finds

Rmðs;tÞ ¼ mrðsÞðerst? 1Þ=rs ¼ mrðsÞWðs;tÞ;

where

ð2Þ

Wðs;tÞ ¼ ðerst? 1Þ=rs:

It is immediately apparent that the distribution of se-

lection coefficients found in the mutant cells at time

t is given by the product of r(s) with the weight factor

W(s, t).

2138D. Pinkel

Page 5

At small values of st, W(s, t) ¼ t, and Rm(s, t) ¼ m tr(s),

indicating the buildup of denovo mutationslinearlywith

time and with a distribution in s given by r(s). As s and/

or t increase, W(s, t) increases dramatically due to the

relative expansion of mutations that occurred early in

thecultures.Figure2bshowstheshapeofW(s,t)for0,

s , 0.1 and times t ¼ 1, 11.7 (the first bottleneck in the

actualexperiment),100,and200populationdoublings.

For plotting convenience these graphs have been nor-

malized to the values of W(s, t) at s ¼ 0.1. As time

progresses, the weight of this function becomes con-

centrated at higher values of s. Thus for almost any

shape of r(s) that one might choose, the width of the

rangeof selectioncoefficientsthat areprominent in the

mutant population will become narrower as time pro-

gresses. For most shapes of r(s) the ‘‘effective’’selection

coefficients will fall within a narrow range near the max-

imum s that is possible. An alternate derivation of Equa-

tion2,whichhastheflexibilitytoaddressmorecomplex

systems, is given in section 1 of the supplemental ma-

terials at http:/ /www.genetics.org/supplemental/.

Given Equation 2, the ratio of the number of mutant

cells with selection coefficients between 0 and some

value of s to the total number of ancestral cells at time t,

denoted by Pm(s, t), can be calculated. Thus

ðs

Integrating over the whole range of s gives Pm(‘, t), the

ratio ofthetotal number ofmutantcellsto the ancestral

cellsattimet.Thefraction ofmutantcellswithselection

coefficients between any values s1and s2is then given by

Qðs1;s2Þ ¼ ½Pmðs2;tÞ ? Pmðs1;tÞ?=Pmð‘;tÞ:

These relationships allow determination of the average

behavior of the system for various assumptions concern-

ing m and r(s). The range of effective selection coef-

ficients and the characteristic times T1and T100, for

which the number of mutant cells are respectively equal

to, or 100 times greater than, the ancestral population

are now compared for two specific choices r(s).

Comparison of uniform and delta-function distri-

butions for r(s): Consider first a uniform distribution

r(s) ¼ 1/smaxfor 0 # s # smaxand 0 otherwise. The

product r(s)W(s, t) is W(s,t)/smaxfor 0#s #smaxand so

has the shape of W(s, t) up to smax, whereupon it drops

to 0. Figure 2b shows the shape of this function, indi-

cating that as time increases the predominant mutants

have selection coefficients progressively closer to smax.

Applying Equations 3 and 4 allows calculation of the

range of s of the effective mutations:

ðs

¼

rsmax

0

z

Pmðs;tÞ ¼

0

Rmðs9;tÞds9 ¼ m

ðs

0

rðs9ÞWðs9;tÞds9:

ð3Þ

ð4Þ

Pmðs;tÞ ¼ m

0

rðs9ÞWðs9;tÞds ¼

ðrst

m

smax

ðs

0

ers9t? 1

rs9

??

ds9

m

ez? 1

??

dz ¼

m

rsmaxIðrstÞ:

ð5Þ

The integral I(rst) can be evaluated numerically in a

straightforward manner. Figure 3a shows a graph of

Log10½I(rst)? along with regression fit for the region 5 ,

rst , 50. Using this fit as an approximation gives:

Log10½Pmðs;tÞ? ¼ Log10½m=ðrsmaxÞ?1Log10½IðrstÞ?

? Log10½m=ðrsmaxÞ?10:39rst ? 0:50

10:0005ðrstÞ21 ...

? Log10½m=smax?10:27st ? 0:34

or equivalently

ð6Þ

Pmðs;tÞ ?0:32me0:9rst

rsmax

:

The use of the linear approximation for the fit to

Log10½I(rst)? is particularly accurate for 7 , rst , 16,

for which the deviation of the approximate value of

Log10½Pm(s, t)? from the true value is ,0.1. This corre-

sponds to times on the order of ?100 to ?230 genera-

tionsfors?0.1.Section4ofthesupplementalmaterials

Figure 3.—Approximate evaluation of the integral I(rst).

(a) The black line shows the numerical evaluation of the inte-

gral in Equation 5 for 0.1 , rst , 50. The red line, coincident

withtheblackline,showsasecond-orderfittothevaluesfor5,

rst , 50. The coefficients of the fit are sensitive to the exact

rangeofthecurvethatisbeingfitted,butthesearerepresenta-

tive.(b)EvaluationofI(rst)for0,rst,3(blackline).Theresult

of integrating a Taylor series expansion of the integrand is

showninred.Theformulaforthiscurveisshownonthegraph

and used in section 4 of the supplemental materials at http:/ /

www.genetics.org/supplemental/.

Analysis of Asexual Competition2139