**Multileveled Selection on Plasmid Replication**

**Johan Paulsson**

**1**

*Department of Molecular Biology, Princeton University, Princeton, New Jersey 08544*

Manuscript received January 9, 2002 Accepted for publication April 15, 2002

ABSTRACT

The replication control genes of bacterial plasmids face selection at two conflicting levels. Plasmid
copies that systematically overreplicate relative to their cell mates have a higher chance of fixing in
descendant cells, but these cells typically have a lower chance of fixing in the population. Apart from
identifying the conflict, this mathematical discussion characterizes the efficiency of the selection levels
and suggests how they drive the evolution of kinetic mechanisms. In particular it is hypothesized that: (1)
tighter replication control is more vulnerable to selfishness; (2)*cis*-acting replication activators are relics
of a conflict where a plasmid outreplicated its intracellular competitors by monopolizing activators; (3)
high-copy plasmids with sloppy replication control arise because intracellular selection favors
overreplica-tion, thereby relieving intercellular selection for lower loss rates; (4) the excessive synthesis of*cis*-acting
replication activators and*trans*-acting inhibitors is the result of an arms race between*cis*selfishness and

*trans*retaliations; (5) site-specific recombination of plasmid dimers is equivalent to self-policing; and (6)
plasmids modify their horizontal transfer to spread without promoting selfishness. It is also discussed how
replication control may be subject to a third level of selection acting on the entire population of
plasmid-containing cells.

## P

LASMIDS are self-replicating gene clusters com- sequences. The two selection levels are first analyzed monly found in the cytoplasm of prokaryotes. They separately and then combined in the study of selection are widely used as cloning vectors but also serve as model conflicts and conflict suppression. The analysis con-systems for replication control (Summers1996;Paul- cludes by discussing lineage selection acting on theen-sson and Ehrenberg 2001) and microbial ecology tire population of plasmid-containing cells.
(Macken*et al.*1994;StewartandLevin1977;

Berg-strom*et al.*2000). By connecting molecular and

ecolog-ical aspects, it is shown here how plasmid replication _{PLASMID REPLICATION CONTROL}
faces selection at two levels. Plasmids that outreplicate

If plasmids maximally exploited their hosts, over- and their intracellular competition stand higher chances of

underreplication below and above the carrying capacity fixing in the descendant cells, but these cells tend to

of the cytoplasm would automatically check random grow more slowly due to the larger plasmid burden.

fluctuations around an average copy number. Natural Previous analyses have dealt with multileveled

selec-plasmids instead define their own carrying capacity by tion on various plasmid-carried genes (Eberhard1990;

encoding functions for autoregulating the initiation of

Mongold1992;Pomiankowski1999;Bergstrom*et al.*

replication. By decreasing the average copy number, 2000;CooperandHeinemann2000), but not with those

replication control reduces the metabolic burden im-that affect replication. Other analyses have inspected

posed on the host, and by suppressing the demographic the interplay between similar selective mechanisms

in-noise around the average, it additionally reduces the volved in the reproduction of viruses (Chao1991, 1994;

loss probability at cell division.

Szathmary 1992; Bonhoefffer and Nowak 1994;

**Kinetics of replication control:** Many plasmids,

in-TurnerandChao1999), prebiotic replicators (

Szath-cluding R1 and ColE1 (Summers1996), regulate

initia-maryandDemeter1987;Maynard-Smithand

Szath-tion of replicaSzath-tion kinetically using plasmid-encoded

mary1995), and cytoplasmic organelles (Hurst*et al.*

activators and inhibitors. Activators can interact *in cis*
1996), but do not deal with plasmid-specific traits. The

or*trans, depending on plasmid, with the origin of *
repli-main purposes of the present work are therefore to

cation to attract DNA polymerase and initiate identify the selective forces acting on plasmid

replica-tion, while inhibitors act*in trans*to disrupt the activator
tion control genes and conjecture their molecular

con-production line. (Cis-acting molecules interact only with
the plasmid copy from which they were produced, while
*trans-acting molecules can interact with any plasmid*

1_{Address for correspondence:}_{Department of Molecular Biology, Princeton}

copy.) The simplest kinetic models are based on the

University, Washington Rd., Princeton, NJ 08544.

E-mail: paulsson@princeton.edu single-rate equation

*dy*

*dt*⫽*y(r(y)*⫺ ), (1)

2

*m*⬇

具*m*典

*i(1*⫺ /k)⬇ 具
*m*典

*i* (4)

(Paulsson and Ehrenberg 2001). Sharper negative
where *y* is the plasmid concentration, function *r(y) is*

feedback can thus more effectively suppress random the per plasmid replication frequency, andis the rate

fluctuations around a given average具*m*典.
constant for dilution due to exponential cell growth,

**Random fluctuations increase the average loss rate:**
here assumed to be independent of*y*(see

intercellu-The probability that all plasmid copies segregate to the

lar selection). Molecularly,*r*depends on the activator

same daughter at cell division depends on the copy synthesis rate that in turn depends on the inhibitor

number and the type of plasmid partitioning. With a concentration. Since inhibitors typically have short

half-perfectly working partition function losses occur only lives and are expressed constitutively from plasmids,

when*m*⫽1. If copies instead segregate independently
their concentration in turn stays proportional to a

to identical daughters—binomial partitioning—the loss
changing*y*(BremerandLin-Chao1986;Brennerand

probability is*Lm*⫽2(1⁄2)*m*. Since cells with lower*m*run

Tomizawa 1991; Merlin and Polisky 1993). This

disproportionally higher risks of giving rise to plasmid-closes the feedback loop: A higher plasmid

concentra-free progeny, random fluctuations around an average tion results in a higher inhibitor concentration, which

具*m*典increase the average loss probability具*L*典(the Jensen
reduces the effective activator synthesis rate and thereby

inequality guarantees that 具*L*典ⱖ *L*具*m*典since*L* is convex
also the per plasmid replication frequency.

in*m). In fact, many naturally arising copy number *
distri-A commonly used kinetic approximation for

inhibi-butions allow for the approximation具*L*典 ⬇2b具*m*典_{, where}
tion mechanisms is the negative Hill function:

1_{⁄}

2ⱕ*b*ⱕ1. Higher*b*reflects broader distributions and

greater loss rates. For Poisson distributions*b*⬇*e*⫺1/2⬇

*r(y)*⫽ *k*

1⫹ (Ky)*i* ⬇*k(Ky)*

⫺*i*_{.} _{(2)}

0.6 and for the Gaussians of Equation 4,*b*decreases with
*i(1*⫺ /k) (appendix). This summarizes the established
The approximation is valid when*k*Ⰷ , as for plasmids

perspective on plasmid replication control: Tighter neg-ColE1 (Brenner andTomizawa 1991;Paulssonand

ative feedback more effectively suppresses random

fluc-Ehrenberg 2001) and R1 (Nordstro¨ mand Wagner

tuations and thereby increases segregational stability for
1994;PaulssonandEhrenberg2001). Parameter*i*is

a given average copy number (Figure 1A).
the Hill coefficient of inhibition;*k*is typically the

maxi-mal activator synthesis rate; and the compounded

con-stant *K* depends on the per plasmid rate of inhibitor _{INTRACELLULAR SELECTION}
synthesis, inhibitor half-life, and the interactions

be-Plasmids with too similar replication control systems tween activators and inhibitors (Paulssonand

Ehren-are unable to stably coexist in heteroplasmid cells; they

berg2001). Because the steady-state concentration is

are *incompatible* (for an excellent review see Novick

1987). Common causes of incompatibility are
*suscepti-y*⫽

*i*

### √

* ⫺*

_{k/}_{1}

*K* ⬇

*i*

### √

**

_{k/}*K* , (3) bility to each other’s replication inhibitors or
competi-tion for rate-limiting activators. Mutacompeti-tions that affect
*K* has no effect on the normalized dynamics of Equa- _{these properties could thus allow their plasmids to }
sys-tion 1 (PaulssonandEhrenberg2001). The tightness _{tematically over- or underreplicate relative to their cell}
of the feedback instead depends on *i* and *k/*: *r* de- _{mates: Replication control is subject to intracellular }
se-creases ⵑ*i(1* ⫺ /k)% when *y* increases 1% above *y* _{lection.}

(appendix). _{Cis}_{and}_{trans}_{: acquiring activators and avoiding }

**inhibi-Replication control checks random fluctuations:** _{tors:}_{Replication control allows a plasmid copy to }
kinet-Since chemical reactions are probabilistic by nature, the ically communicate its presence to the other copies in
plasmid copy number *m* varies randomly from cell to the cell and set its own replication frequency according
cell. Such chemical noise can be modeled using master to the total plasmid concentration. Consequently, some
equations (Paulsson and Ehrenberg 2001) but one mutations affect kinetic properties that are public to all
of its most important determinants is already evident copies (transmutations) while others act on properties
in the deterministic equations. This can be illustrated that are kept private to the mutant copies (cis
muta-by approximating cell growth and plasmid segregation tions).*Trans*mutations are neutral (appendix) in terms
with an intrinsic plasmid half-life and by using*r*(where of intracellular selection since all copies are affected
*y*is replaced by*m*divided by the cell volume) as a birth the same way (Novick 1987; Szathmary 1992). *Cis*
intensity per copy (Paulsson andEhrenberg 2001). mutations that allow a plasmid copy to overreplicate
The total birth and death probabilities within a short relative to its cell mates in contrast have an intracellular
time interval⌬*t*are then approximately*rm*⌬*t*and*m*⌬*t,* advantage (Novick1987). For instance, plasmids such
respectively, and the linear noise approximation (van as pT181 that share activators *in trans* (Novick 1987)

Figure1.—(A) Stationary copy number
distri-butions calculated from master equations (
appen-dix). Parameter *i* determines the sensitivity of
negative feedback and thus the significance of
intrinsic copy number fluctuations. The numbers
represent approximate具*L*典when*L*⫽2(1_{⁄}

2)*m*. (B)

Simplistic burdens (solid line) and losses (dotted
lines) as functions of average copy number 具*m*典.
The sum (dashed lines) of burdens and losses has
a minimum at具*m*典opt. When具*m*典⬍具*m*典optselection

for lower burdens is weak, while when具*m*典⬎具*m*典opt

selection for lower loss rates is weak. The
approxi-mation具*L*典⬇ 具*L*典0is used so that the net growth

rate is0(1⫺(10⫺4具*m*典⫹2*b*具*m*典)).

activators will replicate more frequently. Changes in the not allow for frequency-dependent intracellular selec-tion (r2/r1 is constant).

structure or turnover rates of activators or inhibitors by

contrast affect all copies equally. Plasmids such as R1 **Incompatibility and genetic drift:**If two types of
or-ganisms exploit the same niche in the same way, the
and ColE1 that keep their activators*in cis*are instead

under selection for high activator synthesis rates. Their carrying capacity of the environment checks only fluc-tuations in their total number. Flucfluc-tuations in their activator genes are right next to the origin of replication.

The RNA that promotes replication of ColE1 is still individual numbers instead stand uncorrected and ran-dom drift quickly drives one or the other to fixation. physically attached to its gene when it binds and forms

a replication complex at the origin. For R1, the mRNA In direct analogy, replication control in heteroplasmid cells acts on the weighted sum of plasmid copy numbers of the replication protein is also attached to DNA and

the protein never leaves the plasmid copy from which it rather than the two separately. The inability to sense and correct individual fluctuations leads to greatly increased was made. Mutations affecting the structure or turnover

rates of*trans*inhibitors are still neutral, but their RNA losses;*i.e., heteroplasmid cells give rise to homoplasmid*
segregants at a much higher rate than homoplasmid
or DNA targets are selected for lower inhibitor affinity.

A generalization of the approximation in Equation 2 cells give rise to plasmid-free segregants (Novick1987). The average fraction of homoplasmid-descendant for incompatible Y1 and Y2 plasmids in heteroplasmid

cells—tailor-made for the molecular processes above—is cells in which a plasmid copy eventually is fixed can be
estimated by replacing cell growth and plasmid
*segrega-r*1⫽ *k*1(C1(K1*y*1⫹*K*2*y*2))⫺*i* _{tion by plasmid elimination intensities} _{m}

1 and *m*2

(Paulsson and Ehrenberg 2001) and by assuming

*r*2⫽ *k*2(C2(K1*y*1⫹*K*2*y*2))⫺*i*. (5)

birth intensities*r*1*m*1and*r*2*m*2. With a constant total copy

For plasmid ColE1,*k*is the maximal synthesis rate of _{number}_{m}

T⫽*m*1⫹*m*2, the effective single-copy

substitu-the*cis*activator,*C*depends on the inhibitor’s*cis*target _{tion rates equal the elimination intensity of one type}
sites, and *K* depends on the structure and turnover _{multiplied by the probability that the other type }
repli-rates of the*trans*inhibitor. Since both plasmid types are _{cates first. The ratio between single-copy substitution}
subject to the same cell volume and growth rate,*v*and _{rates is then} _{r}

2/r1 (appendix), which uniquely

deter-, it is convenient to use the condensed notations _{mines fixation fractions. This is equivalent to a}_{Moran}
(1958) model of selection and drift in a haploid
*popula-Qcis*⬅ *i*

### √

*k/C*

tion and has been used by Walsh (1992) to predict

*Qtrans*⬅ *Ki*

### √

/v. (6) fixation rates of organelle genes. If Y1 and Y2differ bya *trans-acting mutation,* *r*2/r1 ⫽ 1 and all copies have

The average copy number in homoplasmid cells (see

the same chance of fixing (appendix). For*cis*mutations
Equation 3) and the ratio between replication

frequen-fixation fractions are harder to calculate, but when*r*2/r1

cies in heteroplasmid cells are then

is independent of *m*1 and *m*2, as in Equation 7, their

ratio (appendix) is the standard

具*m*典 ⬇*Qcis/Qtrans*

*r*2/r1⫽(Q*cis*2*/Qcis*1)*i*. (7) *f*intra2

*f*intra
1

⫽

### 冢

*r*2

*r*1冣
*m*T⫺1

. (8)

Equation 5 is thus simplified in two ways: It assumes a

par-titioning mechanisms, and unequal*m*Tare given in the tor⌬marks differences between X2and X1parameters;

*e.g.,*⌬␥ ⫽ ␥2⫺ ␥1.

appendix.

The steady-state densities of Equation 9 (appendix)
are equal when the difference in mutation rates
bal-INTERCELLULAR SELECTION _{ances differences in losses, growth, and horizontal}

transfer: Plasmids depend entirely on their hosts for

reproduc-tion and are thus under selecreproduc-tion to maximize the net

2⌬ ⫽ ⌬具*L*典 ⫺ ⌬ ⫺ ⌬⌫. (10)
growth rate of plasmid-containing cells. Since copy

num-bers vary statistically from cell to cell it may seem that Without mutations, X2 cells would be outcompeted by

individual cells have individual fitnesses. However, copy X1 cells (or vice versa) when⌬具*L*典 ⫺ ⌬ ⬎ ⌬⌫since

number fluctuations are both epigenetic and transient. their net growth rate per cell is lower at all densities. Selection therefore effectively acts on the net growth This is a version of the Stewart-Levin criterion (Stewart

rate accumulated over a few generations, *i.e., on the* andLevin1977) that normally pertains to competition
distribution associated with a replication control mecha- between plasmid-containing and plasmid-free cells, but
nism rather than on individual fluctuations. here summarizes how the plasmid-containing cells that
**Net growth and genetic drift:**Much of the analysis is _{are disfavored by burdens and losses must compensate}
simplified to inspect the competition between homo- _{with more horizontal transfer.}

plasmid X1and X2cells, containing plasmids Y1and Y2, Deterministic models are practical when all cell types

respectively. Plasmids are thus considered essential to _{exist in high numbers, but since X}_{1} _{and X}_{2} _{cells in}
their hosts and arising heteroplasmid cells are assumed _{Equation 9 coexist only due to mutations, stochastic}
to immediately turn into homoplasmid cells with proba- _{descriptions are more appropriate. With the same }
nota-bilities that are included in the effective mutation and _{tions and assumptions as in the deterministic analysis}
conjugation rates below. This is approximate since sepa- _{(}_{appendix}_{), where}_{n}_{is used for numbers instead of}* _{x}*
ration of plasmids requires cell divisions, but it is suffi-

_{for densities, assume that single-cell substitutions occur}cient for the current purposes. Over evolutionary time

_{as a result of conjugation between cells of different types}one should also expect an accumulation of competing

_{or of birth of one cell multiplied by the probability that}cell types, not just X1 and X2, but this simplification

_{a cell of the other type is eliminated [a}

_{Moran}

_{(1958)}

makes it possible to analytically demonstrate some first _{model with migration]. The ratio between fixation }

prob-principles. _{abilities is then}

Most ecological plasmid models (StewartandLevin

1977;Macken*et al.*1994;Bergstrom*et al.*2000) con- *f*inter
2

*f*inter
1

⫽

### 冢

⌫2⫹(1 ⫺具*L*典2)2

⌫1⫹(1 ⫺具*L*典1)1

### 冣

*n*_{T}⫺1

(11) dense growth, losses, and horizontal transfer into a

de-terministic and continuous rate equation

approxima-(appendix), in direct analogy with Equation 8. tion for changes in cell densities. In close analogy, X1

**Plasmid burdens:**To compete with both plasmid-free
and X2densities are modeled by

and plasmid-containing cells, plasmids are constantly
under intercellular selection to reduce metabolic
*bur-dx*1

*dt* ⫽ [1 ⫺具*L*典11⫺ 2 ⫺ ⌬␥*x*2⫺ (x1,*x*2)]x1⫹ 1*x*2 dens while also considering loss rates and conjugation
frequencies. Burdens depend strongly on copy
*num-dx*2

*dt* ⫽ [2 ⫺具*L*典22⫺ 1 ⫹ ⌬␥*x*1⫺ (x1,*x*2)]x2⫹ 2*x*1, bers, gene expression levels, environmental conditions,_{and the history of plasmid-host coevolution. In spite of}
(9) _{such contingency, a brief account of phenomenological}

features helps put the present analysis in perspective.
with rate parametersfor cell growth,具*L*典for plasmid

Because the low losses at high copy numbers do not
losses,2 _{for mutations (}

1 is the rate from type 2 to

compensate for the high losses at low copy numbers, type 1), and ␥ for transfer—assuming conjugation to

the average loss rate 具*L*典 increases with fluctuations
be proportional to the product of donor and recipient

around an average具*m*典(seeplasmid replication

con-cells (Stewart andLevin 1977;Macken*et al.* 1994).

trol). This argument has permeated the plasmid litera-The increase in horizontal transfer with total population

ture, yet similar questions are never raised for burdens:
*x*T ⫽ *x*1 ⫹ *x*2 is counteracted when larger populations

Do copy fluctuations have a significant impact on the take up a larger total volume. For this reason, and to

average host growth rate? There are two scenarios where
reduce notational complexity,⌫ ⫽ ␥*x*Tis used

through-they should not. First, if the burden responds more or out and can be seen as the maximal conjugation rate

less linearly to fluctuations in copy number, the effect of per donor or recipient cell. The elimination function

up-fluctuations cancels the effect of down-fluctuations.

keeps*x*Tconstant (Macken*et al.*1994) and the

opera-Second, if there is a long phenotypic lag before a change in copy number affects growth, cells effectively integrate over plasmid fluctuations, sensing mainly the average.

By contrast, if the growth rate quickly and nonlinearly cellular selection operates on plasmid burdens, loss rates, and conjugation frequencies, while intracellular responds to plasmid fluctuations, one should expect

fluctuations to also affect the average burden. For in- selection determines the fraction of descendant cells
that are finally affected by a mutation or conjugation
stance, if a high growth rate requires that*m*is above or

below a certain threshold, then plasmids with 具*m*典 on event. If heteroplasmid cells arise with the same
muta-tion rate 0 per plasmid copy, the effective rates per

the right side of the threshold are under selection for

narrow distributions, while plasmids with 具*m*典 on the cell of forming homoplasmid descendants of the other
type are ⫽ 0*m*T*f*intra (Equation 8). Similarly, if the

wrong side are under selection for a different 具*m*典 or

broader distributions. Similarly, if the burden were pro- two plasmids have identical conjugation mechanisms,
the effective conjugation rates areⵑ⌫ ⫽ ⌫0*f*intra.

portional to*m*2_{, the average burden would be }

propor-tional to具*m*2_{典}_{⫽}_{具}_{m}_{典}2_{⫹ }2

*m*, where2*m*is the copy number By combining the expressions for intra- and
intercel-variance. On the other hand, if statistical uncertainty lular genetic drift when mutations are rare and by
as-in the expression of some plasmid gene is advantageous, suming low rates of conjugation and plasmid losses as
randomizing transcription or translation is more likely well as small differences in cell growth rates—typical*in*
than randomizing replication. Phenotypic variability *vivo* parameter values—selfish plasmids are predicted
does not rely on plasmid fluctuations. to reign with higher probability than altruistic plasmids

Because it is speculative if or how copy fluctuations (appendix) approximately when affect growth, most of the analysis does not rely on

detailed assumptions. In some quantitative examples, ⌬具*L*典⫺ ⌬/

⌬ln*r* ⬍
*m*T

*n*T

⫹ ⌫0

. (13)

however, it is assumed that

The approximation allowsto be either1or 2and

⬇ 0(1⫺ *B*具*m*典), (12)

*m*T to be either*m*T1or *m*T 2. When the population size

where0is the growth rate of cells carrying a utopian

*m*Tdiffers greatly between the two plasmids,*m*Tin

Equa-plasmid that can confer,*e.g., antibiotic resistance *

with-tion 13 is closer to the*m*T for the plasmid with higher

out an associated metabolic burden, and*B*represents

intracellular fitness (seeappendix). Equation 13 con-the small burden per plasmid copy, independently of

forms closely withLeigh’s (1983) analysis of individuals fluctuations.

(plasmid copies)*vs.*groups (plasmid-containing cells)
**A trade-off between burdens and losses:**An increase

that stressed three major requirements for group selec-in average copy number generally selec-increases the burden

tion to be effective: that plasmids impose on their hosts but instead reduces

their loss rate. There is thus a trade-off between the two _{1. Each new group should be founded by members}
disadvantages and presumably an optimal average copy _{from few other groups.}

number that maximizes the net growth rate of the plas- _{2. The number of groups should be high compared to}
mid-containing cell. For instance, if 具*L*典 ⫽ 2b具*m*典 _{(see}

the number of individuals per group (nTⰇ *m*T).

plasmid replication control) and ⫽ 0(1⫺*B*具*m*典) _{3. Transfer between groups should be low (}⌫_{0} Ⰶ _{).}

(see above), then (1⫺ 具*L*典) as a function of具*m*典has

Since a daughter cell has a single mother, the first
re-an internal maximum at具*m*典opt (appendix). At higher

quirement is automatically fulfilled. The number of

cop-具*m*典, metabolic burdens are too large, and at lower具*m*典,

ies per cell is also fairly low, ranging from a few to at plasmid losses are too high (Figure 1B). Narrower

distri-most a few hundred, while the number of cells per
butions (lower*b*) similarly come at the price of higher

population can be very high. Finally, conjugation rates burdens (Paulsson and Ehrenberg 2001) but this

tend to be low and some plasmids actively avoid forming analysis focuses on average copy numbers.

heteroplasmid cells with incompatible relatives (see

suppressing conflicts). From this one might expect SELECTION CONFLICTS

intercellular selection to overrule intracellular selection and plasmids to live in reasonable harmony with the Intracellular selection favors replication control

sys-plasmid-containing cell. However, counteracting these tems that allow their plasmids to outreplicate other

plas-effects, simple mutations can result in great intracellular mids. Intercellular selection instead favors control

sys-advantages while the differences in losses and metabolic tems that allow their cells to outgrow competing cell

burdens typically are very small. Intracellular selection types. This section compares the relative strengths of

thus operates with small populations but large selection the two forces and predicts to what extent selfishness

coefficients while intercellular selection operates with can promote an increase in average copy numbers. It

large populations but small selection coefficients. is also proposed how the conflict can cause neutrality

More cells in a given volume imply more encounters to random copy number fluctuations and explain the

and thus more transfer. This is taken into account in
existence of*cis*activators.

the above analysis because the transfer rate is assumed
**The effective level of selection:**The fate of

in Equation 13 is defined by ⌫0 ⫽ ␥0*n*T. The second Parameter⫺*B*ln*b*⬎0 is thus a measure of how sensitively

the intercellular selection responds to changes in具*m*典.
term in the right-hand side of Equation 13 thus increases

with*n*Tand the total right-hand side has a minimum at An estimate of the balance between the selective

forces can be found by using the expressions for*r,*,
the cell population size for which plasmid selfishness is

most efficiently suppressed: and 具*L*典 directly in the genetic drift equations. At the
price of less generality, more transparent results can also
be obtained by using the approximations in Equations
*n*T⫽

### 冪

*m*T

␥0/

. (14)

13–16 that predict the selfish plasmid to be at a net
advantage as long as it is not too selfish,*i.e., when*
At lower *n*T, the intercellular selection process is too

random to efficiently pick up on small selection coeffi- _{⌬}_{具}

*m*典⬍ 2i

⫺*B* ln(b)

### 冢

*m*T

*n*T

⫹ ⌫0

0冣, (17)

cients, and at higher*n*T, the transfer rate is so high that

selfish and altruistic plasmids meet too often for the

where*m*Tis an intermediate between the two plasmids.

altruists to benefit from their strategy. Equation 14 thus

A higher *i*is partially counteracted by a higher⫺ln*b,*
exemplifies how larger cell populations do not

necessar-but the total effect should still be a higher ⫺*i/ln* *b*
ily lead to more placid plasmids but it should be

modi-(appendix). This poses an interesting dilemma. Plas-fied when the conjugation rate saturates or accelerates

mids must code for sensitive control—high*i—to *
effec-at high*n*T.

tively reduce copy number variation in a cell population
**Sensitivity of replication control and selfish deviations**

(Equation 4) and thereby lower the average loss rate at
**from optimality:** By favoring overreplicating plasmids,

cell division. However, higher sensitivity also results in intracellular selection promotes a selfish increase in the

greater payoffs for overreplicating *cis* mutants, raising
average copy number. How large deviations⌬具*m*典from

the question if plasmids can reconcile effective noise

具*m*典optone should expect depends on how the two

selec-suppression with restrained selfishness.
tive forces respond to changes in具*m*典.

**Does the selection conflict generate noisy plasmids?**
At the intracellular level, consider the idealized case

A parasitic increase in the average copy number typically where plasmids replicate as soon as their concentration

leads to lower loss rates and higher metabolic burdens. decreases below a threshold value, but never when

As a consequence, the selective pressure for even lower above. Volume expansion due to cell growth continually

loss rates is relieved while the selection on burdens dilutes plasmids, and when the threshold concentration

intensifies. If the only effect of random fluctuations is is reached, a plasmid copy replicates. This raises the

to increase the loss rate—as is commonly assumed (see inhibitor concentration and blocks further replication

intercellular selection)—parasitically high
aver-attempts. Consequently, if Y2plasmids due to a*cis*

muta-ages should thus result in selective neutrality to noise tion have a slightly higher threshold than Y1plasmids,

suppression and efficiency of replication control. In only Y2 plasmids can ever replicate. Realistic control

other words, even if low average copy numbers and mechanisms would give only a partial advantage to Y1

effective control would allow for the most cost-efficient or Y2plasmids but with higher sensitivity one approaches

plasmid-containing cells, multileveled selection could the threshold situation

instead result in plasmids with high averages but broad distributions. For a quantitative example, again consider

⌬ln*r*⬇ *i*⌬ln具*m*典 ⬇ *i*

具*m*典opt

⌬具*m*典 (15) _{具}_{L}_{典 ⬇}_{2b}具*m*典_{and}_{}_{/}_{}

0⬇1⫺*B*具*m*典. If具*m*典1Ⰷ具*m*典opt, then

the burden is relatively high and the loss rate is relatively
low (Figure 1B). For a competing Y2plasmid with具*m*典2⫽

(Equations 5–7). The second approximation is based

具*m*典1 but broader (b2 ⬎ *b*1) copy number distribution,

on a first-order Taylor expansion around 具*m*典opt. When

具*L*典2⫺具*L*典1could be insignificant even if具*L*典2/具*L*典1is very

*i* is high, a *cis* mutant can thus receive a substantial

high (Figure 1B), as when具*L*典1⫽10⫺10and具*L*典2⫽10⫺8.

intracellular advantage even if it has only a slightly

At the heart of this argument is the assumption that
higher具*m*典.

average loss rates increase with random fluctuations At the intercellular level, selection favors cells that

while average metabolic burdens do not. However, if
better balance metabolic burdens [⬇ 0(1⫺ *B*具*m*典)]

loss rates are very low due to plasmid selfishness, and
and plasmid losses (具*L*典 ⬇ 2b具*m*典_{). If} _{X}

1 cells have an

fluctuations indeed increase the burden (see

intercel-optimal trade-off as outlined inintercellular

selec-lular selection), lowering the burden could in fact

tion, while Y2plasmids deviate⌬具*m*典above具*m*典opt, X2cells

be the primary role of noise suppression. Replication are disadvantaged (Figure 1B) by intercellular selection

control would then not be balancing losses against bur-and a second-order Taylor expansion (appendix)

dens, but burdens against selfishness.
around具*m*典optgives

**Cis****activators—relics of selfishness?** For R1, ColE1,
and similar plasmids, both*cis*and*trans*activators could

⌬具*L*典⫺ ⌬

0

⬇⫺*B* ln(b)

2 ⌬具*m*典

2_{.} _{(16)}

only apparent regulatory difference is a short time delay mids, and site-specific recombination to resolve over-when activators reside in the cytoplasm before binding replicating plasmid multimers.

to plasmids. However, a plasmid that starts to monopo- **Retaliations** **in trans****:** The previous chapter treated
lize its activator molecules—forcing them to act*in cis—* the selection balance between two plasmid types in the
also receives a great intracellular advantage over its cell hypothetical absence of other types. However, rather
mates. If the fraction of activators made from Y1and Y2 than ending in a static compromise between selection

copies are*m*1/mT and*m*2/mT, and Y2copies keep their levels, conflicts can lead to an innovative evolutionary

activators*in cis*but tap into the common pool of *trans* game of moves and countermoves. In particular, selfish
activators as effectively as the Y1copies, Y1and Y2 plas- deviations toward higher*Qcis*and具*m*典⬎具*m*典opt(Equations

mids take fractions*m*2

1/m2T and*m*1*m*2/m2T ⫹ *m*2/mT, re- 6 and 7) would not necessarily be succeeded by a

re-spectively. For ColE1 (Brenner and Tomizawa 1991; _{vertant to lower}_{Q}_{cis}_{, but more likely to higher}_{Q}_{trans}_{that}

PaulssonandEhrenberg2001) and R1 (Nordstro¨ m _{can reduce}_{具}_{m}_{典}_{back toward}_{具}_{m}_{典}_{opt}_{without suffering an}

andWagner1994; Paulsson andEhrenberg 2001), _{intracellular disadvantage. The interplay between the}
the rate of acquiring activators is proportional to the _{two levels of selection can thus lead to an arms race}
momentary plasmid replication frequency so that*r*2/r1⫽ between* _{cis}*selfishness and

*retaliations (Figure 2).*

_{trans}2 ⫹ *m*2/m1. Equation 13 cannot be used directly for _{For instance, the inhibitor target sites are under }

intracel-the balance between intracel-the selective forces because*r*2/r1 _{lular selection to avoid inhibitors, but low-affinity targets}

depends on*m*1 and *m*2, but the fixation fractions are _{provide intercellular selection for more potent }

inhibi-still analytically tractable (appendix) and the*cis*fixation _{tors, amounting to an evolutionary game of }
hide-and-advantage is _{seek. Similarly, the arms race may result in high synthesis}

rates of both the*cis*activators and the*trans*inhibitors,
*f*intra

2

*f*intra
1

⬇ 4*m*T

### √

*m*T

, (18) _{something that has been observed for plasmids ColE1,}
R1, and numerous other plasmids. At some point the
race slows down by the metabolic burden associated
where⬇ 3.14 is the mathematical constant. A single

with overproducing inhibitors and activators (an aspect
Y2copy in a cell with Y1copies thus has a 4*m*T/

### √

*m*Ttimes

of intercellular selection that is ignored above) or by higher chance of being fixed than a single Y1copy in a

entropic effects when most mutations lead to lower pro-cell with Y2copies. This in turn means that Equation 13

moter activities. Chromosomal mutations typically affect
can be used with⌬ln*r*⬇2 ln 2 (appendix).

all plasmid copies in the cell and thus take the role of Selfish changes in replication control should often be

*trans*mutations.
expected to reduce the fitness of the plasmid-containing

**Safe sex:** The evolutionary success of plasmids
de-cell. However,*cis*action does not necessarily affect the

pends directly on conjugation—sex between prokary-copy number distribution in the subsequent

homoplas-otes—whereby plasmids transfer horizontally to new mid cells at all. Activators still have the same structure

cells or even new types of cells (Stewart and Levin

and are synthesized at the same rate; they are only

allo-cated earlier. Parameter⌬具*L*典⫺ ⌬/in Equation 13 1977;Macken*et al.*1994;Bergstrom*et al.*2000).
How-could thus be very low or even negative. In other words, ever, as can be seen in Equations 13 and 17, conjugation
the strong intracellular selective force to privatize activa- that mixes incompatible plasmids also promotes
tors is opposed by a weak—if any—force at the intercel- selfishness, especially in large cell populations. Since
lular level. This may explain why *cis* activators are so selfishness in turn reduces the growth rate of
plasmid-popular in replication control, like RepA of R1 and containing cells, plasmids could benefit in the long run
RNA II of ColE1, but at the same time raises the question by conjugating discriminatorily to cells that are free of
how plasmids like pT181 can share their RepC activators incompatible relatives.

*in trans.* Many plasmids avoid redundant conjugation by

en-coding mechanisms for surface exclusion (Summers

1996) that prevent plasmids from the same exclusion
SUPPRESSING CONFLICTS _{group to enter the cell. Since plasmids of the same}

exclusion group also typically belong to the same incom-Conflicts between levels of selection provide niches

patibility group, this reduces the number of intracellular for suppression mechanisms that protect higher-level

encounters between competing plasmids. A similar ef-units from lower-level selfishness. As demonstrated by,

fect is obtained indirectly by repressing conjugation for
*e.g., tumor suppressor genes, the actual conflict can*

long periods and transiently turning it into full activity then be insignificant compared to the potential conflict.

(Lundquist and Levin 1986). Since repressors need This section discusses three types of mechanisms for

time to accumulate in the recipients, conjugation into
suppressing intracellular selfishness:*trans*retaliations to

a plasmid-free cell can start an avalanche in which most lower the average copy number without suffering an

plasmid-free cells in a population receive the plasmid. intracellular penalty, discriminatory conjugation for

Figure 2.—(A) An intracellular fitness
land-scape (Equation 7) for two replication control
sensitivities. (B) An intercellular fitness landscape
for Y2plasmids when Y1is optimal,*Qcis*1/*Qtrans*1⫽

具*m*典opt, using具*L*典⫽2⫻0.6具*m*典and 1⫺ /0⫽10⫺4

具*m*典so that具*m*典opt⬇18.

when most cells carry the plasmid but increase greatly Summers1996). In other words, multimers are cheaters that gain an intracellular advantage at the cost of an in response to plasmid-free cells.

intercellular disadvantage. Many natural plasmids sup-Both these mechanisms allow plasmids to

epidemi-press cheating by using site-specific recombination to cally sweep through a population of plasmid-free cells

actively resolve multimers back to monomers (Summers

but still keep formation of heteroplasmid cells at a

mini-andSheratt1984). This resembles “self-policing” (

Kel-mum. They could thus play the role of uniparental

in-ler1999), where lower-level selfishness is penalized in heritance of intracellular organelles that similarly allows

favor of a higher-level reproduction rate, or rather “self-effective transmission without pitting copies against

exorcism,” since selfishness is genetically expelled each other (Eberhard 1980; Cosmides and Tooby

rather than just punished. 1981; Eberhard1990;Walsh1992). Previous studies

have instead stressed that surface-exclusion plasmids re-ceive a selfish advantage by shutting out incompatible

A THIRD LEVEL OF SELECTION? relatives (Eberhard 1990; Cooper and Heinemann

2000) and that transitory derepression is metabolically _{In addition to intra- and intercellular selection, }
lin-favorable and avoids extended exposure of phage-sensi- _{eage selection could favor plasmid traits that help the}
tive pili (LundquistandLevin1986;Eberhard1990). _{population of containing cells to fight }
plasmid-These rationales are to the point, but short-term advan- _{free cells. This section discusses how spitefully low loss}
tages support rather than contradict the possibility of _{rates are favored by lineage selection, suppressed by}
long-term protection against intracellular selfishness. _{intercellular selection, and generated by intracellular}

**Policing against multimers:**Plasmid monomers spon- _{selection.}

taneously form multimers through homologous recom- _{Intermittent selection and spitefully low losses:}_{If }
plas-bination. Multimerization is highly unfavorable for plas- _{mids have been essential in the recent history, if they}
mids because it imposes a larger burden on the host _{colonize a new host, or if the plasmid-carrying cell }
ex-and increases the plasmid loss rate (Summers*et al.*1993; _{plores a new environment, it is possible that there are}

Summers 1996), supposedly by reducing the number _{no plasmid-free competing cells. If plasmids are }
burden-of independently segregating copies for a given total _{some, the first arising plasmid-free competitor under}
genetic load. nonselective conditions can initiate a rapid wipeout of
The replication frequency of multimers depends on plasmids from the population. To survive periods
be-the replication control, but for ColE1 be-the effect is fairly tween selective sweeps, plasmids may thus be well served
straightforward. If*j*replication origins are intact, multi- by spitefully low losses,*i.e., a so low* 具*L*典 that the total
merization increases the synthesis rates of both the *cis* effect of losses and metabolic burdens lowers the net
activator and the*trans*inhibitor by a factor*j.*The*trans* growth rate.

effect downregulates replication attempts of monomers For a quantitative example assume that 具*L*典 ⬇ 2 ⫻
and multimers alike while the*cis*effect gives an unequal 0.6具*m*典_{and}⬇

0(1 ⫺10⫺4⫻ 具*m*典), so that具*m*典opt⬇ 18

advantage to multimers. In terms of Equations 5–7 with (appendix). At具*m*典⫽18, then (1 ⫺具*L*典)/0⬇0.998

Y1as monomers and Yjas*j-fold multimers,kj*⫽*jk*1 and and 具*L*典 ⬇ 2 ⫻ 10⫺4 so that plasmid-free competitors

*Kj*⫽*jK*1so that*rj*/r1⫽*j.*Intracellular selection can thus arise quickly even in fairly small populations. If具*m*典⫽

accentuate the multimer problem by inducing runaway 40, then具*L*典 ⬇3⫻10⫺9_{so that plasmid-free competitors}

multimerization as demonstrated and convincingly ar- rarely arise from plasmid-containing cells, but then
in-stead (1 ⫺ 具*L*典)/0 ⬇ 0.996. The 0.2% difference in

effective net growth is selectively significant when the ages to exploit another. However, plasmids may also be
selfish in the hierarchical sense that individual copies
population has⬎103_{individuals, suggesting a selection}

conflict between the individual cell and the population. cheat on the plasmid-containing cell. By inspecting the selective forces acting on plasmid replication control, Though conflicts often are resolved in favor of the

shorter time scale and the lower level of selection, lin- this work suggests how a number of plasmid traits in fact can be traced back to such a hierarchical selection eage selection could in principle be sufficient to favor

plasmid-host clades that sacrifice net growth for lower conflict. The relative simplicity of these mechanisms and the unequaled ease with which plasmids can be

具*L*典. However, just as many putative examples of group

selection have now been explained by lower-level selec- made subject to evolutionary experiments make them
well suited for molecular analyses of multileveled
selec-tion, very low 具*L*典 could also be due to intracellular

selection: *cis* selfishness can decrease loss rates more tion.

than is metabolically justifiable (see selection con- _{I am grateful to R. Kishony, E. C. Cox, C. N. Peterson, M. Ehrenberg,}
flicts). Selfishness of the lower-level unit could thus E. Szathmary, and M. Nowak for comments on the manuscript. This
work was supported by a Lewis-Thomas Fellowship from Princeton

increase the long-term stability of the higher-level unit

University and Bristol-Myers Squibb, the Swedish National Graduate

by overriding the selection for a middle-level unit.

School of Scientific Computing, and a Swedish Science Research

A rigorous treatment of this problem must take

sto-Council grant to Ma˚ns Ehrenberg.

chastics into account. The advantage of very low 具*L*典
heavily relies on the difference between zero and one
competing cell and is easily obscured in mathematical

LITERATURE CITED rate equation models where the fraction of

plasmid-Bengtsson, B. O.,andH. Andersson,1997 The population

genet-containing cells can approach zero, but never quite

ics of plant mitochondrial plasmids. J. Theor. Biol.**188:**163–176.

go extinct. Spatial population structure should also be

Bergstrom, C. T., M. LipsitchandB. R. Levin,2000 Natural

selec-expected to have a large effect since the incentive to _{tion, infectious transfer and the existence conditions for bacterial}

plasmids. Genetics**155:**1505–1519.

suppress competitors is more compelling if one has to

Bonhoeffer, S.,andM. A. Nowak,1994 Intra-host versus inter-host

deal with them in person.

selection—viral strategies of immune function impairment. Proc.
Natl. Acad. Sci. USA**91:**8062–8066.

Bremer, H.,andS. Lin-Chao,1986 Analysis of the physiological

DISCUSSION control of replication of ColE1-type plasmids. J. Theor. Biol.**123:**
453–470.

Natural selection occurs at all levels of biological orga- _{Brenner, M.,} _{and} _{J. Tomizawa,} _{1991} _{Quantitation of }

ColE1-encoded replication elements. Proc. Natl. Acad. Sci. USA **88:**

nization. At higher levels it favors cooperation between

405–409.

lower-level units and at lower levels it favors cheaters

Chao, L.,1991 Levels of selection, evolution of sex in RNA viruses,

that exploit the common good for their own interests _{and the origin of life. J. Theor. Biol.}_{153:}_{229–246.}

Chao, L.,1994 The population genetics of RNA viruses, pp. 233–250

(Keller1999). Ignoring selection conflicts within the

in*The Evolutionary Biology of Viruses*, edited byS. Morse.Raven

genome—instead focusing directly on function—is

con-Press, New York.

venient when all genes reproduce in sync (Hurst *et* _{Cooper, T. F.,}_{and}_{J. A. Heinemann,}_{2000} _{Postsegregational killing}

does not increase plasmid stability but acts to mediate the

*exclu-al.*1996). However, supernumerary (B) chromosomes

sion of competing plasmids. Proc. Natl. Acad. Sci. USA**97:**12643–

(O¨ stergren 1945), meiotic drive genes (Haig and

12648.

Grafen 1991; Lyttle 1991), cytoplasmic organelles _{Cosmides, L. M.,}_{and}_{J. Tooby,}_{1981} _{Cytoplasmic inheritance and}

intragenomic conflict. J. Theor. Biol.**89:**83–129.

(Hurst*et al.*1996), and RNA viruses (Chao1991, 1994;

Eberhard, W. G.,1980 Evolutionary consequences of intracellular

Szathmary 1992; Bonhoefffer and Nowak 1994;

organelle competition. Q. Rev. Biol.**55:**231–249.

TurnerandChao1999) have all demonstrated an abil- _{Eberhard, W. G.,}1990 Evolution in bacterial plasmids and levels
of selection. Q. Rev. Biol.**65:**3–22.

ity to distort transmission frequencies to their

advan-Haig, D.,andA. Grafen,1991 Genetic scrambling as a defense

tage. A molecular function may then serve some genes

against meiotic drive. J. Theor. Biol.**153:**531–558.

at the expense of others. _{Hurst, L. D., A. Atlan}_{and}_{B. O. Bengtsson,}_{1996} _{Genetic }

con-flicts. Q. Rev. Biol.**71:**317–364.

Many analyses of intragenomic conflicts briefly

men-Keller, L.,1999 *Levels of Selection in Evolution.*Princeton University

tion bacterial plasmids, and the few explicit studies

Press, Princeton, NJ.

(Novick 1987; Eberhard 1990; Mongold 1992; _{Leigh, E. G.,} _{1983} _{When does the good of the group override}

the advantage of the individual? Proc. Natl. Acad. Sci. USA**80:**
BengtssonandAndersson1997;Riley1998;

Pomian-2985–2989.

kowski 1999; Bergstrom *et al.* 2000; Cooper and

Lundquist, P. D.,andB. R. Levin,1986 Transitory derepression
Heinemann2000) show that they are far from being _{and the maintenance of conjugative plasmids. Genetics}_{113:}_{483–}

497.

books in an altruistic gene library that cells can borrow

Lyttle, T. W.,1991 Segregation distorters. Annu. Rev. Genet.**25:**

and return at their convenience. By contrast, so-called

511–557.

selfish plasmids can reproduce without conferring ad- _{Macken, C. A., S. A. Levin}_{and}_{R. Waldsta¨tter,}_{1994} _{The dynamics}

of bacteria-plasmid systems. J. Math. Biol.**32:**123–145.

vantages to their hosts and may even encode

toxin-Maynard-Smith, J.,andE. Szathmary,1995 *The Major Transitions*

antidote systems to kill off plasmid-free cells (Riley

*in Evolution.*Oxford University Press, Oxford.

1998). The term “selfish” is then used synonymously _{Merlin, S.,}_{and}_{B. Polisky,}_{1993} _{Analysis of establishment phase}

replication of the plasmid ColE1. J. Mol. Biol.**230:**137–150.

man-Mongold, J. A.,1992 Theoretical implications for the evolution of _{The average plasmid loss rate for binomial }
parti-postsegregational killing by bacterial plasmids. Am. Nat. **139:**

tioning and Poisson distributed copies is

677–689.

Moran, P. A. P.,1958 Random processes in genetics. Proc. Camb.

Philos. Soc.**54:**60–71. _{具}_{L}_{典}_{⫽}

_{兺}

∞
*m*⫽0

21⫺*m*具*m*典
*m*

*m!e*

⫺具*m*典⫽ _{2e}⫺具*m*典/2⬇_{2}⫻_{0.6}具*m*典_{,} _{(A2)}

Nordstro¨ m, K.,andE. G. Wagner,1994 Kinetic aspects of control of plasmid replication by antisense RNA. Trends Biochem. Sci.

**19:**294–300. _{which is approximate also because} _{m}_{⫽} _{0 should be}

Novick, R. P., 1987 Plasmid incompatibility. Microbiol. Rev. **51:**

excluded and the distribution should be normalized:

381–395.

O¨ stergren, G.,1945 Parasitic nature of extra fragment chromo- Only plasmid-containing cells can contribute to the loss

somes. Bot. Not.**2:**157–163. _{rate. For Gaussians the same type of calculation leads to}

Paulsson, J.,andM. Ehrenberg,2000 Random signal fluctuations can reduce random fluctuations in regulated components of

chemical regulatory networks. Phys. Rev. Lett.**23:**5447–5450. _{具}_{L}_{典}_{⫽}

### 冮

∞⫺∞2
1⫺*me*

⫺(*m*⫺具*m*典)2_{/(2}_{}2

*m*)

√22

*m*

*dm* ⫽

{2

*m*⫽具*m*典/*i*}

2(2(ln 2/2*i*)⫺1_{)}具*m*典_{.}

Paulsson, J.,andM. Ehrenberg,2001 Noise in a minimal
regula-tory network: plasmid copy number control. Q. Rev. Biophys.**34:**

1–59. _{(A3)}

Paulsson, J., O. G. BergandM. Ehrenberg,2000 Stochastic

focus-ing: fluctuation-enhanced sensitivity of intracellular regulation. _{This is approximate because discrete copy numbers are}
Proc. Natl. Acad. Sci. USA**97:**7148–7153.

replaced by a continuum and because*m*ⱕ0 should be

Pomiankowski, A.,1999 Intragenomic conflict, pp. 121–152 in

Lev-excluded. The left tail also contributes greatly to 具*L*典

*els of Selection in Evolution*, edited byL. Keller.Princeton

Univer-sity Press, Princeton, NJ. _{but is badly represented in linear noise approximations}

Riley, M. A.,1998 Molecular mechanisms of bacteriocin evolution.

when distributions are broad. The negative binomial—a

Annu. Rev. Genet.**32:**255–278.

distribution over the natural numbers with a shape

pa-Stewart, F. M., andB. R. Levin,1977 The population biology

of bacterial plasmids: a priori conditions for the existence of _{rameter that determines the variance for a given }
aver-mobilizable nonconjugative factors. Genetics**87:**209–228.

age—arises in numerous simple chemical reactions

Summers, D. K., 1996 *The Biology of Plasmids.* Blackwell Science,

(PaulssonandEhrenberg2000, 2001;Paulsson*et al.*

Oxford.

Summers, D. K.,andD. J. Sheratt,1984 Multimerization of high 2000), allows for the same simplification, and supports

copy number plasmids causes instability—ColE1 encodes a deter- _{the same conclusions (not shown).}
minant essential for plasmid monomerization and stability. Cell

**Intracellular selection:**A heteroplasmid cell gives rise

**36:**1097–1103.

Summers, D. K., C. W. BetonandH. L. Withers,1993 Multicopy to homoplasmid descendants over time. For

incompati-plasmid instability—the dimer catastrophe hypothesis. Mol. Mi- _{ble plasmids, the transition is relatively fast so that most}
crobiol.**8:**1031–1038.

cells are homoplasmid already after a few divisions

Szathmary, E.,1992 Viral sex, levels of selection, and the origin

of life. J. Theor. Biol.**159:**99–109. (Novick 1987). The soundest way of predicting the

Szathmary, E., andL. Demeter, 1987 Group selection of early _{fraction of descendants in which a type eventually fixes is}
replicators and the origin of life. J. Theor. Biol.**128:**463–486.

to define a time-continuous Markov process for plasmid

Turner, P. E.,andL. Chao,1999 Prisoner’s dilemma in an RNA

replication during the cell cycle and a stochastic rule

virus. Nature**398:**441–443.

van Kampen, N. G.,1992 *Stochastic Processes in Physics and Chemistry.* _{for how copies are partitioned between daughter cells.}
North-Holland, Amsterdam.

Such models have been used to inspect the quality of

Walsh, J. B.,1992 Intracellular selection, conversion bias, and the

replication control (PaulssonandEhrenberg 2001),

expected substitution rate of organelle genes. Genetics**130:**939–

946. _{but can be solved analytically only in the simplest }

scenar-ios. When they cannot be solved analytically, one must

Communicating editor:M. W. Feldman

resort to either numerical integration of the Markov process or exact Monte Carlo algorithms for simula-tions. Believing that analytical approximations are more

APPENDIX _{informative than more exact numerical solutions when}

details are insufficiently characterized, this work makes Equations are derived in order of appearance.

a number of idealizations. Cell growth and plasmid
**par-Plasmid replication control:**Local steady-state

sensitiv-titioning are replaced by elimination intensities*m*1and

ity is found by differentiating around steady state in

log-*m*2—as if plasmids were degraded rather than

di-log scale. For *r*in (1) and (2), this gives

luted—and the copy number is assumed to be a constant
(ln*r/*ln *y)*|*y*⫽ ⫺*i(1*⫺ /k). (A1) *m*T ⫽ *m*1 ⫹ *m*2. If substitutions occur when a random

copy is eliminated and one of the other type replicates,
High sensitivity thus requires an efficient design (high _{then the substitution rates are}

*i*) and rate constants such that the mechanism can

oper-␣ ⫽ *m*2⫻*r*1*m*1/(r1*m*1⫹ *r*2*m*2)

ate far from saturation (kⰇ ). Molecularly, plasmids

obtain high sensitivity by multimerization or cooperative _{ ⫽ }

*m*1⫻*r*2*m*2/(r1*m*1⫹*r*2*m*2), (A4)

binding of regulatory molecules, multistep schemes

sim-ilar to proofreading, and perhaps also noise-enhanced where*r*1*m*1and*r*2*m*2are total the birth intensities of the

sensitivity: stochastic focusing (Paulsson andEhren- two plasmids, respectively. If substitutions instead occur when a copy replicates and one of the other type is

eliminated [a standardMoran(1958) model for a hap- mutation such that具*m*典2⫽2具*m*典1implies a twofold higher

total mutation rate from Y2to Y1, but a twofold reduction

loid population], the substitution rates are

in the fixation fraction of the invading Y1copy. An

excep-␣ ⫽*r*1*m*1⫻ *m*2/(*m*1⫹ *m*2) _{tion to this rule is if the} _{trans}_{mutation increases the}

average copy number, but not the effective population

 ⫽*r*2*m*2⫻ *m*1/(*m*1⫹ *m*2). (A5)

size, as can be the case for a mutation that increases
The rates determine the time course of the process, but _{both average and variance. For conjugation one should}
the final results—the fixation fractions—are deter- _{also expect effective differences even for} _{trans}_{}
muta-mined only by their ratio␣/. In both (A4) and (A5), _{tions.}

␣/ ⫽ *r*1/r2. Fixation fractions can thus be approxi- _{Intercellular selection:}_{With}⌬* _{s}*⫽ ⌬具

*典 ⫺ ⌬ ⫺ ⌬⌫ ⱖ*

_{L}mated from a birth-and-death process with absorbing _{0, a constant} _{x}

T in (9) leads directly to a quadratic

boundaries and␣ ⫽*r*1and ⫽*r*2. In the simple scenario _{equation for stationary densities with exactly one }

non-that*r*1/r2is constant, they simply follow from a random _{trivial stable solution:}

walk,

*x*1

*x*T

⫽1

2

### 冢

1⫹ 1 ⫹ 2

⌬*s* ⫹

### 冪冢

1⫹ 1⫹ 2

⌬*s*

### 冣

2

⫺42

⌬*s*

### 冣

. (A7)*f*intra

2 ⫽

*r*1/r2⫺1

(r1/r2)*m*T⫺1

and *f*intra

1 ⫽

*r*2/r1⫺1

(r2/r1)*m*T⫺1

, (A6)

so that (8) follows directly. The simplification that *m*T Equation 11 is derived as in (A4–A6) assuming that an

is the same in both types of homoplasmid cells can be X1 cell replaces an X2 cell (and vice versa) with total

relieved but requires additional assumptions of how*m*T intensity␥1*n*1*n*2⫹ 1(1⫺具*L*典1)n1⫻*n*2/nT, where⌫ ⫽ ␥*n*T,

changes with*m*1and*m*2during the competition. To see *i.e., again a*Moran(1958) model for a haploid population.

what effect this can have, assume that when a single Y2 With具*L*典 ⬇2b具*m*典and/0⬇1⫺*B*具*m*典, the net growth

copy arises among Y1 copies, the population size is a rate (1⫺具*L*典)has a single local maximum. Using the

constant *m*T1 and when a single Y1 copy arises among approximation具*L*典⬇ 具*L*典0, leads to

Y2 copies, the population size is a constant *m*T2. The

具*m*典opt⬇ln(⫺*B/(2 lnb))/lnb*

advantage of this simplification is that one can still use

(8) as an approximation where *m*T is taken from the 具*L*典_{opt}⬇⫺*B/lnb,* (A8)

plasmid with highest*r: The fittest plasmid determines*

which breaks down when具*m*典⬍ ⫺ln 2/ln*b*or*B* ⱖ ⫺2
the effective population size. This follows from (A6)

ln*b.*Even fairly precise approximations of the terms具*L*典
and can be more intuitively understood by noting that

orcan also give large relative errors in the difference a more fit individual typically fails to take over due to

⫺具*L*典.
the initial randomness at low numbers, but once it has

**Selection conflicts:**When mutations are so rare that an
accumulated enough, its fixation is almost guaranteed

arising cell type goes to extinction before the next mutant regardless of the number of competitors. A less fit

indi-arises, the population switches between pure populations, vidual instead has to fight against the odds all the way

to fixation. As an effect, when they invade each other’s

X1
2*n*T*f*inter2

→

←_{}

1*n*T*f*1inter

X2⇒*P(X*1)⫽

### 冢

1⫹ 2*f*inter2

1*f*inter1

### 冣

⫺1

, (A9) populations, only the population size of the fitter type

matters, and the problem can be reduced to the one

that led to (A6). where*f*inter_{are the fixation probabilities and}* _{P(X}*
1) is the

Another simplification is that *m*Tis used both as an long-run probability of X1cells. Using ⫽ 0*m*T*f*intra

to-average copy number and effective population size. In gether with (8) and (11) gives *P(X*1) ⫽ 1⁄2 when *f*intra2 /

reality there are bottlenecks because cells at different *f*intra

1 ⫻*m*T2/mT1⫽*f*inter1 */f*inter2 ,*i.e., when*

stages in the cell cycle contain different average copy

numbers and because copy numbers fluctuate randomly _{⫺ ⌬}ln(⌫ ⫹(1⫺具*L*典))

⌬ln*r* ⫽

*m*T⫺1⫹ ⌬ln*m*T/⌬ln*r*

*n*T⫺1

. in single cells. Similarly, unequal partitioning at cell

(A10) division relaxes intracellular selection, while

approxi-mating partitioning with a continuous plasmid

elimina-This is approximate because heteroplasmid cells are as-tion rate can make it seem more severe than it actually

sumed to immediately turn homoplasmid, rather than
is, especially for highly sensitive replication control _{gradually forming different descendants over time. }
Equa-mechanisms. _{tion A10 is also implicit since the conjugation rate depends}

Since a higher 具*m*典 makes it harder for individual _{on intracellular selection,}_{⌫ ⫽ ⌫}

0*f*intra. The more

approxi-copies to fix, plasmids that differ by a *trans* mutation _{mate but clearer (13) follows from}
do not have the same fixation fractions when arising in

each other’s cells. However, the increase in copy num- (mT⫺1)/(nT⫺1)⬇*m*T/nT

ber also results in a proportionally higher total mutation _{⌬}

ln⬇⌬/ rate, so that the effect is canceled when considering

*p*0(∞)⫺*p*0(0)⫽(*m*T⫺1)1

⌬⌫ ⫽ ⌫0⌬*f*intra ⬇⌫0⌬ln*r*

*pm(*∞)⫺*pm(0)*⫽[(*E*⫺1_{⫺}_{1)(2}_{m}

T⫺*m*)⫹(*E*⫺1)(*m*T⫺*m*)]*m*

⌬ln*m*TⰆ*m*T⌬ln*r.* (A11)

*pm*T(∞)⫺*pm*T(0)⫽(*m*T⫹1)*m*T⫺1, (A14)

The first approximation is valid for high effective

popula-tion sizes, the second when growth rate differences are _{where} _{p}

*m*(∞) ⫽ 0 for *m* ⬆ [0, *m*T] and *pm*(0) ⫽ 1 for
small, the third when additionally loss rates and burdens _{some initial}_{m.}_{The fixation fraction}* _{f}*intra

1 is simply*p*0(∞)

are small, the fourth when *r*2/r1 is fairly close to one, _{with}_{P}

*mT⫺*1(0)⫽1 and*f*intra2 is*pm*_{T}(∞) with*p*1(0)⫽1. The

though (r2/r1)*m*Tis either much smaller or larger than one

system has *m*T ⫹ 1 unknowns and *m*T ⫹ 1 equations.

(Equation A6), and the fifth when intracellular selection _{The different} _{}

*m* values can be determined from the
is so efficient that the higher total mutation rate with _{middle equations in (A14), while} _{p}

0(∞) and *pm*_{T} are
higher copy number makes an insignificant logarithmic _{found from the first and last equations, respectively.}
contribution. Parameter*m*Tis the effective population size _{The system is overdetermined because} _{p}

0(∞) ⫽ 1 ⫺

of Y2 rather than Y1 plasmids (see above). Again, small _{p}

*m*T(∞) and solving it leads to the exact

relative errors in individual terms can give large relative

*f*intra

2 ⫽ (mT⫹1)(2mT)⫺1

errors in their difference. A similar approach to predict

total fixation probabilities was taken byWalsh(1992). * _{f}*intra

1 ⫽ (mT!)2(mT⫹ 1)((2mT)!mT)⫺1. (A15)

That ⫺*i/lnb* increases with *i* can be shown for both

Further applying Stirling’s formula gives the virtually
(A3) and the negative binomial as long as *i* is so large

exact (18). By comparison with (8), it follows that the that the approximations are accurate.

fixation advantage*f*intra

2 /fintra1 is approximately the same

The fixation probabilities of *cis vs. trans* activators

as if the intracellular advantage was constant with⌬ln can be derived from any birth-and-death process with

*r*⬇2 ln 2.
absorbing boundaries 0 and *m*T and a ratio between

**Figure legends:**Distributions in Figure 1A are
calcu-replacement rates*r*2/r1⫽2⫹*m*2/(mT⫺*m*2). Dropping

lated numerically by integrating a master equation with
the subscript,*m*⫽*m*2, a simple alternative is

birth-and-death intensities *Cm*1⫺*i*_{and} * _{m, conditioned}*
on

*m*⬎0 (see Equation 2 andPaulssonand

*Ehren-m*⫺12

_{←}

*m*T⫺(

*m*⫺1)→

*m*T⫺*m*

*m*_{←} 2*m*T⫺*m*→

*m*T⫺(*m*⫹1)

*m*⫹ 1. (A12)

berg2000, 2001): This corresponds to the master equations

*p·m*⫽((E⫺1⫺1)Cm1⫺*i*⫹(E⫺1)*m)pm*⫹ *p*1*pm*. (A16)
*p·*0⫽(mT⫺ 1)p1

The nonlinear probability term is uncommon in master
*p·m*⫽[(E⫺1⫺ 1)(2mT⫺*m)*⫹(E ⫺1)(mT ⫺*m)]pm* equations, but must be introduced to get an exact
equa-tion for the condiequa-tioned system (Paulssonand

*Ehren-p·m*_{T}⫽(mT⫹ 1)p*m*_{T}⫺1, (A13)

berg2000). The intuitive reason is that the probability
mass that leaves the system through extinction boosts
where*E*isvan Kampen’s (1992) step operator*Ej _{f}*

_{(n}

_{)}⫽

*f* (n ⫹ *j*). By integrating from 0 to∞, using notation all other probabilities proportionally. Constant *C/*
uniquely determines具*m*典. Approximate solutions of (A16)

*m*⫽兰∞0 *pm*(t)dt, one obtains a linear algebraic equation