Multileveled Selection on Plasmid Replication
Department of Molecular Biology, Princeton University, Princeton, New Jersey 08544
Manuscript received January 9, 2002 Accepted for publication April 15, 2002
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 ofcis-acting replication activators andtrans-acting inhibitors is the result of an arms race betweencisselfishness and
transretaliations; (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.
PLASMIDS 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 the
en-sson and Ehrenberg 2001) and microbial ecology tire population of plasmid-containing cells. (Mackenet al.1994;StewartandLevin1977;
Berg-stromet 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
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
Szath-tion of replicaSzath-tion kinetically using plasmid-encoded
mary1995), and cytoplasmic organelles (Hurstet al.
activators and inhibitors. Activators can interact in cis 1996), but do not deal with plasmid-specific traits. The
ortrans, 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 actin transto 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
1Address for correspondence:Department of Molecular Biology, Princeton
copy.) The simplest kinetic models are based on the
University, Washington Rd., Princeton, NJ 08544.
E-mail: email@example.com single-rate equation
dt⫽y(r(y)⫺ ), (1)
i(1⫺ /k)⬇ 具 m典
(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 ofy(see
intercellu-The probability that all plasmid copies segregate to the
lar selection). Molecularly,rdepends 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,
whenm⫽1. If copies instead segregate independently their concentration in turn stays proportional to a
to identical daughters—binomial partitioning—the loss changingy(BremerandLin-Chao1986;Brennerand
probability isLm⫽2(1⁄2)m. Since cells with lowermrun
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sinceL is convex also the per plasmid replication frequency.
inm). 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:
2ⱕbⱕ1. Higherbreflects broader distributions and
greater loss rates. For Poisson distributionsb⬇e⫺1/2⬇
1⫹ (Ky)i ⬇k(Ky)
0.6 and for the Gaussians of Equation 4,bdecreases with i(1⫺ /k) (appendix). This summarizes the established The approximation is valid whenkⰇ , 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). Parameteriis
a given average copy number (Figure 1A). the Hill coefficient of inhibition;kis 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⫽
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 ⵑi(1 ⫺ /k)% when y increases 1% above y lection.
(appendix). Cisandtrans: 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).Transmutations are neutral (appendix) in terms with an intrinsic plasmid half-life and by usingr(where of intracellular selection since all copies are affected yis replaced bymdivided 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⌬tare then approximatelyrm⌬tandm⌬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whenL⫽2(1⁄
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
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 activatorsin cisare 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 oftransinhibitors 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-r1⫽ k1(C1(K1y1⫹K2y2))⫺i tion by plasmid elimination intensities m
1 and m2
(Paulsson and Ehrenberg 2001) and by assuming
r2⫽ k2(C2(K1y1⫹K2y2))⫺i. (5)
birth intensitiesr1m1andr2m2. With a constant total copy
For plasmid ColE1,kis the maximal synthesis rate of numberm
T⫽m1⫹m2, the effective single-copy
substitu-thecisactivator,Cdepends on the inhibitor’scistarget 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 thetransinhibitor. Since both plasmid types are cates first. The ratio between single-copy substitution subject to the same cell volume and growth rate,vand 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 aMoran (1958) model of selection and drift in a haploid popula-Qcis⬅ i
tion and has been used by Walsh (1992) to predict
√/v. (6) fixation rates of organelle genes. If Y1 and Y2differ by
a trans-acting mutation, r2/r1 ⫽ 1 and all copies have
The average copy number in homoplasmid cells (see
the same chance of fixing (appendix). Forcismutations Equation 3) and the ratio between replication
frequen-fixation fractions are harder to calculate, but whenr2/r1
cies in heteroplasmid cells are then
is independent of m1 and m2, as in Equation 7, their
ratio (appendix) is the standard
r2/r1⫽(Qcis2/Qcis1)i. (7) fintra2
Equation 5 is thus simplified in two ways: It assumes a
par-titioning mechanisms, and unequalmTare given in the tor⌬marks differences between X2and X1parameters;
e.g.,⌬␥ ⫽ ␥2⫺ ␥1.
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 X1 and X2 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), wheren is used for numbers instead ofx 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 [aMoran(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;Mackenet al.1994;Bergstromet al.2000) con- finter 2
(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-dx1
dt ⫽ [1 ⫺具L典11⫺ 2 ⫺ ⌬␥x2⫺ (x1,x2)]x1⫹ 1x2 dens while also considering loss rates and conjugation frequencies. Burdens depend strongly on copy num-dx2
dt ⫽ [2 ⫺具L典22⫺ 1 ⫹ ⌬␥x1⫺ (x1,x2)]x2⫹ 2x1, 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;Mackenet 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: xT ⫽ x1 ⫹ x2 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,⌫ ⫽ ␥xTis 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.
keepsxTconstant (Mackenet 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 thatmis 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 ⫽ 0mTfintra (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⌫ ⫽ ⌫0fintra.
portional tom2, the average burden would be
propor-tional to具m2典⫽具m典2⫹ 2
m, where2mis 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—typicalin 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⫺ ⌬/
⌬lnr ⬍ mT
however, it is assumed that
The approximation allowsto be either1or 2and
⬇ 0(1⫺ B具m典), (12)
mT to be eithermT1or mT 2. When the population size
where0is the growth rate of cells carrying a utopian
mTdiffers greatly between the two plasmids,mTin
Equa-plasmid that can confer,e.g., antibiotic resistance
with-tion 13 is closer to themT for the plasmid with higher
out an associated metabolic burden, andBrepresents
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 mT).
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 (lowerb) 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 ofcisactivators.
the above analysis because the transfer rate is assumed The effective level of selection:The fate of
in Equation 13 is defined by ⌫0 ⫽ ␥0nT. The second Parameter⫺Blnb⬎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
withnTand 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 forr,, 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 nT⫽
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 nT, the intercellular selection process is too
random to efficiently pick up on small selection coeffi- ⌬具
cients, and at highernT, the transfer rate is so high that
selfish and altruistic plasmids meet too often for the
wheremTis an intermediate between the two plasmids.
altruists to benefit from their strategy. Equation 14 thus
A higher iis partially counteracted by a higher⫺lnb, 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—highi—to effec-at highnT.
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 acis
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
⌬lnr⬇ i⌬ln具m典 ⬇ i
⌬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 ⬎ b1) 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, bothcisandtransactivators could
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 actin 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 arem1/mT andm2/mT, and Y2copies keep their levels, conflicts can lead to an innovative evolutionary
activatorsin cisbut 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 higherQcisand具m典⬎具m典opt(Equations
mids take fractionsm2
1/m2T andm1m2/m2T ⫹ m2/mT, re- 6 and 7) would not necessarily be succeeded by a
re-spectively. For ColE1 (Brenner and Tomizawa 1991; vertant to lowerQcis, but more likely to higherQtransthat
PaulssonandEhrenberg2001) and R1 (Nordstro¨ m can reduce具m典back toward具m典optwithout 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 thatr2/r1⫽ betweencisselfishness andtransretaliations (Figure 2).
2 ⫹ m2/m1. Equation 13 cannot be used directly for For instance, the inhibitor target sites are under
intracel-the balance between intracel-the selective forces becauser2/r1 lular selection to avoid inhibitors, but low-affinity targets
depends onm1 and m2, but the fixation fractions are provide intercellular selection for more potent
inhibi-still analytically tractable (appendix) and thecisfixation tors, amounting to an evolutionary game of hide-and-advantage is seek. Similarly, the arms race may result in high synthesis
rates of both thecisactivators and thetransinhibitors, fintra
, (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 4mT/
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⌬lnr⬇2 ln 2 (appendix).
all plasmid copies in the cell and thus take the role of Selfish changes in replication control should often be
transmutations. expected to reduce the fitness of the plasmid-containing
Safe sex: The evolutionary success of plasmids de-cell. However,cisaction 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;Mackenet al.1994;Bergstromet 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:transretaliations 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,Qcis1/Qtrans1⫽
具m典opt, using具L典⫽2⫻0.6具m典and 1⫺ /0⫽10⫺4
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 (Summerset 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. Ifjreplication 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 thetransinhibitor by a factorj.Thetrans growth rate.
effect downregulates replication attempts of monomers For a quantitative example assume that 具L典 ⬇ 2 ⫻ and multimers alike while theciseffect 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 Yjasj-fold multimers,kj⫽jk1 and and 具L典 ⬇ 2 ⫻ 10⫺4 so that plasmid-free competitors
Kj⫽jK1so thatrj/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⫺9so 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⬎103individuals, 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
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(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. USA80: BengtssonandAndersson1997;Riley1998;
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Lundquist, P. D.,andB. R. Levin,1986 Transitory derepression Heinemann2000) show that they are far from being and the maintenance of conjugative plasmids. Genetics113:483–
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Lyttle, T. W.,1991 Segregation distorters. Annu. Rev. Genet.25:
and return at their convenience. By contrast, so-called
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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.
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tioning and Poisson distributed copies is
Moran, P. A. P.,1958 Random processes in genetics. Proc. Camb.
Philos. Soc.54:60–71. 具L典⫽
⫺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:
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⫽
Paulsson, J.,andM. Ehrenberg,2001 Noise in a minimal regula-tory network: plasmid copy number control. Q. Rev. Biophys.34:
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. USA97:7148–7153.
replaced by a continuum and becausemⱕ0 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
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age—arises in numerous simple chemical reactions
Summers, D. K., 1996 The Biology of Plasmids. Blackwell Science,
(PaulssonandEhrenberg2000, 2001;Paulssonet al.
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
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
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. Genetics130:939–
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 intensitiesm1and
ity is found by differentiating around steady state in
log-m2—as if plasmids were degraded rather than
di-log scale. For rin (1) and (2), this gives
luted—and the copy number is assumed to be a constant (lnr/ln y)|y⫽ ⫺i(1⫺ /k). (A1) mT ⫽ m1 ⫹ m2. 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-␣ ⫽ m2⫻r1m1/(r1m1⫹ r2m2)
ate far from saturation (kⰇ ). Molecularly, plasmids
obtain high sensitivity by multimerization or cooperative ␤ ⫽
binding of regulatory molecules, multistep schemes
sim-ilar to proofreading, and perhaps also noise-enhanced wherer1m1andr2m2are 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-␣ ⫽r1m1⫻ m2/(m1⫹ m2) tion to this rule is if the trans mutation increases the
average copy number, but not the effective population
␤ ⫽r2m2⫻ m1/(m1⫹ m2). (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.
␣/␤ ⫽ r1/r2. Fixation fractions can thus be approxi- Intercellular selection:With⌬s⫽ ⌬具L典 ⫺ ⌬ ⫺ ⌬⌫ ⱖ
mated from a birth-and-death process with absorbing 0, a constant x
T in (9) leads directly to a quadratic
boundaries and␣ ⫽r1and␤ ⫽r2. In the simple scenario equation for stationary densities with exactly one
non-thatr1/r2is constant, they simply follow from a random trivial stable solution:
1 ⫹ 2
冣. (A7) fintra
so that (8) follows directly. The simplification that mT 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 howmT intensity␥1n1n2⫹ 1(1⫺具L典1)n1⫻n2/nT, where⌫ ⫽ ␥nT,
changes withm1andm2during the competition. To see i.e., again aMoran(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 mT1 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 mT2. The
advantage of this simplification is that one can still use
(8) as an approximation where mT is taken from the 具L典opt⬇⫺B/lnb, (A8)
plasmid with highestr: The fittest plasmid determines
which breaks down when具m典⬍ ⫺ln 2/lnborB ⱖ ⫺2 the effective population size. This follows from (A6)
lnb.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
, (A9) populations, only the population size of the fitter type
matters, and the problem can be reduced to the one
that led to (A6). wherefinterare the fixation probabilities andP(X 1) is the
Another simplification is that mTis used both as an long-run probability of X1cells. Using ⫽ 0mTfintra
to-average copy number and effective population size. In gether with (8) and (11) gives P(X1) ⫽ 1⁄2 when fintra2 /
reality there are bottlenecks because cells at different fintra
1 ⫻mT2/mT1⫽finter1 /finter2 ,i.e., when
stages in the cell cycle contain different average copy
numbers and because copy numbers fluctuate randomly ⫺ ⌬ln(⌫ ⫹(1⫺具L典))
. 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,⌫ ⫽ ⌫
0fintra. 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)⬇mT/nT
ber also results in a proportionally higher total mutation ⌬
ln⬇⌬/ rate, so that the effect is canceled when considering
⌬⌫ ⫽ ⌫0⌬fintra ⬇⌫0⌬lnr
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, mT] and pm(0) ⫽ 1 for small, the third when additionally loss rates and burdens some initialm.The fixation fractionfintra
1 is simplyp0(∞)
are small, the fourth when r2/r1 is fairly close to one, withP
mT⫺1(0)⫽1 andfintra2 ispmT(∞) withp1(0)⫽1. The
though (r2/r1)mTis either much smaller or larger than one
system has mT ⫹ 1 unknowns and mT ⫹ 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 pmT are higher copy number makes an insignificant logarithmic found from the first and last equations, respectively. contribution. ParametermTis the effective population size The system is overdetermined because p
0(∞) ⫽ 1 ⫺
of Y2 rather than Y1 plasmids (see above). Again, small p
mT(∞) and solving it leads to the exact
relative errors in individual terms can give large relative
2 ⫽ (mT⫹1)(2mT)⫺1
errors in their difference. A similar approach to predict
total fixation probabilities was taken byWalsh(1992). fintra
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.
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 mT and a ratio between
Figure legends:Distributions in Figure 1A are calcu-replacement ratesr2/r1⫽2⫹m2/(mT⫺m2). Dropping
lated numerically by integrating a master equation with the subscript,m⫽m2, a simple alternative is
birth-and-death intensities Cm1⫺iand m, conditioned on m⬎0 (see Equation 2 andPaulssonand Ehren-m⫺12←mT⫺(m⫺1)→
m⫹ 1. (A12)
berg2000, 2001): This corresponds to the master equations
p·m⫽((E⫺1⫺1)Cm1⫺i⫹(E⫺1)m)pm⫹ p1pm. (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·mT⫽(mT⫹ 1)pmT⫺1, (A13)
berg2000). The intuitive reason is that the probability mass that leaves the system through extinction boosts whereEisvan Kampen’s (1992) step operatorEjf(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