Multileveled Selection on Plasmid Replication

Corresponding author: Johan Paulsson, Princeton University, Washington Rd., Princeton, NJ 08544., paulsson{at}princeton.edu (E-mail)

Communicating editor: M. W. FELDMAN

	ABSTRACT

TOP ABSTRACT PLASMID REPLICATION CONTROL INTRACELLULAR SELECTION INTERCELLULAR SELECTION SELECTION CONFLICTS SUPPRESSING CONFLICTS A THIRD LEVEL OF... DISCUSSION APPENDIX LITERATURE CITED

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 overreplication, 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.

PLASMIDS are self-replicating gene clusters commonly found in the cytoplasm of prokaryotes. They are widely used as cloning vectors but also serve as model systems for replication control (SUMMERS 1996 ; PAULSSON and EHRENBERG 2001 ) and microbial ecology (MACKEN et al. 1994 ; STEWART and LEVIN 1977 ; BERGSTROM et al. 2000 ). By connecting molecular and ecological aspects, it is shown here how plasmid replication faces selection at two levels. Plasmids that outreplicate their intracellular competition stand higher chances of fixing in the descendant cells, but these cells tend to grow more slowly due to the larger plasmid burden.

Previous analyses have dealt with multileveled selection on various plasmid-carried genes (EBERHARD 1990 ; MONGOLD 1992 ; POMIANKOWSKI 1999 ; BERGSTROM et al. 2000 ; COOPER and HEINEMANN 2000 ), but not with those that affect replication. Other analyses have inspected the interplay between similar selective mechanisms involved in the reproduction of viruses (CHAO 1991 , CHAO 1994 ; SZATHMARY 1992 ; BONHOEFFER and NOWAK 1994 ; TURNER and CHAO 1999 ), prebiotic replicators (SZATHMARY and DEMETER 1987 ; MAYNARD-SMITH and SZATHMARY 1995 ), and cytoplasmic organelles (HURST et al. 1996 ), but do not deal with plasmid-specific traits. The main purposes of the present work are therefore to identify the selective forces acting on plasmid replication control genes and conjecture their molecular consequences. The two selection levels are first analyzed separately and then combined in the study of selection conflicts and conflict suppression. The analysis concludes by discussing lineage selection acting on the entire population of plasmid-containing cells.

	PLASMID REPLICATION CONTROL

TOP ABSTRACT PLASMID REPLICATION CONTROL INTRACELLULAR SELECTION INTERCELLULAR SELECTION SELECTION CONFLICTS SUPPRESSING CONFLICTS A THIRD LEVEL OF... DISCUSSION APPENDIX LITERATURE CITED

If plasmids maximally exploited their hosts, over- and underreplication below and above the carrying capacity of the cytoplasm would automatically check random fluctuations around an average copy number. Natural plasmids instead define their own carrying capacity by encoding functions for autoregulating the initiation of replication. By decreasing the average copy number, replication control reduces the metabolic burden imposed on the host, and by suppressing the demographic noise around the average, it additionally reduces the loss probability at cell division.

Kinetics of replication control:
Many plasmids, including R1 and ColE1 (SUMMERS 1996 ), regulate initiation of replication kinetically using plasmid-encoded activators and inhibitors. Activators can interact in cis or trans, depending on plasmid, with the origin of replication to attract DNA polymerase and initiate replication, while inhibitors act in trans to disrupt the activator 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 copy.) The simplest kinetic models are based on the single-rate equation

(1)

where y is the plasmid concentration, function r(y) is the per plasmid replication frequency, and µ is the rate constant for dilution due to exponential cell growth, here assumed to be independent of y (see INTERCELLULAR SELECTION). Molecularly, r depends on the activator synthesis rate that in turn depends on the inhibitor concentration. Since inhibitors typically have short half-lives and are expressed constitutively from plasmids, their concentration in turn stays proportional to a changing y (BREMER and LIN-CHAO 1986 ; BRENNER and TOMIZAWA 1991 ; MERLIN and POLISKY 1993 ). This closes the feedback loop: A higher plasmid concentration results in a higher inhibitor concentration, which reduces the effective activator synthesis rate and thereby also the per plasmid replication frequency.

A commonly used kinetic approximation for inhibition mechanisms is the negative Hill function:

(2)

The approximation is valid when k >> µ, as for plasmids ColE1 (BRENNER and TOMIZAWA 1991 ; PAULSSON and EHRENBERG 2001 ) and R1 (NORDSTROM and WAGNER 1994 ; PAULSSON and EHRENBERG 2001 ). Parameter i is the Hill coefficient of inhibition; k is typically the maximal activator synthesis rate; and the compounded constant K depends on the per plasmid rate of inhibitor synthesis, inhibitor half-life, and the interactions between activators and inhibitors (PAULSSON and EHRENBERG 2001 ). Because the steady-state concentration is

(3)

K has no effect on the normalized dynamics of Equation 1 (PAULSSON and EHRENBERG 2001 ). The tightness of the feedback instead depends on i and k/µ: r decreases i(1 - µ/k)% when y increases 1% above (Appendix).

Replication control checks random fluctuations:
Since chemical reactions are probabilistic by nature, the plasmid copy number m varies randomly from cell to cell. Such chemical noise can be modeled using master equations (PAULSSON and EHRENBERG 2001 ) but one of its most important determinants is already evident in the deterministic equations. This can be illustrated by approximating cell growth and plasmid segregation with an intrinsic plasmid half-life and by using r (where y is replaced by m divided by the cell volume) as a birth intensity per copy (PAULSSON and EHRENBERG 2001 ). The total birth and death probabilities within a short time interval t are then approximately rmt and µmt, respectively, and the linear noise approximation (VAN KAMPEN 1992 ) predicts a stationary Gaussian distribution with variance

(4)

(PAULSSON and EHRENBERG 2001 ). Sharper negative feedback can thus more effectively suppress random fluctuations around a given average m.

Random fluctuations increase the average loss rate:
The probability that all plasmid copies segregate to the same daughter at cell division depends on the copy number and the type of plasmid partitioning. With a perfectly working partition function losses occur only when m = 1. If copies instead segregate independently to identical daughters—binomial partitioning—the loss probability is . Since cells with lower m run disproportionally higher risks of giving rise to plasmid-free progeny, random fluctuations around an average m increase the average loss probability L (the Jensen inequality guarantees that L L_m since L is convex in m). In fact, many naturally arising copy number distributions allow for the approximation L 2b^m, where ¹/₂ b 1. Higher b reflects broader distributions and greater loss rates. For Poisson distributions b e^-1/2 0.6 and for the Gaussians of Equation 4, b decreases with i(1 - µ/k) (Appendix). This summarizes the established perspective on plasmid replication control: Tighter negative feedback more effectively suppresses random fluctuations and thereby increases segregational stability for a given average copy number (Fig 1A).

View larger version (22K): In this window In a new window Download PPT slide   Figure 1. (A) Stationary copy number distributions calculated from master equations (Appendix). Parameter i determines the sensitivity of negative feedback and thus the significance of intrinsic copy number fluctuations. The numbers represent approximate . (B) Simplistic burdens (solid line) and losses (dotted lines) as functions of average copy number m. The sum (dashed lines) of burdens and losses has a minimum at mopt. When m < mopt selection for lower burdens is weak, while when m > mopt selection for lower loss rates is weak. The approximation Lµ Lµ0 is used so that the net growth rate is µ0(1 - (10-4m + 2bm)).

	INTRACELLULAR SELECTION

TOP ABSTRACT PLASMID REPLICATION CONTROL INTRACELLULAR SELECTION INTERCELLULAR SELECTION SELECTION CONFLICTS SUPPRESSING CONFLICTS A THIRD LEVEL OF... DISCUSSION APPENDIX LITERATURE CITED

Plasmids with too similar replication control systems are unable to stably coexist in heteroplasmid cells; they are incompatible (for an excellent review see NOVICK 1987 ). Common causes of incompatibility are susceptibility to each other's replication inhibitors or competition for rate-limiting activators. Mutations that affect these properties could thus allow their plasmids to systematically over- or underreplicate relative to their cell mates: Replication control is subject to intracellular selection.

Cis and trans: acquiring activators and avoiding inhibitors:
Replication control allows a plasmid copy to kinetically communicate its presence to the other copies in the cell and set its own replication frequency according to the total plasmid concentration. Consequently, some mutations affect kinetic properties that are public to all copies (trans mutations) while others act on properties that are kept private to the mutant copies (cis mutations). Trans mutations are neutral (Appendix) in terms of intracellular selection since all copies are affected the same way (NOVICK 1987 ; SZATHMARY 1992 ). Cis mutations that allow a plasmid copy to overreplicate relative to its cell mates in contrast have an intracellular advantage (NOVICK 1987 ). For instance, plasmids such as pT181 that share activators in trans (NOVICK 1987 ) have cis-binding sites under intracellular selection for high activator affinity because a copy that attracts more activators will replicate more frequently. Changes in the structure or turnover rates of activators or inhibitors by contrast affect all copies equally. Plasmids such as R1 and ColE1 that keep their activators in cis are instead under selection for high activator synthesis rates. Their activator genes are right next to the origin of replication. The RNA that promotes replication of ColE1 is still physically attached to its gene when it binds and forms a replication complex at the origin. For R1, the mRNA of the replication protein is also attached to DNA and the protein never leaves the plasmid copy from which it was made. Mutations affecting the structure or turnover rates of trans inhibitors are still neutral, but their RNA or DNA targets are selected for lower inhibitor affinity.

A generalization of the approximation in Equation 2 for incompatible Y₁ and Y₂ plasmids in heteroplasmid cells—tailor-made for the molecular processes above—is

(5)

For plasmid ColE1, k is the maximal synthesis rate of the cis activator, C depends on the inhibitor's cis target sites, and K depends on the structure and turnover rates of the trans inhibitor. Since both plasmid types are subject to the same cell volume and growth rate, v and µ, it is convenient to use the condensed notations

(6)

The average copy number in homoplasmid cells (see Equation 3) and the ratio between replication frequencies in heteroplasmid cells are then

(7)

Equation 5 is thus simplified in two ways: It assumes a clear cut between cis and trans mutations and it does not allow for frequency-dependent intracellular selection (r₂/r₁ is constant).

Incompatibility and genetic drift:
If two types of organisms exploit the same niche in the same way, the carrying capacity of the environment checks only fluctuations in their total number. Fluctuations in their individual numbers instead stand uncorrected and random drift quickly drives one or the other to fixation. In direct analogy, replication control in heteroplasmid cells acts on the weighted sum of plasmid copy numbers rather than the two separately. The inability to sense and correct individual fluctuations leads to greatly increased losses; i.e., heteroplasmid cells give rise to homoplasmid segregants at a much higher rate than homoplasmid cells give rise to plasmid-free segregants (NOVICK 1987 ).

The average fraction of homoplasmid-descendant cells in which a plasmid copy eventually is fixed can be estimated by replacing cell growth and plasmid segregation by plasmid elimination intensities µm₁ and µm₂ (PAULSSON and EHRENBERG 2001 ) and by assuming birth intensities r₁m₁ and r₂m₂. With a constant total copy number , the effective single-copy substitution rates equal the elimination intensity of one type multiplied by the probability that the other type replicates first. The ratio between single-copy substitution rates is then r₂/r₁ (Appendix), which uniquely determines fixation fractions. This is equivalent to a MORAN 1958 model of selection and drift in a haploid population and has been used by WALSH 1992 to predict fixation rates of organelle genes. If Y₁ and Y₂ differ by a trans-acting mutation, and all copies have the same chance of fixing (Appendix). For cis mutations fixation fractions are harder to calculate, but when r₂/r₁ is independent of m₁ and m₂, as in Equation 7, their ratio (Appendix) is the standard

(8)

More details on genetic drift including bottlenecks, partitioning mechanisms, and unequal m_T are given in the Appendix

	INTERCELLULAR SELECTION

TOP ABSTRACT PLASMID REPLICATION CONTROL INTRACELLULAR SELECTION INTERCELLULAR SELECTION SELECTION CONFLICTS SUPPRESSING CONFLICTS A THIRD LEVEL OF... DISCUSSION APPENDIX LITERATURE CITED

Plasmids depend entirely on their hosts for reproduction and are thus under selection to maximize the net growth rate of plasmid-containing cells. Since copy numbers vary statistically from cell to cell it may seem that individual cells have individual fitnesses. However, copy number fluctuations are both epigenetic and transient. Selection therefore effectively acts on the net growth rate accumulated over a few generations, i.e., on the distribution associated with a replication control mechanism rather than on individual fluctuations.

Net growth and genetic drift:
Much of the analysis is simplified to inspect the competition between homoplasmid X₁ and X₂ cells, containing plasmids Y₁ and Y₂, respectively. Plasmids are thus considered essential to their hosts and arising heteroplasmid cells are assumed to immediately turn into homoplasmid cells with probabilities that are included in the effective mutation and conjugation rates below. This is approximate since separation of plasmids requires cell divisions, but it is sufficient for the current purposes. Over evolutionary time one should also expect an accumulation of competing cell types, not just X₁ and X₂, but this simplification makes it possible to analytically demonstrate some first principles.

Most ecological plasmid models (STEWART and LEVIN 1977 ; MACKEN et al. 1994 ; BERGSTROM et al. 2000 ) condense growth, losses, and horizontal transfer into a deterministic and continuous rate equation approximation for changes in cell densities. In close analogy, X₁ and X₂ densities are modeled by

(9)

with rate parameters µ for cell growth, Lµ for plasmid losses,² for mutations (₁ is the rate from type 2 to type 1), and for transfer—assuming conjugation to be proportional to the product of donor and recipient cells (STEWART and LEVIN 1977 ; MACKEN et al. 1994 ). The increase in horizontal transfer with total population is counteracted when larger populations take up a larger total volume. For this reason, and to reduce notational complexity, is used throughout and can be seen as the maximal conjugation rate per donor or recipient cell. The elimination function keeps x_T constant (MACKEN et al. 1994 ) and the operator marks differences between X₂ and X₁ parameters; e.g., .

The steady-state densities of Equation 9 (Appendix) are equal when the difference in mutation rates balances differences in losses, growth, and horizontal transfer:

(10)

Without mutations, X₂ cells would be outcompeted by X₁ cells (or vice versa) when Lµ - µ > since their net growth rate per cell is lower at all densities. This is a version of the Stewart-Levin criterion (STEWART and LEVIN 1977 ) that normally pertains to competition between plasmid-containing and plasmid-free cells, but here summarizes how the plasmid-containing cells that are disfavored by burdens and losses must compensate with more horizontal transfer.

Deterministic models are practical when all cell types exist in high numbers, but since X₁ and X₂ cells in Equation 9 coexist only due to mutations, stochastic descriptions are more appropriate. With the same notations and assumptions as in the deterministic analysis (Appendix), where n is used for numbers instead of x for densities, assume that single-cell substitutions occur as a result of conjugation between cells of different types or of birth of one cell multiplied by the probability that a cell of the other type is eliminated [a MORAN 1958 model with migration]. The ratio between fixation probabilities is then

(11)

(Appendix), in direct analogy with Equation 8.

Plasmid burdens:
To compete with both plasmid-free and plasmid-containing cells, plasmids are constantly under intercellular selection to reduce metabolic burdens while also considering loss rates and conjugation frequencies. Burdens depend strongly on copy numbers, gene expression levels, environmental conditions, and the history of plasmid-host coevolution. In spite of such contingency, a brief account of phenomenological features helps put the present analysis in perspective.

Because the low losses at high copy numbers do not compensate for the high losses at low copy numbers, the average loss rate L increases with fluctuations around an average m (see PLASMID REPLICATION CONTROL). This argument has permeated the plasmid literature, yet similar questions are never raised for burdens: Do copy fluctuations have a significant impact on the average host growth rate? There are two scenarios where they should not. First, if the burden responds more or less linearly to fluctuations in copy number, the effect of up-fluctuations cancels the effect of down-fluctuations. 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 responds to plasmid fluctuations, one should expect fluctuations to also affect the average burden. For instance, if a high growth rate requires that m is above or below a certain threshold, then plasmids with m on the right side of the threshold are under selection for narrow distributions, while plasmids with m on the wrong side are under selection for a different m or broader distributions. Similarly, if the burden were proportional to m², the average burden would be proportional to , where ²_m is the copy number variance. On the other hand, if statistical uncertainty in the expression of some plasmid gene is advantageous, randomizing transcription or translation is more likely than randomizing replication. Phenotypic variability does not rely on plasmid fluctuations.

Because it is speculative if or how copy fluctuations affect growth, most of the analysis does not rely on detailed assumptions. In some quantitative examples, however, it is assumed that

(12)

where µ₀ is the growth rate of cells carrying a utopian plasmid that can confer, e.g., antibiotic resistance without an associated metabolic burden, and B represents the small burden per plasmid copy, independently of fluctuations.

A trade-off between burdens and losses:
An increase in average copy number generally increases the burden that plasmids impose on their hosts but instead reduces their loss rate. There is thus a trade-off between the two disadvantages and presumably an optimal average copy number that maximizes the net growth rate of the plasmid-containing cell. For instance, if (see PLASMID REPLICATION CONTROL) and (see above), then (1 - L)µ as a function of m has an internal maximum at m_opt (Appendix). At higher m, metabolic burdens are too large, and at lower m, plasmid losses are too high (Fig 1B). Narrower distributions (lower b) similarly come at the price of higher burdens (PAULSSON and EHRENBERG 2001 ) but this analysis focuses on average copy numbers.

	SELECTION CONFLICTS

TOP ABSTRACT PLASMID REPLICATION CONTROL INTRACELLULAR SELECTION INTERCELLULAR SELECTION SELECTION CONFLICTS SUPPRESSING CONFLICTS A THIRD LEVEL OF... DISCUSSION APPENDIX LITERATURE CITED

Intracellular selection favors replication control systems that allow their plasmids to outreplicate other plasmids. Intercellular selection instead favors control systems that allow their cells to outgrow competing cell types. This section compares the relative strengths of the two forces and predicts to what extent selfishness can promote an increase in average copy numbers. It is also proposed how the conflict can cause neutrality to random copy number fluctuations and explain the existence of cis activators.

The effective level of selection:
The fate of plasmid-containing cells depends on selection at two levels: Intercellular selection operates on plasmid burdens, loss rates, and conjugation frequencies, while intracellular selection determines the fraction of descendant cells that are finally affected by a mutation or conjugation event. If heteroplasmid cells arise with the same mutation rate ₀ per plasmid copy, the effective rates per cell of forming homoplasmid descendants of the other type are (Equation 8). Similarly, if the two plasmids have identical conjugation mechanisms, the effective conjugation rates are .

By combining the expressions for intra- and intercellular genetic drift when mutations are rare and by assuming low rates of conjugation and plasmid losses as well as small differences in cell growth rates—typical in vivo parameter values—selfish plasmids are predicted to reign with higher probability than altruistic plasmids (Appendix) approximately when

(13)

The approximation allows µ to be either µ₁ or µ₂ and m_T to be either m_T1 or m_T2. When the population size m_T differs greatly between the two plasmids, m_T in Equation 13 is closer to the m_T for the plasmid with higher intracellular fitness (see Appendix). Equation 13 conforms closely with LEIGH's (1983) analysis of individuals (plasmid copies) vs. groups (plasmid-containing cells) that stressed three major requirements for group selection to be effective:

1. Each new group should be founded by members from few other groups.

2. The number of groups should be high compared to the number of individuals per group (n_T >> m_T).

3. Transfer between groups should be low (₀ << µ).

Since a daughter cell has a single mother, the first requirement is automatically fulfilled. The number of copies per cell is also fairly low, ranging from a few to at most a few hundred, while the number of cells per population can be very high. Finally, conjugation rates tend to be low and some plasmids actively avoid forming heteroplasmid cells with incompatible relatives (see SUPPRESSING CONFLICTS). From this one might expect intercellular selection to overrule intracellular selection and plasmids to live in reasonable harmony with the plasmid-containing cell. However, counteracting these effects, simple mutations can result in great intracellular advantages while the differences in losses and metabolic burdens typically are very small. Intracellular selection thus operates with small populations but large selection coefficients while intercellular selection operates with large populations but small selection coefficients.

More cells in a given volume imply more encounters and thus more transfer. This is taken into account in the above analysis because the transfer rate is assumed to be ₀n₁n₂ (see INTERCELLULAR SELECTION) where ₀ in Equation 13 is defined by . The second term in the right-hand side of Equation 13 thus increases with n_T and the total right-hand side has a minimum at the cell population size for which plasmid selfishness is most efficiently suppressed:

(14)

At lower n_T, the intercellular selection process is too random to efficiently pick up on small selection coefficients, and at higher n_T, the transfer rate is so high that selfish and altruistic plasmids meet too often for the altruists to benefit from their strategy. Equation 14 thus exemplifies how larger cell populations do not necessarily lead to more placid plasmids but it should be modified when the conjugation rate saturates or accelerates at high n_T.

Sensitivity of replication control and selfish deviations from optimality:
By favoring overreplicating plasmids, intracellular selection promotes a selfish increase in the average copy number. How large deviations m from m_opt one should expect depends on how the two selective forces respond to changes in m.

At the intracellular level, consider the idealized case where plasmids replicate as soon as their concentration decreases below a threshold value, but never when above. Volume expansion due to cell growth continually dilutes plasmids, and when the threshold concentration is reached, a plasmid copy replicates. This raises the inhibitor concentration and blocks further replication attempts. Consequently, if Y₂ plasmids due to a cis mutation have a slightly higher threshold than Y₁ plasmids, only Y₂ plasmids can ever replicate. Realistic control mechanisms would give only a partial advantage to Y₁ or Y₂ plasmids but with higher sensitivity one approaches the threshold situation

(15)

(Equation 5Equation 6Equation 7). The second approximation is based on a first-order Taylor expansion around m_opt. When i is high, a cis mutant can thus receive a substantial intracellular advantage even if it has only a slightly higher m.

At the intercellular level, selection favors cells that better balance metabolic burdens [µ µ₀(1 - Bm)] and plasmid losses (L 2b^m). If X₁ cells have an optimal trade-off as outlined in INTERCELLULAR SELECTION, while Y₂ plasmids deviate m above m_opt, X₂ cells are disadvantaged (Fig 1B) by intercellular selection and a second-order Taylor expansion (Appendix) around m_opt gives

(16)

Parameter -B ln b > 0 is thus a measure of how sensitively the intercellular selection responds to changes in m.

An estimate of the balance between the selective forces can be found by using the expressions for r, µ, and L directly in the genetic drift equations. At the price of less generality, more transparent results can also be obtained by using the approximations in Equation 13Equation 14Equation 15Equation 16 that predict the selfish plasmid to be at a net advantage as long as it is not too selfish, i.e., when

(17)

where m_T is an intermediate between the two plasmids. A higher i is partially counteracted by a higher -ln b, but the total effect should still be a higher -i/ln b (Appendix). This poses an interesting dilemma. Plasmids must code for sensitive control—high i—to effectively reduce copy number variation in a cell population (Equation 4) and thereby lower the average loss rate at cell division. However, higher sensitivity also results in greater payoffs for overreplicating cis mutants, raising the question if plasmids can reconcile effective noise suppression with restrained selfishness.

Does the selection conflict generate noisy plasmids?
A parasitic increase in the average copy number typically leads to lower loss rates and higher metabolic burdens. As a consequence, the selective pressure for even lower loss rates is relieved while the selection on burdens intensifies. If the only effect of random fluctuations is to increase the loss rate—as is commonly assumed (see INTERCELLULAR SELECTION)—parasitically high averages should thus result in selective neutrality to noise suppression and efficiency of replication control. In other words, even if low average copy numbers and effective control would allow for the most cost-efficient plasmid-containing cells, multileveled selection could instead result in plasmids with high averages but broad distributions. For a quantitative example, again consider L 2b^m and µ/µ₀ 1 - Bm. If m₁ >> m_opt, then the burden is relatively high and the loss rate is relatively low (Fig 1B). For a competing Y₂ plasmid with but broader (b₂ > b₁) copy number distribution, L₂ - L₁ could be insignificant even if L₂/L₁ is very high (Fig 1B), as when and .

At the heart of this argument is the assumption that average loss rates increase with random fluctuations while average metabolic burdens do not. However, if loss rates are very low due to plasmid selfishness, and fluctuations indeed increase the burden (see INTERCELLULAR SELECTION), lowering the burden could in fact be the primary role of noise suppression. Replication control would then not be balancing losses against burdens, but burdens against selfishness.

Cis activators—relics of selfishness?
For R1, ColE1, and similar plasmids, both cis and trans activators could result in constitutive attempts to initiate replication. The only apparent regulatory difference is a short time delay when activators reside in the cytoplasm before binding to plasmids. However, a plasmid that starts to monopolize its activator molecules—forcing them to act in cis—also receives a great intracellular advantage over its cell mates. If the fraction of activators made from Y₁ and Y₂ copies are m₁/m_T and m₂/m_T, and Y₂ copies keep their activators in cis but tap into the common pool of trans activators as effectively as the Y₁ copies, Y₁ and Y₂ plasmids take fractions m²₁/m²_T and m₁m₂/m²_T + m₂/m_T, respectively. For ColE1 (BRENNER and TOMIZAWA 1991 ; PAULSSON and EHRENBERG 2001 ) and R1 (NORDSTROM and WAGNER 1994 ; PAULSSON and EHRENBERG 2001 ), the rate of acquiring activators is proportional to the momentary plasmid replication frequency so that . Equation 13 cannot be used directly for the balance between the selective forces because r₂/r₁ depends on m₁ and m₂, but the fixation fractions are still analytically tractable (Appendix) and the cis fixation advantage is

(18)

where 3.14 is the mathematical constant. A single Y₂ copy in a cell with Y₁ copies thus has a 4^mT/ times higher chance of being fixed than a single Y₁ copy in a cell with Y₂ copies. This in turn means that Equation 13 can be used with ln r 2 ln 2 (Appendix).

Selfish changes in replication control should often be expected to reduce the fitness of the plasmid-containing cell. However, cis action does not necessarily affect the copy number distribution in the subsequent homoplasmid cells at all. Activators still have the same structure and are synthesized at the same rate; they are only allocated earlier. Parameter L - µ/µ in Equation 13 could thus be very low or even negative. In other words, the strong intracellular selective force to privatize activators is opposed by a weak—if any—force at the intercellular level. This may explain why cis activators are so popular in replication control, like RepA of R1 and RNA II of ColE1, but at the same time raises the question how plasmids like pT181 can share their RepC activators in trans.

	SUPPRESSING CONFLICTS

TOP ABSTRACT PLASMID REPLICATION CONTROL INTRACELLULAR SELECTION INTERCELLULAR SELECTION SELECTION CONFLICTS SUPPRESSING CONFLICTS A THIRD LEVEL OF... DISCUSSION APPENDIX LITERATURE CITED

Conflicts between levels of selection provide niches for suppression mechanisms that protect higher-level units from lower-level selfishness. As demonstrated by, e.g., tumor suppressor genes, the actual conflict can then be insignificant compared to the potential conflict. This section discusses three types of mechanisms for suppressing intracellular selfishness: trans retaliations to lower the average copy number without suffering an intracellular penalty, discriminatory conjugation for effective horizontal transfer without mixing related plasmids, and site-specific recombination to resolve overreplicating plasmid multimers.

Retaliations in trans:
The previous chapter treated the selection balance between two plasmid types in the hypothetical absence of other types. However, rather than ending in a static compromise between selection levels, conflicts can lead to an innovative evolutionary game of moves and countermoves. In particular, selfish deviations toward higher Q_cis and m > m_opt (Equation 6 and Equation 7) would not necessarily be succeeded by a revertant to lower Q_cis, but more likely to higher Q_trans that can reduce m back toward m_opt without suffering an intracellular disadvantage. The interplay between the two levels of selection can thus lead to an arms race between cis selfishness and trans retaliations (Fig 2). For instance, the inhibitor target sites are under intracellular selection to avoid inhibitors, but low-affinity targets provide intercellular selection for more potent inhibitors, amounting to an evolutionary game of hide-and-seek. Similarly, the arms race may result in high synthesis rates of both the cis activators and the trans inhibitors, 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 with overproducing inhibitors and activators (an aspect of intercellular selection that is ignored above) or by entropic effects when most mutations lead to lower promoter activities. Chromosomal mutations typically affect all plasmid copies in the cell and thus take the role of trans mutations.

View larger version (63K): In this window In a new window Download PPT slide   Figure 2. (A) An intracellular fitness landscape (Equation 7) for two replication control sensitivities. (B) An intercellular fitness landscape for Y2 plasmids when Y1 is optimal, , using so that mopt 18.

Safe sex:
The evolutionary success of plasmids depends directly on conjugation—sex between prokaryotes—whereby plasmids transfer horizontally to new cells or even new types of cells (STEWART and LEVIN 1977 ; MACKEN et al. 1994 ; BERGSTROM et al. 2000 ). However, as can be seen in Equation 13 and Equation 17, conjugation that mixes incompatible plasmids also promotes selfishness, especially in large cell populations. Since selfishness in turn reduces the growth rate of plasmid-containing cells, plasmids could benefit in the long run by conjugating discriminatorily to cells that are free of incompatible relatives.

Many plasmids avoid redundant conjugation by encoding mechanisms for surface exclusion (SUMMERS 1996 ) that prevent plasmids from the same exclusion group to enter the cell. Since plasmids of the same exclusion group also typically belong to the same incompatibility group, this reduces the number of intracellular encounters between competing plasmids. A similar effect is obtained indirectly by repressing conjugation for long periods and transiently turning it into full activity (LUNDQUIST and LEVIN 1986 ). Since repressors need time to accumulate in the recipients, conjugation into a plasmid-free cell can start an avalanche in which most plasmid-free cells in a population receive the plasmid. Consequently, the conjugational activity can be low when most cells carry the plasmid but increase greatly in response to plasmid-free cells.

Both these mechanisms allow plasmids to epidemically sweep through a population of plasmid-free cells but still keep formation of heteroplasmid cells at a minimum. They could thus play the role of uniparental inheritance of intracellular organelles that similarly allows effective transmission without pitting copies against each other (EBERHARD 1980 ; COSMIDES and TOOBY 1981 ; EBERHARD 1990 ; WALSH 1992 ). Previous studies have instead stressed that surface-exclusion plasmids receive a selfish advantage by shutting out incompatible relatives (EBERHARD 1990 ; COOPER and HEINEMANN 2000 ) and that transitory derepression is metabolically favorable and avoids extended exposure of phage-sensitive pili (LUNDQUIST and LEVIN 1986 ; EBERHARD 1990 ). These rationales are to the point, but short-term advantages support rather than contradict the possibility of long-term protection against intracellular selfishness.

Policing against multimers:
Plasmid monomers spontaneously form multimers through homologous recombination. Multimerization is highly unfavorable for plasmids because it imposes a larger burden on the host and increases the plasmid loss rate (SUMMERS et al. 1993 ; SUMMERS 1996 ), supposedly by reducing the number of independently segregating copies for a given total genetic load.

The replication frequency of multimers depends on the replication control, but for ColE1 the effect is fairly straightforward. If j replication origins are intact, multimerization increases the synthesis rates of both the cis activator and the trans inhibitor by a factor j. The trans effect downregulates replication attempts of monomers and multimers alike while the cis effect gives an unequal advantage to multimers. In terms of Equation 5 HREF="#FD6">Equation 6Equation 7 with Y₁ as monomers and Y_j as j-fold multimers, and so that . Intracellular selection can thus accentuate the multimer problem by inducing runaway multimerization as demonstrated and convincingly argued by Summers and co-workers (SUMMERS et al. 1993 ; SUMMERS 1996 ). In other words, multimers are cheaters that gain an intracellular advantage at the cost of an intercellular disadvantage. Many natural plasmids suppress cheating by using site-specific recombination to actively resolve multimers back to monomers (SUMMERS and SHERATT 1984 ). This resembles "self-policing" (KELLER 1999 ), where lower-level selfishness is penalized in favor of a higher-level reproduction rate, or rather "self-exorcism," since selfishness is genetically expelled rather than just punished.

	A THIRD LEVEL OF SELECTION?

TOP ABSTRACT PLASMID REPLICATION CONTROL INTRACELLULAR SELECTION INTERCELLULAR SELECTION SELECTION CONFLICTS SUPPRESSING CONFLICTS A THIRD LEVEL OF... DISCUSSION APPENDIX LITERATURE CITED

In addition to intra- and intercellular selection, lineage selection could favor plasmid traits that help the population of plasmid-containing cells to fight plasmid-free cells. This section discusses how spitefully low loss rates are favored by lineage selection, suppressed by intercellular selection, and generated by intracellular selection.

Intermittent selection and spitefully low losses:
If plasmids have been essential in the recent history, if they colonize a new host, or if the plasmid-carrying cell explores a new environment, it is possible that there are no plasmid-free competing cells. If plasmids are burdensome, the first arising plasmid-free competitor under nonselective conditions can initiate a rapid wipeout of plasmids from the population. To survive periods between selective sweeps, plasmids may thus be well served by spitefully low losses, i.e., a so low L that the total effect of losses and metabolic burdens lowers the net growth rate.

For a quantitative example assume that L 2 x 0.6^m and µ µ₀(1 - 10^-4 x m), so that m_opt 18 (Appendix). At , then (1 - L)µ/µ₀ 0.998 and L 2 x 10^-4 so that plasmid-free competitors arise quickly even in fairly small populations. If , then L 3 x 10^-9 so that plasmid-free competitors rarely arise from plasmid-containing cells, but then instead (1 - L)µ/µ₀ 0.996. The 0.2% difference in effective net growth is selectively significant when the population has >10³ individuals, suggesting a selection conflict between the individual cell and the population.

Though conflicts often are resolved in favor of the shorter time scale and the lower level of selection, lineage selection could in principle be sufficient to favor plasmid-host clades that sacrifice net growth for lower L. However, just as many putative examples of group selection have now been explained by lower-level selection, very low L could also be due to intracellular selection: cis selfishness can decrease loss rates more than is metabolically justifiable (see SELECTION CONFLICTS). Selfishness of the lower-level unit could thus increase the long-term stability of the higher-level unit by overriding the selection for a middle-level unit.

A rigorous treatment of this problem must take stochastics into account. The advantage of very low L heavily relies on the difference between zero and one competing cell and is easily obscured in mathematical rate equation models where the fraction of plasmid-containing cells can approach zero, but never quite go extinct. Spatial population structure should also be expected to have a large effect since the incentive to suppress competitors is more compelling if one has to deal with them in person.

	DISCUSSION

TOP ABSTRACT PLASMID REPLICATION CONTROL INTRACELLULAR SELECTION INTERCELLULAR SELECTION SELECTION CONFLICTS SUPPRESSING CONFLICTS A THIRD LEVEL OF... DISCUSSION APPENDIX LITERATURE CITED

Natural selection occurs at all levels of biological organization. At higher levels it favors cooperation between lower-level units and at lower levels it favors cheaters that exploit the common good for their own interests (KELLER 1999 ). Ignoring selection conflicts within the genome—instead focusing directly on function—is convenient when all genes reproduce in sync (HURST et al. 1996 ). However, supernumerary (B) chromosomes (OSTERGREN 1945 ), meiotic drive genes (HAIG and GRAFEN 1991 ; LYTTLE 1991 ), cytoplasmic organelles (HURST et al. 1996 ), and RNA viruses (CHAO 1991 , CHAO 1994 ; SZATHMARY 1992 ; BONHOEFFER and NOWAK 1994 ; TURNER and CHAO 1999 ) have all demonstrated an ability to distort transmission frequencies to their advantage. A molecular function may then serve some genes at the expense of others.

Many analyses of intragenomic conflicts briefly mention bacterial plasmids, and the few explicit studies (NOVICK 1987 ; EBERHARD 1990 ; MONGOLD 1992 ; BENGTSSON and ANDERSSON 1997 ; RILEY 1998 ; POMIANKOWSKI 1999 ; BERGSTROM et al. 2000 ; COOPER and HEINEMANN 2000 ) show that they are far from being books in an altruistic gene library that cells can borrow and return at their convenience. By contrast, so-called selfish plasmids can reproduce without conferring advantages to their hosts and may even encode toxin-antidote systems to kill off plasmid-free cells (RILEY 1998 ). The term "selfish" is then used synonymously with "parasitic" and relates to how one organism manages to exploit another. However, plasmids may also be selfish in the hierarchical sense that individual copies cheat on the plasmid-containing cell. By inspecting the selective forces acting on plasmid replication control, this work suggests how a number of plasmid traits in fact can be traced back to such a hierarchical selection conflict. The relative simplicity of these mechanisms and the unequaled ease with which plasmids can be made subject to evolutionary experiments make them well suited for molecular analyses of multileveled selection.

	FOOTNOTES

² This is an approximation that works well when L < 10%.

	ACKNOWLEDGMENTS

I am grateful to R. Kishony, E. C. Cox, C. N. Peterson, M. Ehrenberg, E. Szathmary, and M. Nowak for comments on the manuscript. This work was supported by a Lewis-Thomas Fellowship from Princeton University and Bristol-Myers Squibb, the Swedish National Graduate School of Scientific Computing, and a Swedish Science Research Council grant to Måns Ehrenberg.

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

	APPENDIX

TOP ABSTRACT PLASMID REPLICATION CONTROL INTRACELLULAR SELECTION INTERCELLULAR SELECTION SELECTION CONFLICTS SUPPRESSING CONFLICTS A THIRD LEVEL OF... DISCUSSION APPENDIX LITERATURE CITED

Equations are derived in order of appearance.

Plasmid replication control:
Local steady-state sensitivity is found by differentiating around steady state in log-log scale. For r in (1) and (2), this gives

(A1)

High sensitivity thus requires an efficient design (high i) and rate constants such that the mechanism can operate far from saturation (k >> µ). Molecularly, plasmids obtain high sensitivity by multimerization or cooperative binding of regulatory molecules, multistep schemes similar to proofreading, and perhaps also noise-enhanced sensitivity: stochastic focusing (PAULSSON and EHRENBERG 2000 , PAULSSON and EHRENBERG 2001 ; PAULSSON et al. 2000 ).

The average plasmid loss rate for binomial partitioning and Poisson distributed copies is

(A2)

which is approximate also because m = 0 should be excluded and the distribution should be normalized: Only plasmid-containing cells can contribute to the loss rate. For Gaussians the same type of calculation leads to

(A3)

This is approximate because discrete copy numbers are replaced by a continuum and because m 0 should be excluded. The left tail also contributes greatly to L but is badly represented in linear noise approximations when distributions are broad. The negative binomial—a distribution over the natural numbers with a shape parameter that determines the variance for a given average—arises in numerous simple chemical reactions (PAULSSON and EHRENBERG 2000 , PAULSSON and EHRENBERG 2001 ; PAULSSON et al. 2000 ), allows for the same simplification, and supports the same conclusions (not shown).

Intracellular selection:
A heteroplasmid cell gives rise to homoplasmid descendants over time. For incompatible plasmids, the transition is relatively fast so that most cells are homoplasmid already after a few divisions (NOVICK 1987 ). The soundest way of predicting the fraction of descendants in which a type eventually fixes is to define a time-continuous Markov process for plasmid replication during the cell cycle and a stochastic rule for how copies are partitioned between daughter cells. Such models have been used to inspect the quality of replication control (PAULSSON and EHRENBERG 2001 ), but can be solved analytically only in the simplest scenarios. When they cannot be solved analytically, one must resort to either numerical integration of the Markov process or exact Monte Carlo algorithms for simulations. Believing that analytical approximations are more informative than more exact numerical solutions when details are insufficiently characterized, this work makes a number of idealizations. Cell growth and plasmid partitioning are replaced by elimination intensities µm₁ and µm₂—as if plasmids were degraded rather than diluted—and the copy number is assumed to be a constant . If substitutions occur when a random copy is eliminated and one of the other type replicates, then the substitution rates are

(A4)

where r₁m₁ and r₂m₂ are total the birth intensities of the two plasmids, respectively. If substitutions instead occur when a copy replicates and one of the other type is eliminated [a standard MORAN 1958 model for a haploid population], the substitution rates are

(A5)

The rates determine the time course of the process, but the final results—the fixation fractions—are determined only by their ratio /ß. In both (A4) and (A5), . Fixation fractions can thus be approximated from a birth-and-death process with absorbing boundaries and and . In the simple scenario that r₁/r₂ is constant, they simply follow from a random walk,

(A6)

so that (8) follows directly. The simplification that m_T is the same in both types of homoplasmid cells can be relieved but requires additional assumptions of how m_T changes with m₁ and m₂ during the competition. To see what effect this can have, assume that when a single Y₂ copy arises among Y₁ copies, the population size is a constant m_T1 and when a single Y₁ copy arises among Y₂ copies, the population size is a constant m_T2. The advantage of this simplification is that one can still use (8) as an approximation where m_T is taken from the plasmid with highest r: The fittest plasmid determines the effective population size. This follows from (A6) and can be more intuitively understood by noting that a more fit individual typically fails to take over due to the initial randomness at low numbers, but once it has accumulat