Trade-off between segregational stability and metabolic burden: a mathematical model of plasmid ColE1 replication control1

J. Mol. Biol. (1998) 279, 73±88

Trade-off Between Segregational Stability and Metabolic Burden: A Mathematical Model of Plasmid ColE1 Replication Control Johan Paulsson and MaÊns Ehrenberg* Department of Molecular Biology, BMC, Box 590 S-75124, Uppsala, Sweden

A model of ColE1 copy number control has been developed where molecular details of replication are connected both to segregational stability and metabolic burden. Ef®cient replication control reduces copy number variation and increases segregational stability for a given average copy number. Copy number variation is predicted to depend on the type of inhibition mechanism as well as RNA I and RNA II turnover rate constants. It is shown that when both RNA I and RNA II transcription frequencies and the rate constant for degradation of free RNA I are very large, a hyperbolic inhibition mechanism must compensate with a 1.4 times greater average copy number to obtain the same segregational stability as an exponential inhibition mechanism. How sensitively the replication frequency responds to changes in RNA I concentration depends on the type of inhibition mechanism and the number of attempts to form an RNA II replication primer per plasmid and cell cycle. If RNA I is too stable, it will not follow changes in plasmid concentration closely, and when the transcription frequency for RNA I is only slightly higher than for RNA II, RNA I concentration becomes randomized. In both these cases, the proportionality between the single cell RNA I and plasmid concentrations is lost and this impairs copy number control. Thresholds in the rate for degradation of free RNA I as well as in RNA I and RNA II transcription frequencies have been computed, where an increase in these rate constants has a negligible effect on segregational stability but a corresponding decrease leads to segregational disaster. This indicates that there exists a well de®ned optimal set of rate constants where the regulation system works well without excessive metabolic load. A number of new experiments are suggested to address features of particular importance for the evolution of ColE1 copy number control.

*Corresponding author

Keywords: ColE1; plamid; regulation; segregational; burden

# 1998 Academic Press Limited

Introduction Plasmids must propagate in pace with the doubling rate of their hosts to be stably maintained in a cell population. However, even if the average plasmid copy number of an entire exponentially growing bacterial population is at steady state, copy numbers in individual cells of the same age will display statistical variation. This is partly an effect of the random distribution of (ColE1) plasmids to the daughter cells at cell division, but even more important are the spontaneous ¯uctuations that arise because the number of plasmid molecules in single cells is small. The task of copy number control is to reduce copy number variation and thereby increase segregational stability. 0022±2836/98/210073±16 $25.00/0/mb981751

It has been experimentally established that plasmid maintenance does not solely depend on segregational stability. For instance, in environments in which plasmids are dispensable to their host, differential growth between plasmid-free and plasmid-containing cells may reduce long term stability greatly (Wouters et al., 1980; Helling et al., 1981; Bouma & Lenski, 1988). To understand the evolutionary strategy of plasmid ColE1 it is therefore necessary to relate the details of the regulation system to both segregational stability and metabolic load. This has been done in the present analysis where metabolic load is represented by plasmid copy number and turnover (turnover will be used as a collective term for both synthesis and degradation) rates of the regulatory RNA elements. # 1998 Academic Press Limited

74 ColE1 copy number control depends on inhibition of a cis acting RNA replication primer precursor (RNA II) by a trans acting antisense RNA (RNA I) (Tomizawa & Itoh, 1981; Tomizawa 1984; Masukata & Tomizawa, 1986; Figure 1). RNA II transcription is initiated 555 base-pairs (bp) upstream of the origin of replication (ori) and the 108 to 110 bp long RNA I is transcribed from the complementary strand of the RNA II coding region, starting 445 bp upstream of the ori (Morita & Oka, 1979; Tomizawa et al., 1981; Tomizawa & Itoh, 1981; Lacatena & Cesareni, 1981). Replication priming by RNA II is mainly sensitive to RNA I

Segregational and Metabolic Aspects of ColE1

inhibition when transcription proceeds through a particular inhibition window, approximately extending from bases 110 to 360 in the RNA II coding region (Tomizawa, 1986), during which time the RNAs may form a reversible initial ``kissing'' complex (Tomizawa, 1990a,b) that can be converted into a stable duplex. When RNA II is shorter than 110 nucleotides, RNA I binds inef®ciently and when it is longer than 360, RNA I can still bind to RNA II but has very little effect on primer formation (Tomizawa, 1986). A small plasmid encoded polypeptide, the Rom protein, stabilizes the RNA I± RNA II interaction, increasing the

Figure 1. Overview of the replication priming process of plasmid ColE1. Both the preprimer RNA II (bp 0 to 555) and the inhibitor RNA I (bp 2 to 110 from the opposite strand) are constitutively transcribed. RNA I can bind to RNA II and form a stable duplex. If binding occurs during RNA II transcription in an ``inhibition window'', stretching roughly from base 110 to 360 in the RNA II gene, conformational changes in RNA II are triggered and subsequent primer formation is inhibited. Before base 110, RNA I binds ver inef®ciently to RNA II and after base 360, RNA I can still bind, but has no signi®cant effect on primer formation. In the absence of inhibition, RNA II can form an RNA-DNA hybrid at the ori (base 555). The hybrid is then cleaved by RNase H yielding a mature primer for DNA polymerase I.

75

Segregational and Metabolic Aspects of ColE1

probability of duplex formation from an initial kissing complex (Cesareni et al., 1982; Lacatena et al., 1984; Tomizawa & Som, 1984; Tomizawa, 1990b). If an RNA I:RNA II duplex is formed, transcription of RNA II proceeds beyond the ori and does not initiate plasmid replication. In the absence of inhibition, transcription of RNA II through ori results in formation of a stable DNA-RNA hybrid. Subsequently, the RNA II part of the hybrid is cleaved by RNase H, yielding a mature replication primer for DNA polymerase I (Itoh & Tomizawa, 1980). Rapid, constitutive synthesis and degradation of RNA I makes its concentration approximately proportional to plasmid concentration (Brenner & Tomizawa, 1991; Bremer & Lin-Chao, 1996; Merlin & Polisky, 1993). Accordingly, an elevated plasmid concentration is coupled to an elevated RNA I concentration, leading to an increased probability of inhibition of the subsequently transcribed preprimers and consequently to a decrease in plasmid replication frequency. By the same principle, a low plasmid concentration will enhance the plasmid replication frequency, so that any deviation from steady state in copy number will be corrected. The present work predicts how higher segregational stability can be obtained by increasing the average copy number or the turnover of RNA I and RNA II. It further predicts threshold values in all rate constants for RNA I and RNA II turnover where an increase has only a marginal effect on stability but a decrease can be disastrous. This indicates the existence of an optimal design for copy number control systems of ColE1 type, for which a given segregational stability is achieved at a minimal metabolic burden. It is also discussed how such principles of optimality could serve as guidelines for the understanding of plasmid evolution.

Theory Plasmid replication is a stochastic process with events randomly distributed in time. Different control mechanisms and different sets of rate constants have very different capacities to reduce ¯uctuations in plasmid copy number and increase stability. The purpose of the present mathematical analysis is to identify the main sources of statistical variation in single cell copy number and explicitly relate them to segregational stability. The general principles of this work, how ¯uctuations can be eliminated, are also relevant for other regulation systems. Previous stochastic models Many features of ColE1 copy number control can be analyzed without taking statistical variation of plasmid and RNA I molecules into account (e.g. see Ataai & Shuler, 1986; Paulsson et al., 1998). Ehrenberg's (1996) model is to our knowledge the only one that relates ColE1 replication control to segregational stability. The main objective of that work was to discuss the two plausible inhibition

mechanisms, exponential and hyperbolic. To simplify the analysis, Ehrenberg assumed a perfect proportionality between plasmid and RNA I concentrations as well as an active partitioning mechanism (distributing plasmids evenly between the two daughter cells). Under these assumptions the analysis showed how ColE1 replication can become cell cycle speci®c in the hypothetical case that the copy number goes from one to two. It also predicted that segregational stability may increase inde®nitely with increasing RNA II transcription frequency. The present work will also treat the two suggested modes of control, since the in vivo mechanism has not been determined, but our main purpose is to make a cost-bene®t analysis of different strategies of ColE1 copy number control. It is then crucial to take into account that RNA I and plasmid concentrations can be disproportional, that ColE1 is a multi-copy plasmid and that plasmids seem to be randomly distributed between the daughters at cell division. Assumptions on replication control kinetics In a broad range of conditions multivariate models of ColE1 replication control (Brendel & Perelson, 1993) can be simpli®ed to include only plasmid and RNA I as concentration variables (see below for the role of the Rom protein). This has been done in the present work that includes kinetics of plasmid replication, RNA II synthesis, RNA I synthesis and RNA I degradation. The per plasmid rate of ColE1 replication is set by the initiation rate of RNA II transcripts, kII, the probability, Q0, that such a transcript is not inhibited by RNA I, and by r, the probability that an RNA II transcript that escaped inhibition initiates replication (Figure 1 and Table 1; Itoh & Tomizawa, 1980; Tomizawa & Itoh, 1981; Masukata & Tomizawa, 1986). There are two different suggestions in the literature for the inhibition mechanism (Ehrenberg, 1996). If there is a single rate-limiting step in the inhibition process the probability Q0, that replication priming is not inhibited by RNA I, is ``hyperbolic'' and given by equation (A1). If, in contrast, there are many ratelimiting steps Q0 is ``exponential'' and well approximated by equation (A2). The qualitative difference between hyperbolic (equation (A1)) and exponential (equation (A2)) inhibition is illustrated graphically in Figure 2 (Paulsson et al., 1998), where the relative plasmid replication frequency is plotted as a function of the relative RNA I concentration. The Figure shows that for a single step (hyperbolic) inhibition, the relative replication frequency is approximately inversely proportional to RNA I concentration whereas for multiple step (exponential) inhibition, the replication frequency may respond much more sharply to changes in RNA I concentration. It is known that there are at least 250 steps of nucleotide addition in the inhibition window (Tomizawa, 1986), but the number

76

Segregational and Metabolic Aspects of ColE1

Table 1. De®nitions of parameters used Description Variable I dI dI/I ISS hIiM M MSS hMit Rate constant kII kI eI kH r  kII/kH

Number of RNA I molecules Change in I Relative change in I Number of RNA I molecules where plasmid synthesis is at pace with dilution (at a given time) Average number of RNA I molecules when there are M plasmids Number of plasmids Steady state number of plasmids, i.e. the M for which hIiM is also ISS Average number of plasmids at the end of the cell cycle Initiation rate constant of RNA II transcription, i.e. the RNA II transcription frequency Initiation rate constant of RNA I transcription, i.e. the RNA I transcription frequency Rate constant for degradation of free RNA I Growth rate constant of the host cell Probability that a mature primer initiates plasmid replication ('real' priming attempts) multiplied by the number of RNA II transcriptions per plasmid and cell cycle.

Probability r Q0 QSS 0 P(loss) HL EL P(M,t) P(M,I)

Probability that a mature primer finally initiates plasmid replication Probability that replication priming by RNA II is not inhibited by RNA I Q0 for the steady state number of RNA I molecules, ISS Probability that a plasmid containing cell gives rise to a plasmid free segregant at cell division Lower limit in P(loss) for hyperbolic inhibition Lower limit in P(loss) for exponential inhibition Probability that the cell contains M plasmids at time t Probability that the cell contains M plasmids and I RNA I molecules at time t

Other KI Rf dRf dRf/Rf RfSS n(t) t {M,I}

Inhibition constant for RNA I:RNA II complex formation Replication frequency (kII  Q0  r) Change in Rf Relative change in Rf Rf at steady state (ˆ kH) Cell volume at time t Generation time of the host cell, equal to ln(2)/kH State in the cell with M plasmids and I RNA I molecules

of rate-limiting steps of the overall inhibition process has not been determined. It should be noted that there is basically no practical difference between 5 and 250 rate-limiting steps so that exponential inhibition kinetics applies in both cases. RNA I is constitutively transcribed from plasmids with rate constant kI and actively degraded with rate constant eI (Bremer & Lin-Chao, 1986; Brenner & Tomizawa, 1991; Merlin & Polisky, 1993). Free RNA I also forms duplexes with RNA II. Experiments (Tomizawa, 1986) strongly indicate that basically every RNA II eventually binds an RNA I irrespective of variations in RNA I concentration. This means that the total rate of RNA I degradation through duplex formation is determined by the transcription frequency of RNA II. It has also been shown (Tomizawa, 1990a) that RNA I and RNA II initially form a reversible kissing complex that can be stabilized by the Rom protein, increasing the probability of duplex formation (Cesareni et al., 1982; Lacatena et al., 1984; Tomizawa & Som, 1984; Tomizawa, 1990b). If Rom is always present at a saturating concentration, its effect on duplex formation can be included in the inhibition constant (KI, see Table 1 and Appendix). For simplicity, this is assumed in the present analysis and has some experimental support (Tomizawa & Som, 1984; Keasling & Palsson,

1989). However, if the probability of duplex formation from an initial kissing complex is proportional to Rom concentration, and if Rom concentration is nearly proportional to plasmid concentration, then copy number control becomes more ef®cient (Ehrenberg, 1996; Summers, 1996) and Rom must be included as a separate variable. Assumptions on segregation ColE1 replication frequency is determined by RNA I concentration. It is thus the concentration rather than the number of plasmids that is regulated. Since the aim of control is to keep a ®xed copy number at cell division, this type of mechanism works best when the volume of a dividing host cell is constant. Here we have assumed that the volume of the host grows exponentially with rate constant kH from n(0) at the beginning of the cell cycle to 2  n(0) at its end, without variation. ColE1 carries an ef®cient site for multimer resolution, cer (Summers et al., 1985; Chan et al., 1985; Leung et al., 1985), and it is assumed here that all plasmids are present in their monomeric form at cell division (Summers et al., 1993; Patient & Summers, 1993; Boe & Tolker-Nielsen, 1997). Wildtype ColE1 seems to lack a mechanism to actively partition individual plasmids between daughter

77

Segregational and Metabolic Aspects of ColE1

Figure 2. Average number of replications per plasmid and cell cycle as a function of the relative number of RNA I molecules (I/ISS) for hyperbolic and exponential inhibition mechanisms. Both curves are obtained for r  kII/kH ˆ 10. In the upper right region plasmids replicate more than once per copy and cell cycle for elevated RNA I concentrations, leading to runaway replication, and in the lower left region plasmids replicate less than once per copy and cell cycle for low RNA I concentrations and are lost.

(equation (A5)), which is more than 100 times greater. The probability distribution P(M, t), that a cell immediately prior to division (t ˆ t) contains M plasmids, is thus of critical importance for segregational stability (equation (A5)). When plasmid and RNA I concentrations are not assumed to be proportional, P(M, t) must be calculated from the joint probabilities, P(M, I, t), that a single cell of age t carries M plasmids and I RNA I molecules (equation (A4)). The probability distribution P(M, I, t), can be found by modeling the kinetic events that change the number of plasmids or RNA I molecules (Figure A1) as master equations (equation (A3)). The numerical integration of these differential equations is computationally demanding (some simulations required more than 100,000 differential equations to be integrated over numerous cell cycles) so copy number distributions and segregational stability were mainly analyzed for a ®xed average of 16 plasmids at the end of the cell cycle. The average copy number depends on the rate constants of the system and the effective inhibition constant, KI. KI affects average numbers of plasmid and RNA I but has no impact on the dynamics of copy number control (not shown). To study the effects of changing rate constants on the dynamics of copy number control, KI was adjusted to get the same average copy number in all simulations.

Results cells and the probability that a plasmid in a mother cell ends up in a speci®c daughter cell is therefore taken to be 1/2. It is also assumed that plasmids segregate independently of each other. This simpli®cation may seem unjusti®ed since plasmids serve as replication templates for new plasmids, but in the absence of physical connections linking them together the spatial locations of newly replicated plasmids will rapidly be randomized by Brownian motion. The simpli®cations mentioned above allow the analysis to focus on replication control but may lead to an overestimation of segregational stability. Segregational stability as a function of replication control Variation around the average copy number is of crucial importance for P(loss), the probability that a plasmid-containing cell gives rise to a plasmidfree cell at cell division (NordstroÈm et al., 1980; NordstroÈm & Austin, 1989; Summers, 1991, 1996). For example, a cell population in which all cells contain exactly 16 plasmids at the end of the cell cycle has P(loss)  3  10ÿ5 (equation (A5)) but for a hypothetical population where half of the dividing cells contain eight plasmids and the other half 24 (so that the average is still 16) P(loss)  4  10ÿ3

The results will focus on different strategies to avoid copy number ¯uctuations and increase segregational stability. High RNA II transcription frequency increases sensitivity of control and thereby segregational stability Exponential and hyperbolic inhibition mechanisms work very differently in the non-stationary situations characteristic of single cells. The difference in sensitivity is clearly seen by comparing dRf/Rf, the relative change in plasmid replication frequency Rf, with dI/I, the relative change in the number of RNA I molecules I, for both types of inhibition. Comparison is most instructive for deviations close to the ``steady state'' number of RNA I molecules, ISS, where plasmid replication is at pace with the growth of the host cell (this number depends on the cell volume, Appendix). For the hyperbolic inhibition: dRf dI ˆ ÿ ss  …1 ÿ kH =…kII  r†† Rf ss I

…1†

so that the relative change in Rf is as great as the relative change in I only when kH/(kII  r) is very small (so that r  kII/kH is very large, Table 1). Experiments (Lin-Chao & Bremer, 1987; Brenner & Tomizawa, 1991) suggest that the

78

Segregational and Metabolic Aspects of ColE1

value of r  kII/kH is somewhere between 5 and 100. For exponential inhibition the relative change in replication frequency is instead ampli®ed by the term ln(r  kII/kH) compared to the relative change in I:

r  kII/kH (Figure 3(b)). The hyperbolic control limit, HL, corresponds approximately to a Poisson distribution of plasmids (Q0 is inversely proportional to RNA I concentration) and is given by (equations (A15) and (A16)):

dRf dI ˆ ÿ ss  ln…r  kII =kH † Rf ss I

HL  2  eÿ0:5hMit

…2†

A sensitive response in Rf to changes in I is directly re¯ected in narrow plasmid distributions at the end of the cell cycle. Indeed, it is found here that for hyperbolic inhibition P(M, t) moves from a very broad to a Poisson distribution (approximately) with increasing r  kII/kH. For exponential inhibition P(M,t) becomes inde®nitely narrower for increasing values of r  kII/kH, until all probability mass is concentrated at the average copy number hMit. The width of P(M, t) is re¯ected in P(loss) (equation (A5), Figure 3), which decreases with increasing values of r  kII/kH for both hyperbolic and exponential inhibition. The rate constants kI and eI, of RNA I synthesis and degradation, respectively, are very large in Figure 2 so that the RNA I and plasmid concentrations are strictly proportional (see below). P(loss) asymptotically approaches a lower limit for both inhibition mechanisms with increasing

…3†

For exponential inhibition and high values of r  kII/kH, replication control becomes very sharp so that replication is negligible above and very rapid below steady state in RNA I concentration. In this limit all copy number variations are eliminated at the end of the cell cycle and the corresponding limit in P(loss), EL, is given (equation (A16)) by: EL ˆ 2  2eÿ ln…2†hMit

…4†

A perfectly working hyperbolic control system (equation (3)) must thus compensate with a copy number 2  ln(2)  1.4 times higher than that of a perfectly working exponential control system (equation(4)) to obtain the same segregational stability. Correspondingly, HL is more than 20 times higher than EL when hMit ˆ 16 and more than 15,000 times higher when hMit ˆ 50. An exponential mechanism can reduce the metabolic load associated with high copy number signi®cantly and still maintain a high segregational stability

Figure 3. (a) The copy number distribution at the end of the cell cycle for exponential and hyperbolic inhibition. Both curves are obtained for r  kII/kH ˆ 10. The average copy number at the end of the cell cycle is 16 in both panels. (b) P(loss) as a function of r  kII/kH for exponential and hyperbolic inhibition. HL corresponds to the theoretical lower limit in P(loss) for hyperbolic (equation (3)) and EL to the same limit for exponential inhibition (equation (4)).

Segregational and Metabolic Aspects of ColE1

also for a lower and more realistic value of r  kII/ kH (Figure 3(b)). Limited RNA I degradation rate causes a delayed response in plasmid replication frequency The single cell plasmid concentration changes during the cell cycle as a result of replication, continuous dilution and an uneven distribution of plasmids between the two daughters at cell division. If eI, the rate constant for degradation of free RNA I, is not large then RNA I concentration cannot keep up with these changes and is instead determined by previous plasmid concentrations. A suitable parameter describing the RNA I response rate to changes in plasmid concentration is the dimensionless entity (eI ‡ kH)/kH, the rate constant for decrease in RNA I concentration due to degradation (eI) and dilution (kH) of free RNA I, normalized by the growth rate constant of the host cell (kH). In fact, (eI ‡ kH)/kH is found to be the only determinant for how rapidly RNA I concentration adjusts to changes in plasmid concentration. To study the effect of limited eI separately, RNA I synthesis is made very rapid in all simulations so that duplex formation is negligible (see below). P(loss) curves are drawn in Figure 4 as functions of the parameter (eI ‡ kH)/kH for hyperbolic and exponential inhibition kinetics. Simulations were

79 made for different values of the ratio r  kII/kH since this parameter is important for how rapidly plasmid concentration changes (equations (1) and (2)), and they demonstrate very high P(loss) values for both types of inhibition when entity (eI ‡ kH)/ kH is small. As eI increases, P(loss) decreases to a plateau value corresponding to a perfect proportionality between RNA I and plasmid concentrations (Figures 4 and 5(a)). Another plateau value (Figure 5(a)) is reached when eI approaches zero (and entity (eI ‡ kH)/kH consequently approaches 1) where dilution dominates over degradation. For both exponential and hyperbolic inhibition kinetics and all values of r  kII/kH, the simulations show a threshold where an increase in eI has virtually no effect on P(loss) but where a decrease in eI can be disastrous. The scale of Figure 4 makes thresholds dif®cult to observe, but an example where the other rate constants have been chosen from in vivo data (Lin-Chao & Bremer, 1986; Bremer & Lin-Chao, 1986; Brenner & Tomizawa, 1991) is given in Figure 5(a). To understand why P(loss) is affected by eI we compared the average relative deviation from steady state in the number of RNA I molecules, hIiM/ ISS, with the relative deviation in copy number, M/ MSS. It was found that deviations from steady state in RNA I are, on average, smaller than deviations in copy number when eI/kH is small (Figure 5(b) and Discussion).

Figure 4. P(loss) as a function of the normalized rate constant of RNA I-dependent decrease in RNA I concentration, (eI ‡ kH)/kH, (see main text and Table 1) for exponential (a) and hyperbolic (b) inhibition. The average plasmid copy number at the end of the cell cycle is 16 in both panels. The RNA I synthesis rate is so large in both (a) and (b) that decrease in RNA I due to duplex formation is negligible (see main text).

80

Segregational and Metabolic Aspects of ColE1

Figure 5. The same data as the intermediate curve in Figure 4(a) now plotted as a function of the normalized rate constant for active RNA I degradation. The other rate constants were chosen from experimental estimates of in vivo values: r  kII/kH ˆ 10 (somewhere in the range 5 to 100 according to Lin-Chao & Bremer (1986) and Brenner & Tomizawa (1991)) and kI suf®ciently high so that decrease in RNA I due to duplex formation is negligible (Lin-Chao & Bremer, 1986). The value of in vivo RNA I instability is from Brenner & Tomizawa (1991). (b) hIiM/ISS, the average relative number of RNA I molecules for a given number of M of plasmids as a function of M/MSS, the relative number of plasmids. When eI/kH ! 1, plasmid and RNA I become perfectly proportional.

Limited RNA I synthesis rate causes randomization of RNA I concentration Simulations as well as analytical calculations predict that the quality of copy number control is essentially unaffected by the transcription frequency, kI, of RNA I if free RNA I does not disappear through duplex formation with RNA II (not shown). However, experiments (Tomizawa, 1986) strongly indicate that basically every RNA II eventually binds an RNA I and that the duplex is subsequently degraded. Under this assumption the proportion of RNA I that is degraded via duplex formation when there are M plasmids and I RNA I molecules is given by: kII  M kII  kII  M ‡ eI  I kI

…5†

since kII  M is the rate of duplex formation and eI  I is the degradation rate of free RNA I. Equality holds when RNA I concentration equilibrates rapidly to plasmid concentration, so that degradation and synthesis are balanced and kII  M ‡ eI  I  kI  M. This ratio is predicted to be extremely important for the proportionality between plasmid and RNA I concentrations. First, a copy number control

would not work at all if kII 5 kI, since then RNA I must be consumed faster than it is synthesized. Second, when kI is only slightly higher than kII, a new kind of disproportionality between plasmid and RNA I concentrations is observed in the simulations. Even in simulations where RNA I concentration responds very rapidly to changes in plasmid concentration (high eI) a very large variation in the number of RNA I molecules (for a given copy number) is observed unless kII5kI. The impact that variation in RNA I has on the dynamics of copy number control depends on the inhibition mechanism, the average number of RNA I molecules and the other rate constants of the system. To illustrate the dramatic effect that duplex formation may have on segregational stability, an example is given in Figure 6 where kI is varied and all other rate constants are taken from in vivo data (Lin-Chao & Bremer, 1986; Brenner & Tomizawa, 1991).

Discussion Disregarding conjugation, stability in plasmid maintenance depends mainly on (1) the intrinsic segregational stability and (2) the ®tness change that plasmids confer their hosts (Wouters et al.,

Segregational and Metabolic Aspects of ColE1

Figure 6. P(loss) as a function of the ratio, kI/kII, between the transcription inhibition when (eI ‡ kH)/ kH ˆ 20 (see Figure 4(a)) and r  kII/kH ˆ 30 (see Figure 3(b)).

1980; Cooper et al., 1987; Chiang & Bremer, 1988; Proctor, 1994). In a non-selective environment, the ever so rare event of plasmid loss may initialize a rapid washout of plasmids from the population due to a faster growth of plasmid-free cells and in environments where plasmids are essential to the host, a plasmid-host system that can maintain a given segregational stability at a minimal metabolic load will outgrow the competition (Wouters et al., 1980; Modi et al., 1991). To understand the underlying strategy of plasmid control systems it is therefore necessary to show how the molecular details of replication control are related both to segregational stability and to metabolic burden. Determinants of segregational stability The probability P(loss), that a plasmid-containing cell gives rise to a plasmid-free segregant at cell division, is determined by the plasmid copy number distribution at the end of the cell cycle (equation (A5) and Figure 3). The present analysis shows how higher segregational stability can be obtained either by increasing the average copy number or by improving replication control and thereby reducing copy number variation. We have found that major determinants of copy number variation are (1) the type of inhibition mechanism, (2) the RNA II transcription frequency, (3) the RNA I transcription frequency and the rate constant for degradation of free RNA I. Two different suggestions for inhibition mechanisms have been made in the past, hyperbolic (equation (A1); Bremer & Lin-Chao, 1986; Keasling & Palsson, 1989; Brendel & Perelson, 1993) and exponential (equation (A2); Ataai & Shuler, 1986; Brenner & Tomizawa, 1991). Both mechanisms become increasingly ef®cient with an increasing RNA II transcription frequency. However, for

81 hyperbolic control the copy number distribution at the end of the cell cycle can never be narrower than Poisson (Figure 3) even for an in®nite RNA II transcription frequency. For exponential inhibition, in contrast, the copy number distribution becomes inde®nitely narrower with an increasing RNA II transcription frequency (Figure 3). These results are in conformity with Ehrenberg (1996), who modeled low copy number ColE1 plasmids with an active partitioning mechanism. For multi-copy plasmids that are randomly distributed at cell division, there is a lower limit in P(loss) that is set by the average copy number at the end of the cell cycle. This is reached when plasmid and RNA I concentrations are perfectly proportional throughout the cell cycle and the RNA II transcription frequency is very high. However, a maximally ef®cient hyperbolic inhibition mechanism must compensate with a 2  ln(2)  1.4 (equations (3) and (4)) times larger average copy number to obtain the same P(loss) as a maximally ef®cient exponential inhibition mechanism. Related to this asymptotic limit, there is a threshold value in the RNA II transcription frequency such that an increase has a limited effect on P(loss) but a decrease can be devastating (Figure 3). For ef®cient control, plasmid and RNA I concentrations should be nearly proportional throughout the cell cycle. Such a proportionality was a priori assumed in Ehrenberg's model to keep focus on the inhibition mechanism, but must be explicitly investigated to understand the strategies of copy number control. To set the replication frequency adequately, the relative deviation from steady state must be the same for plasmid and RNA I, so that a reduction to 50% in copy number corresponds to a 50% reduction in RNA I. This is not the case when the rate constant for RNA I degradation, eI, is small. A deviation from steady state in copy number is likely to have been closer to steady state in the recent past, simply because these states are more probable. As eI decreases, RNA I concentration depends more and more on the recent history of plasmid concentrations. Consequently, RNA I will often be closer to steady state than plasmid. This is illustrated in Figure 5(b) and explains how small values of eI lead to broad copy number distributions with their corresponding high P(loss) values (Figure 4). There is a threshold where P(loss) increases dramatically if eI is reduced. This threshold is surprisingly independent of the sensitivity of control. Sensitive control means rapid copy number adjustments that require rapid RNA I degradation to keep RNA I and plasmid concentrations proportional, but rapid adjustments also reduce variation in copy number (Figure 3) so that situations where plasmid concentration changes fast in fact become very rare. These two effects of sensitivity practically neutralize each other in the requirement for rapid RNA I degradation (Figure 4).

82 In a macroscopic approach (Paulsson et al., 1998), where stochastic effects were neglected, it was shown that the RNA I synthesis rate constant kI can be replaced by a ``net'' synthesis rate constant kI ÿ kII, where kII is the rate constant for RNA II transcription and therefore for total duplex formation. It was furthermore shown that the value of kI ÿ kII has no effect on the dynamics of control (provided that kI > kII so that replication can be regulated at all). However, when stochastic effects are taken into account this is no longer true since RNA I synthesis and RNA I:RNA II duplex formation are two different random events. The number of RNA I transcription events and the number of duplex formation events in some time intervals display statistical variation around their respective average. This means that there can be huge relative variation in the difference between the two numbers of events. When kI is only slightly higher than kII, there will inevitably be large random ¯uctuations in the number of RNA I molecules, impairing copy number control signi®cantly. It can be shown that the ratio kII/kI is the key parameter determining the randomizing effect of duplex formation on RNA I concentration. When this ratio is much smaller than 1 (kII5kI), the effect is negligible, but when it approaches 1, RNA I concentration becomes completely randomized. Experimental measurements indeed show that the ratio is approximately 0.2 4 kII/kI 4 0.4 in vivo (Lin-Chao & Bremer, 1986; Brenner & Tomizawa, 1991), i.e. in a region where the model predicts randomization to be very limited (Figure 6). If only a fraction, f, of the RNA II molecules eventually form a stable duplex with RNA I, it is instead the ratio f  kII/kI that determines the quality of control. In conclusion, the rate of RNA I synthesis must be much higher than the rate of RNA I disappearance through duplex formation to avoid a randomization of RNA I concentration. It is important to distinguish between the two pathways of RNA I degradation. How rapidly RNA I concentration responds to changes in plasmid concentration is only determined by the rate constant for degradation of free RNA I, even when degradation via duplex formation is dominant. The simple reason is that the degradation rate of free RNA I depends on RNA I concentration, so that more RNA I is degraded when concentration is high and vice versa, while the rate of duplex formation depends on plasmid concentration. In fact, if RNA I could inhibit replication without ever forming a stable duplex with RNA II, the RNA I synthesis rate constant, kI would be expected to affect only the average copy number, but not the dynamics of copy number control (not shown). Likewise, RNA I degradation via duplex formation may randomize RNA I concentration almost completely even when eI is very high. Increasing eI will increase the degradation rate of free RNA I but also the rate of duplex formation since the average copy number is approximately proportional to eI. Randomization is only avoided when RNA I is

Segregational and Metabolic Aspects of ColE1

transcribed at a much higher frequency than RNA II (i.e. kI4kII). ColE1 replication control is set up so that plasmid concentration is the signal that determines plasmid replication frequency. In summary it can be said that the inhibition mechanism and the value of r  kII/kH determine how this signal can be ampli®ed, that eI determines signal delay and that the ratio kII/kI determines the noise level. Optimal copy number control For both exponential and hyperbolic inhibition, segregational stability can be ``bought'' at the expense of an increased average copy number (equations (3) and (4), Figure 7) or by reducing copy number variation by increasing the rate constant for degradation of free RNA I or the RNA I and RNA II transcription frequencies (Figures 3, 4 and 7). Depending on the metabolic load associated with plasmid synthesis versus RNA transcription, there are ``optimal'' parameters of the system where a certain P(loss) is obtained at a minimal metabolic burden to the host cell. It is striking that there are threshold values in the rate constants for degradation of free RNA I and for transcription of RNA I and RNA II such that an increase has virtually no effect on P(loss) but a decrease can be devastating (Figures 3(b), 4, 5(a), 6 and 7). Interestingly, these three rate constants have been experimentally measured (Lin-Chao & Bremer, 1986; Brenner & Tomizawa, 1991) and are all in a region where little can be gained in segregational stability by a further increase, but a great deal is lost by a reduction. Brendel & Perelson (1993) also suggest that instability of free RNA I as well as RNA I and RNA II transcription frequencies are important for copy number control. They argued that these parameters determine the macroscopic copy number, so that, for instance, copy number would be too low if RNA I were too stable. This would be true if all other parameters were kept constant and RNA I stability was increased. However, the average copy number is determined by a number of parameters and a mutation that makes RNA I bind to RNA II with lower af®nity (Tomizawa & Itoh, 1981) could increase the average copy number to almost any level. The present model instead suggests that the importance of rapid RNA I degradation is to ensure a tight coupling between plasmid and RNA I concentrations, that RNA I should be transcribed at a much higher frequency than RNA II to avoid randomization of RNA I concentration and that the importance of high RNA II transcription frequency is to increase sensitivity of inhibition (equations (1) and (2)), i.e. always to improve the dynamics of copy number control. ColE1 plasmids can be essential to the host cell by providing antibiotic resistance, but also because they code for systems that selectively kill plasmidfree cells (Sabik et al., 1983). In all situations where plasmids are essential for the survival of the host,

83

Segregational and Metabolic Aspects of ColE1

Figure 7. P(loss) as a function of the total number of RNA II transcriptions per cell cycle for different copy numbers and for exponential (a) and hyperbolic (b) inhibition. All curves were obtained for the same set of r  kII/kH values, in a range from 2 to 256. The total number of RNA II transcriptions per cell cycle (Appendix) depends on both kII/kH, the normalized RNA II transcription frequency, and hMit, the average copy number at the end of the cell cycle.

the ``net'' growth rate of the plasmid-host system is reduced by the metabolic burden of plasmids, but also by plasmid loss since it results in death for one of the daughter cells. Under such conditions, an optimal set of rate constants must consider both types of growth rate reductions and maximize, by de®nition of optimality (Ehrenberg & Kurland, 1984), the net growth rate of the plasmid-host system. The present model relates the molecular details of copy number control to both metabolic burden (replication of plasmid and transcription of RNA) and P(loss) so that, in principle, well de®ned ``®tness'' landscapes could be computed. Whether the absolute differences in ®tness between different strategies are suf®ciently large to drive evolution will depend on the population dynamics of the host cell, for instance effective population sizes. (It should be pointed out that the simulations may overestimate segregational stability somewhat since sources of variation that are not related to replication control have been neglected; see Theory). It may also be important that selection and evolution of ColE1 copy number control is affected by both cell-cell competition (differential growth) and plasmid-plasmid competition (mutations that give higher copy number may take over the plasmid population).

We are optimistic that the present work will open the door for a quantitative cost-bene®t analysis of ColE1 copy number control, which subsequently can be used to formulate testable hypotheses concerning how it has evolved and how it will adapt to different selective pressures. Such an analysis can also be helpful in the construction of new cloning vectors since it would indicate ways to raise the copy number with maximal increase in segregational stability and minimal reduction in plasmid-host ®tness. Experimental challenges ColE1 is perhaps the most studied and documented replicon, but a number of crucial features of copy number control remain obscure. These concern (1) the number of rate-limiting steps in the inhibition process, (2) the dynamic in¯uence of the Rom protein, (3) the effect of plasmid and RNA synthesis on the growth rate of the host and (4) physiological control of important rate constants. ColE1 has an ef®cient site for multimer resolution (Summers & Sheratt, 1984; Summers et al., 1985; Chan et al., 1985; Leung et al., 1985), indicating that plasmids are predominantly monomeric (Summers & Sheratt, 1984; Patient and Summers,

84 1993; Boe & Tolker-Nielsen, 1997). When the resolution site is deleted, multimers rapidly take over the population (Summers et al., 1993) and this will have very different implications for hyperbolic and exponential inhibition. It has been shown (Paulsson et al., 1998) that for hyperbolic control, a dimer population should have at least as high a copy number as a monomer population. Dimerization thus implicates a greater metabolic burden and greater (or unchanged) segregational stability (equations (A5) and (3)). For exponential inhibition, the number of dimers per cell could be almost twofold lower than the number of monomers. This means that dimerization has a slight impact on metabolic burden, but reduces segregational stability enormously (equations (A5) and (4)). Observations have shown that strains in which multimer resolution is non-functional are extremely unstable (Summers & Sherratt, 1984; Summers et al., 1993; Boe & Tolker-Nielsen, 1997). This is compatible with exponential inhibition. The in¯uence of the Rom protein on the dynamics of copy number control is still unknown but may be signi®cant (Ehrenberg, 1996). Throughout the present analysis we have assumed that the probability that an initial RNA I:RNA II kissing complex is converted to a stable duplex is independent of variations in plasmid concentration, as may be expected if Rom concentration is always saturating (or at least constant; Tomizawa & Som, 1984). If copy number distributions could be measured accurately they could help to reveal the unknown role of the Rom protein by comparing copy number variation when Rom is either absent, overexpressed (saturating) or expressed in in vivo amounts. The precision of such experiments would be higher if only cells about to divide (or at least cells of a certain volume) were taken into account, since copy number distributions averaged over all cell ages would be broad also in those cases where the distribution at a given cell age is narrow. P(loss) could also be measured but confers less information about the regulation system, since maybe not all plasmids are independently distributed at cell division, and effects of copy number variation could be dif®cult to separate from effects of the average copy number unless the latter is measured with very high precision. How the metabolic load of plasmids affects the growth of the host cell is of crucial importance to understand the evolutionary strategy behind copy number control (Summers 1996). This still awaits systematic investigation, and would require growth experiments for a wide variety of ColE1 derivatives of different copy numbers and, if possible, with varying RNA I and RNA II transcription frequencies. The metabolic burden of plasmids depends on auxiliary genes, for instance coding for antibiotic resistance (Wouters et al., 1980; Noack et al., 1981). This is very interesting in itself, but to understand why the in vivo copy number is as it is, it would also be necessary to measure the burden of the basic replicon per se, or larger derivatives

Segregational and Metabolic Aspects of ColE1

where additional genes are silent. From an evolutionary point of view, there is no reason why the total gene expression (apart from transcription of RNA I and RNA II) should be greater for a higher average copy number, since a low copy number could be compensated by stronger promoters to obtain a certain expression level. It has been observed that the host cell can adapt to the plasmid and reduce plasmid related metabolic burden (Bouma & Lenski, 1988), which shows that ®tness changes must be studied over very long periods of time. To retain optimality under changing conditions, it is necessary that the rate constants of the system are under physiological control. Herman et al. (1994) showed that the Rom protein is important in the stringent control of pBR322 replication. This indicates that Rom plays a key role in physiological control of the average copy number. However, the present work shows additionally that physiological control is important to preserve the quality of copy number variation ef®ciently, i.e. to retain an ability to reduce copy number variation ef®ciently. For instance, when the growth rate of the host cell increases, RNA II transcription frequency must also be increased or quality of control will be reduced (equations (1) and (2), Figure 3(b)). If RNA II transcription frequency changes, so does the rate of RNA I degradation due to duplex formation, and this places new requirements on the RNA I transcription frequency to avoid a randomization of the concentration (equation (4) and Figure 6). The degree of randomization depends on the ratio between RNA I and RNA II transcription frequencies, but the effect of randomization on copy number control depends on all rate constants and the inhibition mechanism. Therefore, it is dif®cult to quantitatively predict the optimal change in RNA I transcription without a complete characterization of the system. However, the model predicts that an increased RNA II transcription frequency should optimally lead to an increased RNA I transcription frequency to avoid randomization of RNA I concentration. Several interesting results have come from Bremer and co-workers (Bremer & Lin-Chao, 1986; Lin-Chao &Bremer, 1986; Chiang et al., 1991) showing that the rate constants in fact are under physiological control. The present work, where rate constants are related to segregational stability, suggests that their experimental results have important implications for how ColE1 maintains optimal control under varying external conditions.

Acknowledgments This work was supported by the Swedish Natural Science Council, the National Graduate School of Scienti®c Computing and the Center for Parallel Computers in Stockholm. We thank Drs Kurt NordstroÈm, Leif Abrahamsson and Mats Gustafsson for valuable com-

Segregational and Metabolic Aspects of ColE1 ments. We also thank Drs Otto Berg and E. Gerhart H. Wagner for useful suggestions and for critically reading this manuscript.

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85 Lacatena, R. M., Banner, D. W., Castagnoli, L. & Cesareni, G. (1984). Control of initiation of pMB1 replication: puri®ed Rop protein and RNA I affect primer formation in vitro. Cell, 37, 1009± 1014. Leung, D. W., Chen, E., Cachianes, G. & Goeddel, D. V. (1985). Nucleotide sequence of the partition function of Escherichia coli plasmid ColE1. DNA, 4, 351± 355. Lin-Chao, S. & Bremer, H. (1986). Effect of the bacterial growth rate on replication control of plasmid pBR322 in Escherichia coli. Mol. Gen. Genet. 203, 143± 149. Lin-Chao, S. & Bremer, H. (1987). Activities of the RNAI and RNAII promoters of plasmid pBR322. J. Bacteriol. 169, 1217± 1222. Masukata, H. & Tomizawa, J. (1986). Control of primer formation of ColE1 plasmid replication: conformational change of the primer transcript. Cell, 44, 125± 136. Merlin, S. & Polisky, B. (1993). Analysis of establishment phase replication of the plasmid ColE1. J. Mol. Biol. 230, 137± 150. Modi, R. I., Wilke, C. M., Rosenzweig, R. F. & Adams, J. (1991). Plasmid macro-evolution: selection of deletions during adaptation in a nutrient-limited environment. Genetica, 84, 195± 202. Morita, M. & Oka, A. (1979). The structure of a transcriptional unit on Colicin E1 plasmid. Eur. J. Biochem. 97, 435± 443. Noack, D., Roth, M., Geuther, R., Muller, G., Undisz, K., Hoffmeier, C. & Gaspar, S. (1981). Maintenance and genetic stability of vector plasmids pBR322 and pBR325 in Escherichia coli K12 strains grown in a chemostat. Mol. Gen. Genet. 184, 121± 124. NordstroÈm, K. & Austin, S. J. (1989). Mechanisms that contribute to the stable segregation of plasmids. Annu. Rev. Genet. 23, 37 ±69. NordstroÈm, K., Molin, S. & Aagard-Hansen, H. (1980). Partitioning of plasmid R1 in Escherichia coli. I . Kinetics of loss of plasmid derivatives deleted of the par region. Plasmid, 4, 215± 227. Patient, M. E. & Summers, D. K. (1993). ColE1 multimer formation triggers inhibition of Escherichia coli cell division. Mol. Microbiol. 9, 1089± 1095. Paulsson, J., NordstroÈm, K. & Ehrenberg, M. (1998). Requirements for rapid plasmid ColE1 copy number adjustments ± a mathematical model of inhibition modes and RNA turnover rates. Plasmid, in the press. Proctor, G. N. (1994). Mathematics of microbial plasmid instability and subsequent differential growth of plasmid-free and plasmid-containing cells, relevant to the analysis of experimental colony number data. Plasmid, 32, 101±130. Sabik, J. F., Suit, J. L. & Luria, S. E. (1983). cea-kil operon of the ColE1 plasmid. J. Bacteriol. 153, 1479± 1485. Summers, D. K. (1991). The kinetics of plasmid loss. Trends Biotechnol. 9, 273± 278. Summers, D. K. (1996). Plasmid replication and its control. In The Biology of Plasmids, pp. 31 ± 64, Blackwell Science Ltd, Oxford. Summers, D. K. & Sheratt, D. J. (1984). Multimerization of high copy number mutants causes instability: ColE1 encodes a determinant essential for plasmid monomerization and stability. Cell, 36, 1097± 1103. Summers, D. K., Yaish, S., Archer, J. & Sheratt, D. (1985). Multimer resolution systems of ColE1 and

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ColK: localisation of the crossover site. Mol. Gen. Genet. 201, 334± 338. Summers, D. K., Beton, C. W. & Withers, H. L. (1993). Multicopy plasmid instability: the dimer catastrophe hypothesis. Mol. Microbiol. 8, 1031± 1038. Tomizawa, J. (1984). Control of ColE1 plasmid replication: the process of binding RNA I to the primer transcript. Cell, 38, 861± 870. Tomizawa, J. (1986). Control of ColE1 replication: binding to RNA I to RNA II and inhibition of primer formation. Cell, 47, 89 ± 97. Tomizawa, J. (1990a). Control of ColE1 plasmid replication: intermediates in the binding of RNA I and RNA II. J. Mol. Biol. 212, 683± 694. Tomizawa, J. (1990b). Control of ColE1 plasmid replication: interaction of Rom protein with an unstable complex formed by RNA I and RNA II. J. Mol. Biol. 212, 695± 708. Tomizawa, J. & Itoh, T. (1981). Plasmid ColE1 incompatibility determined by interaction of DNA I with primer transcript. Proc. Natl Acad. Sci. USA, 78, 6096± 6100. Tomizawa, J. & Som, T. (1984). Control of ColE1 plasmid replication: enhancement of binding of RNA I to the primer transcript by the Rom protein. Cell, 38, 871± 878. Tomizawa, J., Itoh, T., Selzer, G. & Som, T. (1981). Inhibition of ColE1 RNA primer formation by a plasmid-speci®ed small RNA. Proc. Natl Acad. Sci. USA, 78, 1421± 1425. Wouters, J. T., Driehuis, F. L., Polaczek, P. J., van Oppenraay, M. L. & van Andel, J. G. (1980). Persistence of the pBR322 Plasmid in Escherichia coli K 12 grown in chemostat cultures. Antonie Van Leeuwenhoek, 46, 353± 362.

Appendix When the inhibition process is dominated by a single rate-limiting step, the probability Q0 that primer formation is not inhibited by RNA I, is ``hyperbolic'' (Ehrenberg, 1996) and given by: Q0 …I; t† ˆ

1

1‡

I KI  n…t†

…A1†

where KI is the effective inhibition constant for RNA I± RNA II interactions (Table 1) and n(t) is the cell volume at time t. The number, I, of RNA I molecules in a single cell divided by n(t) is the single cell RNA I concentration. For multiple step inhibition kinetics, Q0 is instead ``exponential'' (Ehrenberg, 1996) and well approximated by the expression: Q0 …I; t† ˆ e

ÿK

I I n…t†

…A2†

which becomes exact when there are in®nitely many rate limiting steps in the inhibition window.

Figure A1. Transitions to and from a state {M,I} with M plasmids and I RNA I molecules. Transitions occur due to plasmid replication, RNA I synthesis and RNA I degradation events. A plasmid replication event is connected to an RNA I degradation event since RNA II binds an RNA I also when priming is not inhibited.

Transitions to and from a state {M, I}, with M plasmids and I RNA I molecules (Figure A1) occur due to (see Table 1): (1) Plasmid replication events and corresponding RNA I degradation due to duplex formation, which move a state {M ÿ 1,I ‡ 1} with M ÿ 1 plasmids and I ‡ 1 RNA I molecules to a state {M,I} with M plasmids and I RNA I molecules, with rate kII  r  Q0(I ‡ 1,t)  (M ÿ 1). Replication also moves a state {M,I} to a state {M ‡ 1,I ÿ 1} with rate kII  r  Q0(I,t)  M. (2) RNA I synthesis events, which move a state {M,I ÿ 1} to a state {M,I} with rate kI  M. RNA I synthesis also moves a state {M,I} to a state {M,I ‡ 1} with the same rate, kI  M. (3) RNA I degradation events, which move a state {M,I ‡ 1} to a state {M,I} due to active degradation of free RNA I with rate eI  (I ‡ 1), where eI is the rate constant for the ®rst step in active RNA I degradation by RNase E (Lin-Chao & Cohen, 1991), and due to RNA I:RNA II duplex formation events with rate kII  (1 ÿ r  Q0(I ‡ 1,t))  M. The total rate of RNA I:RNA II duplex formation is kII  M but some of these duplex formations are simultaneous to plasmid replication events (see (1)), and instead move a state {M,I ‡ 1} to {M ‡ 1,I}. Degradation also moves a state {M,I} to a state {M,I ‡ 1} with the rate eI  I ‡ kII  (1 ÿ Q0(I,t))  M. The probabilities P(M,I,t) for all states {M,I} at cell ages t can be obtained by numerical integration of the corresponding set of differential equations

87

Segregational and Metabolic Aspects of ColE1

dP…M; I; t† P…M ÿ 1; I ‡ 1; t†  kII  r  Q0 …I ‡ 1; t† ˆ dt  …M ÿ 1† ÿ P…M; I; t†  kII  r  Q0 …I; t†  M ‡ P…M; I ÿ 1; t†  kI  M

…A3†

These differential equations take into account that the rate of change, dP(M,I,t)/dt, of the probability, P(M,I,t) that a cell is in a state {M,I} at time t in the cell cycle, is determined by all probability ¯ows that give positive (arrows pointing inwards in Figure A1) and negative (arrows pointing outwards in Figure A1) contributions to P(M,I,t). The solution to these equations requires that the probability distribution, P(M,I,0), at the beginning of the cell cycle is known. Such initial conditions can be found in an iterative procedure (Ehrenberg, 1996). P(M,t) can be calculated from the joint distribution P(M,I,t) according to: 1 X

P…M; I; t†

…A4†

2  …1=2†M  P…M; t†

…A5†

Iˆ0

and P(loss) according to: P…loss† ˆ

1 X Mˆ1

The steady state number of RNA I molecules, ISS, at a given cell age t is de®ned as the number where plasmid synthesis is at pace with dilution due to cell growth, so that: kII  r  Q0 …I ss ; t†  M ˆ kH  M

…A6†

or equivalently: Qss 0 ˆ kH =…kII  r† QSS 0

…A7† SS

is a short notation for Q0(I ,t). where For hyperbolic inhibition (equation (A1)) this means that: I ss ˆ KI  n…t†  …r  kII =kH ÿ 1†

…A8†

and for exponential inhibition: I ss ˆ KI  n…t†  ln…r  kII =kH †

…A9†

Differentiating equation (A1) with respect to I gives: dQ0 …I; t† …Q0 …I; t††2 ˆÿ dI KI  n…t†

dI  …1 ÿ r  kII =kH † I ss …A11†

 Q0 …I ‡ 1; t††  M ‡ eI  …I ‡ 1†† ÿ P…M; I; t†  …kII  …1 ÿ r

P…M; t† ˆ

dQ0 dI  kH =…r  kII † dI I ss ˆ ÿ ss  ˆ ÿ ss Q0 KI  n…t† KI  n…t† I  kH =…r  kII † ˆ ÿ

ÿ P…M; I; t†  kI  M ‡ P…M; I ‡ 1; t†  …kII  …1 ÿ r

 Q0 …I; t††  M ‡ eI  I†

bolic inhibition, using equations (A7) and (A8), gives:

…A10†

Evaluating the derivative at steady state for hyper-

For exponential inhibition (A2), the derivative is instead given by: dQ0 …I; t† Q0 …I; t† ˆÿ dI KI  n…t†

…A12†

Evaluating this at steady state as above (equations (A7) and (A10)) gives: dQ0 dI dI I ss dI ˆ ÿ ˆ ÿ ss ˆ ÿ  ss ss Q0 KI  v…t† KI  v…t† I I  ln…r  kII =kH †

…A13†

The compounded parameter r  kII/kH is crucial for the quality of copy number control. Since the average number of plasmids is proportional to the cell volume, the average copy number over an entire cell cycle is the same as the average copy number in newborn cells divided by ln(2); hMi0/ ln(2). The average rate of RNA II synthesis over the cell cycle is thus kII  hMi0/ln(2). Multiplying by the generation time, t, of the cell, where t ˆ ln(2)/kH, gives the average total number of RNA II transcriptions per cell cycle, hMi0  kII/kH (Figure 7). kII/kH is thus the average number of RNA II transcriptions that a plasmid in a newborn cell will give rise to. The probability r must be included since not every RNA II can form a primer even in the absence of RNA I. When r  kII/kH is very large, Q0(I,t) can be written as: Q0 …I; t† ˆ

1 KI  v…t† ˆ I I KI  v…t†

…A14†

where the one in the denominator (equation (A1)) can be neglected. When this is the case, and when plasmid and RNA I concentrations are proportional throughout the cell cycle, the copy number is almost exactly Poisson distributed (not shown). Since we are only interested in cells that contain plasmids at the beginning of the cell cycle, the distribution must be normalized to disregard from plasmid-free cells that arise due to plasmid losses. The small deviation from a Poisson distribution comes from this normalization but can be neglected for a high copy number plasmid where P(loss) is very small. The lower limit in P(loss) for hyperbolic inhibition, HL, corresponds to the case when copy numbers are Poisson distributed:

88

Segregational and Metabolic Aspects of ColE1

1 X

hMiM t  eÿhMit HL  2  …1=2†  M! Mˆ1  2  eÿhMit 

M

1 X Mˆ0

ˆ 2  eÿhMit  e

hMit 2



hMit

M

…A15†

2

M! ˆ 2  eÿ

When EL and HL are expressed in the same base equations (A15) and A16)) show that a maximally ef®cient hyperbolic mechanism must compensate with a 2  ln(2)  1.4 times higher copy number to obtain the same segregational stability as a maximally ef®cient exponential mechanism.

hMit 2

When hMit is large ( as in vivo), the approximations are almost perfect. Under the same conditions, exponential inhibition works with extreme precision so that copy number variations at the end of the cell cycle are completely eliminated and the corresponding lower limit in P(loss), EL, is directly given by: EL ˆ 2  …1=2†hMit ˆ 2  eÿ ln…2†hMit

…A16†

References Ehrenberg, M. (1996). Hypothesis: hypersensitive plasmid copy number control for ColE1. Biophys. J. 70, 135± 145. Lin-Chao, S. & Cohen, S. N. (1991). The rate of processing and degradation of antisense RNA I regulates the replication of ColE1-type plasmids in vivo. Cell, 65, 1233± 1242.

Edited by D Draper (Received 27 October 1997; received in revised form 12 February 1998; accepted 13 February 1998)