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A mathematical model for λdv plasmid replication: Analysis of wild-type plasmid
PLASMID
11, 151-165
(1984)
A Mathematical Model for Xdv Plasmid Replication: Analysis of Wild-Type Plasmid SUN
BOK
LEE’
AND
JAMES
E. BAILEY~
Department of Chemical Engineering, California Institute of Technology, Pasadena, Calijbrnia 91125 Received
April
12, 1983; revised
December
2, 1983
A mathematical model for Xdv plasmid replication in a growing single cell of Escherichia coli has been formulated and solved numerically. Quantitative description of the molecular control mechanism for initiation of Xdv replication presumes regulatory functions of repressor and initiator proteins and transcriptional activation of the origin region. Random selection of a single plasmid for activation and replication is assumed, as is regular plasmid segregation to daughter cells. The model is capable of simulating the periodic changes in each regulatory element and the plasmid copy number during the cell cycle. The calculated average copy number, repressor concentration, and timing of plasmid replication agree well with experimental data. The simulated Mv plasmid replication rate is controlled primarily by transcription frequency. Initiation of plasmid replication is not related to variations in the levels of repressor or initiator proteins during the cell cycle. Simulation studies of perturbations in plasmid and repressor segregation indicate that replication regulation of the Xdv plasmid compensates to readjust copy number to normal values in a few generations. Implications of these studies relative to the molecular mechanisms of replication control are discussed.
in view of the extensive genetic and biochemical information available on the parent X phage (Hershey, 1971; Szybalski and Szybalski, 1979). Analysis of the Xdv plasmid replicon can also give a basis for understanding the phage Xreplicon which is now widely used as a cloning vector (Williams and Blattner, 1980; Brammer, 1981). The purpose of this paper is to describe a mathematical model for the replication control of Xdv plasmid in a single E. coli cell. Previously, several conceptual models for control of initiation of DNA synthesis have been proposed (Kolter and Helinski, 1979; Jacob et al., 1963; Pritchard et al., 1969; Sompayrac and Maalee, 1973). However, these hypotheseshave not yet been tested by restating them in mathematical form and determining whether the resulting equations properly describe plasmid copy number, stable maintenance of plasmid copy number, and influences of mutations in the replication regulation sequences.Such questions are considered here and in an accompanying paper (Lee and Bailey, 1984a) for the Xdv system.
Regulation of initiation of plasmid DNA replication is one of the crucial requirements for stable maintenance of plasmid copy number (Rowbury, 1977). Although the molecular mechanisms of replication control have not yet been completely elucidated, fine structural mapping of replication regions and identification of geneproducts involved in replication regulation have been carried out for several plasmids including ColEl (Conrad and Campbell, 1979; Oka et al., 1979; Tomizawa et al. 1981) CloDF13 (Stuitje et al., 1981) RI (Stougaard et al., 198 1; Stougaard et al., 1981), RlOO (Rosen et al., 1980, 1981), and Xdv (Berg, 1974; Matsubara, 1976, 1981). Among these, the plasmid Xdv, a deletion derivative of phage X which replicates autonomously in Escherichia coli (Matsubara and Kaiser, 1968) is a particularly useful model system for analysisof regulation of replication
’ On leave from the Korea Advanced and Technology, Seoul, Korea. ’ To whom correspondence should
Institute
of Science
be addressed. 151
0147-619X/84
$3.00
Copyright 0 1984 by Academic Press, Inc. All rights of reproduction in any form reserved.
152
LEE AND BAILEY
SYSTEM
DESCRIPTION
Formulation of the mathematical model for Xdv replication requires quantitative representations of the interactions between the pertinent regulatory elements and regulatory sites. Since details on current knowledge of this regulation mechanism have been extensively discussed elsewhere (Berg, 1974; Matsubara, 1976, 1981; Hershey, 1971; Szybalski and Szybalski, 1979) the following short summary will only highlight previous experimental information and hypotheses which are employed in formulation of the mathematical model. The minimal replicon of Xdv plasmid (Fig. 1) consists of the following genes and genetic sites: the autorepressor region, which contains the promoter-operator (PROR), autorepressor gene (cro) and terminator (fRZ), and the origin region which carries an initiator gene (0), replication origin (ori), and another initiator gene (P). Some of the Xdv isolates carry, in addition, the leftward promoter-operator (PLOL), lambda repressor gene (cl), and, rarely, the N or Q gene which can affect transcriptional activity. However, in most cases, this second set of genes is inactive due to mutations or deletions. An additional gene, oop, and genetic site, ice, located between t,l and the 0 gene in the hdv replicon, are not considered in the present model since the roles of these sequences in replication of X DNA are not yet clear (see Hobom et al., 1979; Hobom, 1981). The promoter-operator for rightward transcription (PRO,) is the site where RNA polymerase and cro repressor can bind, and at this site the initiation frequency of transcription is determined. The rightward operator consists of three tandem sequences at which cro repressor (designated R in Fig. 1) binds: the sequences are termed ORl, OR2, and OR3. The rightward promoter, PR, overlaps with OR2 and 0,2 as shown in Fig. 1. The cro repressor binds preferentially to 0,3 and, at higher concentrations, it turns off transcription of its own gene, cro, by binding at the O,l and 0~2 sites. The cro repressor is thus subject to negative autogenous control or autorepression. About
FIG. I. The genetic structure of Xdv plasmid replicon. Details are described in the text.
80% of the transcription from PRO, terminates at tRl, the site of transcriptional termination, in the presence of bacterial rho factor (Rosenberg et al., 1978). Transcription of the origin region initiated at PRO, and passing t,l leads to activation of the ori site and to coordinated production of two initiator proteins, 0 and P protein. The gene products 0 and P, long recognized as absolutely essential for initiation of replication, act cooperatively and positively in the initiation process although their exact modes of action are not yet clear. There is evidence suggesting that the 0, P, and several bacterial replication proteins such as the dnaB and dnaK gene products form a complex which acts at the ori site to initiate replication (Furth, Yates and Dove, 1979; Hobom et al., 1979). It has been reported that the 0 protein (designated I in Fig. 1) is unstable in vivo while the P protein is apparently much more stable (Lipinska et al., 1980; Kuypers et al., 1980). In addition to the requirements for initiation and replication proteins and for an intact ori site, replication initiation of hdv plasmid as well as X phage has been shown to depend upon a local transcription event that occurs at or near the origin of replication (so-called transcription activation). It has been demonstrated that replication is inhibited if transcription of the ori region is blocked, even though all the initiator proteins are provided in tram (Dove et al, 197 1; see also Wold et al., 1982). Whether this origin transcription provides the initiator system with a primer RNA for starting DNA synthesis or whether it converts the DNA ori region into an active
KINETIC
MODEL
FOR PLASMID
structure that is competent to interact with initiators has not yet been resolved (see Hoborn, 198 1). It is also possible that these two events can occur simultaneously. In this paper, the second hypothesis is employed. The model can be readily modified to reflect future information on this question. Besides the copy number control mechanisms which determine the timing and initiation frequency of DNA replication, models of replication and segregation are required for the description of the plasmid life cycle. Three possible modes of replication of plasmids by which a constant average number of plasmids per cell can be maintained during cell growth have been proposed by Rownd (1969): (1) Each plasmid copy replicates only once during a cell generation; (2) a master copy of plasmids replicates N times during each generation to produce a doubling of the number of plasmid copies, and (3) the N copies of plasmid exist in a pool and one of the plasmid copies is selected from the pool at random for replication. This process of random selection is repeated N times during each generation, until there has been a doubling of the number of plasmids. Using density-shift experiments, NRI (Rownd, 1969), ColEl (Bazaral and Helinski, 1970) RI (Gustafsson and Nordstrom, 1975), F’ (Gustafsson et al., 1978), and Xdv (Matsubara and Mukai, 1975) plasmids have been shown to follow the random selection mode of replication. Two models for segregation have been proposed (Novick and Hoppensteadt, 1978; Cullum and Broda, 1979): (1) An equal number of plasmid copies are distributed to each daughter cell (regular segregation) and (2) plasmids are partitioned at random on cell division (random segregation). A third model in which some fixed number of plasmids are segregated regularly with the remainder randomly segregated can also be envisioned. Although Xdv probably does not have equipartitioning, for the sake of simplicity, the mathematical model will presume regular segregation of all plasmids as a base case, and the effects of perturbations from regular seg-
REPLICATION
153
regation will then be explored using the mathematical model. The copy number control function can be affected by other factors including plasmid influences on host cell growth rate and oligomerization of the plasmid. The presence of multicopy plasmids may affect the growth rate of the host cell which, in turn, influences the replication of plasmid DNA. In the case of hdv plasmid, significant inhibition of host cell growth apparently does not occur. Matsubara and Kaiser (1968) have reported that the growth rate (as measured by total cell mass or the number of viable cells) of isogenic bacteria with and without Xdv 1 plasmid are equal. Naturally occurring higher oligomers of Xdv plasmid have been described (see Matsubara, 198 1). For example, Matsubara et al. (1975) reported that the monomer genome content of dimeric Xdvl was about 40% higher than the monomeric Xdvl genome content when pure monomer and dimer carriers were isolated and replicated independently. In the present model, which concerns the primary features of the plasmid replication control system, formation of oligomeric Xdv plasmid and concomitant effects on copy number are not considered. Such effects will be described in an accompanying paper (Lee and Bailey, 1984a). KINETIC MODEL
The model is based on the following assumptions: (1) The Xdv plasmid consists of genes between PRO, and P, and all plasmids exist as monomeric forms. (2) The 0 protein acts as a limiting initiator due to its instability. The level of P protein is consequently assumed to be relatively unimportant and is not considered. (3) Replication follows the random selection mode. (4) The ori site of the single plasmid selected for replication is activated by transcription. The initiator protein forms a complex with the activated origin. Activation of the replication complex to a certain critical value is required for the initiation of replication. (5) Segregation is by equal number
154
LEE
AND
partitioning at cell division. (6) The host cell grows in volume exponentially, and the plasmids do not affect the host cell growth rate. First the copy number control function of the Xdv plasmid is considered. The transcription efficiency at PROR is determined by the interactions between RNA polymerase and the cro repressor. Since the promoter overlaps with ORI and 0,2 (Fig. l), repressor bound either to ORl, 0,2, or to both sites will repress PR . The probability that the operon is not repressed can be determined using standard methods of statistical thermodynamics (Cantor and Schimmel, 1980). In Table 1, eight possible configurations of cro repressor at 0, and corresponding statistical weights are listed. If the transcriptional efficiency, 1, is assumed to be proportional to the fraction of PRO, templates at which RNA polymerase can bind (state 0 and state 3 in Table l), 7 can be computed by comparing the relative probability of states 0 and 3 to the sum of the relative probabilities of all possible states. Thus n is given by wo
+
w3
17’7
. c i=o
(1)
w
Assuming that the concentration of operator is much lower than that of repressor (a usual
BAILEY
case) and introducing Eq. (1) gives
statistical weights into
1 +
K3Flo
‘I = 1 + a[Rlo + b[R]; + c[R]; ’ where a=K,+K2+K3,
(3)
b = K,K2 + KzK3 + K3Kl,
(4)
c = K,KzK3.
(5)
K, , K2, and K3 represent the binding affinity of cro repressor at OJ, OR2, and OR3, respectively, and [RIO represents the total repressor concentration. (Equation (2) can also be derived by conventional multiple equilibria calculations, although at the expense of greater algebraic effort.) The rate of transcription of a gene at time t can be expressed as the product of overall transcription rate constant, transcription efficiency, and the gene concentration. Similar expressions for the rate of translation can be formulated. If the mRNA and protein are inactivated by first-order decay processes, the synthesis of cro mRNA and protein gene products are described by
d[mRN& dt
= k$q[G]
- k-,[mRNAIR
- dmRN&, TABLE
dt
- k~p,[Rlo- /ARIo,
Configuration” Statistical
(i)
OR3
OR2
0,’
+
+
5
+
6
+
1
+
(1 A(+) indicates by a cro repressor.
weight
(W,)
1
0 1
2 3 4
f&WI
&lRl
+
KdRl +
+
+ +
+ that
(6)
4Rlo = k$flRNA]R
1
POSSIBLE CONFIGURATIONS OF cro REPRESSOR AT OR AND THE STATISTICAL WEIGHTS CORRESKINDING TO EACH STATE
State
(2)
the corresponding
+
K&lRlz &&lRl* K,&lR12 KX&,lRl’ site is occupied
(7)
where p is the specific growth rate of the host cell. The intracellular concentrations of mRNA and protein, which are now expressed as moles per unit volume of cell, decrease as a consequence of two processes, degradation and dilution by cell growth (Fredrickson, 1974). The concentration of plasmid DNA at time t, [G], is expressed as
where G is the number of plasmid molecules per cell, V is the cell volume at time t, and
KINETIC
MODEL
FOR
PLASMID
NA is Avogadro’s number. For exponential single-cell growth (Pritchard et al., 1969), the change in cell volume during one cell generation can be determined from the following equation dV -=pv,v=v, dt at
1,2,...),
t=m(rz=
155
REPLICATION
where
e _ m*10 [oclo * Since it is assumed that only one plasmid is selected for replication at a time by the random selection mode, [orilo can be determined by the equation (cf. Eq. (8))
(9)
where V0 and 7 represent the cell volume at birth and the doubling time of the host cell, respectively. Transcription of the 0 gene is initiated at PRO, and requires readthrough of an intervening terminator t,l. If the termination efficiency at t,l is defined asf(0
Replication complexes are presumed to be formed by binding of initiator proteins at the activated ori site. In turn, the level of activated replication complexes determines the replication initiation frequency. Hence the following simplest scheme describing the interactions of initiator protein, I, host replication proteins designated Pr, and the activated origin, on*, is used to describe the formation of activated replication complexes, REP: KA I + Pr * 1.Pr, (17) orz* + 1. Pr -’ REP,
d[mRNAlI = k$z( 1 - f)[G] - k-,[mRNA]z dt
where KA is the equilibrium
- ,4mRNAlI, dt
- ~$410 - A.410. ( 11) As mentioned before, transcription of the origin region of the single plasmid selected for replication results in activation of the ori site. Since activation of the origin region is determined by the same transcription frequency as that for 0 protein, the formation of activated origin, [o~z*]~, can be described as ~d[omo = ko,?j(l -f)[orz] dt
- p[orz*]o.
host protein, and KB is the equilibrium association constant for binding of protein complexes to the activated origin. Mass balances for I, Pr, and orz* give [Ilo = [I] + [I - Pr] + [REP]
(19)
[Pr], = [Pr] + [I. Pr] + [REP]
(20)
[orz*lo = [orz*] + [REP].
(12)
(13)
so that Eq. (12) may be rewritten in the form dfl
--$ = eldl
-J-Ml - 0) - CL4 (14)
(21)
Assuming that [Pr] 9 [I +Pr] + [REP] and [REP] < [I] + [I. Pr], Eqs. (17) through (21) imply [REP] = K,K,[I][Pr][ori*]
= ~&Morilo 1 + KoDlo ’
Since the origin regions to be transcribed are either activated or not
[ori] = [ori] - [orP]o,
association con-
(10) stant for the binding of initiator protein and
4110= ki[mRNA]z
-
(18)
(22)
where K
0
=
K&[prlo 1 +KA[~]o'
(23)
K. in Eq. (22) can be treated as constant pro-
156
LEE
AND
vided that a constant fraction of host replication proteins is involved in replication of plasmid DNA. Initiation of plasmid replication and segregation are considered next. It is assumed that plasmid replication initiates when the replication complex level of activation [REP] reaches a critical value [REP],,. If replication of Xdv plasmid DNA occurs at chromosomal replication fork rates (Cooper and Helmstetter, 1968), replication of the entire Xdv molecule will be completed in 0.05 min. Hence it seems reasonable to assume, relative to the cell cycle time, that replication of the selected plasmid is essentially instantaneous once initiation occurs. Thus, if G(0) is the number of plasmid molecules in a newborn cell, the number of plasmids in that cell at time t is given by
BAILEY
no change in component concentrations at cell division. As an initial condition it is assumed that one plasmid molecule is introduced into an exponentially growing host cell: G,(O) = 1.
(28)
DETERMINATION OF MODEL PARAMETERS
All calculations have been performed for doubling time r = 1 h. Under these conditions, the overall transcription rate constant k:, the overall translation rate constant ki, and the initial cell volume V0 have been estimated to be 4.26 M mRNA/M DNA-mitt, 13.8 M protein/M mRNA-min, and 0.5 X IO-l5 liter, respectively (Lee and Barley, 1983). It is assumed G(t) = G(0) + m, here in the absence of more detailed data that (24) these cell-average transcription and translation where m is determined from parameters adequately approximate the parameters for the particular transcription and tget
KINETIC
MODEL
FOR PLASMID
and Johnson et al. (1978, 1979) the binding affinity of cro repressor on each operator was estimated. They reported that the binding affinity is in the range of 108-1010 M-’ and the relative dissociation constants (inverse of binding affinity) on a wild-type operator are OR1:OR20R3 = 8:8: 1 under conditions which approximate physiological condition. The termination efficiency at tRl, f; was assumed to be 0.8 according to Rosenberg et al. (1978). The values of K. and [REP],, were taken as adjustable parameters in the absence of experimental data for their evaluation. These values were adjusted so that the calculated copy number of wild-type Xdv plasmid agrees with experimental data. All of the model parameters for wild-type Xdv plasmid are summarized in Table 2. SIMULATION
Initiation
RESULTS
Frequency of Plasmid
Replication
It is not possible to integrate the simultaneous nonlinear differential equations above
157
REPLICATION
for [mRNA]a, [RIO, [mRNA]r, [Ilo, and 6’ analytically. Consequently, a numerical procedure (a variable time-step Runge-Kutta algorithm with predictor-corrector) was used to accomplish the integration on a computer. This method calculates the solution marching ahead in time in small time increments: values of [mRNAIR, [RI,, [mRNA]r, [Ilo, and 0 at the next time-step are determined from values of these quantities at earlier times using the differential equations of the model. As the integration advances, the [REP] value is calculated simultaneously. Cellular plasmid content is increased by one copy and the transcriptional activation level is reset to zero when [REP] reaches [REP],, as described above. As indicated in Eq. (28) above, the calculation begins with a single plasmid in the cell. The model parameter values listed in Table 2 are used in all of the computations reported here. The condition called here “cyclic state” is reached when each calculated intracellular concentration at cell division is equal to the concentration at the beginning of that cell cycle calculation.
TABLE 2 KINETIC PARAMETERS USED FOR WILD-TYPE Xdv PLASMID Parameters
k’,
Values”
Reference
0.5 X lo-‘r liter
Lee and Bailey (1984b)
4.26 M mRNA/M DNA-min
Lee and Bailey (1984b)
13.8 M protein/M mRNA-min
Lee and Bailey ( 1984b)
0.46 min-’
Pato and von Meyenburg (1975)
0.01 min-’
Matsubara (198 1) Wyatt and Inokuchi ( 1974) Lipinski et al. (1980) I Kuypers et al. (1980)
0.14 min-’ 1.25 X lo* 1.25 x lo* 1.0 x lo9
M-’ M-’
1
M-’
Rosenberg et al. (1978)
0.80 2.4
Takeda et al. (1977) Johnson et al. (1978, 1979)
X lOa M-’
2.0 x lo-‘*
M
-
’ The values of kO,, kz, and V. are those at r = 1 h. Details are given elsewhere (Lee and Bailey, 1984b).
158
LEE AND BAILEY
The trajectories of each intracellular concentration at cyclic state are given in Fig. 2. The number of plasmid DNA molecules and chromosomal DNA content in a single E. coli cell are also illustrated in Fig. 2. Chromosomal DNA contents were calculated using CooperHelmstetter relationships assuming that the time required for a round of chromosome replication is 40 min and the time interval between the termination of replication and subsequent cell division is 20 min (Cooper and Helmstetter, 1968). Simulation results show about 3 X 1O-’ M of cro repressor, which corresponds to an average of 130 monomers per cell, is present in a cell carrying the wildtype Xdv plasmid. This value is comparable to the available experimental data (cf. Lauer et al., 198 1). The following additional features of the simulations are noteworthy: (1) the initiation frequency of replication at cyclic state is equal to the number of plasmid molecules of the newborn cell, (2) there is an approximately equal time interval between successive plasmid replications, and (3) initiation of plasmid replication is not related to variations in the levels of repressor or initiator proteins during the cell cycle. Figure 2 also shows that the replication pattern of plasmid DNA is different from that of chromosomal DNA. A gene
o-0
Time (mid RG. 2. Calculated trajectories of the numbers of plasmid and chromosomal DNA molecules and the autorepressor and initiator protein concentrations during the cell cycle. Arrows on the curves indicate pertinent coordinate axes.
encoded in plasmid DNA could be increased in number throughout the cell cycle while a gene encoded in the chromosome would be doubled at a specific point in the cell cycle. In this model, it is assumed that plasmid replication follows a random selection mode. While density shift experiments have shown Xdv plasmids are selected randomly for replication (Matsubara and Mukai, 1975) these data do not directly prove that the process of random selection is repeated until there has been doubling of the number of plasmids during each generation. In other words, the question arises whether the replication control mechanism determines the initiation frequency of replication so that the number of plasmid molecules at cell division reaches exactly double the number of plasmid copies of the newborn cell. In connection with this discussion, it should be noted that the plasmid copies increase one by one as described in Eq. (24). The simulation results in Fig. 2 show that cyclic-state conditions are achieved for this model: the number of plasmid molecules at cell division is exactly twice that of newborn ceils, which means that the frequency of initiation during each generation (N) is equal to the number of plasmid molecules of the newborn cell (No). This result is consistent with experimental data for Rl and F’ plasmids (Gustafsson et al., 1978). In density-shift experiments, it was also observed that the time interval between successive replications is approximately equal to l/ Nth of a doubling time, where Nis the number of plasmid replications (Rownd, 1969; Gustafsson et al., 1978). The simulation results agree well with this experimental observation. As can be seen in Fig. 2, the time intervals between initiation events in the simulation are approximately equal but increase slightly as the cell age increases. The simulation results show that at cyclic state both repressor and initiator concentration remain nearly constant during the cell cycle (Fig. 2). Since the concentrations of repressor and initiator do not vary significantly during the cell cycle, the rate of origin activation is
KINETIC
MODEL
FOR PLASMID
kept nearly constant throughout the cell cycle due to constant 7 values (see Eqs. (2) and ( 14)). This causes the regular intervals between replication initiation mentioned above. This analysis supports the hypothesis that the rate of Xdv plasmid replication is limited by the frequency of a transcription event at the ori site which in turn is determined by the repressor level. However, the initiation event does not correlate with changes either in repressor concentration or in initiator concentration during the cell cycle. Perturbations in Segregation and Copy Number Control The model assumes that the plasmid copies are distributed in equal numbers at cell division since little is known about Xdv plasmid segregation. It is, however, possible that a newborn cell will not receive exactly No copies of a multicopy plasmid at division of a cell with 2N0 plasmids. To examine the responses of the copy number control function to perturbations in plasmid segregation, the model has been applied to calculate the response of the replication control system when daughter cells receive unequal numbers of plasmids. As an example, in Fig. 3 the changes in plasmid molecules with time are shown when the daughter cell receives only one plasmid copy at the time indicated by the arrow. Plas-
FIG. 3. Changes in the number of plasmid molecules during successive cell generations when a daughter cell receivesonly one plasmid copy at the cell division indicated by the arrow.
REPLICATION
159
mid segregation at subsequent cell divisions is assumed regular. The simulation shows that, during the first generation, the number of plasmid molecules increases significantly, and, after three to four generation times, the plasmid copy number returns to its steady-state value. This calculation indicates that, because of the response of the replication control function, abrupt perturbations in plasmid segregation do not yield permanent alterations in plasmid copy number unless the plasmids are lost completely at division. Previously two simple models for the replication control function have been proposed based on studies of other plasmids (Cullum and Broda, 1979; Nordstrom et al., 1980). One model hypothesizes that the copy number is always set to 2N, in all cells before cell division independent of the initial plasmid number (constant copy number model). The other model presumes that, if a cell contains 2N, plasmid copies at cell division and each daughter cell receives between 1 and 2N0 plasmid copies, exactly N copies are synthesized during a cell cycle regardless of initial copy number (constant initiation frequency model). The present model has been applied to calculate how the Xdv plasmid replication control system reacts to initial copy numbers from 1 to 2N. In Table 3, the calculated copy numbers for these three different models are compared. Since the parameter values listed in Table 2 are used, N in the other models is taken as 25 so that all three models have the same cyclic-state copy number. Simulation results obtained from the present model give values between the constant copy number model and the constant initiation frequency model. However, the constant initiation frequency model agrees better with the simulation results for Xdv plasmids. As discussed earlier, the repressor level determines the initiation frequency of replication by controlling the transcription frequency. When the newborn cell receives fewer copies than N, the repressor level then decreases, which results in an increase in replication initiations in subsequent
160
LEE AND BAILEY TABLE 3
COMPARISON
OF THREE DIFFERENT FUNCTION
REPLICATION MODELS FOR THE RESFQNSESOF REPLICATION TO PERTURBATIONS IN PLASMID SEGREGATION
CONTROL
Number of plasmid molecules at cell division (after one gen.) Number of plasmids per newborn cell (NO)
Constant copy number model”
Constant initiation frequency model”
Computer model b
1 5 10 20 25 30 40 45 49
50 50 50 50 50 50 50 50 50
26 30 35 45 50 55 65 70 74
33 35 39 46 50 54 62 66 70
a The number of initiation events per cell cycle, N, was taken as 25 for comparison. ’ The parameter values listed in Table 2 were used for simulation computation.
cell cycles until cyclic-state conditions are restored. The converse situation applies in the daughter cell receiving more than N plasmid copies as seen in Table 3. Regulation of plasmid replication depends upon several interacting molecular species and genetic loci. Consequently, consideration of perturbations in only one concentration-the plasmid content at birth-does not provide a complete description of the response of the replication regulation system to possible segregation irregularities at cell division. To explore more general perturbations, time trajectories of plasmid DNA concentration and repressor concentration have been calculated assuming different initial values for both concentrations. As before, it is assumed that all intracellular species partition regularly thereafter. The results of these simulations are shown in a phase plane diagram in Fig. 4. Each line is a trajectory of coexisting repressor and plasmid concentrations resulting from a particular choice of initial values of these concentrations. Time is a parameter along these curves, and the arrows indicate the direction of increasing time. For all perturbed initial states considered, the system trajectories, sometimes in
rather complex motions, converge after three or four generation times to the stable limit cycle (heavy closed curve) of the unperturbed
2.6 26 Reprer~~
3.0
3.2
Cwntrdion
3.4 Ol!O%
FIG. 4. Phase plane diagram showing transients in repressor and plasmid concentrations for tire wild-type Xdv replicon. Time is a parameter along each curve, increasing in the direction of the arrows. To facilitate presentation of trends, irregular motions caused by intermittent plasmid synthesis have been smoothed. The dotted region in the upper right-hand comer of the figure shows points along an unsmoothed trajectory.
KINETIC
MODEL
FOR
cyclic-state system. As in the preceding investigation of single component perturbations, the replication initiation control system compensates effectively for abrupt perturbations from the cyclic state provided the daughter cell receives at least one copy of the plasmid. DISCUSSION
In this paper, a mathematical model of Xdv plasmid replication has been developed based on the known molecular control mechanisms. According to the origin activation model employed here, the initiation frequency and timing of Xdv plasmid replication is determined by the transcription frequency at or near the ori site. This model successfully predicts the stable maintenance of plasmid copy number during the cell cycle and the plasmid and repressor level. It is often assumed (initiator accumulation model) that replication begins when a certain amount of an intracellular initiator protein has accumulated (Jacob et al., 1963), or, alternatively (inhibitor dilution model) when the concentration of an inhibitor protein is reduced below a threshold level by an increase in cell volume (Pritchard et al., 1969; Pritchard, 1978). Neither of these models has been tested by formulating the hypotheses involved in mathematical terms and calculating the corresponding replication behavior. In the present model for Xdv plasmid replication, the underlying hypotheses concerning molecular regulation have been expressed in mathematical form, allowing unambiguous calculation of the corresponding molecular interactions and their effects. The calculated maximum variation in repressor concentration is less than 2% during the cell cycle due to the autorepression action of cro repressor (Fig. 2). Under the autorepression system control, the dilution of repressor protein due to cell growth is readily compensated because a decrease in repressor concentration results in an immediate increase of transcription efficiency for repressor message (see Eqs. (2) and (6)). The initiator protein concentration
PLASMID
REPLICATION
161
also does not vary significantly during the cell cycle because the autorepressor and initiator genes belong to a single operon. The inhibitor dilution model also assumes that the initiation of replication would raise inhibitor concentration and reduce the probability of further initiations (Pritchard, 1978). However, if the plasmid replication follows the random selection model so that plasmid replication occurs throughout the cell cycle (see Fig. 2), each new round of plasmid replication does not greatly change the plasmid concentration and, accordingly, the synthesis of inhibitor message. Suppose that the newborn cell has 20 plasmid molecules in a cell. New replication events will increase the number of plasmid molecules to 2 1, which results in only a 5% increase in plasmid concentration. At least for the multicopy plasmid replicons, therefore, variations in inhibitor concentration between successive plasmid replication events would be too small to explain an on-off regulatory action based on differences in inhibitor level. The autorepressor model proposed by Sompayrac and Maaloe ( 1973), which includes the basic features of an autorepression system in conjunction with DNA replication, seems to be closest to the Xdv plasmid replication control mechanism. Sompayrac and Maaloe (1973) presumed, however, that a new round of replication is inhibited by the consumption of initiator protein. This model does not alone explain Xdv replicon function: the Xdv plasmid cannot replicate even in the presence of excess initiator protein unless sufficient origin activation has occurred. Besides this point, if plasmid synthesis occurs during the entire cell cycle in the random selection mode, many cycles of accumulation and consumption of initiator protein would be required according to the Sompayrac and Maaloe model. Such fluctuations in initiator protein concentration seem unlikely based on current information, although the possibility that some mechanism exists to deactivate or decompose the initiator protein rapidly after replication initiation cannot be completely excluded.
162
LEE
AND
Previous models have been proposed primarily to account for chromosomal DNA replication. When DNA replication occurs once and only once per generation as in the case of chromosomal replication at certain growth rates, the doubling of repressor or initiator genes at a discrete time would result in burst syntheses of regulatory gene products. Then the variations in regulatory elements during the cell cycle may be enough to account for the initiation of DNA replication based on the changes in regulatory protein concentrations. However, density shift data for plasmids are in better agreement with the random selection model of replication (Rownd, 1969; Bazaral and Helinski, 1970; Matsubara and Mukai, 1975; Gustafsson and Nordstrom, 1975; Gustafsson et al., 1978). If plasmid replication follows the random selection model, it is unlikely that previous conceptual models for chromosome replication can make quantitative predictions about the frequency and timing of plasmid replication initiation. In connection with this discussion, it is of interest to note the experimental results of Leonard et al. (1982). They found a fundamental difference between replication control of the chromosome and the oriC-containing plasmid (minichromosome) in synchronously growing populations of E. coli B/r, although the reason for such a difference is not apparent. Their experimental data shows that chromosome replication begins at a specific time during the cell cycle while the minichromosome replicates throughout the cell cycle. The patterns of radioactivity incorporation into chromosome and minichromosome are very close to the simulation results shown in Fig. 2. In view of the work of Leonard et al. ( 1982), it appears that the mode of replication of extrachromosomal DNA differs from that of the chromosome. It has been reported that most naturally occurring plasmids are maintained stably in host cells. On this basis, precise regulation of initiation of DNA replication and regular segregation of plasmids into daughter cells at cell division is sometimes inferred. The present
BAILEY
model for Xdv plasmid replication shows, however, that the molecular replication initiation control system rapidly corrects perturbations arising from irregular segregation. Consequently, a plasmid may maintain a stable copy number although segregation is not regular. Further analysis of this property requires a different, stochastic modeling approach which includes the possibility of irregular segregation in successive cell divisions. It is clear that improved understanding of the relative influences of replication controls and segregation properties on overall plasmid stability will require experimental methods with more sensitivity and resolution than is provided by conventional tests for a plasmid selection marker. In this paper the general behavior of the replication control function for the Xdv plasmid replicon has been described using a mathematical model. Further specific tests of the present model and extensions to analyze mutant plasmids are provided in a companion paper (Lee and Bailey, 1984b). The practice of formulating hypotheses in mathematical form and utilizing the resulting model to design experimental tests of the hypotheses and to correlate and organize data has proven invaluable in many venues of science. With continuing refinement of experimental methods in molecular biology and rapid advances in computer technology, such syntheses of experimental ingenuity and mathematical structure should contribute to new discoveries and advances in molecular biology and biotechnology. APPENDIX:
NOMENCLATURE
a, b, c constants defined in Eqs. (3)-(5) f termination efficiency at t,l G number of plasmid molecules per cell [G] plasmid DNA concentration I initiator protein [I] concentration of initiator protein rate conkO, overall transcription stant
KINETIC
MODEL
FOR PLASMID
decay rate constant for initiator mRNA k!, decay rate constant for repressor mRNA kz overall translation rate constant k!, decay rate constant for initiator protein k$, decay rate constant for repressor protein K, binding affinity of cro repressor at OR1 K2 binding affinity of cro repressor at 0,2 K3 binding affinity of cro repressor at 0,s KA association constant in Eq. (17) KS association constant in Eq. (18) K0 constant defined in Eq. (23) m an integer [mRNA], mRNA ”concentration of initiator gene bRNAlix mRNA concentration of repressor gene n an integer or the number of generations N number of plasmid replication initiation events during the cell cycle No number of plasmid molecules per newborn cell Avogadro’s number concentration of plasmid origin or-i* activated origin [ori*] concentration of activated origin replication proteins of host cell [P; concentration of replication proteins of host cell repressor protein concentration of repressor protein REP replication complex [REP] concentration of replication complex of repliIREPL ___ critical concentration cation complex t time tj* time of the jth replication initiation
V V. Wi
k!,
ro;
REPLICATION
163
cell volume cell volume of newborn cell statistical weights defined in Table 1
Greek v
transcription efficiency defined in Eq. (2) p cellular specific growth rate 7 doubling time 0 dimensionless degree of origin activation defined in Eq. (15) Subscript 0
total concentration ACKNOWLEDGMENTS
This work was supported by the National Science Foundation, the Energy Conversion and Utilization Technology (ECUT) program of the Department of Energy, and the Korea Science and Engineering Foundation (KOSEF). Discussions with Judith L. Campbell contrib uted to this research. REFERENCES M., AND HELINSKI, D. R. (1970). Replication of a bacterial plasmid and an episome in Escherichia
BAZARAL,
coli. Biochemistry
9, 399-406.
BERG, D. E. (1974). Genes from phage X essential for Xdv plasmids. Virology 62, 224-233. BRAMMER, W. J. (1982). Vectors based on bacteriophage lambda. In “Genetic Engineering 3” (R. Williamson, ed.), pp. 53-8 1. Academic Press, New York. CANTOR, C. R., AND SCHIMMEL, P. R. (1980). “Biophysical Chemistry Part III. The Behavior of Biological Macromolecules,” p, 1245. Freeman, San Francisco. CONRAD, S. E., AND CAMPBELL, J. L. (1979). Role of plasmid-coded RNA and ribonuclease III in plasmid DNA replication. Cell 18, 6 1-7 1. COOPER, S., AND HELMSTETTER, C. E. ( 1968). Chromosome replication and the division cycleofEscherichia coli B/r. J. Mol. Biol. 31, 519-540. CULLUM, J., AND BRODA, P. (1979). Rate of segregation due to plasmid incompatability. Genet. Res. 33, 61-^ IY. DOVE, W., KNOKUCHI,H., AND STEVENS,W. (197 1). Rep lication control in phage lambda. In “The Bacteriophage Lambda” (A. D. Hershey, cd.), pp. 747-77 1. Cold Spring Harbor Laboratory, Cold Spring Harbor, N. Y. FREDRICKSON,A. G. (1976). Formulation of structured growth models. Biotechnol. Bioeng. 18, 148 1- 1486. FURTH, M. E., YATES, J. L., AND DOVE, W. F. (1979). Positive and negative control of bacteriophage lambda DNA replication. Cold Spring Harbor Synzp. @ant. Biol. 43, 147-153.
164
LEE
AND
GUSTAF~~ON, P., AND NORDSTROM, J. (1975). Random replication of the stringent plasmid R 1 in Escherichia coli K-12. J. Bacterial. 123, 443-448. GUSTAFSSON,P., NORDSTROM, K., AND PERRAM, J. W. ( 1978). Selection and timing of replication of plasmids Rl drd-19 and FIac in Escherichia coli. Plasmid 1, 187-203. HERSHEY,D. (ed.) (197 1). “The Bacteriophage Lambda.” Cold Spring Harbor Laboratory, Cold Spring Harbor, N. Y. HOBOM, G. (1981). Replication signals in prokaryotic DNA. Curr. Top Microbial. Immunol. 94/95,93-142. HOBOM, G., GROSSCHEDE,R., LUSKY, M. SCHERER,G., SCHWARZ, E., AND KOSSEL, H. (1979). Functional analysis of the replicator structure of lambdoid bacteriophage DNA. Cold Spring Harbor Symp. Quant. Biol. 43, 165-178. JACOB,F., BRENNER, S., AND CUZIN, F. (1963). On the regulation of DNA replication in bacteria. Cold Spring Harbor
Symp.
Quant.
Biol.
28, 329-348.
JOHNSON,A. D., MEYER, B. J., AND I%ASHNE, M. (1978). Mechanism of action of the cro protein of bacteriophage X. Proc. Natl. Acad.
Sci. USA 75, 1783-1787.
JOHNSON,A. D., MEYER, B. J., AND I?ASHNE, M. (1979). Interactions between DNA-bound repressors govern regulation by the X phage repressor. Proc. Natl. Acad. Sci. LISA 76, 5061-5065. KOLTER, R., AND HELINSKI, D. R. (1979). Regulation of initiation of DNA replication. Annu. Rev. Genet. 13, 355-391. KUYPERS,B., REISER,W., AND KLEI?$ A. (1980). Cloning of the replication gene 0 of E. coli bacteriophage lambda and its expression under the control of the luc promoter. Gene 10, 195-203. LAUER, G., PASTRANA, P., SHERLEY, J., AND P~ASHNE, M. ( 198 1). Construction of overproducers of the bacteriophage 434 repressor and cro proteins. J. Mol. Appl. Genet. 1, 139-147. LEE, S. B., AND BAILEY, J. E. (1984a). A mathematical model for Xdv plasmid replication. Analysis of copy number mutants. Plasmid 11, 166-177. LEE, S. B., AND BAILEY, J. E. (1984b). Analysis of growth rate effectson productivity of recombinant Escherichia coli populations using molecular mechanism models. Biotechnol.
Bioeng.
26, 66-73.
LEONARD, A. C., HUCUL, J. A., AND HELMSTETTER, C. E. (1982). Kinetics of minichromosome replication in Escherichia coli B/r. J. Bacterial. 149, 499-507. LIPINSKA, B., PODHAJSKA, A., AND TAYLOR, K. (1980). Synthesis and decay of X DNA replication proteins in minicells. B&hem. Biophys. Res. Commun. 92, 120126. MATSUBARA, K. (1976). Genetic structure of a replicon of plasmid Xdv. J. Mol. Biol. 102, 427-439. MATSLJBARA, K. (1981). Replication control system in lambda dv. Plasmid 5, 32-52. MATSUBARA, K., AND KAISER, A. D. (1968). Xdv: An
BAILEY
autonomously replicating DNA fragment. Co/d Spring Harbor
Symp.
Quant.
Biol.
33, 769-775.
MATSUBARA, K., AND MUKAI, T. (1975). Mode of replication of plasmid hdv. J. B&hem. Japan 17, 373382. MATSUBARA, K., TAKAGI, Y., AND MUKAI, T. (1975). In vitro construction of different oligomeric forms of Xdv DNA and studies on their transforming activities. J. Viral. 16, 479-485. NORDSTROM, K., MOLIN,
S., AND AAGAARD-HANSEN, H. (1980). Partitioning of plasmid Rl in Escherichia co/i. I. Kinetics of loss of plasmid derivatives deleted of the par region. Plasmid 4, 2 15-227. NOVICK, R. P., AND HOPPENSTEADT, F. C. (1978). On plasmid incompatibility. Plasmid 1, 421-434. OKA, A., NOMURA, N., MORITA, M., SUGISAKA, H., SUGIMOTO, K., AND TAKANAMI, M. (1979). Nucleotide sequence of small ColEl derivatives: Structure of the regions essential for autonomous replication and colicin El immunity. Mol. Gen. Genet. 172, 151-159. PATO, M. L., AND VON MEYENBURG, K. (1970). Residual RNA synthesis in Escherichia coli after inhibition of initiation of transcription by rifampicin. Cold Spring Harbor
Symp.
Quant.
Biol.
35, 497-504.
PRITCHARD,R. H., BARTH, P. T., AND COLLINS, J. (1969). Control of DNA synthesis in bacteria. Symp. Sot. Gen. Microbial.
19, 263-297.
PRITCHARD, R. H. (1978). Control of DNA replication in bacteria. In “DNA Synthesis-Present and Future” (I. Molineux and M. Kohiyama, eds.) pp. l-26. Plenum, New York. ROSEN,J., RYDER, T., INOKUCHI, H., OHTSUBO, H., AND OHTSUBO, E. (1980). Genes and sites involved in replication and incompatibility of an RlOO plasmid derivative based on nucleotide sequence analysis. Mol. Gen. Genet.
179, 527-537.
ROSEN,J., RYDER, T., OHTXJBO, H., AND OHTSUEQ E. (198 1). Role of RNA transcripts in replication incompatibility and copy number control in antibiotic resistance plasmid derivatives. Nature (London) 290, 794797. ROSENBERG,M., COURT, D., SHIMATAKE, H., BRADY, C., AND WULFF, D. L. (1978). The relationship between function and DNA sequence in an intercistronic regulatory region in phage X. Nature (London) 272,414423. ROWSURY,R. J. (1977). Bacterial plasmids with particular reference to their replication and transfer properties. Progr. Biophys. Mol. Biol. 31, 27 l-3 17. ROWND, R. (1969). Replication of a bacterial episome under relaxed control. J. Mol. Biol. 44, 387-402. SOMPAYRAC,L., AND MAAL.@E,0. (1973). Autorepressor model for control of DNA replication. Nature New Biol. 241, 133-135. STOUGAARD,P., MOLIN, S., AND NORDGTRISM,K. (1981). RNAs involved in copy-number control and incom-
KINETIC
MODEL
FOR PLASMID
patibility of plasmid Rl. Proc. Nat!. Acad. Sci. USA 78,6008-6012. STOUGAARD, P., MOLIN, S., NORDSTROM, K., AND HANSEN,F. G. ( 198 1). The nucleotide sequence of the rep lication control region of the resistance plasmid Rl drd19. Mol. Gen. Genet. 181, 116-122. STUITJE, A. R., SPELT, C. E., VELTKAMP, E., AND NIJKAMP, H. J. J. (198 1). Identification of mutations affecting replication control of plasmid CloDF13. Nature @ondon) 290,264-267.
SZYBALSKI, E. H., AND SZYBALSKI, W. (1979). A comprehensive molecular map of bacteriophage lambda. Gene 7,2 17-270. TAKEDA, Y., FOLKMANIS, A., AND ECHO=, H. (1977). Cro regulatory protein specified by bacteriophage X. J. Biol. Chem. 252, 6177-6183.
REPLICATION
165
TOMIZAWA, J., ITOH, T., SEUER, G., AND SOM, T. (198 1). Inhibition of ColE 1 RNA primer formation by a plasmid-specified small RNA. Proc. Nat/. Acad. Sci. USA 78, 1421-1425. WILLIAMS, B. G., AND BLATTNER, F. R. (1980). Bacteriophage X vectors for DNA cloning. In “Genetic Engineering: Principles and Methods” (J. K. Setlow and A. Hollaender, eds.), Vol. 2, pp. 201-281. Plenum, New York. WOLD, M. S., MALLORY, J. B., ROBERTS, J. D., LEBOWITZ, J. H., AND MCMACK, R. (1982). Initiation of bacteriophage X DNA replication in vitro with purified X replication proteins. Proc. Natl. Acad. Sci. USA 79, 6176-6180. WYATT, W. W., AND INOKUCHI, H. (1974). Stability of lambda 0 and P replication functions. Virology 58, 313-315.
11, 151-165
(1984)
A Mathematical Model for Xdv Plasmid Replication: Analysis of Wild-Type Plasmid SUN
BOK
LEE’
AND
JAMES
E. BAILEY~
Department of Chemical Engineering, California Institute of Technology, Pasadena, Calijbrnia 91125 Received
April
12, 1983; revised
December
2, 1983
A mathematical model for Xdv plasmid replication in a growing single cell of Escherichia coli has been formulated and solved numerically. Quantitative description of the molecular control mechanism for initiation of Xdv replication presumes regulatory functions of repressor and initiator proteins and transcriptional activation of the origin region. Random selection of a single plasmid for activation and replication is assumed, as is regular plasmid segregation to daughter cells. The model is capable of simulating the periodic changes in each regulatory element and the plasmid copy number during the cell cycle. The calculated average copy number, repressor concentration, and timing of plasmid replication agree well with experimental data. The simulated Mv plasmid replication rate is controlled primarily by transcription frequency. Initiation of plasmid replication is not related to variations in the levels of repressor or initiator proteins during the cell cycle. Simulation studies of perturbations in plasmid and repressor segregation indicate that replication regulation of the Xdv plasmid compensates to readjust copy number to normal values in a few generations. Implications of these studies relative to the molecular mechanisms of replication control are discussed.
in view of the extensive genetic and biochemical information available on the parent X phage (Hershey, 1971; Szybalski and Szybalski, 1979). Analysis of the Xdv plasmid replicon can also give a basis for understanding the phage Xreplicon which is now widely used as a cloning vector (Williams and Blattner, 1980; Brammer, 1981). The purpose of this paper is to describe a mathematical model for the replication control of Xdv plasmid in a single E. coli cell. Previously, several conceptual models for control of initiation of DNA synthesis have been proposed (Kolter and Helinski, 1979; Jacob et al., 1963; Pritchard et al., 1969; Sompayrac and Maalee, 1973). However, these hypotheseshave not yet been tested by restating them in mathematical form and determining whether the resulting equations properly describe plasmid copy number, stable maintenance of plasmid copy number, and influences of mutations in the replication regulation sequences.Such questions are considered here and in an accompanying paper (Lee and Bailey, 1984a) for the Xdv system.
Regulation of initiation of plasmid DNA replication is one of the crucial requirements for stable maintenance of plasmid copy number (Rowbury, 1977). Although the molecular mechanisms of replication control have not yet been completely elucidated, fine structural mapping of replication regions and identification of geneproducts involved in replication regulation have been carried out for several plasmids including ColEl (Conrad and Campbell, 1979; Oka et al., 1979; Tomizawa et al. 1981) CloDF13 (Stuitje et al., 1981) RI (Stougaard et al., 198 1; Stougaard et al., 1981), RlOO (Rosen et al., 1980, 1981), and Xdv (Berg, 1974; Matsubara, 1976, 1981). Among these, the plasmid Xdv, a deletion derivative of phage X which replicates autonomously in Escherichia coli (Matsubara and Kaiser, 1968) is a particularly useful model system for analysisof regulation of replication
’ On leave from the Korea Advanced and Technology, Seoul, Korea. ’ To whom correspondence should
Institute
of Science
be addressed. 151
0147-619X/84
$3.00
Copyright 0 1984 by Academic Press, Inc. All rights of reproduction in any form reserved.
152
LEE AND BAILEY
SYSTEM
DESCRIPTION
Formulation of the mathematical model for Xdv replication requires quantitative representations of the interactions between the pertinent regulatory elements and regulatory sites. Since details on current knowledge of this regulation mechanism have been extensively discussed elsewhere (Berg, 1974; Matsubara, 1976, 1981; Hershey, 1971; Szybalski and Szybalski, 1979) the following short summary will only highlight previous experimental information and hypotheses which are employed in formulation of the mathematical model. The minimal replicon of Xdv plasmid (Fig. 1) consists of the following genes and genetic sites: the autorepressor region, which contains the promoter-operator (PROR), autorepressor gene (cro) and terminator (fRZ), and the origin region which carries an initiator gene (0), replication origin (ori), and another initiator gene (P). Some of the Xdv isolates carry, in addition, the leftward promoter-operator (PLOL), lambda repressor gene (cl), and, rarely, the N or Q gene which can affect transcriptional activity. However, in most cases, this second set of genes is inactive due to mutations or deletions. An additional gene, oop, and genetic site, ice, located between t,l and the 0 gene in the hdv replicon, are not considered in the present model since the roles of these sequences in replication of X DNA are not yet clear (see Hobom et al., 1979; Hobom, 1981). The promoter-operator for rightward transcription (PRO,) is the site where RNA polymerase and cro repressor can bind, and at this site the initiation frequency of transcription is determined. The rightward operator consists of three tandem sequences at which cro repressor (designated R in Fig. 1) binds: the sequences are termed ORl, OR2, and OR3. The rightward promoter, PR, overlaps with OR2 and 0,2 as shown in Fig. 1. The cro repressor binds preferentially to 0,3 and, at higher concentrations, it turns off transcription of its own gene, cro, by binding at the O,l and 0~2 sites. The cro repressor is thus subject to negative autogenous control or autorepression. About
FIG. I. The genetic structure of Xdv plasmid replicon. Details are described in the text.
80% of the transcription from PRO, terminates at tRl, the site of transcriptional termination, in the presence of bacterial rho factor (Rosenberg et al., 1978). Transcription of the origin region initiated at PRO, and passing t,l leads to activation of the ori site and to coordinated production of two initiator proteins, 0 and P protein. The gene products 0 and P, long recognized as absolutely essential for initiation of replication, act cooperatively and positively in the initiation process although their exact modes of action are not yet clear. There is evidence suggesting that the 0, P, and several bacterial replication proteins such as the dnaB and dnaK gene products form a complex which acts at the ori site to initiate replication (Furth, Yates and Dove, 1979; Hobom et al., 1979). It has been reported that the 0 protein (designated I in Fig. 1) is unstable in vivo while the P protein is apparently much more stable (Lipinska et al., 1980; Kuypers et al., 1980). In addition to the requirements for initiation and replication proteins and for an intact ori site, replication initiation of hdv plasmid as well as X phage has been shown to depend upon a local transcription event that occurs at or near the origin of replication (so-called transcription activation). It has been demonstrated that replication is inhibited if transcription of the ori region is blocked, even though all the initiator proteins are provided in tram (Dove et al, 197 1; see also Wold et al., 1982). Whether this origin transcription provides the initiator system with a primer RNA for starting DNA synthesis or whether it converts the DNA ori region into an active
KINETIC
MODEL
FOR PLASMID
structure that is competent to interact with initiators has not yet been resolved (see Hoborn, 198 1). It is also possible that these two events can occur simultaneously. In this paper, the second hypothesis is employed. The model can be readily modified to reflect future information on this question. Besides the copy number control mechanisms which determine the timing and initiation frequency of DNA replication, models of replication and segregation are required for the description of the plasmid life cycle. Three possible modes of replication of plasmids by which a constant average number of plasmids per cell can be maintained during cell growth have been proposed by Rownd (1969): (1) Each plasmid copy replicates only once during a cell generation; (2) a master copy of plasmids replicates N times during each generation to produce a doubling of the number of plasmid copies, and (3) the N copies of plasmid exist in a pool and one of the plasmid copies is selected from the pool at random for replication. This process of random selection is repeated N times during each generation, until there has been a doubling of the number of plasmids. Using density-shift experiments, NRI (Rownd, 1969), ColEl (Bazaral and Helinski, 1970) RI (Gustafsson and Nordstrom, 1975), F’ (Gustafsson et al., 1978), and Xdv (Matsubara and Mukai, 1975) plasmids have been shown to follow the random selection mode of replication. Two models for segregation have been proposed (Novick and Hoppensteadt, 1978; Cullum and Broda, 1979): (1) An equal number of plasmid copies are distributed to each daughter cell (regular segregation) and (2) plasmids are partitioned at random on cell division (random segregation). A third model in which some fixed number of plasmids are segregated regularly with the remainder randomly segregated can also be envisioned. Although Xdv probably does not have equipartitioning, for the sake of simplicity, the mathematical model will presume regular segregation of all plasmids as a base case, and the effects of perturbations from regular seg-
REPLICATION
153
regation will then be explored using the mathematical model. The copy number control function can be affected by other factors including plasmid influences on host cell growth rate and oligomerization of the plasmid. The presence of multicopy plasmids may affect the growth rate of the host cell which, in turn, influences the replication of plasmid DNA. In the case of hdv plasmid, significant inhibition of host cell growth apparently does not occur. Matsubara and Kaiser (1968) have reported that the growth rate (as measured by total cell mass or the number of viable cells) of isogenic bacteria with and without Xdv 1 plasmid are equal. Naturally occurring higher oligomers of Xdv plasmid have been described (see Matsubara, 198 1). For example, Matsubara et al. (1975) reported that the monomer genome content of dimeric Xdvl was about 40% higher than the monomeric Xdvl genome content when pure monomer and dimer carriers were isolated and replicated independently. In the present model, which concerns the primary features of the plasmid replication control system, formation of oligomeric Xdv plasmid and concomitant effects on copy number are not considered. Such effects will be described in an accompanying paper (Lee and Bailey, 1984a). KINETIC MODEL
The model is based on the following assumptions: (1) The Xdv plasmid consists of genes between PRO, and P, and all plasmids exist as monomeric forms. (2) The 0 protein acts as a limiting initiator due to its instability. The level of P protein is consequently assumed to be relatively unimportant and is not considered. (3) Replication follows the random selection mode. (4) The ori site of the single plasmid selected for replication is activated by transcription. The initiator protein forms a complex with the activated origin. Activation of the replication complex to a certain critical value is required for the initiation of replication. (5) Segregation is by equal number
154
LEE
AND
partitioning at cell division. (6) The host cell grows in volume exponentially, and the plasmids do not affect the host cell growth rate. First the copy number control function of the Xdv plasmid is considered. The transcription efficiency at PROR is determined by the interactions between RNA polymerase and the cro repressor. Since the promoter overlaps with ORI and 0,2 (Fig. l), repressor bound either to ORl, 0,2, or to both sites will repress PR . The probability that the operon is not repressed can be determined using standard methods of statistical thermodynamics (Cantor and Schimmel, 1980). In Table 1, eight possible configurations of cro repressor at 0, and corresponding statistical weights are listed. If the transcriptional efficiency, 1, is assumed to be proportional to the fraction of PRO, templates at which RNA polymerase can bind (state 0 and state 3 in Table l), 7 can be computed by comparing the relative probability of states 0 and 3 to the sum of the relative probabilities of all possible states. Thus n is given by wo
+
w3
17’7
. c i=o
(1)
w
Assuming that the concentration of operator is much lower than that of repressor (a usual
BAILEY
case) and introducing Eq. (1) gives
statistical weights into
1 +
K3Flo
‘I = 1 + a[Rlo + b[R]; + c[R]; ’ where a=K,+K2+K3,
(3)
b = K,K2 + KzK3 + K3Kl,
(4)
c = K,KzK3.
(5)
K, , K2, and K3 represent the binding affinity of cro repressor at OJ, OR2, and OR3, respectively, and [RIO represents the total repressor concentration. (Equation (2) can also be derived by conventional multiple equilibria calculations, although at the expense of greater algebraic effort.) The rate of transcription of a gene at time t can be expressed as the product of overall transcription rate constant, transcription efficiency, and the gene concentration. Similar expressions for the rate of translation can be formulated. If the mRNA and protein are inactivated by first-order decay processes, the synthesis of cro mRNA and protein gene products are described by
d[mRN& dt
= k$q[G]
- k-,[mRNAIR
- dmRN&, TABLE
dt
- k~p,[Rlo- /ARIo,
Configuration” Statistical
(i)
OR3
OR2
0,’
+
+
5
+
6
+
1
+
(1 A(+) indicates by a cro repressor.
weight
(W,)
1
0 1
2 3 4
f&WI
&lRl
+
KdRl +
+
+ +
+ that
(6)
4Rlo = k$flRNA]R
1
POSSIBLE CONFIGURATIONS OF cro REPRESSOR AT OR AND THE STATISTICAL WEIGHTS CORRESKINDING TO EACH STATE
State
(2)
the corresponding
+
K&lRlz &&lRl* K,&lR12 KX&,lRl’ site is occupied
(7)
where p is the specific growth rate of the host cell. The intracellular concentrations of mRNA and protein, which are now expressed as moles per unit volume of cell, decrease as a consequence of two processes, degradation and dilution by cell growth (Fredrickson, 1974). The concentration of plasmid DNA at time t, [G], is expressed as
where G is the number of plasmid molecules per cell, V is the cell volume at time t, and
KINETIC
MODEL
FOR
PLASMID
NA is Avogadro’s number. For exponential single-cell growth (Pritchard et al., 1969), the change in cell volume during one cell generation can be determined from the following equation dV -=pv,v=v, dt at
1,2,...),
t=m(rz=
155
REPLICATION
where
e _ m*10 [oclo * Since it is assumed that only one plasmid is selected for replication at a time by the random selection mode, [orilo can be determined by the equation (cf. Eq. (8))
(9)
where V0 and 7 represent the cell volume at birth and the doubling time of the host cell, respectively. Transcription of the 0 gene is initiated at PRO, and requires readthrough of an intervening terminator t,l. If the termination efficiency at t,l is defined asf(0
Replication complexes are presumed to be formed by binding of initiator proteins at the activated ori site. In turn, the level of activated replication complexes determines the replication initiation frequency. Hence the following simplest scheme describing the interactions of initiator protein, I, host replication proteins designated Pr, and the activated origin, on*, is used to describe the formation of activated replication complexes, REP: KA I + Pr * 1.Pr, (17) orz* + 1. Pr -’ REP,
d[mRNAlI = k$z( 1 - f)[G] - k-,[mRNA]z dt
where KA is the equilibrium
- ,4mRNAlI, dt
- ~$410 - A.410. ( 11) As mentioned before, transcription of the origin region of the single plasmid selected for replication results in activation of the ori site. Since activation of the origin region is determined by the same transcription frequency as that for 0 protein, the formation of activated origin, [o~z*]~, can be described as ~d[omo = ko,?j(l -f)[orz] dt
- p[orz*]o.
host protein, and KB is the equilibrium association constant for binding of protein complexes to the activated origin. Mass balances for I, Pr, and orz* give [Ilo = [I] + [I - Pr] + [REP]
(19)
[Pr], = [Pr] + [I. Pr] + [REP]
(20)
[orz*lo = [orz*] + [REP].
(12)
(13)
so that Eq. (12) may be rewritten in the form dfl
--$ = eldl
-J-Ml - 0) - CL4 (14)
(21)
Assuming that [Pr] 9 [I +Pr] + [REP] and [REP] < [I] + [I. Pr], Eqs. (17) through (21) imply [REP] = K,K,[I][Pr][ori*]
= ~&Morilo 1 + KoDlo ’
Since the origin regions to be transcribed are either activated or not
[ori] = [ori] - [orP]o,
association con-
(10) stant for the binding of initiator protein and
4110= ki[mRNA]z
-
(18)
(22)
where K
0
=
K&[prlo 1 +KA[~]o'
(23)
K. in Eq. (22) can be treated as constant pro-
156
LEE
AND
vided that a constant fraction of host replication proteins is involved in replication of plasmid DNA. Initiation of plasmid replication and segregation are considered next. It is assumed that plasmid replication initiates when the replication complex level of activation [REP] reaches a critical value [REP],,. If replication of Xdv plasmid DNA occurs at chromosomal replication fork rates (Cooper and Helmstetter, 1968), replication of the entire Xdv molecule will be completed in 0.05 min. Hence it seems reasonable to assume, relative to the cell cycle time, that replication of the selected plasmid is essentially instantaneous once initiation occurs. Thus, if G(0) is the number of plasmid molecules in a newborn cell, the number of plasmids in that cell at time t is given by
BAILEY
no change in component concentrations at cell division. As an initial condition it is assumed that one plasmid molecule is introduced into an exponentially growing host cell: G,(O) = 1.
(28)
DETERMINATION OF MODEL PARAMETERS
All calculations have been performed for doubling time r = 1 h. Under these conditions, the overall transcription rate constant k:, the overall translation rate constant ki, and the initial cell volume V0 have been estimated to be 4.26 M mRNA/M DNA-mitt, 13.8 M protein/M mRNA-min, and 0.5 X IO-l5 liter, respectively (Lee and Barley, 1983). It is assumed G(t) = G(0) + m, here in the absence of more detailed data that (24) these cell-average transcription and translation where m is determined from parameters adequately approximate the parameters for the particular transcription and tget
KINETIC
MODEL
FOR PLASMID
and Johnson et al. (1978, 1979) the binding affinity of cro repressor on each operator was estimated. They reported that the binding affinity is in the range of 108-1010 M-’ and the relative dissociation constants (inverse of binding affinity) on a wild-type operator are OR1:OR20R3 = 8:8: 1 under conditions which approximate physiological condition. The termination efficiency at tRl, f; was assumed to be 0.8 according to Rosenberg et al. (1978). The values of K. and [REP],, were taken as adjustable parameters in the absence of experimental data for their evaluation. These values were adjusted so that the calculated copy number of wild-type Xdv plasmid agrees with experimental data. All of the model parameters for wild-type Xdv plasmid are summarized in Table 2. SIMULATION
Initiation
RESULTS
Frequency of Plasmid
Replication
It is not possible to integrate the simultaneous nonlinear differential equations above
157
REPLICATION
for [mRNA]a, [RIO, [mRNA]r, [Ilo, and 6’ analytically. Consequently, a numerical procedure (a variable time-step Runge-Kutta algorithm with predictor-corrector) was used to accomplish the integration on a computer. This method calculates the solution marching ahead in time in small time increments: values of [mRNAIR, [RI,, [mRNA]r, [Ilo, and 0 at the next time-step are determined from values of these quantities at earlier times using the differential equations of the model. As the integration advances, the [REP] value is calculated simultaneously. Cellular plasmid content is increased by one copy and the transcriptional activation level is reset to zero when [REP] reaches [REP],, as described above. As indicated in Eq. (28) above, the calculation begins with a single plasmid in the cell. The model parameter values listed in Table 2 are used in all of the computations reported here. The condition called here “cyclic state” is reached when each calculated intracellular concentration at cell division is equal to the concentration at the beginning of that cell cycle calculation.
TABLE 2 KINETIC PARAMETERS USED FOR WILD-TYPE Xdv PLASMID Parameters
k’,
Values”
Reference
0.5 X lo-‘r liter
Lee and Bailey (1984b)
4.26 M mRNA/M DNA-min
Lee and Bailey (1984b)
13.8 M protein/M mRNA-min
Lee and Bailey ( 1984b)
0.46 min-’
Pato and von Meyenburg (1975)
0.01 min-’
Matsubara (198 1) Wyatt and Inokuchi ( 1974) Lipinski et al. (1980) I Kuypers et al. (1980)
0.14 min-’ 1.25 X lo* 1.25 x lo* 1.0 x lo9
M-’ M-’
1
M-’
Rosenberg et al. (1978)
0.80 2.4
Takeda et al. (1977) Johnson et al. (1978, 1979)
X lOa M-’
2.0 x lo-‘*
M
-
’ The values of kO,, kz, and V. are those at r = 1 h. Details are given elsewhere (Lee and Bailey, 1984b).
158
LEE AND BAILEY
The trajectories of each intracellular concentration at cyclic state are given in Fig. 2. The number of plasmid DNA molecules and chromosomal DNA content in a single E. coli cell are also illustrated in Fig. 2. Chromosomal DNA contents were calculated using CooperHelmstetter relationships assuming that the time required for a round of chromosome replication is 40 min and the time interval between the termination of replication and subsequent cell division is 20 min (Cooper and Helmstetter, 1968). Simulation results show about 3 X 1O-’ M of cro repressor, which corresponds to an average of 130 monomers per cell, is present in a cell carrying the wildtype Xdv plasmid. This value is comparable to the available experimental data (cf. Lauer et al., 198 1). The following additional features of the simulations are noteworthy: (1) the initiation frequency of replication at cyclic state is equal to the number of plasmid molecules of the newborn cell, (2) there is an approximately equal time interval between successive plasmid replications, and (3) initiation of plasmid replication is not related to variations in the levels of repressor or initiator proteins during the cell cycle. Figure 2 also shows that the replication pattern of plasmid DNA is different from that of chromosomal DNA. A gene
o-0
Time (mid RG. 2. Calculated trajectories of the numbers of plasmid and chromosomal DNA molecules and the autorepressor and initiator protein concentrations during the cell cycle. Arrows on the curves indicate pertinent coordinate axes.
encoded in plasmid DNA could be increased in number throughout the cell cycle while a gene encoded in the chromosome would be doubled at a specific point in the cell cycle. In this model, it is assumed that plasmid replication follows a random selection mode. While density shift experiments have shown Xdv plasmids are selected randomly for replication (Matsubara and Mukai, 1975) these data do not directly prove that the process of random selection is repeated until there has been doubling of the number of plasmids during each generation. In other words, the question arises whether the replication control mechanism determines the initiation frequency of replication so that the number of plasmid molecules at cell division reaches exactly double the number of plasmid copies of the newborn cell. In connection with this discussion, it should be noted that the plasmid copies increase one by one as described in Eq. (24). The simulation results in Fig. 2 show that cyclic-state conditions are achieved for this model: the number of plasmid molecules at cell division is exactly twice that of newborn ceils, which means that the frequency of initiation during each generation (N) is equal to the number of plasmid molecules of the newborn cell (No). This result is consistent with experimental data for Rl and F’ plasmids (Gustafsson et al., 1978). In density-shift experiments, it was also observed that the time interval between successive replications is approximately equal to l/ Nth of a doubling time, where Nis the number of plasmid replications (Rownd, 1969; Gustafsson et al., 1978). The simulation results agree well with this experimental observation. As can be seen in Fig. 2, the time intervals between initiation events in the simulation are approximately equal but increase slightly as the cell age increases. The simulation results show that at cyclic state both repressor and initiator concentration remain nearly constant during the cell cycle (Fig. 2). Since the concentrations of repressor and initiator do not vary significantly during the cell cycle, the rate of origin activation is
KINETIC
MODEL
FOR PLASMID
kept nearly constant throughout the cell cycle due to constant 7 values (see Eqs. (2) and ( 14)). This causes the regular intervals between replication initiation mentioned above. This analysis supports the hypothesis that the rate of Xdv plasmid replication is limited by the frequency of a transcription event at the ori site which in turn is determined by the repressor level. However, the initiation event does not correlate with changes either in repressor concentration or in initiator concentration during the cell cycle. Perturbations in Segregation and Copy Number Control The model assumes that the plasmid copies are distributed in equal numbers at cell division since little is known about Xdv plasmid segregation. It is, however, possible that a newborn cell will not receive exactly No copies of a multicopy plasmid at division of a cell with 2N0 plasmids. To examine the responses of the copy number control function to perturbations in plasmid segregation, the model has been applied to calculate the response of the replication control system when daughter cells receive unequal numbers of plasmids. As an example, in Fig. 3 the changes in plasmid molecules with time are shown when the daughter cell receives only one plasmid copy at the time indicated by the arrow. Plas-
FIG. 3. Changes in the number of plasmid molecules during successive cell generations when a daughter cell receivesonly one plasmid copy at the cell division indicated by the arrow.
REPLICATION
159
mid segregation at subsequent cell divisions is assumed regular. The simulation shows that, during the first generation, the number of plasmid molecules increases significantly, and, after three to four generation times, the plasmid copy number returns to its steady-state value. This calculation indicates that, because of the response of the replication control function, abrupt perturbations in plasmid segregation do not yield permanent alterations in plasmid copy number unless the plasmids are lost completely at division. Previously two simple models for the replication control function have been proposed based on studies of other plasmids (Cullum and Broda, 1979; Nordstrom et al., 1980). One model hypothesizes that the copy number is always set to 2N, in all cells before cell division independent of the initial plasmid number (constant copy number model). The other model presumes that, if a cell contains 2N, plasmid copies at cell division and each daughter cell receives between 1 and 2N0 plasmid copies, exactly N copies are synthesized during a cell cycle regardless of initial copy number (constant initiation frequency model). The present model has been applied to calculate how the Xdv plasmid replication control system reacts to initial copy numbers from 1 to 2N. In Table 3, the calculated copy numbers for these three different models are compared. Since the parameter values listed in Table 2 are used, N in the other models is taken as 25 so that all three models have the same cyclic-state copy number. Simulation results obtained from the present model give values between the constant copy number model and the constant initiation frequency model. However, the constant initiation frequency model agrees better with the simulation results for Xdv plasmids. As discussed earlier, the repressor level determines the initiation frequency of replication by controlling the transcription frequency. When the newborn cell receives fewer copies than N, the repressor level then decreases, which results in an increase in replication initiations in subsequent
160
LEE AND BAILEY TABLE 3
COMPARISON
OF THREE DIFFERENT FUNCTION
REPLICATION MODELS FOR THE RESFQNSESOF REPLICATION TO PERTURBATIONS IN PLASMID SEGREGATION
CONTROL
Number of plasmid molecules at cell division (after one gen.) Number of plasmids per newborn cell (NO)
Constant copy number model”
Constant initiation frequency model”
Computer model b
1 5 10 20 25 30 40 45 49
50 50 50 50 50 50 50 50 50
26 30 35 45 50 55 65 70 74
33 35 39 46 50 54 62 66 70
a The number of initiation events per cell cycle, N, was taken as 25 for comparison. ’ The parameter values listed in Table 2 were used for simulation computation.
cell cycles until cyclic-state conditions are restored. The converse situation applies in the daughter cell receiving more than N plasmid copies as seen in Table 3. Regulation of plasmid replication depends upon several interacting molecular species and genetic loci. Consequently, consideration of perturbations in only one concentration-the plasmid content at birth-does not provide a complete description of the response of the replication regulation system to possible segregation irregularities at cell division. To explore more general perturbations, time trajectories of plasmid DNA concentration and repressor concentration have been calculated assuming different initial values for both concentrations. As before, it is assumed that all intracellular species partition regularly thereafter. The results of these simulations are shown in a phase plane diagram in Fig. 4. Each line is a trajectory of coexisting repressor and plasmid concentrations resulting from a particular choice of initial values of these concentrations. Time is a parameter along these curves, and the arrows indicate the direction of increasing time. For all perturbed initial states considered, the system trajectories, sometimes in
rather complex motions, converge after three or four generation times to the stable limit cycle (heavy closed curve) of the unperturbed
2.6 26 Reprer~~
3.0
3.2
Cwntrdion
3.4 Ol!O%
FIG. 4. Phase plane diagram showing transients in repressor and plasmid concentrations for tire wild-type Xdv replicon. Time is a parameter along each curve, increasing in the direction of the arrows. To facilitate presentation of trends, irregular motions caused by intermittent plasmid synthesis have been smoothed. The dotted region in the upper right-hand comer of the figure shows points along an unsmoothed trajectory.
KINETIC
MODEL
FOR
cyclic-state system. As in the preceding investigation of single component perturbations, the replication initiation control system compensates effectively for abrupt perturbations from the cyclic state provided the daughter cell receives at least one copy of the plasmid. DISCUSSION
In this paper, a mathematical model of Xdv plasmid replication has been developed based on the known molecular control mechanisms. According to the origin activation model employed here, the initiation frequency and timing of Xdv plasmid replication is determined by the transcription frequency at or near the ori site. This model successfully predicts the stable maintenance of plasmid copy number during the cell cycle and the plasmid and repressor level. It is often assumed (initiator accumulation model) that replication begins when a certain amount of an intracellular initiator protein has accumulated (Jacob et al., 1963), or, alternatively (inhibitor dilution model) when the concentration of an inhibitor protein is reduced below a threshold level by an increase in cell volume (Pritchard et al., 1969; Pritchard, 1978). Neither of these models has been tested by formulating the hypotheses involved in mathematical terms and calculating the corresponding replication behavior. In the present model for Xdv plasmid replication, the underlying hypotheses concerning molecular regulation have been expressed in mathematical form, allowing unambiguous calculation of the corresponding molecular interactions and their effects. The calculated maximum variation in repressor concentration is less than 2% during the cell cycle due to the autorepression action of cro repressor (Fig. 2). Under the autorepression system control, the dilution of repressor protein due to cell growth is readily compensated because a decrease in repressor concentration results in an immediate increase of transcription efficiency for repressor message (see Eqs. (2) and (6)). The initiator protein concentration
PLASMID
REPLICATION
161
also does not vary significantly during the cell cycle because the autorepressor and initiator genes belong to a single operon. The inhibitor dilution model also assumes that the initiation of replication would raise inhibitor concentration and reduce the probability of further initiations (Pritchard, 1978). However, if the plasmid replication follows the random selection model so that plasmid replication occurs throughout the cell cycle (see Fig. 2), each new round of plasmid replication does not greatly change the plasmid concentration and, accordingly, the synthesis of inhibitor message. Suppose that the newborn cell has 20 plasmid molecules in a cell. New replication events will increase the number of plasmid molecules to 2 1, which results in only a 5% increase in plasmid concentration. At least for the multicopy plasmid replicons, therefore, variations in inhibitor concentration between successive plasmid replication events would be too small to explain an on-off regulatory action based on differences in inhibitor level. The autorepressor model proposed by Sompayrac and Maaloe ( 1973), which includes the basic features of an autorepression system in conjunction with DNA replication, seems to be closest to the Xdv plasmid replication control mechanism. Sompayrac and Maaloe (1973) presumed, however, that a new round of replication is inhibited by the consumption of initiator protein. This model does not alone explain Xdv replicon function: the Xdv plasmid cannot replicate even in the presence of excess initiator protein unless sufficient origin activation has occurred. Besides this point, if plasmid synthesis occurs during the entire cell cycle in the random selection mode, many cycles of accumulation and consumption of initiator protein would be required according to the Sompayrac and Maaloe model. Such fluctuations in initiator protein concentration seem unlikely based on current information, although the possibility that some mechanism exists to deactivate or decompose the initiator protein rapidly after replication initiation cannot be completely excluded.
162
LEE
AND
Previous models have been proposed primarily to account for chromosomal DNA replication. When DNA replication occurs once and only once per generation as in the case of chromosomal replication at certain growth rates, the doubling of repressor or initiator genes at a discrete time would result in burst syntheses of regulatory gene products. Then the variations in regulatory elements during the cell cycle may be enough to account for the initiation of DNA replication based on the changes in regulatory protein concentrations. However, density shift data for plasmids are in better agreement with the random selection model of replication (Rownd, 1969; Bazaral and Helinski, 1970; Matsubara and Mukai, 1975; Gustafsson and Nordstrom, 1975; Gustafsson et al., 1978). If plasmid replication follows the random selection model, it is unlikely that previous conceptual models for chromosome replication can make quantitative predictions about the frequency and timing of plasmid replication initiation. In connection with this discussion, it is of interest to note the experimental results of Leonard et al. (1982). They found a fundamental difference between replication control of the chromosome and the oriC-containing plasmid (minichromosome) in synchronously growing populations of E. coli B/r, although the reason for such a difference is not apparent. Their experimental data shows that chromosome replication begins at a specific time during the cell cycle while the minichromosome replicates throughout the cell cycle. The patterns of radioactivity incorporation into chromosome and minichromosome are very close to the simulation results shown in Fig. 2. In view of the work of Leonard et al. ( 1982), it appears that the mode of replication of extrachromosomal DNA differs from that of the chromosome. It has been reported that most naturally occurring plasmids are maintained stably in host cells. On this basis, precise regulation of initiation of DNA replication and regular segregation of plasmids into daughter cells at cell division is sometimes inferred. The present
BAILEY
model for Xdv plasmid replication shows, however, that the molecular replication initiation control system rapidly corrects perturbations arising from irregular segregation. Consequently, a plasmid may maintain a stable copy number although segregation is not regular. Further analysis of this property requires a different, stochastic modeling approach which includes the possibility of irregular segregation in successive cell divisions. It is clear that improved understanding of the relative influences of replication controls and segregation properties on overall plasmid stability will require experimental methods with more sensitivity and resolution than is provided by conventional tests for a plasmid selection marker. In this paper the general behavior of the replication control function for the Xdv plasmid replicon has been described using a mathematical model. Further specific tests of the present model and extensions to analyze mutant plasmids are provided in a companion paper (Lee and Bailey, 1984b). The practice of formulating hypotheses in mathematical form and utilizing the resulting model to design experimental tests of the hypotheses and to correlate and organize data has proven invaluable in many venues of science. With continuing refinement of experimental methods in molecular biology and rapid advances in computer technology, such syntheses of experimental ingenuity and mathematical structure should contribute to new discoveries and advances in molecular biology and biotechnology. APPENDIX:
NOMENCLATURE
a, b, c constants defined in Eqs. (3)-(5) f termination efficiency at t,l G number of plasmid molecules per cell [G] plasmid DNA concentration I initiator protein [I] concentration of initiator protein rate conkO, overall transcription stant
KINETIC
MODEL
FOR PLASMID
decay rate constant for initiator mRNA k!, decay rate constant for repressor mRNA kz overall translation rate constant k!, decay rate constant for initiator protein k$, decay rate constant for repressor protein K, binding affinity of cro repressor at OR1 K2 binding affinity of cro repressor at 0,2 K3 binding affinity of cro repressor at 0,s KA association constant in Eq. (17) KS association constant in Eq. (18) K0 constant defined in Eq. (23) m an integer [mRNA], mRNA ”concentration of initiator gene bRNAlix mRNA concentration of repressor gene n an integer or the number of generations N number of plasmid replication initiation events during the cell cycle No number of plasmid molecules per newborn cell Avogadro’s number concentration of plasmid origin or-i* activated origin [ori*] concentration of activated origin replication proteins of host cell [P; concentration of replication proteins of host cell repressor protein concentration of repressor protein REP replication complex [REP] concentration of replication complex of repliIREPL ___ critical concentration cation complex t time tj* time of the jth replication initiation
V V. Wi
k!,
ro;
REPLICATION
163
cell volume cell volume of newborn cell statistical weights defined in Table 1
Greek v
transcription efficiency defined in Eq. (2) p cellular specific growth rate 7 doubling time 0 dimensionless degree of origin activation defined in Eq. (15) Subscript 0
total concentration ACKNOWLEDGMENTS
This work was supported by the National Science Foundation, the Energy Conversion and Utilization Technology (ECUT) program of the Department of Energy, and the Korea Science and Engineering Foundation (KOSEF). Discussions with Judith L. Campbell contrib uted to this research. REFERENCES M., AND HELINSKI, D. R. (1970). Replication of a bacterial plasmid and an episome in Escherichia
BAZARAL,
coli. Biochemistry
9, 399-406.
BERG, D. E. (1974). Genes from phage X essential for Xdv plasmids. Virology 62, 224-233. BRAMMER, W. J. (1982). Vectors based on bacteriophage lambda. In “Genetic Engineering 3” (R. Williamson, ed.), pp. 53-8 1. Academic Press, New York. CANTOR, C. R., AND SCHIMMEL, P. R. (1980). “Biophysical Chemistry Part III. The Behavior of Biological Macromolecules,” p, 1245. Freeman, San Francisco. CONRAD, S. E., AND CAMPBELL, J. L. (1979). Role of plasmid-coded RNA and ribonuclease III in plasmid DNA replication. Cell 18, 6 1-7 1. COOPER, S., AND HELMSTETTER, C. E. ( 1968). Chromosome replication and the division cycleofEscherichia coli B/r. J. Mol. Biol. 31, 519-540. CULLUM, J., AND BRODA, P. (1979). Rate of segregation due to plasmid incompatability. Genet. Res. 33, 61-^ IY. DOVE, W., KNOKUCHI,H., AND STEVENS,W. (197 1). Rep lication control in phage lambda. In “The Bacteriophage Lambda” (A. D. Hershey, cd.), pp. 747-77 1. Cold Spring Harbor Laboratory, Cold Spring Harbor, N. Y. FREDRICKSON,A. G. (1976). Formulation of structured growth models. Biotechnol. Bioeng. 18, 148 1- 1486. FURTH, M. E., YATES, J. L., AND DOVE, W. F. (1979). Positive and negative control of bacteriophage lambda DNA replication. Cold Spring Harbor Synzp. @ant. Biol. 43, 147-153.
164
LEE
AND
GUSTAF~~ON, P., AND NORDSTROM, J. (1975). Random replication of the stringent plasmid R 1 in Escherichia coli K-12. J. Bacterial. 123, 443-448. GUSTAFSSON,P., NORDSTROM, K., AND PERRAM, J. W. ( 1978). Selection and timing of replication of plasmids Rl drd-19 and FIac in Escherichia coli. Plasmid 1, 187-203. HERSHEY,D. (ed.) (197 1). “The Bacteriophage Lambda.” Cold Spring Harbor Laboratory, Cold Spring Harbor, N. Y. HOBOM, G. (1981). Replication signals in prokaryotic DNA. Curr. Top Microbial. Immunol. 94/95,93-142. HOBOM, G., GROSSCHEDE,R., LUSKY, M. SCHERER,G., SCHWARZ, E., AND KOSSEL, H. (1979). Functional analysis of the replicator structure of lambdoid bacteriophage DNA. Cold Spring Harbor Symp. Quant. Biol. 43, 165-178. JACOB,F., BRENNER, S., AND CUZIN, F. (1963). On the regulation of DNA replication in bacteria. Cold Spring Harbor
Symp.
Quant.
Biol.
28, 329-348.
JOHNSON,A. D., MEYER, B. J., AND I%ASHNE, M. (1978). Mechanism of action of the cro protein of bacteriophage X. Proc. Natl. Acad.
Sci. USA 75, 1783-1787.
JOHNSON,A. D., MEYER, B. J., AND I?ASHNE, M. (1979). Interactions between DNA-bound repressors govern regulation by the X phage repressor. Proc. Natl. Acad. Sci. LISA 76, 5061-5065. KOLTER, R., AND HELINSKI, D. R. (1979). Regulation of initiation of DNA replication. Annu. Rev. Genet. 13, 355-391. KUYPERS,B., REISER,W., AND KLEI?$ A. (1980). Cloning of the replication gene 0 of E. coli bacteriophage lambda and its expression under the control of the luc promoter. Gene 10, 195-203. LAUER, G., PASTRANA, P., SHERLEY, J., AND P~ASHNE, M. ( 198 1). Construction of overproducers of the bacteriophage 434 repressor and cro proteins. J. Mol. Appl. Genet. 1, 139-147. LEE, S. B., AND BAILEY, J. E. (1984a). A mathematical model for Xdv plasmid replication. Analysis of copy number mutants. Plasmid 11, 166-177. LEE, S. B., AND BAILEY, J. E. (1984b). Analysis of growth rate effectson productivity of recombinant Escherichia coli populations using molecular mechanism models. Biotechnol.
Bioeng.
26, 66-73.
LEONARD, A. C., HUCUL, J. A., AND HELMSTETTER, C. E. (1982). Kinetics of minichromosome replication in Escherichia coli B/r. J. Bacterial. 149, 499-507. LIPINSKA, B., PODHAJSKA, A., AND TAYLOR, K. (1980). Synthesis and decay of X DNA replication proteins in minicells. B&hem. Biophys. Res. Commun. 92, 120126. MATSUBARA, K. (1976). Genetic structure of a replicon of plasmid Xdv. J. Mol. Biol. 102, 427-439. MATSLJBARA, K. (1981). Replication control system in lambda dv. Plasmid 5, 32-52. MATSUBARA, K., AND KAISER, A. D. (1968). Xdv: An
BAILEY
autonomously replicating DNA fragment. Co/d Spring Harbor
Symp.
Quant.
Biol.
33, 769-775.
MATSUBARA, K., AND MUKAI, T. (1975). Mode of replication of plasmid hdv. J. B&hem. Japan 17, 373382. MATSUBARA, K., TAKAGI, Y., AND MUKAI, T. (1975). In vitro construction of different oligomeric forms of Xdv DNA and studies on their transforming activities. J. Viral. 16, 479-485. NORDSTROM, K., MOLIN,
S., AND AAGAARD-HANSEN, H. (1980). Partitioning of plasmid Rl in Escherichia co/i. I. Kinetics of loss of plasmid derivatives deleted of the par region. Plasmid 4, 2 15-227. NOVICK, R. P., AND HOPPENSTEADT, F. C. (1978). On plasmid incompatibility. Plasmid 1, 421-434. OKA, A., NOMURA, N., MORITA, M., SUGISAKA, H., SUGIMOTO, K., AND TAKANAMI, M. (1979). Nucleotide sequence of small ColEl derivatives: Structure of the regions essential for autonomous replication and colicin El immunity. Mol. Gen. Genet. 172, 151-159. PATO, M. L., AND VON MEYENBURG, K. (1970). Residual RNA synthesis in Escherichia coli after inhibition of initiation of transcription by rifampicin. Cold Spring Harbor
Symp.
Quant.
Biol.
35, 497-504.
PRITCHARD,R. H., BARTH, P. T., AND COLLINS, J. (1969). Control of DNA synthesis in bacteria. Symp. Sot. Gen. Microbial.
19, 263-297.
PRITCHARD, R. H. (1978). Control of DNA replication in bacteria. In “DNA Synthesis-Present and Future” (I. Molineux and M. Kohiyama, eds.) pp. l-26. Plenum, New York. ROSEN,J., RYDER, T., INOKUCHI, H., OHTSUBO, H., AND OHTSUBO, E. (1980). Genes and sites involved in replication and incompatibility of an RlOO plasmid derivative based on nucleotide sequence analysis. Mol. Gen. Genet.
179, 527-537.
ROSEN,J., RYDER, T., OHTXJBO, H., AND OHTSUEQ E. (198 1). Role of RNA transcripts in replication incompatibility and copy number control in antibiotic resistance plasmid derivatives. Nature (London) 290, 794797. ROSENBERG,M., COURT, D., SHIMATAKE, H., BRADY, C., AND WULFF, D. L. (1978). The relationship between function and DNA sequence in an intercistronic regulatory region in phage X. Nature (London) 272,414423. ROWSURY,R. J. (1977). Bacterial plasmids with particular reference to their replication and transfer properties. Progr. Biophys. Mol. Biol. 31, 27 l-3 17. ROWND, R. (1969). Replication of a bacterial episome under relaxed control. J. Mol. Biol. 44, 387-402. SOMPAYRAC,L., AND MAAL.@E,0. (1973). Autorepressor model for control of DNA replication. Nature New Biol. 241, 133-135. STOUGAARD,P., MOLIN, S., AND NORDGTRISM,K. (1981). RNAs involved in copy-number control and incom-
KINETIC
MODEL
FOR PLASMID
patibility of plasmid Rl. Proc. Nat!. Acad. Sci. USA 78,6008-6012. STOUGAARD, P., MOLIN, S., NORDSTROM, K., AND HANSEN,F. G. ( 198 1). The nucleotide sequence of the rep lication control region of the resistance plasmid Rl drd19. Mol. Gen. Genet. 181, 116-122. STUITJE, A. R., SPELT, C. E., VELTKAMP, E., AND NIJKAMP, H. J. J. (198 1). Identification of mutations affecting replication control of plasmid CloDF13. Nature @ondon) 290,264-267.
SZYBALSKI, E. H., AND SZYBALSKI, W. (1979). A comprehensive molecular map of bacteriophage lambda. Gene 7,2 17-270. TAKEDA, Y., FOLKMANIS, A., AND ECHO=, H. (1977). Cro regulatory protein specified by bacteriophage X. J. Biol. Chem. 252, 6177-6183.
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TOMIZAWA, J., ITOH, T., SEUER, G., AND SOM, T. (198 1). Inhibition of ColE 1 RNA primer formation by a plasmid-specified small RNA. Proc. Nat/. Acad. Sci. USA 78, 1421-1425. WILLIAMS, B. G., AND BLATTNER, F. R. (1980). Bacteriophage X vectors for DNA cloning. In “Genetic Engineering: Principles and Methods” (J. K. Setlow and A. Hollaender, eds.), Vol. 2, pp. 201-281. Plenum, New York. WOLD, M. S., MALLORY, J. B., ROBERTS, J. D., LEBOWITZ, J. H., AND MCMACK, R. (1982). Initiation of bacteriophage X DNA replication in vitro with purified X replication proteins. Proc. Natl. Acad. Sci. USA 79, 6176-6180. WYATT, W. W., AND INOKUCHI, H. (1974). Stability of lambda 0 and P replication functions. Virology 58, 313-315.