A mathematical model for λdv plasmid replication: Analysis of copy number mutants

PLASMID

11, 166-177

(1984)

A Mathematical Model for Xdv Plasmid Replication: Analysis of Copy Number Mutants SUN BOK LEE' AND JAMES E.BAILEY~ Department

of Chemical

Engineering,

California

Institute

of Technology,

Pasadena,

California

91125

Received April 12, 1983; revised December 2, 1983 A mathematical model based on the molecular control mechanisms for Xdv plasmid replication in a single Escherichia coli cell has been applied to simulate replication of mutant Xdv plasmids. Model simulations of changes in repressor level and copy number resulting from mutations in the promoter-operator PROR region are consistent with experimental data. Calculated effectson Xdv plasmid copy number of oligomer formation and of alternations in termination efficiency at t,l also agree with experiment. The model has been employed to simulate the influence of cro mutants and of cro and t,J double mutants on copy number and stable maintenance of Xdv plasmid copy number. The genetic structure included in formulation of the replicon model provides a framework for relating changes in specific genetic loci on the plasmid with resulting alterations in host-plasmid system function.

Replication of plasmids is known to be perturbations in segregation for the wild-type controlled in a strict manner giving a stable plasmid. average number of plasmid molecules per cell In this paper, the model has been extended in an exponentially growing population of to simulate the effects of mutations or alterbacteria (Broda, 1979). The number of copies ations in the regulatory functions. Model simulation of replication of mutant plasmids proper cell, however, can be altered by genetic changes in either the regulatory sites or the vides a quantitative analysis of copy number regulatory genes of the plasmid. In many cases, mutants as well as further tests of the proposed analyses of mutant plasmids have provided a model. In particular, the present paper deals basis for understanding the biological mechwith the influences of altered nucleotide seanism of replication control function at the quences in the operator, of alterations in termolecular level. mination efficiency, and of plasmid oligoIn a previous paper (Lee and Bailey, 1984a), merization on plasmid copy number and rea mathematical model has been developed for pressor levels. The calculated effects of these replication control of the hdv plasmid based genetic alterations in the Xdv replicon are on the molecular regulatory mechanism. In- compared with available experimental data. cluded in this model are mathematical de- In addition, some phenomena predicted by scriptions of transcription, translation, and model simulations are presented and disdeactivation of primary regulatory species and cussed. of their interactions with the hdv replicon. KINETIC MODEL This hdv replication model has successfully described stable maintenance of plasmid copy A brief summary of the model equations number and simulates plasmid levels, the timdescribing Xdv plasmid replication is given ing of plasmid replication, and responses to below; details of model development and supporting evidence for major features of the model are described in the companion paper ’ On leave from the Korea Advanced Institute of Science (Lee and Bailey, 1984a). Synthesis of repressor and Technology, Seoul, Korea. and initiator proteins via transcription and * To whom correspondence should be addressed. 0147-619X/84

$3.00

Copyright 0 1984 by Academic FTeu, Inc. All rigbu of reproduction in any form reserved.

166

SIMULATION

OF COPY MUTANT

translation are described by the following equations:3 Repressor synthesis:

167

PLASMIDS

Single cells are presumed to increase in volume exponentially:

(8)

d[mRNAlR = kO,n[G] - k-,[mRNAIR dt

- dmRNA]R,

(1)

4Rlo = kE[mRNAIR dt

A single plasmid copy is presumed to be synthesized instantaneously whenever the replication complex activation level, [REP], reaches a critical threshold value [REP],,. [REP] is given by

- k$,[Rlo - ~[Rlo. (2) Initiator

[REP] =

synthesis:

d[mRNAlI = kzn( 1 - f)[G] - k-,[mRNA]i dt

- AmRNAlI,

(3)

d[Ilo = kz[mRNA], dt

- k!JIlo - dI10. (4) Transcriptional activation of the origin of a single plasmid selected at random for replication is presumed to follow the kinetics: d8

-& = ch(l

-f)U

- 0) - d-4

(5)

where fi is a normalized activation level of the plasmid origin of replication. The transcription efficiency of the PR promoter, 0, in Eqs. (l), (3), and (5) has the form 1 + K3[Rlo

’ = 1 + a[Rlo + b[R]?, + c[R];

(6)

where a = K, + K2 + K3, b = KIK2 + K2K3 + K, K3, and c = K, K2K3. K, , K2, and K3 are the cro repressor binding affinities to the three tandem operators &I, OR2, and 0,3, respectively. The concentration of plasmid genes [G] in Eqs. ( 1) and (3) can be determined from

El = &-

A

.

’ Nomenclature from Lee and Bailey (1984a).

~Ko[Uob~i3o 1 + Ko[Ilo ’

(9)

At each time tj during the cell cycle when [REP] reaches [REP],,, the number of plasmids G is increased by one, and 0 is reset to zero corresponding to random selection of another plasmid for replication. For a specific set of parameter values corresponding to a particular replicon genotype (parameters for the wild-type Xdv replicon are given in Lee and Bailey, 1984a), differential Eqs. (1) through (5) and (8) are integrated numerically using a Runge-Kutta algorithm, and [REP] and [G] are calculated also through the cell cycle using the rules just described. When elapsed time reaches the cell doubling time 7, plasmids and all plasmid replication regulating components are equally partitioned to daughter cells. The calculation continues through several simulated generations until a regular, repetitive, cyclic state is achieved. In this state, the trajectories of all concentrations are exactly repeated from one cell cycle to the next. SIMULATION Mutations

RESULTS

in the Promoter-Operator

PRO,

Recently, Murotsu and Matsubara (1980) reported that the copy number of Xdv plasmids depends on the strength of the rightward promoter-operator PROR. They measured the plasmid copy number and the cro repressor level for four different types of operator (see Table 1). Also, the operator region nucleotide sequence has been determined for each case (see Fig. 1).

168

LEE AND BAILEY TABLE 1

EFFECT OF MUTATIONSOFTHEPROMOTER-OPERATOR P,O, ONTHEUOREPRESSOR LEVELS AND PLASMID COPY NUMBER Repressor level (units/mg protein) CaSe

Plasmid

PROR

Relative affinity for repressor binding’

1 2 3 4

XdvBB 1 XdvBC2 hdvl XdvAJ5

Wild type virC34 vlv3 virC34virR 18

1 0.18 0.06 0.02

Copy number

Expt.

Calb

Expt. a

Calb

0.14 0.28 1.60 2.54

0.14 0.27 1.23 2.33

25 27 85 125

25 38 87 122

a Experimental data were taken from Murotsu and Matsubara ( 1980). b Except for the following parameters, the same values as listed in Table 2 of the accompanying paper (Lee and Bailey, 1984a) were used for calculation. Case 2, K$’ = l/5 KY’; Case 3, Ki”“’ = l/l0 KY’, KY* = l/20 KY’; Case 4, Kc;“’ = I/250 KY’, Ky” = l/5 KF. Superscripts mut and wf denote the mutant and wild type, respectively. Calculated repressor contents were converted to activity units using the factor 0.14 units/mg protein obtained from the measured wild-type repressor activity (Murotsu and Matsubara, 1980) and the calculated wild-type repressor content (Lee and Bailey, 1984a).

It is assumed here that a change in the nucleotide sequence of an operator subdomain (say ORI) implies an alteration in the binding affinity of repressor for that subdomain (K1).4 The known mutation sites for each Xdv form thus define the particular model parameters which can and must be modified to describe the mutation’s effect on plasmid replication. For XdvBC2, Xdvl, and XdvAJ5, one, two, and one binding atfmity parameters, respectively, should be modified. Although there are two PRO, mutations in XdvAJ5, the vi&34 mutation is identical to the mutation in XdvBC2 and must therefore be described by the same change in repressor binding affinity K2. Since it is not currently possible to predict the shift in protein-DNA binding affinity which results from a particular nucleotide sequence alteration, binding affinities for the mutant operators were estimated by trial and error. This estimation process was guided by the apparent repressor binding aflinities reported by Flashman ( 1975) and Murotsu and Matsubara (1980), and by comparison of the calculated copy number and repressor levels with experimentally measured values. 4 Nomenclature used here is taken from Lee and Bailey (1984a).

As shown in Table 1, reasonable modifications of appropriate KI and K2 values provide good agreement with experimental data. It is significant that both intracellular cro repressor level and plasmid copy number are well approximated simultaneously by means of adjustment of only those parameters directly affected by specific nucleotide sequence alterations. It is also important that the mathematical modeling framework relates successfully changes in one element of the plasmid replication control system with the overall quantitative performance of the system as manifested by the copy number. Table 1 shows that both repressor level and copy number of Xdv plasmid are dependent on PROR structure, complicating analysis of the relationship between cellular plasmid and autorepressor content. Using the hdv plasmid replication mathematical model, this complication can be resolved readily, and another perspective for comparing model and experiment can be derived. Earlier simulations indicate that temporal changes in messenger RNA and protein concentrations during the cell cycle are insignificant in the cyclic state (see Lee and Bailey, 1984a). Under these conditions, simplified approximate descriptions of repressor message and protein levels are

SIMULATION

FOR31

OF COPY MUTANT

169

PLASMIDS

r------w-I

f--w

-1

ACGTTAAATCTATCACCGCAAGGGATAAATATCTAACACCGTGCGTGTTGACTATTTTACCTCTGGCGGTGATAATGGTTGC... a vl

? virc34

! virR18

t v3

FIG. 1. Nucleotide sequences of PROR mutants (taken from Flashman, 1978: Ohlendorf et al., 1982).

provided by the following steady-state material balances: kzs[G]

- k-,[mRNA]R - p[mRNAIR

ki[mRNAIR Eliminating yields

x 0,

(10)

- kr$[R10 - p[R]o = 0.

(11)

[mRNA]n

[RIOL yGj-

from these equations

+ d(kf$ + PI k”m k”P

9

(12)

which indicates that the transcription efficiency at PRO, is approximately proportional to the number of repressor molecules per plasmid. Using relationship ( 12), PRO, transcription efficiencies can be estimated from [RI, and [G] measurements done by Murotsu and Matsubara (1980) for eight different systems. One series of experiments involves cells carrying the four plasmids listed in Table 1, and the other experiments used cells containing pBR322-derived chimeric plasmids which contain the PRO, and cro gene sequences of Xdv DNA. Ideally, if there is no interference in transcription at PRO, in the chimeric plas-

mid, the transcription efficiency should be equal for a given PRO, sequence in both situations. Such equality is not observed (see Table 2), but, except for the Case 1 - Case 2 change, the qualitative trends are similar for the two series of plasmids. For the XdvvirC34: :pBR322 chimerae, ncal deviates from the general trends observed for Xdv replicon plasmids. This anomalous result may be due to experimental errors. Experimental data of Murotsu and Matsubara (1980) show that the copy number of the chimeric plasmids fluctuates from 24 to 29. If it is assumed that the copy numbers of Xdv: :pBR322 chimerae are actually identical independent of PRO, mutation, the qcalvalues for the chimeric plasmids are 1.OO, 1.OO, I .23, and 2.25 for cases 1 through 4, respectively. Independent estimates of the transcription efficiencies of wild-type and mutant promoteroperators are available from model Eq. (6) based on repressor-operator interaction. The theoretical transcription efficiencies q&or in Table 2 were calculated from Eq. (6) using the [RIO level computed by the full model. Repressor binding affinity values employed for mutant operators were those indicated in Ta-

TABLE 2 COMPARISONOFRELATIVETRANSCRWTIONEWICIENCY ~CALCULATEDFROM EXPERIMENTALDATAWITHTHEORETICALVALUES"

Case

Plasmid

1 2 3 4

hdvBB 1 XdvBC2 Xdv 1 XdvAJ5

PRO, Wild type virC34

vlv3 virC34virR

18

Xdv Replicon

pBR322 Replicon

mle‘x

1.00 1.85 3.36 3.63

1.00 0.86 1.13 2.16

1.00 1.28 2.50 3.40

’ Expressed as a fraction of wild-type plasmid transcription efficiency. b Calculated from the experimental data of Murotsu and Matsubara (1980).

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LEE AND BAILEY

ble 1. The theoretical values are in the same range as the values calculated from experimental data and show the same trends. The preceding analyses for PRO, mutant plasmids indicate that the limiting factor for the replication initiation of Xdv plasmid is the transcription frequency at the ori region rather than the repressor concentration itself. The replication control function of mutant plasmids determines the transcription frequency so that the repressor level and plasmid copy number match the efficiency of the promoteroperator. Efect of Oligomer Formation Plasmid Copy Number

on the

Several bacterial plasmids have been reported to exist in oligomeric states in which the genome of the plasmid is represented an integer number of times in a single closed duplex: e.g., ColEl plasmid in Proteus mirabilis (Bazaral and Helinski, 1968) pMB9 (Bedbrook and Ausubel, 1976), CloDF13 (Stuitje et al., 1978), pRDK 101, pACYC 184 (James et al., 1982) and Xdv plasmid in Escherichia coli (Hobom and Hogness, 1974; Chow et al., 1974). In the case of Xdv plasmid it is known that the copy number is determined not only by the plasmid genetic structure, but also by the oligomeric state of the plasmid (Mats&u-a et al., 1975; Matsubara and Otsuji, 1978). In order to further test the mathematical model and its underlying hypotheses, the copy number for different oligomeric plasmid forms

has been calculated from the model equations. Only one of the multiple origin regions in the oligomeric plasmid is presumed to be involved in plasmid replication (see Matsubara, 198 1). Also, in this simulation computation, it is assumed that the concentration of genes contained in the plasmid can be expressed as the product of the number of genes per plasmid molecule and the plasmid DNA concentration. With these assumptions, previous model equations (Lee and Bailey, 1984a) can be used for the simulation of oligomerization effects by modifying one equation: the gene concentration encoded in an oligomeric plasmid is now expressed as

PI = n&&--)

.

(13)

where nG is the number of monomer units per plasmid genome. For simulations of oligomerit mutant plasmid replication, the modified cro repressor binding affinities listed in Table 1 were used. When the plasmids propagate as higher oligomeric forms, simulation results indicate a decrease in the number of plasmid replication initiations during the cell cycle, which is equivalent in the cyclic state to the number of plasmid genomes per newborn cell (see Table 3). This trend is independent of the genetic structure of the plasmid replicon. The calculated copy number of Xdvl plasmid from the model (Cases 3 in Table 3) agrees well

TABLE 3 SIMULATEDEFFECTOFOLIGOMERIZATIONONPLASMIDCOPYNUMBER Plasmid molecules per cell” Monomer CiW

Plasmid

1 2 3 4

XdvBB 1 hdvBC2 Xdv 1 hdvAJ5

PROR

oki=

Wild type virc34 vlv3 virC34virR

18

’ The parameters listed in Table 1 were used for calculation.

1)

25 38 a7 122

Dimer

h=2) 19 29 66 93

Trimer ok = 3) 16 25 57 80

SIMULATION

OF COPY MUTANT

with the experimental data of Matsubara and co-workers (Matsubara et al., 1975; Matsubara and Otsuji, 1978). They have reported that the copy number of monomeric Xdv 1 plasmid is about 79-95 while that of dimeric form is about 55-66 when the pure monomeric and dimeric plasmids are constructed in vitro and the resulting molecules are replicated independently. The plasmids imm434dv and imm2 ldv both include cro gene and promoter-operator PRO, functionally analogous to but structurally different from those sequences in Xdv DNA (Grosschedl and Schwarz, 1979; Matsubara, 198 1). Calculations listed in Table 3 for the Xdv replicon model indicate that the effect of oligomerization on plasmid copy number is largely independent of cro repressor binding affinity at PROR. Combined, these features suggest strongly that the trend of decreasing copy number (as oligomer units) with increase of plasmid oligomerization calculated from the Xdv replication model should also obtain for imm434dv and imm2 ldv. Such results have been observed experimentally by Matsubara and Otsuji (1978). Alterations

in Termination

Eficiency

at t,l

Although Xdv plasmids carrying t,l mutation are not available at present, the effect of termination efficiency on Xdv plasmid replication can be analyzed from experimental studies of the influence of the N gene product

171

PLASMIDS

on plasmid synthesis. The product of the N gene is known to modify transcription in such a way that RNA polymerase reads through the termination site tRl (Rosenberg et al., 1978). According to experimental results of Berg and Kellenberger-Gujer ( 1974) heteroimmune phage Wimm434 stimulates the replication of Xdv021 dimeric plasmid by a factor of 3.5 on the average. The effect of N gene products made by such a superinfecting phage can be simulated using the Xdv replicon model by changing the parameterf(O if < 1) which characterizes the termination efficiency at &I. For comparison with data for the XdvO21 plasmid, it is important to note that Xdv021 carries the same vlv3 mutation in PROR as Xdv 1, requiring use of modified K, and K2 parameters from Table 1 in the model calculations. Examples of simulation results are shown in Table 4. Presuming an eightfold reduction in termination efficiency by N gene product (f = 0.1) the calculated copy numbers of XdvBBl and Xdvl are, irrespective of oligomeric states, 3- and 2.5-fold higher than that of the control case (normal termination efficiency at tRl), respectively. In the case off = 0 (complete prevention of termination), the simulation results show 3.2- and 2.7-fold stimulation for the replication of XdvBB 1 and Xdvl, respectively, relative to the control. Since the N gene product essentially completely prevents transcription termination at t,l (Rosenberg et al., 1978), comparison of

TABLE 4

Repressor level (xl@ M)

Copy number Case

Plasmid

1 2 3 4 5 6

XdvBB 1 hdvBB 1 XdvBB 1 Xdvl Xdvl Xdv 1

p&l Wild type Wild type Wild type vlv3 vlv3 vlv3

Termination efficiency (f)

Monomer

Dimer

Monomer

Dimer

0.8 0.1 0.0 0.8 0.1 0.0

25 74 80 87 221 235

19 57 61 66 168 179

0.31 0.46 0.47 2.77 3.79 3.88

0.37 0.53 0.54 3.19 4.38 4.47

172

LEE AND BAILEY

the results of simulation computation with experimental data shows reasonable agreement. Simulation results also show that alterations in termination efficiency greatly affect the plasmid copy number without concomitant influence on repressor synthesis (Table 4). This occurs because the synthesis of cro message is not influenced by termination of message at tRl, while initiator synthesis and origin activation results only from readthrough transcripts passing t,l. Further Analysis of Copy Number Mutants Plasm,d iVolecuks

Since the mathematical model of Xdv plasmid replication is genetically structured, the model may be applied to simulate the prop erties of a variety of replication mutants. Effects of genetic alterations in any regulatory element of plasmid replication are studied by modifying the specific parameter(s) which characterize interaction(s) of that regulatory element with other components of the replication control system. The model is next utilized to simulate replication of mutant plasmids for which detailed experimental data are not yet available. Simulation computations for cro mutants as well as for PR mutants have been conducted. In these calculations, it was assumed that cro mutation alters the binding affinities for all three binding sites with equal probability and that the promoter mutation affects the transcription rate constant. The resulting cro repressor levels and plasmid content of newborn cells are plotted in Fig. 2, where the same quantities for the wild-type and 0, mutants considered above are also shown. All of the (repressor concentration-plasmid level) points so calculated lie on a common locus. This indicates that, independent of the site of the specific mutation affecting the transcription efficiency at PROR, the autorepressor control system provides a one-to-one correspondence between repressor level and copy number. The effects of mutations in multiple sites of the Xdv replicon can also be investigated using the mathematical model. Curve b in

per Newborn Cell

FIG. 2. Relationship between cro repressor level and plasmid copy number for various types of mutant plasmids. Termination efficiencies for curves a and b are 0.8 (wildtype 1~1) and zero, respectively. The symbols X, 0, 0, and A represent the wild-type PRO, and cro, cro mutant, PR mutant, and 0, mutant, respectively. For simulation, it was assumed that all plasmids are perpetuated as monomers.

Fig. 2 shows repressor level and newborn cell plasmid content for Xdv plasmids with t,l and various autorepressor system mutations. As before, the t,l mutation effect is represented here by takingf = 0 in the replication mathematical model. As noted above when considering mutations in t,J only, the effect of such genetic change in plasmids also containing autorepressor mutations is to increase copy number for a given repressor concentration. Results of many simulations of multiple mutant Xdv plasmids are summarized in Fig. 3. There, contour lines of various copy numbers are shown for a wide spectrum of cro and t,l double mutants. In this figure, f is the transcription termination efficiency at tRl and K is the ratio of mutant plasmid cro repressor binding affinity relative to that of the wildtype plasmid (i.e., Kp”’ = K&“‘; i = 1, 2, 3; KY”’ values are shown for reference on the figure). For (A K) values in the stippled region in Fig. 3, stable maintenance of plasmid copy number does not occur due to tight repression

SIMULATION

0

0.2

04

0.6

Of3

OF COPY MUTANT PLASMIDS

IO5 IO

173

experimental observations, it appears that the success of Xdv plasmid formation is related closely to the transcription frequency at the origin region and is determined by the resulting effect of plasmid synthesis on host cell growth. When the plasmid copy number is sufficiently large, the cells carrying such plasmids cannot be detected due to negligible growth or death of the host cell, even though plasmids are present in the cells. DISCUSSION

FIG. 3. Contour lines of various copy numbers. K and frepresent the ratio of mutant plasmid cro repressor binding atlinity relative to that of wild-type plasmid and the termination efficiency at tRI, respectively. K3 values are also shown for reference. Stable plasmid maintenance does not occur for mutant plasmids characterized by parameter combinations in the stippled region.

and/or termination of most messages. The combination of smallSand small K values also will not produce stable plasmid copy number maintenance since extremely large plasmid copy number may be expected to cause cell death or sufficient growth inhibition that colonies do not appear. Only those plasmids for which the kinetic parameters lie below the stippled region and sufficiently far from the lower left corner of Fig. 3 can be perpetuated in the host cell. A series of X phage derivatives carrying various types of cro mutations have been tested for their ability to give rise to plasmid Xdv (Matsubara and Takeda, 1975; Matsubara, 1976). Among the phage tested, tof 2, tof 5 and tof 6 mutants, which overproduced rightward transcripts, were unable to produce Xdv (Matsubara and Takeda, 1975). In the case of tof tn mutant phage, plasmids were produced at 3O”C, but no plasmids were obtained at temperatures above 40°C (Matsubara, 1976). It was also observed that the plasmids carrying virCcv3 and cl7 mutation in PRO, and &I, respectively, cannot be formed (Matsubara, 1974). The virCcv3 mutation shows weakest PRO, strength (Matsubara, 1974), and the cl 7 mutation creates a new promoter in the terminator region (Rosenberg et al., 1978). By comparing the simulation results (Fig. 3) with

Biochemical and genetic analysis of mutants have been employed as a powerful technique for experimental study of cellular processes at the molecular level. Analogously, the analysis of mutant systems using a mathematical model can be a powerful tool for evaluating and improving the model and its underlying hypotheses. In this paper, the previous mathematical model, which successfully described the general behavior of the wild-type Xdv plasmid replicon, has been applied for simulation of copy-number mutants. Quantitative predictions for the mutant plasmids were generally consistent with experimental measurements. Mathematical descriptions of cellular processes in terms of multiple cellular chemical species are often termed “structured” (Fredrickson et al., 1970). The Xdv plasmid replication model developed in this work and its immediate predecessor may be called genetically structured. By describing the key chemical interactions which determine replication function at the molecular level, the conceptual framework on which the model is based defines a one-to-one correspondence between the site of a particular change in nucleotide sequence and one or a very few model parameters. For example, a mutation in OR1 potentially alters K1, while a mutation in cro may change the repressor protein and thereby KI , Kz, and K3. While the magnitude of the parameter modification caused by mutation may be unknown, it is very significant that the parameters which can be changed are defined completely. The model equations then provide the required vehicle for relating the particular

174

LEE

AND

parameter change associated with a corresponding mutation with quantitative effects of the mutation on a complex system, here propagation of Xdv plasmid in E. coli. Genetically structured mathematical models should be increasingly important in relating quantitatively DNA sequences to the implied properties of cells. A central requirement in further development of this approach is a means to relate changes in nucleotide sequence not only with particular parameters but also with the numerical values of those parameters. Ideally, for evaluating the model by comparison with experiment, it is desirable to have independent information, experimental or theoretical, on the nucleotide sequence-parameter value relationship. Such information is not available, however, for the Xdv PROR mutants considered above, necessitating parameter adjustments to match calculated overall properties, such as copy number, with experiment. Success of the modeling exercise then depends on the agreement of other, simultaneous model results such as repressor level or oligomerization effects, without further parameter change. The above model for Xdv replication clearly satisfies such criteria. Recent advances in experimental technique and knowledge of protein-nucleic acid interactions will contribute to improved estimates of the model parameters for the mutant plasmids. For several cases of PRO, and t,l mutants such as vir23, vcl, vc3 and vs326 (Johnson et al., 1979; Ohlendorf et al., 1982), and tin and c~lc (Rosenberg et al., 1978; Court et al., 1980), the changes in parameter values due to mutations have been determined experimentally. Further experimental and modeling studies of Xdv plasmids carrying such mutations would further test the validity of the present mathematical model. Eventually, a theoretical basis may exist for evaluating parameter value changes directly from the nucleotide sequence change (see, for example, Ohlendorfet al., 1982). Such a molecular theory, in concert with genetically structural models, will establish quantitative mappings from nucleotide sequence to cell function.

BAILEY

Evenintheabsenceofsuchageneraltheory of intermolecular actions and of more extensive experimental data, a genetically structured model can be employed to evaluate major features and trends that may be expected from genetic manipulation. In this paper, it has been shown that the plasmid copy number and the repressor level have a one-to-one correspondence independent of the types of mutation affecting the transcription efficiency at PRO,. Also, the model simulation results indicate that only those plasmids whose promoter strength and termination efficiencies are in a certain limited domain are maintained at stable copy numbers. Proof of these predictions awaits additional experimental investigations using well-defined multiple mutant plasmids. The proposed mathematical model for the Xdv plasmid replicon successfully simulates experimental data as shown in this work and in companion studies (Lee and Bailey, 1984a,b). The model is based on the following two major hypotheses: random selection of plasmid for replication (Matsubara and Mukai, 1975) and transcriptional activation of the origin region (Dove et al., 197 1). Next, possible mechanisms underlying these two hypotheses will be discussed, taking into consideration the model features and the simulation results. So far, the molecular mechanism of the random selection mode of plasmid replication is unknown. A possible mechanism can be proposed based on the simulation results of this model and information available from the literature. The initiator 0 protein is known to be metabolically unstable and present in low amounts in the cell (Kuypers et al., 1980; Wold et al., 1982). According to the model simulation, about 9 X lop9 M of 0 protein is present in a cell carrying wild-type Xdv plasmid, which indicates an average of about four molecules of active 0 protein in a cell during the cell cycle. Tsurimoto and Matsubara ( 198 1) have proposed that eight monomers of the 0 protein would be required to cover four tandem repeats in the ori site (but the inner two repeats are primarily covered). Even making allowances for uncertainties in the

SIMULATION

OF COPY MUTANT

simulation caused by approximate values of the model parameters, it appears that a limited amount of active 0 protein is present in the cell. Consequently, only one or a few plasmid ori DNA regions can be bound to 0 protein at a time. Interestingly, the ?r protein of R6K plasmid, another initiator protein with molecular weight and NH*-terminal domain similar to those of 0 protein, also shows apparent instability (see Germino and Bastia, 1982). R 100 and the related plasmids R 1 and R6 also encode initiator proteins called Rep Al or Rep A, and the size and basic nature of these Rep proteins are similar to those of ?r protein (Rosen et al., 1980; Ryder et al., 1982). However, since the stability of these Rep proteins is unknown, it is uncertain whether these systems also evidence small intracellular initiator levels adequate only for binding to one or a few plasmids. It is interesting to note that the random selection mode of plasmid replication seems to be more reasonable than the democratic mode of replication from the viewpoint of cellular physiology of host-plasmid interactions. If the plasmid replication takes place one by one throughout the cell cycle with relatively regular time intervals, only a small, nearly constant fraction of replication proteins of the host cell may be required for plasmid replication, as assumed in the present model. However, if plasmid replication follows the democratic mode of replication and thus the number of plasmid molecules doubles once per generation at some discrete time, the simultaneous replication of multicopy plasmid molecules could affect the chromosome replication adversely. This in turn could adversely affect plasmid replication due to reduction in synthesis of host cell proteins and precursors required for plasmid replication. In the case of Xdv plasmid, for instance, P protein is known to form a complex with the dnaB gene product of E. coli (Furth et al., 1979; Wold et al., 1982) which is also required for chromosome replication. Since only small amounts of dnaB protein are present in a host cell, about 20 molecules per cell (Kornberg, 1980), it may be impossible to replicate simulta-

PLASMIDS

175

neously all copies of even a wild-type Xdv plasmid (copy number x25). On the other hand, experimental observation indicates that the Xdvl mutant plasmid (copy number ~85) can be maintained and replicated without significant inhibition of host ceil growth (Matsubara and Kaiser, 1968), supporting the random selection model. Although the random selection process for plasmid synthesis can minimize certain interference with host-cell metabolic activity, extremely large numbers of plasmid molecules may seriously affect the growth rate of host cells. As an example, host cells carrying the XdvAF18 plasmid with In mutation in the CTO repressor gene were inviable and the rate of chromosomal DNA synthesis declined progressively when the temperature was raised to 42°C presumably due to lower repressor binding affinity at elevated temperatures (Matsubara, 1976). The simulation results shown in Fig. 3 indicate that proper combinations of promoter strength and termination efficiencies are required for stable maintenance of plasmid copy number. In connection with this discussion, it is of interest to note that the cloning of foreign genes under control of a strong promoter is sometimes impossible and requires proper termination of transcript for the successful expression of the cloned genes (e.g., Gentz et al., 1981; Luk and Szybalski, 1982). It seems likely that both the ability to form Xdv plasmid from various types of mutant phages and the success of gene cloning with a strong promoter can be explained based upon effects of plasmid functions on host cell macromolecular synthesis and metabolism. Further analysis of this aspect, however, requires a detailed model for host-plasmid interactions and additional experimental data. Sufficient information on mechanism and intermolecular interaction parameters from molecular biology investigations is now available to enable detailed and quantitative mathematical description of certain biological functions. Germinating from initial models of limited scope, such as the Xdv replicon system considered here, additional features can be in-

176

LEE AND BAILEY

corporated to expand the genetic framework included and the biological features calculated. Quantitative description of molecular control mechanisms should lead ultimately to predictive capability for function in different host-vector systems, a scenario with major implications for application of molecular biotechnology. ACKNOWLEDGMENTS This work was supported by the Korea Science and Engineering Foundation (KOSEF), the National Science Foundation, and the Energy Conversion and Utilization Technology (ECUT) program of the Department of Energy.

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