J. theor. Biol. (1979) 81, 105-122

A Kinetic Model for Interaction of Regulatory Proteins and RNA Polymerase with the Control Region of the lac Operon of Escherichiu cofi W _ MANDECK& Institute of Biochemistry and Biophysics, Polish Academy of Sciences, ul. Rakowiecka 02-532 Warsaw, Poland

36,

(Received 7 June 1977, and in revised ,form 3 1 May 1979) The proposed model deals with kinetic aspects of the interaction of repressor, CRP and RNA polymerase with the control region of the lactose operon and is formulated as a system of linear differential equations. Several variants of the model are considered. They differ in the assumed mechanisms which limit expression of the operon (due to diffusion of the molecules of polymerase to the promoter and/or due to a specific interaction of polymerase and promoter) and in the existence or non-existence of an indirect interaction between the molecules of repressor and CRP, when they are bound to the control region, An analysis of the model provides a unified interpretation for several phenomena connected with regulation of the lactose operon, in particular, for the dependence of expression on concentrations of regulatory proteins and for different patterns of expression in vivo and in vitro for a class of promoter mutations.

1. Introduction Expression of the lactose operon is regulated by interaction of three proteins-/act repressor, CRP and RNA polymerase-with their specific DNA binding sites in the lac control region, i.e. operator, CRP binding site and promoter, respectively. Binding of the repressor to the operator prevents transcription of the adjacent structural genes. In the presence of an inducer, such as IPTG, the repressor is transformed into its less active form that binds weakly to the operator. The association constant for this binding in vitro is three orders of magnitude lower than that for the repressor-inducer complex (Jobe, Riggs & Bourgeois, 1972; Barkley, Riggs, Jobe & Bourgeois, t Abbreviations used : lac operon : lactose operon, IPTG : isopropyl-B-D-thiogalactopyranoside, CAMP: adenosine 3’,5’-cyclic monophosphate. CRP: CAMP receptor protein, active CRP: complex of CAMP and CRP. 105 0022-5193/79/210105+ 18 %02.00/O ~0 1979 Academic Press Inc. (London) Ltd.

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1975). Therefore, in an induced cell the operator is free during a significant fraction of the life time of the cell and enhanced expression of the operon is observed. This type of negative control leads to a decrease of expression when the concentration of repressor is increased [for reviews on lac repressor, see Bourgeois & Pfahl (1976) and Miiller-Hill (1975)]. The lac operon is also positively controlled by active CRP (Eron & Block, 1971; de Crombrugghe, Chen, Gottesman & Pastan, 1971). Both genetic evidence (Beckwith, Grodzicker & Arditti, 1972) and the nucleotide sequence of the lac regulatory region (Gilbert & Maxam, 1973; Maizels, 1973; Dickson, Ableson, Barnes & Reznikoff, 1975; Dickson, Ableson, Johnson, Reznikoff & Barnes, 1977) show that CRP stimulates transcription through specific binding to a specific site close to the promoter. This effect was demonstrated in vitro by Majors (1975a, h). The coordinate action of repressor, inducer and active CRP results in a variation of the relative rate of expression in vivo from 1 to 104-lo5 depending on their concentrations. Although understanding of the regulation of the lac expression is relatively advanced, several aspects are still obscure. The problem to be dealt with in this paper is the mechanism that combine the action of repressor, active CRP and RNA polymerase to give the observed expression of the lac operon. The problem was recently mentioned by Sanglier & Nicolis (1976) and Van Dedem & Moo-Young (1975). In the following the model for kinetics of the interaction of the regulatory proteins and RNA polymerase will first be formulated. The analysis of the model required knowledge of the parameter values. Since several parameters were of unknown numerical values, different postulates concerning the regulatory mechanisms could be adopted at this point. These postulates led to the definition of three variants of the model. It has been investigated which variants provide results that show satisfying agreement with available experimental data on : (i) the relative expression rates in viva for the extreme concentrations of repressor, inducer and active CRP (Jacob & Monod, 1961; Hopkins, 1974); (ii) th e d e p en de rice of expression on repressor concentration (J. H. Miller, cited in Gilbert & Mtiller-Hill, 1970; Jobe, Sadler & Bourgeois, 1974); (iii) the parameter values for the interaction of repressor and operator, namely, for the kinetic constants in vitro (Barkley et al., 1975), for reaction constants in vivo (Kao-Huang et al., 1977) and for concentration of repressor unbound to DNA (Kao-Huang et al., 1977; Lin & Riggs, 1975); (iv) different patterns of expression in vivo and in vitro for the class III promoter mutations, e.g. U1/5 mutation (Eron & Block, 1971); (v) the hyperbolic dependence of expression on CAMP concentration (Perlman et al., 1970; Piovant & Lazdunski, 1975).

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2. The Model

The following general assumptions, usually admitted at this point (e.g. von Hippel, Revzin, Gross & Wang, 1974) define the class of models which will be considered here: (1) the molecules of repressor, CRP and RNA polymerase can exist in any of three states: bound to their specific sites in the lac control region, free in the intracellular space or bound to non-operator DNA ; (2) when bound to their specific site the molecules can influence the binding of the other molecules, e.g. repressor bound to the operator prevents binding of RNA polymerase ; (3) association and dissociation processes are governed by the usual mass action kinetic equations. Mass action description is correct when the considered reactions are diffusion controlled, no matter what is the dimension of the space in which diffusion takes place. Therefore, the presented approach is valid for both conventional three-dimensional diffusion of the regulatory proteins and RNA polymerase in the intracellular space, and one-dimensional diffusion of these molecules along DNA. The likelihood of the latter in the case of repressor-operator recognition was demonstrated by Richter & Eigen (1974) and Berg & Blomberg (1976). Since the kinds of experimental evidence available are limited further assumptions have been chosen so as to allow a relatively comprehensive formulation of the model with the lowest possible number of parameters. The state of the control region is defined as (sc, sa). The first element refers to CRP (C) and the second to repressor (R). When the value of sc is C, CRP is bound to the regulatory region and when it is s-, CRP is unbound. Likewise, the repressor can be bound to operator, sa, or the operator can be free, s-. For example, in the state (-, R) only repressor is bound to the operator. Furthermore, the state T is distinguished in which RNA polymerase is bound to the regulatory region and can initiate transcription. The various states are shown in Table 1.t There are potentially numerous transitions between the various states. For instance, to go from state (- , R) to (- , - ), repressor dissociates from the operator and the dissociation constants d,, ‘d, describe this reaction. However, several transitions can be excluded, i.e. their kinetic constants can be assumed to equal 0. The formally possible transition to the same state is t Concentrations will be expressed uniformly throughout the paper in numbers of molecules per cell ; it is assumed that the volume of the E. coli cell is 1 pm3, and that there is one copy of the lac operon per cell. From the definition of molar concentration one can easily calculate that 1 molec cell- 1 = 1.66 x 10e9 t4.

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TABLE

1

Transition matrix Resulting state

(-. -1

1 (-.-I

2c

-)

3 (-,R)

Initial state (-, RI

d,

4 ‘dz

a2R 7

alC

(C, R)

T

k.3

alC a2R ‘a,R

4 (CR) 5T

(C, - )

‘a2 R

k,P

kzP

Elements of the matrix are the dissociation rate constants or products of the association rate constant and the concentration of the respective regulatory protein or RNA polymerase. If two parameters are given for a single transition the one indexed with i refers to the situation with saturating amounts of inducer present in the cell. Points denote transitions which are not considered (see text). The following parameter values were adopted in the model : variant A: a, = 5 s-l, a,=5s-‘, b, = 05 s-‘, d, = d3 = 1O-3 SK’, d, = ZOs-‘, ‘d, = id, = 1 s-‘, k,P = 5x 10-3s-‘, k,P = 0.1 s-l, k, = 10~‘, C = 15, R = 0.2. variant B: d, = d, = 2x lo-“ s-l, id2 = id, = 0.2 SC’, k,P = 0.5 s-l, k, = 0.1 s-l, the other coefficients as in A. Variant B: a2 = ia = 5 s-l, i d3 = 2 s-l, the other coefficients as in B.

equivalent to the remaining in the given state and the rate constant for this transition equals 0. Similarly, the preferential simultaneous association or dissociation of two proteins to, or from, the control region can be neglected. It is also assumed that RNA polymerase leaves the regulatory region free, immediately after starting the transcription. Hence the matrix of transitions, the elements of which are dissociation rate constants or products of the association rate constant and the concentration of the respective regulatory protein, assumes the form shown in Table 1. The transition matrix is denoted as: A = (a,) where i, j = 1,2, . . ., 5 number the states of the control region, as defined in Table 1. The concentration vector of the states of the regulatory region is denoted as x = (xi), i = 1,2, . . ., 5. An auxiliary matrix B is defined as:

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b, = aij- 6, i aji, j=l

It can be easily checked that if the concentrations of the regulatory proteins are constant matrix B generates a system of linear differential equations with constant coefficients :

dx, Bx dt

(1)

which describes the kinetics of the lac regulatory system. When concentrations of the states are found, the number M of lac mRNA chains present in the cell can be calculated from the equation :

where z1 is roughly the half life of the lac mRNA molecule and k, is the rate constant at which RNA polymerase leaves the regulatory region (see also Table 1). The number of /?-galactosidase molecules, Z, is determined by the equation : 1 dz -=&f-Z (3) dt 52 where a is the number of enzyme molecules translated during one time unit from one mRNA chain and t2 is the cell division time. The delays due to transcription and translation, although formally not included in equations (2) and (3) have to be considered when the results of simulation are interpreted (see Results). After adding both sides of equation (1) the following expression is obtained :

$ (itIxi)=O. Since it is assumed that one copy of the lac operon is present in the cell,

The steady state is given by the following system of linear equations: Bx = 0

(4) Jjl

xi=

l.

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Obviously, one obtains from equations (2) and (3) that in the steady state M = z1 k,x, and Z = crr,M. In some special cases the solution of the set of equations (4) leads to concise expressions for the rate of mRNA chain initiation. This is the case when every single reversible reaction considered in Table 1 (i.e. each reaction given by the upper left 4 x 4 submatrix of the transition matrix) is in equilibrium. Then, through the direct solution of system (4), one can obtain the following expression for the amount of the enzyme present in the cell : P(k +k,c) ’ - (1 +r)(l +c)+p(k, where y=- Ra2 4 ’

Ca c= I 4

and

+k,c)

(5)

p = c. k3

R, C and P are the concentrations of free molecules of repressor, active CRP and polymerase, respectively, and Y, c and p are dimensionless concentrations of these proteins. The other symbols are defined in Table 1. Equation (5) may also be derived through the method known from enzyme kinetics [see, e.g. Segel (1975)], i.e. transforming the equilibrium kinetic equations for the reactions considered. Several qualitative conclusions which can be drawn from equation (5) do not depend upon the parameter values adopted. According to equation (5), expression of the genetic system considered changes hyperbolically with changes in concentration of any regulatory protein. As far as repressor and CRP are concerned this seems to be in agreement with experimental evidence. The dependence of the expression on repressor concentration can be satisfactorily approximated by a hyperbola (Fig. I). Gene expression was found to be a hyperbolic function of CAMP concentration (Perlman er al., 1970; Piovant & Lazdunski, 1975). The assumption that the concentration of active CRP is proportional to the concentration of CAMP, as suggested by Epstein, Rothman-Denes & Hesse (1975), gives also the hyperbolic dependence of expression on active CRP concentration. In a general case it is more convenient to solve system (4) numerically, because the composite form of the general analytical solution is quite complex. To simulate the induction or time changes of gene expression computations are necessary. Therefore, a simulation program was developed in FORTRAN and applied to the problem. Results of the simulation and the accuracy of the model largely depend on parameter values adopted. Efforts were made to apply parameter values as known from experimental studies. However, when experimental data were not sufficient or were lacking crude estimations had to be done.

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Ill

10-l

IO-i 0 (/-l(wild

I

I

I

IO

50

f/q)

(/‘q)

type)

1

I

1

I

FIG. 1. Dependence of the steady state enzyme level Z (arbitrary units) on relative repressor concentration R/R,, (R,,-concentration of repressor in the wild type cell). Mutations producing indicated amounts of repressor are given in brackets. Experimental data for the induced level (a) are taken from J. H. Miller, cited in Gilbert & Miiller-Hill(l970). for the basal level (b&from Jobe et al. (1974). The solid line presents the results of simulation (variant B). The other variants (A and B’) lead to results that do not differ more than ST;,from the results of variant B. The unit of length on the abscissa is equal to the square root of the concentration of repressor.

The total number of lac repressors is about 10 molec-’ cell (Jobe et al., 1974). However, a great majority of them-at least 98% (Lin & Riggs, 1975) or at least 93% (Kao-Huang et al., 1977bare bound to non-operator DNA. Therefore, the number of free repressors assumed in the mode1 is 0.2 molec cell - ‘. The total number of CRP molecules is about 700 per cell (Anderson et al., 1971). Since data on concentration of free CRP in the cell are not available, it is tentatively assumed that the fraction of CRP molecules bound to non-

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specific DNA is the same as in the case of repressor, hence the number of free CRP molecules equals 15 per cell. Kinetic constants for repressor-operator interaction were measured in different reaction conditions by Barkley et al. (1975). The association constant of repressor and operator was found to be in the range 3.10”3. 1013 M- ’ and that of the repressor-inducer complex and operator-in the range 3. 109-3. lo’* M- ‘. Kao-Huang et al. (1977) studied the nonspecific DNA binding of the lac repressor in a mini cell strain of E. coli and concluded that the association constants of the two forms of repressor and operator are in vivo greater or equal to lo’* M- ’ and lo9 M-I, respectively. The values of association constants in the model are based on the above results and are in the range of 3.10’*-1.5. 1013 M-I for repressor-operator interaction and 3. lOa-1.5. 10” M-~ for the interaction of the repressorinducer complex with operator. The kinetic rate constants were adopted in the model so as to preserve agreement with the results of the papers quoted. For interaction of active CRP with the regulatory region arbitrary assumptions could not be avoided. Experiments of Piovant & Lazdunski (1975) on the induction of catabolite sensitive genes suggest that gene expression is maximal when the regulatory region is occupied by CRP throughout most of the life time of the operon. This, together with the tentative assumption that the association rate constant is equal to that for the repressor-operator interaction, allows kinetic constants to be proposed for the interaction of active CRP and its specific site. Relatively little is known on the kinetics of RNA polymerase interaction with the lac promoter. In vitro experiments of Majors (1975b) showed that time needed for formation of RNA polymerase-lac promoter complex is Gene expression

Relative rates of expression (fully induced level = 100) for variant:

Components present in the cell: Active CRP

Repressor

TABLE 2 under different conditions

Inducer

-

-

-

+ + +

+ + + +

+ +

t The data are based on Jacob & Monod

A

B

B

70 1100 0.07 1.1 63 1000

66 1100 0.015 1.1 45 1000

68 1120 0,015 1.1 12 loo0

(1961)

and Miiller-Hill

(pers. comm.).

Experiment? 70 1200 0.07 1 70 1000

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about 4 min. The respective value for the T7 early promoter is 15-20 s [Hinkle & Chamberlin (1972); see also review by Chamberlin (1974)]. On the other hand, the kinetics of appearance of the first /I-galactosidase molecules after in viva induction indicated that the RNA polymeraseelac promoter complexes must be formed in about 5 s (Dalbow & Young, 1975). The latter value was used in this paper. Therefore, the product of the association rate constant of RNA polymerase and regulatory region activated by CRP and the concentration of polymerase (i.e. k,P) was assumed to equal 0.14.2 s-i. Other kinetic constants, k, P (ki-association rate constant of polymerase and non-activated promoter) and k, (rate constant at which polymerase leaves the regulatory region) were chosen in such a way that the results obtained were consistent with the experimental evidence on the expression of the lac operon under different conditions (Table 2). The complete set of parameter values is presented in Table 1 and is discussed in the following section. 3. Results

The analysis of experimental data presented above showed that a certain degree of variation of parameter values is admissible. Let us consider a simple criterion for classification of the sets of the parameter values. The criterion will be based on the relation between numerical values of two terms in equation (5) for the induced cells. The parameter values satisfying condition (1 +‘r)(l

+c) $ p(k, +k,c)

(6)

give rise to variant A of the model. Since the steady state rate of expression in this variant, on the basis of equation (5), is equal to: p(k, +k,c) z - (1 +r)(l +c)

(71

a characteristic feature of variant A is the proportionality between the rate of P-galactosidase synthesis and the concentration of RNA polymerase. The proportionality was often assumed a priori in theoretical papers (e.g. Dalbow & Bremer, 1975; Sanglier & Nicolis, 1976: Van Dedem & MooYoung, 1975). The parameter values which do not satisfy condition (6) lead to variant B of the model. In this variant the induced expression of the operon weakly depends on polymerase concentration being close to saturation. It also appears to be justified from a biological point of view to introduce into variant I? an additional assumption concerning the influence of CRP on the

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interaction between the repressor-inducer complex and operator, when both CRP and repressor are bound to the regulatory region. This variant is denoted as variant B’. The three variants of the model are considered below. I’uriant A

Some quantitative relations for this variant can be obtained immediately. To make equation (7) consistent with the dependence of expression on the repressor concentration (Fig. 1) one has to use the values: and

ir = 0.1 for the fully induced level (8)

r = lo3 for the basal level expression

(W. Mandecki, Ph.D. satisfied in viva:

thesis). Hence the following

relation

should

be

R.K -= lo4 ‘R ‘K where K is the association constant for interaction of repressor and operator, and index i specifies the parameter value in the induced cell. The proportionality constant (104) is an order of magnitude greater than expected from experiments in vitro; inducers such as IPTG decrease by about 103-fold the association constant for repressor and operator (Barkley et al., 1975). The same inducers do not affect the affinity of repressor for nonoperator DNA (Lin & Riggs, 1972, 1974) nor do they affect the concentration of free repressor. Thus either the situation in vivo is different from the expected one or the assumptions inherent to the variant A should be modified (see variant B’). Variant B

The value of dimensionless concentration of the repressor-inducer complex given in equation (8) (ir = ‘R . ‘K = 0.1) is obtained when the concentration of free repressor is 02 molec cell-’ (0.3. lo-’ M) and the association constant for the repressor-inducer complex and operator is 3.10* M- ‘. Since the values of ‘K found in vitro may be under special conditions as much as two orders of magnitude greater than the value mentioned (Barkley et al., 1975), the possibility should not be discarded that ir is significantly greater than 0.1. If so, it is impossible to maintain an agreement of the dependence of expression on repressor concentration [equation (5)] with experimental data (Fig. 1) in the framework of variant A. For instance, for r = 1 one gets from equation (5) that the lO-fold decrease in repressor concentration should lead to the about IO-fold decrease in the expression rate, while the measured effect is about 50% decrease (Fig. 1).

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FIG. 2. Relation between the steady state enzyme level Z (arbitrary units) and RNA polymerase concentration P. k, is the association rate constant of RNA polymerase and nonactivated promoter. The numerical value of k,P in viva was assumed in the model to equal 5. 10m3 s-‘. a-results of variant A, b-results of variants B and B (the difference between the curves for these two variants was not more than 2%).

When one agrees with the general formulation of the model, one has to assume a mechanism limiting the expression of the lac genes due to a specific interaction of RNA polymerase and promoter. If so, condition (6) is not satisfied and the steady state rate of expression is given by equation (5). TABLE 3 Summarized results of the model

Numerical value in variant Parameter Proportion of time during which operator is free (with transcription) Proportion of time during which operator is free (no transcription) Proportion of time during which CRP site is occupied (no transcription) Time interval between two initiations

Definition

A

B

B

x1+x,

09 I

0.18

0.18

I, +x, ifP=O

0.91

0.65

0.17

ifP=O

0.79

0.79

0.79

(k,x,)-’

14

14

14

x-2 + xq

The parameter values presented apply to the fully induced case.

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Such interaction has the following implications : (1) the operator is free of repressor, a relatively small fraction of the time (Table 3), (2) there is a deviation from proportionality between this fraction and the rate of expression in highly induced cells, and (3) the rate of expression and RNA polymerase concentration are not linearly related (Fig. 2). variant B Both models, A and B, require that relation (8) be satisfied. However, it is possible to avoid this condition and to be more consistent with the in vitro evidence by introducing the assumption that CRP, when bound to its site in the lac regulatory region, decreases indirectly the value of ‘K about lo-fold. Then, the proportionality constant in equation (9) can be lo3 for the situation when CRP is unbound to the regulatory region. It is characteristic for this variant of the model that any single reversible reaction defined in Table 1 is not in equilibrium although all concentrations are constant in the steady state. Simulated expression of /?-galactosidase in different conditions (when saturating amounts of IPTG and CAMP are present in the cell, or not) is compared with the experimental evidence in Table 2. The comparison shows a satisfying agreement of the results of computations based on the three variants of the model with experimental data. The formulation of the model [equations (l), (2) and (3)] allowed also a simulation of the kinetics of the system, i.e. an investigation of the time dependence of gene expression on parameter values. Particular attention was paid to the time course of enzyme synthesis immediately after induction of the lac system. To be consistent with the experimental data of Dalbow & Young (1975) it was assumed that at a given time the cell, which is initially in the repressed state, is fully induced, i.e. the molecules of repressor from that time onward can bind only weakly to the operator (for parameter values, see Table 1). The output variables are the concentration x5 of the state T of the regulatory region, the number M of lac mRNA molecules and the number Z of molecules of the enzyme present in the cell. The results of the simulation are given in Fig. 3. The time changes of the measurable variable-the amount Z of the enzyme-for different variants of the model are very close to each other. Therefore, the time characteristics of the induction do not allow to distinguish which variant of the model is the most likely. Elongation and translation of mRNA causes a delay between the transcription initiation and the appearance of the first molecule of the pgalactosidase. The delay corresponds to the time needed for transcription and translation of lac mRNA. Since these processes are simultaneous, the first molecules of the enzyme are synthetized after the time needed for

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5-

0

IO

20

30

40

50

Time (5 )

FIG. 3. Simulation of induction of the lac system. In this simulation it was assumed that the system, initially repressed, was induced in time t = 0 by addition of saturating amounts of inducer. l-amount Z of the enzyme present in the cell, Z-number M of mRNA chains.3-concentration x5 of the state T of the regulatory region. The letter A, B or B’ indicates the variant of the model which is the source of the results presented. The time changes of 2. M and x5 are. in fact. not simultaneous due to delays caused by transcription and translation.

transcription, usually within 80-100 s after induction (Dalbow & Young, 1975). The first molecules of mRNA are synthetized in about 80s after induction (Kennel & Riezmann, 1977). These delays displace the curves of the time dependence of, lac mRNA and P-galactosidase concentrations (Fig. 3) along the x-axis by the indicated time intervals. The model made it possible to simulate expression of mutants of the lac regulatory region by varying appropriate parameter values. The following mutations can be described within the framework of the model: (1) Operator constiturive murations (Jacob & Monod, 1961). Strains carrying these mutations were characterized as possessing the high basal level of P-galactosidase, which results from an increased dissociation rate constant of repressor and mutated operator (Jobe et al., 1974). Obviously, expression of such mutants can be easily simulated in the model when values of coefficients dz and d, (dissociation rate constants for interaction of repressor and operator, see Table 1) are increased. (2) CRP independent (or partially independent) mutations, or so called class I promoter mutations (Beckwith et al., 1972). Two ways of simulation are possible : (a) increase of the dissociation rate constant for active CRP and its site (d,/a, in Table 1) or (b) assumption that, due to mutation in the CRP site, the stimulatory effect of active CRP is not transmitted to the promoter.

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Interpretation (b) requires that the value of coefficient k, (affinity of RNA polymerase to the regulatory region associated with active CRP) be low and approximately equal to that of kl (affinity of RNA polymerase to the unbound regulatory region). For certain mutations which are deletions in the CRP site, e.g. LI (Dickson et al., 1975), interpretation (a) is probably true, while for the others, e.g. point mutations in the CRP binding site (Dickson et al., 1975; Dickson et al., 1977), situation is less clear and the possibility (b) cannot be excluded on the basis of the results of simulation and the present experimental evidence. (3) Class II promoter mutarions (Hopkins, 1974). These mutations result in about lO-fold lower expression of the lac operon, but the sensitivity of the mutants to stimulation by active CRP is maintained. Again, two ways of simulation are possible: either affinity of RNA polymerase to promoter in both states of the regulatory region (CRP bound or unbound) is proportionally decreased, i.e. coefficients k, and k, (Table 1) are decreased but their ratio is preserved, or minimum time between two initiations (roughly equal to k;l) is increased about lO-fold as compared to the time between two succeeding initiations in the wild type strain. Any of these interpretations cannot be discarded. (4) Class III promoter mutations. Mutations of this class partially or completely (W’S) relieve the requirement for active CRP (Beckwith et al., 1972; Silverstone, Arditti & Magasanik, 1970). The mutants produce in viva normal levels of lac mRNA and fi-galactosidase, while in vitro they synthetize several-fold more lac mRNA than the wild type strain, e.g. IO-30-fold more in the case of UVS (Eron & Block, 1971). This intriguing characteristics of such mutants can be simulated in the model. One should realize that in the framework of the model the only possible reason for such different patterns of expression of UV5 and the wild type strain is a difference in concentration (activity) of RNA polymerase in vivo and in vitro. The concentration of RNA polymerase used in vitro usually gives maximum expression rate and, in total value is greater than in the cell (about 5 . IO3 molecules per cell in vivo, Ishihama, Taketo, Saitoh & Fukuda, 1976; and 5 . lo4 molecules in the volume of the cell (1 um3) in vitro, Majors, 1975). If this is so, then the correct characteristics of UV5 will be obtained when it is assumed in variants B or B’ that: (a) gene expression in the mutant is limited by diffusion of RNA polymerase to promoter, i.e. kyutmt 9 kyld type and (b) expression is independent of CRP, i.e. affinities of promoter to RNA polymerase are equal irrespectively of whether CRP is bound to the regulatory region or not (k, s k,) and/or association rate constants of active CRP and its site (a,/d,) is negligibly small. The set of parameter values that fits well to the pattern of expression of UV5 mutant is : k, = k, = 0.15 s- ‘, k, = 2 s-r, and other

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parameter values as in Table 1, variant B. Limitation by diffusion of polymerase to promoter, and not by a specific interaction of RNA polymerase and promoter as for the wild type strain in variants B or B’, makes possible the expression of the lac genes in the mutant to be increased when the concentration of poiymerase is increased and this is supposed to be the case in vitro. 4. Discussion

Three variants of the model for interaction of the regulatory proteins and RNA polymerase with the lac control region were considered. The short characteristics of these variants is as follows. Variants A and B require that the ratio of association constants for repressor and operator, and for repressor-inducer complex and operator, be lo-fold higher than that calculated from experimental data in vitro. This discrepancy does not occur in variant B’. Furthermore, variants B and B’ give correct results for a broader interval of kinetic constants’ values than variant A. In addition, variants B and B allow a plausible interpretation for the pattern of expression for the class III promoter mutants. Therefore, variant B’ of the model agrees best with available experimental evidence. The most important assumptions in variant B’ are the following: (1) limitation of expression of the lac operon by a specific interaction of RNA polymerase and promoter. and (2) an indirect interaction between active CRP and the repressorinducer complex. One should realize that these postulates were introduced to provide a unified interpretation for both in vivo and in vitro experiments. Their results are not always fully comparable-hence a limitation of the method. Nevertheless, the presented theoretical approach-a state model of the lac regulatory region-has been shown to be a justified approximation and can be applied to other regulatory systems. Some attention should be paid to the mode of action of active CRP. It was assumed in all three variants of the model that the binding of this complex influences the recognition of the lac promoter by RNA polymerase as proposed by Burd, Wartell, Dogson & Wells (1975) and Dickson et al. (1975). However, it was postulated in variant B that active CRP also increases the dissociation rate constant of the repressor-inducer complex and the operator. The mechanism for such an action of CRP could rely on induction of conformational changes in the operator DNA caused by the binding of CRP to the regulatory region. The postulate of an indirect interaction of active CRP and repressor implies that the ratio of association constants for repressor and operator, and for repressor-inducer complex and operator (K/‘K), is increased several-fold

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by presence of CRP. Therefore, it provides an explanation of why there is a difference between the ratio based on in vitro measurements and the one postulated for in viva conditions (see also Results). The limitation of gene expression due to a specific interaction of RNA polymerase and promoter, assumed in variants B and B’, may have biological significance since it stabilizes gene expression. As shown by the results of the simulation. the fully induced luc system is virtually independent of small fluctuations in the concentration of repressor, active CRP or RNA polymerase; a 100% increase in one of these concentrations produces only a l&20% change in the rate of expression(Figs 1 and 2). The observable variable in the model is the concentration of /Igalactosidase. This concentration is linked indirectly with the rate of transcription initiation and an intermediate step is the translation of lac mRNA. It was assumed that the efficiency of mRNA translation [a in equation (3)] does not depend on the state of the system (e.g. instantaneous lac mRNA concentration). Therefore, in the steady state, the concentration of P-galactosidase is proportional to the number of mRNA chain initiations per time unit. This assumption seems to be justified since translational control has not been detected for the lac system. As far as other systems are concerned, the only convincing case of translational control is a phage RNA translation in vitro (Weissman et al., 1973). However, this RNA possesses complex secondary structure, which may be absent in bacterial mRNA. The model can be verified by precise measurements of: (1) kinetic constants for interaction of repressor, active CRP and RNA polymerase with the lac regulatory region and (2) concentrations of fractions of these molecules unbound to non-specific DNA. Until now, such measurements have been carried out for the lac repressor (Barkley et al., 1975; Lin & Riggs, 1975; Kao-Huang et al., 1977) but, in fact, even these measurements define only one side limits for association constants and for concentration of free repressor. For active CRP and RNA polymerase only crude estimates are possible at present. The crucial experiment for verification of the postulate of active CRP effect on the binding of repressor-inducer complex to operator would be the measurement of the repressor-operator association constant in the presence of IPTG, CRP and CAMP. Finally, there is a possibility of confirming the limitation of the induced lac expression through the study of the dependence of expression on RNA polymerase concentration in vivo using the method of temperature shift in thermosensitive mutants of RNA polymerase genes (Mandecki KCWild, 1979). The author is indebted to Professor K. L. Wierzchowski for making it possible to complete this work, and to Professor T. Klopotowski for his continuous interest. Helpful suggestions of Professor M. A. Savageau and Drs A. Zag&ski

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Rabczenko during preparation of the manuscript are gratefully acknowledged. The author thanks also Professor B. Miiller-Hill and Dr H. Kneser for supplying data and stimulating criticism. This work was supported by the Polish Academy of Sciences within the project 09.7. REFERENCES ANDERSON, W.B., SCHNEIDER, A.B., EMMER, M., PERLMAN. R. L. & PASTAN. I. (1971). J. biol. Chem. 246, 5929. BARKLEY, M. D., RIGGS, A. D., JOBE, A. & BOURGEOIS, S. (1975). Eiochem. 14, 1700. BECKWITH. J., GRODZICKER, T. & ARDITTI, R. (1972). J. mol. Biol. 69, 155. BERG, 0. G. & BLOMBERG, C. (1976). Biophys. Chem. 4, 367. BOURGEOIS, S. & PFAHL. M. (1976). Adv. Prot. Chem. 30, 1. BIIIID, J. F., WARTELL, R. M., DOGSON, J. B. & WELLS, R. D. (1975). J. biol. Chem. 250, 5109. CHAMBERLM, M. J. (1974). Ann. Rev, Biochem. 43, 721. DALBOW, D. G. & BREMER, H. (1975). Biochem. 150, 1. DALBOW. D. G. & YOUNG, R. Y. (1975). Biochem. J. 150, 13. DE CROMBRUGGHE, B., CHEN, B., GOTESMAN, M. & PASTAN, I. (1971). Nature new Biol. 230,37. DICKSON, R. C.. ABLESON, J., BARNES, W. M. & REZNIKOFF, W. S. (1975). Science 187, 27. DICKSON, R. C., ABLESON, J., JOHNSON, P., REZNIKOFF, W. S. & BARNES. W. M. (1977). J. mol. Biol. 111, 65. EPSTEIN, W., ROX-HMAN-DENES, L. B. & HESSE, J. (1975). Proc. nutn Acad. Sci. U.S.A. 72,230O. ERON, L. & BLOCK, R. (1971). Proc. natn Acad. Sci. U.S.A. 68, 1828. GILBERT, W. & MAXAM, A. (1973). Proc. natn Acad. Sci. U.S.A. 70, 3581. GILBERT, W. & MILLER-HILL, R. (1970). In The Lactose Operon (J. R. Beckwith & D. Zipster. eds), pp. 93-l 11. New York: Cold Spring Harbor Laboratory. HINKLE, D. & CHAMBERLM, M. J. (1972). J. mol. Biol. 70, 187. HIRSH, J. & SCHLEIF, R. (1973). J. mol. Biol. 80, 433. HOPKINS, J. D. (1974). J. mol. Eiol. 87, 715. ISHIHAMA, A., TAKETO, M., SAITOH, T. & FUKLJDA, R. (1976). In RNA Polymerase (R. Losick & M. Chamberlin, eds), pp. 485502. New York: Cold Spring Harbor Laboratory. JACOB, F. & MONOD, J. (1961). Cold Spring Harb. Symp. quant. Biol. 26, 193. JOBE, A.. RIGGS, A. & BOURGEOIS, S. (1972). J. mol. Biol. 64, 181. JOBE, A., SADLER, J. R. & BO~RGEQIS, S. (1974). J. mol. Biol. 85, 231. KAO-HUANG, Y., REVZIN. A., BUTLER, A. P., O’CONNOR. P., NOBLE, D. W. & VON HIPPEL, P. H. (1977). Proc. natn Acad. Sci. U.S.A. 74, 4228. KENNEL. D. & RIEZMAN, H. (1977). J. mol. Biol. 114, 1. LIN, S. Y. & RIGGS. A. D. (1972). J. mol. Biol. 72, 671. LIN. S. Y. & RIGGS, A. D. (1974). Proc. natn Acad. Sri. U.S.A. 71, 947. LIN, S. Y. & RIGGS, A. D. (1075). Cell 4, 107. MAIZEIS, N. (1973). Proc. nom Acad. Sci. U.S.A. 70, 3585. MAJORS, J. (1975a). Nature 256, 672. MAJORS, J. (1975b). Proc. natn Acad. Sci. U.S.A. 72,4394. MANDECKI, W. & WILD, J. (1979). Molec. gen. Genet. 173, 339. MANDECKI, W. (1979). Ph. D. thesis: “Model regulacji ekspresji operonu laktozy Escherichia coli”, Institute of Biochemistry and Biophysics, Polish Academy of Sciences. MILLER-HILL, B. (1975). Prog. biophys. mol. Biol. 30, 227. PERLMAN, R., CHEN, B., DE CROMBRLJGCHE, B., EMMER, M., GOITESMAN, M., VARMUS, H. & PASTAN, I. (1970). Cold Spring Harb. Symp. quant. Biol. 35, 419. PIOVANT, M. & LAZDUNSKI, C. (1975). Biochem. 14, 1821. REZNIKOFF, W. (1972). Ann. Rev. Genet. 6, 133. RICHTER. P. H. & EIGEN. M. (1974). Biophys. Chem. 2, 255.

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