J. Mol. Rid. (1985) 181, 211-230

The 0, Control System of Bacteriophage Lambda A Physical-Chemical Model for Gene Regulation

Madeline A. Shea and Gary K. Ackersf-

Department of Biology and McCollum,-Pratt Institute The Johns Hopkins University Baltimore, Md 21218, U.S.A.

(Received 30 August 1983, and in revised form 31 July 1984)

A quantitative model has been developed for processes in the bacteriophage lambda that control the switchover from lysogenic to lytic modes of growth. These processes include the interactions of cI repressor and cro proteins at the three DNA s&es of the right operator, OR, the binding of RNA polymerase at promot8ers P, and P,,, the synthesis of CI repressor and cro proteins, and the degradative action of recA during induction of lysis. The model is comprised of two major physical-chemical components: (1) a statistical thermodynamic theory for relative probabilities of the various molecular configurations of the control system; and (2) a kinetic model for the coupling of these probabilities to functional events, including synthesis of regulatory proteins cI and cro. Using independently evaluated interaction constants and rate parameters, the model was found capable of predicting essential physiological characteristics of the system over an extended time. Sufficiency of the model to predict known physiological properties lends credence to the physical-chemical assumptions used in its construction.

Several major physiological characteristics were found to arise as “system properties” through the non-linear, time-dependent, feedback-modulated combinations of molecular interactions prescribed by the model. These include: (1) maint’enance of the lysogenic state in the absence of recA-mediated CT repressor degradation; (2) induction of lys& and the phenomenon of subinduction; and (3) autogeneous negative control of cro.

We have used the model to determine the roles, within the composite system, of several key molecular processes previously characterized by studies in vitro. These include: (1) co-operativity in c1 repressor binding to DNA; (2) interactions between repressors and RNA polymerase (positive control); and (3) the monomer-dimer association of cI repressor molecules. A major role of c1 repressor co-operativity is found to be that of guaranteeing stability of the lysogenic state against minor changes in cl repressor levels within the cell. The role of positive control seems to be that of providing for a peaked, rather than monotonic, dependence of P,, activity on c1 repressor level. while permitting P, activity to br a step function. The model correlates an immense body of studies in vivo and i?a vitro, and it makes testable predictions about molecular phenomena as well as physiological characteristics of bacteriophage lambda. The approach developed in this study can be extended to include more features of the lambda system and to t)reat, other syst’ems of gene regulation.

1. Introduction events through the interactions of regulator\-

Molecular mechanisms for the regulation of proteins with specific DNA sequenc&, e.g.

gene expression continue to be of major interest repressors binding at operator sites (Johnson et al..

in molecular biology and also now comprise an area 1981): and (2) those based on feedback control at

of rapidly increasing biophysical study. From work the translational level (von Hippel 8t Fairfield,

on prokaryotic systems, several general types of 1983; Kowalczykowski et al., 1981; Lonberg et al..

regulatory mechanisms have emerged including: 1981; Newport et al., 1981). In both types of

(1) those based on the control of transcriptional system. the roles of protein-protein interact’ions between the nucleic acid-bound proteins have

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t Author to whom all correq)ondence should be addressed.

0022-2846~85102021 l-20 $0X00/0

emerged as major elements of control. The bacteriophage lambda embodies a complex

family of interlocking regulatory mechanisms. of 211

fl 1985 Academic Press Inc.. (Lor~ci~m) Ltd.

the first type, that cbontrol: (1 ) processes of infec4ion Mid establishment of tjhr lysogenif* state: (2) maintenance of prophage in the lysogenic state: and (3) the induction of lysis in response t’o environmental “signals” (for reviews, see Herskowitz & Hagen, 1980; Hendrix. 1983). Processes in the first category depend primarily on I)roteins cTT and C-ITT and the promoter P,,: these I)rocesses are not, included in t,he model presented here. The latter two sets of processes depend critically upon the interactions of three proteins (~1 repressor, cro protein, and RNA polymerase) at I)rZA sites of the operator 0, and it,s associated promoters. P, and P,, (see Fig. 1) (Ptashnr & Hopkins, 1968: Risen et nl., 1970; Johnson ut nl.. 1981). The induction sequence is initiated by proteolytic f4eavage of cl repressor molecules mediated by a fourth protein, recA (Tomizawa $ Ogawa. 1967: Roberts et al.. 1978), in response to a signal that triggers the inducible DNA-repair functions (Bailone rt al., 1979: Sussman rt al., 1978). Principal components of the control system to be considered in this paper are depict,ed in Figure 1. along with a summary of their actions. The ultimate goals of this elaborate swit’ching rnechan- ism appear to be: (1) stability of the Iysogenic state (l’,, on. P, off) against minor fluctuations in physiological concentrations of regulatory proteins (T’tashne. 1978): (2) efficient switchorer to the lytic stat)e (PRM off. T-‘, on) in response to a decisive signal (i.e. one that surpasses a critical t,hreshold (,Johnson ef al.. 1981)), leading to; (3) transcription of early genes during a brief time interval. after Lvhich: (4) the control system is irreversibly “locked out” of its Iysogenic (*onfigurations.

In a previous paper we described a static equilibrium model for the role of cl repressor in maintaining the Iyogenic stat,e through it)s binding and co-operative Interactions at the right operator 0, (Ackers rt nl., 1982). Here. we have extended that model to include effects of the other regulat’ory proteins and to predict the time-dependent behavior of the more complex system during both tysogenic growth and the induct’ion of tysis. A preliminary report of this work has been presented (Shea & Ackrrs, 1983).

A primary goal of the work described here was to determine whether the actual time-dependent biology of the tarnbda control system can be explained by the physical interaet,ions of regulatory proteins with the lambda DSA and wit,h each other. To answer this question, we have developed a quant,itative model based on a particular view and set of assumptions regarding the physical nature of the motecutat interact,ions responsible fo1 regulation. Our model has t,wo major components: (I) a statistical t)hermodynarnic theory f’or the relative probabitit,ies of the various molecular configurations of the 0, control system: and (2) a kinet’ic model for the coupling of these probabilities to the a&vit’ies of promoters I’, and P,, and the net’ production of regulatory proteins CT and cro. The structure and viewpoint of this model represent

ActIons of CI repressor Action of N

Extends rlghtward transcrlptlon

Actions of cro

I. Blocks PR

’ 2 Blocks PRM

Figure 1. Principal elements of the lambda right operator control system and actions of regulatory proteins (1 repressor. cro and N. OR is situated between the genes for ~1 repressor and cro proteins: their promoters, PRM and PR. respectively, overlap opposite pnds of’ 0, and diverge from it. The operator consists of’ 3 17-base-pair protein binding sites designated (),I. 0,P. 0,:s: these a,rr separated bv “spacers” of 7 and 6 base-pairs. rrspec,tively. ‘The dimeric forms of cro and ~1 repressor bind to these sites in R “tight” reversible mannel according to a hierarchy of int,rinsic affinities give11 in Table I. Thus OR may exist, in many microscopic states. depending on which proteins are bound at which sites. At a given concentration of dimers. the probability of’ binding is determinrd by the intrinsic free enrrgies ot binding to each site, and also by co-operative int,rra&ons that occur between adjacently bound regulatory prot’eins in certain combinations. (The rules govrrnmg thest, microsf~opif~ states are a major f*omponent of our mofirl.)

I’rinvipal ahions of t,hr oI)erator-bolllltl regulat,or~ ijroteins are indicated. The 2 repressors cl and (‘ro act ~tIltagonistic~tl1~ at 0, to control promotrr af*t.ivky anfl thus. the prophage life cycle. The fundamental require- ment for maintrnanfBc of t,he lysogenic st,ate is t,hat the ~Jrf~tnoters I’, and P, (rontrolling CI’O. ,V and ot,hrr earl?; genes) be turned off while P,, remains highly active: at 0,. ~1 taxc*luttes RX.4 polvmerase from PR by binding at, sites ORI and 0,d. i‘he efficient and irreversible switc*hover to the lytk state occurs as N result, of the rec.-\-nwdiatefl degradation of ~1 rrl)ressor. The resulting drcrrasr in repressor bound at (),I and O,:! (drrcpression of I’,) leads to the synthesis of era prot~ein. whkh in t,urn satnratrs OR3 and Ixevludrs initiat,ion of transvript,ion at I’ KM: cro then accumulat,es to a level such that it, binds to (),I mfl OR2 and represses P, as ~~11. These intrravtions of CT rryxwx~r and cro prf&ins at 0, Hervt to cx)ntrol tllr wllular levels of CI and cro dimrrs. and t,hcveby maintain the Iysopenic. stat’e or allow a brief int,erval of S tr.ansvril)tion during the irreversible indurtion of Iysis. (Thr X Ixotrin leads to Iytic growth of the phage and cell Iysis b>- allowing early lytiv grlw ester&d trausf*ri~)tiori. rightwart beyonfl c/w anfi Irftward Iwyonfl ,Y.)

one of the simplest possible int,erpretations. in terms of the physics of the system. t’hat would be compatible with t’he purely qualitative picture of the control mechanism. as presently understood (see Fig. 1). In order to test whet’her t’hese concepts are sufficient to predict t’he known physiological “system behavior“, a mat’hemat,ical formulation of the model was used to simulate the time-dependent behavior for both the Igsogenic st.ate and for the

Physical-Chemical Model of 1 Regulation 213

induction of lysis. In carrying out the simulations. we have utilized independent estimates of the interaction and rate parameters as determined from studies in vitro and in viva conducted by other investigators. These parameters and the methods used for their evaluation are summarized in the Appendix.

The significance of quantitative models such as this is threefold. (1) The test for sufficiency of the model can disprove or lend credence to its assumptions regarding the physical nature of the processes being modelled. (2) A mathematical formulation of these processes in terms of physical principles provides unparalleled precision in defining the concepts and mechanisms under consideration. One can question the correctness (or wisdom) of what is being said about the system: but not it’s meaning. (3) The time-dependent behavior in a complex dynamic system, replete with reciprocating feedback loops such as lambda (see Fig. 1) is not readily predictable from even an accurate knowledge of the behavior of the isolat,ed parts or subsystems. A molecular “systems approach” such as that proposed here permit,s one to assess the relationships between behavior of the whole system and that of its component parts. In this way, one can determine which characteristics of the components are “context-independent” and which characteristics of the biology arise as “system properties “. A major goal of this study was therefore to evaluate the roles played by particular types of molecular interact’ions in the time- dependent composite system, including the roles of repressor co-operativity, repressor dimerization, and “positive control” interactions bebween c1 repressor and RNA polymerase.

2. Formulation of the Model

In t’his section we describe how our model was generated and which features of the bacteriophage lambda system were incorporated. Section 3 summarizes our rationale for certain features of the model. The Appendix describes the numerical evaluation of model parameters from studies of lambda lysogens and isolated proteins in vitro and in uizw.

(a) Thermodynamic description of interactions

Several types of interactions that will be of interest throughout this paper are illustrated in Figure 2. This diagram depicts the reversible monomer-dimer assembly of c1 repressor molecules simultaneously in equilibrium with the three operator sites OR1, 0,2 and 0,3 (Ackers et al.. 1982,1983). The repressor dimers bind tightly to the t,hree DNA sites and also interact co-operatively with other CT repressor dimers bound at adjacent sites (Johnson rt al., 1979). The terms co- operativity or co-operative interaction will be used throughout this paper to denote a difference between the Cgibbs free energy of binding a given

PR OR3 OR2 j--f 0,I / cro

pRM

‘RM (b)

PR _ cl-o

Figure 2. Interactions of c1 repressor molecules (O-0) at the right operator of the bacteriophage lambda genome. (a) Active (DNA-binding) c1 repressor dimers are in equilibrium with inactive monomers as shown (Sauer. 1979). The dimers bind specifically and reversibly to each of the 3 operator sit,es as shown by the arrows. Binding of repressor alone can result in configurations 1, 2. 3. 4, 10, 11, 12 and 29 as listed in Table 2. Repressor dimers at adjacent sites are capable of co-operative interaction. (b) and (c) Types of co-operative interaction believed to occur between adjacently bound repressor dimers (configurations 29 and 1%. respectively. of Table 2).

pair of protein molecules at adjacent DNA sites (e.g. 0,l and 0,2) and the sum of intrinsic free energies (AG1+AG,) for their binding independently at the same two sit)es. Thus. the free energy of binding two CI repressor dimers adjacently at OR1 and 0,2 is AG, +AG,+AG,z. where the first two terms are the intrinsic free energies for binding at sites OR1 and 0,2, respectively, and AG,, is the free energy of co- operative interaction. The reference state for all free energies is the phage genome with no regulatory proteins bound. In theory, the values of AGlz and AG,, may be positive or negative, diminishing or enhancing occupancy of the interacting site. respectively. In Figure 1, the carboxy-terminal domains of the repressor molecules are depicted as interacting with each other to provide a source of co-operative free energy. Although strong evidence suggests such interactions as a major source of the co-operativity (Pabo et al., 1979; Johnson et al.. 1979; Sauer et al., 1979), we emphasize that the actual definition used here is purely thermo- dynamic. Numerical values of these parameters are given in Table 1.

(b) Assumptions

Assumptions (1) to (7) as given below were used to generate the microscopic molecular con-

Table 1 Interaction free mergies for repressor and cro binding at 0, (37°C’)

Free energy (kcal) AG, AG, AG, AGI, A(&

0,.repressor -11.7kO.03 - 10.1 kO.05 - 10.1 kO.03 -l.Q+O.OA -2.0+0.06

OR-w0 - 10.8 +O.Ol - 10.8kO.01 -12~1~0~01 0.0 0.0

Resolved from analysis of footprinting titration data (Johnson el al., 1979: Johnson, 1980) according to methods described previously (Ackers et al., 1982); conditions: 0.2 x-KCl. AC,, AG, and AG3 each represent the free energy of binding cro or rI repressor to the respective sub-subscripted site of the operator when no other proteins are bound. The co-operativny terms AC, x 2 and AG,, represent the additional energy for simultaneous binding to 2 adjacent operator sites over that obtained by filling the same sites individually. See Ackers ft trl. (1982) and Brenowitz et al. (1985) for discussion of the method.

figurations in which 0, may exist. A number of these assumptions are well-established facts and representative references to their sources are given. Tn section 3, we present’ our rationale for several of the assumptions that may be less obvious.

(1) The three binding sites are specific. non- overlapping, DNA sequences; each s&e binds one protein at a time (Humayan et al., 1977a,b; Johnson, 1980; Meyer et al., 1975; Ptashne et al.. 1976; Folkmanis et al., 1976).

(2) Repressor and cro prot’eins are bound at) the operator sites in their dimeric forms only. Repressor monomers are in equilibrium wit’h dimers (Sauer rt al., 1979; Chadwick et al., 1973; Johnson et al.. 1978:

Takeda et al., 1977; Folkmanis et al.. 1976). (3) There are no co-operative int,eractions

between cro dimers and any other proteins, including adjacent DNA-bound (‘I‘0 dimrrs (<Johnson et al.. 1979).

(4) At an operator with three bound repressor dimers. the only co-operative interact)ions are between repressor dimers at’ OR1 and O,2 (Johnson et al., 1979).

(5) RNA polymerase occupying the promoter PRM prevents repressor or cro from binding at 0,3 and c!ice versa (Maurer et al.. 1980; Meyer et al.. 1975:

Meyer & Ptashne, 1980). (6) RNA polymerase occupying the promo&r P,

prevents repressor or cro from binding t’o (),I and 0,2 and vice versa (Meyer et al., 1975.1980).

(7) RNA polymerase may occupy Pa and PRM simultaneously (Meyer et al., 1980: ,Johnson, 1980); there appears to be no interaction between adjacent RNA polymerasr molecules.

These rules for binding, co-operative int)eraction and exclusion determine the possible combinations of repressor, cro and R,NA polymerase molecules bound at the right operator. The resulting set of 40 configura,tions are given in Table 2. We recognize that ot*her configurat8ions may be physically possible; however, none is support)ed by exist)ing data. Also given in Table 2 arc relative values of the total free energy for t’he operator in each configurat,ion. The determination of t)hese free energies is based upon assumption (8). This assumption, which is not abstracted from known it, ciao informat,ion. forms a major cornerstone of our model.

(8) Occupancy of operator sites is determined by

equilibrium probabilities.

statist’ical t hertnodynamic

For each of the 40 microscopic configurations (Table 2). we formulated an expression for its probability as a function of protein concentration according t’o principles of statist,ical thermo- dynamics (cf. Hill; 1960). The probability of an operator in configuration s may be written in general :

cc,> = - exp (- AG,/RT) x [Rz]‘[Cz]j[RNAP]k

c exp (-AG,/RT) x [ R,]‘[C,]j[RNAP]k’ 0) s,i.j,k

where AB, is the Gibbs free energy of the s configurat,ion (Table 2), R is the gas constant, T is the absolute temperature, [R2], [C,] and [RNAP] are the concent’rations of unbound repressor dimers. cro dimers and RNA polymerase molecules. Indices i, J’ and lz indicate, respectively, the number of each protein bound to an operator in the s configuration. The summation of s is taken from 1 through 40, and i, j and k have the values (0, 1, 2 or 3) appropriat)e to t’he species s. This probabilit’g, (c,), represents the fractional contribution of configuration s (having free energy AG,) to t)he overall population for a particular set of protein concentrations: ]R2], [C,] and [RNAP]. The free energy AG, is a sum of all of the free energies of binding to the operator sites plus co-operative interaction terms appropriate for a given configuration s (see Table 2). These energetic contributions were determined from experimentIs in do and in /,izjo (see Appendix and Table 1).

With t)hese and the remaining assumptions of the model, we calculated the distribution of the various configurations, the promot’er activities and con- centrations of repressor and cro as a function of time and changing cellular conditions.

(9) Transcription initiation is the rate-limiting st,ep in cro and CT repressor synthesis (Ytashne et al..

1980): it is proportionaJ to the product, of two terms: (I) the fractional saturation of the corresponding promoter by RNA polvmerase; and (2) the rat,e constant that governs the ‘isomerization of RNA polymerasr from closed to open complex (JIKlure. 1980).

(10) The fractional saturation of Y, or PRM is determined by the rapid equilibrium between RNA polymerase in solution and that bound to the

Physical-Chemical Model of 1 Regulation

Table 2

215

ConJigurationa of 0, and their total free energies

Configurationt Component free Total free 8 0,3 0,2 0,1 energy terms energy, AC, (kcal)

Non-liganded species 1 0 0 0

Singly liganded species 2 0 0 R 3 0 R 0 4 R 0 0

; 0 0 C 0 c 0 7 c” 0 0

ii RNAP 0 0 0 RNAP

Doubly liganded species 10 0 R-- R 11 R 0 R 12 R-- R 0 13 0 c c 14 c 0 C 15 C C 0 16 RNAP RNBP 17 0 C R 18 0 R C 19 R 0 C 20 t: 0 R 21 R C 0 22 R 0

$23 ii RNAP $24 RNAP R 0

RNAP 0 ;; (’

R RNAP

$27 RNAP C 0 $28 RNAP 0 c

Triply liganded species 29 R R-- R 30 c C C 31 (’ R-- R 32 R c R 33 R-- R C 34 R C C 35 ( R C

1;; ( (1 R RNAP R-- R

$38 RNAP C C $39 RNAP C R

$40 RNAP R C

Reference state 0.0

-11.7 - 10.1 -10.1 - 10.8 - 10.8 -12.1 -11.5 - 12.5

AG, +AG,+AQ,, -23.7 AC, +AG, -21.8 AG,+AG,+AG,, - 22.2 AC,. +AG,. -21.6 AC,.+AG,. -22.9 AC,. + 80,. -22.9 AC, + AC,, - 24.0 AC, + AC,. -22.5 AG,. fAG2 -20.9 AG1.+AG3 -20.9 AC, + AG,. - 23.8 AC,. +AG, -20.9 AG, + AC,. - 22.2 AGR +AG, -22.6 AG, + AGsM -21,6 AG, + AC,, -23.2 80, +AG,. - 24.6 AC,. + AC,, - 22.3 AC,. + AG,, -22.3

AC, +AG, +AG, +AG,, AG,.+AG,.+AG,. AG, +AG, +AG3.+AG,z AG1 +AG,.+AG, AG,,+AG, +AG, +AG,, AG,.+AGZ,+AG3 AG,.+AG, +AG,, AC, + AG,. + AG,. AG, +AG, +AGsM+AG,, AG,.+AG,.+AG,, AC, +AG,.+AG,, AG,,+AG, +AG,

- 33.8 -33.7 -35.8 - 32.6 -33.0 -31.7 -33.0 - 34.6 - 35.2 -33.1 - 34.0 - 32.4

t R, cl dimer; C, cro dimer; RNAP, RNA polymerase; 0, empty site, (- -) indicates. for adjacent R dimers, a co-operative interaction, primed subscripts indicate cro intrinsic free energy.

$ Transcriptionally active configuration.

promoter in t)he closed form (Hawley & X&lure, 1982).

(11) RNA polymerase initiates transcription only

when it isomerizes into the open form (Chamberlin, 1974). In zliz!o, this reaction is essentially uni- directional.

(12) Repressor exerts autogenous positive control when bound to 0,2 (Meyer & Ptashne, 1980); it stimulates the rate of RNA polymerase isomerization from closed to open complex bound at P RM (Hawley &, McClure, 1982). Cont’ributing configurations are 24, 37 and 40 (Table 21.

(13) During induction of lysis, recA catalyzes cleavage of repressor monomers. However, there is no evidence for a specific inducible degradative

pathway for cro protein (Roberts et al.. 1978: Crow1 et al., 1981).

(c) Rates of production for cl repressor and cro proteins

The assumptions listed above allow us to describe the rates of change in c1 repressor and cro protein configurations and concentrations under any set of conditions by the following expressions: dR dt = {@NAP at PRM)I~PR~~ +

@NAP at PR~MPRM2)4t - [RILI (2) dCz dt = (RNAP at P,)k,,A,,.

where 12 is the number of cl repressor monomers cbquivalent to concentration (nM); (RNAP at PRM)1 is the probability of RNAP at P,, and repressor at, ~,d (i.e. (c& + (c3,) + (cbO) (subscripts refer to Table 2); (RNAP at P,,)* is the probability of MAP at P,, and no repressor at 0,2 (i.e. (cs) + fd ,+ ((25) .+ (c27) t <cm> +J%3) + <cd); ‘PRM, IS the stimulated lsomerlzat,lon rate at, P,,

(open complexes/s); kPRM2 is the basal isomerization rate at P,, (open complexes/s): A, is t,he proportionality constant, relating monomers of repressor synthesized per open complex formed at PRM (R/open complex); le,, is the degradative rat’e for repressor monomers (l/s); C, is the number of cro protein dimers); (RNAP at. PR) is the probability of RNAP at P,. = (cg) + (c16) + ((-23) + Cc,,); bR is the isomerization rate at P, (open complexes/s); A,, is the proportionality constant for dimers of cro synthesized per open complex formed at P, (c2/open complex).

These non-linear equations and t.heir underlying assumptions formed the basis of the time-course simulations. which are represented below. Numerical integration yielded the cellular con- centrations of repressor and cro proteins. The integrations were conducted with initial conditions corresponding to physiological levels of the prot,eins in a monolgsogen. The equations were used with or without t,he terms representing specific degradat.ion of repressor monomers: it. was included when simulat,ing the induction of lysis. Xon-specific degradat,ion was not included in any of t.he cbalculations present.ed here.

3. Rationale for 0, Assumptions

Here we review briefly the experimental evidence for certain of the assumptions used in formulating t.he model. Tn particular. we will focus on our interpretat.ion of data that, provided critical constraints on the mechanistic representation of processes described by equations (1). (2) and (3). The Appendix describes t)he determination of numerical values for parameters used in the simulations.

(a) Protein synthe8is

Factors influencing rates of prokaryotic protein synthesis act at the levels of gene transcription, messenger RNA translat’ion and protein processing. These factors include “strengths“ of promoters. a,ct,iva,tors and terminator sites. rat,es Of transcription. affinit.ies of ribosome binding sites. rat,es of translation and life-times of mRNA transcripts, limiting levels of essential amino acids of specific transfer RNAs and post,-t.ranslat,ional protein modifications. In ma.nx systems, however, a restricted view of the kinet,lc complexity of t,he cellular metabolism machinery is a,dequate because either a single process or a few processes dominate regulation of protein synt,hesis under particular cellular conditions or during particular periods in the cell cycle. For the bact,eriophage lambda.

t,ranscript,ion initiation is the rat’e-limiting step governing t,he extent of repressor and (bra ~ynt hrbsis after the establishment of‘ a Iysogen and during induction of lysis (Johnson. 1980: Meyer. 1!)79). IVc postulat,e (section 2) that the net rat<’ of’ rcapr<‘ssor or cro synthesis can be relat.ed t,o the frequency of transcription initiation by a composite rxprcssion representing several processes (assumptions (IO) to (12)). A general formulation of this c>oncept will in elaborated here because it provided the hasis f’or numerical evaluation of parameters llSt4 in calculating protein levels (see Appendix) a,ccaording to equations (2) and (3).

The following expression for thr net ratcb of protein synthesis ma.) be applied to art! combination of operator. promoter and transcribed gene, regardless of whether such an arrangement occurs naturally in tht> cell or only after manipu- lation in vitro.

dpyttein = (RNAP at Promoter)kp,omote,AProtein: (4)

where (R’XAP at Promoter) is the frac:tional saturation of t,he promoter by R’NA polymerase, a function of the concentrations of RNA polymerase. repressor and activator proteins; their Gibbs frcAt> energies of binding and interaction: the association const,ants of the species of regulator)- proteins that bind to DNA (determined at temperature and salt. condit.ions equivalent to those in, Gw): L+romoter is t,he rat,e of RSA polymerase isomerization from closed to open complex (i.e. formation cut’ t,ratl-

scriptionally act.ive RS,4P). a fun&on of the promoter sequence catalysts, etc. The product of t’hese two variables will be referred to as “promoter activity”. AProtein is the proportionality coefficient t,hat represents the number of protein molecules synthesized per open complex formed. This constant, may be viewed as the product: d, x .4,, x A 111, where A, represents transcriptional processes that depend on [RNAP], length of mRNA trans- cript. efficiency of stop signal, promoter-specific number of natural abort’ive initiations, stability of mRXA and RNA binding properties of regulatory proteins; il,, represents translation factors that include ribosome binding sites, limiting amino acids and rare codons. and A,,, represents post-transla- tional processes that’ include amino acid modifica- t’ion, protrase activation, etc.

Tn this trcatmrnt. the pro(“ss’s c,otltjrolling transcription init,iation are separat.ed into lu’c) energetic classes. those that drtertn ine t h(, probability ot’ oc~cupan(~y of’ iI promot.~~r I)y RN.-\ polymerase and t,hose that govrrtl thrb taLca of isomerizatioli into opetl f:ornplf~x ti~rrnatiori. Thr first. &MS is governed largeIF b,v t.hc. ~~quilibriutn bet.wrrn RNA polymerase free in solution and RNA polymerase bound as a closed cotnpk~s al t tit, promoter. The s~c~~nd class is irifluettc-c,tl Iargel>~ h>. factors such as neighboring stimulator!- prot’t*in--- protein cwntact s. and t Ilt’ sttucd urfb illl(l conformation of the USA (e.g. dire&m of helicit J and supercoiling). Chemical and physical factors,

Physical-Chemical Model of 1 Reg&ation 217

such as ionic strength and temperature, are known to influence both classes.

The experiment’al justification for this separation of variables relies on the work of McClure and associates using the method of abortive initiation (McClure, 1980) to study transcription initiation properties. Their original analysis of the catalytic pathway showed that RNA polymerase is in rapid equilibrium with promoter sites in the form of a “closed complex” and that it undergoes a series of kinetically controlled steps that lead to the formation of an “open complex”. It is this open form that is catalytically active in the initiation of transcription. The& experimental results (Hawley & McClure, 1982) and those of Meyer & Ptashne (1980) have provided a basis for independent quantitative assessment of these processses. Specifically, their studies of RNA polymerase binding in t,he presence of repressor protein permitted resolution of the effects of repressor on the separate processes of ligand binding and catalysis of isomerizat,ion.

As its name suggests, CI repressor bound at OR represses promoter P, by rendering the probability (RNAP at PR) negligible. However. while repressing P, at’ lysogenic concentrations. it’ exerts positive control over its own synthesis (see Ptashne ef al., 1976). Hawley & McClure (1982) thoroughly explored the mechanism of P,, activation in &TO and directly demonstrated that t,he protein-protein contact, identified by Pabo & Sauer (1983) must serve exclusively to cat,alyze the rat,e of isomerization of closed to open complex (Izpr,,,t,,) rather than t,o increase t’he ext’ent of RNA polymerase binding ((RNAP at Promoter): see Hochschild et al.. 1983; Hawley & McClure. 1983). Hence the 0, configurations given in Table 2 show no co-operative binding interaction between repressor and RNA polymerase. Instead, we account for the catalytic effect of repressor bound at 0,2 upon P,, activity by summing separately the fractional probabilities of configurations 24, 37 and 40 (calculated according to eqn (1)) and multiplying them by the higher value of J&,,,,.,,~~ to represent a faster rate of isomerization into open complex. The total rate of repressor synthesis (eqn (2)) is described as being proportional to the sum of two promoter activities representing the (phenomenologically similar, but mechanistically distinct) basal and stimulated rates of CT tran- scription initiation (assumption (13)). Thus. the total ant’ivity of P,, ((RNAP at PRM)l/“PRM1 + (RNAP at PRM)Z&MZ) is seen to be modulated by repressor binding and catalysis to provide for tnaximal P,, activity at, lysogenic levels of repressor. This is in accord with the observations reported by M , . ever & Ptashne (1980: and see Appendix).

(b) Repressor degradation

To model the induction of lysis, calculations of repressor Irv~~ls include a term representing a

9

concentration-dependent degradation of monomers. This reaction is believed to involve a complex of proteins cI, recA, single-stranded DNA and ATP (Sussman et al., 1978). At each time, t’he concen- tration of monomers was calculated from the tot)al repressor concentration and KDssn, the dimer- monomer dissociation constant (Sauer. 1979). The rate of degradation was determined from the work of Phizicky & Roberts (1980; see Appendix). The product of these two values was subtracted from the calculated increment of newly synthesized repressor to yield the next change in repressor level. (The effects of higher and lower values of KDssn are discussed in section 5(e).)

4. Computational Methods

All calculations were performed on a Hewlett- Packard 1000 System. Programs were written to allow the variation of all model parameters, the stepsize of calculation and overall time period. At each stage of development, the validity (internal consistency) of the program was t’ested by performing calculations for simple limiting cases, as outlined below.

The function that was used to calculate fractional saturation of any site or sites for a particular set of protein concentra,tions was tested by: (1) perform- ing certain of the calculations by hand calculator; (2) comparing the sum of binding ext’ents of each individual protein to the total extent; and (3) setting [C,] = [RNAP] = 0 and comparing the results to calculations using the previous eight- configuration model for repressor alone (Ackers et al., 1982). For a given protein, setting it’s free energies of binding equal to zero was equivalent to setting its concentration to zero.

The numerical integration program that calculated the changing protein levels was tested in several ways. First, t’he integration algorithm was shown to converge for decreasing stepsize. Second. when both synthetic rate constants of equation (2) were set equal to zero, the expected exponential solutions for degradation by a concentration- dependent protease resulted. The distribution of configurations was checked by summing the fractional contributions of all species to ascertain that their probabilities totalled loo?,,.

5. Results

(a) Simu,lation of the lysogrnic stnte

(i) Protein levels during lysogenic growth

To model lysogenic growth. k,, was set bo zero so that equation (2) represented repressor synthesis alone. The predicted changing concentrat,ions of repressor and cro proteins over a one-hour period are shown in Figure 3(a). The amount of repressor increases sufficiently to allow the daughter cells to maintain the lysogenic state (see Appendix). Under these conditions, the production of cro protein was predicted by the model to be negligible (Fig. 3(a)),

cI monomer _________________________________ _____________-m-m-- __________-_---. cro

I

0~010 (b)

b's ,/

0 IO 20 30 40 50 60

Time (min)

Figure 3. Maintenance of the Iysogenic state calculated ovt‘r a 1 h period for a wild-type lysogen. (a) Levels of rrprrssor (monomer. dimer and t’otal) and cro dimer cA(~ulated by numrrical integration of equations (2) and (3). using parameter values Wed in Tables 1 and 3. k~& are expressed as concentrations (1lM) which are interc~onrrrtiblr with numbers of molecules because we do not account for volume changes during cell growth and division and the concentration of one molecule in a volume equal to that of an E. coli cell is equivalent to 1 nM. Repressor is seen to double almost linearly as it maintains repression of P, (controlling cro trans&ption) and stimulat,es its own synthesis from P,,. et-o is essentially non-existent. These simulations do not

account for normal cellular non-specific mRNA and protein degradation. (b) Rat,e of open complex formation during lysogenic growth. Promoter activity: (REAP at Promoter)lr~,O,,,,,. = open complex/s. P,, stimulated: @NAP at PRdlkPRM1, max. of 6.45 x 10-3. at t = 0 min. P,, basal: (RNAP at PRM)ZkPRMZ. max. of 6 x IO-‘. at f = 0 min. P,, total: P,, stimulated + P,, basal, max. of 6.5 x 10-3, at t = 0 min. P,: (RSAP at P,)kFR. max. of 7.2 x 10-4, at t = 0 min.

as it is known t,o he in a Iysogen. This result was found to he independent of the CI’O synthesis rate. c~hosen over wide ranges (calculations not shown). These simulations ignore any mRNA or non-specific protein degradation: the (or0 level would he even loner if such degradation were included in cqtation (3).

(ii) I’rottrofrr occupnncy and afhity its the lysoyen,ic

?stntP

A promoter in equilibrium with regulatory proteins may exist in t,hree functional states: vacant. repressed. or occupied by RSA polyrnerase. ;-\t each point during the simulation. the relative probabilities of all configurations were calculated

a~c~ortlirip to equation ( I ); promoter occupancy \\‘ilh tlrterminctl hy summing I he fractional prol)ahilitiw of’ the appropriatje mic*roscopic* vontigttrations. It should he noted t,hai k\-~~ls of promok oc~~~~pan~~~ I)y TZSAd polq’n~rase art’ not the c*orrcct indicators of’ the rate of synthesis, heoaust they r.efl~~~t only. the equilibrium prohahility of closed c~otnples fortnation. Thus. in some caasrs it was I~OIY tneaningfitl to exarninr changes it) protnotrr activity. i.e. thr net rate of open complex formatiotl (<RSr\P at P,,)k Promoter). This was rspeciall~ obvious for rT transcription initiation. where qua1 values of (RSAP at PRM),ota, muld (Borrespond to dramatically different values of d[R]/dt. This is hecause the relative contributions of (KSAP at PRM), ittld (RNAAP at PR,)z differed k)y a fwtor ot 1 1. the ratio of’ the isomerization tatr c*ortst,ants t2 PRMl aid kPRMZ (Maurer rt cl/.. 1980: J’lry~ B T’tashnc. 1980: Hawlr~ Kr Mc(%tre. 198%: and see ;\ppen(lix).

As shown in Figure 3(b). thr a,ctivity of PRM rc~tnaittc~tl almost. constant during Iysogenic* growth. while I’, was tnactlve: P,, was on. while P, was off. Under initial conditions, t’he calculated degree of repression of P, was 950,, whereas the repression of P RM was 2W,. The results presented in Figure 3 are consistent wit’h the repression curves we reported earlier based on calculations of the equilibrium binding isotherms in t)he absence of RNX poly- merase (Ackers Pt (11.. 1982).

I’sing equation (1). we det’erminrd the tiistri- hution at each int’erval during t)ht: simulations of the Iysogenic state and induction of lysis (ralrulations not shown). Although as many as a dozen (*onfigurations were significantly populated (> 5°0) during the tiff>-cycle of the cell. onI;\. a frw were appreciahl,v abundant during any one phase.

During normal lysogenic growth. the most abundntlt csonfigurat,ions of 0, were combinations of repressor and RNA polymerase act’ing to repress cm

transcription from P, and stimulate c1 transcrip- tdon from P,,. This is indicated in Figure 3(h) by the calculations of promoter acti+v.

(i) I’rotritt let& during inductiotl of lysis

At the onset of intluc%ion of I>SA repair func+iotrs, repressor monomers are srlrctivelv degraded in a reaction mediated h)- reca=\. Thus. t;) simulate induction of lysis. liRd was srt equal tIo t,he value cited in Table 3.

(Talculat’ed levels of regulatory proteins repressor and (xro for a normal Iysogen are shown in Figure l(a). As in the calculations ot’ Iysogenic growth, no mRSX or non-specitic protein degradation is rcprcsrnted in 1 hr c7tlculat.ioris shown. The level of’ repressor dimers (regulator! form) is seen to deerease itnmrdiatelv (Fig. 1(a)). However. the net rate of cat-0 prod&ion was still very low and the effecat was revrrsibk unt,il the

Physical-Chemical Model of I Regulation 219

Table 3 Model parameters for dynamic calculations of OR behavior

P&repressor P&r0

RNAP intrinsic energy - 11.5 + 0.5 kcal/mol - 12.5 f 0.5 kcal/mol Basal ~Plomoter 0.001 s- 1 0~014+0~003 s-l Stimulated kPromo,er 0~011+0XK~3s-’ Proportionality constant 11.0 monomersjoct 6.0 dimers/oct Rate of degradation 0.0065 s - ’ Lysogenie protein concentration 200 nM OnM Dimer dissociation constant 2OnM RNA polymerase concentration 30 nM 30 nM

Step&e/time increment for numerical integration calculations = 10 s. t Protein molecules/open complex.

,010

005

0 IO 20 30 40 50 60

Time (min)

Figure 4. Induction of lysis calculated over a 1 h period for a wild-type lysogen. (a) Repressor and cro protein levels; CI monomers and dimers are in equilibrium (see Fig. 2). Specific recA-mediated degradation of repressor monomers begins at t = 0 s and continues until all repressor molecules have been cleaved. The consequent depletion of competent (operator binding) cl dimers is completed in approximat.ely 30 min: the resulting protein fragments do not bind effectively at any operator &es to exclude or stimulate RNAP. The rate of cro product,ion is slow until total repressor has reached approx. 80?,; inart,ivation. These simulations do not account for normal cellular non-specific mRiVA and protein degra,dation. (b) Rate of open complex formation during induction of lysis. Promoter activity: (REAP at Promoter)kp,,,t,,, = open complex/s. P,, stimulated: (RNAP at PRM)IkPRM1z max. of 6.45 x 10e3. at t = 0 min. _ P,, basal: (RRAP at P,,),k,,,,, max. of 2.5 x 10e4. at t = 20 min. P,, total: PRM stimulated + P,, basal. max. of 6.5 x 10-3, at t = 0 min. P,: (RNAP at P,)k,,, max. of 8.2 x 10Y3. at t = 36 min.

concentration of repressor dropped to about 20yo of its original value: this phenomenon of subinduction has been observed and studied experimentally bl Devoret’ and co-workers (Bailone et al.. 1979). V\Te emphasize that, while the calculated decay of cl parallels their results, they report the production of lytic cent,ers rather than cro levels as a gauge of the progress of lytic induction. We know of no quantitative study of cro levels after lysogenic induction; thus. the calculated cro levels shown in Figure 4 represent current in viz10 predictions of the these levels by the model.

(ii) Promoter occupancy and activity

During induction of lysis, the activity of P,, dropped concomitantly with the repressor level, while P, became active. A low level of cro. sufficient to turn off P,,, was reached in 25 minutes. During the period when both repressor and cro levels were low, P, was most active. However, the activity of P, declined within a few minutes as the (are level increased and served to reduce its own synthesis (autogeneous negative control). This sequence would result in the required turnoff of delayed early lytic genes. Again, this calculation is a prediction in both senses described above.

(iii) Distribution of operator conjgurations

The dominant states of 0, in a normal lysogen changed dramatically during the induction of lysis (calculations not shown). Initially. t,he abundant configurations were the same as those during maintenance of the lysogenic state. However, as soon as repressor degradation began. some cro was synthesized and bound to its highest affinity site. 0,3. The competition for control of 0, began. After 15 minutes, the configuration CR-R had a 2lqi,, likelihood; it has the functional significance ot t,urning off both genes. As repressor was furthe degraded, species such as COR and CCR rearhed their maximal values; these also repress both genes. After 25 minutes. the repressor level was below loo/, of its lysogenic value and t.he only abundant configurations were those which repressed P,, or activat,ed P,. ,4t, later times, several of those represented era bound at 0,1 or 0,2, t.hus blocuking transcription from P, as well. Aft.er 25 minutes. the

(*alculated frac%ional saturation by cro n-as %1”,, at YR and 730, at 1’ RM; after one hour, the values IV~IY-’ Wf, and 9W,. respectively. It is obvious that small changes in the protein concentrations near a critical point may cause major changes in the distribution of prevalent c*onfigurat~ions.

(c) The role of co-operative repr~swr bindiny

(lo-operative interactions between bound ~1 repressor dimers stabilize the configurat’ions that prevent RNA polymerase from binding at P,. Figure 5(a) shows calculated protein levels during lysogenic growth by a “non-co-operative prophage”: repressor monomers were not degraded and the free energy of each configuration containing repressor was det’ermined solely by the intrinsic binding energies of the proteins for the respective sites (i.e. AG12 = AUz3 = 0 kca.1: 1 kcal = 1.184 k.J). It was found that t’he repressor level remained c~losr to the initial lysogenic value of NO n>l. wherras substantial productSion of cro protein occurred without) delay. The level of cro was sufficient to prevent transcript,ion from PRM and consequent accumulatIion of CI repressor. The reduced oc~cupanq of O,% by repressor resulted in less stimulat,ion of f-1 transcription and greater oc~c~upancy of PR by R?\‘A polymerasr.

Figure 5(b) shows the pattern of transc+ptional activity corresponding to these protein levels. Here the states with adjacent)ly bound cl repressor were no longer as energetically favorable and no longer dominat~ed the distribution. For instancae. two configurations uith the same stoichiometry, R,OR, and OR’-R, now also had the same energy and thus the same probability at any particular set of protein c*onrentrat,ionx. Functionally, they are very different: only ROR interferes with RSA polymrr- ase binding to PRM, while they both interfere with its binding to P,.

I>urinp tht> induction of lysis of this “non- c*o-oprrat,ivr” prophage. there was no lap itI wo production (see Fig. B(a)). The primary effect of the absence of co-operative repressor-ditner inter- actions is seen in the early minutes: reprrssot exceeded Woo inactivation in 19 minutes rather t,han d4 (see Fig. i(a) for comparison to wild-t)-pr). .At that point. the level of cro had risen steadily to the value reached after 69 minutes in a normal Iysogen: the virus would be committed to ITtic growth becbause of parallel S protein production. The early lap in cro production that is necessary for the c~alcnlations to agrrc with the observed subinduc%ion phenomenon in clip (SW Hailone rt nl., 1!)79) did not) occur without repressor--1’ef)I’essoI’ u)- operatirity (Fig. 6(a)). However. (bra still repressed its own synthesis at high c~orlcrntratiolls.

The distribution of configurations for a non- co-opcrat’ive prophagc duritjg indrrc*t.ion of I)-sis showed tha,t 0,R was quickly tilled by cro (Fig. B(b)). After only six minutes. the extent of rcprcssion of PKM by (bra was So,,. whereas in a

0.010 (b)

0 IO 20 30 40 50

Time (min)

Figure 5. Maintenance of the I\;sogenie state c&ulatrtl over a I h period for a non-co-operative prophagr. Bound repressor dimers are incapable of c*o-operativr int,er- act,ions. The energy of rach configurat’ion is determined solely by the intrinsic binding free energies. (b) Repressor and cro protein levels. Although the initial ~oncrrttrations and rate caonstants are the same as in Fig. 3(a). the loss of 2 kcal stabilizing interaction energy is sufficient to lower t,he probability of configurations that repress P,. Thus, substantial (TO accumulates L. while repressor is synthesized at a reduced rate because of P,, repression by cro and reduced levels of st,imulated PRM ac*tivity. ((REAP at P,,),). Xotr that cro increases almost linearly without any lag period. These simulations do not account for normal cellular non-specific mItliA and protein degradation. (b) Rate of open complex format,ion during lysogenic growth. Promoter activity: (RSAP at. Promoter)&.,,,,,,,. = open complex/s. PRM stimulated: (REAP at PRM)lkPRM1. max. of 2.3 x 10Y3. at t = 0 min. P,, basal: (REAP at PRM)2kPRM. max. of 4.5 x 10e4. ;Lt t = 0 min. PRM total: P,, stimulated + P,, basal. max. of 2.7 x 10Y3. at t = 0 min. P,: (REAP at’ PR)kPR, max. value of 5.1 x 10-j at t = 0 min.

normal lysogen (Fig. 1) it’ was only IW,,. The configurat’ion CR,-R, prevalent during t,he switc*h- over in a normal lysogen, was less favorable in free energy by ;Z kcal and peaked at only S,?z;, in three minutes rather than 18”/) in 15 minutes. Th(x constellations of dominant lysogenic and lytic growth configurations were unchanged but the course of their t,ransition was altered, beginning 12 minutes earlier. Presumably. development of lytic centers would be hastened correspondingly. (As shown in Figure 6, it is unlikely that sucah a prophagr c*ould maintain lysogenic growth.)

Physical-Chemical Model of ;1 Regulation 221

(b) 1

0 IO 20 30 40 50 60

Time (min) Time (mm)

Figure 6. Induction of IJ;sis calculat’ed over a I h period f’or a non-co-operatlve prophage. The bound repressor dimrrs are incapable of co-operative inter- artions. The energy of each configuration is determined solely by the inbrinsic binding free energies. (a) Repressor and cro protein levels. Specific recA-mediated degradat,ion of repressor begins at t = 0 s and rontinues until all repressor monomers have been cleaved. rendering them ineffective for repression of P, or stimulation of P,,. Ahhough the initial concent,rations and rate constant are the same as in Fig. 5(a), P, is derepressed in the presencae of a higher concentration of repressor than normal. The repressor level drops more quickly because of reduced stimulated P,, activity; degradation is complete in 20 min. There is no longer a lag in cro production. as had been seen in Fig. 5(a). These simulations do not. account for normal cellular non-specific mRI\‘A and protein degradation. (b) Rate of open complex formation during Iysogenic growth. Promoter activity: (REAP at Promoter)k,,,,,,,,. = open complex/s. P,, stimulabed: (REAP at P,M)1kp,M,, max. of 2.3 x 10-3. at t = 0 min. P RM basal: (REAP at’ P&,/C,,,, max. of 4.5~ 10-4. at I = 0 min. P,, total: P,, stimulat,ed + P,, basal. max. of 2.7 x 10K3. at t = 0 min. I’,: (RNAP at P&k,,. max. value of 8.5 x 10Y3. at I = 19 min.

Figure 7. (‘omparison of wild-type and non co-operative prophage behavior. Levels of total repressor (plotted in monomer units) and cro dimers are calculated for each case as t,he+v were in Figs 3(a), 4(a). 5(a) and 6(a). (a) Maintenance of the lysogenic state (i.e. no CI degradation) calculated over a 1 h period for a normal lysogen (rontinuous lines) and a non-coo-operating prophage (broken lines). Acrumulation of cro dimrrs could lead to establishment of the anti-immune phenotype if this non-co-operativit~ werr restricted to ~1 interactions t,o 0, and interactions at 0, were normal. (b) Induction of lysis calculated over a I h period for P normal Iysogen (continuous lines) and a non-c.o-opelativ~ prophage (broken lines). Degradation of cl pro(,eetls mart‘ quickI>, in t,hr latter case becaause it is ~ount~rhalancrti I,!, weaker cT synthesis than that in a tvild-type lysoyrr~ (see (a). broken curves). However. of greater sipnibc*anc*r is the absence of a lag ~)eriod in cro s?nthrsis in~mrtiiatt4~~ following the onset of cI degradation in the non- co-operative prophage, showing that the initial c~onditionz; (corresponding to Iysopeny) are unstable irl the i1lwnc.r of cI co-opei-ativity.

repressor molecules bound anti intclracLt,ed normall? with each other at OR but failed to stimulate RKA polymerase at PRM (kPRMI = kPRMZ = 0401 s - ‘), the amount of repressor in a lysogen barely increa,sed. The estrnt of cro production was slightly higher than in a normal Ivsogen but not nearly so high as in the case without repressor co-operativity

(Fig. 6(a)).

Thus. we see that during lysogenic growth and the swit’chover to Ipt,ic growth, a dominant element, of’ control is c*learly the co-operativity between repressors hound at’ adjacent sites and. in particular. at sites 1 and 2 in OR.

(d) Thr r& ofpositiae control

Ll’e have simulated the effects of the absence of autogeneous positive control of cI repressor as shown in Figure 8. Jt was found that when

400

_______________------ 200

c cro non-co-operative

5 ______________--- -*------ _____- -.

c ut 9 9,

(b)

0 to 20 30 40 50 60

We also carried out simulations for a. system where CT repressor was lacking both co-oprrative intera&ions and autogeneous positive wntrol properties (i.e. AG,, = A(:,, = 0 and k PRMl = k pRM2 = 0401 5-l). In that lysogenic case. repressor reached a lower value and cro reached a

NO PRM stimulation ____________________---------------.

(bi

0 IO 20 30 40 xl 60

Time (min)

Figure 8. Comparison between behavior of a wild-type lysogen and one lacking positive control of PRM. (I’,, total = {(REAP at P,,)1 +(RXAP at’ PRM)Z]kPRM2). (a) Maintenance of the lysogenic state calculated o\~r a I h period for a wild-typr (continuous lines) and unstimulated lysogrn (broken lines). The levrl of CI remains constant (no net synthesis). This modification renders the prophage stable over a sin&= generation time becausr I’, is oE, however. aRer several clell divisions have occurred, the remaining ~1 rrpressor will IW insufficient to maintain the lysogenic state (calculations llot shown). (b) Induction of lysis calculated over a I h period for a wil&type lysogen (continuous lines) and one’ lacking positive control of I’,,. Qualitatively. thtb chararteristics of thr induction process are idrnt,ic+al.

higher value than if the system were altered by either effect alone. However, the loss of repressor- repressor co-operativity definitely had the most devastating effect on lysogenic growth.

Our model predicted that induction of lysis without positive cont,rol was essentially identical to induction of lysis by a normal prophage (see Fig. 8(b)); P, reached maximal activity after 20 ratht>r than 27 minutes. The subinduct’ion phenomenon was intact,

The fact that’ repressor can exert positive control over it,s transcription from P,, but represses P,, at high levels provides for a peaked level of P,, ac%ivitjy. although P,, occupants is a monotoni- caally decareasing function of addItIona ~1. Positive control is known to compensate for the weakness of I’ RM rtllative to I’, and the absence of a high affinity ribosome binding site on the c1 mRn’A originating at) P,, (Ptashne, 1978): our calculations showed that it prolonged the lag period before committed

Iyt ic growth b\- c*ombating degradation slightI> more c$ti\c~tively. The caal(*ulations suggest t*hat pohitivfl c~~ntrol mutants of VI repressor thal no longer stimulate RNAP will. however. still 1~ abkb to maintain tht immune Iysogenic stat<, for sclvtaral generations (whereas phage t,hat have lost (Y- operativtl interactions will not’). This is consistent wit,h experimental results obtained during t’he isolation of positive cont’rol mut’ants reported I, (iuarentfb rt f/l. (198%).

We investigated the importance of thr repressor monomer-dimer equilibrium at in llitv lysogenic concentrations by altering Ko,,,, the dimer dissociation const,ant. from its value of 20 nlcl (Sauer. 1979). Values covering four orders of magnitude were used in simulations of lysogenic gr0wt.h and lytica induction for wild-type opclrators (not shown). Changes of an order of magnitude in K DSSll exerted little effect on the protein levels for the lysogenic state (e.g. at KDssn = 200 11x1, cro accumulated t’o levels lower t,han 20 nM over 1 h).

However, even small changes of the dissociation consta.nt in either direction had dramatic effects on the dynamic characteristics of the induction of lysis. (These also reflect the lysogen’s ability to withstand normal variations in repressor concrn- tration.) Because of the distinct, roles of the oligomers (dimers regulate transcription initiation, while monomers are specifically degraded), the system is very sensitive to the equilibrium between them. During the induction of lysis. t’he monomer- dimer equilibrium apparently serves t’o buffer the competition bet’ween degradation of monomers by recA and synthesis of monomers stimulated by CI dimers bound at 0,2. This dynamic: buffering was reflect,ed in the systjem‘s lag period before substantial synthesis of cro and the ensuing commitment to lytic growth (Fig. 5). It corresponds t’o observations in viva that a lysogen can recover from partial deplet’ion of repressor (Bailone et nl., 1979). This phenomenon of subinduction is one of several features that were not, incorporated explic+tly into the model. but appeared in the simulations of prot’eiri synthesis as direct con- s’quenct’s of the contributing molecular inter- actions. Simulations using altered values of t,he free energies of co-operativity (SW Fig. 6) or the dimer dissociation const,ant (cha,lculations not shown) also poignantI> demonst~rat~rtl t,he importance of repressor--repressor interactions.

6. Discussion

The analysis and calculations presented here illustrate a method whereby quant)itat’ivr theorirs of gene regulation ma,y be formulated and tested. The model developed in this study embodies a particular view regarding the physical-chemical

Physical-Chemical Model of i Regulation - --

nature of the lambda OR conbrol syst’em and the dominant physical processes by which it operates in viuo. It should be noted that other types of dominant processes can, in principle, be envisioned. For example, it is possible that molecular states of the operator are determined by purely kinetic events in such a way that assumption (8), which forms a major cornerstone of the present model, would be invalid. Our goal has been to formulate the simplest physical mechanism consistent with all known information regarding properties of the system and its component parts in viva and in vitro. We believe that) the use of statistical thermo- dynamic probabilit,ies as determinants of operator states, and our method of coupling them to transcriptional events, satisfv this criterion. The model is sensitive to variations in values of its parameters and is clearly capable of predicting the known physiological behavior of the 0, control system, i.e. the results obtained from it are found to be in quantitative as well as qualitative agreement with physiological characteristics of lambda Iysogens known from an immense body of studies in ~ivo and in vitro.

For such a complex dynamic system whose components are available in such small yuantit,ies. it is unusual to be able t,o devise a realistic model wit’h any hope of determining a unique set of values for the necessary pa,rameters. This is possible here only because most of the parameters have been determined independently. We also found that, observations in &lo placed severe constraints on t#he possible values for parameters that were difficult to measure directly (see Appendix).

The model required specification of ten energies of interaction. four rate constants, three initial concentrations. and one dimerization energy. Out of the ten free energies, eight were known for cro and repressor from DNase protection titrations. The rate constants included one for specific degradation and three for RXA polymerase isomerization into open complex. Studies of proteolysis by recA in Go and in ritro fixed the rate of degradation of repressor. The proportionality constants for protein synthesis were known least accurately. The lysogenic protein concent,rations were known tnost accurately for repressor and cro; for RKA polymrr- ase. we used a value that was wit’hin the experimentally resolved range that provided an effective c+oncentration at promoters. To test the sensitivity of thrl model, most) parameters were varied independent Iy and in possibly correlated sets of constants, such as ACP,,mo,er and kPromoter (see Appendix). Thus, we have adjusted very few parameters and it is unlikely t)hat) we have discovered coincidentally a set of values t’hat are consistent with the known physiological characteristics.

The model predicted several physiological features wit’hout their explicit incorporat)ion into the initial conditions or relations that prescribe the syst’em. These included the phenomenon of suh- induc+ion (see sections 6(b) and (e)) and the

autogeneous negative control of cro. 150th are known to be of great physiological significance (Eisen et al., 1970: Folkmanis et al., 1977) and have been demonstrated repeatedly in ~it7o (Takeda et aZ., 1977: Johnson, 1980). Our result,s show the importance of specific features of molecular inter- actions such as co-operative binding of repressor dimers. and demonstrate that the binding affinities of regulatory proteins and their catalytic effect,s a,re sufEcient to explain consequent cellular behavior.

Tn some cases, the consequences of dist’inct mechanisms of control were explored and c*larifirtl through use of the tnodel. Par example. \VP modrlled the molecular interaction responsible fot autogeneous positive cont,rol by repressor as either an increase in the isomerization rate at P,,, or the likelihood of RXX polymerasr binding there. Although &her model could account for somr observations. the catalytic interacation \vas tnost consistent with t,he mechanism believed t.o operate ir/ r*ilro (Hawley 8 SlrClure. 198%).

Thcl mo&l l)rrsrntrd here cbotltains tickvera I approximations. lt does not acc*ount for cahangrs in volume during cellular growth or cahanyes in gene dosage as induction proceeds. \Ve note that c~4lul;tr division halts during induction of the SOS functions: however. once replication bryin~. there are multiple copies of the operat’or rrgions and the

genes that the>- csontrol. Although this rffect ivelj. dilutes regulatory proteins. net synthesis increastls becbause of the rise in gene copy num her. Kec~ause of’ the general form of the equations used to describe cellular pro(*esses. this and other litnitations of thrs tnodel ma!’ be addressed in a si ruightfor\vwrd manner in the future.

In this paper. we hare focused on exploring the effects of modifying several ke\- processes whew the defecats do not necessarily correspond to t host, of a particular mutation already identified. Thr results highlight.: (1) the importance of co-operative. site- specific binding of regulat’ory proteins to multiple operat,or sites: at~d (2) the complex rffec.ts of an individual eletnent such as a protein like ~1 reprt5sor that rlxerts both positivt) and nepativta control over transcription through its binding and cat’alytic properties. The assumptions and predictions of our model all require flirt h(kr t&iny: fortunat’ely, the calculations make vrry sl)ecific quantitative statements about the rspe&Yl behavior in I*~LYI and thus. the simulated results can be trsted against obserratiotis of both mutant and wild-type phage. Results of these (*al(~ul;ltions \vilI be presented (4sewherr (>I. A. Shea K- (:. Ii. .Icakrrs. unpublished results).

Appendix

Evaluation of Model Parameters

Here we summarize the information and illtrr- pretations used to obt#ain the physical constants required b\- the model.

(a) FWP rnrrgles of fh~ op~f~tor cmfiguratiom

(i) Repressor and cro

We det,ermined t,he intrinsic free energies of binding and co-operative interaction from I)Sase protection data obtained in z:itro by Johnson et al. (1979). These free energies were resolved by least- squares analysis of their dat!a in terms of the equations we presented earlier (Ackers rt al.. 1982). Values are listed in Table 1.

The operator configurations given in Table 2 show occupancy of 0, sites by repressor and cro dimers. While repressor dimers dissociate apprecia,bly to monomers at or near the concentrations that obtain in ciao in a lysogenic cell (Chadwick et nl., 1973: Sauer. 1979). cro prot’ein remains completely dimeric even at concentrations several orders of magnitude lower (Johnson cf crl.. 1978: Folkmanis rf nl.. 1976). Thus. we treat (‘ro solely as a dimer.

( ii ) R,TA pdy m wzw

The free energies assigned to K’NA polymerase binding at I’, and P,, were determined primarily from t,he experimental results presented by Ptashne and co-workers (Ma,urer et al., 1980; Meyer it al.. 1980; Meyer & Ptashne, 1980) with reference to those of Hawley & McClure (1980,1982). A more extensive description of the parameter estimation method and results will be presented elsewhere (M. A. Shea &, G. K. Ackers, unpublished results): a summary follows.

As described by equation (4). formation of a (+omplex capable of initiating gene transcript,ion proceeds through two distinct steps: (1) free RNA polymerase binds to a promoter in rapid equi- libnum to form a closed complex; and (2) at some finite rate. t’he closed complex isomerizes irreversibly int,o an open complex capable of initiating transcription (McClure, 1980; Chamberlin, 1974). At P, and P,,, the probabilit,y of closed c*omplex formation (see Table 2 for a list of these caonfigurations) depends on the free concentrat,ions of three ligands, [R,], [C,] and [RNAP], and on t,heir free energies of interaction and co-operativity acc*ording to equat,ion (1). The basal value of k promoter (km and kPRM2) is det.ermined by factors intrinsic to the promoter sequence: in the case of I’ RM. the speed may be increased to kslimu,a,edPromo,er (kPRMI) by the presence of cT at, OR2 (Meyer & Ptashnr. 1980: Hawley 8r McClure, 1982); thus, total PRM act,ivitjy is the sum of basal and stimulated activity. There are no known activators of I’,. Evaluation of these rat)es will be considered in se&ion (b) of this Appendix.

To resolve the parameters governing closed cnotnplex formation. we analyzed results of promot,er rc,gulat ion obt,ained %rL Go by Meyer. Maurer and I’tashnr (for a review, see Ptashne of nl., 1980): O&W% fusions that removed cT and CTO from 0, c~ontrol wllowrd synthesis of P-galactosidase to

indica,tr I’, or PRM acativit’y (depending oil thr construc%ion): cl or cro was supplied by a plasmid carried in t,hr same lysogen. 13~ inferring that changes in final P-galact,osidase activit!; reflected differences in the rate of transcript.ion Initiation at P, and P,, in these OR-I’,-laci! or OR-PRM-/nc% fusion phages, it was possible to calculat,r relative promoter activities under identical conditions and thus, to compare the effects of mutations in OR or

the presence of cl or cro on promoter activity. Some of t,heir findings on the effects of cl on PR a.nd P,, activity are given as ratios of promoter activity in Table 4 to indicate the nature of t’hr reduced clata that, w-e analvzed.

Mathematical expressions for these quantities were formulated according to t’he mode1 of proteill- DXA interactions at, 0, as prescribed by our assumptions (1) to (12). The model gives rise to 40 configurat,ions (Table 2); however, only a subset, of these a,rc pertinent t,o the studies of 0, that were c~onduct~rd in the absence of cro (listed in Table 4). The remaining 15 configurations that describe all of t.he possible int,eractions of cl and RNA polymerase at. 0, arv givm by 000, OOR, ORO, ROO. TWO. OP. OR-R. K-RO. ROR, PP, RP, POR. PRO. R-RR and PR-R. where 0 represents an empty sit,e. R, is a bound cl dimer, and 1’ is RNA polymerase bound as a closed complex. These are t)he only species that may contribute to the repression and ac%ivity of I’, and P,, according tjo our simple model of gene regulation at 0,. In .thr absencae of both cl and cro. only four equilibrium configurat)ions of OR are possible: 000. 1’00, 0)’ and PP. Correspondingly. the expressions for caloscd complex probahilit,y were simplified because the probabilit,y of a configuration. givt*n by equation (I), was calculat,ed for a species index .s ranging from 1 through 15 or 1 through 4, rather t,han 40. Thus, they afforded compact algebraic trea,t’ment.

Under the conditions of some of the studies, only a few of t’he OR configurations would he populaW appreciably or functionally significant: thus, the f>xact t>xprrssions fotn cxlosrd complex formation were further simplified t’o obtain the approximate analytical expressions given in Table 413. Roth thr

exact and approximate expressions wc’rc’ rJ\-aluatrd to resolw a. set of free energies that \vtlrc simultaneously consist.ent with all of thr st rtdies iti

z:i%‘o. The cro and cl interaction const,ants ww fixed at thP values listed in Table 1. and thr Ivsogenica concentrations of proteins were as list*etl iii “l’al;lt~ 3. All of t’he relationships except (a), (e). and (g) in Table 4 allowed independent evaluat#ion of the frrac energies of AG, and A&,, because rate constant’s or t,heir rat,ios cancelled.

Many of the ratios of promot,er activitit%s listed in Table 4 (*over a range of values brc~~usr of variations in the available data (Maurer PI trl., 19X0; Meyer rf (11.. 1980): WV averaged thr set of free energies meeting t.hesr crit,rria a.nd rou~~tlrd oft’ t.o t,he nearest 0.5 kcal. Thus, the values of’ --- 11.5 Ii& for P,, and - 12.5 kcal for P, are less prrcsise than those det)ermined from t’he DXasr I)rotrc+ion

Physical-Chemical Model of ,? Regulation 225

Table 4 A. Promoter activity ratios derived from studies in vivo

Numerator Denominator

Promoter OR [at Promo&r 0, ICI1 Ratio

a P RM wt 200 IlM P RM wt 0 7-11:” 1) P RM wt 200 nM P RM wt 20-40~~ 10-201

ii P RM 0,3- 200 nM P RM 0,3- 20-40 /lM 2-3: P RM 0,3- 200 nM P RM wt 200 nM l.l-1.251

e PR wt OnM P RM wt OnM 12-20$ f PR wt 200 nM P RM wt 200 nM 0~02-0~04~ g P RM wt 200 nM P RM wt R Inax 14: h P, wt OnM PII wt 400 nM 75~1005

t Represents the estimate of total ~1 level expressed in monomer units. $ Maurer et al. (1980). $ Meyer et al. (1980). 11 Meyer & Ptashne (1980). wt. wild-type.

R. Approximate analytical expressions for ratios

a k,,,,/k,,,, (K,P+ l)/(K,P+ 1 +K,R) = 7-l 1

b (K,R’+K,P)/(K,R+K,P+l) = lo-20

c (K,P+ 1 +K,R’)/(K,P+ 1) = 2-3

d (K,P+l+K,R)/(K,P+l) = 1.2

e b&u,, ((1 +&,P)/W’){KJ’I(l +&P)j = 12-20 f (K,P+ l)(K,P+ l)/K,K2K,,R2(K,P+ 1 +K,R) = 0.02-0.04

!4 10-s (KsP(K,P+ 1)/0.4)“3 = R,,,

h K,K2K,,4RZ(K,P+1+K,2R)/(K~P+1)(K,P+1) = 75-100

Equilibrium constants in these expressions are simply related to the interaction free energies by the standard formula K = exp (- AC/RT): P = 30 nM, R = 80 nM dimer, R’ = 10 to 20 pi%-dimer.

titration studies of c1 and cro. They should be viewed as effective free energies that pertain to conditions in vivo and are consistent with the studies of c1 and cro conducted under physiological conditions in vitro (Johnson et al., 1979).

In Figure 9(a), promoter occupancy is calculated for these values as a function of free RNA polymerase level in the absence of c1 or WO. A broken line indicates the 30 nM level used in calculations of protein levels (Figs 3 t,o 8); P, occupancy is 0.95 and PRM occupancy is 0.80. Assumption (7) of our model of prot’ein-DNA interactions at 0, stipulates no interaction between closed complexes at P, and PRM; however, we note that if there were even slight negative co- operativity (e.g. + 1 kcal), the occupancy of P,, would be most affected because it is the weaker of the two sites (Ackers et al.. 1983). At 30n~-RNA polymerase, the occupancies would be 0.88 at P, and 0.48 at P,, (M. A. Shea & CJ. K. Ackers, unpublished results).

In a lysogen, there is never an instance in which the level of both cro and CI is zero. To indicate the changes that occur. promoter occupancy was calculated as a function of c1 level for an RNA polymerase concentration of 30 n&f using the free energies resolved from the studies in viva (see Fig. 9(b)). The separate contributions of con- figurations having c1 bound to and absent from OR2 are shown as P,, stimulated and P,, basal. As t,he level of cI increases, P, is repressed but P,,

cont,inues to be occupied at a high level, even though the binding affinity of RNA polymerase for P, is higher than that for P,,. This reflects the competitive nature of c1 and RNA polymerase binding. (If there were anti-co-operative inter- actions between RNA polymerase binding at P, and P,,, the increase in [cI] would actually introduce a peak in the P,, occupancy curve, an increase of 5O”/b over the level in the absence of cl, because c1 binding at P, would disrupt the anti- co-operativity and effectively increase the binding constant at PRM (M. A. Shea & G. K. Ackers, unpublished results).) In Figure 9(c), total promoter occupancy is shown for two additional set,s of AG, and AGRM values to show how this property changes for free energy values within the error limits that we have reported. Note that the maximum of the P,, stimulated curve changes position because of the altered competition between CI (at 0,I and 0,2) and RNA polymerase at P,.

We not,e that most of the values determined from our analysis of the results obtained irr viva and presented by Meyer et al. (1980) are in good agreement with the values presented by Hawley B McClure (1982) as determined from studies of P, and P,, in vitro under different conditions. The only exception is AG,,; our value is lower than theirs by 1.5 kcal. Although no single set’ of data placed sufficient constraints on the parameters for a unique determination, the intersect,ion of the possible sets proved to be small.

-II -10 -9 -8 -7 -6

Log [RNAP]

-9 -8 -7 -6 -5

Log [CI total]

-9 -8 -7 -6 -5

Log [CI total]

Figure 9. Promoter occupancy as a function of cl and RN\;A polymerase levels. Unless noted otherwise. intrr- action energies are as listed in Table 1 for repressor and in Table 3 for RNA4 polpmrrase. (a) The cnlrulated probability of closed complex formation (occupancy of promoters) in t’he absence of CI or cro proteins. The broken line indicates [RNAP] = 30 n;M: (RNAP at PR): (cg) + (ct6) = (OP) + (PP) = 0.95, (RNAP at P&: (ce) + (cl& = (POO) + (PP) = 0.80. (b) Effect of cl on the fractional probability of RNA polymerase occupancy of P, and P RM; [RKA polymerase] = 30 nv. P,, stimulated: (REAP at P,,), = (c& + (cj7) + (cbO), = 0.33 at [cI] = 174 mvt. P,, basal: (RNAI’ at PR,), = Cc*) + (C16) + ((.25) + (C28) + ((.x3) + (ca9), = 0.70 at [cI] = 1 nnf. P,,: (RNAP at PRM),o,a, = (RNAP at P,,),+(RNAP at P,,)Z, = 0.79 at [cl] = 1 nM. I’,: (RNAP at PR) = (c9) + (clh) + (cZ3) + (cZ6), = 0.95 at [CT] = 1 nwt. The continuous vertical line indicates lysogenir [cI] = 200 KVI. (c) Effect of ~1 on total P, and P,, occupancy for 3 s&s of AG, and AC&, values: - 13. - 12 kcal (-- - ---~): - 12.5. - 11.5 kcal as listed in Table 3 (----); - 1%. -11 kcal (- -). [REAP] = 30 tIM. The continuous vertical line indicates lysogenir [cI] = 200 nM.

(I)) lsonwrization r&e crt ponrotrrs

Promoter activitjv is defined in equation (1) to he the product of (R%AP ate Protnoter) wit)h X.Pro,,,oter. where this r&e constant reflects the speed of irreversible isomerization from closed to open complex. This is a simple formulation that, does not explicitly account for the lifetime of open c~otnplrs

or the sped of’ vtiaitt ittit ialiott. It’ il pl’c)tttOt~~t’ is actiratjed by ot hrt. ligantls or physival~ riiviroirtlrrtttal wndi~ions. eithw otw or 1~0th of these tertrth may 1w affwt~~d. tlrltt~ttding 011 1 Irc~ mechanism of wtivaticttt. S(~vrr;tI c,sl)(‘t’irrit,trt;tI studies have shown that cl ac+vaies PKM: it ;L~J~WIY.

to pritnarily cause an increase in X.Promolcr (whtbti hound at 0,2: Meyer & l’tashw. I!##: Haule\, S- SIvClurv. 1982). Thus. \v(l twltrirrcl two t’tttios of raits wnst~atits to tlrwrihr this system: X,,,,,jX.l,RM2 atitl k,,/k,,,,. Thr approach was itietrticxl to that outlined ahove for rrsolution of’ AG, trtltl A(irRM f’rom t ht. t~slwssions listed in Table -t. Tttc twolvc~l rate ratios were 11 and 11, resprc*tivc~ly; 1 tiv\~ \vt’rv related to ahs0lul.e rate cotist ants hy tletinitrg /cPRMZ as t’he unit ra,te antl rounding up the vaIu(~ of t tttl dt~termittation of’ kPRMZ it/ rifro 1,~ Hit\~It~~ k Mc(“lurr (19X2) to WOO1 per sec~)ntl ‘to c~nrptrlrsiw that only the two ratios art l<trox\t~ t’rottt our analysis. Thus, the twwl\-~(1 ratios t)~(~tttttr~ \.altttbh ot 0~001 s-t, 0.01 1 s-t and 0.014 s--’ t’cw I, iPRM2. “PRMI

and k,,. where Hawley & Mc(‘Ittrt~ (l!W) hatI reported WOO07 s-t. 0~0078 SC’ wtitl 0.010 h I: ttottt that the ratios of’ valurs intlt~ltrttdetrtl~ tl(~rivt~tl from trsults oht,ttinrd it/ /.ivr~ and t how rrltortwl filr studies in vifrr, aw the samt~.

In Figuw IO(a). promotc~r ac+i\-it)- is c~alc~trl;tt~tl using thew values as a filtict.iott of’ fwtl RN;! pol~mtww lcvrl (thtl hrokrti litit, indiwtw SO t~axi- RSA pal\-tnwase level). By cwnparison wit It Figittv S(a). it IS c~lear that activity in the ;tt)sc~nc~v of’ wgulatory proteins is sitnlJl\- l~rol~ortiotr;tl to tttc. prohahility of closed c~~tnplrx format ion. I ti Figure, IO(h), promoter activity c~aiculatrd as a futtct ion of’ rT lrwl is shown for :I 3011M l(~VPI 01’ RN=\ polymeraw. Thv separate cwntrihutiotts of’ the O, configurations without ~1 at OR2 (PRM basal) and thonr with ( PRM stimulated) i\rr stro\vtr to sptrtr distitivt rattgw of thta (,I Irvt~ls. *it low rc~lwssor c,oncrntrations. 1 here LV~LS no stittiulatioti 01’ l’,nz: t>hus. PRM it(*tivity Was simply l)t’OpOrtiOt~iLI t 0 t Oti1l OWL~~~~II~~~ of RNA polytneras~. Thrrv wits also no competition for RXA pal\-mrrase ttindittg t 0 PR itt low rc~pressor It~vels. ,411 itrcwase itt wptwsor c~oticent’ratiol1~11 increased OR2 ownpanc~ 1)~. CI ;~ti thus the rate of c*losrd c~~tnplrx isomc~riz;tt,iott at. I’,,: Fipuw 10(h) shows that the ac.ti\-it>. of’ PRM incwawd by a maximal factor of nitw at Iysogrtiic. wpressor Ivvels. Thus. the cwmt)iwtl t&v.ts of exc~luded hitiding and catalysis were sc’w t 0 ltrovitlt~ for maximal CT t rattwription at lysogtv~ic~ IVVPIS of’ reprrssor (1 hrx vwt ica.1 linv indivat (1s 200 trwf~l level). ;\t Itighc~r Ievc~ls ot’ ~1. t hr l)rottrc,t(*r ~vas repressed by CI occwpanvv of O,:S. This agtws wit It c~xperimrnt~al results obtt;ined it1 c*iw (Jlv?c,t, it r/l.. Ic)XO. see Fig. ti therein; Meyrr & f’tashttc~. 1980) and may he judged against rat.ios (‘. f iltl(l g givrw in Tahlc 1A.

Figure 10(c) shows the c+fe:cts of’ chatigw iti t’tw energy on promo&r acti\-it>-. The maximal ac+ivit> of both prornot~ers and the position of tlrca [Jrxk IjRM activity both change slightly. Howevw. t ttv total promotrr activity is more srrisitiw t 0 C’tliLtlgC’S itt

Physical-Chemical Model of 1 Regulation 227

(a)

-II -10 -9 -8 -7 -6

t.og [RNAP]

E P

-9 -8 -7 -6 -5

a Log [cI total]

-9 -8 -7 -6 -5

Log [cl total]

Figure 10. Promoter activit,g. Unless noted otherwise, interaction energies are as listed in Table 1 for repressor and Table 3 for RNA polymerase. (a) Calculated P, and P,, activity in the absence of CI and cro. P, activity: {((c,) + (c,&) = ((OP)+(PP))}k,,, P,, activity: {((cd + (cl&) = ((Poe) + W)))~RM~. (b) Promoter activity: (RNAF at Promoter),&,,,,,,. = (open complex/s). P,, stimulated: (RNAP at’ PRM)lkPRMI; max. of 6.5x lo-’ at 174nm. P,, basal: (R’NAP at pRM)*kPRM*; 7.9 X 10e4 at [cI] = 1 nM. P,, total: P,, stimulated + P,, basal: 6.5 x 10F3 at [CT] = 174 nM. P, total: (REAP at PR)kPR; 1.3 x lo-* s-l at [d] = 1 nM. The continuous vertical line indicat,es lysogenic [CT] = 200 nx. (c) Effect of c1 on total P, and P,, activity ior 3 sets of AG, and AG,, values: - 13. - 12 keal (-- - --); - 12% - 11.5 kcal (-- -): - 12. - 11 kcal (- -). [Rh’APj = 30 rm. The continuous vertical line indicates lysogenic [cl] = 200 nM.

k Promokr~ This is shown in Figure 11 for P, and I’,, by var-\ring thests rat>ios by 5Oo& above and below the resolved values. The posit’ion of maximal I’,, activity does not change. although the maxima1 value increases. Similarly. at very low CT levels, P, ac%ivity reaches different plateaus but, as the CT level increases, the activity funnels down to a narrowly defined value. (Note that Fig. 8. which shows protein levels for a positive control mutant, indicates t,he system response to a kpRMl/kPRM2 ratio of 1.)

Sensitivit!~: The 0, network is clearly sensitive to the relative magnitudes of parameters describing

21-+--.., p I- 18

----I ..,R

\. ‘\ 14 1. ;\

(b)

-9 0 -8.0 -7.0 -6.0 -5.0

Log [cI] total

Figure 11. P, and P,, activity a.s a function of CT level and ~~~~~~~~~ ratios. Llnless noted otherwise. interaction energies are as listed in Table 1 for q’rp’ssor and in Table 3 for Rlu’A polymerasr. (a) Variations in k ‘k ratios: 18. 14. PRMl’ PRM2 11 (wt), 8. 5.5. (b) Variations in kpR/kpRM, ratios: 21. 18, 14 (wt), 10. 7. (AG, = -12.5, AG,, = - 11.5; kpRM2 = 0.001 open complexes/s.

RX=2 polymerase characteristics. However. it was generally found that, for any values within the error limits listed in Table 3, the qualitative behavior of the predicted lysogenic and lytic behavior is the same. (1) During lysogenic growth no cro is made (i.e. P, is off) and P,, is on. (2) During induction of lysis, CI is degraded in half an hour and there is a pause before commit’ted lytic growt,h (I’,, off. P, on); ultimately, cro turns P, off (early genes off; the tof function of cro). This is shown ‘in Figure 12, where lysogenic and lytic protein levels are calculated for variations in the free energy of RNA polymerase binding.

(c) Proportionality constm& for protein synthesis

According to our model, the rate of repressor or cro synthesis is equal to the product’ of P,, or P, act’ivity. respectively. and the synthetic proportionality constant, ilp,,,,in. This constant, reflects the effective number of open complexes that initiate transcription and the number of protein molecules made per transcript,. It has dimensions of protein molecules/open complex. To evaluate A,! the constant for repressor, we assumed that the number of repressor molecules should double over a

0 IO 20 30 40 50 60

Time (min)

Figure 12. Comparison of wild-type behavior for 3 srt,s of RNA polymerase interaction energies (AG,, AG,,). - 134. -124 kcal (-- - -); -12.5. - 11.5 kcal as listed in Table 3 (--- ~): -12.0. -11.0 kcal (- - - -). I-nless noted otherwise. interaction energies are as listed in Table 1 for repressor and Table 3 for RNA polymerase. ,\I1 other constants are given in Table 3. (a) Maintenance of the Iysogenic state calculated over a 1 h period. (I)) lnduct,ion of Ivsis calculated over a 1 h period. No non-spe(sific mR?Ft\ or protein degradation w-as introducrd.

generation time in a lysogenic cell. The absolute value that we chose to use for all of our subsequent modeling WtlS A, = 11 monomers/P,, open csomplex. There are no precise determinations of caellular cro levels. Because control of c1 and cro levels occurs primarily at the transcriptional level. we arbitrarily chose A,, = 6 dimers/P, open complex as a value close to il, (5.5 dimers/P,, open complex). We ascert,ained that the qualitative behavior during lysogenic and lptic growth was the SiLlllP for a range of svnt’hetic constant’s (caalculations not, shown).

(d) Rate ?;f’ repressor deyradation

Equation (2) was formulated to include specific cl repressor degradation catalyzed by recA according t,o the model of Phizicky & Robert,s (1980). They showed that recA catalyzes a concentration- dependent degradation of CT monomers in vitro in the presence of single-stranded DNA and ATP. To determine the rate and extent’ of this degradation. we simulated their data for repressor monomer drgradat)ion as a function of initial repressor

concentration using the h’nSsn value ot’ t’0 n>l (Sauer. 1979): comparison of VitlUt5 of mid-pha>cl slopes n-it h the experimentally dt~trrmirie~l (‘urv(’ showrd t,hat the degradation rate ir/ ritrci \~as

approximat,ely 0.0065 per stwmd.

The studies of repressor inactivation ir/ rlic~) reported by Bailone rt nl. (1979) provided another quantitative limit on the range and characteristicss of repressor degradation. They showed that thcx

cellular inducer destroyed almost all of the

lysogenic repressor in approximately 30 minutes. (Alt,hough t,hrir studies showed a lag in the onset of

repressor degradation of about 1Omin. the simulations presented in section 6 of the main text assumed a maximal protease activity from ttitl beginning at f = 0 s,) Throughout this paper, we used a value of 04065js because of its caonsistency tvith hot h the dat’a of Phizicky & Roberts (1980). and the observed repressor levels during induction of’ lysis in virw (Bailone rt al.. 1979).

(cs) India1 concentrnfion of p-&ins

Our estimates for the amounts of repressor, cro

and RNA polymerase found in a normal lysogen are shown in Table 3. During Ipsogenic growth. the only active lambda promoter is P,, controlling CT (Kourilsky et al.. 1971) and rex (Howard, 1967). The amount of repressor in solution has been determined to be approximately 200 molecules/cell (in monomer units) or 200 IlM (Riechardt & Kaiser, 1971; Levine et al., 1979).

Based on a dissociation constant of 20 nM (Sauer. 1979), 200 nM t,otal represents an X0 nM con- centration of dimers. Although this value may underestimate the t,otal amount) of cellular repressor by- omitting some that may he bound non- specifically to DNA, this estimat,e is close t)o the value that is significant for calculaGng the equilibrium distribution of repressor molecules bound t’o the right) opera,tor (see discussion of non- specific binding by Ackers et al., 1982; Sauer. 1979).

To t*est, t,he sensitivit,y of the model to /Ro], the init*ial eoncrntration of repressor. we varied its value in 25 nM increment)s hetwern 0 n&l and 400 nM. and in 100 nM increments hetwern 100 UJI

and 1000 nMM. During Iysogenic growth (i.e. no degradation of repressor); essentially no cro was produced for [R,] values equal t)o or greater than 75 n>f (calculations not shown). I)uring inductjion of Iysis (i.e. repressor degradat’ion by ret>A). repressor was not degraded sufficiently wit,hin one hour for substantial cro or X to be produced if / R, 1 was near 1000 nM. These result!s are consistent with the observat’ions in lrino reported by Eailonr r>t rrl. (1979). who observed that) repressor at half t,hr normal Iysogenic level (i.e. 100 nM) could maintain the lysogenic state. while a hyperimmune phage (containing a repressor level S-fold higher than normal) could not be induced. Tf we allowed degradation of repressor monomers t,o occur in the absence of repressor catalysis of RNAP isomerization. the swit’chover occurred at, earlier

Physical-Chemical Model of A Regulation 229

times but still allowed for subinduction at [R,] equal t’o or greater than 75 nM; at lower initial concentrations, there was no lag in cro or S protein production (calculations not shown).

cro protein and mRNA for cro is not apparent in a lysogen (Kourilsky et al., 1970); therefore. the initial cro concentration was always set to zero.

The concentration of free RNA polymerase in the cell is not known exactly; however, there are reasonable limits that may be imposed. The value of 30 t1M was chosen on the basis of studies by McClure and co-workers of abortive init’iation at lambda promoters (McClure, 1983). It remained a, fixed value throughout our calculations. regardless of other cellular conditions. However, by inspection of Figures 9(a) and 10(a): it, is possible to see how changes in the free concentration of RSA polymerase may modulate gene transcript’ion at times during the cell cycle by reducing or increasing promoter activity. These changes will be affected by the actions of repressors and activators as indicated in parts (b) and (d) of Figures 9 and 10 for the OR control system.

This work is part of a dissertation (M.A.S.) submitted to The <Johns Hopkins University (Department of Biophysics) in partial fulfilment of the requirements for the Doctorate of Philosophy. It has been supported b>- research grant GM24486 and Predoctoral Training grant (:M0731 from the Xational Institutes of Health.

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