J. Mol. Rid. (1985) 181, 211-230
The 0, Control
System of Bacteriophage
A Physical-Chemical
Lambda
Model for Gene Regulation
Madeline A. Shea and Gary K. AckersfDepartment
of Biology and McCollum,-Pratt
The Johns Hopkins University
Baltimore,
Md 21218, U.S.A.
(Received 30 August
Institute
1983, and in revised form 31 July
1984)
A quantitative model has been developed for processesin the bacteriophage lambda that
control the switchover from lysogenic to lytic modes of growth. These processesinclude 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 promot8ersP, 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 CTrepressor 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?avitro,
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’emsof gene
regulation.
1. Introduction
events through the interactions of regulator\proteins with specific DNA sequenc&, e.g.
repressors binding at operator sites (Johnson et al..
1981): and (2) those based on feedback control at
the translational level (von Hippel 8t Fairfield,
1983; Kowalczykowski
et al., 1981; Lonberg
et al..
1981; Newport et al., 1981). In both types of
system. the roles of protein-protein interact’ions
between the nucleic acid-bound proteins have
Molecular mechanisms for the regulation of
gene expression continue to be of major interest
in molecular biology and also now comprise an area
of rapidly increasing biophysical study. From work
on prokaryotic systems, several general types of
regulatory mechanisms have emerged including:
(1) those based on the control of transcriptional
___---t Author
addressed.
to whom all correq)ondence
emerged
as major
elements
of control.
The bacteriophage lambda embodies a complex
family of interlocking regulatory mechanisms. of
should be
211
0022-2846~85102021
l-20
$0X00/0
fl
1985
Academic
Press
Inc..
(Lor~ci~m)
Ltd.
the first type, that cbontrol: (1 ) processes of infec4ion
tjhr
lysogenif*
state:
Mid
establishment
of
(2) maintenance
of prophage
in the lysogenic state:
and
(3) the induction
of lysis in response
t’o
reviews,
see
environmental
“signals”
(for
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
rnechanism 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
Action
of CI repressor
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
I’rinvipal
ahions
of t,hr oI)erator-bolllltl
ijroteins
are indicated. The 2 repressors
~tIltagonistic~tl1~
at 0, to control
promotrr
of
our mofirl.)
regulat,or~
cl and (‘ro act
af*t.ivky
anfl
thus. the prophage life cycle. The fundamental
requirement 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
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 processesin 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 assessthe 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 timedependent 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 cooperativity or co-operative interaction will be used
throughout this paper to denote a difference
between the Cgibbsfree energy of binding a given
of
1 Regulation
OR3
213
OR2
j--f
0,I
/
PR
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 monomersas
shown (Sauer. 1979). The dimers bind specifically and
reversibly to each of the 3 operator sit,esasshownby 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 capableof
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 cooperative 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 thermodynamic. Numerical values of these parameters are
given in Table 1.
(b) Assumptions
to
Assumptions (1) to (7) as given below were used
generate the microscopic molecular con-
Table 1
Interaction
Free energy
0,.repressor
OR-w0
(kcal)
free mergies
for repressor
and cro binding
AG,
AG,
AG,
-11.7kO.03
- 10.8 +O.Ol
- 10.1 kO.05
- 10.8kO.01
- 10.1 kO.03
-12~1~0~01
Resolved
from analysis of footprinting
titration
data (Johnson
el al.,
methods described previously
(Ackers et al., 1982); conditions:
0.2 x-KCl.
the free energy of binding cro or rI repressor to the respective
sub-subscripted
the
proteins are bound. The co-operativny
terms AC,x 2 and AG,, represent
binding to 2 adjacent operator
sites over that obtained
by filling the same
(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 lessobvious.
(1) The three binding sites are specific. nonoverlapping, 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
(<Johnsonet 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
at 0, (37°C’)
AGI,
-l.Q+O.OA
0.0
A(&
-2.0+0.06
0.0
1979: Johnson,
1980) according
to
AC,, AG, and AG3 each represent
site of the operator
when no other
additional
energy for simultaneous
sites individually.
See Ackers ft trl.
equilibrium
statist’ical
t hertnodynamic
probabilities.
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 thermodynamics (cf. Hill; 1960). The probability
of
an operator in configuration s may be written in
general :
AG,/RT)
cc,>=-c expexp(- (-AG,/RT)
x [Rz]‘[Cz]j[RNAP]k
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
co-operative
interaction
terms
plus
appropriate for a given configuration s (see
Table 2). These energetic contributions
were
determined from experimentIs in do and in /,izjo
(seeAppendix and Table 1).
With t)heseand the remaining assumptions of the
model, we calculated the distribution of the various
configurations, the promot’er activities and concentrations of repressor and cro as a function of
time and changing cellular conditions.
(9) Transcription initiation is the rate-limiting
st,epin cro and CTrepressor synthesis (Ytashne et al..
1980): it is proportionaJ to the product, of two
saturation
of the
terms: (I) the fractional
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
215
Table 2
ConJigurationa
8
Configurationt
0,3
0,2
Non-liganded
1
0
Singly
2
3
4
;
7
ii
of 0, and their total free energies
Component
free
energy terms
0,1
Total free
energy, AC, (kcal)
species
liganded
0
0
R
00
c”
0
RNAP
0
0
species
0
R
0
Reference
state
0.0
R
0
0
0C
0
0
-11.7
- 10.1
-10.1
- 10.8
- 10.8
-12.1
-11.5
- 12.5
c0
0
0
RNAP
Doubly
10
11
12
13
14
15
16
17
18
19
20
21
22
$23
$24
liganded
0
R
R-0
c
C
RNAP
0
0
R
t:
R
;;
ii
RNAP
RNAP
(’
$27
$28
RNAP
RNAP
Triply
29
30
31
32
33
34
35
1;;
$38
$39
$40
liganded
R
c
(’
R
R-R
(
(
RNAP
RNAP
RNAP
RNAP
species
R-0
R
c
0
C
RNBP
C
R
0
0
C
R
RNAP
R
0
RNAP
C
0
species
R-C
R-c
R
C
R
(1
R-C
C
R
R
R
0
c
C
0
0
c
AG, +AG,+AQ,,
AC, +AG,
AG,+AG,+AG,,
AC,. +AG,.
AC,.+AG,.
AC,. + 80,.
AC, + AC,,
AC, + AC,.
AG,. fAG2
AG1.+AG3
AC, + AG,.
AC,. +AG,
AG, + AC,.
AGR +AG,
AG, + AGsM
AG, + AC,,
80, +AG,.
AC,. + AC,,
AC,. + AG,,
-23.7
-21.8
- 22.2
-21.6
-22.9
-22.9
- 24.0
-22.5
-20.9
-20.9
- 23.8
-20.9
- 22.2
-22.6
-21,6
-23.2
- 24.6
- 22.3
-22.3
R
C
R
R
C
C
C
R
R
C
R
C
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
R
C
C
R
0
0
0
R
t R, cl dimer; C, cro dimer; RNAP, RNA polymerase;
for adjacent R dimers, a co-operative
interaction,
primed
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
unidirectional.
(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
PRM (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
0, empty
subscripts
site, (- -) indicates.
indicate cro intrinsic
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
dCz
dt = (RNAP
at P,)k,,A,,.
(2)
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); kPRM2is 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) +
rate at P,
((-23) + Cc,,); bR is the isomerization
(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 concentrations
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
synthesis
ma.) be applied
to art!
protein
combination of operator. promoter and transcribed
gene, regardless of whether such an arrangement
occurs naturally in tht> cell or only after manipulation 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,ratlscriptionally
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 transcript. 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-translational processes that’ include amino acid modificat’ion, protrase activation, etc.
Tn this trcatmrnt.
the pro(“ss’s
c,otltjrolling
transcription
init,iation
are separat.ed into lu’c)
classes.
energetic
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--cwntact
s.
protein
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
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
repressespromoter P, by rendering the probability
(RNAP
at PR) negligible. However.
while
repressing P, at’ lysogenic concentrations. it’ exerts
positive control over its own synthesis (seePtashne
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
exclusively
serve
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
activities
representing
the
two
promoter
(phenomenologically similar, but mechanistically
distinct) basal and stimulated rates of CT transcription 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 Mever
,.
& 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
217
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 concentration of monomers was calculated from the tot)al
repressor concentration and KDssn, the dimermonomer 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
discussedin section 5(e).)
4. Computational
Methods
All calculations were performed on a HewlettPackard 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) performing 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 eightconfiguration 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 concentrationdependent 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)),
a~c~ortliripto 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
fortnation.
tneaningfitl
b's
0
PRM),
ittld
(RNAAP
at PR,)z differed k)y a fwtor
ot
1 1. the ratio of’ the isomerization
tatr c*ortst,ants
t2PRMl 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.
,/
IO
20
30
Time
40
50
60
(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 numbersof moleculesbecausewe 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’Osynthesis rate.
c~hosenover 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
(3).
(ii) I’rottrofrr
of closed c~otnples
cI monomer_____________-m-m-__________-_---.activity. i.e. thr net rate of open complex formatiotl
_________________________________
(<RSr\P
at P,,)k Promoter).This was rspeciall~
cro
I
obvious for rT transcription initiation. where qua1
values of (RSAP at PRM),ota,muld (Borrespondto
(b)
dramatically different values of d[R]/dt. This is
hecause the relative contributions
of (KSAP
at
0~010
cqtation
prohahility
Thus.
in some caasrs it was I~OIY
to exarninr
changes
it) protnotrr
Under initial conditions, t’he calculated degree of
repression of P, was 950,, whereas the repression of
PRM 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 polymerase (Ackers Pt (11..1982).
I’sing equation (1). we det’erminrd the tiistrihution 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,ionsof 0, were combinations of
repressor and RNA polymerase act’ing to repress cm
transcription from P, and stimulate c1 transcriptdon from P,,. This is indicated in Figure 3(h) by
the calculations of promoter acti+v.
(i) I’rotritt
At
func+iotrs,
and
afhity
its the lysoyen,ic
during
repressor
inductiotl
of lysis
intluc%ion of
monomers
I>SA
are
repair
srlrctivelv
degraded in a reaction mediated h)- reca=\.Thus. t;)
simulate induction of lysis. liRdwas srt equal tIo t,he
value cited in Table 3.
(Talculat’ed levels of regulatory proteins repressor
and
occupnncy
let&
the onset of
(xro
for
a
normal
Iysogen
are
shown
in
;-\t each point during the simulation. the relative
Figure l(a). As in the calculations ot’ Iysogenic
growth,
no mRSX
or non-specitic protein
degradation is rcprcsrnted in 1hr 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
probabilities
very
?stntP
A promoter in equilibrium with regulatory
proteins may exist in t,hree functional states:
vacant.
repressed.
of all
or occupied
by RSA
configurations
were
polyrnerase.
calculated
low
and
the
effecat was
revrrsibk
unt,il
the
Physical-Chemical
Model of I Regulation
219
Table 3
Model parameters for dynamic calculations
RNAP intrinsic energy
Basal ~Plomoter
Stimulated kPromo,er
Proportionality
P&repressor
P&r0
- 11.5 + 0.5 kcal/mol
0.001 s- 1
0~011+0XK~3s-’
- 12.5 f 0.5 kcal/mol
0~014+0~003 s-l
constant
11.0 monomersjoct
Rate of degradation
Lysogenie protein concentration
Dimer dissociation constant
RNA polymerase concentration
Step&e/time
increment
of OR behavior
0.0065 s - ’
200 nM
6.0 dimers/oct
OnM
2OnM
30 nM
for numerical
integration
calculations
30nM
= 10 s.
t Protein molecules/open complex.
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,ersrather 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 viz10predictions of the
these levels by the model.
60
(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 sensesdescribed above.
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:
max. of 6.45 x 10e3. at t = 0 min.
(RNAP at PRM)IkPRM1z
_
max. of 2.5 x 10e4. at
P,, basal:(RRAP at P,,),k,,,,,
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.
(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
,010
005
0
IO
20
30
Time
40
50
(min)
(*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
bindiny
repr~swr
(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
growth
by
a
“non-co-operative
lysogenic
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 polymrrase binding
to PRM, while they both interfere
with
its binding
to P,.
I>urinp
tht> induction
of lysis of this “nonc*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
interactions
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 nonco-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,eract,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.
= open complex/s.
PRM stimulated:
Promoter)&.,,,,,,,.
(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*hover 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
400
_______________-----200
5
c
ut
9
9,
c
cro
non-co-operative
-.
______________---*------_____-
(b) 1
0
IO
20
30
Time
40
50
60
(min)
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
interartions. The energy of each configuration
is determined
solely by the inbrinsic binding free energies. (a) Repressor
recA-mediated
and
cro
protein
levels.
Specific
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
is complete
reduced stimulated P,, activity; degradation
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.
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
(b)
0
to
20
30
40
50
60
Time (mm)
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+vwere 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 nonco-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)).
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 = kpRM2 = 0401 5-l). In that lysogenic case.
repressor reached a lower value and cro reached a
Iyt ic growth b\- c*ombating degradation slightI>
more c$ti\c~tively. The caal(*ulations suggest t*hat
NO
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
PRM stimulation
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%).
(bi
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).
0
IO
20
30
40
xl
60
However, even small changes of the dissociation
Time(min)
consta.nt in either direction had dramatic effects on
Figure 8. Comparison between behavior of a wild-type
the dynamic characteristics of the induction of
lysogen and one lacking positive control of PRM. (I’,,
lysis. (These also reflect the lysogen’s ability to
total = {(REAP at P,,)1 +(RXAP at’ PRM)Z]kPRM2). withstand normal variations in repressor concrn(a) Maintenance
of the lysogenic state calculated o\~r a
tration.)
Because of the distinct, roles of the
I h period
for a wild-typr
(continuous
lines) and
oligomers (dimers regulate transcription initiation,
unstimulated
lysogrn (broken lines). The levrl of CI
while monomers are specifically degraded), the
remains constant (no net synthesis). This modification
system is very sensitive to the equilibrium between
renders the prophage stable over a sin&= generation time
them. During the induction of lysis. t’he monomerbecausr I’, is oE, however. aRer several clell divisions
dimer equilibrium apparently serves t’o buffer the
have occurred,
the remaining
~1 rrpressor
will IW
insufficient to maintain the lysogenic state (calculations
competition bet’ween degradation of monomers by
llot shown). (b) Induction
of lysis calculated over a I h
recA and synthesis of monomers stimulated by CI
period for a wil&type
lysogen (continuous lines) and one’
dimers bound at 0,2. This dynamic: buffering was
lacking
positive
control
of I’,,.
Qualitatively.
thtb
reflect,ed in the systjem‘s lag period before
chararteristics
of thr induction process are idrnt,ic+al.
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
higher value than if the system were altered by
from partial deplet’ion of repressor (Bailone et nl.,
either effect alone. However, the loss of repressor1979). This phenomenon of subinduction is one of
repressor co-operativity definitely had the most
several features that were not, incorporated
devastating effect on lysogenic growth.
explic+tly into the model. but appeared in the
Our model predicted that induction of lysis
simulations of prot’eiri synthesis as direct conwithout positive cont,rol was essentially identical to
s’quenct’s of the contributing molecular interinduction of lysis by a normal prophage (see
actions. Simulations using altered values of t,he free
Fig. 8(b)); P, reached maximal activity after 20
energies of co-operativity (SW Fig. 6) or the dimer
ratht>r than 27 minutes. The subinduct’ion
dissociation const,ant (cha,lculationsnot shown) also
phenomenon was intact,
demonst~rat~rtl t,he importance of
poignantI>
The fact that’ repressor can exert positive control
repressor--repressorinteractions.
over it,s transcription from P,, but repressesP,, at
high levels provides for a peaked level of P,,
ac%ivitjy. although P,, occupants is a monotoni6. Discussion
caally decareasingfunction of addItIona ~1. Positive
control is known to compensate for the weakness of
The analysis and calculations presented here
I’ RMrtllative to I’, and the absence of a high affinity
illustrate a method whereby quant)itat’ivr theorirs
ribosome binding site on the c1 mRn’A originating
of gene regulation ma,y be formulated and tested.
at) P,, (Ptashne, 1978): our calculations showed
The model developed in this study embodies a
that it prolonged the lag period before committed
particular view regarding the physical-chemical
--
Physical-Chemical
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
thermodynamic 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
were
known
least accurately.
The
synthesis
lysogenic protein concent,rations
were known tnost
accurately for repressor and cro; for RKA polymrrase. 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
very few
Appendix).
Thus, we have adjusted
parameters
and it is unlikely
t)hat) we have
discovered coincidentally
a set of values t’hat are
with
the
known
physiological
consistent
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 suhinduc+ion
(see sections
6(b) and (e)) and the
Model of i Regulation
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 interactions 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%).
l)rrsrntrd
here cbotltains tickvera I
Thcl mo&l
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.
sitespecific 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:
the calculations
make vrry sl)ecific
fortunat’ely,
quantitative
statements
about
the
rspe&Yl
behavior in I*~LYIand 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 illtrrpretations
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 leastsquares 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 equilibnum 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
kpromoter(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
cnotnplexformation. 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 CTOfrom 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-eanalvzed.
Mathematical expressions for these quantities
were formulated according to t’he mode1of proteillDXA 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~rdin 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
absencaeof 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&,, becauserate 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 prrcsisethan
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
Promoter
a
1)
P RM
P RM
P RM
P RM
PR
PR
P RM
P,
ii
e
f
g
h
Denominator
OR
wt
wt
200
200
0,30,3-
200
200
P RM
P RM
P RM
P RM
P RM
P RM
P RM
IlM
nM
nM
nM
OnM
200 nM
wt
wt
wt
wt
200
nM
OnM
t Represents
the estimate of total
$ Maurer et al. (1980).
$ Meyer et al. (1980).
11Meyer & Ptashne (1980).
wt. wild-type.
a
k,,,,/k,,,,
b
c
(K,R’+K,P)/(K,R+K,P+l)
(K,P+
1 +K,R’)/(K,P+
d
e
f
(K,P+l+K,R)/(K,P+l)
(K,P+
l)/(K,P+
wt
wt
0
20-40~~
20-40 /lM
200 nM
OnM
200 nM
R Inax
400nM
wt
expressions
1 +K,R)
ICI1
wt
wt
wt
wt
in monomer
analytical
0,
0,3-
PII
~1 level expressed
R. Approximate
!4
h
Promo&r
[at
Ratio
7-11:”
10-201
2-3:
l.l-1.251
12-20$
0~02-0~04~
14:
75~1005
units.
for ratios
= 7-l 1
= lo-20
1) = 2-3
= 1.2
b&u,,
((1 +&,P)/W’){KJ’I(l
+&P)j
= 12-20
(K,P+
l)(K,P+
l)/K,K2K,,R2(K,P+
1 +K,R)
= 0.02-0.04
10-s (KsP(K,P+
1)/0.4)“3
= R,,,
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
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 cooperativity (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 configurations 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,,
by the
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 interactions 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 antico-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,sof AG,
and AGRMvalues 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
-9
-7
[CI
-6
-5
total]
-7
-8
Log
-6
[RNAP]
-8
Log
-9
-7
-8
Log
-6
-5
[CI total]
Figure 9. Promoter occupancy as a function of cl and
RN\;A polymerase levels. Unless noted otherwise. intrraction 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
the product
of (R%AP
ate Protnoter)
wit)h
where
this r&e
constant
reflects
the
irreversible
isomerization
from
closed
complex. This is a simple formulation
that,
explicitly
account for the lifetime
of open
(1) to he
X.Pro,,,oter.
speed of
to open
does not
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 1Irc~
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\,
SSIvClurv. 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; 1tiv\~ \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
where
Hawley
& Mc(‘Ittrt~
(l!W)
hatI
and
k,,.
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~axiRSA 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.
1here 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
t.og
-8
-7
-6
[RNAP]
I-
14
E
-9
P
a
-8
-7
Log [cI total]
-6
(b)
21-+--.., p
..,R
18
----I
\. ‘\
1. ;\
-5
-9 0
-8.0
-7.0
Log
-9
-8
-7
Log
-6
-5
[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]= 200nx. (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 indicateslysogenic[cl] = 200nM.
k Promokr~
thests rat>ios by 5Oo& above
and
[cI] total
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)
below
the resolved values. The posit’ion of maximal I’,,
activity does not change. although the maxima1
value increases. Similarly. at very low CTlevels, 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 responseto a kpRMl/kPRM2
ratio
of 1.)
Sensitivit!~:
The 0, network is clearly sensitive to
the relative magnitudes of parameters describing
-5.0
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
(a) Variations
in
Table 3 for
Rlu’A
polymerasr.
k PRMl’ ‘k PRM2 ratios: 18. 14. 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.
This is shown in Figure 11 for P, and I’,,
by var-\ring
-6.0
Proportionality
protein
constm&
synthesis
for
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
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
0
IO
20
30
Time
40
50
60
(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
WtlS
A, = 11
monomers/P,,
open
modeling
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 concentrationdependent
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
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 concentration
of dimers.
Although
this value
may
underestimate
the t,otal amount) of cellular repressor
by- omitting
some that
may
he bound
nonspecifically
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 nonspecific 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
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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