Molecular Biology of the Cell
Vol. 6, 1367-1380, October 1995
Computer Analysis of the Binding Reactions Leading
to a Transmembrane Receptor-linked Multiprotein
Complex Involved in Bacterial Chemotaxis
Dennis Bray*t and Robert B. Bourrett
tDepartment of Zoology, University of Cambridge, Cambridge CB2 3EJ, United Kingdom; and
tDepartment of Microbiology and Immunology, University of North Carolina, Chapel Hill,
North Carolina 27599-7290
Submitted May 15, 1995; Accepted August 1, 1995
Monitoring Editor: Lucy Shapiro
The chemotactic response of bacteria is mediated by complexes containing two molecules
each of a transmembrane receptor and the intracellular signaling proteins CheA and
CheW. Mutants in which one or the other of the proteins of this complex are absent,
inactive, or expressed at elevated amounts show altered chemotactic behavior and the
phenotypes are difficult to interpret for some overexpression mutants. We have examined
the possibility that these unexpected phenotypes might arise from the binding steps
that lead to active complex formation. A limited genetic algorithm was used to search for
sets of binding reactions and associated binding constants expected to give mutant
phenotypes in accord with experimental data. Different sets of binding equilibria and
different assumptions about the activity of particular receptor complexes were tried.
Computer analysis demonstrated that it is possible to obtain sets of binding equilibria
consistent with the observed phenotypes and provided a simple explanation for these
phenotypes in terms of the distribution of active and inactive complexes formed under
various conditions. Optimization methods of this kind offer a unique way to analyze
reactions taking place inside living cells based on behavioral data.
INTRODUCTION
The signal transduction network that controls bacterial
chemotaxis employs a cascade of transient protein
phosphorylation and dephosphorylation reactions to
transmit information about the external environment
to the flagellar motor (for reviews see Bourret et al.,
1991; Stock et al., 1991). As is the case for many eukaryotic
signaling pathways (see for example Pawson
and Schlessinger, 1993), the intracellular signals in
bacterial chemotaxis originate in transmembrane receptor-linked
multiprotein complexes that form on the
inner face of the plasma membrane (Gegner et al.,
1992; Maddock and Shapiro, 1993; Schuster et al.,
1993). The complex responsible for chemotaxis to aspartate,
in particular, contains the dimeric aspartatebinding
receptor Tar (TT)Y (Milligan and Koshland,
1988) and two cytoplasmic proteins, CheW (W), which
is a monomer (Gegner and Dahlquist, 1991), and CheA
(AA), which is a dimer under most conditions (Gegner
and Dahlquist, 1991). The complex is believed to have
the composition TTWWAA (Gegner et al., 1992).
The ligand binding state of TT controls the autophosphorylation
rate of AA in the complex (Borkovich
et al., 1989; Borkovich and Simon, 1990; Ninfa et al.,
1991) and hence the flow of phosphoryl groups
through the signal transduction pathway (Figure 1).
The pathway branches with the transfer of phosphoryl
groups from CheA-P to CheY (Y) and CheB (B) (Hess
et al., 1988b; Wylie et al., 1988). Y can also autophosphorylate
using small molecule phosphodonors such
Abbreviations used: AA, CheA dimer; B, CheB; Bp, phosphorylated
CheB; CCW, counterclockwise; CW, clockwise; R, CheR;
TT, Tar dimer; W, CheW monomer; Y, CheY; Yp, phosphory-*
Corresponding author. lated CheY; Z, CheZ.
i 1995 by The American Society for Cell Biology 1367
D. Bray and R.B. Bourret
Figure 1. Schematic view of
the receptor-associated comT1TTT
plex and its role in the
transduction of chemotactic
signals. The chemotactic stim|z>'AADPnn
r > r ulus, in this case the attractant
tWW)ATP W ) W ) aspartate or the repellent
nickel ions, bind specifically to
A_/> the transmembrane receptor
A A A A Tar (TT) causing a conformation
change. This is transmit*§P
ted across the plasma memP
brane to a complex of proteins
including CheW (W) and
B ~~~~~~~~~~~~~~~~~~CheAdimers (AA). The "signal"
or conformational change
/(POAc > jF \ + p. causes a change in the rate of
,+Pi > 1 AA autophosphorylation; AA
HOAc +_ 1 autophosphorylation inz
creases in response to repelB
LJl 1 11lents and decreases in response
to attractants. The
phosphoryl group is transferred
from the protein complex
to CheY (Y) or CheB (B).
Phosphorylated CheY then
diffuses to the flagellar motor
and changes the direction of
its rotation. In this way, the
Clockwise detection of chemotactic attractant
or repellent produces
a change in swimming behavior
of the bacterium. Note that other complexes formed from IT, W, and/or AA presumably exist and participate in various reactions as
catalogued in Table 1 but are not shown here. The methylation reactions of the adaptation pathway catalyzed by CheR and CheB are not
included in the simulation and are also not shown here. The simulation includes the possibility of ligand binding to all complexes that include
the receptor TT, but the effects of ligand binding are not explored in this manuscript.
as acetyl phosphate (Lukat et al., 1992). In the excitation
pathway, phosphorylated CheY (Yp) binds to the
FliM protein in the switch at the base of the flagellar
motor to change flagellar rotation from the default
counterclockwise (CCW) direction to clockwise (CW)
and thus control bacterial swimming behavior (Barak
and Eisenbach, 1992; Welch et al., 1993, 1994). In the
adaptation pathway, phosphorylated CheB (Bp) adjusts
receptor methylation state in a phosphorylationdependent
manner to further regulate receptor signaling
strength (Lupas and Stock, 1989; Borkovich et al.,
1992). The signaling species turn over rapidly to accurately
represent changing external conditions. Yp and
Bp spontaneously autodephosphorylate (Hess et al.,
1988b; Wylie et al., 1988), and dephosphorylation of Yp
is further stimulated by the CheZ (Z) protein (Hess et
al., 1988a).
Mutant bacteria in which TT, W, and AA, separately
or together, are removed or expressed in higher than
normal amounts have previously been isolated and
their swimming behavior has been characterized (Parkinson,
1978; Stewart et al., 1988; Liu and Parkinson,
1989; Sanders et al., 1989). Biochemical studies provide
information on some of the crucial binding steps in
formation of the active complex (Gegner and Dahlquist,
1991; McNally and Matsumura, 1991; Gegner et
al., 1992; Bourret et al., 1993; Schuster et al., 1993;
Swanson et al, 1993a,b; Wolfe et al., 1994) but the
pathway of its formation and the dissociation constants
of intermediate complexes are not known.
We have used a computer-based approach to try to
deduce the binding steps involved in the formation of
TTWWAA from TT, W, and AA. Briefly, the procedure
entailed testing the performance of different networks
by numerical integration for both wild-type
and mutant genotypes and changing the dissociation
constants until the performance corresponded with
experimental data. In this way we obtained sets of
binding steps compatible with the phenotype of mutants
with altered levels of TT, W, or AA. Note that
Escherichia coli and Salmonella typhimurium each contain
multiple transmembrane receptor species (reviewed
in Hazelbauer et al., 1990) but for simplicity,
we have considered the case where TT is the only type
of receptor present. Thus changing levels of TT in the
model corresponds to changing amounts of the entire
receptor population in a real bacterium. The mutants
chosen to provide boundary conditions for the optimization
procedure included strains with overexpressed
TT or W, which give the same swimming
behavior as those in which TT or W have been deleted.
This seemingly paradoxical result has been suggested
Molecular Biology of the Cell1368
Analysis of Binding Equilibria
to arise from an accumulation of inactive complexes
(Gegner et al., 1992; Swanson et al., 1993a). Some of the
networks tested included the complexes TTAA and
TTWAA, which have not been fully tested experimentally.
The possibility that TTWAA has full autophosphorylation
activity comparable to TTWWAA was
also examined.
MATERIALS AND METHODS
Formation of the active complex TTWWAA is not affected by binding
of ligand to receptor, nor by AA autophosphorylation (Gegner
et al., 1992). We were therefore able to model the binding interactions
leading to formation of this complex separately from the rapid
signaling changes occurring with ligand binding. Simulation of the
response of a bacterium with a specific genotype entailed two steps.
First, binding interactions between TT, W, and AA and their oligomeric
complexes were allowed to come to equilibrium (representing
the state of a bacterium before an experiment to measure its chemotactic
performance). Then the concentration of TTWWAA produced
by this set of binding interactions was used to predict the
unstimulated swimming behavior of the bacterium using the computer-based
simulation of the phosphorylation cascade described
previously (Bray et al., 1993).
Binding Interactions Leading to 1TWWAA
Each network of binding interactions between TI, W, AA, and their
complexes was tested to see if it could produce the desired steady
state concentrations of TIWWAA for a range of genotypes. Testing
entailed an optimization process in which Kd values were varied
until the absolute difference between simulated TTWWAA concentrations
and target TTVWWAA concentrations-the "cost" value"reached
an acceptably small value. Various optimization procedures
were examined, including variants of the Simplex and
Simulated Annealing methods (Press et al., 1992). The method found
empirically to generate solutions with the least expenditure of CPU
time was a limited genetic algorithm method used previously to
simulate the evolution of a small network of cell signaling proteins
(Bray and Lay, 1994a). In this procedure, random "mutational"
changes are made in the dissociation constants of each binding step,
each set of changes being followed by a selection step. Mutations
were simulated by a series of random numbers between 0 and 1,
generated by the routine ranl described by Press et al. (1992). These
numbers were used to first select one of the dissociation (off) rates of
the network, and second, to specify a factor (which ranged linearly
from 0.6-1.4) by which this rate was to be multiplied. The dissociation
constant of this binding step was then deduced using the
relation:
Kd = offrate
on rate
using a constant value for the on rate, as detailed below.
A starting network, characterized by a set of initial dissociation
constants, was allowed to produce "offspring" networks by an
alternating sequence of groups of five mutations each followed by a
selection step. At each mutation, the dissociation rate (off rate) of the
binding step was multiplied by the factor:
(ran + 0.5)P
where ran is a random number between 0 and 1, and p changed
linearly from 3.0 to 1.3 as the optimization progresses. Networks
were subjected to 300 sequential selection steps in this fashion and
the network closest to the target was stored as an "isolate." Five
independent isolates were collected from each starting network and
the one with the lowest cost was then used as a founder network for
another series of mutations. The entire process was repeated for a
total of three rounds and the Kd values of the network with the
lowest overall cost were recorded. The same optimization procedure,
starting with different sets of initial Kd values, was then
allowed to run automatically for a period of time, sometimes in
excess of a week or more, and the solutions obtained were then
analyzed as described in this paper.
Dissociation constants provided by this optimization procedure
were assessed by numerical integration. Each individual binding
step is represented by a differential equation of the form:
d[AB] = [A] x [B] x ON RATE - [AB] x koff
where A and B represent the two species to the left of an equation
in Table 3, and AB is the corresponding binary complex. k0ff is the
rate of dissociation of the AB complex (in units of sec-1) and
ON_RATE is the rate of association, which we have taken as 1 x 106
M-1s-1 for all binding interactions-a typical diffusion limited value
for protein interactions (Northrup and Erickson, 1992). The method
of integration was, in most cases, a modified Euler integration with
adaptive step size developed specifically for this problem (Bray and
Lay, 1994b). This was chosen because it required a smaller number
of integrations to achieve a solution than conventional methods
such as forth-order Runge Kutta or variable step Runge Kutta
algorithms (Press et al., 1992). Each integration provided an estimate
of the final, steady state concentration of the state of the system. The
concentration of the complex TTWWAA (and in some cases
TTWAA) was then compared with target concentrations deduced
from the swimming behavior of the bacterium.
Prediction of Swimming Behavior
The relationship between the swimming behavior of the bacterium
and the intracellular concentration of TTWWAA depends on the
rate of transfer of phosphate groups from the TTWWAA complex to
Y and on the influence of Yp on flagellar rotation. An existing model
of signaling reactions in bacterial chemotaxis (Bray et al., 1993)
provided the basis for evaluating swimming performance for any
specified genotype or stimulus condition. Behavior is normally
quantitated as the rotational bias, which is the fraction of time the
flagella spend in counterclockwise rotation. The current version of
the program differs from the original in several ways:
1) The binding interactions between TT, W, and AA are modeled as
described in this paper.
2) Autophosphorylation of AA was permitted in all complexes
containing AA rather than just in free AA alone.
3) Several rate constants were changed due to recent experimental
evidence (Table 1).
4) Autophosphorylation of Y by acetyl phosphate is included (Lukat
et al., 1992).
5) The bias of the flagellar switch is calculated simply from the
concentration of Yp and a Hill coefficient of 5.5 (Kuo and Koshland,
1989) rather than a complicated model of switch binding.
The active complex of TT, W, and AA (TTWWAA) exhibits a rate
of autophosphorylation of AA some 680-fold greater than that of
free AA (Stewart, personal communication). The rate of phosphorylation
of AA in complexes containing W is the same as that of free
AA under conditions of saturating ATP concentration (McNally and
Matsumura, 1991). The question of whether the intermediate complex
TTWAA forms in appreciable amounts and whether its activity
resembles that of AA or of TTWWAA has not been addressed
experimentally and was examined in this study.
Intracellular Yp and 1T'WWAA Concentrations
A crucial link between the concentration of TTWWAA and the
swimming behavior of the bacterium is provided by the intracellular
concentration of Yp. This protein interacts with the switch complex
of the flagellar motor and, if present in sufficiently high quanVol.
6, October 1995 1369
D. Bray and R.B. Bourret
Table 1. Rates of phosphorylation reactions used in the simulation
Reaction Rate constant Reference
Autophosphorylation of CheA by ATPa,b
AA -- AAp, 2.5 x 10-2 s-1 Tawa and Stewart, 1944
WAA -- WAAp,
WWAA WWAAp,
TTWAA TTWAAp
TTAA -- TTAAp
Receptor-stimulated autophosphorylation of CheA by ATPa
TTWWAA -* TTWWAAp 1.7 x 101 s-5 Stewart, personal communication
Phosphotransfer from CheAp to CheYc
AAp + Y -> AA + Yp, 3 x i07 M- s- Stewart, personal communication
WAAp + Y -> WAA + Yp,
WWAAp + Y WWAA + Yp,
TTWAAp + Y TTWAA + Yp,
TTWWAAp + Y -> TTWWAA + Yp,
TTAAp + Y -> TTAA + Yp
Autophosphorylation of CheY by acetyl phosphatea
Y -> Yp 8.8 X 10251 Lukat et al., 1992
Autodephosphorylation of CheYp
Yp -* Y 3.7x 10- 51 Lukat et al., 1991
CheZ-stimulated dephosphorylation of CheYp
Yp + Z -->Y + Z 5X105 M-1s-1 Lukat et al., 1991
Phosphotransfer from CheAp to CheBc
AAp + B -> AA + Bp 6 x 106 M- s1 Stewart, personal communication
WAAp + B -* WAA + Bp,
WWAAp + B WWAA + Bp,
TTWAAp + B TTWAA + Bp,
TTWWAAp + B -> TTWWAA + Bp,
TTAAp + B -- TTAA + Bp
Autodephosphorylation of CheBpd
Bp -* B 3.5 x 10-ls-1 Stewart, 1993
a
The simulation represents autophosphorylation reactions as pseudo-first order reactions with a correction of the form [phosphodonor]/(Km
+ [phosphodonor]) applied to these rate constants (the concentrations of ATP and acetyl phosphate are unlikely to change in the few
seconds it takes for the swimming response to reach a stable state). Parameters used were [ATP] = 3 x 10-3 M (Bochner and Ames, 1982),
Km,ATP = 3 x 10-4 M (Wylie et al., 1988; McNally and Matsumura, 1991; Tawa and Stewart, 1994), [acetyl phosphate] = 10' M (Pruss and
Wolfe, 1994), and Km,acetyl phosphate = 7 x 10-4 M (Lukat et al., 1992).
b AA autophosphorylation rates have been well characterized for free AA and for the TTWWAA ternary complex. The rates for autophosphorylation
of complexes containing TT and/or W in addition to AA are assumed to be the same as for free AA. WWAA autophosphorylation
is known to exhibit the same Vmax as AA (McNally and Matsumura, 1991). Our simulation does not include the fact that WWAA
has a much lower Km for ATP than AA (McNally and Matsumura, 1991); however, this simplification has little consequence at physiological
ATP concentration, which is essentially saturating for both reactions. The possibility that communication between TT and AA in the absence
of W may result in reduced autophosphorylation (Ames and Parkinson, 1994) for TTAA is not included in our simulation. In a few
simulations, the autophosphorylation rate of TTWAA was assumed to be like that of TTWWAA, as noted in the text.
c Rates have been measured experimentally only for phosphotransfer from free AAp to Y or B. The rates for phosphotransfer from complexes
containing TT and/or W in addition to AAp are assumed to be the same.
d The rate constant for autodephosphorylation of Bp is 7 x 10-1 s-1 at 35°C (Stewart, 1993). The simulation used half this value (Stewart,
personal communication), to be consistent with the other rate constants, which were measured at 20-25°C.
tities, causes the default counterclockwise rotation of the motor to
change to a clockwise rotation (Barak and Eisenbach, 1992; Welch et
al., 1993, 1994). The concentration of Yp in the cytoplasm of a
wild-type unstimulated bacterium is consequently an important
reference point for any calculation of rates of phosphorylation and
dephosphorylation. Because there are at present no direct experimental
measurements of this concentration, we estimated this value
using our computer model. The basis for the calculation was that a
mutant lacking TT, W, and Z (T-W-Z-) has a rotational bias close
to wild type (Liu and Parkinson, 1989). We therefore determined the
concentration of Yp generated by the network of phosphorylation
and dephosphorylation reactions involving only AA, B, and Y,
using experimentally determined values of protein concentrations
(see Bray et al., 1993) and rate constants (Table 1). There is more Y
than B in the cell (DeFranco and Koshland, 1981; Stock et al., 1985;
Kuo and Koshland, 1987; Stock et al., 1991) and phosphotransfer
from AAp to Y is faster than to B (Table 1). Thus, the presence or
absence of B makes only an -5% difference in the concentration of
Yp in this simulation (our unpublished data), and B can removed
from this reaction network, leaving only AA and Y. In this case, a
simple kinetic equation can then be written for the steady state
formation and breakdown of Yp and solved algebraically, without
using the computer simulation (Figure 2). This leads to a quadratic
equation for Yp with only one reasonable solution, close to 1.6 ,tM.
Molecular Biology of the Cell1370
Analysis of Binding Equilibria
Given that a cell has wild-type bias at this calculated concentration
of Yp, and a known Hill coefficient governs interactions between Yp
and the switch, it is also possible to derive a simple equation
relating bias to Yp, namely:
(ypHill)
2.333(SetYpHIl') + ypHill
where the exponent Hill is the Hill coefficient and SetYp is the Yp
concentration in an unstimulated wild-type cell. Note that when
there is no Yp, the simulated bias is l (i.e., CCW rotation); as Yp
increases, the simulated bias approaches 0 (i.e., CW rotation), and
when Yp = SetYp, the simulated bias is 0.7 (i.e., wild-type value); all
as expected.
The value of Yp can also be used to predict the steady state
concentration of the active complex TTWWAA in an unstimulated
wild-type bacterium. In such a bacterium, the increased rate of
formation of Yp due to the activity of the complex must be balanced
by the increased loss of Yp due to Z activity. Current estimates of
the rates of these two reactions lead to a concentration of TlNWWAA
close to 1 ,uM. Precise target values used for wild-type and mutant
strains were obtained by a process of titration using the chemotactic
simulation (Table 2). The above equation can be used to determine
the Yp concentration that corresponds to a given bias; one can then
work backward to deduce the TTWWAA concentration that yields
the desired value of Yp.
Network Evaluation
To summarize at this point: each individual network (that is, each
set of binding equilibria and their dissociation constants) was evaluated
by calculating the concentration of the complex TTWWAA (or
in some cases TTWAA plus TTWWAA) the network would produce
given specific starting levels of TT, W, and AA. This was done for
each of the mutant genotypes in the test set (Table 2). The steady
state concentrations of the complexes reached were then compared
with the target values predicted from the steady state, unstimulated,
swimming behavior for each mutant. Absolute differences between
the simulated and target TTWWAA concentrations for each of the
mutant types were added together and the sum was used as a cost
value. Cost values were consequently smaller the more closely the
performance of the simulation matched test criteria, and a cost less
than 10'7 M ensured that the bacterial strains had the desired
swimming behavior.
RESULTS
10-Reaction Network
There are a number of experimental observations
concerning the binding interactions between TT,
W and AA as follows:
1) Both TT and AA normally exist in solution as
dimeric species whereas W is monomeric (Milligan
and Koshland, 1988; Gegner and Dahlquist, 1991).
2) W and AA form a complex WWAA (Gegner and
Dahlquist, 1991; McNally and Matsumura, 1991).
3) W binds to TT (Gegner et al., 1992).
4) A ternary complex forms between TT, W, and AA
and has a probable composition of TTWWAA (Gegner
et al., 1992).
5) Ternary complex formation is not affected by
ligand binding, phosphorylation, or the phosphotransfer
substrates Y or B (Gegner et al., 1992).
6) Binding of AA to TT depends on W, which forms
a link between the two (Gegner et al., 1992) and
therefore opens the possibility of the complex
TTWAA.
These observations can be used to formulate pathways
for formation of the TTWWAA complex. There
are 10 theoretically possible binding reactions between
TT, W, and AA given the following constraints: 1) TT
and AA maintain their dimeric state, 2) TT and AA
interact only through W, and 3) no complex contains
more than one TT, one AA, or two W molecules (Table
3, reactions #1-10).
The network derived from binding steps 1-10 in
Table 3 has a set of 10 coupled binding constants
each of which can take a range of values. The number
of independent variables is less than this, however,
because of thermodynamic constraints. The
difference in free energy between any two states of
the system is fixed. Therefore, the free energy
changes in a series of binding equilibria that connect
these two states must always add to this value
whatever the individual steps between the states.
For this reason, there are only six independent variables
in this network, and the other four Kd values
can be deduced (Figure 3).
Multiple attempts were made to optimize the network
on the basis of the set of test criteria shown in
Table 2. A range of starting Kd values was used,
Figure 2. Phosphotransfer reactions
used to estimate the steady state con- AAp Y
centration of Yp. The reactions affect- k = 0.023
ing Yp in the mutant strain T-W-Z-,
which has approximately wild-type k2 = 3x 107M-1 S-1
swimming bias, are shown except for Kl k2 k3 11
formation and destruction of Bp. Reac- k3 = 0.037 stions
shown are as follows: 1) autophosphorylation
of AA to AAp, 2) k4 = 1.24 x 103s-1
phosphotransfer of phosphate from AA Yp
AAp to Y, 3) autodephosphorylation of
Yp to produce Y, and 4) phosphorylation of Y by transfer from acetyl phosphate. Reaction rates kl-k4 are derived from Table 1 and take into
account phosphodonor concentration. The total concentration of AA-containing species was taken to be 2.5 JIM and that of Y-containing
species was taken to be 10 ,uM (see Bray et al., 1993). Algebraic solution of equations derived from the above reactions gives a steady state
concentration of Yp of 1.63 ,uM.
Vol. 6, October 1995 1371
D. Bray and R.B. Bourret
Table 2. Criteria used for network optimization
Experimental observations Simulation targets
Genotypea Rotationb References Bias [TTWWAA] (M)c
Wildtype Wildtype Block et al., 1982 0.70 1.13 x 10-6
10 T CCW Liu and Parkinson, 1989 >0.95 <8 x 10-8
10 W CCW Liu and Parkinson, 1989; Sanders et al., >0.95 <8 x 108
1989
10 T, 10 W Wildtype Liu and Parkinson, 1989 0.70 1.13 x 10-6
10 A Wildtype/ Parkinson, personal communication; -0.8 1.02 x 10-6
CCW Bourret, unpublished results
10 A, 10 W CCW Liu and Parkinson, 1989; Bray et al., 1993 >0.95 <8 x 10-8
a A genotype of "10 W" indicates a strain in which CheW is expressed at 10 times the normal level, with other genes being expressed at the
wild-type level. Note that the extent of overproduction was not measured experimentally; it is assumed to be 10-fold for the purposes of
simulation.
b Direction of flagellar rotation in strains overproducing the indicated proteins. "Wildtype" indicates the mix of CCW and CW rotation
characteristic of a wild-type cell.
c The concentration of the active complex needed to produce the desired bias was estimated by a form of titration in which the swimming
behavior of a range of networks with different TTWWAA concentrations was evaluated by the chemotactic simulation.
with individual Kd values ranging systematically, in
powers of 10, between 10-5 M and 10-8 M, thereby
encompassing the usual range of dissociation constants
for protein associations in cells. Individual
tests typically involved a total of 3,000 selection
steps, each of which entailed a numerical integration
that might take 106 steps or more. Because
individual tests might take several hours of CPU
time on a Macintosh Quadra 950 or a Sun SPARC
workstation, the systematic evaluation of some networks
took a week or more to complete. Eventually,
sets of Kd values that gave cost values less than 10-8
M were obtained; one such set is given in Table 3.
When these Kd values were installed in the full simulation,
they produced phenotypes that corresponded
to the experimentally determined test values.
The Kd values obtained in successive runs and
from different starting points showed significant
variation, reflecting the random nature of the selection
process and the fact that a finite cost value was
used as a target. When solutions with cost values
less than 10-M were compared, however, individual
Kd values were seen to be clustered around
typical values: average values obtained for the 10Table
3. Examples of sets of Kd values derived by optimization to meet the target criteria of Table 2
Kd (M)
Sevenreaction
10-Reaction 12-Reaction 12*-Reaction
Binding stepa network network network networkb
1 TT + W < TTW 1.57 x 10-9 5.18 x 10-1o 3.65 x 10-9 4.17 x 10-8
2 TTW + W <-> TTWW 4.79 x 10-8 2.23 x 10-8 5.11 x 10-8 1.95 x 10-7
3 W + AA +- WAA 4.28 x 10-9 2.16 x 10-9 8.94 x 10-9 6.22 x 10-8
4 WAA + W WWAA 1.23 x 10-7 7.25 x 10-8 1.02 x 10-7 1.06 x 10-6
5 IT + WAA TTWAA 7.16 x 10-6 2.97 x 10-4 5.21 x 10-8
6 1T7W + AA 1TWAA 2.99 x 10-5 7.27 x 10-4 7.77 x 10-8
7 TTWAA + W TTWWAA - 6.10 x 10-11 7.87 x 10-12 3.44 x 10-6
8 TTW + WAA 1 TTWWAA 8.36 x 10-7 8.43 x 10-7 6.40 x 10-7 4.30 x 10-6
9 TTWW + AA TTWWAA 7.47 x 10-8 8.16 x 10-8 1.12 x 10-7 1.37 x 10-6
10 1T + WWAA TTWWAA 1.07 x 10-8 6.02 x 10-9 2.29 x 10-8 1.69 X 10-7
11 1T + AA-TTAA 3.93 x 10-5 1.74 X 10-7
12 TTAA+ W TTWAA 6.76x 10-8 1.86x 10-8
a
Candidate binding steps used in the optimization routine. IT, W, and AA are the starting components. TTWWAA is the active complex.
Various networks were built using binding steps drawn from this list and tested by the optimization procedure.
b
TTWAA is endowed with activity equivalent to TTWWAA in the 12*-reaction network.
Molecular Biology of the Cell1372
Analysis of Binding Equilibria
TTWWAA
TTWW, AA 7 TT, WWAA
TTWAA, W TTW,WVAA l
92
96 94g5\
TTW, W, AA TT, W, WAA
91 93
TT, 2W, AA
Figure. 3. The network of 10 binding steps from Table 3 shown
together with their free energy changes. Each state of the system
comprises a set of protein species, such as TTW plus WAA (TTW,
WAA), which can be converted to one or more other states by
reversible binding. The starting state is (TT, 2W, AA) and the active
complex is TTWWAA. Ten different binding equilibria exist connecting
these various states, each of which has an associated change
in standard Gibbs energy g. Because the difference in free energy of
any two states is fixed (albeit unknown), whatever the pathway of
binding steps from one to the other, their individual free energy
changes (g values) must add up to the same value. This constraint
means that the free energy changes of some steps can be deduced
from those of other steps, as follows: g2 = g3 + g8 - g9; g4 = g1 +
98 - g10; g6 = g3 + g5 - g1; g7 = g1 + g8 - g5. Because the free
energy change of a reaction is related to its dissociation constant
(Kd) by g = RTlnKd, the above expression for g2 is equivalent to the
following:
K2)=Kd'(3) X Kd(8)
Kd(9)Kd(2) =d3 Kd(9)
and similar relationships can be written for Kd(4), Kd(6), and Kd(7).
reaction network, together with the standard error
of the mean, are given in Table 4.
7-Reaction Network
Among the various solutions that the optimization
routine produced for the 10-reaction network, the Kd
values for the binding steps involved in the formation
and destruction of TTWAA frequently took on extreme
values compared with the Kd values for other
binding steps. The two binding steps leading to
TTWAA (TTW + AA = TTWAA and TT + WAA =
TTWAA) both had relatively large Kd values compared
with other steps whereas step #7 (TTWAA + W
= TTWWAA) in which TTWAA is converted to the
active complex had a Kd lower than any other (Table
4). The consequence of these Kd values was that the
concentration of TTWAA at equilibrium was usually
very small. We therefore deleted these reactions from
the network and attempted to obtain solutions to the
test criteria in a new network composed of seven
reactions and lacking TTWAA entirely. This effort was
successful (a sample solution is given in Table 3),
indicating that it is not necessary to form TTWAA to
meet the target criteria.
No systematic attempt has been made to determine
the smallest possible network that permits a solution
to the constraints listed in Table 2. However, solutions
were found to a network with six reactions derived
from the 7-reaction network by deleting reaction #8
(our unpublished data).
12-Reaction Network
Although direct binding of TT and AA has not been
observed (Gegner et al., 1992) there is genetic and
biochemical evidence that TT can regulate AA activity
in the absence of W (Liu and Parkinson, 1989; Ames
and Parkinson, 1994). A network permitting formation
of TTAA was therefore constructed by adding reactions
#11 and 12 (Table 3) to the 10-reaction network.
Because of thermodynamic constraints, 5 of the 12 Kd
values could be derived from the others, leaving seven
independent variables to be optimized. Multiple solutions
were obtained by optimization, one of which is
given in Table 3.
High TTWAA Activity
The carboxy terminal domain of AA is necessary for
regulation of CheA kinase activity by TT and W (Bourret
et al., 1993). An AA heterodimer in which one
subunit is missing the C-terminal regulatory domain
should form at most one functional connection to TT.
The properties of such heterodimers in the presence of
TT and W have been extensively investigated, but it
remains unclear whether TT can regulate AA under
these circumstances (Swanson et al., 1993a; Wolfe et al.,
1994).
The analogous species in our simulation is TTWAA,
which has not been observed experimentally. To see
whether computer simulation could shed some light
on this debate, the effect of assigning the same high
level of CheA phosphorylation to TTWAA as TTWWAA
was explored for the 10- and 12-reaction networks.
The optimization routine was asked to find Kd
values that would generate the Table 2 target values
for [TTWWAA] with [TTWAA + TTWWAA] instead.
Solutions were obtained for the 12*-reaction (see Table
3), the lowest cost value achieved being 4.7 x 10-8 M.
No solution was found for the 10-reaction case.
Simulation of 1T, W, or AA Overproduction
The simulation was used to predict flagellar rotational
bias as a function of TT, W, or AA concentration
for each of the four networks described in
Vol. 6, October 1995 1373
D. Bray and R.B. Bourret
Table 4. Average Kd values obtained in solutions to the 7-reaction and 10-reaction networks
Kd (M)
Binding step 7-reaction 10-reaction
1 T+ W -TTW 1.00 ± 0.16 X 10-9 8.06 ± 0.93 x 10-10
2 TTW + W-> TTWW 6.68 ± 0.80 x 10-8 5.42 ± 0.63 x 10-8
3 W + AA <-WAA 2.96 ± 0.24 x 10-9 3.40 ± 0.30 x 10-9
4 WAA + W *WWAA 1.88 ± 0.21 x 10-7 1.63 ± 0.21 x 10-7
5 TT + WAA < TTWAA 2.62 ± 0.08 x 10-6
6 TTW + AA <-> TTWAA 1.06 ± 0.64 x 10-3
7 TTWAA + W <-> TTWWAA 1.03 ± 0.17 x 10-1°
8 TTW + WAA <-> TTWWAA 9.33 ± 0.32 x 10-7 8.73 ± 0.33 x 10-7
9 TTWW + AA <-> TTWWAA 6.58 ± 0.51 X 10-8 7.71 ± 0.51 x 10-8
10 TT + WWAA -FTTWWAA 1.04 ± 0.15 x 10-8 9.95 ± 1.26 x 10-9
Optimizations were performed repeatedly on these networks using different sets of starting Kd values. Values obtained for the binding
reactions are shown together with the standard error of the mean. Note that because Kd values for each binding step were averaged
independently, the values given in this table-in contrast to those given in Table 3-do not in themselves constitute a solution to the
optimization. That is, a network of reactions possessing the above Kd values will not give the correct chemotactic behavior, nor will it conform
to the thermodynamic constraints discussed in Figure 3.
Table 3 (Figure 4). The optimization routine ensures
that all networks generate the target bias of -0.7 at
wild-type protein concentration and the appropriate
bias (-1 for TT or W and -0.8 for AA) at 10 x
wild-type protein concentration. The predicted bias
at other protein concentrations reveals several interesting
features.
First, the absence of TT, W, or AA is predicted to
result in a bias of 1, consistent with experimental
observation (Parkinson, 1978; Liu and Parkinson,
1989). Second, bias is much more sensitive to TT, W, or
AA concentration and reaches much lower values in
the case where TTWAA has high activity (12*-reaction
A B
network) than when it does not (7-, 10-, and 12-reaction
networks). This may be an experimentally testable
prediction. The three networks with low TTWAA
activity make essentially indistinguishable predictions.
Third, all networks predict that bias differs from
1 only over a very narrow range of W concentrations.
Finally, one might expect that increasing the concentration
of the AA kinase would in turn lead to an
increase in Yp concentration and hence a decrease in
bias. Instead, all networks make the counterintuitive
prediction that increasing AA leads to a bias of 1,
which implies that Yp actually decreases when AA
increases.
C
0 5 10 15 20
[Tar] (x wildtype)
5 10 15
[CheW] (x wildtype)
20 5 10 15
[CheA] (x wildtype)
Figure 4. Predicted effect of overexpression of TT, W, or AA on rotational bias for each of the four networks whose Kd values are given in
Table 3: 7-reaction, O; 10-reaction, O; 12-reaction, 0; 12*-reaction, A. The circle, diamond, and square plots in the W panel superimpose.
Molecular Biology of the Cell
20
1374
Analysis of Binding Equilibria
The solutions listed in Table 3 were also used to
predict the concentrations of the various proteins
and oligomeric species involved in formation of the
receptor complex. This analysis revealed a number
of general features of the solutions (illustrated in
Figure 5 for the case of the 12-member binding
network). In a wild-type organism, the most abundant
protein species was the active complex TTWWAA.
Expression of TT in elevated amounts led to
wt
W++
Figure 5. Concentrations of oligomers containing TT, W, and AA
expected for different genotypes. The Kd values listed in Table 3 for
the 12-reaction network were used to predict the steady state concentrations
of TT, W, and AA and of the oligomeric complexes
formed from these proteins. Overexpression of TT and/or W was at
10 times wild-type concentration. Concentrations are shown as vertical
bars. The active complex (TTWWAA) is shown in black, the
others in gray. Truncated bars, topped with -, indicate a concentration
greater than 2 ,uM.
Vol. 6, October 1995
a reduction of this active complex and formation of
TT-containing species-chiefly TT itself and TTW.
Evidently, therefore, the smooth phenotype observed
in this mutant may be explained by the sequestration
of W into an inactive complex (TTW)
thereby reducing the amount available to form
TTWWAA. Similarly, in mutants in which W is
overexpressed, the large amounts of TTWW and
WWAA produced serve to reduce the amounts of
TT and AA available for active complex formation.
Co-overexpression of both TT and W results primarily
in formation of complexes containing both proteins
(TTW and TTWW) and restores essentially
wild-type levels of the active complex.
Dissociation Constants
The dissociation constants of two individual steps
have been measured experimentally with purified
proteins: a dissociation constant of 17 ,tM was obtained
for the reaction W + AA < > WAA (Gegner and
Dahlquist, 1991), and a dissociation constant of -10
,tM was obtained for TT + W < > TTW (Gegner et al.,
1992). It was therefore important to examine the dissociation
constants predicted for these particular binding
steps by computer simulation. All of the networks
examined give dissociation constants three to four
orders of magnitude smaller than the experimental
values (Table 5). Attempts to "clamp" the two binding
steps to the experimental values during the simulation
were unsuccessful and failed to produce solutions: the
largest dissociation constants achieved for these two
reactions were 4.2 X 10-8 M and 6.2 x 10-8 M, respectively,
obtained with the 12*-network.
The concentrations of TT, W, or AA in the mutant
bacteria used to define the boundary conditions for
the optimization routine have not been measured experimentally
and were arbitrarily set at 10 times the
wild-type concentration (Table 2). We also ran a series
of optimizations in which proteins were assumed to
be overexpressed 30-fold instead of 10-fold, and found
that this reduced the discrepancy between experimental
and simulated Kd values by a factor of - 10. The
largest Kd values obtained in this series were 5.2 x
10-7 M for the binding of TT and W and 7.8 x 10-7 M
for the binding of W and AA (our unpublished data).
The basis on which we selected values of intracellular
protein concentrations, and one possible explanation
for the very striking discrepancy between simulated
and experimental Kd values, are provided in the DISCUSSION
section.
DISCUSSION
Multiprotein Complex Formation
Many proteins in living cells exist as part of multimeric
aggregates, such as enzyme complexes, prote-
1375
D. Bray and R.B. Bourret
Table 5. Comparison of experimental and simulated Kd values for W binding to TT or AA
Kd (M)
Maximum simulated valueb
Experimental 7-reaction 10-reaction 12*-reaction
Binding step valuea network network network
#1 TT + W TTW -1 x 10-5 3.12 x 10-9 7.00 x 10-9 4.20 x 10-8
#3 W + AA WAA 1.7 x 10-5 2.78 x 10-9 9.65 x 10-9 6.21 x 10-8
a References are Gegner et al. (1992) for reaction #1 and Gegner and Dahlquist (1991) for reaction #3.
b
The different networks are composed of the binding steps listed in Table 3. The largest Kd values generated in multiple solutions for each
network are given in this table.
asomes, and centrioles. More specifically, multiprotein
complexes are also a common feature of eukaryotic
signal transduction pathways, including those mediated
by binding domains such as SH2 and SH3, (Cohen
et al., 1995; Pawson, 1995), G proteins (Clapham
and Neer, 1993; Conklin and Bourne, 1993), the MAP
kinase cascade (Choi et al., 1994; Marcus et al., 1994),
the retinoblastoma protein (Welch and Wang, 1995),
or compartmentalized serine/threonine kinases (Mochly-Rosen,
1995). The aggregates form by the diffusion-limited
association of their component proteins
and thus their concentration in the cell-and hence the
level of their biological activity-is in general thought
to be controlled by the law of mass action. In this case,
if the number of component proteins present and the
affinity with which they bind to each other were
known, then it might be possible to calculate the concentration
of the active multiprotein complex. In reality,
however, for anything other than complexes with
very few components, the equilibrium state is given
by a set of nonlinear simultaneous equations that cannot
be solved analytically. Furthermore, even if all of
the dissociation constants had been measured under
defined conditions, which is almost never the case,
then there is still the problem of estimating their true
value in the crowded conditions of the interior of a
living cell.
In the present study we have attempted to use optimization
techniques to learn how one set of proteins
associates in the living cell. Bacterial chemotaxis is
arguably the best understood system of intracellular
signaling and is the particular case explored here.
Many mutant bacteria have been isolated in which one
or more of the proteins of the TTWWAA complex
have been deleted or overexpressed and the consequences
for chemotactic behavior have been measured
experimentally. This provides a set of boundary conditions
for our optimization scheme, and values for
binding constants that reproduce the mutant phenotypes
can then be estimated. This approach could in
the future be applicable to other signal pathways. For
example, overexpression of the CD4 receptor reduces
activation of the p56lck tyrosine kinase in thymocytes,
apparently by disrupting complex formation (Nakayama
et al., 1993). Similarly, overexpression of a
fragment of the retinoblastoma protein results in inhibition
of activity of the full length protein, again apparently
by disrupting complex formation (Welch and
Wang, 1995).
Simulated Overexpression of Chemotaxis Proteins
Overexpression of the TT, W, or AA proteins leads to
phenotypes that are unexpected and could not have
been easily predicted. Overexpression of TT gives the
same smooth phenotype as deletion of TT (Liu and
Parkinson, 1989); overexpression or deletion of W
gives the same, smooth swimming, phenotype (Parkinson,
1978; Liu and Parkinson, 1989; Sanders et al.,
1989); overexpression of TT and W together, in contrast
to their individual overexpression, restores the
wild-type swimming pattern (Liu and Parkinson,
1989). Overproduction of AA is less well characterized,
but the results of our unpublished studies show
that it gives a slightly higher bias than wild type bias.
The optimization scheme leads to a computer simulation
that faithfully reproduces each of these behaviors
by changing the distribution of TT, W, and AA among
the various possible complexes (and hence the amount
of the active TTWWAA species) under different circumstances.
The AA overproduction phenotype is
particularly noteworthy. One might intuitively expect
that increasing amounts of the CheA kinase would
lead to more Yp and hence a reduced bias; in fact this
is exactly what a previous, simpler version of the
simulation predicted (Bray et al., 1993). In contrast, the
current simulation correctly predicts that increasing
AA increases the bias (Figure 4), a result which stems
from the large difference in autophosphorylation activity
between TTWWAA and other complexes containing
AA.
Molecular Biology of the Cell1376
Analysis of Binding Equilibria
In all simulated networks, overexpression of W had
a more potent effect in increasing bias than did in
overproduction of TT (Figure 4). There is some evidence
that this is reproduced in vivo, because overexpression
of W in a Z- background gives a CCW phenotype
whereas overexpression of receptor in the
same background results in wild-type swimming behavior
(Liu and Parkinson, 1989). The solutions obtained
were also consistent with other experimental
evidence, including the phenotypes of 23 other mutants
that have been studied experimentally and that
were listed in a previous publication (Bray et al., 1993).
Potential Role of TTWAA
There is experimental evidence that a complex containing
only one link between TT and AA may be fully
functional; however, related experimental results cannot
be easily explained by this model (Wolfe et al.,
1994). The alternate approach of computer simulation
has provided some information about the putative
TTWAA complex. The existence of such a complex is
not necessary to account for the mutant phenotypes
listed in Table 2 (e.g. see 7-reaction network of Table
3). The TTWAA complex could nevertheless exist: it
might, for example, be needed to fulfill requirements
not demanded in this version of the simulation. Alternatively,
TTWAA might be formed gratuitously, as a
necessary by-product of the binding steps that lead to
TTWWAA.
The computer simulation indicates that if TTWAA
does exist and has the high autophosphorylation activity
characteristic of TTWWAA, then binding steps
in addition to those of the 10-reaction network (see
Table 3) are necessary to account for the mutant phenotypes
of Table 2. This is because overexpression of
TT leads to loss of TTWWAA activity (Liu and Parkinson,
1989). In the 10-reaction network, excess TT
pulls W out of other complexes to form TTW. This
reduces TTWWAA concentration but leads to an increase
in free AA and hence TTWAA. If TTWWAA
and TTWAA are of comparable activity, then the excess
TT phenotype cannot result. A likely solution
based on available experimental evidence (Liu and
Parkinson, 1989; Ames and Parkinson, 1994) is to postulate
direct binding between TT and AA in spite of
the failure to observe such binding in vitro (Gegner et
al., 1992). In the 12-reaction network, excess TT also
leads to high levels of TTW, but now some of the AA
freed from complexes with W can be sequestered in
a nonproductive TTAA complex, unavailable for
TTWAA formation (Figure 5).
Finally, a network in which TTWAA has high autophosphorylation
activity is predicted to generate very
different behavior than networks with low TTWAA
activity when TT, W, or AA concentrations are
changed (Figure 4). Computer simulation thus suggests
an experimental approach that could indicate
whether TTWAA has high or low activity, i.e., a careful
examination of bias as a function of TT, W, or AA
expression. In fact, Sanders et al. (1989) have reported
that "at no level of CheW expression tested did the
cells tumble more frequently than wild-type cells."
This result appears to be consistent with low but not
high TTWAA activity in our simulations (Figure 4).
Specifically, the 10- and 12-reaction networks predict
that the minimum bias occurs at wild-type W concentration,
and only the 12*-network has a bias < 0.7 at
any W concentration.
Binding Constants
An important discrepancy between theory and experiment
emerged in this study. For all of the networks
examined, the predicted dissociation constants for two
reactions, TT + W <-- TTW and W + AA <-> WAA,
were much smaller than the experimentally measured
values, indicating a much tighter binding. This discrepancy
could arise from a variety of sources. The
rates of phosphorylation and dephosphorylation reactions
or the binding constants for protein/protein interactions
could be very different in the cell than measured
in vitro due to protein modifications or
association with other factors presently not identified.
Another possibility is that the effective concentration
of the species W or AA in the vicinity of TT might be
much higher than predicted simply by dividing the
number of molecules by the volume of the cell. A more
accurate simulation would have to include any restriction
of particular proteins to subcompartments within
the cytoplasm. The higher the concentration of TT, W,
and AA in the cytoplasm, the weaker the binding
would need to be to produce a specific concentration
of TTWWAA complex. Note that when we increased
the dosage of overexpression. mutants (thereby increasing
the concentration of TT, W, and AA in these
mutants) then the Kd values we obtained were larger.
A major contribution to the -discrepancy between
experimental and simulated Kd values is likely to be
made by macromolecular crowding. The cytoplasm of
living cells is a highly concentrated aqueous solution
of large and small molecules and E. coli has been
estimated to contain collectively about 340 g/liter of
total RNA and protein (Zimmerman and Trach, 1991).
Steric repulsion between macromolecules at these
high concentrations causes profound changes in a variety
of biochemical processes, including the association
of proteins into macromolecular complexes (Minton,
1981). Thus, a recent review on macromolecular
crowding states that " . . . under conditions of volume
occupancy comparable to those found in vivo ....
macromolecular associations are predicted to exceed
those in solution by as much as several orders of
magnitude" (Zimmerman and Minton, 1994). In other
Vol. 6, October 1995 1377
D. Bray and R.B. Bourret
words, if the experimental measurements of binding
between CheW, CheA, and Tar had been made under
conditions more closely approximating the interior of
the cell, then the Kd values obtained would have been
closer to those obtained in our simulation.
Additional Network Solutions
The range of solutions obtained for both 10- and 12reaction
networks indicated that the system was overdetermined;
that is, there appear to be more variable
parameters than are needed to find a solution. Some of
the Kd values obtained in the solutions were also
consistently very large (a weak binding), suggesting
that these particular reactions might be of relatively
little importance in the cell. This led us to explore
simpler networks with seven binding steps (Table 3)
and another set of reactions with only six binding
steps has been found. Other solutions may well exist,
and it will be interesting to search for the smallest
network of binding steps that can meet the behavioral
boundary conditions.
It should be noted, however, that a blind insistence
on the smallest possible number of binding steps will
not necessarily lead to the simplest solution or to one
more likely to exist in the living cell. Indeed, Occam's
razor could lead to a system of 12 reactions and its
associated set of oligomeric intermediates, depending
upon how the cut is made, because it is produced by
the reiteration of binding steps known to exist. If TT
binds to W, then why should an oligomeric complex
containing TT not also bind to W, or a complex containing
W not bind to TT? Because we do not have
direct evidence for the number and types of intermediate
oligomeric that exist in the cell, the most important
conclusion that can be reached is that some solutions,
at least, exist. That is, that sets of binding
reactions can be constructed that fulfill the observed
phenotypes.
Conclusion
The principal conclusions of this study are as follows:
1) It is possible to find by a process of optimization
sets of binding equilibria between TT, W, and AA that
fulfill test criteria based on the known phenotypes of
specific mutants overexpressing TT, W, or AA.
2) In these computer-based solutions, the smooth
swimming of mutants with overexpressed TT or W
results from sequestration of other components of the
ternary complex into inactive complexes.
3) Formation of the partial complex TTWAA is not
essential to fulfill the test criteria. However, if TTWAA
is formed, then its level of autophosphorylation activity
will influence which types of binding equilibria are
possible.
4) The Kd values of binding equilibria predicted by
computer-based optimization are all several orders of
magnitude smaller (implying tighter binding) than the
values of two binding steps that have been measured
experimentally. At least part of this discrepancy may
be attributed to the high concentration of macromolecules
in the cell cytoplasm, which causes major increases
in the strength of association between protein
molecules.
Optimization methods provide a potentially powerful
method to analyze intracellular signaling reactions.
As illustrated in the present study, they allow rates of
reactions and binding constants in complex networks
to be selected on the basis of the final phenotype they
confer on the cell. In this way one can endeavor to
reconcile experimental evidence obtained from biochemical
studies of purified proteins with that obtained
from the physiological performance of the entire
cell. Evidently, in view of the large discrepancies
in binding constants that exist between theory and
experiment, we are still a long way from having an
accurate picture of the molecular events that carry
signals to the flagellar motor. However, our present
results do show that a set of physically reasonable
binding constants can be found using this approach
that are able to fulfill one set of mutant criteria. There
seems no reason in principle why the acquisition of
further data from in vivo and in vitro sources together
with the use of more powerful computers should not
provide an increasingly accurate picture of complex
networks of intracellular signaling reactions.
ACKNOWLEDGMENTS
We thank Julian Lewis, John Albery, Heinrik Kacser, and Stephen
Lay for advice, and Rick Stewart and Alan Wolfe for comments on
the manuscript. We also thank Rick Stewart and Sandy Johnson for
providing us with their unpublished data. This work was supported
by a grant from the Medical Research Council to D.B.
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