Proc. Nati. Acad. Sci. USA Vol. 75, No. 11, pp. 5260-5263, November 1978 Chemistry
Interacting enzyme systems at steady state: Further Monte Carlo calculations on two-state molecules (Ising systems/critical behavior/phase transitions/enzyme flux/cooperativity)
TERRELL L. HILL AND YI-DER CHEN Laboratory of Molecular Biology, National Institute of Arthritis, Metabolism and Digestive Diseases, National Institutes of Health, Bethesda, Maryland 20014
Contributed by Terrell L. Hill, August 16, 1978
ABSTRACT In this work, Monte Carlo calculations were made on a 10 X 10 lattice of two-state steady-state enzyme molecules in two special cases for which the Bra-Williams (mean field) approximation had earlier produced some very interesting phase-transition properties. The Monte Carlo results proved to be similar to Bragg-Williams in some respects but not in others. The discrepancies are attributed primarily to: (i) inadequate treatment by Bragg-Williams of strong negative cooperativity; and (ii) the finite size of the 10 X 10 lattice used in the exact calculations. This paper (the ninth in a series) is a supplement to two earlier papers (1, 2). In the first (1), the Monte Carlo method was employed to study a 10 X 10 lattice (with periodic boundary conditions) of two-state, interacting enzyme molecules. Equilibrium and steady-state properties were examined in some detail for a few special cases with rather conservative choices of parameters. In the second paper (2), the Bragg-Williams (BW) or mean field approximation was applied to the same class of systems, but a much wider variety of parameter choices could be used because of the relative simplicity of BW calculations. In particular, two BW special cases with the kinetic (transition-state) parameters fa = fa = -1/2 exhibited unusually interesting critical behavior. It is the purpose of the present paper to examine (as promised in ref. 2) these same two special cases by using Monte Carlo calculations on a 10 X 10 lattice. Review of model and notation Because this work is a direct continuation of the two earlier papers (1, 2), we give only a very condensed review of the model and basic notation. We consider a lattice of M identical, two-state enzyme molecules. The two states are labeled 1 and 2. There are nearest-neighbor interactions between the molecules. These interactions perturb the rate constants and hence alter the kinetics of the system. Each molecule has z nearest neighbors in the lattice. The particular lattice we consider in this paper is a 10 X 10 square lattice with periodic boundary conditions. Thus, M = 100 and z = 4. The unperturbed firstorder rate constants are designated ao for 1 2 and (0 for 2 1 in, say, the counterclockwise direction around the two-state cycle and i'o for 1 - 2 and a' for 2 - 1 in the clockwise direction. The thermodynamic force X driving this cycle is given by eX/kT = aofto/a;,Bo (3). This force is not affected by neighbor interactions. We define a variable x as (ao + 3'o)/(i3o + a'). This variable governs the relative population of the two states; at equilibrium (X = 0), x is an "activity." We now select (o as a reference rate constant and set go- 1. Thus, from here on, the other rate constants and the flux J become dimensionless quantities that are measured in units of ibo. Hence x is now (a0 + 3,')/(1 + a') and eX/kT = ao/afoi -F.
The rate constants a0 and do may be expressed in terms of x and F as follows: ao = do
ao(1 + a'0)xF/(l + aoF)
=
(1 +
a')xl(l + aof)
[1
We shall be interested, below, in 0 and 1 as functions of x, with F held constant, where 0 P2 = N2/M is the steady-state probability that any given molecule is in state 2. We consider now the perturbed rate constants for a particular molecule (in either state) that has (instantaneously) a2 nearest neighbors in state 2 and a, = z - a2 nearest neighbors in state 1. The possible values of a2 are 0, 1, . . . , z. From equations 11 of ref. 4, the perturbed rate constants are a = a0 exp[fa(Wle - w)/kT] a' = a', exp[(1 - fa)(w2e -Wie)/kT] f
=
fo exp[f#(w2e- wie)/kT]
0'=i exp[(l - f,)(Wile-we)IkT]
[2]
where e refers to the instantaneous environment of the given molecule (namely, a2 molecules in state 2 and z - a2 molecules in state 1), fa and f, are constants (usually but not necessarily between 0 and 1), and wie is the interaction free energy between the given molecule in state i and e. Hence Wle-to& = aj(W11 - W12) + a2(W12 - W22) = Z(W1I W12) + a2(2wl2 W1 - W22), [3] where wq is the interaction free energy between two nearest neighbors in states i and j (we take W12 = W21). Eqs. 2 then become -
-
a
=
ao(rua2)a,
# = ,(rua2)"f,
=
a'O(rua2fa-l
(3' =
F'(3tUa2)l-fe,
a'
[4]
where ys - e-wij/kT, r = (Yl2/Yll)Z, U = YlY22/YI2. [5] The final expressions for the perturbed rate constants, for arbitrary a2, are obtained on substituting flo = 1 and Eqs. 1 into Eqs. 4. Case 1: Continuous approach toward a one-way cycle We consider here the case (2) fa = fe = -1/2, r = 1, and a0 = F-1/2. The other unperturbed rate constants are then ao = x, i,'0 = aox, and (0 = 1. Thus both a'0 -*0 and #3O -O 0 as F a), producing a one-way cycle. We are interested in the complete range F = 1 (equilibrium) to F = ao. The negative values of fa and fin lead to unconventional results (2).
The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertisement" in accordance with 18 U. S. C. §1734 solely to indicate this fact.
Abbreviation: BW, Bragg-Williams. 5260
Chemistry:
Proc. Natl. Acad. Sci. USA 75 (1978)
Hill and Chen
5261
0.4
0.2 -4
-2
0
+2
+6
+4
Inx
FIG. 1. 0 as a function of In x, from equilibrium (F = 1) to a oneway cycle (F = a).
In Fig. 1, we show 0(ln x) for several values of F and u = 0.220. For clarity, in all the figures of this paper we have drawn smooth curves through the "experimental" (Monte Carlo) points. The typical amount of scatter of the points can be seen from the examples in ref. 1. We have generally discarded the first 5000 transitions (transient) in a Monte Carlo simulation run and used an additional 20,000 to 100,000 transitions (depending on the magnitude of the fluctuations) at steady state to obtain time averages. In a few cases, we have used up to 500,000 steady-state transitions. Returning to Fig. 1, we have chosen u = 0.220 because this is the reciprocal of u = 4.55. In ref. 1, we found that u _ 4.55 is the 10 X 10 critical value of u at equilibrium (F = 1) or at any value of F if fa + ff = 1 (quasi-equilibrium). Also, in ref. 2, it was pointed out that the transformation wq, fa, fjy -Wi -fa, -ft3 in a oneway-cycle system will leave @(x) unchanged. Hence, even though u = 0.220 corresponds to repulsive forces (i.e., u < 1) between neighboring molecules in the same state (11 or 22), the O(ln x) curve for F = co in Fig. 1 is a critical curve. In fact, it is the same curve shown in figure 12 of ref. 1. For u < 0.22 and
M/O'|
I
u=
et'
I
1 06
\U1
5
1.0
I~~~~10
lo
0.22 F -12
r=
0.8
F=a
0.6
0.0
0.4
0.2
0.4
0.6
0.8
0
FIG. 3. Reciprocal of variance U2 in N2 (number of molecules in state 2) as a function of 0, for the same example as in Fig. 1.
C,0.0 0.-.
.
.
.
10
FIG. 2. Nearest-neighbor correlation function cl(8) for the same example as in Fig. 1.
F = co, there will be a phase transition between molecules with repulsive forces! This odd steady-state property is a consequence of the negative fa and fdj. At the other extreme, the F = 1 case in Fig. 1 is the equilibrium Ising curve for a 10 X 10 system with repulsive interactions (antiferromagnet; negative cooperativity). The switch from F = 1 to F = co in this steady-state example corresponds to changing the sign but not the magnitude of the intermolecular forces in an equilibrium system (u = 0.220 u = 4.55;
antiferromagnet - ferromagnet; negative - positive cooperativity). The other curves in Fig. 1 provide the expected transition between the two extreme cases. The exact 10 X 10 F = 1 and F = X curves would be symmetrical about 0 = 1/2, as the figure suggests. Fig. 2 shows the nearest-neighbor correlation function (1) Cl as a function of 0 for the same example. Also included (broken curve) for comparison is cI(8) = (1 - 20)2 for a (random) system with no neighbor interactions. The exact curves for F
Chemistry:
5262
0.1 -
-10l
Hill And Chen
1
F= ,
I,
-4
-2
,
,
I>
0
+2
-0.2~~~~~~~02
,
+4
Proc. Nati. Acad. Sci. USA 75 (1978)
+6
In x
FIG. 4. Flux J(ln x) corresponding to Fig. 1.
= 1 and F = co (10 X 10 system) would again be symmetrical about 0 = 1/2 (as in the u = 1 curve). The F = X (critical) curve exhibits positive correlation (cooperativity) compared with u = 1, while the F = 1 curve shows the opposite effect. The mean value c1 = 0.524 at 0 = 1/2, in the F = Xo case, was already reported in ref. 1. The F = 106 curve in Fig. 2 shows positive cooperativity up to about 0 = 1/2 and negative cooperativity for larger 0. Very similar comments can be made about Fig. 3, which gives M/Iu2 as a function of 0, where a2 is the variance in N2. In the u = 1 (random) case, MI a2 = 1/0(1 - 0). Fluctuations are seen to be relatively small in the F = 1 (antiferromagnetic) case. We already found in ref. 1, for the F = co case at 0 = 1/2, MIa = 1/5.96 = 0.168. Fig. 4 contains flux curves and corresponds to Fig. 1. In many of the examples in this paper, either Ja (i.e., calculated from a, a' transitions) or Jo was found to be very erratic because of particular terms (large rate constants, small probabilities) in equations 9 of ref. 1; the nonerratic member of the pair was always used. The F = X curve in Fig. 4 is the same as J(x) in figure 12 of ref. 1. It is not hard to prove that this should be so. This critical curve would have a cusp if M were infinite. At the other extreme, the F = 1 flux curve is the abscissa (i.e., there is no net mean flux at equilibrium, except for fluctuations). We have included J(ln x) for F = 2 and 10 in order to see the nature of the transition between the F = 1 and F = Xo cases. It is easy to show that, if u >1, then J - uas x - o. This accounts for the asymptotic value J -p 0.0484 seen in the figure. Figs. 1-4 do not correspond directly to any of the BW figures in ref. 2 for this same example. On the other hand, Fig. 5 here is the exact analogue of figure 6 in ref. 2. Note, however, that u = y1/2 (y is used in ref. 2). In Fig. 5 we keep F constant at 106 but vary u. In the BW approximation (figure 6 of ref. 2): (i) when y is continuously reduced (repulsive forces) below y = 1, a critical curve is passed and a phase transition is then observed that
lUlU
0.0
0.2
0.4
0.6
0.8
1.0
FIG. 6. Correlation function cl(@) corresponding to Fig. 5.
persists no matter how small y is (the vertical extent of the "loop" decreases and the loop moves towards ln x -a C as y 0); and (ii) when y is increased above y = 1 (attractive forces), there is negative cooperativity for small 0 and positive cooperativity for large 0 (leading to a critical point and phase transition behavior in the usual way). The exact (i.e., Monte Carlo) behavior of the 10 X 10 system (Fig. 5) is somewhat different. The differences are due primarily to the fact that the BW approximation does not handle strong negative cooperativity realistically enough (e.g., the BW approximation, as we are using it, does not allow, as it should, for a lattice with regularly alternating states 1 and 2 when 0 '/2). In addition, of course, it should be kept in mind that the 10 X 10 system is finite and the BW system infinite (M = aD). To understand the behavior of the 10 X 10 system, Fig. 5 should be examined together with Fig. 6 (correlation function, c,). When u is decreased from 1 to 0.5, positive cooperativity is observed for all 0. But this effect is reversed for 0 >0.22 (Fig. 6) when u = 0.22, and in fact there is negative cooperativity for 0 greater than about 0.5. At u = 0.1 and, especially, u = 0.04, negative Cooperativity completely dominates (in the otter case, near 0=1/2, the lattice is almost completely ordered-ne., with alternating states). The initial positive cooperativity (as u is
0.6-
F = 106 r= 1
0.04
0.4
10
J
-4
-2 0.5
0.21
+2
=
0. 5
+4
+
+4
+6
18 . 0.22 -4
1 -2
+2
0 Inx
FIG. 5.
0(ln x) for F = 106 and various values of u.
FIG. 7. Flux J(ln x) corresponding to Fig. 5.
Chemistry:
J 0.2
0.1
Proc. Natl. Acad. Sci. USA 75 (1978)
Hill and Chen
5263
J
10
-
0.8 -
u = 20
1510
0.6 0.40.2
0.2
0.4
0.6
1.0
0.8
1.2
0.6 x
x
FIG. 8. @(x) and J(x) at F = 20 in the case/a = f= a/2,o = 1, r = 0.05. See text.
FIG. 9. Same as Fig. 8 but for u = 15. See text.
decreased below 1) does not persist long enough to produce critical behavior (unlike BW). The same is true (not shown) at F = 108 but it is not true at F = c, as already mentioned in connection with Fig. 1: u = 0.22 is the 10 X 10 critical value of u and a phase transition occurs for u < 0.22. For u > 1, the exact behavior is rather similar to BW. There is a phase transition for large enough u. Also, if we compare u = 1 and u = 10 in Fig. 6, we see that, in the u = 10 case, there is slight negative cooperativity for 0 < 0.26 and strong positive cooperativity for large 0. The above paragraph refers to F = 106. At F = the situation is quite different. Both exact and BW (see figure 9 of ref. 2) systems exhibit only negative cooperativity when u > 1 (attractive forces). No first-order phase transition is possible. Actually, the curve labeled F = 1 in Fig. 1 is relevant here: it is also the F = c, u = 4.55 curve (the same is true in Figs. 2 and 3). (Analogously, as already mentioned, the F = o curve in Fig. 1 is also the F = 1, u = 4.55 curve.) The corresponding flux curves in Fig. 7 should be compared with figure 8 of ref. 2 (BW). Again the behavior is qualitatively similar for u > 1 but rather different for u < 1, especially for u = 0.1 and 0.04 (strong negative cooperativity). The peak fluxes in Fig. 7, when u < 0.5, coincide approximately with the
Case 2: Two-step phase transition in BW approximation The second interesting example studied in ref. 2 (BW) had, for parameters, fao = =-1/2, a' = 1, r = 0.05 and F = 1, 4, 20, ao. Several values of y > 1 were used in each case. It was found for F = 4 and 20 (but not F = 1 and a)) that two-step phase transitions occur in some circumstances (see figures 11 and 12 of ref. 2). To make a long story short, we tried (Monte Carlo) a large number of combinations of u and F values, in the intervals 1 < u < 50 and 1 < F < 400, without finding any twostep phase transitions. Of course, this could well be a consequence of the use of a finite 10 X 10 system. Figs. 8 and 9 show some typical results. Despite the absence of two-step transitions, the flux still exhibits the off-on "switch" effect noted in connection with figures. 15 and 16 of ref. 2. For example, in the F = 20 and 25 cases in Fig. 9, the flux is "on" over a relatively small range in x (or a ligand concentration) and is otherwise "off."
00,
minima in
as
x
akc.
c1
in
Fig.
6. As
already noted,
when
u
Interacting enzyme systems at steady state: Further Monte Carlo calculations on two-state molecules (Ising systems/critical behavior/phase transitions/enzyme flux/cooperativity)
TERRELL L. HILL AND YI-DER CHEN Laboratory of Molecular Biology, National Institute of Arthritis, Metabolism and Digestive Diseases, National Institutes of Health, Bethesda, Maryland 20014
Contributed by Terrell L. Hill, August 16, 1978
ABSTRACT In this work, Monte Carlo calculations were made on a 10 X 10 lattice of two-state steady-state enzyme molecules in two special cases for which the Bra-Williams (mean field) approximation had earlier produced some very interesting phase-transition properties. The Monte Carlo results proved to be similar to Bragg-Williams in some respects but not in others. The discrepancies are attributed primarily to: (i) inadequate treatment by Bragg-Williams of strong negative cooperativity; and (ii) the finite size of the 10 X 10 lattice used in the exact calculations. This paper (the ninth in a series) is a supplement to two earlier papers (1, 2). In the first (1), the Monte Carlo method was employed to study a 10 X 10 lattice (with periodic boundary conditions) of two-state, interacting enzyme molecules. Equilibrium and steady-state properties were examined in some detail for a few special cases with rather conservative choices of parameters. In the second paper (2), the Bragg-Williams (BW) or mean field approximation was applied to the same class of systems, but a much wider variety of parameter choices could be used because of the relative simplicity of BW calculations. In particular, two BW special cases with the kinetic (transition-state) parameters fa = fa = -1/2 exhibited unusually interesting critical behavior. It is the purpose of the present paper to examine (as promised in ref. 2) these same two special cases by using Monte Carlo calculations on a 10 X 10 lattice. Review of model and notation Because this work is a direct continuation of the two earlier papers (1, 2), we give only a very condensed review of the model and basic notation. We consider a lattice of M identical, two-state enzyme molecules. The two states are labeled 1 and 2. There are nearest-neighbor interactions between the molecules. These interactions perturb the rate constants and hence alter the kinetics of the system. Each molecule has z nearest neighbors in the lattice. The particular lattice we consider in this paper is a 10 X 10 square lattice with periodic boundary conditions. Thus, M = 100 and z = 4. The unperturbed firstorder rate constants are designated ao for 1 2 and (0 for 2 1 in, say, the counterclockwise direction around the two-state cycle and i'o for 1 - 2 and a' for 2 - 1 in the clockwise direction. The thermodynamic force X driving this cycle is given by eX/kT = aofto/a;,Bo (3). This force is not affected by neighbor interactions. We define a variable x as (ao + 3'o)/(i3o + a'). This variable governs the relative population of the two states; at equilibrium (X = 0), x is an "activity." We now select (o as a reference rate constant and set go- 1. Thus, from here on, the other rate constants and the flux J become dimensionless quantities that are measured in units of ibo. Hence x is now (a0 + 3,')/(1 + a') and eX/kT = ao/afoi -F.
The rate constants a0 and do may be expressed in terms of x and F as follows: ao = do
ao(1 + a'0)xF/(l + aoF)
=
(1 +
a')xl(l + aof)
[1
We shall be interested, below, in 0 and 1 as functions of x, with F held constant, where 0 P2 = N2/M is the steady-state probability that any given molecule is in state 2. We consider now the perturbed rate constants for a particular molecule (in either state) that has (instantaneously) a2 nearest neighbors in state 2 and a, = z - a2 nearest neighbors in state 1. The possible values of a2 are 0, 1, . . . , z. From equations 11 of ref. 4, the perturbed rate constants are a = a0 exp[fa(Wle - w)/kT] a' = a', exp[(1 - fa)(w2e -Wie)/kT] f
=
fo exp[f#(w2e- wie)/kT]
0'=i exp[(l - f,)(Wile-we)IkT]
[2]
where e refers to the instantaneous environment of the given molecule (namely, a2 molecules in state 2 and z - a2 molecules in state 1), fa and f, are constants (usually but not necessarily between 0 and 1), and wie is the interaction free energy between the given molecule in state i and e. Hence Wle-to& = aj(W11 - W12) + a2(W12 - W22) = Z(W1I W12) + a2(2wl2 W1 - W22), [3] where wq is the interaction free energy between two nearest neighbors in states i and j (we take W12 = W21). Eqs. 2 then become -
-
a
=
ao(rua2)a,
# = ,(rua2)"f,
=
a'O(rua2fa-l
(3' =
F'(3tUa2)l-fe,
a'
[4]
where ys - e-wij/kT, r = (Yl2/Yll)Z, U = YlY22/YI2. [5] The final expressions for the perturbed rate constants, for arbitrary a2, are obtained on substituting flo = 1 and Eqs. 1 into Eqs. 4. Case 1: Continuous approach toward a one-way cycle We consider here the case (2) fa = fe = -1/2, r = 1, and a0 = F-1/2. The other unperturbed rate constants are then ao = x, i,'0 = aox, and (0 = 1. Thus both a'0 -*0 and #3O -O 0 as F a), producing a one-way cycle. We are interested in the complete range F = 1 (equilibrium) to F = ao. The negative values of fa and fin lead to unconventional results (2).
The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertisement" in accordance with 18 U. S. C. §1734 solely to indicate this fact.
Abbreviation: BW, Bragg-Williams. 5260
Chemistry:
Proc. Natl. Acad. Sci. USA 75 (1978)
Hill and Chen
5261
0.4
0.2 -4
-2
0
+2
+6
+4
Inx
FIG. 1. 0 as a function of In x, from equilibrium (F = 1) to a oneway cycle (F = a).
In Fig. 1, we show 0(ln x) for several values of F and u = 0.220. For clarity, in all the figures of this paper we have drawn smooth curves through the "experimental" (Monte Carlo) points. The typical amount of scatter of the points can be seen from the examples in ref. 1. We have generally discarded the first 5000 transitions (transient) in a Monte Carlo simulation run and used an additional 20,000 to 100,000 transitions (depending on the magnitude of the fluctuations) at steady state to obtain time averages. In a few cases, we have used up to 500,000 steady-state transitions. Returning to Fig. 1, we have chosen u = 0.220 because this is the reciprocal of u = 4.55. In ref. 1, we found that u _ 4.55 is the 10 X 10 critical value of u at equilibrium (F = 1) or at any value of F if fa + ff = 1 (quasi-equilibrium). Also, in ref. 2, it was pointed out that the transformation wq, fa, fjy -Wi -fa, -ft3 in a oneway-cycle system will leave @(x) unchanged. Hence, even though u = 0.220 corresponds to repulsive forces (i.e., u < 1) between neighboring molecules in the same state (11 or 22), the O(ln x) curve for F = co in Fig. 1 is a critical curve. In fact, it is the same curve shown in figure 12 of ref. 1. For u < 0.22 and
M/O'|
I
u=
et'
I
1 06
\U1
5
1.0
I~~~~10
lo
0.22 F -12
r=
0.8
F=a
0.6
0.0
0.4
0.2
0.4
0.6
0.8
0
FIG. 3. Reciprocal of variance U2 in N2 (number of molecules in state 2) as a function of 0, for the same example as in Fig. 1.
C,0.0 0.-.
.
.
.
10
FIG. 2. Nearest-neighbor correlation function cl(8) for the same example as in Fig. 1.
F = co, there will be a phase transition between molecules with repulsive forces! This odd steady-state property is a consequence of the negative fa and fdj. At the other extreme, the F = 1 case in Fig. 1 is the equilibrium Ising curve for a 10 X 10 system with repulsive interactions (antiferromagnet; negative cooperativity). The switch from F = 1 to F = co in this steady-state example corresponds to changing the sign but not the magnitude of the intermolecular forces in an equilibrium system (u = 0.220 u = 4.55;
antiferromagnet - ferromagnet; negative - positive cooperativity). The other curves in Fig. 1 provide the expected transition between the two extreme cases. The exact 10 X 10 F = 1 and F = X curves would be symmetrical about 0 = 1/2, as the figure suggests. Fig. 2 shows the nearest-neighbor correlation function (1) Cl as a function of 0 for the same example. Also included (broken curve) for comparison is cI(8) = (1 - 20)2 for a (random) system with no neighbor interactions. The exact curves for F
Chemistry:
5262
0.1 -
-10l
Hill And Chen
1
F= ,
I,
-4
-2
,
,
I>
0
+2
-0.2~~~~~~~02
,
+4
Proc. Nati. Acad. Sci. USA 75 (1978)
+6
In x
FIG. 4. Flux J(ln x) corresponding to Fig. 1.
= 1 and F = co (10 X 10 system) would again be symmetrical about 0 = 1/2 (as in the u = 1 curve). The F = X (critical) curve exhibits positive correlation (cooperativity) compared with u = 1, while the F = 1 curve shows the opposite effect. The mean value c1 = 0.524 at 0 = 1/2, in the F = Xo case, was already reported in ref. 1. The F = 106 curve in Fig. 2 shows positive cooperativity up to about 0 = 1/2 and negative cooperativity for larger 0. Very similar comments can be made about Fig. 3, which gives M/Iu2 as a function of 0, where a2 is the variance in N2. In the u = 1 (random) case, MI a2 = 1/0(1 - 0). Fluctuations are seen to be relatively small in the F = 1 (antiferromagnetic) case. We already found in ref. 1, for the F = co case at 0 = 1/2, MIa = 1/5.96 = 0.168. Fig. 4 contains flux curves and corresponds to Fig. 1. In many of the examples in this paper, either Ja (i.e., calculated from a, a' transitions) or Jo was found to be very erratic because of particular terms (large rate constants, small probabilities) in equations 9 of ref. 1; the nonerratic member of the pair was always used. The F = X curve in Fig. 4 is the same as J(x) in figure 12 of ref. 1. It is not hard to prove that this should be so. This critical curve would have a cusp if M were infinite. At the other extreme, the F = 1 flux curve is the abscissa (i.e., there is no net mean flux at equilibrium, except for fluctuations). We have included J(ln x) for F = 2 and 10 in order to see the nature of the transition between the F = 1 and F = Xo cases. It is easy to show that, if u >1, then J - uas x - o. This accounts for the asymptotic value J -p 0.0484 seen in the figure. Figs. 1-4 do not correspond directly to any of the BW figures in ref. 2 for this same example. On the other hand, Fig. 5 here is the exact analogue of figure 6 in ref. 2. Note, however, that u = y1/2 (y is used in ref. 2). In Fig. 5 we keep F constant at 106 but vary u. In the BW approximation (figure 6 of ref. 2): (i) when y is continuously reduced (repulsive forces) below y = 1, a critical curve is passed and a phase transition is then observed that
lUlU
0.0
0.2
0.4
0.6
0.8
1.0
FIG. 6. Correlation function cl(@) corresponding to Fig. 5.
persists no matter how small y is (the vertical extent of the "loop" decreases and the loop moves towards ln x -a C as y 0); and (ii) when y is increased above y = 1 (attractive forces), there is negative cooperativity for small 0 and positive cooperativity for large 0 (leading to a critical point and phase transition behavior in the usual way). The exact (i.e., Monte Carlo) behavior of the 10 X 10 system (Fig. 5) is somewhat different. The differences are due primarily to the fact that the BW approximation does not handle strong negative cooperativity realistically enough (e.g., the BW approximation, as we are using it, does not allow, as it should, for a lattice with regularly alternating states 1 and 2 when 0 '/2). In addition, of course, it should be kept in mind that the 10 X 10 system is finite and the BW system infinite (M = aD). To understand the behavior of the 10 X 10 system, Fig. 5 should be examined together with Fig. 6 (correlation function, c,). When u is decreased from 1 to 0.5, positive cooperativity is observed for all 0. But this effect is reversed for 0 >0.22 (Fig. 6) when u = 0.22, and in fact there is negative cooperativity for 0 greater than about 0.5. At u = 0.1 and, especially, u = 0.04, negative Cooperativity completely dominates (in the otter case, near 0=1/2, the lattice is almost completely ordered-ne., with alternating states). The initial positive cooperativity (as u is
0.6-
F = 106 r= 1
0.04
0.4
10
J
-4
-2 0.5
0.21
+2
=
0. 5
+4
+
+4
+6
18 . 0.22 -4
1 -2
+2
0 Inx
FIG. 5.
0(ln x) for F = 106 and various values of u.
FIG. 7. Flux J(ln x) corresponding to Fig. 5.
Chemistry:
J 0.2
0.1
Proc. Natl. Acad. Sci. USA 75 (1978)
Hill and Chen
5263
J
10
-
0.8 -
u = 20
1510
0.6 0.40.2
0.2
0.4
0.6
1.0
0.8
1.2
0.6 x
x
FIG. 8. @(x) and J(x) at F = 20 in the case/a = f= a/2,o = 1, r = 0.05. See text.
FIG. 9. Same as Fig. 8 but for u = 15. See text.
decreased below 1) does not persist long enough to produce critical behavior (unlike BW). The same is true (not shown) at F = 108 but it is not true at F = c, as already mentioned in connection with Fig. 1: u = 0.22 is the 10 X 10 critical value of u and a phase transition occurs for u < 0.22. For u > 1, the exact behavior is rather similar to BW. There is a phase transition for large enough u. Also, if we compare u = 1 and u = 10 in Fig. 6, we see that, in the u = 10 case, there is slight negative cooperativity for 0 < 0.26 and strong positive cooperativity for large 0. The above paragraph refers to F = 106. At F = the situation is quite different. Both exact and BW (see figure 9 of ref. 2) systems exhibit only negative cooperativity when u > 1 (attractive forces). No first-order phase transition is possible. Actually, the curve labeled F = 1 in Fig. 1 is relevant here: it is also the F = c, u = 4.55 curve (the same is true in Figs. 2 and 3). (Analogously, as already mentioned, the F = o curve in Fig. 1 is also the F = 1, u = 4.55 curve.) The corresponding flux curves in Fig. 7 should be compared with figure 8 of ref. 2 (BW). Again the behavior is qualitatively similar for u > 1 but rather different for u < 1, especially for u = 0.1 and 0.04 (strong negative cooperativity). The peak fluxes in Fig. 7, when u < 0.5, coincide approximately with the
Case 2: Two-step phase transition in BW approximation The second interesting example studied in ref. 2 (BW) had, for parameters, fao = =-1/2, a' = 1, r = 0.05 and F = 1, 4, 20, ao. Several values of y > 1 were used in each case. It was found for F = 4 and 20 (but not F = 1 and a)) that two-step phase transitions occur in some circumstances (see figures 11 and 12 of ref. 2). To make a long story short, we tried (Monte Carlo) a large number of combinations of u and F values, in the intervals 1 < u < 50 and 1 < F < 400, without finding any twostep phase transitions. Of course, this could well be a consequence of the use of a finite 10 X 10 system. Figs. 8 and 9 show some typical results. Despite the absence of two-step transitions, the flux still exhibits the off-on "switch" effect noted in connection with figures. 15 and 16 of ref. 2. For example, in the F = 20 and 25 cases in Fig. 9, the flux is "on" over a relatively small range in x (or a ligand concentration) and is otherwise "off."
00,
minima in
as
x
akc.
c1
in
Fig.
6. As
already noted,
when
u
