J . Phys. Chem. 1989, 93, 3605-3612
+
case of BH ethylene, the stabilized product may well involve a trigonal CBC type structure.
IV. Conclusions Ten reactions of BH with various inorganic oxides and hydrocarbons are reported. A summary of these rate constants is given in Table IV. We have proposed reaction mechanisms where enough supporting evidence is available. We find that BH reacts readily with the oxides studied, presumably leading to formation of various boron oxyhydrides. In several cases the most likely product channel leads to HBO or HOBO. This implies that any hydrides produced in boron combustion will be rapidly partially oxidized and fed into the known’ B/O/H/C chemistry scheme as HBO or HOBO. These species can be detected by infrared absorption, and, in principle, our proposed mechanisms could be confirmed. Our production method for BH, 193-nm photolysis of BH3C0, is not a clean system (at
3605
least 70% BH3 is produced which is also readily oxidized), and we are not able to study the products of these reactions at this time. BH displays the expected electrophilic behavior in its reactions with unsaturated hydrocarbons. The association reaction of BH with H, appears to proceed over a slight barrier. Detailed calculations on the potential surface for this reaction as well as an experimental study of the temperature dependence of the reaction rate are under way in our laboratories.
Acknowledgment. We thank Dr. Mike Page of the Laboratory for Computational Physics, NRL, for providing us the results of his calculations prior to publication and for many helpful discussions. Registry No. BH, 13766-26-2; CO, 630-08-0; H2, 1333-74-0; NO, 10102-43-9; H20, 7732-18-5; 0 2 , 7782-44-7; COZ, 124-38-9; CH4, 7482-8; C3H8, 74-98-6; C2H4, 74-84-0; (CH3)2C=C(CH3)2, 27416-06-4.
King-Altman-Hill Diagram Method for Open Systems Douglas Poland Department of Chemistry, The Johns Hopkins University, Baltimore, Maryland 21 218 (Received: August 8, 1988)
The diagram method of King and Altman and Hill to calculate the steady-state populations of species connected by first-order reactions is extended to open systems. It is shown that a system of species, linked by first-order or pseudo-first-order reactions, with zero-order sources and first-order sinks is mathematically equivalent, with respect to the diagrams required to calculate the nonconserved species distribution, to a closed system of species in equilibrium with a fictitious reservoir. As with closed systems, one can interpret nonequilibrium steady states in open systems as a complex of interlocked cycles with appropriate steady-state cycle currents.
Introduction About 30 years ago, King and Altman’ published a simple method that utilized diagrams (or graphs) to calculate the steady-state concentrations of intermediates in enzyme-catalyzed reactions. Later, HillZ rediscovered the method and extended it, applying it to many models in physical biochemistry. An important application3 of the King-Altman-Hill (KAH, for short) approach is the determination, in a simple pictorial manner, of the functional form for the conversion of substrate to product and the transport of substances across membranes for many complex kinetic schemes. The reactions considered by King and Altman and by Hill are first-order conversions, for example, between the different forms of an enzyme or protein El E2 E3 (1)
- -
or second-order reactions, such as the binding of substrate to an enzyme or the dissociation of the enzyme-substrate complex to give enzyme and product (where we use the convention that the rate constant above the double arrow refers to the forward reaction).
- k3
ki
E
+ S ki
ES
k4
E
+P
(2)
If the interconversion of the enzyme forms is rapid compared to the time scale for significant change to occur in S or P, then one
can treat the reactions in eq 2 as pseudo-first-order reactions (3) where k,’ = k l [ S ]and k4/ = k , [ P ] . The fact that S and P can have nonequilibrium values is the reason that the system can exist in a nonequilibrium steady state. In the present paper, we will extend the KAH method to include the case of open systems. We will first review the method for closed systems and then outline the extensions of the method required to treat open systems.
KAH Method for Closed Systems Consider a set of species that are coupled by reversible firstorder or pseudo-first-order reactions k!,
i-j
-
where we take kij to represent the rate constant for the process i j . One can represent a set of n species, all of which are interconnected by reactions of the form of eq 4, by a graph, such as one of those illustrated in Figure 1 for n = 2-6. If we let x i be the mole fraction (or probability) of species i and introduce the vector of mole fractions, x, and the matrix of rate constants K X
( I ) King, E. L.; Altman, C. J . Phys. Chem. 1956, 60, 1375. (2) (a) Hill, T. L. Free Energy Transduction in Biology; Academic Press: New York, 1977. (b) Hill, T. L. Cooperatiuity Theory in Biochemistry:
Springer-Verlag: New York, 1985. (3) Poland, D. J . Phys. Chem., following paper in this issue.
0022-3654/89/2093-3605$01.50/0
(4)
k,,
=
(Xi, X2,
..., X,)
K =
(~jj)
where K~~
= - k .lJ. ( i # j )
K~~
= x k i j (i # j )
then the kinetics are given by the single equation 0 1989 American Chemical Society
(5)
3606
The Journal of Physical Chemistry, Vol. 93, No. 9, 1989 dx/dt = -xK
(6)
AX = x - X,
(7)
Poland n
Defining where the subscript "s" refers to the steady state, eq 6 has the solution Ax(?) = Ax(0) exp(-Kt)
(8) A simple numerical method for obtaining Ax(t) that avoids the calculation of the eigenvectors of K is given in Appendix A. At the steady state, dx/dt = 0 (where 0 is the null vector), and one has from eq 6 x,K = 0
(9)
n
which is not sufficient to calculate the values of xi since these quantities are not independent. That is, the species probabilities must obey the conservation relation n
EXi =
1
i= I
One can combine eq 9 and 10 as follows. Define the vector uj as uj = (0, 0, ...( 1, ..., 0) (11) Le., a null vector except for a "1" in thejth element. If one then replaces the jth column of K with the vector (all values of 1)
;)
v+=(
giving the matrix K,, then the combination of eq 9 and 10 reads xsKJ = uJ
U
Figure 1. Schematic representation of n species connected reversibly by first-order reactions for n = 2-6. These schemes represent the general case where every species is connected to every other species.
King and Altmanl recognized that eq 16 had a simple physical interpretation in terms of diagrams or graphs. In particular, recognizing that det Kj is independent of j , one sees that eq 16 has a partition function like structure (using Hill's notation)
where
(13)
which immediately yields x,. x, = uJK,-I
c = i(Cj)
(14)
J=I
King and Altman' noted that if one uses Cramer's rule,4 one can obtain the elements of x, as the ratio of determinants xj(s) = lKjl/lKjl
(15)
where K,, is K with thejth column replaced by v+ and thejth row replaced by uJ. For example, for the case of four coupled species, K I would have the structure
(i
1
0
0
0
K42
K43
K44
::: ::: 1::)
where one can see that the matrix can be blocked along the diagonal into a 3 X 3 matrix and a 1 X 1 matrix (the 1 element). When a matrix can be blocked in this manner, the determinant of the matrix is the product of the determinants of the diagonal blocks. Since the determinant of the 1 X 1 matrix is simply unity, one has IK1,1 =
I
K22
K23
K24
K32
K33
K34
K42
K43
K44
I
Defining the diminished matrix KJ,', which represents K with the j t h row and j t h column deleted (because the designation as to which species is number 1 is arbitrary, it does not matter that our example used the first row and first column), one has XJS) = lKj,'l/lKjl
(16)
(4) Ayres, F. Theory and Problems of Matrices; Schaum Publishing: New York, 1962.
(19)
The combination of symbols in eq 19 is unfortunate, but the same notation in eq 17 is very simple and useful. One notes in passing that eq 17 holds for a nonequilibrium steady state and, as a special case, true equilibrium. Since at equilibrium one has
where qj is the molecular partition function for species j , eq 17 must evolve into the above relation as the system approaches true equilibrium. Exactly how this happens is outlined in Appendix B. Since Kjjl is diminished [reduced to ( n - 1) X (n - 1) from n X n ] Z j will be a sum of terms each containing n - 1 factors (rate constants). As an example, consider the general set of three coupled species. Then, e.g., the quantity Z3is given by
The graphical representation of these terms is illustrated in Figure 2. One sees that C3contains the sum of all two-step processes that lead to species 3. King and Altman generalized this result is the sum of products of ( n - 1)k. (b) The as follows: (a) signs of all of the terms appearing in C j are positive. (c) Each species appears as the origin in ki,j only once in a given product (in other words, multiple paths can converge on any species, but multiple paths leading to species j cannot diverge from a given species). (d) None of the paths can represent closed circuits. A simple graphical way to generate all of the paths can be obtained by constructing a modified Cayley tree, or branched
cj
King-Altman-Hill
Diagram Method for Open Systems
The Journal of Physical Chemistry, Vol. 93, No. 9, 1989 3607
I
/
uu
Figure 2. Diagrammatic illustration of paths generated in the quantity
uu
\
x3required to calculate the steady-state probability of species 3.
Figure 4. Branching tree graph for species 5 in a system of five species
reversibly connected to each other (as shown in Figure 1).
& 2
2
b
U
Figure 5. Schematic representation of n species (for n = 2-6) connected in a cycle. These graphs are a special case of the more general reaction
schemes shown in Figure 1. TABLE I: Number of Diagrams, 0, Appearing in Systems Figure 3. (a) Branching tree graph from which all diagrams required to calculate the steady-state probability of species 4 (interconnected in a general manner with three other species as shown in the upper square) can be traced. (b) Three sample diagrams traced from the branching tree graph of Figure 3a.
graph. For example, consider the case of four coupled species. All paths leading to species 4 can then be traced on the tree graph shown in Figure 3a. Several such paths are illustrated in Figure 3b; there are a total of 16 allowed paths or 16 terms in C4(and in all the other for the case n = 4). One must remember that in any path each species number can be used only once. The tree graph for n = 5 is shown in Figure 4 with species 5 as the target; the graphs for the other species would be similar. The trees shown in Figures 3 and 4 represent the most general cases for n = 4 and n = 5 , respectively; in special cases, some reactions might be absent and then the trees would be correspondingly simplified. The question arises as to how many terms in general there will for various n. Since all of the terms in are positive, be in the number of terms is easily calculated by setting all of the k,/
xi
xi
E, for Various
an
n
an
general tree cyclic path
2 3
4 5
1 3 16
3 4
125
5
general tree 1296 16807 262144
n 6 7 8
cyclic path 6 7 8
= 1 and evaluating the determinant. Then R, is the general number of terms. For example
R2 =
,'I I,'
=3
n, =
3
-1
-1
= 16
and so on. The values of R, are listed in Table I for various n; one sees that the number of terms appearing in increases very rapidly with n. For cyclic reaction schemes, such as illustrated is much less; these numbers in Figure 5, the number of terms in 1, are also indicated in Table I.
xj
3608 The Journal of Physical Chemistry, Vol. 93, No. 9, 1989
Poland
I
Figure 6. Schematic illustration of an open system. The sources and sinks are completely independent of the chemistry going an in the system,
7 y
Y &
y
(1)-(2)
kz1
xR=
Eyre 7. .Kinetic scheme for a reaction in an open system. The ol values
represent zcrc-order source terms, while the w values are first-orderrate constants for the irreversible removal of the appropriate species.
mika
*
%kl2
+
mimz
R m 8. Diaerams for terms arisine in the treatment of the o ~ e swtem n .shim in Figure 7. Here. R is treaied as fictitious reservoir reversibly coupled to each spccies. .
I
UhS ww fgn oprn sy%im We now consider open systems such as that illustrated in Figure
6 where there is a source and a sink. We will assume that the source rate for species x I is independent of x, (Le., zero order) and that the sink rate is proportional to xl (first order). Thus, one has the general reaction scheme shown in Figure 7. We emphasize that there need he no thermodynamic relation between the source
1.
and the sink rate constants, aIand wl, respectively. They literally can be set arbitrarily by external means. As a special case, the system might he in contact, via a semipermeable membrane, with a reservoir, and in that case, one would have
e,
xi
reservoir
0,
While this is often what people mean when they say that a system is open, we want to treat the general case where the state of the species external to the system is irrelevant to the problem and a, and w1 are completely independent. The kinetic equations for the scheme shown in Figure 7 are dx,/dt = al
+ (-k+I + k2lx2) - wlxl
dx,/dt = 02 -k (klzxl
- k21x2)- w2x2
(21)
One can easily generalize the above equations to the case of n species interconnected in a general manner, as shown in Figure I , each species having source (a,)and sink (w,) terms. We introduce the source vector a and the sink matrix W a
=
((YI.
a2,
..., a")
The general kinetic equation is then given by dx/dt = a - xK - xW
Figure 9. Graphs for open systems where n species (n = 2-4) are con-
nected in general to arbitrary sinks and sources (left-hand graphs) and the equilibrium steady-state model (right-hand graphs) where all of the sinks and sources are combined into a single fictitious reservoir. R. of the concentrations: Le., eq IO does not hold. For the example shown in Figure 7, one has
(23)
At the steady state, one has dx/dt = 0 giving
x,(K
+ W) = a
(24)
+ W)-l
(25)
Solving eq 24 for x, one has x, = a(K
As with the closed system, one can alternatively use Cramer's
rule to obtain the elements of x, xj(S)
= I(K
+ W)jI/I(K + W)l
+
(26)
where (K + W), is the matrix K W with the j t h row replaced by a. For the open system one does not have the conservation
On expanding the determinants, one finds the graphical representation shown in Figure 8, where we have interpreted the open system shown in Figure 7 as two species in contact with a common reservoir, R, as shown in Figure 9a. It must he emphasized that this is a mathematical equivalence only; there is not literally a single reservoir acting as a common source and sink for all of the species. This same interpretation holds for systems with .more
King-Altman-Hill
The Journal of Physical Chemistry, Vol. 93, No. 9, 1989 3609
Diagram Method for Open Systems
than two species, as illustrated for the cases n = 3 and n = 4 in parts b and c, respectively, of Figure 9. Again, the mathematics of eq 26 can be interpreted as if the n species were in equilibrium with a single reservoir R; the diagrams that contribute to X i in the open system n R are exactly equivalent to those for a closed system of n 1 species; Le., with respect to the diagrams, R is just another species. Note, however, that in the closed system the x values are conserved (eq lo), while that is not the case for the open system (Le., in our example, El + Ez # C R ) . To show, in general, that an open system has the same diagrams contributing to the E, as does the appropriate closed system, consider the system of three linked species illustrated in Figure 9b. If one treats this system as an open system, as in eq 23, then one obtains the following matrix required to construct
+
+
x3
;;:(( =
M‘ = K33
+ k13 + k l R ) -k21
(k21
+ k23 + k2R)
-k23)
kR2
-kR3
where we have set ai = kRi and w j = kIR. Now if we treat R just as an additional species in a closed system, we obtain (using eq 15) M = (K + W)3 = (kl2
+ k13
+ klR)
(2..
-kIR
-k12
+ k23 + kZR)
(k21
-kRZ
-k2R (kR1
+ kR2 + kR3)
x,v+ = 1
(32)
Defining the matrix Kj as K with the jth row replaced by v+, one obtains a combination of eq 9 and 32 xsKj = uj
(33)
where uj is defined by eq 11. Equation 33 reflects the steady-state conditions that dx,/dt = 0 and the special condition that xR = 1. Again, by Cramer’s rule, one has
xj = lKjjl/lKjl
(34)
where, again Kjj is the matrix K with thejth column replaced by v+ and t h e j t h row replaced by up Replacing the j t h column and row, respectively, by v+ and ui is equivalent to diminishing the matrix by eliminating the j t h column and row, giving Kii. Hence, one has
-kl3
-k12
then eq 30 is in general given by
)
xj
= lKjj’l/lKjl
(35)
As with the corresponding quantity in eq 15, one observes that Kj is independent of j (it does not matter which column in K is replaced by v+). Hence, one can replace the Rth column, and the determinant is equivalent to removing the Rth row and column thus giving E R . One has
xj =
Ej/ER
(36)
We maintain that the diagrams obtained by expanding the two determinants are identical, that is det M = det M‘ To see that this is so (without actually expanding the determinants), one notes that IlMIZIl = M’
where
where
Thus, we again derive the basic equations for our open system. It should be emphasized that one cannot interpret an open system as a closed system with an added species R representing a fictitious reservoir if one wants to correctly describe the kinetics; Le., eq 6 does not describe the kinetics since xR is in fact at unit concentration throughout (Le., the source reactions are really zero order). One must use eq 23 to describe the actual kinetics. At the steady state, one can introduce the idea of the equivalent closed system and consistently combine eq 9 and 30 into one equivalent relation, eq 33, leading to eq 35.
I,=
1 0
(o
0 1 0
1
0 0
0
1
-) (: : ;) 12=
Since det (IIMI2I1)= (det I,)(det M)(det 12)(det I , ) and det I, = -1
det I2 = 1
one obtains the desired result, det M = det M’. One can generalize the argument to any number of species. Thus, the open systems shown in Figure 9 can be interpreted in terms of an equivalent closed system with a single fictitious reservoir, R. Although we have shown that the mathematics of the steady state in the open system is equivalent to that for the appropriate closed system, one would like to understand why this is so. Suppose that we assume that this is indeed so and write down the kinetic equations for a closed system of species 1, 2, 3, and R. Then, the kinetic equations are given by eq 6, and at the steady state, we have the condition of eq 9, h K = 0. Now, unlike a true closed system, the conservation relation of eq 10 does not hold, but rather one has the alternate condition
=1 (30) that is, the source reactions are really zero order (they are described as first order in eq 6 since there R is just another species); one can make the reactions the equivalent of zero order at the steady state by forcing xR to equal 1. Introducing the modification of the column vector v+ of eq 12 XR
ER
Cj = det Kj/
= det
KRR’
(37)
Exj # 1
(38)
and one notes that XR
=
CR/ER
= 1
Cycles and Cycle Currents Hill has greatly extended the use of diagrams to calculate steady-state probabilities. In particular, he has shown how to use diagrams to calculate steady-state cycle current^.^ As an important example of a closed system, consider the standard Michaelis-Mented scheme for conversion of substrate, S, to product, P, via an enzyme-substrate complex, ES: E
k,
k,
c+
++
+ S k2 ES
ks
EP
k4
*-,
E
+P
As with the example given in eq 1, the factors S and P can be incorporated into pseudo-first-order rate constants kl’ = kl [SI and k i = k6[P]. The scheme of eq 39 can be interpreted as a cycle as shown in Figure 10a (using pseudo-first-order rate constants; hence S and P are not shown explicitly), with bond currents (between successive species) and the cycle current indicated. The bond currents are J1
= k,[E][S] - k,[ES] J3
J2
= k,[ES]
- k,[EP]
= k5[EP1 - k6[EI[Pl
In terms of the above currents, the kinetics are simply
.+=(!)
(39)
k6
( 5 ) See: Reference 2a, Chapter 2.
(6) Michaelis, L.; Menten, M. L. Biochem. Z.1913, 49, 333.
(40)
3610
The Journal of Physical Chemistry, Vol. 93, No. 9, 1989 d[E]/dt = -JI + J3 d[ES]/dt = Jl - J2 d[EP]/dt = J2 - J3
(41)
AA R
J3
At the steady state, one has a single cycle current
J = J1 = 52 = J3
Poland
(42)
1
J2
Hill has shown that the cycle current can be written as J = [Elo(n+[Sl - n - [ P I I / C
(43)
where
n+= klk2k3 II-= k2k,k6 (44) Le., n+is the product of rate constants taken counterclockwise around the cycle and n-is the product taken clockwise. is
J2
2
/
Figure 10. Illustration of bond currents ( J , , J2,J 3 )and the cycle current (J)for (a) an enzyme-catalyzed reaction in a closed system and (b) an open reaction between two species and a fictitious reservoir. All systems with nonequilibrium steady states can be represented graphically as cycles.
the sum of diagrams given by eq 19; [E], is the original concentration of enzyme. At equilibrium
n+[S], = n-[P],
K =
n+/n-
(45)
where K is the equilibrium constant for the overall reaction
s-P
(46)
In a closed system, the existence of a nonequilibrium steady state is possible because S and P can have values different from S, and P, (of course the system will slowly relax to the equilibrium state, Le., S S, and P PJ; we assume that there is a single well-defined temperature so that the values of the rate constants kij(T) are fixed. In an open system as illustrated in Figure lob, one also has a cycle current given in a similar manner (if there are pseudofirst-order rate constants, one could factor out the concentration terms explicitly as in eq 43; in an open system, one need not have pseudo-first-order reactions to have a nonequilibrium steady state)
-
-
J =
In+- n-)/CR
= (alkl2w2 - a2k214/(WlW2 + Ulk21 + w 2 k d
J1 1
2
2
J
J2
3
J1 (b)
(47)
which, for the example shown in Figure lob, becomes J
@ A EJ2
(48)
Figure 11. Illustration of bond currents and cycle currents for various kinetic schemes in open systems. The graphs in (a) represent models where, in a linear chain of reactions, only the first and the last species are connected to the reservoir. In the models illustrated in (b) all of the species in a linear chain are connected to the reservoir.
In the open system, there is no thermodynamic equilibrium state that the system is moving toward since there is no necessary relation between c y 1 , wI,a2,and w2. If one sets a2 = 0 and w l = 0, then one has the special case
J = a
(49)
Hill' has shown that at the steady state there is a simple relation between the number,f, of independent currents in a graph, the number of lines, L, and the number of species, n. The relationship is f=L-n+l
(50)
For simple cyclic systems, such as those shown in Figure 1 la, there is only one current. Such a system is the case of a chain where only the first and last species are connected to the outside, e.g. R-1-2-3-4-R If the species are connected to each other in a linear chain, with each connected to the outside, then one has the graphs shown in Figure 1 lb. The number of steady-state currents in that case is
f=n-2 The independent currents are shown as JI,J2, and J3 in Figure 11b; the other currents, labeled a in the middle graph and a and
d C
(7) See: Reference 2a, p 14. Equation 50 was derived for closed systems, but it also applies to open systems where R is counted as one of the species in n. The derivation is simple. Given L reactions, one can write down the current equations for n - 1 independent reactions (no reaction for R); the n - 1 species are independent because there is no conservation relation for an open system. Thus, the number of independent currents is L - ( n - 1) = L - n + 1. In a closed system there are n reactions, but only n - 1 are independent, because there is a conservation relation; there is no species R in that case.
J2
J2 (b)
Figure 12. Illustration of bond currents and cycle currents for various kinetic schemes in open systems. The graphs in (a) show the cycle diagrams for the central graph in Figure 11b. The graphs in (b) represent reactions linked in a cycle (as shown in Figure 5 ) where all of the species are linked to the reservoir.
King-Altman-Hill
Diagram Method for Open Systems
The Journal of Physical Chemistry, Vol. 93, No. 9, 1989 3611
E:
ES A then one has the graph shown in Figure 13b. The steady-state populations are
(a)
Figure 13. Illustration of two enzyme reactions where the total amount of enzyme is variable.
b in the right-hand graph, are related to the independent currents by simple conservation laws (the sum of the currents into a junction equals the sum of the currents out):
+ JI
J,
Jb + J2 = J3
= J2
The arrows in Figure 1 1 b are just for convenience in defining the above equations; any current can be positive or negative, so the arrows simply indicate the sign conventions used in each graph. The more complicated graphs of Figure 1 l b can also be analyzed in terms of cycle currents. The three possible cycles for the middle graph in Figure 11 b are illustrated in Figure 12a. One has the following relations between the bond currents and the cycle currents.
Again, the arrows define the sign convention. The cycle currents are given by analogues of eq 47 JI
=
- a3k32k21w1)/xR
(alk12k23w3
The extra factors in the above equations for JIIand JIII[the factors k32 w3 and k 1 2 ul,respectively] represent the sum of single reactions leading into the appropriate cycles.8 The case where one has a simple cycle of reactions with each species connected to the outside is illustrated in Figure 12b. The independent currents are labeled J 1 ,etc., and the interior currents again are obtained from the independent currents by conservation laws. For example, for the left-hand graph in Figure 12b one has
+
+
J , = Jb
J,
J1
+ Jb = J3
53
A E Z E
+S
-
k3
ES - E
k2
Appendix A Simple Matrix Calculation of First-Order Kinetics. The kinetics of the general closed system are given by eq 8, which can be written in terms of the eigenvalues of the matrix K:
+P
[ESI, = 4 k i [ S I
+ k 4 [ P I ) / u ( k 2 + k3)
If we modify the scheme to read (8) See: Reference 2a, p 21. (9) At the end of ref 3 a system that is open with respect to substrate and product is briefly considered.
e-X21,..., e-'J)A
Ax(?) =
(A- 1)
(Since E x , = 1, one of the eigenvalues must be zero.) The calculation of A requires knowledge of the eigenvectors of K. This calculation can be circumvented as follows. One takes the mth derivative of Ax using eq 8
AX,,,
E
(dmAx/dtm)o = Ax(0) (-K)m
('4-2)
where the derivative is evaluated at t = 0. One can also form the same mth derivative from eq A-1: (dmAx/dtm)o =
([-X1]"',
[-X2]",
..., [-X,]")A = I,A
(A-3)
Letting m vary from zero to n - 1 one can use the corresponding vectors in (A-2) to construct the matrix of derivatives
D = [Ax,]
k4
This scheme has the diagram shown in Figure 13a. There will be a steady-state current in the cycle due to the fact that S and P do not have their equilibrium values, but since the connection between E and the outside (R) is a dead-end, there can be no steady-state current in that bond (there can only be a steady-state current if a bond is involved in one or more cycles). The steady-state values of E and ES are
[El, = n/u
Conclusion We have shown that, with a few minor modifications, the KAH diagram method can be applied to open systems. To calculate steady-state populations, one need only introduce a single fictitious reservoir that is coupled to each species (even though the sinks and sources are in reality completely independent). The diagrams used to calculate the steady-state populations are then exactly the same as those for a closed system with the reservoir as one of the species. The only difference in the procedure for treating open systems is that the normalization factor is different ( E Ris used in eq 36 rather than in eq 17), reflecting the fact that the species are not conserved in an open system. Although one can use eq 14 or eq 25 for closed and open systems, respectively, to obtain the numerical values of the species concentrations at the steady state, the diagram method is very important because it allows one to determine very simply the dependence of the cycle currents on such quantities as substrate and product concentration in enzyme-catalyzed reactions. This topic is explored in the following paper.
+ J , = J2
We close by examining a simple enzyme-catalyzed reaction that is open with respect to the various enzyme species? First, consider the case
kl
In this system there are two independent currents. The three currents shown in Figure 13b are related by the equation J 1 - J2 = JR. The inner current, with the reservoir, is simply JR = CY.
(A-4)
Similarly, one can construct the matrix of powers of the eigenvalues
L = [Iml
('4-5)
One then has the equivalence relation
D = LA or A = L-ID
(A-6)
Thus, given the matrix of rate constants, K, and the eigenvalues of K, one can simply obtain the matrix A required in eq A-1 for the kinetics.
3612 The Journal of Physical Chemistry, Vol. 93, No. 9, 1989
Poland
k 23 k32
(b)
Figure 14. Three species linked via reversible reactions (a) in a linear fashion and (b) in a cyclic fashion. Both models have the same equilibrium probability distribution, but they have different steady-state probability distributions.
Appendix B Relationship between Steady-State and Equilibrium Probability Distributions. Consider a closed system consisting of three species in equilibrium via first-order reactions kl3
kll
1-2-3 kii
k,i
At equilibrium the probability, say, of species 2 is given by P2
=
42 41
+ 42 + 43
-
42/41
+ 42/41 + 43/41
-
K12
K13
k12/k21
=
-
(B-1) k2lk32 + k l Z k 3 2 + k12k23 Now consider the same system at a nonequilibrium steady state. According to the KAH method, one will have different steady-state populations depending on whether one has the scheme of Figure 14a (I = linear scheme) or of Figure 14b (I1 = cyclic scheme). Labeling the various diagrams as shown in Figure 15, one has the two sets of partition function like probabilities
"=
= (a1
(11)
c3'k21k13
Figure 15. Diagrams that arise in the calculation of probability distributions for the two kinetic schemes shown in Figure 14.
That is, the probability distributions are the same for I and I1 as long as there is some path between the three species. One sees that the probability distribution p,(I) is exactly the same as that obtained from the standard equilibrium expression as shown, for example, in eq B-1 for p 2 ( I ) . At the steady-state
-
(B-5)
ai = bi = ci
(B-6)
P L I ) # Pi(II)
The question is how pi(I) pi(II) as the steady state relaxes to the true equilibrium state. One way this might happen is for the following condition to hold true:
k12k23/k21k32
k12k32
(1) = uI/A
b3 'k23k13
+ K12 + K13
where qi is the molecular partition function for species i and KO is the equilibrium constant for the reaction i j . The Kij are related to the rate constants k , in the standard fashion:
K12 = One then has
a3 'k12k23
We note that the six k , in the problem can all be independent at the steady state; at equilibrium, there is the thermodynamic relation k12 k23k31
=
k l 3k32k21
(B-7)
meaning that only one k, is dependent. Thus, eq B-6, in general, will not hold at equilibrium. One can write pi(II) in the form
Pi(II) = (ai
+ a , ) / ( A + S)
(B-8)
where
+ bl + c , ) / ( A + B + C)
ai=bi+ci
S=B+C
(B-9)
Then, in order for pi(II) to equal p,(I), one must have a J A = (ai
where A =
+ 0 2 + a3 C=
C]
B = bl
+ c2 + c3
+ b2 + b3
01
(B-3)
Now, at true equilibrium, we have
Pi(I) = Pi(II)
+ a J / ( A + S)
(B-4)
bi
+ ci = ( B + C)a,/A
(B-10)
which is the actual manner in which the two probability distributions become equivalent at equilibrium.
