Kinetics of synthesis and/or conformational changes of biological macromolecules

BIOPOLY MERS VOL. 4, PP. 3-15 (1966)

Kinetics of Synthesis and/or Conformational Changes of Biological Macromolecules

-1. C. PIPKIN, Division of Applied Mathematics, and J. H. GIBBS, Department of Chemistry, Brown University, Providence,

Rhode Island

Synopsis

The analogy, in both the thermodynamics and the kinetics, of reversible polymerizations on templates (in the case wherein these are catalyzed by exoenzymes) to helix-coil transitions (in the case where these proceed only from one end of each macromolecular chain) is presented. A suggestion, based on this analogy, is made concerning the possible nature of biological control of synthesis of macromolecules (enzyme induction and repression). The equations governing the kinetics of these one-dimensional cooperative processes are presented and their solutions discussed.

Introduction

The processes to be treated in this paper are cooperative in the most ex- treme sense available to a one-dimensional system of finite length. That is, they are those which may be viewed, either exactly or in good approximation, as proceeding only at one moving point. This is analogous to the progress, a t a single two-dimensional surface, of a first-order phase transition in a three-dimensional system.

One example of a process which, in reasonable approximation for moder- ate chain lengths, may be viewed as occurring in this fashion is the transition from a-helix to random coil (and vice versa) displayed by individual polypeptide molecular chains in dilute solution. In this system the process of conversion to random coil originating at any given position in the interior of the helix tends to be suppressed, relative to that commencing a t an end of the helix, since release of the first amino acid from the helical posture via the former mechanism requires expenditure of energy sufficient to rupture three adjacent hydrogen bonds whereas release of each amino acid via the latter process requires only enough energy to break one. Of course, if the helix is sufficiently long, processes starting at loci in the helix interior may dominate the transition to random coil by virtue of the overwhelming number of such interior loci in this case. However, helix lengths in proteins appear not to be long enough for this, and, even though there is apparently some cooperative interaction between helical sections in certain proteins, as is indicated by the sharpness of their reversible denaturation

3

4 A. C . PIPKIN AND J. H. GIBBS

transitions, that part of denaturation which is attributable to transforma- tion from helix to random coil can probably be discussed profitably in the context of the approximation in question.

Another example is the helix-coil transition displayed by individual nucleic acid molecules in dilute solution. In this case a disordering originating in the interior of the helix tends to be suppressed, relative to one starting at an end, because the former leads to a random ring with only small entropy, whereas the latter leads to two random chains with a larger entropy. Just as in the polypeptide case, however, disorderings originating in the interior of a helix become significant if the molecule is long enough.

The remarks above refer to transitions in vitro. In vivo, nucleic acid transitions from double helix to random coil and the reverse are apparently coupled to syntheses of either RNA chains (transcription) or new DNA chains (replication) with one or both of the original DNA chains, released from the double helix at least in part, acting as a template, via Watson- Crick base-pairing properties, for the appropriate sequencing of the monomers being polymerized. That is, the normal conditions in the living cell are such that the helix is much more stable than the random coil and can only be disrupted by coupling to a chemical event (polymerization) involving a decrease in free energy sufficient to offset the increase in free energy associated with the disruption of the double helix. At the end of the whole process, of course, the original helix is either restored (in the case of transcription) or replaced with two new ones (in the case of replication). The important point for the present purposes, however, is that in both cases (transcription and replication) the polymerization which supplies the (negative) free energy change during the process is catalyzed by an exo- enzyme, i.e., one which catalyzes only polymerizations progressing sequentially from one end.

Such polymerizations, proceeding only from one end (i.e., via the succes- sive addition of monomers only, as opposed to the formation and condensation of oligomers) , are themselves examples of the most pronounced type of one-dimensional cooperative process, irrespective of whether or not they involve either collateral helix-coil transitions of templates or templates in any fashion whatsoever. The peculiarities of the kinetics of free-radical (vinyl) polymerizations, as contrasted with those of condensation polymerizations that are not enzymatically catalyzed, have long been recog- nized, as has the requirement of microscopic reversibility with regard to the mechanism of establishment of thermal free radical polymerization-depolymerization equilibria, but the direct analogy of these free-radical polymerizations to physical one-dimensional cooperative processes has apparently been overlooked.

To point up this analogy between helix-coil transitions and reversible polymerizations by exoenzymes, it is only necessary t o inspect the distribution functions describing the equilibria as functions of temperature, for example, in the two cases. For the purely physical helix-coil transition in the case under discussion, in which states involving randomly coiled regions

KINETICS OF BIOLOGICAL MACROMOLECULES 5

that do not extend continuously to one particular chain end are excluded, this distribution function is’

K

i = l Ni/C N , = gie-ic’kT/Q(T,K)

= [r“-’(l - r ) ] / ( l - rR) (1)

where the substitution r = ge-‘/kT has been used. €is an energy and g a degeneracy associated with the release of a given monomer unit from the helical posture (all units between it and the end of the molecular system being released also), kT has its usual meaning, K is the total number of units in the individual polymer molecular system, and Q is the partition function for the canonical ensemble of these systems.

For the polymerization-depolymerization equilibrium, the individual polymer molecular system is open with regard to the number of monomer units that are incorporated in it. Therefore, if each polymer molecule alone is taken as the individual system in the statistical mechanical ensemble of such systems, this ensemble is grand-canonical with respect to the monomeric units of the polymer chains. The distribution function is thus

K

i = l Ni/C N i = ‘e” i /kTQ(T, i ) /X(T,K,~)

e p i / k T --ir/kT e / X

= (Xr)i/C (Xr)’

- - K

i = l

= (kr)i-l(l - Xr)/[l - ( X T ) ~ ] (2)

where the substitution h = erlkT has been made in addition to that which introduces r. The quantities g and E in the expression for r have, of course, significances analogous to but different from those in the previous case. Naturally E is essentially the energy change associated with the chemical reaction in which one monomer unit is added to the polymer chain, and g is related to the corresponding entropy change. j~ is the chemical potential of the monomer reservoir. In this case K is the maximum number of monomeric units that can be incorporated into the polymer chain and is finite and equal to the number of binding sites on the template if a template is involved.

A sharp “transition” is thus yielded in this case, just as in the purely physical process of the helix-coil transition.

The sharpness of this chemical “transition” may conceivably be the basis of the control mechanisms governing replication, transcription, and perhaps translation (protein synthesis on messenger RNA templates, a process also known to progress from one chain end2). If, for example, in the case

,? is the grand canonical partition function.

6 A. C. PIPKIN AND J. H. GIBBS

of transcription, release of RNA chains from the DNA template should be a slow step as compared with individual polymerization and depolymerization steps, the distribution of chain lengths of RNA appearing in the free solution as a consequence of this slow release would be a direct re- flection of the quasi-equilibrium set up by the faster processes occurring on the template. These chains would tend to be either completed K-mers (active complete messengers) or monomers, as a consequence of the sharpness of the transition on the template according to which intermediate chain lengths are unlikely at equilibrium except over an extremely narrow range of conditions. Now this restricted equilibrium existing only on the template could be markedly affected (thrown to either the pure monomer or pure completed K-mer side) by only a minor change in conditions; this, of course, is the very meaning of the sharpness of the transition.

Furthermore, if the equilibrium indeed exists as a restricted equilibrium involving only processes on the template as a consequence of slow release from the template, a minor change in conditions, and its consequent marked effect on the monomer-polymer equilibrium, could be specific for a particular template (gene or, more correctly, cistron), the only requirement being that the agent causing this minor change in conditions be one that alters relevant conditions in one (or a few) cistron(s) only. A “repressor,” capable of binding to the nucleotide sequence of a particular cistron and which would have to be stripped off the cistron sequentially if polymerization were to proceed on the cistron acting as template, would, of course, meet these requirements.

Unfortunately, there are available at present neither adequate thermodynamic data describing these polymerizations nor adequate data concerning conditions in vivo from which one might be able to infer whether or not any of the polymerizations associated with replication, transcription, or translation of genetic messages are indeed reversible under conditions in vivo.

The equilibrium properties of the various types of “transitions,’ mentioned here have been elucidated previously,’ as outlined above. The kinetics of polymerization on templates, considered as irreversible, have also been treated.3 The discussion given above would suggest that the general kinetics of these processes, with allowance for reversibility, may be of significance.

Basic Equations and Illustrative Special Cases For each type of process (“transition”) mentioned in the introduction,

we consider an ensemble of identical systems subject to identical environmental thermodynamic conditions, these conditions being chosen such that the process will occur. In the case, for example, of reversible polymerization on templates, a system would be a template plus whatever number of monomer units have been polymerized on it, and the relevant environmental conditions would be temperature, concentration of unpolymerized and unabsorbed monomers in the free solution, etc. The reservoir (solu-


tion) that establishes these environmental conditions is assumed to be sufficiently large that these conditions do not change significantly during the process.

Now we enumerate the states available to each system as 1, 2, . . ., K . Thus, in the case of polymerization on a template, containing K sites, state k would correspond to degree of polymerization k , the largest attainable degree of polymerization being K; in the case of the helix-coil transition a polymer molecule in state k would contain k segments in its randomly coiled region and K - k in its helical region.

We let Nk(t ) be the fraction of systems that are in state k at time t. Since we shall be primarily interested in cases for which the initial condition is

NI (0) = 1

N* (0) = 0 for 2 I k 5 K , (3)

Nk(t) also represents the fraction of systems in which the process has led, at time t, to an excess of forward over backward steps equal to k.

Now the flow of each of the types of process in question is analogous to diffusion of particles on one-dimensional lattices, one particle to each lattice. Thus each “system” would consist of K particle sites and one particle. Equivalently, one may imagine particle systems of the ensemble all diffusing on a common (one-dimensional) lattice, provided only that the individual lattice sites are assumed unsaturable. This is the picture that we shall utilize in the ensuing discussion.

N,(t) is thus viewed as the density of diffusing material at position k at time t, k being an integer. The flux, or rate of transfer, from position k to position k + 1 is Qk(t). Conservation requires that

The flux &(t) is given by

= PNk(t) - nNk+l(t> (5)

We suppose that the time has been scaled in such a way that

p + q = l (6)

We are interested in cases in which p > q > 0. The fact that we take p to be larger than p means that transfer toward higher k is more likely than transfer toward lower k.

< k < a,

the problem of main interest involves only the sites k = 1, 2, . . . , K , where K is some large integer. If there is no transfer from k = 1 to k = 0, or from k = K to k = K + 1, then

Although we can treat eqs. (4) and (5 ) as valid for all k, -

&o(O = &K(O = 0 (7)


[End conditions other than those of eq. (7) may also be of interest.] We are interested in the solution of eqs. (4) and (5 ) , with the end conditions (7) and initial conditions (3).

This problem, and similar problems, can be solved exactly in a variety of forms. Montrol14 has considered the initial-value problem for eqs. (4) and (5) on the infinite lattice - Q, < k < a. Montroll’s method of solution (see Appendix B) can easily be extended to the problem posed by eqs. (3)-(7), and provides a form of solution which is useful for direct under- standing of the nature of the process. In Appendix D we exhibit the exact solution obtained in this manner. Taka& gives the matrix exponential solution and the solution in terms of normal modes of relaxation (Appendix A), as well as the solution in Montroll’s form for the special case K = (Appendix C).

In spite of the availability of exact solutions, the nature of the solution is most easily understood by considering special casf s and approximate solutions. In the present section we consider the special case of equilibrium and the time-dependent behavior in the case p = 1, q = 0. In the follow- ing section we discuss the time-dependent behavior of the mean and variance of the distribution, in semiquantitative terms, for q # 0.

With Qk( a) = 0, the equilibrium distribution is found from eq. (5) to be of the form Nn( m) = C ( P / ~ ) ~ . The constant C is found from the conservation condition

The equilibrium distribution Nk( Q, ) has been presented above.

which follows from eq. (4), with the end conditions (7) and initial conditions (3). The equilibrium distribution is then

The point of main interest here is that the densities Nk( a) are negligibly small unless k is close to K , provided that ( ~ / q ) ~ is large. Furthermore, ( ~ / q ) ~ is large even if p is only slightly larger than q, provided that K is sufficiently large. The condition ( ~ / q ) ~ >> 1 is satisfied provided that ( p - q)K >> 1. In the next section we show that the latter form of the criterion has significance in time-dependent cases as well.

To provide some initial insight into the evolution of the distribution in time, we consider the especially simple case p = 1, q = 0, in which transfer takes place only toward higher states k. In this case, the time-dependent distribution is a Poisson distribution3 with parameter t, for all states except the final state k = K:

N,(t) = e - P ’ / ( k - I)! 1 5 k 5 K - 1 (10)


When t is small enough for NK(t) to be negligible, the mean and variance of the distribution (10) are respectively (k) = 1 + t and u2 = t. Thus, the mean moves toward higher values at unit rate, and the distribution spreads as it moves, the width of the distribution being proportional to t”2. This translation of the distribution continues until NK(t ) begins to become noticeably different from zero, i.e., until (k) + u is of the order of K . This occurs at a time of the order of K - K”’, or simply K , if K is large. The main process of translation of (k) toward its final value, K , is complete by the time T = K . At larger times, the distribution settles into its equilibrium form without any further large change in (k).

Qualitative Description of Solution

The most important features of the distribution can be described in terms of the moments (kp) :

ri

(kP) = 2 kPN,(t) k = 1

The mean (k) and the variance u2 = (k2) -- ( k ) 2 are of particular interest.

(ko) = 1, weobtain Equation (8) states that (ko) = 1. Froin eqs. (4), (5), and (7) with

d(k)/dt = p - Q + ~Ni(t) - p N ~ ( t ) (13)

By similarly forming an expression for the rate of change of (k2), and taking eq. (13) into account, we obtain

du2/dt = 1 - ~Nl ( t ) (2 (k ) - 1) - pNK(t)(2K + 1 - 2(k)) (14)

It is convenient to divide the process into three stages in time. First, there is a relatively short initial interval during which N1(t) decays toward zero from its initial value Nl(0) = 1. There is then a relatively long int&- mediate period during which Nl( t ) is negligibly small and NK( t ) is not yet noticeably large. The final decay toward equilibrium begins when NK(t ) starts to become important.

From eq. (13) it follows that initially, when N1 = 1 and NK = 0, the mean increases at the rate p from its initial value (k) = 1. According to eq. (14), the variance also increases at the rate p initially. When (k) - 1 is of the order of u, the distribution begins to lose contact with the site k = 1, i.e., Nl(t) begins to become negligibly small. The precise behavior of the distribution during the initial interval can be studied by using the matrix exponential solution (Appendix A). .

The most important changes in the distribution occur during the intermediate period, when both N1 and N K are negligible. Equations (13) and (14) show that during this period, the mean grows at the constant rate p - q, and the variance increases at unit rate. The initial rate of increase of (k) is p , larger than the steady rate of transfer p - Q, because transfer toward lower k from the state k = 1 is not allowed. The cumulative


effect of this higher initial rate is to make (k) larger than 1 + ( p - q)t during the intermediate period:

( k ) = 1 + (P - q)t + q / ( p - q) (15) [The fact that q / ( p - q) is the appropriate displacement is shown in Appendix C.] Similarly, the end condition at k = 1 initially impedes spreading of the distribution and makes the initial rate of increase of 0 2

equal to p , smaller than the steady rate, unity. The cumulative effect is to make u2 smaller than t during the intermediate period (Appendix C) :

u2 t - 3 p q / ( p - q ) 2 (16) If the mean grows to a value of t h t order of K at the rate p - q, then the

time required is of the order of T = K / ( p - q) . This estimate, and the preceding qualitative description, is based on the assumption that the distribution changes mainly by steady transfer, with some small spreading. Of course, when p = q = l /2 , this estimate is meaningless, as are the approximations (15) and (16). When p is very close to q, the mean increases mainly because the distribution spreads. To obtain a criterion for change mainly by spreading, we first note that if u2 always increased at unit rate, then u would be of the order of K by the time t = K2. During this time, the mean would increase by steady transfer to a value of the order of ( p - q)K2. If the latter value is small in comparison to u = K , then the distribution has changed mainly by spreading. We thus obtain the criterion ( p - q)K << 1 for change by spreading, and the estimate T = K 2 for the reaction time in this case. If, on the other hand, the opposite condition ( p - q)K >> 1 is satisfied, our previous estimate of T can be expected to be reasonable:

K / ( P - 4) If (P - a>K >> 1 (17) T - {KZ If ( p - q)K << 1

As we mentioned above, the case ( p - q)K >> 1 is the case of main interest, because it is in this case that the reaction can be regarded as approaching completion, with negligible equilibrium densities N k ( a) when k is not close to K . It is for this case that the intermediate-period approximations (15) and (16) are valid.

In the final approach to equilibrium, the site k = K is noticeably occupied and the previous estimates based on neglect of N,(t) are no longer appli- cable. The rate of increase of the mean tends to zero as ( k ) approaches its final value, and the variance begins to decrease as the distribution packs into the high-k states [assuming that ( p - q)K >> 11. This final period begins when ( k ) is smaller than K By an amount of the order of u. With T = K / ( p - q) as a first approximation to this time, and u2 - t during the intermediate period, we have u2 - K / ( p - q) as an estimate of the maximum value of 0 2 . Then, as a second approximation, the final period begins at a time of the order of

T - c~,,,(p - q)-' = K ( p - q)-'[l - K-"'(p - 9)-"'3 (18)


M =

Thus, with K ( p - q) >> 1, the final decay toward equilibrium begins at a time which is smaller than T = K / ( p - q) by an amount which is short in comparison to T.

In summary, we have found that when K ( p - q) >> 1, the process is mainly a steady shift of the distribution Nn(t) toward higher values of k at the rate p - q, with some spreading. The initial higher rate of transfer and lower rate of spread occupies an interval whose length is independent of K , and not large. The final decay toward equilibrium, after the distribution has reached the highest sites k , occupies an interval which is short in comparison to K / ( p - q) .

APPENDIX A

Matrix Exponential and Normal Modes of Relaxation

The results in this section can be found in the book by T a k a ~ s . ~ Let N(t) be the vector with components N,(t) (1 5 k 5 K ) , and let M

P - - 1 Q O 0 P - l q . .

. .

P -1 4

By eliminating &(t) from the system of equations (4), (5 ) , and (7) we obtain

N'(t) = MN(t) (A-2)

and the solution is

N(t) = eMtN(0)

where (A-3)

Here Mo is I, the unit matrix. (A-4), is useful when t is small.

left-eigenvectors of the matrix M :

The solution in the form of eqs. (A-3),

Let - A a , u("), and do) be respectively the eigenvalues and right- and

(A-5) The eigenvalues are

Xo = 0, X, = 1 - 2(pq)"' cos (cr?r/K) a = 1,2,. . ., K - 1 (A-6)


The components u k ( . ) and vk(a) of the eigenvectors are given by

~ ~ ( 0 . 0 = (1 - r-Z)r2K/(r2K - I), ~ ~ ( 0 ) = 1

u k ( a ) = rk sin [ k ? r a / ~ ] - r k - 1 sin [(k - l ) ? r a / ~ ]

(A-7)

and, f o r a = 1,2, ..., K - 1,

(A-8)

uk(a) = ( 2 p / X & ) r - z k u k ( a ) (A-9)

and

where

r2 = P / q (A-10)

The eigenvectors satisfy the mutual orthonormality relations

The spectral representation of the matrix M = (Mii) is

(A-11)

(A-12)

By using eq. (A-12) in (A-3), and taking eq. (A-11) into account, we obtain

N(t) = do) [v(O),N(O) ] + c exp { - X a t ] I I ( ~ ) [v'"',N(O)] (A-13) K - 1

a = l

The solution in the form (A-13) is somewhat misleading because it might lead one to suppose that the distribution is close to equilibrium when Xlt > 1. However, in the cases ( p - q)K >> 1 which are of main interest, the solution (A-13) has the form of a very large number of very small terms (when Xlt > l), and neglect of the exponentially decaying terms is not justified. Alternatively, one might suppose that the distribution is close to equilibrium when exp { - X 2 t ] is small in comparison to exp { - A d ) , ie., when (A2 - Xl)t is large. This is true, but affords a poor estimate of the time required for completion of the process. The relaxation times l / X a , given in eq. (A-6), are so closely spaced that (A2 - X1)t becomes large only long after the distribution has come to an approximate equilibrium. The relaxation times l / X a do not in fact have any great significance in relation to the gross behavior of the distribution.

APPENDIX B

Fundamental Distribution

By using eqs. (5 ) and (6) in eq. (4), we obtain

Nk'(t) = pNk-l(t) - Nk(0 + Q N k + l ( O


We will treat eq. (B-1) as satisfied for all k, - m < k < - . shown how eq. (B-1) can be solved. defined by

Montrol14 has Let G(z,t) be the generating function

m

G(z,t) = c zkN,(t) - a

Then, by multiplying eq. (B-1) by zn and summing over all k, we obtain an equation for G which can be integrated immediately, and we find that

G(z,t) = G(z,O) exp { - t + t (pz + d z ) } (B-3)

We use the expansion6

exp ( t ( p z + q/z> 1 = 2 ~ ( 2 4 ~ t > ( m ) k r2 = p/q ( ~ - 4 ) - m

where I , is the Bessel function of imaginary argument of the first kind, of order k. By using eqs. (B-4) and (B-2) in eq. (B-3), we obtain

where the fundamental distribution Fn(t) is defined by

for r 2 = p/q. The corresponding fluxes &(t) are

F k ( t ) is the distribution which arises from the initial conditions No(0) = 1, As a function of k, F,-,(t)

Montrol14 In particular, the mean is

The identity I-, = In implies

Nk(0) = 0 (k # 0) , with no end conditions. has the same form as F,(t), with the origin shifted to k = j . has derived various useful properties of Fk(t ) . (k) = ( p - q)t and the variance is (r2 = t. that

r - V & = rkF-,(t) (B-8)

APPENDIX C

Special Case K = w

In the problems involving a finite lattice k = 1, 2, . . ., K (or a semi- infinite lattice 1 5 k < -), the solution may be representable in the form (B-5), with coefficients N,(O) so chosen as to satisfy the appropriate end conditions. The image densities N,(O) ( j # 1, 2, . . ., K ) . are at our dis- posal, the real densities N,(O) 0’ = 1, 2, . . ., K ) being prescribed by the initial conditions, eq. (3).


111 the problem of eqs. (3)-(7), at times sufficiently small that N K ( t ) is still negligible, the condition Q K ( f ) = 0 is of no importance. The distribution is then negligibly different from thc distribution for the case K = 0 3 .

In the latter case, only the condition &(t) = 0 need be considered. From (B-7), with (B-8), we find that Qo(t) = 0 provided that

Q-j(o)r’ = --Qj(o)r-J (C-1) From the given initial data, eq. (3) with eq. (5) we obtain

Qi(0) = P = 0 k 2 2 (C-2)

and it then follows from eq. (C-1) that the fictitious initial fluxes for k 5 0 are given by

Qo(O1 = 0

&do> = 0 k 5 -2 (C-3)

Q-l(O) = -P+

The image densities N8(0) for k 5 0 are found by using eqs. (C-2) and (C-3) in eq. (5)

No(0) = r--2

N-,(O) = - r - y 1 - r-2) k 2 1 ((3-4) The solution (B-5) is then

m

N,(t) = F,-&) + r-ZF,(t) - (1 - r-2) c r-2iF,+j(t) (C-5) i = 1

Takacs5 has obtained this result by a different method. The sum of the initial image densities (C4) is zero, and the sum of the

densities on the sites k 5 0 remains equal to zero for all time because the flux Qo is zero. The image distribution tends to move toward higher k at the rate p - q, but diffusion past k = 0 is prevented. The initially negative portion of the distribution moves toward k = 0, where it cancels out the initially positive density at k = 0. Thus, the image densities Nn(t) ( k 5 0) tend to zero as time progresses, and the moments of this distribution also tend to zero. At sufficiently large times, the moments of the real distribution (1 5 k < a) are approximately equal to the moments of the full distribution (- m < lc < m ) :

m m

(kP) = c k P N k ( t ) iz c kPN,(t) t + a, p > q (C-6) 1 - m

The moments of the full distribution (- m < k < m) are easily evaluated by using the generating function (B-2), with the initial densities given by eqs. (3) and (C4). The resulting values of (k) and uz provide the intermediate-period approximations given in eqs. (15) and (IS), respectively.


APPENDIX D

Exact Solution, K Finite

For completeness, we record the solution of eq. (B-1) with initial con- By using methods similar to those ditions (3) and boundary conditions (7).

employed in Appendix C we obtain

- (1 - T - ~ ) C ~ ~ ' Z 7 , - " 2 R + j ( t ) J m 5 0 (D-3) j = 1

and Qx(")(t) = P 2 " ( k - m2K)t-'Fk-m2K(t) (D-4)

The term N,(O)(t) is the distribution obtained in Appendix C by ignoring the condition QK(t ) = 0. The sum N,(O)(t) + Nn(l)( t ) satisfies the condition Q K ( t ) = 0, but does not satisfy Qo(t) = 0 exactly. This two-term approximation can be used to study the approach to equilibrium. The remaining partial distributions NL(")(t), which are relatively unimportant, can be interpreted as arising from reflections of the distribution from one or anotherof theendsk = 1 andk = K.

in part by the Advanced Research Projects Agency. This work waa supported in part by U. S. Public Health Service Grant GM-10906 and

References 1. Gibbs, J. H., and DiMareio, E. A., J. Chem. Phys., 30,271 (1959). 2. Dinteis, H. M., and P. M. Knopf, in Znfomnatbnal Macromolecules, H. J. Vogel,

et al., Eds., Academic Press, New York, 1963, p. 376. 3. Simha, R., J. M. Zimmerman, and J. Moaconin, J. Chem. Phys., 39, 1239 (1963). 4. Montroll, E. W., J . Math. Phys., 25, 37 (1946). 5. Takacs, L., Zntroduction to the Theory of Queues, Oxford, New York, 1962, pp. 12-26. 6. Whittaker, E. T., and G. N. Watson, Modern Analysis, Cambridge, 1935, Chap. 17.

Received June 1, 1965 The irreversible case has also been discussed by Cantor,

Tinoco, and Peller [Biopolymers, 2, 51 (1964)l. For the more general case considered here, in which allowance is made for irreversibility, a discussion of the relaxation times has also been given in an article which appeared after the present work was completed, by Lumry, Legare, and Miller [Bwpolymers, 2,489 (1964)l.

Note Added in Proof.

See our Appendix A.

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Kinetics of synthesis and/or conformational changes of biological macromolecules