April, 1962
GROWTHON BRANCHED CHAINPRIMER AND COPOLYMERIZATION
68 ti
ON ENZYME CATALYZED EQUILIBRIUM 11. GROWTH ON A BRANCHED CHAIN PRIMER AXD COPOLYMERIZATION BY LEONARD PELLER~ Department of Chemistry, University of Wisconsin, Madison, Wisconsin Received September 6 , 1961
The distribution etatistics a t equilibrium are obtained for growth on a branched chain primer. The treatment is applied to the addition of glucose units to a limit dextrin of glycogen. An analysis of some existing data points to only a fraction of the available glucose residues, which possess an unesterified hydroxyl in the 4-position, as being capable of participating in the equilibrium. A matrix formulation for copolymerization statistics is applied to the equilibrium copolymerization of two nucleoside diphosphates. Inasmuch as copolymerization as well as homopolymerization involves phosphodieeter bonds between the 3’ and 5’-hydroxyl groups of the ribose rings, the chief effect which might argue against random copolymerization would arise from preferential interactions between the purine and pyrimidine bases. The available experimental evidence for the primer-initiated synthesis of single chain polyribonucleotides seems t o support the inference of random eopolymerization at equilibrium.
which represents a glycogen species. The region Introduction I n the preceding communication3 we presented a within the dashed curve is presumed not to be susdiscussion of the equilibrium molecular weight dis- ceptible to enzymatic attack and contains all the tributions which might be anticipated for certain glucose units designated by u. The region external enzyme-catalyzed reversible polymerizations. We to this dashed curve consists of glucose units which wish to extend some of the arguments presented are reactive and are designated by the symbol i. there for linear homopolymerizations to two other It is to be noted that the number of growing chains, j , is less than the total number of branches. This polymerization processes. non-equivalence is supported by an analysis of some More specifical1,ywe will consider the equilibrium experimental data which will be presented below. polymerization b;y phosphorylases where growth For a particular branched chain primer charoccurs on some pre-existing branched chain primer. acterized by a specified number of unreactive gluThe best known e.xample of this process is provided by studies with glucose-1-phosphate as a monomer cose residues and growing chains, polymerization involves only a change in the number of reactive source and glycogen as t’hebranched chain primerunits. Hence, for the system at equilibrium simple the enzyme being a polysaccharide pho~phorylase.~ mass action considerations suffice to relate m u i j to The presence of 1,6-linkages, the “branches,” in muoj, the molarity of the primer with no reactive this primer doubtless accounts in part for the inability of the enzyme, which catalyzes only the units (the limit dextrin). We can express this formation of 1,4-linkages, to degrade the primer relation in the following form analogous to eq. 13 beyond a “limit dextrin” under phosphorolytic of the preceding paper ~nuij= (muo,)gijp’ (1) conditions. The capacity of polynucleotide phosphorylases where p = XK. The appropriate combinatorial t’o synthesize copolymers from various nucleoside factor, gij, for this type of polymerization has been disphospha,tes5 encourages an examination of given by Schulz’a and also by Schaefgen and equilibrium binary copolymerization along certain F l ~ r y , ~namely b simple statistical lines. A detailed treatment of ( i + j - IJ! gij = “most probable”’ copolymerizations will be preil(j - I ) ! sented elsewhere.e Equation 2 simply counts the number of ways of A. Linear Polymerization on a Branched Chain distributing i glucose residues among j chains Primer with no restrictions on the number per chain. From eq. 1 and 2, we can write a simple expresThe primer sptxies in this instance is presumed to possess a number of viable chain ends in a single ion^^^^^ for the mole fraction of the species (uij) molecule. These mould constitute some fraction (see below) of the total number of chains ending in unesterified hydroxyl groups in the 4-position of k=O the pyranose ring for glycogen as the branched chain primer. We designate the molarity of a The summation is over all species with the same given polymeric species as mujj. The basis for this number of unreactive units and the same number of sort of indexing can be seen by reference to Fig. 1, growing chains. Analogously to the treatment of polymerization (1) Presentedin part ,zt t h e 138th n’ational Meeting of the American with a single chain primer, there are three conservaChemical Society, New York. N. Y . , September 11-16, 1960. tion conditions. (2) National Institutes of Health, Bethesda 14, Maryland. (3) L. Pellpr and L.Barnett J . Phys. Chem.. 6 6 , 680 (1962). (4) D. H. Brown and C. F. Cori in ”The Enzymes,” Ed. by P. D. Boyer, H. Lardy, and K. MyrbBck, Academic Press, New York, N. Y . , 2nd Ed., 1959, Yol. 5 , pp. 207-229. (5) M. Grunberg-Manago, ref. 4,pp. 257-280. ( 6 ) L. Peller, J . Chem. Phys., in press.
k(mukj)
(u,k,j)
+ (mxp)
=
C
k(mukj)O
(u,k,j)
+ (mxp)o
(4a)
(7) (a) G. V. Sohulz, Z . p h y s z k . Chem., B46,25 (1939); (b) J. R. Sohaefgen and P. J. Flory, J . A m . Chem. Soo., 7 0 , 2709 (1948).
LEONARD PELLER
686
Vol. 66
It should be noted that eq. 6 and 7 reduce to eq. 18 and 19, respectively, of the preceding article on assuming no polydispersity in u,ie., ( u ) = ~ u and requiring the functionality of the primer to be unity, Le., ($0 = 1. The above conditions are appropriate for a single chain primer which adds monomeric units to one end. The customary index of the polydispersity 6 = w / v is given by 6 = PWO
t
+
- P ) + jPI2?
[(I - P)
(u?02
Fig. 1.-Schematic drawing of a glycogen-like branched chain primer with chain ends susceptible to phosphorylase action indicated by arrows. R designates the terminal reducing group.
+ P a] ($0
(8)
When p = 0, then 6 = (u2j/(uj2,or the polydispersity is characterized by the number of unreactive glucose residues. When p -+ 1, then 6 -+ l/(j)o ( j z ) ~ / ( j ) ~Inz .this regard it is to be noted that (mxp) ( m p ) = (mxp)o (TnP)O (4b) if the average functionality of the glycogen primer, m m (J')o, is appreciably greater than unity and there is (mukJ) (mukj)O (4c) little polydispersity in this parameter so that ( j z ) o k=O k=O ($2 then 6 -+ 1 as p 1. This narrowing of the It is to be noted that there will be an equation like distribution with increasing functionality was (4c) applying to each primer species characterized pointed out some time ago for the condensation by a given value of u and 1. polymerization of a bifunctional monomer on a From the above conservation relations and eq. 3 multifunctional center all of whose reactive groups for the distribution of species, we can derive the are identicaL7&@However, from a variety of studies useful result it appears likely that glycogen is quite polydisperse with respect to the number of chains susceptible to A=,+ (k?O phosphorylate action.8 1- P (do It was pointed out in the previous paper3 that [1 - m a ) (??zxp)o (u k?o ( 5 ) most studies of phosphorylase catalyzed polymeri+ 1' (u k ) (mukj)O (j)O zation on polysaccharide primers were performed (u,kA under such conditions that X apparently was indeI n the absence of any detailed information coii- pendent of the concentration of primer. This cerning the initial distribution of species with dif- can be attributed to a small concentration of availferent values of u and j , summations over these able chain ends t o initiate growth relative to the indices can only be accomplished formally. ( ) glucose-1-phosphate p r e ~ e n t . ~In such circumindicates the number average of a particular quan- stances p must be very close to unity and a reliable tity. It is to be noted that owing to the nature estimate of v' from a knowledge of K and a measureof the polymerization process (u)= ( u )and ~ ( j ) = ment of X would require an unattainable precision ( j ) owhere the subscript zero refers to initial values in the determination of these two quantities. of these averages. Reference to eq. 17 of the preSmall extents of polymerization with p subceding paper s h o w that p/(l - p) = v', the mean stantially less than one provide some opportunity extension of a chain, and (kjo/(j)o = y o ' , the initial t o estimate v' in the above fashion. I n order to number average degree of extension. I n terms of obtain a small degree of polymerization for this these averages and the parameter p we can calcu- limited thermodynamic equilibrium, the concenlate the number average degree of polymerization tration of primer chain ends must be of comparable magnitude to the initial concentration of monomer ( V I as source. It also is significant that this condition should have the effect of minimizing the relative (u (mukJ) importance of any direct reaction of the monomeric P v = uj k=O Ez ( k u) = (4" -(Jh sources. This arises from the fact that a chain end 1 - P (mukr) of a primer grows by addition of glucose units at a uj k = O rate orders of magnitude faster than the rate of (6) dimer formation from glucose-1-phosphate at the The weight average degree of polymerization (a) same concentration.1° A few experiments of this sort seem t o have been then is given by performed some time ago by Hestrin, employing m both glycogen and the limit dextrin of glycogen as
+
+
+
-+
[
5
+
2
u j k=O
I-
+
+
+
+
(8) S Erlander and D. French, J . Polgmer Sei., ZO, 7 (1956). (9) L. Peller, Biochim. et B i o p h y e . Acta, 47, 61 (1961). (10) D. H. Brown, B. Illingworth, and C. F. Cori, Proc. Natl. Acad. Sci. U . A'., 47, 479 (1961).
GROWTH ON BRAXCHED CHAINPRIMER AKD COPOLPMERIZATIOX
April, 1962
primers.ll The latter is formed by exhaustive phosphorolysis of glycogen. We have taken his meager data for a given sample of the limit dextrin and calculated the quantity in brackets on the right-hand side of eq. 5 . v' = p/(l - p ) has been calculated from his values of X and K = 3.0 obtained from values of 1 / X a t the same pH = 7.3 but under conditions where p + 1 . 1 2 A plot of v' against the quantity in brackets on the right-hand side of eq. 6 is shown in Fig. 2 . It is seen that the average degrees of extension of the chains indeed are quite small as indicated by the range of ordinates. A reasonably good straight line can be drawn through the three points comprising the available data on the system. The line practically passes through the origin, i . e . , = 0, as is to be expected for this initially highly phosphorolyzed sample. The reciprocal of the slope of the line ((j)o/ (u IC),,) which is a measure of the fraction of the total number of glucose units originally present in the primer which can initiate polymerization equals 0.041. As these data are of course quite fragmentary and of necessity imprecise, it does not seem advisable to attach t o o great a significance to the above results. However, two conclusions should be emphasized. First, the results can be harmonized readily without invoking an equilibrium constant for bond formation at small degrees of polymerization which differs from that for large degrees of p~lymerization.~J~ Secondly, a chemical analysis of the limit dextrin revealed that 0.14 of the glucose units had free hydroxyl groups a t the 4-position in the pyranose ring.ll If all these groups were to initiate chains the slope of the line in Fig. 2 mould have to be smaller by more than a factor of three. An imprecision in the data sufficient to account for t h k seems unlikely. Alternatively, one must concede that only a fraction (less than of those glucose residues possessing unesterified hydroxy groups in the 4-position can participate in this inhibited equilibrium. The remainder of these residues may lie on chains which are so short that kinetic barriers for their reaction still exist. This type of interpretation is intended to be conveyed by Fig. 1, where a number of short chains are located within the dashed curve. Obviously, further careful kinetic and equilibrium studies of a more searching nature are in order for this system.
+
Be Copolymerization At the outset we will restrict the treatment by assuming that the equilibrium constant for the formation of a bond of a given type is independent of the length of the chain. For binary copolymerizations, it is apparent that there will be four equilibrium const ants ( K x x , etc.) which must be introduced corresponding to the four reactions Kxx
_ .
_ .
X+XP,-XX+P
(Sa!
KXY Y +XP,-YX+P
(9b)
(11) S. Hestrin, J . Bz'ol. Chem., 179, 943 (1949). (12) W. E. Trevelyan, P. F. Mann, and J. S.H a r m o n , Arch. Biod i e m . Rzophys., 39, 419 (1952). (13) 31. Cohn, ref 4, p. 192.
687
1.0
\ *
0 Fig. 2.-A
I.o
2.0
43
3.0
lot of the average chain extension ( v') against
(h where = ( G P ) ~and +k) (1 + (GT)o = ( G T ) ~ .Points calculated using data taken from
(' -
(nzxp)~
X)
X
(u
Urcj
(muki)O
ref. 12.
KYX
-X + Y P ~ - X Y + P
(9c)
KYY -Y+YPz-YY+P
(9d)
XP and YP represent two different nucleoside diphosphates. Proceeding to define a set of sequential probabilities of bond formation for steps of the above type, we have a t equilibrium these relations between these probabilities and the above equilibrium constant s6 pxx = XxKxx PXY =
XXKXY
PYX =
XYKYX
PYY =
XYKYY
(loa) (lob) (10c) ( W
pxx is the probability, an X follows an X in the chain, etc., while Following an inductive procedure essentially equivalent to that employed in deriving the Ising partition function for a one-dimensional ~ y s t e m , l ~ ~ l 5 it is possible to show that the probability of occurrence of an r-mer (P,.), i.e., its mole fraction, is given by the product of three matrices4 P, = (1
1)P?--1(6XP
6YP)T
(11)
where
are, respectively, the mole fractions of the aucleoside diphosphates XP and YP, respectively, as mxp and my? are the molar concentrations of these species and M is the total molar concentration of r-mers (1 5 T < a). ( 6 x p 6 ~ pis) ~the transpose of the indicated row matrix, a column matrix of two elements. The above expression with the p's expressed by eq. loa-10d applies to the equilib(14) H. A. Kramers and G. H. Wannier, Phvs. Rev., 60, 252 (1941). (15) (a) E. W.Montroll, J . Chem. Phys., 9, 706 (1941); (b) E. N. Laesettre and 3. P. Howe, (bid., 9, 747 (1941).
Vol. 66
LEONARD PELLER
688
rium distribution where direct reaction of the monomeric sources, XP and YP, is presumed to occur. The binary copolymerization thus envisioned is analogous to the homopolymerization described in section A of the previous paper. Furthermore the above relations between the sequential probabilities and the relevant equilibrium constants presuppose a numbering of the chain beginning from the end bearing an unreacted phosphate group. Some explanation is in order concerning the definition of the r-mer whose mole fraction is given by eq. 11. P, is the total probability of all species containing from 0 to r X units and r to 0 Y units, respectively. Therefore we have the relation i=O
where Pis-i is the probability of occurrence of a species composed of i X units and (r - i) Y units. Moreover, Pi.r-i is itself the sum over all possible combinations consistent with the specified number of units of each type. The subsequent discussion in terms of r-mer rather than in terms of species described in more “detail” as regards their composition and the arrangement of their constituent units obviates the construction of awkward combinatorial factors. From the definition of Pr as a probability, it follows that m
P,
=
(1
Pr-l
1) [,:I
r=l
1
(QXP 6 ~ p = ) 1~
(13a)
Recognizing that the sum in brackets in eq. 13a represents a convergent infinite geometric series in the matrix P, one obtains, utilizing the properties of such series,lGthe result
2
pr-1
= (1
- P)--*
r=l
This is of course identical to eq. 5b of the preceding paper with the subscript zero denoting initial values of the indicated quantities. Equations 13-15 serve to define the concentra, and M at equilibtion variables, viz., mxp, ~ Y Pmp, rium in terms of the four equilibrium constants and the initial conditions. Hence, the number and weight average degrees of polymerization also are defined although convenient expressions for these quantities are better found via probabilities of a different type6 than those given by eq. loa-10d. A particularly germane point concerning copolymerization involves the nature of the sequence of X and Y units to be anticipated. Following an inductive procedure of the type utilized in obtaining eq. 11, it can be shown6 that the average number of times an X unit follows a like unit in the sequence (( X - X )) is given by a
(X
- X)
=
pxx __ [( 1 I) (1 bpxx
- PI-1 ( P X P 6YP)TI
-
(16)
with analogous expressions for (Y Y), etc. The partial derivative in this equation is understood to be taken holding 6x and @y as well as all the sequential probabilities other than pxx constant. The quantity in brackets is of course equal to the left side of eq. 13b. Calculating the relevant derivatives, one obtains For a system in thermodynamic equilibrium and described by chemical reactions as in eq. 9a-9d, then
It immediately is apparent that X is simply the equilibrium constant for the reaction
-YX + -XY
-YY + -XX
(18)
where 1 is the unit matrix of order 2. On deter- The nature of the copolymerization can be dismining the inverse of 1 - P, premulliplying it by cerned by considering the magnitude of X. For (1 l), and postmultiplying it by ( P x P ~ ~ Y PX) ~>, >1, there will be a strong preference for bonds between like monomeric units or a tendency for eq. 13a becomes block copolymerization; while for X <
