WEIGHTDISTRIBUTION OF LINEAR AND MULTICHAIN POLYAMINO ACIDS 6175 Dec. 5 , 1955 MOLECULAR [CONTRIBUTION FROM
THE
DEPARTMENT O F BIOPHYSICS, WEIZMANN
INSTITUTE OF SCIENCE]
Molecular Weight Distribution of Linear and Multichain Polyamino Acids. Statistical Analysis BY EPHRAIM KATCHALSKI, MATATIAHU GEHATIA AND MICHAEL SELA RECEIVED JUNE 20, 1955 It is shown that a statistical analysis of the molecular weight distribution of linear polyamino acids leads to results identical with those derived kinetically. The statistical treatment is extended t o multichain polyamino acids synthesized by polymerizing N-carboxy-a-amino acid anhydrides in the presence of a multifunctional initiator. A general formula giving the size distribution of multichain molecules containing growing as well as terminated side chains is derived. For multichain polyamino acids obtained in the absence of a termination reaction, a sharp molecular weight distribution is predicted. It is demonstrated that the maximal value of the number average degree of polymerization which is attained in the presence of a termination reaction, when a b-functional initiator is used, is b times larger than that which is attained when the polymerization is started by a monofunctional initiator. The fact that the ratio between the weight average and number average degrees of polymerization of a multichain polymer composed of branched molecules containing terminated side chains only approaches 1, as the number of the functional groups of the initiator, b, is increased, shows that the molecular weight distribution is sharp when b is large.
I n a previous article1 a theoretical analysis was given for the kinetics of polymerization of N-carboxy-a-amino acid anhydrides, assuming an initiation reaction (a) with a specific rate constant k1, a propagation reaction (b) with a specific rate constant kz, and termination reactions (c) and (d) with the rate constants kS and kq, respectively. Formulas were derived for the molecular weight distribution and average degrees of polymerization of the still growing chains, characterized by free terminal amino groups, and of the terminated chains, characterized by terminal carboxyl groups.
+ M ki XAH + COz kz XAJH + M +XA,+iH + COz XA,H + M ka_ XA,COHNCH(R)COOH X H 4- M kt XCOHNCH(R)COOH XH
--f
(c)
I
A = *OC-CH-NH.
c=o
I n the present article the molecular weight distribution of the linear chains, growing and terminated, will be deduced statistically. The reactivity of the monofunctional initiator, XH, will be assumed to be similar t o that of a growing chain (Le., K I = kz and KB = k4). The statistical treatment will be extended to multichain polyamino acids, synthesized recently by polymerizing N-carboxy-aamino acid anhydrides in the presence of multifunctional initiators, XHb. It will be assumed that each multichain molecule may contain growing, as well as terminated chains, and that it may, therefore, be represented by the formula 293
(-4m.H)b-Q
x> 1. This expression is similar to equation mers such as polyvinylaniine and polyvinyl alcohol 37, since (k'v') is the average degree of polymeriza- may serve as possible high molecular weight multition of the multifunctional initiator used. By an functional initiators with a molecular weight distrianalogous calculation i t can be shown that when the bution of their own. Although our discussion of the molecular weight same multifunctional initiator is used, but the polymerization reaction leading to the formation of the distribution was devoted entirely to multichain multichain polymer is composed of a propagation polyamino acids, other multichain polymers may as well as a termination reaction, the number aver- be included in the same category. Thus from the age and the weight average degrees of polymeriza- study of the polymerization of ethylene oxide" it tion of a polymer containing terminated chains seems that its polymerization initiated by polyalcoonly will be given approximately by (k'v') (kz/K3), hol (e.g., polyvinyl alcohol), mill lead to the formawhen kz >> k3, v m and k'v' >> 1. The last tion of multichain polymers with a molecular size expression is similar to equations 47 and 48 derived distribution similar to that given for multichain for a homogeneous initiator. polyamino acids obtained in the absence of a termiEquations 34 and 34a show that, in the absence nation reaction. of a termination reaction, polymers with high mo(7! R.R . Becker and M. A . Stahmann, J. Bioi. Chein.. 204, 746 lecular weights may be obtained by making use of (1951). initiators with a great number of functional groups, ( 8 ) E. Wessely, K . Riedl and H.Tuppy, Monatsh., 81, 861 (1950). (9) R. R . Becker and M. A . Stahmann, J . B i d . Chcm., 204, 737 and by continuing the polymerization for long periods in the presence of an excess of N-carboxy-a- (1953). (10) H. Fraenkel-Conrat, Biochcm. Biophys. A c t a , 10, 180 (1953). amino acid anhydride. I n the presence of a termi(11) H. Hibbert and S. 2. Perry, Can. J . Rescorch, 8 , 102 (1933); nation reaction the polymerization is brought to P. J. Flory, THISJOURNAL, 62, 1561 (1940). 0.
2 .&*v:;
'1
52 52
--f
OF LINEAR AND MULTICHAIN POLYAMINO ACIDS 6181 WEIGHTDISTRIBUTION Dec. 5, 1955 MOLECULAR
5
Appendix I When the polymerization reaction is brought to completion and the multichain molecules contain terminated chains only, the total molar concentram
tion of the terminated multichains,
c.L\'bj, j=b
m
j1Vbj
=
(XHb)ub(k,/k3)
(IO*)
j=b
By a treatment similar to the one given above we m
obtain for
j'lvbj j= b
m
nr2@n, for the limiting case k1 >> k~
The value of
Hence
(Y*)
Equation lo*, identical with eq. 45, is finally obtained on inserting (5*) and (9%)into (8*).
obviously
equals the initial concentration (XHb)"of the multifunctional initiator. Equation 44 is thus self evident. It is instructive, however, to show that (43a) satisfies this equation. Defining a new function, ani, eq. 1*, and inserting it into (43a), one obtains eq. 2*
ni&; = (XHb)ol'b(kz/ka)
ni = 1
nz = 1
2
Nbj =
b
5
11
(3*)
ani
and v + m may be derived from eq. 35 of our previous article'
j = b nlnz ...n b i = l
j=b
9-
The identity given in eq. 4" can be easily verified
n,z+n, = (XHt,)oL'b2(kz/k3)'
(12*)
1
ns
Equation 13* identical with (46) is finally obtained oninserting (5*), (9*)and (12*) into (11*) m
jiAYb7 = (XH,)ob(b
where ni
-b
= 1 , 2 , 3 . .. j
3=b
+1
ni = 1 , 2 , 3 . ,, m
ni = b, b
+ 1, b + 2 . . .
b
b
ni = j
m
From our previous article' (cf. eq. 2Gc there'! it can be deduced that as v -+
eq. 51a, one obtains eq. 14* bi
- a)*
=
ul
Lr2cd ni
m
ani =
(j*)
(SHb)o"b
(13*)
Appendix I1 Defining new functions, Ul, C2 and 1;1, equations 15*, 16* and 17*, respectively, and introducing into
r=l
i=l
+ 1)(kz/k8)'
e-(b-!)(fia
n;=l
(14*)
np
tkdv
(15*)
Finally, from equations 5*, 4* and 3* we obtain eq. 6* identical with (44) m
C Nbi = (XHb)O j- b
(6*)
b
ni for j and (2*) for
Putting
Nbj
one obtains
i=l
Differentiating (14*) with respect to v one finds where ni = I ,2,3 . . .j - b
b
+ 1 and En, = j .
Pro-
i=l
ceeding with a treatment analogous to the one given above and taking into consideration the symmetry of the terms on the right-hand side of the last equation, (7*) becomes
Eq. 15* becomes on differentiation with respect to
_ dU1-dv
-(b
- gXkz +
Y
k3)Ul
Thus The value of ni-1 + m may
and for v previous paper'.
for the limiting case k~ >> kS be derived from eq. 33 of our
Equation 19* proves the equality of the first term on the right side of (18*11 with the corresponding term on the right side of (52).
E. KATCHALSKI, Itl. GEHATIA AXD 11.SBL.~
6182
Differentiation of ( l G * ) with respect to v gives
where F, =
dv d U2 = ( b - g ) k 2
nl. .no
(20*)
d1, L;? dv =
s
y
0
Vol. $7
(k2V)nr--l
e-(kz+kdu
(n, - l ) !
dv
Because of symmetry (each tzi and n, equals 1, 2, 3, . . . j - g I), equal terms are obtained when the second summation, carried out over i = 1 to g, written explicitly. Hence
+ s
(kzvjni-1 f i r
"e-(kn+ka)u
(ni
- I)! r = 1
F, (24+)
r#t
fi
In this equation
7z1
+ 1, . . . ( j - 1) and
= g, g
i=l
Equation 24" may be written in thc form
("8
tiz =
1,2, . . . ( j - g).
G1 nl...no
As the sum over ni in the last equations represents
Hence
% ! c3= dv
)
[
(b
k n ( b - g)C;
1,
- g)kzv
-
- 1 -tjlni U,= k 2 ( b - g)AV~~j--R$~ (21*)
?Zl...
i=l
P
ill
...?a,;
C, = g e - ( k a + k d ) v c2ddY
e'
j - l - z ' n , r-1
____
n i . . .1zg
j
Iir#ni
-
1
-
(b .-)!
-g
+ 1)
:ie-(kz+ks)u
r=l v#i
e'
L(b
F, (20-j
nl.. ?io ?zr#iz ~~
Equation 21* proves the equality of the second term of the right side of (1s")with the corresponding term of the kinetic equation 52. Differentiation of (17*) with respect to v gives
I t should be noted that the total number of variables n, in the last equation is only (g - 1). From (15*), (26*) and (14*) it finally can be shown that the last term of (18*)is
where II' denotes a product of all the integrals corresponding to the integral values r = 1, 2 , . . . g, except for the particular integral for which r = i. In the following a prime (') will denote operations in which a specified term is omitted. From (22*) and (1G*) we obtain
Equations 19*,21*, 27* and 1S* prove that (Sla) satisfies the kinetic equation 52. Acknowledgment.-This investigation was supported by a research grant (PHS G-3677) irom the National Institutes of Health, Public Health Service.
