April 5, 1955
POLYMERIZATION KINETICS OF N-CARBOXY-CY-AMINO ACIDANHYDRIDES
to the expression of the catalytic properties of metals. Acknowledgments.-This investigation was supported by a grant from the Rockefeller Founda-
1925
tion and by a contract between the Office of Naval Research, Department of the Navy, and Northwestern University, NR 124-054. EVANSTON, ILLINOIS
[CONTRIBUTION FROM THE DEPARTMENT O F BIOPHYSICS, WEIZMANN INSTITUTE
O F SCIENCE]
Theoretical Analysis of the Polymerization Kinetics of N-Carboxy-a-amino Acid Anhydrides BY EPHRAIM KATCHALSKI, YECHIEL SHALITIN AND MATATIAHU GEHATIA RECEIVED OCTOBER 7, 1954
A theoretical analysis is given for the kinetics of polymerization of N-carboxy-a-amino acid anhydrides, assuming an initiation reaction (a), with a specific rate constant kl, a propagation reaction (b), with a specific rate constant kp, and termination reactions (c) and (d), with the respective specific rate constants k 3 and k,. Formulas were derived for the rate of polymerization and for the rate of carbon dioxide evolution as well as 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 peptide chains, characterized by terminal carboxyl groups. From the general treatment three special cases were derived. Case I, kl = k p ; ks = kc = 0. The formulas given for this case were shown to agree with those derived by Floryz for the polymerization of ethylene oxide as well as with those of Ballard and Barnfords deduced for the polymerization of N-carboxy-DL-phenylalanine anhydride and N-carboxy-DL-leucine anhydride initiated by preformed polymer. Case 11, kl # kz; k3 = k , = 0. The formulas for this case were found identical with those of Breitenbach and Allingera derived for the polymerization of Ncarboxy-DL-phenylalanine anhydride initiated by amines of different basicity. Case 111, kl = k,; k3 = k,. The variation with time of the molecular weight distribution of the growing and terminated chains was calculated assuming particular values for k2 and ks. I t was shown that while the molecular weights of the terminated chains are spread over a wide range, the molecular weights of the unterminated chains are concentrated in the high molecular weight fraction of the polymer. Possible methods for the evaluation of k1, kz, kl and k, are suggested and the conditions necessary for the preparation of polya-amino acids of high molecular weight are discussed.
In a previous article' it has been shown by Sela and Berger that the polymerization of N-carboxy-aamino acid anhydrides initiated by amines or alcohols (XH, X = RIRzN or RO) involves not only initiation and propagation reactions (a) and (b), but also a termination reaction (c) (see below). In the initiation reaction the X-residue of the initiator attaches itself to the C-5 carbonyl of the oxazolidine-2,5-dione, carbon dioxide is evolved, and a free amino group appears. During propagation new peptide bonds are formed as a result of the reaction of the free amino groups of the growing chains with the C-5 carbonyl groups of the anhydride. There is, however, no change in the total number of the amino groups. The termination reaction (c) causes the disappearance of the free amino groups of the growing chains as a result of their interaction with the C-2 carbonyl groups and the formation of peptides with terminal carboxyl groups adjacent to an urea link.' It is obvious that the catalyst may be eliminated from the system by analogous termination reaction (d). Initiation : R
R
XH
I + O=C-cH-NH I
0 -
ki
I co
-+ p"\r--& -+ C -ox
+ cop (a)
Propagation : R X--(OC-LH--NH)rH
+ o=y-
k2
R
I X--(OC--I]H--NH)j+l-H
+ COI
(b)
(1) M. Sela and A. Berger, THISJOURNAL, '77, 1893 (1955). also M. Sela and A. Berger, ibid., '76, 6350 (1953).
Cf.
Termination : R
R X-(OC--dH--NH),-H
+ O=c-dH--NH
+ kr I c=o
I
0 -
R
R
X-(oC-CH-~H)i-COSH-dH-cooH XH
I + O=C-cH-NH I
o-----co
(c)
R
R
I
kc + X-cOxH-JH-cOO~
(dl
I n the following a general kinetic analysis of a polymerization, proceeding according to the scheme described above, will be given. From the general treatment some limiting cases of special interest will be derived. General Case.-Here it will be assumed that the polymerization is determined by four different constants KI, Kz, k3 and K4, of the reactions (a), (b), (c) and (d), respectively. The concentration of the N-carboxy-a-amino acid anhydride is denoted by (M), the concentration of the catalyst by (XH), the concentration of the "growing monomer," consisting of one amino acid residue and containing a free terminal amino group by Nl*, and the concentration of a growing peptide composed of j amino acid residues and containing a free terminal amino group by N,*. The rate of disappearance of catalyst is given by eq. 1, the rate of formation of a growing monomer by eq. 2, and the rate of formation of a growing j-mer by eq. 3.
E. KATCHALSKI, Y . SHALITIN
1926
+
dNj* __ = k z ( M ) N * i - l - (kp kz)(M)Nj* (3) dt j = 2 , 3 , 4 . . . . . . 33
9
The value of
Cjn'V,* for any integral number n j = 1
so1
D
C p1\ri*
(M)dt; i.e., dv = (M)dt
=
j = 1
the above equations become, respectively
-
VOl. 77
M. GEHATIA
is (cf. Appendix eq. 5*)
Introducing a new variable v =
AND
9)
= (k,
(4) (5)
dNj* __ = kzN*j-1 dv
- ( k q + k3)Nj* j = 2 , 3 , 4 ,.....
(6) a
Integration of (4) then gives ( X H ) = (XH)oe-(k~+k4)u
(&)'[eg ]
~~(XH)oe-(kz+lir)v
+ k4)(XH)
where q
=
(e-,,
(13)
e-aygj
- k 2 / a and the differential operator
designates differentiation of the term in
) ;9(
square brackets with respect to q, and subsequent multiplication by p; the exponent ( ) " designates that this process is to be applied n times. For the particular case n = 2
(7)
where (XH)ois the initial concentration of catalyst. Substituting (XH) from (7) into (5) and solving for Nl*, one obtains
Letting a = ( K 1 equation becomes
+ kd) - (k2 + ks), the above
N1* = $(XH)oe-(kz+kdv k
(14) __ The
weight average degree of polymerization, (14) and (11)
P,*, can be derived from rn
(1 -
e-av)
j2Ni *
(8)
*P,
- j
--=
Solving equations 6, the following expression is obtained for N,"
= 1
5
jN,*
j = I
kzu[(l -
( a - kd + kpu)(a+ + 2a] + a( a + k ? ) X (1 ( a + k2)kzu + a ( 1 - e-(a + k2)
+k2)~)
e-(a
k 2 ) ~ )
or
When e - ( a + k 2 ) > a, (15) be-
The concentration of the free terminal amino groups of the polymer a t any instant t is given by 13
Nj*,
and can be evaluated from (9)
j = 1
Equation 10 shows that a t large values of v the decrease in the number of free amino groups will be determined mainly by the termination constant
&.
The number of amino acid residues in the growing polymer is given by -0
CjNj* j =I
=
+
molecular weight distribution curve of the peptide fraction containing free amino groups is rather sharp even in the general case under consideration. N-Carboxy-a-amino acid anhydride is consumed in each of the reactions in the polymerization scheme, (a), (b), (c) and (d). The over-all rate of polymerization is thus given by
+
k 1 (XH)oe-ka*[(u kq)kzu (a kd2 a( 1 - e - @ + W Y ) ] ( 1 1 )
+
I n most cases the second term will be negligible
in comparison with k 2 v , so that P,*will approach P,* a t large values of k q v . This indicates that the
__ and
the number average degree of polymerization, Pn*by
;.e.
13
Substituting the values of (XH) and
Nj* j = 1
given by equations 7 and 10, and integrating with respect to v from v = 0 to v = v, one obtains At large values of v, when e-(a + k 2 ) y > k2, the number average degree of polymerization at the times when e - k l v k 2 / k 1 ,so that P 2 FC: k p . (111) kl = k2, k3 = kr.-In this case a termination reaction occurs and the reactivity of the initiator is equal to that of the growing chains. The formulas to be deduced will hold for the polymerization of N-carboxy-a-amino acid anhydride, initiated by an amide or ester of the corresponding amino acid or peptide, and terminated by reaction (c). The concentration of the catalyst is derived from (7) (XH) = (XH), e-@z
+k4v
2
0
0
+
Nl * = kz( XH)oe-(kz
Aa)v
lim
~
- e-ov a
12 10 w
0
a+O
(XH)OkPve-(ki
Vi* = (XH)oe-(kz + k r ) u ( k 2 ) i limy1 = ( a' J. = l
50
+
(74
=
40
Fig. 1.-Molecular weight distribution on a number basis assuming k2 = 0.1 liter mole-' sec.-l and k3 = 0.003 liter mole-' see.-' The curves 1*, 2*, 3* and 4* (full line) give the molecular weight distribution of the growing k 3 ) v = 10, 20, 30 and chains a t times corresponding to ( k z 40, respectively, while the curves 1, 2, 3 and 4 (dotted line) give the molecular weight distribution of the terminated chains a t the same times.
and the concentration of the growing chains from (8) and (9) 1
10 20 30 Degree of polymerization, j .
+k4
y
(812)
1)~'f"( a v ) m = ~
m!
d 8 X h 0
R e
Y
a-0
( X H ) ~ e-(kz ( ~ jl j =2,3,4
+
kt)v
(gc)
...... =
Equation 9c shows that the molecular weight distribution of the growing chains is determined by a Poisson function multiplied by e-kau. The time P a t which the molar concentration of a growing chain of a given degree of polymerization j reaches its maximal value is obtained by determining the value of Y a t which b N j * / b v = 0. One finds F =j/(k2 k3). The degree of polymerization j of the growing chain having the highest molar concentration a t a given time v is obtained by determining the value of j a t which bNj*/bj = 0. It is found that:= k2v.
+
2
0 10 15 20 25 30 35 Degree of polymerization, j . Fig. 2.-Molecular weight distribution on a number basis assuming KZ = 0.1 liter mole-' sec.-l and ks = 0.01 liter mole-' set.-' The curves 1*, 2*, 3*, 4* and 5* (full line) give the molecular weight distribution of the growing k 3 ) u = 5, 10, 15, 20 chains a t times corresponding to (kz and 25, respectively, while the curves 1 , 2 , 3 , 4 and 5 (dotted line) give the molecular weight distribution of the terminated chains at the same times. 0
5
+
E. KATCHALSKI, Y . SHALITIN AND M. GEHATIA
1930
while in Fig. 2 it was assumed that kz = 0.1 liter mole-' sec.-l and k8 = 0.01 liter mole-' sec.-l. The total concentration of the free amino groups of the growing chains derived from (10)
Vol. 77
As the rate of carbon dioxide evolution differs in the present case from the rate of anhydride disappearance, it is of interest to compare these two values. By dividing (16) by (19) one obtains
D
N,* = (XH)oe-ksv (1 - e-kzv)
(1Oc)
j=1
Equation 1Oc shows that the total number of free amino groups in the polymer increases in the first stages of the reaction and decreases as the polymerization proceeds. The increase in concentration of the "2-groups is governed by the term 1 containing the initiation constant kl = k2, while the decrease in the "2-groups is determined by the factor e-ksu, containing the termination constant ks. From (11) one obtains
Equation 29 shows that in the present case the rate of anhydride consumption is proportional t o the rate of carbon dioxide evolution. Integration of (29) gives
;.e.
D
j N i * = (XH)okpve-kIu
(Ilc)
j = l
A comparison of eq. I l a , given for the case where no termination occurs, with eq. l l c , shows that in the former case the weight of the polymer increases gradually with v , whereas in the present case the weight of the fraction composed of growing chains decreases after some time as a result of the termination reaction. Equations l l c and 1Oc enable the calculation of the number average degree of polymerization of the growing chains (cf. also eq. 12)
It is of interest to note that the value of p,* in this case is identical with its value when no termination takes place (cf. eq. 12a).
When the polymerization is brought to completion, (M) = Oand (M)o/(COz) = 1
+ ks/ks
The difference, [(M)o - (M)] - (COz), of course gives the concentration of the carboxyl groups present in the reaction mixture. Thus (COOH) =
k
(Cot)
(31)
The concentration of terminated chains composed of one amino acid residue is derived from (24)
The concentration N i of the terminated chains composed of j amino acid residues is obtained by finding lim Nj from (25) a+O
11
jziVj*is obtained from (14)
The expression
j = 1
5
The value of the weight average degree of polymerization P ?= 1 k2v, obtained from (15) or from (14c) and (1IC),is also identical with its value when no termination occurs (eq. 15a). The rate of carbon dioxide evolution in this case is obtained by inserting the values of (XH) and
+
2
Nj*
j = 2 , 3 , 4 . . . . . .XI
j 2 N i * = ( X H ) o k g v ( l 4-k2v)e-k3~ (14c)
j = 1
from equations 7c and lOc, respectively
j = l
into (20) d(Coy) -dv
E
k2(XH),,e-ksv
(204
A comparison of (19a) with (20c) shows that when the polymerization is characterized only by a propagation reaction, the rate of carbon dioxide evolution with respect to Y is proportional to the initial concentration of catalyst and is independent of the time v ; in the present case, however, the rate of gas evolution remains proportional to the initial concentration of catalyst, but as a result of the termination reaction diminishes with the time Y. On integration (20c) gives
( c o ~=) k2 (xI-I)~(I - e-krv) ki
(21c)
The dotted lines in Fig. 1 and Fig. 2 illustrate the molecular weight distribution of the terminated chains on a number basis, using for k2 and ks the same values as specified for the growing chains. The Nj values given were calculated numerically from (25c) after transformation into an incomplete Gamma functions2 Figure 3 gives the molecular weight distribution of the growing chains and terminated chains on a weight basis, for the case specified in Fig. 1. Figure 4, on the other hand, gives the corresponding molecular weight distribution for the case specified in Fig. 2. The total number of terminated chains is obtained from (26) I)
N, = ( x H ) ~ ( I- e-kau)
(26~)
j = l 33
when e-kav
1, and the terminated polymer is investigated a t a v value a t which kz/k3 >> kzv/ eksu- 1 then = kJk3. The terminated polymer resembles in this respect many of the polymers obtained by addition polymerization. n
j z N j required for the evalua-
The expression
0
10 20 30 40 50 Degree of polymerization, j . Fig. 3.-Molecular weight distribution on weight basis of the growing chains (full line) and of the terminated chains (dotted line) for the cases specified in Fig. 1.
j = 1
I
p,is obtained from (24c) and (27) (n = 2) [ ( l - e-kau )(ki2 + 3k2k3 + 2k22) j2N, = klve-kav ( 2 k 2 + 3kzk3 + kZ2k3v)] (35)
I
I
I
tion of n 1=1
Hence P, = k32 +
3k2k3 + 2kz2
- k3ve-k:~( 2 k 2 + 3kzks + k22k3v) (1
k3 [(kz
+
- e-kav)
- ( k2k3ve-kav 1 - e-ksv)
1
(36)
At large enough values of v the fractions appearing in the numerator and denominator may be neglected, and
If kz/k3 >> 1, -
= 2 ( k z / k 3 ) . It follows that = 2Pn, a relation characteristic for polymers formed by polycondensation or by addition polymerization. The application to the reaction mixture of the usual techniques for the isolation of the peptide fraction will obviously lead to a mixture containing unterminated as well as terminated chains. As the separation of these two types of peptide chains may be difficult, the molecular weight distribution of the total polymer is of interest, particularly wherever physicochemical studies of the unfractionated polymer are carried out. The formulas for such a distribution may be derived from the equations given above. T o illustrate this (Nj* Nj) and (jNj* j N j ) were calculated as function of j , a t different times v, for a polymerization characterized by the
+
+
5
10 15 20 25 30 35 40 Degree of polymerization, j . Fig. 4.-Molecular weight distribution on weight basis of the growing chains (full line) and of the terminated chains (dotted line) for the cases specified in Fig. 2. 0
rate constants, kz and k3, specified in Fig. 1. The results are represented in Figs. 5 and 6.
Discussion In the equations derived above the variable v = (M)dt was introduced instead of the time variable t. The value of v a t any given time may be obtained from a graph representing (M) as a function of t. (M) can be determined experimentally using the titration method for the determination of Ncarboxy-a-amino acid anhydrides,' or, in the special cases where no termination reaction occurs, by subtracting the total amount of the carbon dioxide (7) A. Berger, M. Sela and E. Katchalski, Anal. Chcm., 2S, 1554
(1953).
E. KATCHALSKI, Y . SHALITIN AND M. GEHATIA
1932 I
12
I
I
I
1
VOl. 77
has been done in certain cases for the initiation and propagation constants some representative numerical values will be given. When the polymerization is of first order with respect to the monomer (cf. case I ) , i.e., when the termination reaction is negligible and the reactivity of the initiator is the same as that of the growing peptide chains, the propagation constant kz is easily calculated from the experimental data by one of the conventional kinetic techniques. Thus Eallard and Bamfordj deduced a kt value of 6.0 X 7630
lo3 X e-RT liter mole-' see.-' for the polymerization of r\'-carboxy-DL-phenylalanine anhydride in
nitrobenzene which is initiated by preformed polymer. For the polymerization of the same anhydride in nitrobenzene initiated by N-ethylglycinediethylamide Breitenbach and Allinger6 give kz = 6490
10 20 30 40 50 Degree of polymerization, j . Fig. 5.-Molecular weight distribution of the total number of chains, growing and terminated, for the cases specified in F i g . 1.
0
2.97 X IO3 X e-RT liter mole-'sec.-l The significant effect of the solvent on the propagation constant may be exemplified by the polymerization of N-carboxy-DL-leucine anhydride initiated by the corresponding preformed polymer and carried out at 45' in nitrobenzene and o-nitroanisole, respect i ~ e l y . In ~ the case of the first solvent kz = 8.72 X IOb2 liter mole-' see.-', and in the case of the second solvent kz = 5.83 X lo'* liter mole-' see.-' The initiation constant kl may be calculated, even in the general case, from the rate of carbon dioxide evolution when a relatively large amount of catalyst is used (cf. eq. 19). Equation 19 shows that kl determines the rate of carbon dioxide evolution particularly at the beginning of the reaction D
N,*
when (XH) = (XH)oand
