Vol. 6, No. 2, March-April 1973
Probability Distributions of Branched Random-Flight Chains
this increase in conformational freedom can be ascertained from the magnitude of a circular dichroism band in the 225-240-nm region. Since we have observed this band in the spectrum of polymers with high proline and glycine contents and a t elevated temperatures, it is unlikely that it is unique to the (a-helicalpolypeptides. The very weak band seen in the spectrum of poly(G1y-Pro-Pro) has been rationalized either because of the restricted conformational space available to this residue or the inherent weakness of the n-r* transition in poly( imides). The random form of a number of polypeptides has been shown to give rise to a two-banded circular dichroism. A weak band a t -225 in addition to the stronger band centered a t -200 nm. The intensity of the former band appears to depend on the glycine and proline content of the polypeptide with higher proportions of these residues decreasing the circular dichroism. On the other hand, the
285
presence of glycine tends to reduce the 200-nm band whilst the presence of proline increases this band. In addition Piez and Sherman20 have already noted a dependence between the magnitude of the -200-nm trough and imino acid content which is further substantiated by the results reported here for poly( Gly-Pro-Pro). The effects have been interpreted in terms of the symmetry of the glycine residue and the stereochemical restrictions imposed by the proline residue. Acknowledgments The authors express appreciation to Dr. Kovacs for providing the poly(G1u-Ala) used in this study and to Miss A. Ferszt for her technical assistance. We acknowledge support through the National Institute of Dental Research Program Project No. DE 02587 and a postdoctoral fellowship.
Probability Distributions for the Radii of Gyration of Short, Branched, Random-Flight Chains Santosh K. Gupta a n d W. C. Forsman* School of Chemical Engineering and Laboratory tor Research on the Structure of Matter, University of Pennsylvania, Philadelphia, Pennsylvania 191 74. Received October 19, 1972
ABSTRACT: We have computed t h e probability distributions for the dimensionless radii of gyration, €3, of some typical, short, branched random-flight chains using a formulation published previously. As expected, t h e results show t h a t chains with branches located nearer to the central segment are more compact t h a n those with branches near t h e ends. We also found t h a t the probability of observing chains with values of ( 3 between about 1.0 and 1.5 increased markedly with branching, a t t h e expense of the distribution a t higher values of 5 3 , while t h e probability of observing chains with values of 5 3 below about 1.0 was relatively unchanged.
In a recent publjcation,l we presented a general treatment of random-flight statistics both in the presence and absence of intramolecular interactions, and indicated how it could account for the effects of branching. We have since obtained numerical solutions for the one-, two-, and three-dimensional distribution functions of the unperturbed radius of gyration for some typical, short, branched chains and have observed that these results are consistent with our present intuitive notions on how the distributions would differ from those of linear chains of an equal number of statistical segments. The notation is the same as that used previously.1,2 We define the dimensionless radii of gyration and its components as3
5;Z
=
n2p2(SxZ + S,2)
(1) S. K . Gupta and W. (2. Forsman, Macromolecules, 5,779 (1972). (2) S. K . Gupta and W . (2. Forsman, J . Chem. Phys., 55,2594 (1971). (3) R. F. Hoffman and W. C. Forsman, J . Chem. Phys., 50,2316 (1969).
+
where S,, (Sx2 S y 2 ) l i 2 , and S represent the one-, two-, and three-dimensional radii of gyration respectively, pz = (3/2nP), 12 being the mean-square length of a statistical segment and n, the number of statistical segments per chain. The normalized distribution functions can then be written as
where j = 1, 2, 3, and where the method for generating the matrix D(4) for linear and branched chains has been described previously,l and i = ( -1)112. Equation 2 may be simplified in terms of real and imaginary parts, the latter being zero due to symmetry about 4 = 0. The simplified equation is analogous to eq 3 of reference 2 and can be integrated numerically. Results for the linear chain and five typical branched chains, shown in Figure 1, each having ten statistical segments, are shown in Figures 2 and 3. The integrations
286
Gupta. Forsman
Macromolecules
linear -
Table I Values of the Parameter ga for Various Branched ChainsC
branched
Chain No. (Figure 1)
I
Linear Branched I Branched I1 Branched I11 Branched V
II.
m
XZ
L +
&?anal
ggraph
1.000 0.964 0,909 0.836 0.637
0.988* 0.964 0.914 0.820 0.627
a g = ( ~ 3 Z ) b r a n c h e d / ( 5 3 2 ) l , , , e a r . The analytic values were obtained from reference 5 . b ( E 3 2 ) for this case is 2.685 compared with the theoretical value of 2.695. C The theoretical values in this table also correspond t o the results of solc and Stockmayer.8
only 5 3 , the dimensionless polar radius of gyration are presented here since these are more commonly used. The one-and two-dimensional results can be supplied on request. Results for all possible branched chains for n = 4 are also available but since these lead to similar conclusions, they are not presented here. We also evaluated the zeroth and second moments of 53 from interpolated values of the distributions and Table I lists ( 5 3 2 ) along with the theoretical values obtained by modifying the equations of Forsman5 for short chains. The areas under the curves were very close to one, thus indicating correct normalization. The agreement between the analytic and graphical values of the second moment of (3 leads us to believe that the distributions are correct. Since the radius of gyration is a measure of the average size of the molecule, lines corresponding to branched chains I, 11, and I11 in Figures 2 and 3 suggest that a molecule with branches attached to the central segments is
8 7
P
Figure 1 . The linear chain a n d five branched chains with n = 10. I
I
I 05
10
I
I
15
20
15-
1.0
-
” 0-Iu
3
05-
0
0
25
t3
Figure 2. Distribution of
53
for linear and branched chains: (--)
were performed using a Simpson’s rule subroutine with a n interval of integration of 0.025. It was observed that the upper limit of integration could easily be reduced from infinity to about 4 = 25 since the contribution from above the latter value of was negligible.4 The distributions of
linear, ( - - - - - ) branched I, and (--)
more compact than those molecules in which the branches are located near the end segments. This seems intuitively correct since the central segments lie closer to the center of mass of the chain than do the end segments6,’ and the
$J
(4) S. K . Gupta, Ph.D. Dissertation, Cniversity of Pennsylvania (1972).
branched 11.
( 6 ) W . C . Forsman, Macromolecules, 1, 343 (1968). ( 6 ) P. Debye and F. Bueche, J . Chem. Phys., 20,1337 (1952) ( 7 ) A . Isihara. J. Phys. SOC.Jap., 5,201 (1950).
Vol. 6, No. 2, March-April 1973
Probability Distributions of Branched Random-Flight Chains 287
(3
Figure 3. Distribution of (---.--) branched V .
$3
for linear and branched chains (continued):
presence of branches near the former would tend to make the segment density near the center of mass higher than a t the outside-thus increasing the value of WO(E3) a t lower values of E 3 . The other curves in Figures 2 and 3 are consistent with this explanation, curve V for a cruciform chain being the most compact. An especially interesting observation from Figures 2 and 3 is that the distribution functions for values of E 3 less than about 0.8 are only slightly effected by branching. This means that in an assembly of polymer chains, the effect of branching is to increase the average number of molecules with values of (3 between about 1.0 and 1.5 a t
(-1
linear,
(----)
branched 111,
(-
-)
branched IV, and
the expense of those with higher values of &, with the number of molecules with small values of the polar radius being relatively unchanged.8 Acknowledgment. This work was supported in part by the National Science Foundation through a grant to the Laboratory for Research on the Structure of Matter a t the University of Pennsylvania, and by the Petroleum Research Fund, administered by the American Chemical Society. (8) K. solc and W. H. Stockmayer, presented at the International Symposium on Macromolecules, Helsinki, Finland, July 2-7, 1972.
