JOURNAL OF POLYMER SCIENCE: PART C NO. 15, PP. 303-306 (1960)

Abstract

Conformation of Polypeptides in the Helix-Coil Transition Region

The statistical mechanical theory of helix-coil transition in polypeptide has been developed by many workers; we refer here to Lifson and Roig.' The size and shape of a polypeptide in the helix-coil transition region are of fundamental importance for studies of solution properties such as osmotic virial coefficients, viscosity, and light scattering. Rlechanical properties are also connected with these quantities. Nagai2 attempted to calculate the mean-square end-to-end distance, (R2), and obtained the interesting result that (R2) has a shallow minimum under the appropriate condition at the transition region.

In this report we derive an expression valid over all the transition region by a method a little different from that of Nagai. We adopt an inter- rupted helix model of a polypeptide chain, where random coil parts and rodlike helix parts are connected alternatively by universal joints arid are represented by vectors 11, lr , . . . carrying nl, n2, . . amino residues, re- sped ively. Then we have

1 i < i

whcre

and

( ( l ( . lJ ) ) for 2 # ..i

(Z12) = an1, (Z?) = an3 for coil parts (3)

(Z22) = pnZ2, (Z42) = ,f3n42 for helix parts (4)

Next we have to take the averages of eq. ( 2 ) over all the possible partitions of n into nl, n2, n3, . The first term of eq. (2 ) can he written in terms of hclix fraction 8, as

. .

((721 + 723 + . . .I) = 4 1 - 0) (5) 303

304, S. SAI‘T6, M. G6, AND M. OCIIIAI

N = 1500 I .5 v = 0.0141

0 .5 -

I I

-0.10 -0.05 0 0.05 0.10

N = 1500

v = 0.0141 1.5 1 ,

x to3

0 .5 1 \ I I

-0.10 -0.05 0 0.05 0.10 In W

Fig. 1. Number of roil parts n(1 - 0) versiis In w.

6 - xi05

5 -

4 -

3 -

2 -

6 - N = 1 5 0 0

V = 0.014 I xi05

5 -

4 -

3 -

2 -

I -

-0.10 -0.05 0 0.05 0.10 -0.10 - 0 : O S 0 0:05 0 : l O ~

In W

Fig. 2 . Relation between (o and In w.

The average of the second term is accomplished in two steps

(n2 + n? + . . .) = c (n2 + n4’ + . . + n2p2)m,p f(m,p> (6) m,p

where we put

n2 + n4 + . . . + nzp = tn (7) and (. . . ),,5,p means the average taken by keeping nz and p constant; j ( m , p ) is the probability that the number of amino residues in the helix state is 772, and they are grouped into p parts. It is given by using Lifson- Roig’s partition function 2’ in the form

f (m,p) = w m - 2 p v‘p c(m,p)/Z’ (8)

where tc, t i , and v‘ are the notations used by Lifson and Roig; that is, ‘LO is the factor proportional to the probability of an inbrachain hydrogen bonding for the helix formation, v is the factor for the probability that an

I’OLYPEPrIDES IN HELIX-COIL TRASSITION 13F:GION 30.5

V = 0.0141 - 9 I

-0.10 -0:OS 0 0105 0 : l O I n W

Fig. 3. Effect of .I/@ on chain dimension (R2).

amino residue is in the helix state of the internal rotations but does not participate in hydrogen bonding. The notation v’ is used to specify thc amino residue just lying at the begiririirig of a hclix part.

On the other hand, me can obtain3

Thus we have

(11)

(la)

2 b In 2‘ b In Z + 4 - b In to

3 In 2’ b In TI’

a In 2 b In I U

e = (m) = (m - 2 p ) + 2 ( p ) = -~ + 2 -~

2 is the value of 2’ at 2, = u’. The calculated values of n(1 - e), cp and (R2)/p are given in Figures 1, 2, arid 3, respectively, with n = 1500, v =

v’ = 0.0141. We find that when cy/P is larger than about 100 a shallow minimum in (R2) appears at the transition region. For PBLG molecules

is rstimatcd by Kurata and Stockmayer4 to be 7.88 x 10-15. If we adopt these numerical values, (R2) increases monotone with increase of the helix fraction. The calculation of the fourth moment (R4) is now in progress and will be reported elsewhere.

is equal to (1.5 A)2 and

306 N. SAIT6, M. G6, AND M. OCIIIAI

References 1. S. Lifson and A. Iioig, J. Chem. Phys., 34, 1963 (1961). 2. K. Nagai, J. Chem. Phys., 34,887 (1961). 3. B. H. Zimm and W. H. Stockmayer, J . Chem. Phys., 17, 1301 (1949). 4. M. Kurata and W. H. Stockmayer, Fortschr. Hochpolymer. Forsch., 3, 196 (1963). ;i. P. J. Flory and W. G. Miller, J . Mol . Bzol., 15, 284 (1966).

Discussion P. J. Flory (Stanford University): Dr. W. G. Miller, rerently in our laboratory, rar-

ried out similar calculations of ( ~ 2 ) for polypeptides using the method previously illus- trated by me (see ref. 5 above). The assumption of Gaussian chain is not used, but we also obtain a minimum in ( ~ 2 ) as the system passes through the helix-coil transition. This feature, originally predicted by Dr. Nagai, appears to be well confirmed by our vari- ous theories.

N. Saito: Whether or not (9) has a minimum a t the transition region depends on the parameters used in the calculation, especially on the ratio of a lp . I think it is neressary to see if the same parameters are used in both calculations. I do not think the non- Gaussian treatment for coil parts gives rise to a minimum, although the Gaussian ap- proximation may lead to some error.