32 Yamakawa, Yoshizaki
Macromolecules
hydroxyprolyl residues, The details of these will be published elsewhere.
Acknowledgment. This work was partly supported by a research grant from Department of Science and Technology, Government of India, and PL-480-USPHS Grant NO. 01-126-N. References and Notes L. Paulincr. R. B. Corev. and H. R. Branson. Proc. Natl. Acad. Sci. U.S.X., 37, 205 (f951). L. Pauling and R. B. Corev, Proc. Natl. Acad. Sci. U.S.A., 39, 253 (19531. G. N. Ramachandran and G. Kartha, Nature (London),176, 593 (1955). G. N. Ramachandran and V. Sasisekharan, Adu. Protein Chem., 23, 283 (1968). F. H. C. Crick and A. Rich, Nature (London),176, 780 (1955). V. Sasisekharan, Acta Crystallogr., 12, 897 (1959). E. Bendetti in “Proceedincrs of the Fifth American Peutide Symposium”,M. Goodman A d J. Meienhofer, Eds., Wi1ey;New York, N.Y.. 1977, D 257. E. M. Bradbury, L. Brown, A. R. Downie, A. Elliott, R. D. B. Fraser, and W. E. Hanby, J . Mol. Biol., 5 , 230 (1962). C. Ramakrishnan and N. Prasad, Int. J . Protein Res., 3, 209 (1971). C. H. Bamford, L. Brown, A. Elliott, W. E. Hanby, and I. F. Trotter, Nature (London), 171, 1149 (1953). C. H. Bamford, L. Brown, E. M. Cant, A. Elliott, W. E. Hanby, and B. R. Malcolm, Nature (London),176, 396 (1955). B. Lotz, J . Mol. Biol.,87, 169 (1974).
(13) F. J. Padden and H. D. Keith, J . Appl. Phys., 36, 2987 (1965). (14) T. Miyazawa in “Poly-U-Amino Acids”, G. D. Fasman, Ed., Marcel Dekker, New York, N.Y., 1967, Chapter 2. (15) F. Colonna-Cesari, S.Premilat, and B. Lotz, J . Mol. Biol.,87, 181 (1974). (16) G. N. Ramachandran. C. Ramakrishnan. and C. M. Venkatachalam in “Conformation of Biopolymers”, Vol. 2, G. N. Ramachandran, Ed., Academic Press, New York, N.Y., 1967, D 429. (17) R. A. Scott and H. A. Scheraga, J. Chem. Phys., 45,2091 (1966). (18) C. M. Venkatachalam, B. J. Price, and S.Krimm, Macromolecules, 7, 212 (1974). (19) P. M. Cowan and S.McGavin, Nature (London),176,501 (1955). (20) V. Sasisekharan, J . Polym. Sci., 47, 373 (1960). (21) W. Traub and U. Shmueli in “Aspects of Protein Structure”, G. N. Ramachandran, Ed., Academic Press, New York, N.Y., 1963, p 81. (22) W. Traub and U. Shmueli, Nature (London),198,1165(1963). (23) I. Z. Steinberg, W. F. Harrington, A. Berger, M. Sela, and E. Katchalaski, J . Am. Chem. SOC.,82, 5263 (1960). (24) P. De Santis, E. Giglio, A. M. Liquori, and A. Ripamonti, Nature (London),201, 456 (1965). (25) P. R. Schimmel and P. J. Flory, Proc. Natl. Acad. Sci. U.S.A., 58, 52 (1967). (26) N. G6 and H. A. Scheraga, Macromolecules, 3, 188 (1970). (27) W. L. Mattice, K. Nishikawa, and T. Ooi, Macromolecules, 6, 443 (1973). (28) G. Schilling and Kricheldorf,Makromol. Chem., 178,885 (1977). (29) W. L. Mattice and L. Mandelkern,Biochemistry, 10,1926 (1971). (30) D. A. Torchia, Biochemistry, 11, 1462 (1972). (31) D. A. Torchia and J. R. Lyerla, Jr., Biopolymers, 13,97 (1974). (32) D. A. Rabenold, W. L. Mattice, and L. Mandelkern, Macromolecules, 7, 43 (1974).
Transport Coefficients of Helical Wormlike Chains. 2. Translational Friction Coefficient Hiromi Yamakawa* and Takenao Yoshizaki Department of Polymer Chemistry, Kyoto University, Kyoto, Japan.. Receiued October 4 , 1978
ABSTRACT: The translational friction coefficient of the helical wormlike chain is evaluated by an application of the Oseen-Burgers procedure of hydrodynamics to the cylinder model. The Oseen hydrodynamic interaction tensor is preaveraged, and therefore there is no need of consideration of the effect of the coupling between translational and rotational motions. A useful empirical interpolation formula is also derived to be valid for any possible values of the model parameters. Some salient aspects of the behavior of the sedimentation coefficient are discussed on the basis of the numerical results. In particular, it is pointed out that even if the proportionality of the sedimentation coefficient to the square root of the molecular weight is experimentally observed over a wide range, the chain may not always be regarded as a random coil but may possibly be characterized as a helical wormlike chain.
In a previous paper,’ part 1of this series, a study of the steady-state transport coefficients of helical wormlike (HW) chains2i3(without excluded volume) was initiated. The evaluation of the translational diffusion (or friction) coefficient and intrinsic viscosity of the characteristic regular helix,3 i.e., one of the three extreme forms of the model chain corresponding to its minimum configurational energy, was carried out by an application of the OseenBurgers procedure of hydrodynamics4to cylinder models. In this paper, we proceed to evaluate the translational friction coefficient of HW chains along the same line. As discussed in detail previously,l there are two fundamental problems to be considered. One is concerned with the steady-state transport length scales for the cylinder model, and the other with the possible effect of the coupling between translational and rotational motions of skew bodies. The length scales to be adopted must be of the same order as or larger than those associated with the 0024-9297179122 12-0032$01.oo/o
equilibrium chain configurations. This should always be kept in mind when an analysis of experimental data is made by the use of theoretical expressions for the transport coefficients derived in the present and later papers. As for the second problem, it was shown that if the Oseen hydrodynamic interaction tensor is preaveraged in the Kirkwood-Riseman ~ c h e m ethe , ~ coupling does not affect the translational friction coefficient of the regular helix a t all, while the effect on its intrinsic viscosity may be negligibly small except when the number of its turns is very small. It is clear that the skewness is partly destroyed for the HW chain having internal degrees of freedom, and moreover, the evaluation for this chain is carried out with the preaveraged Oseen tensor as in the case of the Kratky-Porod (KP) wormlike chain.4 Therefore, we do not consider the coupling in the present and later papers. The mean reciprocal distance between two contour points of the chain, which is necessary for the evaluation 0 1979 American Chemical Society
Transport Coefficients of Helical Wormlike Chains 33
Vol. 12, No. I , J a n u a r y - F e b r u a r y 1979 with the preaveraged Oseen tensor, has already been obtained in closed form for several cases of the values of the model pararnetem6 Thus, in section I, we derive a semianalytical expression for the translational friction coefficient by the well-established method. In section 11, we consider the associated K P chain,3 i.e., the K P chain whose Kuhn length is the same as that of the HW chain under consideration. This serves to derive in section I11 an empirical interpolation formula for the translational friction coefficient of the latter which is valid for any possible values of the model parameters. In section IV, we discuss some salient aspects of the behavior of the sedimentation coefficient on the basis of the numerical results. An analysis of experimental data is deferred to the next paper, in which the intrinsic viscosity is studied.
Basic Equations Until the molecular weight is explicitly introduced, the equilibrium average configuration of the HW chain of a given contour length may be described by four parameters: the constant curvature K~ and torsion T O of the characteristic helix, the Kuhn length h-' of the K P chain as defined as the HW chain with K~ = 0, and Poisson's ratio u. Most of the numerical computations were carried out for Q = 0, and therefore we consider this particular case also in this paper. (Note that the moments of the endto-end distance are rather insensitive to the change in u . ) The HW cylinder as the hydrodynamic model is defined by further introducing the total length L and the diameter d of the cylinder. It reduces to the regular helical cylinder studied in part 1, in the limit X 0, X-l being a measure of chain stiffness. In what follows, all lengths are measured in units of A-' unless specified otherwise. It is sometimes the parameters y and convenient to use instead of K~ and io v defined by
-
y
= U
T,(K,'
=
(KO'
+ +
T~')-~/'
(1)
as in part 1. The parameter y determines the form of the characteristic regular helix apart from its size, i.e., the ratio h f p of its pitch h to radius p (whether the reduced or unreduced lengths are used), and u/2n is equal to the number of its turns contained in the unreduced contour length equal to X-l (in the unit contour length when the unreduced lengths are used). Note that the (reduced) parameters KO, T ~and , u increase to infinity as X is decreased to zero. Now, we suppose that the HW cylinder possesses the uniform translational velocity U in a solvent with viscosity coefficient 170, whose unperturbed flow field is nonexistent, and we replace the cylinder by a frictional force distribution f ( x ) per unit length along the cylinder axis as a function of the contour distance x (0 5 x 5 L ) from one end, following the Oseen-Burgers procedure. If the Oseen tensor is preaveraged, there is no need of consideration of the coupling effect arising from the chain skewness, so that the configurational average ( f ( x ) )of the frictional force satisfies formally the same integral equation as in the case of the K P ~ y l i n d e r , ~
( ) denotes the configurational average and also the average over r, The mean total frictional force (F)and the translational friction coefficient ,7 are given by L
(F)= ZU =
( f ( n ) )d r
0
(4)
If the solution of eq 2 in the Kirkwood-Riseman approximation is substituted into eq 4, we obtain for Z L
3 ~ 1 7 & / E = L - l S ( L - t ) K ( t )d t
(5)
0
with t = lx - yI. Thus, the problem is to evaluate the kernel K ( t ) = (IR - r1-l) and the integral in eq 5. Statistical mechanically, the HW chain of contour length t may be described completely by the Green's function G(R,u,aluo,%;t),Le., the conditional distribution function of the radius vector R(t) = R of the contour, the unit tangent vector u(t) = u , and the unit mean curvature vector a(t) = a when R(0) = 0, u(0) = uo, and a(0) = 80.2,7,8 The kernel, or the mean reciprocal distance, may then be evaluated from
K ( t ) = ( 2 n ) - 1 1 d R l ' d r l R - rl-'G(Rluo,ao;t) (6) with G(Rluo,ao;t)= lG(R,u,aluo,ao;t)d u d a
(7)
where the integral over r is subject to the condition, r.uo = 0 (8) The set of unit vectors a, b ( = u X a), and u defines a localized Cartesian coordinate system affixed to the chain, whose axes coincide with the local principal axes of inertia of the chain as an elastic wire.8 It is then clear that K ( t ) is independent of the orientation of the initial localized system (ao,bo,uo),and also that the directions of r and ao, which are perpendicular to uo, are independent of each other, r being just the normal radius vector introduced in the application of the Oseen-Burgers procedure to the hydrodynamic cylinder model. We therefore have
K ( t ) = ( 2 a ) V ' l d R l ' d r I R - rl-'G(Rluo=e,;t)
(9)
where e, is the unit vector in the direction of the z axis of an external Cartesian coordinate system, and
An approximate expression for the mean reciprocal distance given by eq 9 has already been found to be (eq 66 of ref 6)
K(t) = 5
(t'
3
2
+ Y4d2)-'/'(1 + iCfioti + iC= l j=1 Cfijd2jti) =l 2
for t Iu1
i
K ( t ) = ( 6 / ~ c ~ t ) ' "CBij E d2j(c_t)-'+ i=oj = o 0
2
with with
fl0
K(x,Y)= K ( ( x- Y O = (IR - r1-l)
(3)
where R is the distance between the contour points x and y, r is the normal radius vector from the contour point x to the cylinder surface, so that Irl = d/2, and in eq 3 the
= 1/3,
B11 = -0.125, B , = 1, Bzz = 0.0140625
(12)
where h(x) is the unit step function defined by h ( x ) = 1 for x 2 0 and h ( x ) = 0 for x < 0; and ul,uz, f i 0 (i = 2-5), f i j (i = 1-3; J = 1, 21, Blo, Bzo,BZ1, E,, and q are constants
34 Yamakawa, Yoshizaki
Macromolecules
independent of t and d but dependent on K~ and ro and to be determined numerically to give a good approximation to K ( t ) (for the details, see ref 6). c, is the Kuhn length of the HW chain (for u = 0) and is given by28
systematically for more cases of the values of K~ and ro. Before proceeding to do this, in the next section we examine the behavior of E for the associated Kp chain whose Kuhn length is equal to c, and which corresponds to the base of the triangle in our hypothesis on polymer chain configurations (see Figure 1 of ref 3). In what follows, it is convenient to designate the lefthand side of eq 5 or 14 by s,
s = 37rq&/E
(21)
which is related to the experimentally observable (unreduced) translational diffusion coefficient D and sedimentation coefficient s by
D = AkTS/31rq& S
(22)
= m(1- ~ ~ o ) S / ~ T ~ & N A
(23)
where k is the Boltzmann constant, T the absolute temperature, N A the Avogadro number, M the molecular weight of the solute, d its partial specific volume, and po the mass density of the solvent. Note that S is proportional to s.
a
1
-
i-r
+ 7/2
i -F
+ 7/2 1
9
i+9
L2
L(i - F
+ 9/2)
-
61
L(i - r
]Li-r+7/2 -
+ 9/2)
] ) ali-r+7/2
(16)
The Associated KP Chain An expression for any average quantity for the associated K P chain may be obtained from the corresponding expression for the KP chain by further reducing all (reduced) lengths 1 by c,, i.e., by replacing 1 by cm-% Therefore, the kernel Ka-Kp(t,d),or the mean reciprocal distance, as a function of t and d for the associated K P chain may be obtained from the kernel KKp(t,d) for the K P chain as Ka-Kp(t,d) = C,-lK~p(C,-lt,C,-ld)
(24)
By the use of eq 37 of ref 4 for KKP(t,d)in eq 24, we then find, from eq 5, for Sa-KP, i.e., the quantity 8 defined by eq 21 with Ea-KPfor the associated K P chain "
i - 1/2
L(i - 3/2)
]
Sa-KP(L,d;ol)= Fl(L,d;Ll) + h ( L - ol)F3(L,d;al) ull/z-i) (17)
with
foo
(25)
where F1 and F3 are defined by eq 15 and 17 with Boo,Bll, BZ2,f m , and Ll being given by eq 12,18, and 19 but with u1 and the remaining coefficients by (TI =
2.278~,,
=1
f30 fll
f21 fS1
fzo = 0.1130~,-~,
f l 0 =1/3cm-l,
= -0.02447~,-~,
f40
= 0.04080~,-~,
= -0.4736~,-~, = 0.009666~,-~,
Blo = -0.025~,',
f12 f22
f32
= f50 = O,
= 0.004898~,-~,
= -0.002270~,-~,
= 0.0002060~,-~,
B20 = -O.016295cm4, B21= 0 . 0 6 5 6 2 5 ~(26) ~~
Note that u in eq 37 of ref 4 for the K P chain is equal to u1 = 2.278 (with c, = l),and therefore that in the use of this equation in eq 24, the inequalities t > (T or t < (T for the KP chain should be replaced by t > g1 or t < u1 with u1 = 2 . 2 7 8 ~ ~ . As seen from eq 5 and 24, the S a - ~ pgiven by eq 25 is equal to the quantity SKp(L,d;(T1)for the K P chain with c,-lL and c,-'d in place of L and d , respectively, Equation 14 for E is semianalytical since the involved constants fi;, Ei;, Bi;, etc., have been evaluated only for several cases of the values of K~ and T ~ and , moreover, it is not convenient for practical use because of its complexity. Therefore, we reconstruct a simpler and completely analytical interpolation formula for E approximately on the basis of its values calculated from eq 14. For this purpose, the constants above must be evaluated
Sa-Kp( L ,d ;( T i ) = s ~ (C p,-lL
C , ,-ld;
(Ti)
(27)
where ul = 2.278 on the right-hand side of eq 27, so that reduces to sKPwhen K~ = 0, and therefore when c, = 1. Note that the previous results for the KP chain given in ref 4 may be obtained by expanding the Sa-Kp given by eq 25 with c, = 1 in powers of d l L and d , assuming that d / L
