Dynamic light-scattering studies of internal motions in DNA. I. Applicability of the rouse-zimm model

Dynamic Light-Scattering Studies of Internal Motions in DNA. I. Applicability of the

Rouse-Zimm Model

S.-C. LIN and J. MICHAEL SCHURR, Department of Chemistry, University o f Washington, Seattle, Washington 98195

Synopsis

The intermediate scattering function G ( K , t ) for any polymer model obeying a linear separable Langevin equation can be expressed in terms of the eigenvalues and eigenvectors of its normal coordinate transformation. An algorithm for the exact numerical evaluation of G ( K , t ) for linear Rouse-Zimm chains in the presence of hydrodynamic interaction has been developed. The computed G ( K , t ) 2 were fit to C ( t ) = A exp(-t/TA) + B, and apparent diffusion coefficients calculated according to Dapp = 1/(27,4K2). G(K, t )2 was surprisingly well-fit by single-exponential decays, especially a t both small and large values of Kb, where K is the scattering vector and b the root-mean-squared subunit extension. Plots of D,,, vs K 2 in- variably showed a sigmoidal rise from Do a t K 2 = 0 up to a constant plateau value a t large K2b2.

Analytical expressions for G(K,t) , exact in the limit of short times, were obtained for circular Rouse-Zimm chains with and without hydrodynamic interaction, and also for free-draining linear chains, and in addition for the independent-segment-mean-force (ISMF) model. The predicted behavior for G(K, t ) a t large Kb (or KRc) was found in all cases to be single-exponential with ~ / T A a K 2 a t large Kb, in agreement with the computational results. A simple procedure for estimating all parameters of the Rouse-Zimm model from a plot of Dapp vs K 2 is proposed.

Experimental data for both native and pH-denatured calf-thymus DNA in 1.OM NaCl with and without EDTA clearly exhibit plateau behavior of D,,, a t large values of K , in harmony with the present Rouse-Zimm and ISMF theories, and in sharp contrast to previous predictions based on the Rouse-Zimm model.

INTRODUCTION

A question that has intrigued numerous in~es t iga tors l -~~ in recent years is how internal motions of large, flexible macromolecules are manifested in the intensity autocorrelation functions of polarized Rayleigh scattered light, and what information about intramolecular dynamics can be ex- tracted from the experimental data. Progress in this area has been im- peded somewhat by the following dilemma. On one hand, previous theoretical estimate^^,^,^ of the spectral width for the (circular) Rouse-Zimm

predicted relations of the form r a K4, and K 3 , for the free- draining, and nondraining, limits, respectively, when KR, >> 1, where K is the scattering wave vector and RG is the radius of gyration. On the other hand, it has been found e ~ p e r i m e n t a l l y ~ ~ for calf-thymus DNA (Mr g 11 X lo6) that 7-l (or I') a K 2 a t suff iciently large values of KRG >> 1. Es- sentially, similar behavior has also been observed for polystyrene ( M , =

Biopolymers, Vol. 17,425-461 (1978) (c) 1978 .John Wiley & Sons, Inc. 0006-3525/78/0017-0425$01 .OO

426 LIN AND SCHURR

9 X lo6) in benzene34 and polystyrene (M, = 9.5 X lo6) in cy~lohexane.~~ Indeed, it has been found for a number of dilute gels, for which RG is macroscopic in dimension, that 7-1 a K 2 at all observed scattering an- g l e ~ . ~ ~ ~ ~ Indeed, a recent inelastic neutron scattering s t ~ d y 4 ~ of polyte- trahydrofuran and its perdeutero analogue also appears to indicate r a K 2 in the limit of large KRc >> 1. Moreover, Lee and Schurrg have proposed a simple model, the independent-segment mean-force (ISMF) model, for polymer dynamics, which has the virtue that its intensity autocorrelation function, or square of its intermediate scattering function, can be explicitly evaluated. It was found that a t large KRG >> 1 essentially the entire relaxing amplitude decayed in exponential fashion with a rate proportional to K2. However, that model omits completely the linear topological connectivity of the segments that characterizes real polymers, and as a consequence fails to predict correctly the dependence of its longest (and only) internal mode Langevin relaxation time (defined below) on molecular weight.9J7B0

Resolution of the above dilemma is the principal aim of this paper. First, it was realized that an ISMF model comprised of two segments diffusing about their center-of-mass is essentially completely equivalent to a free- draining Rouse chain consisting of 2 beads, both models exhibiting a one- term Debye-type relaxation of the dynamic viscosity with the same time constant, for example. Following the approximate procedure of Dubois- Violette and de G e n n e ~ ~ . ~ and Silbey and Deutch,6 the Fourier transform of the intermediate scattering function for the ISMF model was expanded, and its spectral width estimated from its curvature in the limit w - 0. The resulting prediction that a K4 at large KRc >> 1 is contrary to the exact result for the same model. As shown below, the difficulty with the earlier treatments3i4p6 stems from taking the w - 0, or t - 00 limit. The amplitude of the very slowly relaxing components goes rapidly to zero as KRG becomes large, so that the overwhelming bulk of the relaxation occurs not at long times, or small frequencies, but rather at short times, or high frequencies. To resolve fully the question we have carried out both exact numerical calculations and considerable analytical development of the intermediate scattering function for the Rouse-Zimm model, as well as numerous ex- periments on DNA solutions, only a few of which will be reported here.

The plan of the paper is as follows. A general formulation of the intermediate scattering function for any system exhibiting a linear separable Langevin equation is briefly derived. Then our computational algorithm employed for the linear Rouse-Zimm model is developed, and details of the numerical calculations are given. Analytical expressions for the intermediate scattering function valid at short times are obtained for both the circular and free-draining linear Rouse-Zimm models. The exact result for the ISMF model, including the contribution of segmental motions to center-of-mass diffusion, is also described. A discussion is given of the value and validity of the translational diffusion coefficient obtained from the Rouse-Zimm model. After a brief description of experimental materials

LIGHT-SCATTERING STUDIES IN DNA. I 427

and methods, a discussion of pertinent results in relation to the theoretical developments is given. Finally, a simple algorithm is formulated for estimating the parameters of the linear Rouse-Zimm model.

The emphasis in this paper is on the behavior of the intensity autocorrelation function when KRG >> 1, rather than the KRG < 1 situation that has been the subject of numerous previous works.

GENERAL LANGEVIN THEORY

The Langevin equations for a Rouse-Zimm chain comprised of N + 1 beads connected by springs may be written in the form6,45

= F,(t) n = 1,2, . . ., N + 1 (1)

where r, and Fn(t) denote, respectively, the instantaneous position of the nth bead and the fluctuating force upon it, each component of which is assumed to be an independent Gaussian random variable with a white power spectrum, uncorrelated with fluctuating forces on other beads. The quantities m, f, and g are the appropriate mass, friction factor, and spring constant, respectively, for the bead-spring units. For a linear chain the matrices H a n d A are the usual hydrodynamic interaction and force constant matrices of Zimm. Thus, for a linear chain

1 -1 0 o . . - o -1 2 -1 o. . .

0 -1 2 - 1 s . - 0 1

A =

and

Hmk hm,k 4- f[(67~~)l/~qbIrn - k)1/2]-1 (3)

where q is the solvent viscosity and b = ( 3 k ~ T / g ) l / ~ is the root-mean-square spring extension, or separation between beads. After transformation to normal coordinates

a set of uncoupled Langevin equations for the normal modes results. Be- cause A, and therefore also HA, possesses a vanishing eigenvalue A1 = 0 associated with the uniform mode, the inertial term must be retained for that particular normal coordinate. Thus, the transformed Langevin equations take the form

dbl dt

m - + fb1 = Gl(t)

dPl g dt f -+ - Alpl = Gl(t) I! = 2,3, . . ., N + 1

428 LIN AND SCHURR

where the inertial terms have been neglected for all normal modes with nonvanishing eigenvalue Al of the product matrix H*A (which is diagonalized by the simularity transformation Q-lHA Q). The quantity Gl( t ) - - n=l (Q-l)lnFn(t) is the fluctuating force on the lth normal coordinate.

If the Zth normal coordinate (1 # 1) has the value pl (0 ) at t = 0, then the ensemble average (over all trajectories of the fluctuating force) of its subsequent extension is just

( p i ( t ) > = pl(0)e-t/Tr (6) where 71 3 f/gAl is the Langevin relaxation time for the lth normal mode. The subsequent mean-squared fluctuation about the relaxing mean-extension can be s h o ~ n ~ ~ , ~ ~ to be

(7) which relaxes to its steady-state value with a time constant equal to one-half the Langevin relaxation time, but precisely equal to the dynamic viscosity relaxation time defined in Eq. 62 of Zimm.37

Let [ 1 , 71, and {l denote, respecitvely, the x , y, and z components of the lth normal mode. Then, it follows directly from the linearity of the Langevin equation and the Gaussian random character of the fluctuating force that the conditional probability P([l(t)l&(O)) at any time t is a Gaussian with half-width given by Eq. (7). The initial probability distribution P(,$l(O)) can be found from the conditional probability in the limit of very long times, and is also a Gaussian. The product of the initial and conditional probabilities for the lth mode (1 # 1) is, therefore,

P(t;i ( 0 ) ) - p(t i ( t ) I &(o))d5i(O)dEi(t) = exp(-51(0)~/2df)

- z N + 1

((pl(t) - ( p l ( t ) ) I 2 ) = ( (p1(0) l2) (1 - e-2t/T9

expi- [&( t ) - t l ( O ) exp(-t/rl)I2/2dH(1 - exp(-2t/r1))1 [ 2 4 1 - exp(-2t/q)]1/2

X

x d t l ( O ) d t ( t ) (8)

where df’ = ( [ ~ ( 0 ) ~ ) is the mean-squared extension of the lth normal mode in the x-direction which will be explicitly evaluated below.

For the 1 = 1 mode the correlation function (@l(O)+l(t) is first determined by multiplying both sides of Eq. (5a) by bl(0) and taking the ensemble average under the assumption that ($l(O).Gl(t)) = 0. Upon solving the resultant differential equation, one finds

(bl(0) .Pl(t)) = (Ib1(0)12)e-ft/m (9)

which may be employed in the linear response relation

to obtain the diffusion coefficient along the 1 = 1 normal coordinate. The


quantity ( Ibl(0) 12) may be evaluated by expanding bl(0) in terms of lab- oratory coordinates. Thus,

N+l N+l

p = l q = l (1b1(0)l2) = C C (Q-l)lp(Q-’)lq(+p(O) . k q ( 0 ) ) (11)

A curious feature of this development is that the result of Zimm37 for the center-of-mass diffusion coefficient (observed at very small scattering vectors K ) can only be obtained provided that the following relation is assumed to hold:

where Tpq = [ (6~3)”~7b Ip - q I lI2]--1 is the configuration-averaged hydrodynamic interaction coefficient, which gives the solvent velocity perturbation at the qth bead per unit force exerted on the solvent fluid by the p t h bead. The velocity perturbation at the qth bead at t = 0 due to the p t h bead is, then, ( f fp(0))Tpq, and that represents the part of i , ( O ) that is correlated with f p ( 0 ) , thus giving rise to Eq. (12). Evidently, the hydrodynamic interaction in the form employed here constitutes a velocity- dependent potential, so that the bead velocities are indeed correlated, contrary to what would be expected of a rigorous statistical mechanical treatment. In any event, there results finally

In view of the fact that motion along the 1 = 1 normal coordinate is simple translational diffusion, as is evident from the Langevin equation (5a), the conditional probability P({l(t) 1 &(O)) is the usual Gaussian with diffusion coefficient D1. The initial probability distribution for the 1 = 1, or uniform, mode is simply a constant [L 8,” (Q-l)ln]-l over the allowed range

N+ 1 N + 1 - ( L / 2 ) C (Q-’)ln 5 41(0) 5 (~512) C ( Q - l ) l n

n = l n= 1

of &(O), where L is the x dimension of the illuminated scattering volume. Thus, the product of initial and conditional probabilities for the 1 = 1 mode is given by

P(tl(O))P(tl(t) I tl(O))dtl(O)dtl(t)

The intermediate scattering function for a single Rouse-Zimm chain is defined by

430 LIN AND SCHURR

For simplicity K is here taken to lie along the x axis. Simply expressing the r, (0) and r, ( t ) in terms of the normal coordinates using

and an analogous equation for r,(t), and utilizing Eqs. (8) and (14) for the probability distributions, there results after integration over d&(O)d(l ( t ) the final expression

n=l p = l

where use has been made of the fact that Qpl = Qnl for all n,p (because the first column of Q is the uniform mode eigenvector). The sum is the well- known expression derived by Pecoral and SaitB and ItB5 for the internal modes of the Rouse-Zimm model. It is apparent from this development, however, that that expression applies to any system for which the Langevin equations of the normal coordinates have the form of Eqs. (5a) and (5b), in particular when they are linear, and when the x, y, and z components of the equations of motion are separable. Indeed, Eq. (17) will also be applied to the independent-segment, mean-force model below. It is obvious from Eq. (17) that the apparent diffusion coefficient in the limit of small K is given by

where, the second equality serves to define YO in precisely the manner employed by Zimm.37 Indeed, this expression for DO is identical (apart from the replacement of N by N + 1) to that in Eqs. (97a) and (97b) of Zimm for the translational diffusion coefficient of the polymer. Here, the choice of normalization for the 1 = 1, or uniform eigenvector, was such that

N+ 1

n= 1 C Qni2 = 1.0

Hence, Qnl = ( N + 1)-lI2.

NUMERICAL COMPUTATIONS OF G(K, t ) FOR A LINEAR ROUSE-ZIMM CHAIN

A complete evaluation of Eq. (17) requires determination of the 71 = f l (9Al ) and Qnl, which are derived from solutions to the eigenvalue problem, and also d? and DO E QPl2D1. In this section, DO is not computed, but an


appropriate experimental value is assumed. Owing to artificiality of the Rouse-Zimm ghost model of the polymer, the internal modes and translational diffusion probably do not stand on the same footing, and should not, perhaps, be compared. The sole resistance to translational diffusion is solvent friction, while the resistance to segment motions embodied in f undoubtedly contains contributions from polymer-polymer interactions, as well as from simple solvent friction. This problem will be discussed in greater detail below.

The Eigenvalues and Eigenvectors

The initial step is the solution of the eigenvalue problem

HAal = Alal (20) Although both Hand A are symmetric, the nonsymmetric character of the product matrix HA has in the past caused some difficulties, necessitating the application of perturbation techniques, for example, to solve Eq. (20). It is possible to formulate the problem in a way that requires the diago- nalization only of a symmetric matrix. This procedure offers two advan- tages: (1) any of the readily available computer programs, such as those employing the Householder-Givens Sturm-sequence algorithm, for rapid and accurate determination of the eigenvalues and eigenvectors of symmetric matrices may be employed, and (2) the determination of d: can be reduced to a triviality, as indicated below.

Let c be the real orthogonal matrix that diagonalizes the symmetric matrix A by similarity transformation

(c-'A c)im = A(O&,, (21) The eigenvalues of A are known to be

A(1) = 2 [ 1 - cos (%)I = 4sin2 ( ) (1 - l ) a

2(N + 1) 1 = 1,2,. . ., N + 1 (22)

and the columns of c, which are the eigenvectors of A, have elements of the known form

1 (1 - 1)a 2 N + 1

c , ~ = Ni cos [ ( m - -) -1 m = 1,2,. . ., N + 1 (23)

for 1 = 1,2, . . ., N + 1. The normalization constants can be either numerically computed, using

or analytically summed, as described below, to give N1 = ( N + 1)-1'2 and N I = (2/(N + 1))-1/2 for 1 L 2.

The matrix All2 is constructed according to the relation

432 LIN AND SCHURR

p = l q = l

where use has been made of the property ( c - ~ ) ~ ~ = cnq. Clearly, this matrix satisfies the relation A112A112 = A. Moreover, All2 is symmetric, since

= C C CnqA(q)1/26q,pCmp = (A1’2)nm 9 P

Multiplying Eq. (20) from the left by All2 gives

(26)

A112HAal = A 1/2HA 1/2A1/2, = Al ( A (27)

The matrix A1/2HA1j2 is obviously symmetric, as

( ~ 1 / 2 ~ ~ 1 / 2 ) T = ( A I / P ) T H T ( A ~ / ~ ) T = ~ 1 / 2 ~ ~ 1 / 2

The eigenvalue problem has now been reduced to one of the form

where the elements bml, m = 1,. . .$ N + 1, of bl are related to those awl, or Qml, of a1 by the relation

(A1/’HA1/’)bl = A1 bl (28)

Because A1/2HA1/2 is symmetric, its eigenvectors b are orthogonal. The bl are usually also obtained in normalized form with most standard di- agonalization routines. It is apparent from Eq. (29) that the bl and al cannot, in general, both be simultaneously normalized. The choice usually made in the past was to normalize the al, or columns of Q. In fact, we have indicated such a normalization for the 1 = 1 column already. However, for the internal modes 1 = 2,3,. . ., N + 1, we choose instead to retain the normalization of the bl, and let the al, or columns of Q, remain unnormalized. This gives an enormous simplification in the evaluation of df as noted below.

Equation (28) is solved numerically, then, for the eigenvalues Al and the orthonormal eigenvectors bl. The Qnl can then be obtained by the following procedure. Employing Eq. (25) for ( A 1/2)mn in Eq. (29) above gives, after multiplication on the left by (c - l )km and summing on m,

Because A(1)ll2 = 0, it will be understood that Eq. (30) holds only for k L 2. In that case, one may then multiply both sides of Eq. (30) by A(k)-1 /2 (which is not singular for k 2 2) and Cjk and sum on k from k = 2 to k = N + 1, giving


k = 2 m = l k=2 n = l

= Qj l - ( Q I ) where ( Q l ) denotes the average of the elements in the 1 th column of Q. In the second step on the rhs, the k = 1 term has been included in the sum and simultaneously subtracted outside. Thus, the quantity Qj l - ( Q l ) is computed from known quantities on the lhs. The Qjl are determined simply by arranging the numbers Qjl - ( Q l ) as a column vector and taking the matrix product

C ( H A ) j m ( Q m l - ( Q I ) ) = AlQjl- O ( Q l ) (32)

The constant part, which effectively represents a uniform mode contribution, is stripped-off because of its vanishing eigenvalue. Simply dividing the product in Eq. (32) by Al gives Qj l . It should be noted, perhaps, that ( Ql ) = 0 at the outset for all of the odd-symmetry modes, 1 = 2,4,6, . . . . A subsequent matrix multiplication of the form of Eq. (32) was employed to test the constancy of the (eigenvector) elements Qjl of the columns of Q.

m

The Evaluation of

Returning to the evaluation of df’, we note that the potential energy associated with an arbitrary configuration of a linear chain is given by

g N + l N + l N + 1 N+1

2 1 = 1 k = l m = l n = l = - C C P L * Pk C C QITmAmnQnk

where the expansion (16) has been employed for rm, r n . The final sum on m and n may be written in the form

(34) where use has been made of Eq. (29) and the orthonormality property of the b vectors. Moreover, the I = 1, k = 1 terms in the sum may be omitted, because the first column of Q or the first row of QT, corresponding to the uniform mode, is an eigenvector of the matrix A with zero eigenvalue. In terms of the normal coordinates, the potential energy may be expressed as

( Q T A & ) l k = ( & T A 1 ’ 2 A 1 ’ 2 & ) l k = b7. bk = & k

434 LIN AND SCHURR

and the probability distribution

is evidently a product of Gaussians of mean-square displacement

for each normal mode with I 2 2. It should be emphasized that this result is based on retention of the normalization of the bl and, consequently, forfeit of the normalization of the al, or of the columns of Q.

In summary, then, our algorithm consists of the following main steps: (1) construct A1/2 using Eq. (25) ; ( 2 ) construct A1/2HA1/2 and obtain its eigenvalues A1 and orthonormal eigenvectors bl using any convenient di- agonalization program for symmetric matrices. We employed a modified Householder-Givens Sturm-sequence routine; (3) determine for each I 1 2 the quantities Qjl - ( Q l ) , j = 1, . . ., N + 1, using Eq. (31 ) ; ( 4 ) strip-off the constant part by matrix multiplication with HA, as indicated in Eq. ( 3 2 ) , to obtain the Q j l , j = 1, . . ., N + 1, for each 1 2 2; ( 5 ) insert the 71 = f / (gAl ) , Qml, Qnl, and df = kBT/g directly into Eq. (17), and replace Qp12D1 by the assumed DO.

Correlation functions G(K,t)2 were computed at a series of regularly spaced times t . Then the computed correlation functions were fitted to a single-exponential curve of the form C ( t ) = A exp(-t/rA) + B using a locally written three-parameter (A,& 7 A ) nonlinear least-squares routine, after which both computed and best-fit curves were coplotted on the same graph. At both very small and very large values of K 2 the correlation functions were essentially perfect single-exponentials. Even in the intermediate range, the decay was fit surprisingly closely by the single-exponential relaxation. The apparent diffusion coefficient at any K can be defined in terms of the best-fit relaxation time by Dapp

Plots of Dapp vs K2 are shown in Fig. 1 for three different values of b = [ 3 k ~ T / g ] ~ / ~ when hydrodynamic interaction is present, and in Fig. 2 for the same parameters when it is absent (i.e., H,, = Jm,). The parameters N + 1 = 10 and f = 1.63 X dyne cm-l sec were chosen simply to be rep- resentative of calf-thymus DNA, as determined by prior fitting of experimental data to the independent-segment mean-force model described preyiously,3° and reconsidered below. These values do not represent op- timum parameters for the Rouse-Zimm model for any particular set of data. The quantity DO was taken to be 2.0 X lo-*, and provides only the K2 = 0 baseline for the contribution of internal motions. Choosing another value of DO simply effects a vertical translation of the entire curve.

In Fig. 3 the quantity Dapp is plotted vs K2b2 for each of three different

1 / 2 r ~ K ~ .


7.00-

- 0 w 0) k 6 . 0 0 - * x 0

1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00 KSQ I U N I T IN 1 0 ~ x 1 0 C i I ~ * I - 2 1 1

Fig. 1. Theoretical apparent diffusion coefficients DT = Dapp = ( 2 ~ a I P - l computed for the Rouse-Zimm model (with hydrodynamic interaction) vs K2. From top to bottom the pertinent b values are B b = 4000,3000, and 2000 A, respectively, which in turn imply g values of G 1 g = 0.764 X 1.385 X and 3.056 X dyne/cm, respectively. Other parameters are Do = 2.0 X cm2/sec, and F = f = 1.63 X loT6 dyne/cm sec. The temper- ature is assumed to be T = 298'K.

values of N + 1 (5,10,15), and each of three different b values (4000,3000, 2000 A), all in the free-draining limit. The curves for different b values, but the same N + 1, are superposed, showing that G(K,t) depends on b only through the parameter K2b2, in accord with the analytic result derived below. However, this superposition feature is not preserved in the presence of hydrodynamic interaction, as indicated in Figs. 4 and 5, where Dapp is plotted vs K2b2 for N + 1 = 5 and N + 1 = 10, respectively.

It should be noted that the mean-squared end-to-end distance is ( S 2 ) = Nb2, and the radius of gyration is RG = [ (S2 ) /6]1 /2 = [Nb2/6]1/2 = (3/2)1/2b for N + 1 = 10. It is an unusual feature of these DNAs with relatively few effective segments that RG is roughly comparable to b. Perhaps related to this is the fact that the effective diffusion coefficient for internal motion, Dseg E k ~ T / f = 2.5 X 10-8 at 25"C, is in the same range as DO for translation of the entire molecule.

Evaluation of G( K, t ) at Small Times for the Rouse-Zimm Circular Chain

For a circular chain of N + 1 bead-spring units, the A matrix is that of Eq. ( 2 ) with the insertion of -1 at the (1, N + 1) and (N + 1,l) positions. The Hmatrix has the form

Hn,m = am,n + Tmn (38)

436 LIN AND SCHURR

oo& 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00

KSQ I U N I T I N 1 0 ~ x 1 0 CMrrl-211

Fig. 2. Theoretical apparent diffusion coefficients DT = Dapp = (2TAK2)-' computed for the free-draining Rouse-Zimm model (without hydrodynamic interaction) vs K2. From top to bottom the pertinent b values are B = b = 4000,3000, and 2000 nm, respectively, which in turn imply g values of G = g = 0.764 X 1.385 X and 3.056 X dyne/cm. Other parameters are as in Fig. 1.

where Tmn = f [ ( 6 ~ ~ ) ~ / ~ q b I q ( N + 1 - q ) / ( N + 1)11/2]-1 = T(lq1) and q I m - n I I N + 1. The evident modification in Hmn takes account of the fact that the mean-square displacement between two beads connected by two Gaussian chains of length bp and b(N + 1 - p ) is b l p ( N + 1 - p ) /

Because of the symmetry of the circular chain, both the A matrix and the HA matrix are diagonalized by the complex normal coordinate transformation

(39)

In this case, the matrix Qnl ( N + 1)-ll2 exp[-i27rnZ/(N +1)] is unitary. Moreover, in this instance the uniform mode corresponds to 1 = N + 1, rather than 1 = 1, as before, and its associated quantities PN+1, Qn,N+1 and ( Q - l ) ~ + l , ~ are purely real.

It is easily shown by direct evaluation of &-'€€A& that the eigenvalues All for the circular chain are

( N + 1)(1/?

N+ 1

n= 1 pl = ( N + 1)-1/2 C ei2?rln/(N+l)rn

where the eigenvalue of the circular A matrix is ~tr=2(1-cos-----) 2T1 =4sin2(-) id

N + 1 N + 1


-6 .00 - u W

\ N

ps .00 - U

0 16.00 32.00 48.00 64.00 80.00 96.00 112.00 i m . o o 144.00 160.00

KSQxBSQ

Fig. 3. Theoretical apparent diffusion coefficients DT = Dapp = (2r,4K2)-' computed for the free-draining Rouse-Zimm model vs KSQ*SSQ K2b2. From top to bottom the values of N + 1 are N + 1 = 15,10, and 5, respectively. For the circles (a), b = 4000 A; for the triangles (A), b = 3000 A; and for the asterisks (*), b = 2000 A. Other parameters are as in Fig. 1.

For the case of a complex normal mode transformation the form of Eq. (17) must be altered slightly. Noting first that

where the superscripts r and i refer to the real and imaginary parts, respectively, the procedure outlined previously can be carried through for both real and imaginary parts separately. This gives for the internal mode contribution in Eq. (17) the result

K2 N

2 1=1 X exp (- - C (~ii)~[(QiL)~ + (Q;l12 - 2QilQ;le-T/Ti])] (43)

The (d { )2 = ( ([ i (0))2) and (df)2 = ( ( [ i (0) )2) may not in this case be simply determined from Eq. (37), based, as it is, on Eq. (34). However, the po-

438 LIN AND SCHURR

I '0 16.00 32.00 48.00 64.00 80.00 96.W 112.00 128.00 144.00 160.00

KSQuBSQ Fig. 4. Theoretical apparent diffusion coefficients DT 5 Dapp = (2T.4K2)-' computed for

the Rouse-Zimm model (with hydrodynamic interaction) vs KSQ'SSQ 5 K2b2 for N + 1 = 10. For the circles (a), b = 4000 A; for the triangles (A), b = 3000A; and for the asterisks (*), b = 2000 A. Other parameters are as in Fig. 1.

tential energy is readily evaluated anew to give g N + l N + l *

U h r 2 , . . ., rN+d = 2 C C rm-Amnrn m = l n = l

g N+l N+l N+l N+l

2 1=1 k = l m = l n = l = - C C P f ' P k C C QfmTAmnQnk

g N+l * I = - c [ ( p i ) 2 + (pi121 4 sin2 (-) 2 1=1 N+l (44)

from which it is readily deduced, following the arguments leading from Eq. (35) to Eq. (37), that

(45) Inserting the appropriate real and imaginary parts into Eq. (43) and

(d i )2 = (df)2 = kBT/[g 4 sin2(al/(N + l))] expanding e-7/rl for 7 << 71 leads to

N+l N+l K 2 kBT N 1 Cint(7) = c c exp [ ---- c

n = l p=l N + 1 4g i=1 sin2[d/(N + l)] (46) N + l

where

71-1 = @sin2 (""> [ 1 + c N cos (-) 2*lq ~ ( l q l ) ] (47) f N + l q=i N + 1


- 6.00- U W 0 \ N

z s . 0 0 - U - m - z4.00- 0 5 c

3 s3.00 - c 0

2.00

01 0 16.00 32.00 48.00 64.00 80.00 96.00 112.00 128.00 144.00 160.00

KSQ+BSQ

Fig. 5. Theoretical apparent diffusion coefficients DT = Dapp = ( 2 ~ , 4 K ~ ) - ' computed for the Rouse-Zimm model (with hydrodynamic interaction) vs KSQ*SSQ = K 2 b 2 . For the circles (O), b = 4000 8; for the triangles (A), b = 3000 A; and for the asterisks (*), b = 2000 8. Other parameters are as in Fig. 1.

The sums in Eq. (46) may be performed exactly using the following relations:

1 N cos[2alrl(N+ I)] N + 1 i=1 sin2[dl(N + I)] c

) (50) 1 N + 1 r ( N + l - r ) -

12 2(N + 1) 12(N + 1)

the last of which was derived recently by E i ~ h i n g e r ~ ~ and applies for r L 0. Using Eq. (50), one finds

1 N 1 - cos[2alr/(N+ I)] c N + 11=1 sin2[all(N + I)] (51)

= 2 ( N + 1 ) (-) r (1--) r N + l N + 1

440 LIN AND SCHURR

Using Eqs. (46)-(49) and (51) and setting r = In - P I , there results

which may be rearranged and incorporated in Eq. (17) to give, finally,

G(K, t ) = ( N + 1) exp

X - r (1 - :)] exp [ -K2 - kBT ~ T( l r l ) t ] ] (53) N + l N + l f 2 where ( T ) N 3 ( N + 1)-l Z&1 T( ) q 1 ) is essentially the average value of T( I q I ) over all values of q .

It is a t once apparent that for sufficiently large values of K2, such that K 2 k ~ T / ( 2 g ) >> 1.0, the amplitudes of all terms in the r sum will be negligible; so that essentially all of the scattering amplitude derives from the e x p ( - K 2 k ~ T t / f ] term in braces. Evidently, then, G(K,t) is predicted to be essentially single-exponential in character with a relaxation rate proportional to K 2 in the limit of large K 2 - m, contrary to the earlier pre- d i c t i o n ~ ~ , ~ , ~ regarding the w - 0 limit of the Fourier transform of G(K,t). In fact, the w - 0 limit corresponds to very long times t - m, in which limit the surviving part of G(K,t) at large K 2 might well obey the 7-l- K3, K4 predictions of duBois-Violette and de G e n n e ~ , ~ ~ ~ and Silbey and Deutch.6 However, at sufficiently large values of K 2 that part surviving at long times, corresponding to the r sum of Eq. (53), evidently exhibits only an insig- nificant amplitude that comprises a negligible fraction of the total relaxation. Thus, those earlier conclusions regarding the large K 2 - m, small w - 0 limit are, in fact, irrelevant to the interpretation of observed correlation functions at large values of K2, where in fact the overwhelming bulk of the relaxation occurs at short times with a relaxation time

7 = {K2[oo + ( k e T / f ) ( ( N / N + 1) - ( T ) N ) ] ) - ~ It should be noted that k ~ T l ( 2 g ) = b2/6, so that the large K2 limit is

reached when K2b2/6 >> 1.0, or when fluctuations in the dimensions of the individual springs, or Kuhn ~ e g m e n t s , 3 ~ , ~ ~ . ~ ~ become resolved.

This large K2b2/6 >> 1.0 condition suffices also to insure that relaxation

LIGHT-SCATTERING STUDIES IN DNA. I 44 1

of the exp[-K2k~Tt / f ] term occurs in a time short compared to the smallest of the 71 relaxation times, 7,in. It is apparent from its definition following Eq. (38) that T( I r I ) declines slowly from its maximum value at r = 1 with increasing r . It will be noted below that the maximum value of T ( 1 r I ) should never exceed about 0.59, to preserve physically reasonable behavior of the model. It is evident from Eq. (47), then, that the minimum 71 occurs for 1 z (N + 1)/2, in which case the cosine factor in the q sum alternates between +1.0 and -1.0 on successive terms, thereby effecting considerable cancellation of those terms, so that they may be ignored. In that event, ~ , i , N f /4g. Thus, the ratio of the shortest 71 to [K2keT/ f ] - l is approximately

b2 7min K2- kBT - - K2- (K2keT/f)- ' - 4g 12 (54)

which is, indeed, much greater than 1.0, when the condition K2b2/6 >> 1.0 is satisfied.

The temporal behavior of any term in the r sum of Eq. (53) may be ex- amined by comparing the ratio Z, of its decay rate to (K2kBT/ f ) . Thus,

where a = f/6a7 is the effective hydrodynamic radius of the beads. Perico et a1.I0 have shown for the linear Rouse-Zimm model that unphysical behavior, in the form of discontinuous variation of the phase $ ( 1 ) = d / (N + 1) of the maximum eigenvalue as a function of alb, is obtained unless alb I 0.43. Thus, the maximum physically allowed value of T ( ) r I ) is 0.59 and of Z, is 0.30. Evidently, the most rapidly relaxing terms in the r sum, those with the smallest index r, relax at a rate generally less than 30% of K2kBT/f. With increasing r the relaxation rates of these terms decline, and their corresponding amplitudes likewise decrease. However, for times longer than the various 71 the expression (53) ceases to be valid.

In the free-draining limit H,, = 6 , , and T,, = T ( I n - rn I ) = 0. In this case, Eq. (53) reduces to

lzBT ) t ] (exp (-K"t) f

G(K, t ) = (N + 1) exp -K2 DO - [ ( f ( N + 1) N + r=l C e ~ p [ - K ~ ~ ( N + l ) ~ ( l - & ) ] } 2g N + l (56)

where the second term in braces is now a constant in time.

Evaluation of G(K, t ) at Small Times for the Free-Draining Linear Chain

For the free-draining linear chain H = 1 and the corresponding eigenvalues and normalized eigenvectors of H A = A are given by Eqs. (22) and

442 LIN AND SCHURR

(23) above with the explicit normalization constants N1= (N + 1)-1/2, and N L = (2/(N + 1))1/2, 1 2 2, obtained by the application of trigonometric identities and also Eq. (49) above.

Owing to the real orthogonal property of c = Q (in this case), as well as the normalization of its columns, Eq. (34) now gives ( Q T A Q)lk = 4 sin2 ( ( I - 1)7r/2(N + 1)). Therefore, at short times T << 71 the internal mode ccmtribution to Eq. (17) is

N+l N+l Cint(7) = c c

n = l p = l

The pertinent quantities in the sum may be put in the form 1 (I - l)x QEl = ~ + 1 [ cos [ (2n - 1) -1 + I]

N + l with a corresponding expression for QEL, and also

N + l -2QnlQpi = - - [cos ( ( n + p - 1)

( N + l

N + l - "71 + cos (n - p ) (59)

The terms independent of T may be summed by extending the summation range and rearranging slightly, as follows:

N+1 cos[s(l - l)?r/(N + 1) f 1 c N + 1 1=2 sin2[(1 - l)?r/2(N+ I)] 1

(60)

wherein + applies only for s odd, and - for s even and where M + 1 = 2 ( N + l ) , a n d s =2n-1,or2p-1,orO,orn+p-1,orn-p,asthecase may be. The sum on the rhs of Eq. (60) can be performed with the aid of Eq. (50) above. The terms containing 7/71 can be easily summed using Eq. (48) above. The result is

1 cos[2sLx/(M+ l)] f 1 c -- - M + 1 ~ = 1 sin2[Lx/(M+ l)]

n = l p= l

which can be rearranged and incorporated in Eq. (17) to give

G(K,t) = exp [ -K2 (Do - k ~ T ) t] ( N + 1) (exp f ( N + 1)

N + l N + l N + l exp[-K2(kBT/g)ln - p l ] + c c n = l p= l


which is indeed very similar to the expression (56) for the circular chain. The appearance of kBT/g in Eq. (62) in place of the kBT/2g in Eq. (56) is undoubtedly related to the fact that the radius of gyration of the linear chain is twice that of its corresponding circular counterpart.

G( K,t) for the Independent-Segment, Mean-Force Model

The occurrence of Do - kBT/f ( N + 1) instead of DO in Eqs. (56) and (62) implies that the total contribution of internal segment motions k B T / f ( N + 1) to the diffusion coefficient of the center-of-mass (at large K 2 ) must first be subtracted from Do before superposing, or adding on, the diffusion coefficients of the individual segments kBT/f . Such a constraint was not manifested in the previous resultsg for the independent-segment, mean- force (EMF') model, because the translational motion was (erroneously) assumed to be unaffected by fluctuations in the distribution of segments. This flaw is most easily rectified using a Langevin equation approach similar to that outlined earlier.

If the instantaneous position of the center of mass is denoted by N+ 1

R = ( N + 1)-l C r, n= 1

then the Langevin equation for the nth segment in this model may be written in the form

d2r, dr, N+1

dt dt m=l m-+ f - + g C Bnmrm = F,(t)

where the matrix elements B,, are defined by

B,, = a,, - 1/ (N + 1) (65) The matrix B is evidently diagonalized by the unitary normal coordinate transformation

n= 1

because by direct calculation

(Q-lBQ)kn = akn - ak,nak,N+l (67) Thus, the eigenvalues are A1 = 1.0,l = 1, . . ., N and AN+^ = 0.

The Langevin equations for the normal modes are identical to Eqs. (5a) and (5b), with A1 = 1.0,l 1 2, so that Eq. (17) is directly applicable, provided that values for Clf and DO are available.

The potential energy associated with an arbitrary configuration of the N + 1 segments in this model is simply

L m = l n = l

444 LIN AND SCHURR

where use has been made of the unitary property Qk1= (Q-I)lm and Eq. (67). Following the arguments leading from Eq. (35) to (37), it may be deduced that

df = kBT/g (69) Substituting ( Q - ' ) N + ~ , ~ and ( & - l ) ~ + l , ~ , which are purely real, in place

of (Q-l)lP and (Q-I)Iq for the uniform mode in Eq. (18) also results in

Do = kBT/f(N + 1) (70) Because the normal coordinate transformation in this case is complex,

the internal mode contribution has the form of Eq. (43), which now becomes

where TO = f/g = (R&/3)/(k~T/f), and R8 = 3ksT/g is the radius of gyration for this model, as employed previously. The sum may be evaluated using Eq. (48). Upon incorporating Eq. (71) into Eq. (17) we have the result

N+l N+l G(K,t) = e-KzDot C C

n = l p= l

X exp[-K2(k~T/g)](N + 1) { e ~ p [ + K ~ ( k ~ T / g ) e - ~ l ~ ~ ] + N)

This may be compared with the corresponding result of the previous treatment (of Ref 9) obtained by combining equations 13 and 15 of that work (with no = N -t l ) ,

G(K,t) = e-K2DOte-Kz(kBT/g)(N + 1) { e+K'(k~T/g)e-t/~o + N ) (73)

The extra factor in Eq. (72) arising from the contribution of segment motions to that of the center of mass is easily identified.

In the limit of small times t << TO, the function exp[-tlq] may be expanded to first order in Eq. (72), resulting in

G(K,~) = exp [ - ~ 2 ( D ~ - kBT ) t ] (N + 1) f (N+ 1) t ) + N exp ( - ~ 2 ,)) kBT (74)

The evident similarity between this expression and those for the free-


draining circular and linear Rouse-Zimm chains, Eqs. (56) and (62) , respectively, might be the reason why it has apparently been so successful in accounting for the observed dependence of 7-l vs K2 for polymers of very large radius of gyration, and also for macroscopic gels. It should be noted that kBT/g = R $ / 3 in this case, where R$ is the radius of gyration of the entire molecule, while for the linear Rouse-Zimm chain kBT/g = b2/3, where b2 is the mean-squared spring extension, or end-to-end separation of the Kuhn segments.

It is, perhaps, worth noting that the ISMF model first proposed by Lee and Schurr for the internal segment dynamics of polymers and gels is, in essence, equivalent to that subsequently employed for gels under the name “harmonically bound particle” by Carlson and FraserZ6 and Wun and C a r l ~ o n . ~ ~ However, those authors obtained an incorrect result, because they failed to perform the average indicated in Eq. (15) over the initial distribution of &(O) , as is explicitly indicated in Eq. (8) of this work, or Ref. 26a. McAdam et al.40 likewise proposed the harmonically bound particle model to interpret gel data in a qualitative way, but they apparently did not obtain a complete evaluation of the contribution of internal modes to G ( K , t ) , such as that in Eqs. (72) or (73) , for example.

The Value of DO If, for the moment, it is assumed that center-of-mass diffusion and in-

ternal-mode dynamics both rest on the same footing, then DO ought to be calculated rigorously from the model. In fact, by direct calculation using (Q- l )p ,~+l , as defined in Eq. (39), in place of (Q-l)pl (i.e., for the uniform mode) in Eq. (18) one finds for the circular chain

which is precisely the quantity subtracted from DO in Eq. (53‘). Thus, rigorous adherence to the model for translational diffusion would, in effect, eliminate (i.e., set equal to 1.0) the first factor in Eq. (53), or in Eq. (56) for the free-draining case, or in Eq. (62) for the linear chain, or in Eq. (74) for the ISMF model.

( T ) N may be explicitly determined for the circular chain upon replacing the sum in Eq. (75) by an integral. Setting z = q - ( N + 1)/2, one obtains from the sequel to Eq. (38)

alb ( N + 1)112

5 ( 1 . 3 8 ) ~ = (4.34) - ( N + 1)1/2

446 LIN AND SCHURR

wherein B = ( N + 1)/2, and N has been assumed sufficiently large that cos-l(f(N - 1)/(N + 1)) = 0 or a, depending on the sign of the argument. Finally,

for the circular chain. As discussed earlier, it may often be the case that the experimentally

observed DO f DtZ. For this reason, subsequent analyses of the data will be carried out under each of the following two schemes: (1) Rigorous conformity of all data to the Rouse-Zimm model is assumed, so that DO = 0:'. (2) It is allowed that Do # D t Z , thus introducing another adjustable parameter into the model. The actual methods of analysis under each of these schemes are described in detail below.

MATERIALS AND METHODS Calf-thymus DNA, lot No. 35A614, was purchased from Worthington

Biochemical Corp., Freehold, N.J. The fibers were dissolved in 1M NaCl containing 0.05M Na2C03 as buffering agent and, where noted, 0.01M EDTA. The salt solutions were prepared using double-distilled water, and filtered under gravity flow through an unsupported 1.2 p Millipore filter, as described p r e v i o u ~ l y . ~ ~ , ~ ~ The DNA solutions were further filtered in the same manner directly into the scattering cell, which had just been rinsed with either filtered double-distilled water or salt-solution.

Sedimentation coefficients were determined for several samples at various pHs by Professor D. Teller. Molecular weights were calculated from the sedimentation constants together with the translational diffusion coefficients (determined by dynamic light scattering at small angles) using the relation52

R T s D ( l - E p )

M, =

TABLE I Calf-Thymus DNA Size Parameters

PH 1.1 20,wa D20,w x 108 M ,

11.05 23.3' 1.3 10.2 x 106

11.45 21.8 1.75 4.7 x 106

11.25 24.7 1.3 10.6 X lo6 11.35 29.3 1.4 11.7 X lo6

12.30 23-24 2.0-2.2 (4.0-4.4) X lo6

a Units of [s] are svedbergs (10-13 sec). b Units of D are cm2/sec.


The partial specific volume was taken to be U = 0.53 cm3/g for native53 and 0.29 cm3/g for pH-denatured DNA.S4 Samples were prepared and titrated to pH 8.3 or 10.0 and stored a t 5OC. For these molecular-weight measurements, the DNA was further titrated up to the appropriate pH 2 11, and s was determined two days later. Determination of D took place either on the same day as s, or about 18 hr earlier. The resulting data and computed molecular weights are given in Table I. The denaturation transition near pH 11.4 is clearly manifested in the molecular weights. It is not known whether the apparent rise in M,. of native DNA, as the transition is ap- proached, is truly significant. If so, it may be related to the more con- spicuous apparent aggregation detected earlier in thermal denaturation of both calf-thymus51 and X DNA55 by dynamic light-scattering techniques. The comparatively high molecular weight observed for the pH-denatured form suggests that single-strand breaks do not, in fact, abound in this DNA.

Autocorrelation functions were determined using previously described a p p a r a t ~ s , 5 ~ * ~ ~ , ~ ~ together with a Saicor 43A digital correlator on loan from Professor K. Schmitz. Clipped correlation functions were determined in the analogue mode and transferred directly to the PDP-12 computer via the A-D converter, followed by display on the computer scope to confirm the transfer, and to facilitate selection of the proper starting point for the fitting program. Fitting to a three-parameter single-exponential curve of the form C ( t ) = A exp[ - t / r~ ] + B was accomplished by a nonlinear least-squares assembler code with an optional number of data points up to 512, as well as optional starting and ending points. Additional data input via teletype for (1) time-delay between points, (2) wavelength of scattering light, (3) scattering angle, and (4) index of refraction was employed to calculate the reciprocal relaxation time rA1 and its standard deviation, as well as the apparent diffusion coefficient Dapp = 1 / ( 2 r ~ K ~ ) . A t the conclusion of the fitting program the first 256 points of both raw data and fitted curves were simultaneously displayed on the scope, with a continuous sense-switch operation to remove (or restore) the fitted curve from (or to) the display. These DNA data were generally fit from the n = 2 to n = 256 points, requiring about a minute of computing time. The computed standard deviation for any single correlation function was employed, along with the visual display, primarily as a guide to the quality of the single- exponential fit, rather than as a measure of the absolute error in the reciprocal relaxation time, which in our experience is more realistically judged from the variation between correlation functions obtained on successive trials. Thus, we do not report the “statistical errors,’’ but leave the absolute errors to be inferred from the apparent regularity of the plots of Dapp vs K 2 (see Figs. 8-11).

The high ionic strength (1M NaC1) was employed to suppress the anomalous slow relaxations generally observed for DNAs of this size and concentration a t lower ionic strengths.30-32,51,55,57 Indeed, no sign of the anomalous slow relaxations was detected in these samples.

448 LIN AND SCHURR

A 240.00

25o.w

' 0 25.00 50.00 75.00 1W.00 125.00 150.00 175.00 2W.W 225.00 ZSO.00 OELRY TIME (UNIT IN 5 MICRO SECI

(b) Fig. 6. (a) Clipped intensity autocorrelation function C(TAU) = C(T) vs T (in units of 200

psec) at 0 = 20° for 0.04 mg/ml calf-thymus DNA in (1M NaCl + 0.05M NaZC02), pH 11.25 at t = 22OC without EDTA. (b) C(TAU) = C ( r ) vs 7 (in units of 5 fisec) for the same sample at 0 = 110'. The solid lines indicate the least-squares fits to C(T) = A exp (-&A) + B.


300.00.

250.00 -

200 .oo.

150.00 -

100.00 -

'0 25.00 50.00 75.00 100.00 125.00 150.00 175.00 200.00 225.00 25OSIL7 DELAY TIME [ U N I T I N 200 VICRO SEC)

'0 25.00 50.00 75.00 100.00 125.00 150.00 175.00 zOO.00 725.00 250.00 OELRY TIM I U N I T I N 10 MICRO SECl

(b)

Fig. 7. (a) Clipped intensity autocorrelation function C(TAU) = C(7) vs 7 (in units of 200 psec) at 8 = 20° for 0.04 mg/ml calf-thymus DNA in (1M NaCl + 0.05M Na2C03). pH 12.21, at t = 22°C without EDTA. (b) C(TAU) 1 C(7) vs 7 (in units of 10 psc) for the Same samples at 8 = 80°. The solid lines indicate the least-squares fits to C(7) = A exp (-7/7.4) + B.

450 LIN AND SCHURR

0‘

Fig. 8. Observed DT = Dapp = (2raK2)-’ vs K 2 for 0.04 mg/ml calf-thymus DNA in (1M

1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00 KSQ (UNIT I N lOttlO CM**I-211

NaCl + 0.05M Na2C03), pH 10.30, at T = 22°C without EDTA.

It should be noted that the developing autocorrelation functions were monitored visually on a continuous basis to guard against Tyndall scattering from dust in the illuminated region. The observation of Tyndall scattering, manifested by an upward shift of the “baseline,” prompted rejection of the accumulated data and a restart of the correlation process. Each correlation function was developed for 10-20 min prior to transfer to the computer. Typical correlation functions and their single-exponential fits for both low- and high-scattering angles are presented for native DNA at pH 11.25 in Fig. 6 and for denatured DNA at pH 12.21 in Fig. 7. Finally, plots of Dapp vs K2 are given in Figs. 8 and 9 for the extremes of the pH range investigated without EDTA, and in Figs. 10 and 11 with EDTA.

The solid lines in Figs. 8-11 were obtained by visually fitting the data on the computer scope, using an interactive program with manual parameter adjustment, to the ISMF formula (19) of the previous treatment? valid at short times,

where ACF(T) is the intensity autocorrelation function, x 2 = K2R&/3 and TO = ( R & / 3 ) / ( k ~ T / f ) . This relation is also obtained from Eq. (74) here by setting ACF(7) = G ( K , T ) ~ , D = DO - kBT/(N + l)f and N + 1 = no.

The calcium ion concentration in the solutions without EDT was determined to be [Ca++] z 7.5 X 10-4M.


.70

.6O

-

, --,A 1.00 2.00 3.00 4.00 5.00 6.00 1.00 8.00 9.00 1o.m

KSO I U N I l I N 10*+10 C i I r r l - Z i )

Fig. 9. Observed D T = Dapp = ( ~ T A K ~ ) - ' vs K 2 for 0.04 mg/ml calf-thymus DNA in (1M

ooL

NaCl + 0.05M Na&03), pH 12.50, a t t = 22°C without EDTA.

DISCUSSION Comparison of the Models and the Data

I t is evident from a comparison of Figs. 1 and 2 with 8-11 that the Rouse-Zimm model with or without hydrodynamic interaction, and with the appropriate parameters, predicts curves with qualitatively the same shape as either the experimental data or the ISMF model, in particular exhibiting a plateau value of the apparent diffusion coefficient a t large values of K2. This is in accord with the predictions of Eq. (62) for the free-draining linear chain, and also Eqs. (53) and (56) for the circular chain. However, the Rouse-Zimm model appears to exhibit definitely less "S"- shaped character a t small values of K 2 than either the data or the ISMF model. Solely because of this feature, the ISMF model appears to be in slightly better agreement with these data at small K 2 than the Rouse-Zimm model, a conclusion reached earlier from somewhat different considerations by Schmidt et al.31332 The problem of which model best fits data will require more precise measurements at very low angles for its final resolution. An independent determination of R%, from a static light-scattering Zimm plot, for example, would also facilitate discrimination between the two models.

The plateau value of Dapp at large K 2 is both a distinctive and readily measured parameter containing information about the rate of intramolecular segmental Brownian motion. For the free-draining models including circular [Eq. (5611, linear [Eq. (62)], and ISMF [Eq. (74)], the difference between the plateau value and Do is

452 LIN AND SCHURR

L.03 2.w 3.00 4.m 5.00 6.00 7.m 8.m 9.00 10.00 KSQ ( UNIT I N lot10 C I d - 2 1 I

Fig. 10. Observed DT E Dapp = (2r&!)-' vs K 2 for 0.03 mg/ml calf-thymus DNA in (1M NaCl + 0.05M Na2C03), pH 9.60, at t = 22OC with EDTA.

Dpht - DO = ( N / N + l)(kBT/f) (80) which is quite close to D,, = kBT/f when N >> 1. However, as is apparent from Fig. 1, the hydrodynamic interaction is a complicating feature that acts to diminish D,ht - DO, an effect that grows more pronounced with decreasing b at fixed f and N + 1, or with increasing magnitude of the matrix elements H,, [see Eq. (3)]. This effect is also clearly manifested for the circular chain in Eq. (53). Despite this complication, a complete determination of the model parameters may prove possible, even in the presence of hydrodynamic interaction, as suggested below.

Estimation of Parameters of the Rouse-Zimm Model There are three obvious experimental parameters represented in Figs.

8-11, namely, DO, Dpkt, and the midpoint value KK, where D,, is half-way between DO and D,ht. These parameters are listed in Table 11. As noted above, the analysis, or parameter estimation, can be carried out under either of two schemes. However, certain procedures are common to both, and these will be taken up first.

It is evident from Figs. 3-5 that D,, is at its midpoint, half-way between Do and Dplat, when Kkb2 = 8 for either free-draining or nondraining models. Thus, setting b2 = 8/Kf should provide an accurate estimate of b in any case.

It is apparent from Eq. (53) for the circular chain that

Dplat - Do = -- ltBT - W ( b , N + 1) f N + l


1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00 KSQ [ U N I T I N 10110 CMru(-21 1

Fig. 11. Observed DT = Dapp = (27aK2)-' vs K 2 for 0.03 mg/ml calf-thymus DNA in (1M NaCl + 0.05M Na2C03), pH 12.50, a t t = 22OC with EDTA.

TABLE I1 Experimental Parameters for Calf-Thymus DNA a t 22-23OC

Do Dplet K2, PH (cm2 sec-') (cm2 sec-') (cm+)

Without EDTA 10.3 1.4 X 4.2 X 1.2 x 10'0 12.5 1.88 X 3.48 X 1.95 X 1O'O With EDTA 9.6 1.5 X 4.05 X 2.95 X 10'O

12.5 2.2 x 10-8 5.0 X low8 3.08 X 1O'O

N + l brAl 5 10 15 30

2000 0.961 X 0.856 X 0.768 X 0.610 X 3000 0.652 X 0.580 X 0.520 x 0.412 X 10W8 4000 0.491 X 0.437 X 0.391 X lo-* 0.310 X

where W(b,N + 1) = ( K B T / ~ ) ( T ) N is actually independent o f f (or a ) . Moreover, from Eqs. (76) and (77) one has, when N + 1 is sufficiently large,

W(b,N+ 1) Fb- l (N+ 1)-ll2 (82) where F = 1.06 X cm3/sec is a numerical constant for the circular

454 LIN AND SCHURR

TABLE IV F 5 b(N + 1)li2W(b,N + 1)

N + 1 b(A) 5 10 15 30

2000 4.30 x 10-13 5.41 x 10-13 5.95 x 10-13 6.68 X 13000 4.37 x 10-13 5.50 x 10-13 6.04 x 10-13 6.77 x 10-13 4000 4.39 X 5.53 x 10-13 6.05 x 10-13 6.79 x 10-13

chain independent of any parameters of the model (e.g., f, b, N + l ) , although it is proportional to T/q. The hypothesis is now advanced that equations of the form of Eqs. (81) and (82) hold also for the linear chain in the presence of hydrodynamic interaction when N + 1 is sufficiently large. Values of W(b,N + 1) = (kBT/f)(N/(N + 1) - (D,lat - DO) obtained from the computed curves in Figs. 1,4, and 5, and also from some additional computations are presented in Table 111, and corresponding values of F = b(N + 1)l12W(b,N + 1) are tabulated in Table IV. It is at once apparent that the assumed b-l dependence of W is nearly correct and that the assumed ( N + 1)-ll2 dependence of W is only approximately correct down to N + 1 = 5. The “universal” value of F applicable to the linear chain for very large N should be quite close to that obtained for N + 1 = 30 in the last column of Table IV, that is F = 6.7 X cm3/sec. This value is somewhat smaller than that noted above (1.06 X 10-l2) for the circular chain, a difference attributable to the larger (by a factor two) radius of gyration, hence weaker hydrodynamic shielding, in the linear model.

The two analysis schemes are detailed below. (1) In the “rigorous” interpretation the Rouse-Zimm model contains only

three parameters, f (or kBT/f), b (or g = 3 k ~ T / b ~ ) , and N + 1. In this scheme it is necessary to assume (for the linear chain) not only equations of the form of Eqs. (81) and (82), but also one of the form of Eq. (77), that is,

which gives in conjunction with Eqs. (81) and (82) Dplat = kBT/f. This latter relation could, in analogy with Eqs. (53), (56), and (62), have been assumed directly, and employed with Eq. (81) to infer Eq. (83). The critical point is that one additional relation besides Eqs. (81) and (82) must be assumed in this scheme.

The parameter b is easily determined by setting b = (8/K2,)l12. Then, kBT/f = Dplat is likewise easily obtained. Finally, Eq. (83), with F determined from Table IV (approximately), is employed to determine N + 1


TABLE V Model Parameters for Calf-Thymus DNA a t 22-23°C. Scheme 1

b k B Tlf g f PH (nm) (cm2/sec) N + 1 (dyne/cm) (dyne cm-’ sec)

Without EDTA

12.50 200 3.5 x 10-8 5.2 3.0 x 10-4 1.16 x With EDTA

10.30 260 4.2 X 7.0 1.8 x 10-4 9.69 x 10-7

9.6 165 4.05 X 9.5 4.6 x 10-4 1.01 x 10-6 12.50 161 5.0X 6.0 4.8 x 10-4 8.14 x 10-7

from DO. Equation (83) may be solved to give the expression

in terms of F and experimental data. If ( N + 1) 5 15, values of F may have to be interpolated from Table IV and a few cycles of iteration completed, using Eq. (84) to estimate N + 1 and Table IV to readjust F , etc., until N + 1 converges to a stationary value. Values of the parameters determined according to this scheme for the various experimental situations using F = 5.5 X

At this point there are no further adjustable parameters, so that other measured quantities such as the radius of gyration Re = [( N + 1)/6]1/2 b are left as predictions of the model. Direct observation of a substantially different Re, by static light scattering for example, would invalidate the “rigorous” Rouse-Zimm model.

(2) In this scheme it is not required that Do = Dfz, so that another parameter is introduced into the model. Thus, there are now four parameters to determine, Do, f, b, and N + 1. As before, b is determined by setting b = (8/K;). DO is of course taken directly from the data. Equations (81) and (82) may be employed to determine k B T / f from Dplat, DO, b , and N + 1, using an appropriate value of F from Table IV. However, N + 1 must first be determined in this case from an independent determination of Re = ( ( N + 1)/6)1/2b, or by some other means. Thus, this scheme requires one less assumption, but necessitates instead an extra experimental datum. Unfortunately, we do not have here such an independent determination of Re. However, for illustrative purposes we may employ an estimated value of N + 1. Native DNA with M, = 11 X lo6 contains about 2 X lo4 base pairs, and has a countour length of 7 X lo4 A. If the persistence length is taken49 to be about 700 A, then there are 100 persistence lengths in the polymer. This implies about N + 1 = 10 Kuhn segments, if we assume a minimum of 10 persistance lengths per Kuhn segment to provide an approximately Gaussian distribution of end-to-end distances for each. Pa- rameters estimated according to scheme (2) with N + l = 10 are given in Table VI.

The values of hBTlf for native DNA from schemes (1) and (2) are not very

are presented in Table V.

TA

BL

E V

I M

odel

Par

amet

ers

for C

alf-

Thym

us D

NA

at 2

2-23

OC.

Sc

hem

e 2

Rou

se-Z

imm

Mod

el

ISM

F M

odel

keT

/f

g f

Do

RG

ks

T/f

* 5

b PH

N +

1 (n

m)

(cm

z/se

c)

(dyn

e/cm

) (d

yne

cm-'

sec)

(c

mV

sec)

N

+ 1

(nm

) (c

mz/

sec)

w

cn W

ithou

t ED

TA

12.5

0 10

20

0 2.

7 X

3.

0 x

10-4

1.

5 x

1.9

x 10

-8

10

180

1.8

X

With

ED

TA

12.5

0 10

16

1 3.

6 X

4.

8 x

10-4

1.

1 x

10-6

2.

2 x

10-8

5

130

3.6

X

10.3

0 10

26

0 3.

9 x

10-8

1.

8 x

10-4

1.

0 x

10-6

1.

4 X

7

240

3.1

X

0

X d z

9.60

10

16

5 4.

0 X

4.

6 x

10-4

1.

0 x

10-6

1.

5 X

5

160

3.2

X

z


different, simply because the corresponding values of N + 1 are rather similar in each case. On the other hand, for pH-denatured DNA the kBT/f values estimated by the two schemes differ substantially, as do also the N + 1 values. Still, the relative changes in kBT/f upon addition of EDTA are comparable in either scheme. Thus, the conclusion remains that removal of divalent cations by EDTA produces a marked increase in kBT/ f for pH-denatured DNA, but effects relatively little change in kBT/ f for native DNA. At the same time, removal of divalent cations by EDTA produces also a very substantial decrease in b for native DNA, and also a significant, though smaller, decrease for pH-denatured DNA.

Intercomparison of Model Parameters

It is interesting to compare the parameters estimated using scheme (2) above for the Rouse-Zimm model (with hydrodynamic interaction) to those obtained for the ISMF model. The solid curves in Figs. 8-11 are computed from the ISMF model employing the parameters indicated also in Table VI. The values of DO are the same for either model. The quantity Re for the ISMF model is evidently comparable to b for the Rouse-Zimm model, for which RG = b. Thus, the critical dimension for plateau behavior, although it is similar in value for either model, is subject to substantially different interpretation in the two cases. In the ISMF model, it is the radius of gyration for the entire molecule; and in the Rouse-Zimm model, it is the root-mean-squared segment extension.

Again, the relative changes in kBT/f and Re produced by removal of divalent cations (with EDTA) are qualitatively the same for the ISMF model as for the Rouse-Zimm model, provided that it is b which is compared with Re.

Longest Relaxation Time Estimated for the Free-Draining Rouse Model

The Langevin relaxation times of the various normal modes are in principle readily calculable for each of the molecules represented in Table V using the relation r, = f / ( g A l ) . Unfortunately, Al in the presence of hydrodynamic interaction can only be obtained by numerical diagonali- zation of the H A matrix. However, we may simply estimate the longest relaxation time ~2 for the free-draining Rouse model using A2 = 4 sin2(T/ 2(N + 1)) [see Eq. (22)].

Using Eq. (go), one finds for native DNA at pH 10.30 without E D T A that k B T / f = 3.1 X cm2/sec, hence f = 1.3 X dyne/cm sec. Moreover, using b2 = 8/K%, the value b = 260 nm is obtained, as found also in Table V. With these data one estimates 7 2 = 74 msec.

Following the same procedure, one finds for native DNA at pH 9.6 with E D T A that kBT/f = 2.8 X hence f = 1.45 X loe6. Here also b = 165 nm is the same as in Table V. In this case we estimate r2 = 42 msec.

458 LIN AND SCHURR

The longest molecular relaxation time50 manifested by a similar DNA, dissolved in 0.02 ionic strength phosphate buffer, pH 7.6, containing EDTA, in the second-order (electric field) autocorrelation function g!#( T) of the zero-angle depolarized scattered light was determined previously51 to be about 16-18 msec. Moreover, the dynamic viscosity relaxation time estimated for that same DNA from its intrinsic viscosity, using equation 79 of Zimm, appropriate for a free-draining Rouse model, was 23 msec. It has been shown45 that the molecular relaxation times appearing in &’(T) (which is proportional to the square-root of the relaxing part of the intensity autocorrelation function) are equal to one-half the corresponding Langevin relaxation times for a Rouse-Zimm model polymer. Thus, the Langevin times inferred from the previous zero-angle depolarized scattered light data are 32-36 msec, and that inferred45 from the intrinsic viscosity data and Zimm’s Eq. (79) for the free-draining model is 46 msec. However, a substantial (-40%) decrease in relaxation time was found previously to occur upon increasing the ionic strength from 0.02 to 0.1M. Presumably this trend would continue, if the ionic strength were further increased to l.OM, so that the relaxation times expected for DNA at pH 7.6 in 1.OM salt with EDTA might be considerably shorter, yet. In fact, we have obtained polarized scattered light data analogous to those in Figs. 8-11 that indicate a dramatic reduction in the estimated longest (free-draining) relaxation time for these DNAs in 1.OM NaCl when the pH is decreased into the neutral range. Our studies of the variation of the different parameters with pH will be reserved for a future communication. At the moment we are unable to compare the relaxation times computed here with values determined from the depolarized scattered light, or estimated from the intrinsic viscosity, under the same conditions of ionic strength. Nonetheless, the times are a t least in the correct range, and the trends in the right direc- tions.

Applicability of the Rouse-Zimm Model to Real DNA Macromolecules

It is doubtful whether Rouse or Zimm ever expected, or intended, a complete determination of the parameters of the model. Such a determination raises some nontrivial questions regarding the physical signifi- cance of g and f . A possible qualitative interpretation of these observations is the following. Whenever a substantial subsection of the polymer is ap- preciably extended, then local random motions of its constituent elements usually result in a contraction of the extended dimension. Such Brownian motion trajectories of the composite system are simply much more numerous than those that lead to further extension. The existence of a stretching force constant g is intimately related to the occurrence of rapid local diffusive motions of those chain elements comprising the subsection. Moreover, an actual contraction of the subsection dimension arises only as the net consequence of a very large number of these rapid random dis-


placements. Included among the possible random motions of the constituent elements are also those, such as crankshaft motions and rotation and bending of arcs and loops, that produce no change in the dimensions of any subsection containing those elements. The relaxation of sufficiently short wavelength fluctuations is evidently dominated by these rapid local random diffusive displacements of the chain elements, and neither subsection dimensional changes nor long-range connectivity of the polymer plays a significant role. On the other hand, relaxation of long-wavelength distortions of the polymer molecule requires concerted contraction (or extension) of many subsections, so that in this case the long-range connectivity and subsection dimensional changes predominate. The parameter b represents an intermediate fluctuation wavelength in the cross-over region. For much shorter wavelengths the local diffusive motions of chain elements (including crankshaft motions and rotation and bending of arcs and loops) dominate the relaxation, while for much longer wavelength motions the subsection dimensional changes and long-range connectivity are most important. In this scheme, then, g is simply a spring constant chosen to provide an rms extension comparable to fluctuation wavelengths in the cross-over region. In this case, clearly, b is the physically more meaningful parameter. An increase in persistence length should increase the range of fluctuation wavelengths that are relaxed predomi- nantly by the local diffusive motions, so b should increase. Thus, b is expected to provide a sensitive indication of changes in the persistence length, which in turn monitors the static flexibility of the chain. Of course, at- tractions or repulsions between elements of different subsections will act to decrease or increase, respectively, the persistence length from the value characteristic of the static flexibility alone.

The hypothetical diffusion coefficient kBT/ f is that manifested by the random local displacements when relaxing short-wavelength fluctuations. It reflects the diffusive mobility of the chain elements, which in turn depends upon both the dynamic flexibility of the chain and nonhydrodynamic contributions to the impedance of local element motions. Attractive or repulsive interactions between elements of different subsections also will act to diminish or enhance, respectively, the diffusive mobility in the same way as for unconnected molecules.59

If it is assumed that interactions between elements in different subsections are negligible at the prevailing 1.OM ionic strength, then one may conclude that divalent cations (no EDTA) act to substantially increase the persistence lengths, or decrease the static flexibilities, of both native and pH-denatured DNAs in comparison with values observed in their absence (i.e., with EDTA). For pH-denatured DNA, the divalent cations produce also a corresponding decrease in k B T / f , indicating either a substantial decrease in the dynamic flexibility, or an increase in the internal impedance of element motions, or both. However, for native DNA the change in kBT/f induced by the divalent cations is rather small, indicating only very little change in the diffusive mobility, despite the substantial change in static flexibility, as manifested in b.

460 LIN AND SCHURR

One might account for these observations in the following way. The single-strand pH-denatured DNA simply becomes more rigid, both in the equilibrium, or static, and dynamic senses, so that b increases and kBT/f decreases. If it is supposed, however, that native DNA does not bend uniformly along its length, but instead executes most of its bending at lo- calized regions, or joints, or kinksFO along the chain, then the effect of divalent cations on b implies that these bending regions either decrease in number, or become much more rigid. That would account for the decrease in static flexibility. However, if linear diffusion or hopping of these bending regions, or joints, along the chain contributes substantially to the local random diffusive displacements of the chain elements, and if this mobility of the joints themselves were either unimpaired or somewhat enhanced by the divalent cations, then the resultant dynamic flexibility might be altered only very little, as observed.

Although the explanation offered is highly speculative, the notion that in native DNA the static and dynamic flexibilities do not undergo the same change in the presence of divalent cations in probably correct. Alternative explanations would require that a relatively low concentration of divalent cations promote either a marked decrease in nonhydrodynamic internal friction, or the intercession of long-range repulsive forces between chain elements to enhance the rate of internal Brownian motion, both of which are unappealing in this instance because of the high ionic strength that should act to suppress electrical interactions of any kind.

It now seems likely that variation of the Rouse-Zimm parameters with conditions in solution may provide unique insights into the mechanisms of both static and dynamic. bending of macromolecules. That would be considerable compensation for the fact that the model itself does not physically coincide with any known real molecules.

References 1. Pecora, R. (1965) J . Chem. Phys. 43,1562-1564. 2. Pecora, R. (1968) J. Chem. Phys. 49,1032-1035. 3. Dubois-Violette, E. & deGennes, P.-G. (1967) Physics 3,181-198. 4. deGennes, P.-G. (1967) Physics 3.37-45. 5. SaitB, N. & ItB, S. (1968) J . Phys. SOC. Jpn. 25,1446-1451. 6. Silbey, R. & Deutch, J. M. (1972) J. Chem. Phys. 57,5010-5011. 7. Fujime, S. (1970) J. Phys. Soc. Jpn. 29,751-759. 8. Fijime, S., Maruyama, M. & Asakura, S. (1972) J. Mol. Biol. 68,347-359. 9. Lee, W. I. & Schurr, J. M. (1973) Chem. Phys. Lett. 23,603-607.

10. Perico, A., Piaggio, P. & Cuniberti, C. (1975) J. Chem. Phys. 62,2690-2695. 11. Reed, T. F. & Frederick, J. E. (1971) Macromolecules 4,72-79. 12. Kramer, 0. & Frederick, J. E. (1972) Macromolecules 5,69-75. 13. Huang, W. & Frederick, J. E. (1973) J. Chem. Phys. 58,4022-4023. 14. Huang, W. & Frederick, J. E. (1974) Macromolecules 7,3639. 15. King, T. A., Knox, A. & McAdam, J. D. G. (1973) Chem. Phys. Lett. 19,351-354. 16. King, T. A., Knox, A. & McAdam, J. D. G. (1974) J. Polym. Sci., Polym. Symp. 44,

17. McAdam, J. D. G. & King, T. A. (1974) Chem. Phys. 6,109-116. 18. McAdam, J. D. G. & King, T. A. (1974) Chem. Phys. Lett. 28.90-92.

195-202.


19. Tanaka, T . (1974) J. Phys. SOC. Jpn. 37,574. 20. Tagami, Y. & Pecora, R. (1969) J. Chem. Phys. 51,3293-3298. 21. Pecora, R. & Tagami, Y. (1969) J . Chem. Phys. 51,329S3305. 22. Saleh, B. & Hendrix, J. (1976) Chem. Phys. 12.25-30. 23. Fujime, S. & Ishiwata, S. (1971) J. Mol. Bid. 62,251-265. 24. Ishiwata, S. & Fujime, S. (1972) J . Mol. Bid. 68,511-522. 25. Fraser, A. B., Eisenberg, E., Kielly, W. W. & Carlson, F. D. (1975) Biochemistry 14,

26. Carlson, F. S. & Fraser, A. B. (1974) J. Mol. B id . 89,273-281. 26a. Carlson, F. D. (1975) J. Mol. Biol. 95,139. 27. Maeda, T. & Fujime, S. (1976) “Quasielastic Light-Scattering from Muscle F-Actin,”

28. Wun, K. L. & Prins, W. (1975) Biopolymers 14,111-117. 29. Jolly, D. & Eisenberg, H. (1975) Biopolymers 14,111-117. 30. Schurr, J. M. (1976) Q. Reu. Biophys. 9,109-134. (From a lecture presented a t the

31. Schmidt, R. L., Boyle, J. A. & Mayo, J. A. (1977) Biopolymers 16,317-326. 32. Schmidt, R. L., Whitehorn, M. A. & Mayo, J . A. (1977) Biopolymers 16,327-340. 33. Ramsay-Shaw, B. S. & Schmitz, K. S. (1976) Biopolymers 15,2313-2315. 34. Graff, R., Marange, C., Munch, J. P. & Candau, S. (1974) “Rayleigh Spectroscopy of

Intramolecular Relaxation of Polymeric Coils,” preprint. 35. Hendrix, J. Saleh, B., Gnadig, K. & De Maeyer, L. (1977) Polymer 18,lO-14. 36. Rouse, P. E. (1953) J . Chem. Phys. 21,1272-1280. 37. Zimm, B. H. (1956) J . Chem. Phys. 24,269-278. 38. Tanaka, T., Hocker, L. 0. & Benedek, G. B. (1973) J. Chem. Phys. 59,5151-5159. 39. King, T . A,, Knox, A. & McAdam, J . D. G. (1974) J . Polym. Sci., Polym. Symp. 44,

40. McAdam, J. D. G., King, T. A. & Knox, A. (1974) Chem. Phys. Lett. 26,64-68. 41. Munch, J. P., Candau, S., Duplessix, R., Picot, C., Herz, S. & Benoit, H. (1974) J. Phys.

42. Munch, J. P., Candau, S., Duplessix, R., Picot, C., Herz, S. & Benoit, H. (1976) J. Polym.

43. Wun, K. L. & Carlson, F. D. (1975) Macromolecules 8,190-194. 44. Allen, G., Ghosh, R., Higgins, J. S., Cotton, J . P., Farnoux, B., Jannik, G. & Weill, G.

45. Schurr, J. M. (1976) Biopolymers 16,461-464. 46. Uhlenbeck, G. E. & Ornstein, L. S. (1930) Phys. Reu. 36,823-841. 47. Eichinger, B. E. (1974) Macromolecules 5,496-505. 48. Kuhn, W. (1934) Kolloid-2. 68, 2-15. 49. Hays, J. B., Magar, M. E. & Zimm, B. H. (1969) Biopolymers 8,531-536. 50. Schurr, J . M. & Schmitz, K. S. (1973) Biopolymers 12,1021-1045. 51. Schmitz, K. S. & Schurr, J. M. (1973) Biopolymers 12,1543-1564. 52. Tanford, C. (1961) Physical Chemistry of Macromolecules, Wiley, New York, p.

53. Cahen, G. & Eisenberg, H. (1968) Biopolymers 6,1077-1100. 54. Fu, J. C. C. & Gruenwendel(l976) Biopolymers 15,265-282. 55. Lee, W. I., Schmitz, K. S., Lin, S.X. & Schurr, J. M. (1977) Biopolymers 16, 583-

56. Lee, W. I. & Schurr, J. M. (1974) Biopolymers 13,903-908. 57. Schmidt, R. L. (1973) Biopolymers 12,1427-1430. 59. Lee, W. I. & Schurr, J. M. (1976) Chem. Phys. Lett . 38,71-74. 60. Crick, F. H. C. & Klug, A. (1975) Nature 255,530-533.

2207-2214.

preprint.

Symp. Macromol. Dynamics, 5th Int. Biophys. Congr., Copenhagen, Aug. 1975.)

195-202.

(Paris) Lett . 35:L, 239-241.

Sci., Polym. Phys. Ed. 14,1097-1109.

(1976) Chem. Phys. Lett. 38,577-581.

380.

589.

Received February 28,1977 Accepted May 23,1977

Documents

Dynamic light-scattering studies of internal motions in DNA. I. Applicability of the rouse-zimm model