hydrodynamic properties of macromolecular complexes. iv. intrinsic viscosity theory, with...

Hydrodynamic Properties of Macromolecular Complexes. IV. Intrinsic Viscosity Theory, with

Applications to Once-Broken Rods and Multisubunit Proteins

JOSB GARCIA DE LA TORRE* and VICTOR A. BLOOMFIELD, Department of Biochemistry, University of Minnesota, St . Paul,

Minnesota 55108

Synopsis

We have extended our previous theories of the translational and rotational frictional properties of multisubunit complexes to calculate the intrinsic viscosity of such structures. Our theory is similar to those recently constructed by McCammon and Deutch, and by Na- kajima and Wada, in that it uses a modified hydrodynamic interaction tensor and solves the system of simultaneous interaction equations by digital computation rather than by successive approximations. However, there are some differences in the formulation and averaging of these equations. Extensive numerical comparison is made between this theory and others that are available-associated with the names of Hearst and Tagami, Abdel-Khalik and Bird, and Tsuda-using as a basis exact results for prolate ellipsoids of revolution. For large axial ratios, only our theory asymptotically approaches the correct limit; but for small axial ratios, only the Tsuda “shell-model” theory is adequate, because the other theories neglect the preponderant influence of the sphere located at the center of rotation. Intrinsic viscosities, translational frictional coefficients, and Scheraga-Mandelkern parameters, are tabulated for a large number of polygonal and polyhedral subunit structures, with up to eight elements, using both our theory and Tsuda’s. Particular application is made to hemerythrin and aspartate transcarbamylase. Finally, the viscosities and friction coefficients of once-broken rods are calculated and compared with an approximate theory by Wilenski.

INTRODUCTION

In the three first papers of this ~er ies , l -~ we have developed a general theory for calculating translational and rotational hydrodynamic properties of complex, rigid macromolecules. Similar work has been done indepen- dently by McCammon and Deutch4 and by Nakajima and Wada.5 All of these studies share the following physical basis: (1) the shape of the macromolecule is modeled as an array of spherical beads, or subunits, which in turn act as frictional centers6-9; (2) the modified hydrodynamic interaction tensorslJOJ1 are used instead of the original version by Oseen12; and (3) the system of hydrodynamic interaction equations within the framework of the Kirkwood and Riseman theory6p7 is solved for the forces by digital computation.

* Present address: Departamento de Quimica Fisica, Universidad de Extremadura, Badajoz, Spain.

Biopolymers, Vol. 17,1605-1627 (1978) 0 1978 John Wiley & Sons, Inc. 0006-3525/78/0017-1605$01.00

1606 GARCIA DE LA TORRE AND BLOOMFIELD

The problem of the zero-shear intrinsic viscosity, [TI, has already been studied by McCammon and Deutch4 and Nakajima and Wada.5 The former authors formulated a correct method to compute [T] for a given orientation of the particle, but their procedure to obtain the orientational average, namely, averaging [T] over three mutually perpendicular directions, is not fully correct. More recently, Nakajima and Wada5 have carried out the rigorous orientational average, but they introduced a small error in the derivation of [TI, as we will show. Further, their calculation of the center of frictional resistance requires successive numerical approximations. In this paper, we formulate a new version of the theory for calculating [T] which is more rigorous and simpler than those proposed before.

As a first step, we apply our method to bead models for ellipsoids, since for this geometry the hydrodynamic properties are exactly known a priori. The formalism is then used to calculate frictional (sedimentation or diffusion) coefficients and intrinsic viscosities for a number of structures akin to those commonly adopted by globular, multisubunit proteins. Two complex proteins, hemerythrin and aspartate transcarbamylase, are studied in detail. Finally, as an example of a quite asymmetric structure, we have computed the hydrodynamic properties of a once-broken rod and compared our results with those from previous works.

Throughout this paper we compare our formalism with several simpler theories which were proposed several years ago. As will be discussed later, although these theories are not as general and rigorous as that proposed here, they may be useful in certain circumstances.

THEORY

Geometry

Throughout our derivations, we use the following notation: (1) For any vector a, aT indicates a row vector, while a is a column vector. If M is a tensor, MT represent its transpose. (2) aTb denotes the scalar product of vectors a and b. (3) abT is the dyadic product, which gives a tensor (abT),p = (a),(b)p (4) Ma or aTM is the ordinary dot product, the result being a column or row vector, respectively. ( 5 ) aa is the (Y component of vector a. (6) Map is the cup component of tensor M, with C Y , ~ = x,y,z.

Let us represent the particle as an array of N spherical elements of radius C T ~ , whose Cartesian coordinates in a particle-fixed system of coordinates, xyz , centered at an arbitrary point P are (x;,yi,z;) = RT. If 0 is the center of hydrodynamic resistance, a second system of coordinates, x’y’z’, is defined at 0, with its axes parallel to those of xyz. DT = (Dr,DY,DZ) is the vector joining P and 0, and the Da’s are measured in the particle-fixed, xyz system, so that, for instance, x i = xi - D”.

To specify the orientation of the velocity field, we define a third system, xNy”z” , also centered at 0, in such a way that the unperturbed fluid flow is in the x” direction with velocity gradient g along y”. Then the unper-

MACROMOLECULAR COMPLEXES. IV 1607

turbed velocity of the solvent at the center of the element i,vp, is given by13

(1)

where ex, ey, e, are the unit vectors along the axes x ”,y”,z”, respectively, as shown in Fig. 1. rT = ( x ~ , y ~ , z ~ ) is the vector between the ith element and the center of resistance, and can be expressed in terms of Ri and D:

(2)

A is the tensor that transforms x’y’z’ coordinates into x ” y ” z ” ; its com-

vp = g(ri T ey)ex

ri = A(Ri - D)

ponents depend on the Euler angles between the two systems14:

cos+cos$ - cosesin&in+ sin+cos4 - cosOsin&os+

sinesin$

-sin+sin4 + cosOcos&os+ cos+sinO (3) cos+sin$ + cosOcos$sin+ sin+sinO

-sinOcos$ cose Under the action of the velocity field, Eq. (l), the molecule will rotate, so that the velocity of the ith element will be13

1 1 2

ui = -g(ri X e,) (4)

The unperturbed relative velocity of the ith element with respect to the solvent is then

2“

Z’

z

Fig. 1. Coordinate systems used in hydrodynamic calculations.


where

1 ui - V? = --g(y;ex + xrey)

- --gEri = --gEA(Ri - D)

2 1 1 2 2

-

Solution of the Hydrodynamic Interaction Equations

Within the framework of the Kirkwood-Riseman theory, the intrinsic viscosity is given by13

where the angular brackets denote averaging over all the possible orien- tations of the particle with respect to the viscosity field. The Fi's, forces exerted by the frictional elements on the solvent, are the solution of the hydrodynamic interaction equationslJ3:

(8)

where the prime indicates omission of the term with i = j ; {i is the Stokes Law frictional coefficient of the ith element,

{i = 6 ~ ~ l o ~ i (9) qo being the solvent viscosity and ai being the radius of the ith element. T:, is the modified Oseen tensor, which determines hydrodynamic interaction between two elements of different size1:

N

j= 1 Fi + {i C' T:jFj = {i(ui - v!), i = 1, . . . , N

T!. 1J = p..I 41 + q..Rr.R!T LJ LJ 1J (10) where

pij = ( 8 ~ ~ & j ) - l [ l + (up + r~7)/3R$]

Qij = (8~~&$)- ' [1 - (u? + u ~ ) / R $ ]

(11)

(12) In Eq. (lo), I is the third-rank unit tensor and

(13) R!. r . - r. = AR.. LJ J 1 1J

Note that the double-prime symbol is used to represent vectors or tensor


in the x”y”z” system, while unprimed notation corresponds to the particle-fixed system, xyz:

T ZJ - p . . I ZJ + q. .R. .RT ZJ ZJ LJ (15) Combining Eqs. (10) and (13)-( 15), we get

T‘!. ZJ = AT..AT ZJ (16) and, using Eqs. (5b) and (16), the set of interaction equations (8) can be rewritten as

(17) 1 C QyjFj = --gEA(Ri - D){i

i 2 where

Q‘!. LJ = 6 . . I LJ + (1 - 6..)pTr. U 1 11 (18) Equation (17) is a system of linear equations whose unknowns and coefficients are third-rank vectors and tensors, respectively. Then, the solution of Eq. (17) involves the inversion of a 3N-dimensional “supermatrix”, Q”, which is formed by three-dimensional “boxes” Q;j. Further details of how to set up and manipulate this supermatrix have been given by McCammon and D e ~ t c h . ~ Then, the solution of Eq. (17) for Fi is

By analogy, we can define Qi; tensors and a supermatrix Q in the particle- fixed Cartesian system:

Q . . ZJ = 6. . I ZJ + (1 - 6ij){iTij (20) It is easy to show that

(Q”-’)ij = A ( Q - ’ ) ~ ~ A ~ so that if we denote for simplicity

s.. ( 0 - I ) . . ZJ LJ

the final expression for Fi is

(21 1

1 N Fi = - - g C {jASijATEA(Rj - D) (23) 2 j = 1

By substitution of Eqs. (2) and (23) in Eq. (7), we get NA N N

I171 = - 2 C {j( (Ri - D)TATeye,TASi,ATEA(R, - D)) (24)

Performing the multiplication of the three-dimensional tensors and vectors in Eq. (24), we obtain

2Mqo i = l j=1


Under the physical conditions of the viscosity experiment, the particle will be tumbling around the center of hydrodynamic resistance. Therefore, the orientational average is an average over the Euler angles. Performing the integration over the intervals 13 = 0 to 7r, J / = 0 to 2?r, d = 0 to 27r with the differential (87r2)-4inB dOdJ/dd, we have

' 0 d -, if a = p = y = 6 15

(. . .) = 1 15

, if a = y, p = 6, a # p --

i f a = p , y = 6 , a # y o r a = 6 , P = y , a # P

, 0, otherwise

Thus, the final expression for the intrinsic viscosity is

The derivation we have presented so far is very similar to that previously presented by Nakajima and Wada.5 In fact, our Eq. (23) is the same as Eq. (57) in their paper. However, we feel that there is an error in their procedure of obtaining [q]. Nakajima and Wada formulated the energy dissi- pation at element i, Wi, as the product of the force Fi and relative velocity, Wi = Fi.(ui - vp); while the correct value is Wi = Fi.vY,vp being the unperturbed velocity of the solvent. See the works by Batchelor15 and Yamakawal3 for additional details on this point. As a result of this discrepancy, Nakajima and Wada obtained E instead of e, eT in Eq. (24) and the second term on the right-hand side of our Eq. (27) is missing in their final expression for [q] [Eq. (37) in Ref. 51.

Our analysis goes further to obtain explicitly the center of hydrodynamic resistance. At such a point the intrinsic viscosity must be a minimum, so that, imposing the conditions

b[q] /bDa = 0 a = x,y,z (28) upon Eq. (27) , we get, for instance,

DX {j(SS? + 6 S g + 6 S f f ) + D, i j i j

{j(Sfy + Syi")


and two other equations which can be obtained from Eq. (29) by cyclic permutation of x,y,z. Therefore, we have a system of three linear equations whose unknowns are of three particle-fixed coordinates of the center of resistance, (D x,Dy,D ").

The full inversion of the hydrodynamic interaction supermatrix Q also allows a rigorous calculation of the translational frictional coefficient. With the notation used in this paper, the interaction equations in Ref. 1 are

C QijFj = {iu (30) j

where u is the unperturbed solvent velocity. The solution of Eq. (30) for the Fi's is

Fi = ( F {jsij) u (31)

so that the translational frictional tensor, E , is given by

E = {.S.. I V (32) i j

and the translational frictional coefficient is'

f = 3/Tr[E1] (33)

(34)

The so-called shielding tensors1J6 Gi are defined by the relationship

Fi = ri Gi u so that from Eq. (31) we have

G . = {TI C {.S.. 1 1 J [J

j

and Z can be written in terms of the Gi's:

(35)

(36)

Linear Structures

When the centers of all the frictional elements lie on a straight line, the above equations are much simpler. If z is the particle axis, then Ri = (O,O,ziIT and D = (O,O,d)? Now the interaction tensors are diagonal:

(37)

(38)

( T i j ) x x = ( T i j ) y y = (8~v&i j ) - ' [ I + (US! + ~?)/3R;2]

(Tij)zz = ( 4 ~ ~ & j ) - ' [ l - (US! + ~ 7 ) / 3 R ; ; ] and the components of the supermatrix Q will be diagonal tensors, also:

( Q . . ) CJ X X = ( Q . . ) [J YY = 6 . . V + (1 - a,-){ . V

( Q i j ) z z = 6 i j + (1 - Gij){j(pij + qijzizj)

JPU .. (39)

(40)

dimensional tensors, Q”” and Qzz, defined by

( Q a a ) i j = QF (Y = X,Z

so that

Sy = (Qaa-1 ) i j (Y = x,z

and S$ = 0 when a # 0. Then, Eq. (29) gives


Therefore, the inversion of the Q is reduced to the inversion of two N-

(41)

and d can be obtained by minimization of Eq. (43) or by direct application of Eq. (29), the result being

The Z and Gi tensors are diagonal in this case, and the relationships between them and Si,, as formulated in Eqs. (31)-(361, are now valid for the scalar, acy components.

Other Theories

The computations required to evaluate Eq. (27) may be rather burden- some when a large number of subunits is considered. During the past few years, attempts have been made by several authors to get expressions for the intrinsic viscosity of particles having arbitrary shape. These expressions are usually much simpler than ours, so that it is worthwhile to include them with ours in the comparison with theoretical predictions and experimental data.

Some years ago, Hearst and Tagami17J8 applied the general theory of Kirkwood and Riseman to the calculation of [v] for particles in which the distribution of frictional elements has cylindrical symmetry. After a minor modification to include the case of beads of different size, the equations of Hearst and Tagami read

where


3 ZiZ) + XI.) Rij

p44 = 67~ [ + CC’ aiaj ( A + B 2(A+B)’ i j

x .x. (XI - x’.)’yy’. + 4(x: - x ‘ . ) ( y ; - (50) x (”+ Rij R 11 8.

Here, x‘y’z’ are the Cartesian coordinates in a system whose origin is the center of resistance, the z’ axis being the particle’s axis of cylindrical symmetry.

A similar study was later made by T s ~ d a . ’ ~ - ~ ~ Both Hearst and Tagami’s and Tsuda’s calculations are based on a version of the Kirkwood theory which, as pointed out by several authors, is not entirely correct. (See, for example, Ref. 22 and other papers cited therein.) There is no symmetry restriction in Tsuda’s derivation, so that his formula, adapted for nonidentical beads,

is supposed to be valid for any geometry. Here, the ri in (52) is the distance from bead i to the center of resistance and cosaij = ri rj1rir-j.

The main difficulty with Eqs. (45) and (52) is that they, as well as the rigorous formalism we have developed in the previous section, give [77] = 0 for the intrinsic viscosity of a single sphere. To overcome this problem, Tsuda20*21 made an application of the “shell-model” ideas of Bloomfield et aL6: the spherical surface of the beads is replaced by a shell of very small frictional elements. The resulting expression for [77] is too lengthy to be reproduced here, and can be found in Tsuda’s original work [Eqs. (62)-(64) and (58) in Ref. 21 or Eqs. (29)-(32) in Ref. 201.

A different approach has recently been introduced by Abdel-Khalik and Bird.23 Beginning with the hydrodynamic theory of Curtiss et al.,24 these authors obtained an expression that, after correction for nonidentical beads, reads


The shielding coefficients, Gi, were assigned by Abdel-Khalik and Bird23 in a somewhat empirical way, as several authors have pointed 0 ~ t . ~ 9 ~ ~ ~ ~ In a recent application of our theory for translational motion, Schmitz16 proposed that the trace of the shielding tensors Gi could be an adequate measure for the scalar shielding coefficient, as needed for Eq. (53):

1 3

Gi - Tr(Gi) (54)

At first glance, this approximation seems attractive because it would allow the calculation of [q] as a secondary result of the solution of the hydrodynamic interaction equations for the translational problem by the Gauss- Siedel iterative method, which for high N is much less time-consuming than the full inversion of the supermatrix Q.

Since Eq. (53) also fails in the limit of the single sphere, Abdel-Khalik and Bird23 proposed another kind of shell-model correction whose result for an assembly of spherical elements is

A simple inspection of Eq. (55) reveals an important deficiency: [q] does not depend on the interbead distances, Rij, which means that hydrodynamic interaction between beads has been neglected. One can imagine a number of different assemblies, each composed of N beads with the same set of ri values. All of those assemblies would be calculated to have the same intrinsic viscosity regardless of their completely different shapes. We can therefore anticipate that Eq. (55) will lead to quite erroneous results.

Different expressions for [q], from the theories we have just reviewed, imply different positions of the center of resistance to be used in each of them. We have not derived the center for each theory because most of the applications that we shall present do not require it. It is, in any case, a matter of simple algebra to minimize [q] according to Eq. (28). This procedure could also be used to obtain the center of rotational r e ~ i s t a n c e ~ > ~ ~ from Hearst’s expression for the rotational diffusion coefficient.26

APPLICATIONS

In the presentation of results that follows, the following notation will be used for the different theories:

HT: T1: T s ~ d a , l ~ - ~ l Eq. (52) T2: Al:

A2: GB:

Hearst and Tagami,17J8 Eqs. (45)-(51)

Tsuda, shell-model Eqs. (58),(62)-(64) in Ref. 21 Abdel-Khalik and Bird,23 Eq. (53), with shielding coefficients, Eq. (54). Abdel-Khalik and Bird,23 Eq. (55) This work, Eqs. (27) or (43)


Ellipsoids

Since prolate ellipsoids of revolutions are among the very few particles whose hydrodynamic properties are exactly known, they can be used as a good test of the performance of the different theories. In paper I of this series,l we found that the solution of the hydrodynamic interaction equations for the so-called volume-corrected subunit model for ellipsoids gives values for the translational frictional coefficient in agreement with the correct ones within a 2% deviation. In paper 11,2 a similar calculation for rotational diffusion showed the model to yield the correct behavior in the limit of high axial ratio.

We have calculated the intrinsic viscosity for the volume-corrected model [Fig. l(b) in paper I] using the different theoretical approaches that have been described above. In Fig. 2 the results are plotted as [ s ] / [ ~ ] ~ l l i ~ vs the axial ratio alb. [9lel1ip is the intrinsic viscosity of the actual ellipsoidal particle computed from the exact equations given by Jeffrey,27 Simha,28 and S ~ h e r a g a . ~ ~ Two regions can be distinguished in Fig. 2. For low alb, HT, T1, A l , and GB values deviate markedly from [17le1lip (in fact, these theories predict [s] = 0 for a lb = 1, the single-sphere limit), while T2 gives a rather good approximation, within a 10% deviation. Method A2 yields the correct limit at a lb = 1, but diverges strongly as a lb increases. For high a lb , the GB curve is the only one that correctly predicts the asymptotic value ( [ s ] ~ ~ l [ q ] ~ l l i ~ - 1.00 when a lb - a), and HT and T1 give a few percent discrepancy. On the other hand, T2, despite being the best for low a lb , now shows a 15% deviation. Values from method A1 are in total dis- agreement over the whole range of alb.

This situation is very similar to that previously found for rotational diffusion,2 and the reason is the same: where there is a sphere placed at the center of resistance, its contribution to the total value of the hydrody-

I I

a / b

Fig. 2. Ratio of the intrinsic viscosity of the volume-corrected prolate ellipsoidal model, [n], calculated according to the different theories, to the exact value for the ellipsoid (Refs. 27-29), [wleiiip: (-) GB, (- -1 HT, (- 0 - 0 -1 T1, (- + - + -) T2, (+ + + +) A l , (. . . .) A2 (see text).


namic property, as calculated using the theories in which the frictional forces are supposed to act at the centers of the beads, is zero. This is the case for HT, T1, Al, and GB. On the other hand, Tsuda’s20,21 shell-model method T2 fails for very elongated particles composed of many beads, such as ellipsoidal models with high alb. We first detected the failure of the shell-model equations for translational friction of ellipsoids in paper I of this series, and the anomalous asymptotic behavior of curve T2 corresponds to the equivalent situation for intrinsic viscosity.

Multisubunit Structures Multisubunit proteins are often composed of a few identical subunits

arranged in a rather regular, symmetrical pattern. Measurements of hydrodynamic properties can be useful in determining the shape of the multisubunit aggregate, or, if the shape is known from electron microscopy observations, the experimental values are of high interest to check the predictions of the theory.

Several plausible geometries, mostly regular polygons and polyhedrons, are represented in Fig. 3. Spherical elements of radius u, placed at the corners, represent the subunits in the real particle and are the frictional elements in the theory. Since all the structures are symmetric, the center of resistance coincides with the center of mass.

The intrinsic viscosities of some of the structures in Fig. 3, as obtained from the different theories, are listed in Table I. A comparison between the values of [7,4 for a given structure reveals the following facts: (1)

2

e 6 3 ‘ m 6 4 p 6 5 @ 71 1 2 3 - *66

Fig. 3. Multisubunit structures. The corners of the figures (dots) are the centers of the spherical subunits, whose radius is u. All the straight segments connecting dots, have the same length, 20, and represent the line that joins the centers of touching subunits. When the subunits in a given particle are not equivalent, labels 1,2,3, . . . are used. Note that the first digit of the structure number corresponds to the number of subunits.


TABLE I Intrinsic Viscosities, Expressed in the Form [s]iVf/N~8u~, as Obtained from the Different

Theories, and Shielding Coefficients, Eq. (54), of Some of the Structures in Fig. 3

Struc- [ q ] M / 8 N ~ o ~ ture G I G2 GB HT T1 T2 A1 A2

41 0.707 5.42 6.36 5.30 7.31 1.35 15.7 44 0.603 0.115 7.12 5.55 6.86 9.22 2.84 20.9 45 0.642 0.353 8.63 8.29 8.28 10.60 4.82 31.4 52 0.539 -0.043 9.78 13.33 9.57 12.03 3.39 27.5 61 0.357 11.01 -36.3 10.55 13.09 3.36 39.3 64 0.435 0.179 10.41 8.50 9.71 12.22 3.56 36.6 67a 0.384 13.82 11.80 13.75 16.39 3.62 39.3 71 0.375 -0.103 11.01 -36.3 10.55 12.97 3.53 40.6 72 0.420 -0.198 13.82 11.80 13.75 15.95 3.96 40.6 81 0.262 12.39 30.6 12.56 14.30 2.55 41.9

a This structure is obtained from No. 72 by withdrawal of the central subunit, the other ones being kept in place.

Methods A1 and A2 yield values which are consistently much lower and much higher, respectively, than the others. The inadequacies of method A2 have been described above, and the failure of method A1 must be due to using shielding coefficients that are derived for a translational (homo- geneous) velocity field. (2) Method HT, in some cases, and method T1, in all cases, give values which are more or less close to those obtained from method GB (within a 10% deviation). The reason for this close agreement is that the theories H T and T1, as well as GB, are based on the Kirkwood- Riseman formalism, the difference being that HT and T1 actually correspond to the first-order perturbation solution of the system of hydrodynamic interaction equations (8), and, when hydrodynamic interaction is not too strong, the theories can give reasonable estimates. As pointed out above, method H T is only strictly valid for structures having cylindrical symmetry, such as 32,45,55, and 66 in Fig. 3. (3) Method T2 yields values which are slightly higher than those from GB, and the relative difference between them, for a given structure, decreases as the number of subunits increases. This fact is closely related to our comments on the ellipsoidal model calculations: in method GB, the local effects on the spherical surface of the elements are neglected, and if a particle is modeled using only a few subunits, the diameter of the subunits is comparable to the particle’s dimensions, so that one can expect the GB values to be low in such cases.

The shielding tensors, Gi, and coefficients, G;, have a theoretical interest in themselves. When a subunit is closely surrounded by others, its contribution to the total value of a hydrodynamic property ought to be very small, due to hydrodynamic shielding. Schmitz16 proposed Eq. (54) for G; to measure the shielding effect and reasoned that strong shielding should correspond to Gi = 0, weak shielding to Gi N 1, and that in intermediate situations G; must lie between these two limits. However, he found that for strongly shielded beads, the solution of the interaction equations, coupled with the modified Oseen tensor may give negative, unphysical


values for Gi. This discrepancy was attributed to the fact that the modified tensor, Eqs. (10)-(12), is a truncated series expansion in powers of ai/Rjj; when Rij is comparable to ai or aj, further terms would be necessary. The values reported in Table I confirm Schmitz’s observations. For structures 52 (centered tetrahedron), 71 (centered hexagon), and 72 (centered octa- hedron), the shielding coefficient of the central bead, Gz, is negative. Nonetheless, we have found that the removal of the highly shielded central bead results in a very small change of the hydrodynamic property, as should physically be the case (compare 61 with 71, or 67 with 72; see f values in Table 11). We therefore conclude that the overestimation of Gz is somehow compensated by a underestimation of the Gi’s of the other beads, and the total value off or [7] is not greatly influenced by this imperfection of the theory.

Values of the translational frictional coefficients, f , and intrinsic viscosities, [7], for all the multisubunit structures in Fig. 3 are listed in Table 11. The f’s, or related translational quantities, of some structures in Fig. 3 have already been reported by several a u t h o r ~ , ~ O - ~ ~ but their values are

TABLE I1 Translational Frictional Coefficients for the Multisubunit Structures in Fig. 3, and Their Intrinsic Viscosities and Scheraga-Mandelkern Parameters Calculated from Eq. (27) (GB)

and from the Shell-Model Equation of Tsuda (T2)

Struc- ture

11 21 31 32 42 43 44 45 51 52 53 54 55 61 62 63 64 65 66 67a 71 72 81 82 83

f l6~t)ou

1.OOO 1.333 1.509 1.669 1.624 1.697 1.919 1.968 1.922 2.115 2.054 1.752 2.252 2.129 1.925 1.851 2.089 2.061 2.521 2.252 2.133 2.322 2.163 2.093 2.543

1.309 3.354 5.188 6.289 6.763 7.137 9.228

9.905 10.60

12.03 11.29

16.60 13.09 10.50

12.23 11.81 24.52 16.39 12.98 15.95 14.31 13.06 21.66

8.303

9.662

2.112 2.167 2.214 2.135 2.247 2.191 2.110 2.155 2.157 2.092 2.109 2.231 2.187 2.137 2.196 2.221 2.129 2.133 2.224 2.126 2.126 2093 2.166 2.172 2.116

0 1.676 3.425 4.304 5.073 5.390 7.120 8.638 7.897 9.784 9.122 6.748

14.78 11.01 8.631 8.071

10.42 10.33 23.02 13.82 11.01 13.83 12.38 11.45 19.42

0 1.720 1.928 1.881 2.042 1.995 1.935 2.012 2.000 1.952 1.964 2.082 2.104 2.018 2.057 2.092 2.018 2.040 2.178 2.009 2.013 1.995 2.064 2.078 2.040

a See footnote in Table I.


subject to revision because they used the approximate Kirkwood formula. Our paper is the first one that tabulates intrinsic viscosities for such structures. A convenient way to combine translational and viscosity data is the Scheraga-Mandelkern parameter /3,34 which, as formulated by McCammon and Deutch: reads

P = ~ ~ [ ~ ] ~ i ~ 3 ~ 0 / ~ i 0 0 ~ i ~ 3 f (56)

Both the T2 and GB results for [q] have been included in Table 11. Most of the subunit structures we are considering here are highly compact and are composed of only a few subunits, so that the particle's dimensions are not too much greater than the intersubunit distances. These are the typical conditions for the failure of the GB formalism, as seen in the ellipsoid viscosity calculations; while the shell-model formalisms, as T2, work quite well in these cases. Indeed, if one looks at the P values obtained using the GB method, they are found to be consistently smaller than the theoretical lower limit for a single sphere P = 2.112 X 106. Although it has recently been shown by McCammon et al.35 that /3 values lower than the sphere limit can be anticipated, it is hard to believe that this is also valid for structures as elongated as No. 55, for which GB method gives

The data in Table I1 should be useful for interpretation of experimental data on multisubunit proteins. Since the actual protein subunits are not exactly spherical and the shape of the equivalent polygon or polyhedron may be somewhat nonregular, we think that it is best to compare relative quantities, such as the Scheraga-Mandelkern parameter P or other com- binations:

= 2.104 X lo6.

or

where so and Do are the sedimentation and translational diffusion coefficients, and n and m indicate the number of monomeric subunits in the aggregate. The monomer molecular weight is M I .

Hemerythrin

Hemerythrin has an octameric form with s$,, = 6.7S, [q] = 3.64 cm3/g, U = 0.735 cm3/g; while for the monomeric subunit^$'^,^ = 1.95s and M1 = 107,000/8 = 1.34 X lo4 dalt0ns.3673~ As pointed out by Kl0tz,3~ the subunits are equivalent, so that there are only three possible arrangements: octagon (structure No. 83 in Fig. 3 and Table 11), cube (No. 81), and square antiprism (No. 82); and the octagon is the least probable because of interaction


- m

/

considerations. From the experimental data, sgsp = 3.44; while, from Eq. (57) and Table 11, sdsl = 3.70,382, and 3.14 for a cube, an antiprism, or an octagon, respectively. This result is consistent with Klotz’s assignments, but the numerical fit is not good. Combining the experimental s&w and [7] with M8 = 107,000, we get p = 2.24 X 106, which is rather higher than all the calculated values, as seen in Table 11; (3’s for cube and antiprism, calculated using method T2, are closest.

A

Aspartate Transcarbamylase (ATCase) ATCase is a good example of an enzyme composed of several subunits

arranged in quite a complicated geometry. Its physicochemical properties have been determined by Gerhart and S ~ h a c h m a n , ~ ~ and Rosenbusch and Weber39: M = 3.0 X 105, U = 0.74, s!o,w = 11.7S, [7] = 4.5 cm3/g, and /3 = 2.11 X lo6. ATCase has two catalytic subunits, each composed of three polypeptide chains, and three regulatory subunits, each having two chains. On the basis of the electron microscopic images obtained by Richards and Williams,40 Cohlberg et al.41 proposed a model in which the catalytic subunits appear as flattened trigonal prisms, parallel to each other and with an empty region in between. The three regulatory subunits look like broken cylinders, or “arms” connecting the regulatory subunits in an an- tiparallel way. The reader is referred to Fig. 8 in Ref. 41 for more details.

We used 54 spherical elements to build a bead model whose shape is very similar to the one proposed by Cohlberg et al.41 This model is shown in Fig. 4, where the dimensions are those given by Richards and Williams40 and Cohlberg et al.41 but slightly modified to pack and connect the different building blocks. From the Cartesian coordinates of the spheres and considering that the structure has a center of symmetry which must coincide

I - 9 nm I

Catalytic subunit -, -- -

- 10 nm

- Fig. 4. Bead model for ATCase according to Richards and Williams (Ref. 40) and Cohlberg

et al. (Ref. 41). Bead diameters are 2 nm, so that the overall dimensions are close to those obtained by electron microscopy. The separation between the catalytic subunits could not be obtained experimentally, and we have estimated 2 nm.


with the hydrodynamic center, we evaluated the frictional properties as usual. The values for f/67rqou and [ q ] M / 8 N ~ u ~ were transformed, using u = 1.0 nm and the experimental M and E, to give . s $ ' ~ , ~ = 12.4S, [q] = 3.1 cm3/g, @ = 2.08 X lo6 from GB formalism, and [q] = 2.6 cm3/g, /3 = 1.99 X lo6 using T2 equation.

For a structure having 54 frictional elements whose radii are, approxi- mately, one-tenth of the overall dimensions, we can expect the GB formalism to be more accurate than in the cases studied above. In fact, we have found that GB gives a better agreement with the experimental data on ATCase than T2 does. The difference between calculated (3.1 cm3/g) and observed (4.5 cm3/g) [q] is not so great if one takes into account that [77] depends, roughly speaking, upon the third power of the dimensions. Agreement would be excellent (calculated [q] = 4.5 cm3/g, s $ ' ~ , ~ = 11.0s) if u were increased by 13% to 1.13 nm, well within experimental error in its electron microscopic determination.

The values of the Scheraga-Mandelkern parameter, /3, deserve further comment. The experimental result is around the impenetrable sphere limit, 2.112 X lo6, and the theoretical values are even lower. As pointed out by McCammon et al.,37 this type of situation can be explained in terms of particle porosity. Looking at Fig. 8 in Ref. 41 or Fig. 4 in this paper, it is evident that ATCase can be regarded as a highly porous particle, since the solvent can flow rather freely in several ways: along the symmetry axis, i.e., through the holes at the center of the catalytic subunits and across the empty space between them in any direction. Although neither GB nor T2 theories are completely reliable, the general trend of p, as observed for the ATCase model and most of the structures in Table 11, seems to confirm the hypothesis of McCammon et al.37 on the porosity effect.

Once-Broken Rods

The interest of the once-broken rod model in macromolecular hydrodynamics is twofold. From the experimental point of view, there are important polymers to which the model can be applied, namely, those pre- senting a rigid helical structure but having a small region of discontinuity. Poly(y-benzyl-L-glutamate) (PBLG) containing a flexible trimethylene diamine unit is the most common example of this situation. Short frag- ments of DNA with a small denaturated loop could be another example. Once-broken rods are also present as parts of biological macrocomplexes: the T-even bacteriophages, for instance, have long fibers whose configu- ration is a long broken rod with a fixed angle, x, between the joined half- fibers.

From the theoretical point of view, the broken-rod model has two in- teresting aspects: if x is fixed, the model represents a rigid, rather asymmetric particle; and if the breaking region acts as a flexible joint, the model has only two degrees of freedom, and a detailed study of the flexibility effect is therefore possible.


Starting from the paper by Hearst and Stockmayer,42 several studies on the hydrodynamics of the once-broken rod have been made over the past 15 The most recent one by Wilemski6 seems to be also the most rigorous. Wilemski has used a simplified version of the Kirkwood-Riseman formalism to obtain explicit formulas for [q] and the sedimentation coefficient, s, as functions of a, the fraction of total rod length on one side of the break, and x, as well as the corresponding averages over a and x.

There are, however, some inconveniences in Wilemski’s paper and the previous ones. The model that was used is that depicted in Fig. 5(b), where it is shown that only one bead is involved in the joint; so that, when x is small enough, there is overlapping between beads close to the joint. This situation is hydrodynamically inconsistent, and in the x = 0 limit, it leads to singularities because the interarm distances Ri, -+ 0. In addition, there are a number of approximations in the theory presented by Wilemski, and their total effect on the results is not clear: [q] is calculated using perturbation methods rather than a full solution of the interaction equations; the center of resistance is assumed to coincide with the center of mass; the unmodified Oseen tensor is used, although it is not a good approximation when Ri, is comparable to ai + a, (near the joint); f is calculated by means of the classical Kirkwood formula; the final expressions for f and [q] are only valid for high axial ratios and so on.

The great advantage of Wilemski’s equations is their simplicity. Our theory, although having fewer approximations, does not allow for a complete study of the hydrodynamics of the model because of computer time and memory problems when the number of elements N is high. However, it can be applied to a few simple cases in order to check the accuracy of Wil- emski’s results.

The geometry that we propose for the once-broken rod is shown in Fig. 5(a). If L and b are, respectively, the total contour length and diameter of the rod, the model will have N = L/b touching, identical beads with ai = b/2. The number of beads in each arm are n and N - n, respectively, and we define a = n/N. It is important to note that when the angle x approaches to zero, the model reduces to a side-by-side aggregate of two rods of lengths aL and (1 - a)L, and for x = 0 all the RGl’s are finite.

t Z

N - I

I

X X

( a ) (b) Fig. 5. (a) Bead model for broken rod, as proposed in this paper. (b) Model used in previous

works, showing bead overlapping and short distances for a small x .


From the Cartesian coordinates as indicated in Fig. l(a), one can suc- cessively compute the Tij’s [Eq. (15)], Qij’s [Eq. (2011, Gij’s [Eq. (22)], D from the solution of Eq. (29), and finally [v] using Eq. (29). From the inversion of Q, the translational frictional coefficient, f , can be obtained by means of Eqs. (33)-(36). We have performed this calculation for N Llb = 32, which is not too small and actually falls in the range of experimentally studied samples of PBLG. The x and a were varied along the intervals 0 - 7r and 0 - 1, respectively. Other quantities of interest are the arm-length ( a ) and angular (x) averages off and [v], defined by

and 1

((- - -))a = J (- - - ) d a 0

Rather than f , we have averaged f-l because this is proportional to the measured sedimentation or diffusion coefficient. The integrals in Eqs. (60) and (61) were evaluated from tabulated values using Simpson’s rule.

It should be noted that N = 32 is high enough to make the bead’size rather small when compared with that of the whole particle. Therefore, the deficiencies of the GB formalism can be expected to disappear in this case and the results should be reliable.

For comparison, the analytical equations of Wilemski were evaluated for f and [7] using the following equations in his paper (Ref. 47): f , from (V.1) and (V.lO)-(V.l2); (f-l)a from (V.1) and (V.19)-(V.22); (f-l)x, averaging (V.l) with Simpson formula; [v] from (IV.l), (IV.lO), (A.16), and (A.18); ( [ v ] ) ~ from (IV.l) and (IV.14); and ( [ T ] ) ~ from (IV.l) and (IV. 12).

The results are plotted in Figs. 6 and 7 as values reduced with respect to those for rigid rods (x = ?r or a = 0). Subscripts GB and W are used to distinguish between GB method and Wilemski equations. Subscript R applies to the rigid rod. The advantage of this presentation in terms of ratios is that they are rather insensitive to the actual value of N , and the deviations of the broken rod with respect to the rigid one as represented in Figs. 6 and 7 for N = 32 can be considered as the upper limit of those expected for N > 32.

Individual values of [v] and f can be obtained from Figs. 6 and 7 using the following results for the rigid rod:

h.f [ ~]R,GB/NA b = 1638 h . f [ v ] ~ , ~ / N ~ b ~ = 1462

f R , G B / ~ T V ~ ~ = 8.058

f ~ , ~ / 3 * v 0 b = 7.915


Fig. 6. Values of [&B/[V]R,GB (continuous lines) and [V]W/[q]R,W (discontinuous lines) for N = 32. The numbers attached to the curves represent the values of a. The curves corresponding to the a average have not been drawn because they are very close to those for a = 0.250.

The percentage differences in intrinsic viscosity between the theories are shown in Table I11 as functions of (Y and x. Near the rigid-rod limit (low a, high x) the differences are about 11%, reflecting the discrete character of our model against Wilemski’s, which is strictly valid only for high N. As x decreases or (Y increases, the hydrodynamic interaction between different arms becomes more important, and the difference increases because our model accounts for hydrodynamic interaction more rigorously. Surpris- ingly, however, the difference decreases in the region of very low x, where it should be greatest because Wilemski’s model gives Rg’ (interarm) - in this region. This situation may be due to fortuitous cancellation of several approximations in Wilemski’s theory. Fortunately, it prevents his equations from failure. We made a similar comparison of the results for f . In the region of very low x, f $ diverges asymptotically to a, as it must. On the other hand, the difference between fkl and f& is very small (<2%) for x > 30° over the entire range of a.

Since 10-20% deviations in [7] correspond to 34% differences in particle dimensions, and since deviations are even smaller for f , we conclude that Wilemski’s equations provide a very good approach to the hydrodynamic behavior of once-broken rods, being simple and useful at the same time.

A final remark should be made. As seen in Fig. 7, the change in frictional coefficient with respect to the rigid rod is less than 1% for any a when x >


1 I I

0 60 120 180 x ( degrees)

Fig. 7. Values of fR,GB/fGB (continuous lines) and fR ,w/ fw (dotted lines) for N = 32. The numbers attached to the curves represent the values of a. As found for In], the n-averaged values are very close to the ones for a = 0.250.

TABLE I11 Percent Difference Between Intrinsic Viscosities Calculated from the Wilemski Equations

and the GB Method ( N = 32)

100 ([?Ice - [P]W)/[T]CB a x = O " 30" 60" 90' 120" 150' 180" Averagea

0.125 0 5 8 10 10 11 11 9 0.250 1 9 12 13 12 11 11 12 0.375 9 20 20 17 14 12 11 16 0.500 2 26 24 19 15 12 11 18 Averageb 4 11 13 13 12 11 11 -

a 100 (("I1GB)x - ([?]W),)/([?]CB)x. 100 (([VICS), - ([?]W)m)/([?]GB)w

90". In physical systems, one can often expect some sort of repulsion which will keep x from taking low values, so that f will be very similar to f~ regardless of a and x. In other words, the change of frictional coefficient upon a broken-unbroken transition ought not to be appreciable unless there is an attractive interaction between the two arms. Fortunately, the changes in intrinsic viscosity are greater as shown in Fig. 6.

DISCUSSION AND CONCLUSIONS

Most of the major results of this paper have been discussed in the previous sections. We merely list them here as a summary for the perhaps bewildered, and certainly fatigued, reader: correction of the Nakajima- Wada5 equation for [q] ; analytical determination of the center of resistance;


inadequacies of the theory of Abdel-Khalik and Bird23; comparison of the different theories when applied to prolate ellipsoids; the shielding effect, its numerical characterization and effect on hydrodynamic properties; comparison of our theory with Tsuda’s when applied to multisubunit structures; interpretation of low values of the Scheraga-Mandelkern parameter in terms of porosity; analysis of the Wilem~ki*~ theory for broken rods; and hydrodynamic differences between broken and rigid rods.

Additional comments seems appropriate on the two principal defects of our theory. First, numerical inversion of the hydrodynamic interaction equations is very cumbersome, so that it prohibits application of the theory to structures with very high N. Neither the Gauss-Seidel method used in papers I and I1 of this series nor the full matrix inversion employed here are practical when N > 100. However, the symmetry properties of some macromolecular complexes or the peculiar properties of the supermatrix Q could be taken into account to simplify the numerical work (note that Qij goes to zero as Rij increases, thus Q is somewhat “pseudodiagonal”).

Second, our theory fails in the cases of rotation and viscosity in the single-sphere limit or when the model is made up of only a few beads. This difficulty became apparent in paper I1 of this series and its source is well understood, although further work is necessary to overcome the problem. The velocity field is not the same at each point on the surface of a bead, nor is the distance to the center-of-resistance constant. The calculations reported in this series have been based on the “subunit” model; it now seems that the “shell” model8 has to be reconsidered (see Ref. 1 for differences between these models). This observation is confirmed by the considerable success of the shell-model equations of T ~ u d a l ~ - ~ ’ for viscosity.

Finally, we give some guides for practical use of thes theories. Table I contains valuable information on f and [q] of polygonal and polyhedral multisubunit complexes. Other structures can be studied as follows. If the structure contains only a few subunits (say N < 7), use method T2 for [ q ] . If N is higher and the arrangement is not too compact, [q] should be obtained using method GB except when extensive computer calculation is to be avoided. In such a case, use method T2 for compact particles and T1 for elongated ones. The f can be obtained by method GB or as explained in Ref. 1; the result will be the same, and the choice depends on N and the geometry. When N is high, the Kirkwood7 formula (identical beads) or the shell-model equation8 can give good results.

This research was supported in part by NSF Grant PCM 75-22728 and NIH Grant GM 17855. J.G.T. was supported by a postdoctoral fellowship from the Committee of Cultural Exchange between the USA and Spain.

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3411.

Received September 9,1977 Accepted October 21,1977

hydrodynamic properties of macromolecular complexes. iv. intrinsic viscosity theory, with...

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