[8] The translational friction coefficient of proteins

104

INTERACTIONS

[8]

f = M(1 - ~p)/Ns

(2)

or the Svedberg equation

where D is the translational diffusion coefficient, k is Boltzmann's constant, T the absolute temperature, N is Avogadro's number, p the density o f the solution, ~ the partial specific volume, M the molecular weight, and s the sedimentation coefficient o f the protein. Rather than reporting the absolute magnitude o f f , it is customary to report a friction ratiof/fo, where f0 is the friction coefficient of a sphere of equal molecular weight M and partial specific volume v. f0 is obtained from Stokes' law as

fo = (6~r~o)(3Mv/4zrN) 1/3

(3)

where */0 is the viscosity of the solvent. Oncley 1 developed a method that separated the frictional ratio into two factors

fifo = (f/fe)(fe/fo),

(4)

Protein shapes were modeled as rotational ellipsoids and one factor (fe/fo) was related to the axial ratio p. For rotational ellipsoids p = b/a, where b is the equatorial radius and a is the semiaxis o f revolution. The axial ratio can be calculated from Perrin's formulas 2 (see also Oberbeck a and Herzog et al. 4): For a prolate ellipsoid (p < 1) fe

(1 -- p2)1/2 fo = p2/a In {[1 + (1 -- p2)l/2]/p}

(5)

For an oblate ellipsoid (p > l) fe

(p2 __ 1)1/2 = p2/a arctan (p2 _ 1)l/Z

(6)

The other factor (f/fe) is attributed to hydration through f i f e = (1 + W/pV) I/a

(7)

where w is a specific hydration in grams o f water per gram of protein. Oncley 1 and several authors after him were careful in pointing out that without independent measurements of either of the factors, the other 1 j. L. Oncley, Ann. N.Y. Acad. Sci. 41, 121 (1941). 2 F. Perrin, J. Phys. Radium (Paris) [7] 7, 1 (1936). a A. Oberbeck, J. Reine Angew. Math. (Crelle's J.) 81, 62 (1876). 4 R. O. Herzog, R. Illig, and H. Kudar, Z. Phys. Chem. Abt. A 167, 329 (1934).

[8]

FRICTION COEFFICIENT OF PROTEINS

105

could not be validly computed. It has become a custom to assume a specific hydration of proteins of 0.25 H20 per gram of protein and report resulting axial ratios of equivalent rotational ellipsoids. Frequently the axial ratios resulting from such a treatment are absurd in the light of the present knowledge of protein structure. Unfortunately, the situation is often no better when viscosity data are considered at the same time. 5 The reason why these absurd axial ratios are accepted into the modern literature is that they are rarely interpreted further, i.e., they are deadend numbers. It appears, however, that if a more rigorous treatment of the problem would be available, friction coefficients could indeed provide insights into protein structure in solution. Molecules with molecular weights smaller than about 2000 have frictional ratios smaller than 1; i.e., they diffuse and sediment faster than spheres of equal M and ~. This means that these molecules do not experience the solvent as a continuum. 6-s Such molecules most likely move through solvent by a "jump and wait" mechanism for a significant fraction of their net movement, taking advantage of cavities in the liquid. 9 For proteins with molecular weights in excess of 5000, however, motion of molecules in solution follows a continuous flow pattern. These motions correspond to particles moving at very low Reynolds number (< 10 -4) in a continuous medium. An excellent treatise on the properties of parlicle motion at low Reynolds number is that of Happel and Brenner, " L o w Reynolds Number Hydrodynamics." 10 From a hydrodynamic viewpoint, proteins are rigid, impermeable, and incompressible. 11.12Thus protein atoms not exposed to solvent, as well as any solvent molecules trapped within the structfire, do not contribute to friction. The friction properties of proteins are dictated by their surface exposed to solvent. The surface of proteins can be described as a shell of beads consisting of atomic groups, such as --CH3, - - C H 2 - - , --NH2, etc. If these groups are probed by the solvent molecules, then the pearl necklace model of polymers of Kirkwood 13 and the shell model of Bloomfield et al. 14 should be applicable at this atomic resolution. 5 H. A. Scheraga and L. Mandelkern, J. Am. Chem. Soc. 75, 179 (1953). e S. N i t and W. D. Stein, J. Chem. Phys. 55, 1598 (1971). r A. Poison, J. Phys. Colloid Chem. 54, 649 (1950). s D. C. Teller and C. de Hahn, Pac. Slope Biochem. Conf. Publ. 17, 31 (1975). 9 j. H. Hildebrand and R. H. L a m o r e a u x , Proc. Natl. Acad. Sci. U.S.A. 69, 3428 (1972). 10 j. Happel and H. Brenner, " L o w Reynolds N u m b e r H y d r o d y n a m i c s , " 2nd ed. N o o r d h o f f Int., G r r n i n g e n , L e y d e n , 1973. H M. H. Klapper, Biochim. Biophys. Acta 229, 557 (1971). 12 F. M. Richards, J. Mol. Biol. 82, 1 (1974). 13 j. G. Kirkwood, Recl. Tray. Chim. Pays-Bas 68, 649 (1949), 14 V. Bloomfield, W. O. Dalton, and K. E. Van Holde, Biopolymers 5, 135 (1967).

106

INTERACTIONS

[8]

Moleculor weight (x10 -3) 20 30 40 50 60 70

lsl

. . . . . . .

/

for Spheres SIope=O 012

~' 12~-

t

,"'2

/

~II

/ / / / c A D //00 a

/,,V

~.~

,"

Observed S~ope=00TO

'21 s 0 ~

/

I

I

t

t

I

2

4

6

8

10 12 14 16 18

M 2/3

I

I

I

( X lO -2 )

FIG. 1. Relationship of sedimentation coefficient and molecular weight of monomeric proteins according to Eq. (8) and (9) (From D. C. Teller, E. Swanson, and C. de Hahn, unpublished results).

In the work of Bloomfield and co-workers, the beads were conceived as subunits of protein oligomers, viruses, and the like. However, in order to test the calculations, Bloomfield e t al. 14constructed shell models of objects with known hydrodynamic behavior and extrapolated the results to infinite numbers of beads of infinitesimal radius. Recent theoretical advances 15,16prompted in part by the work of Bloomfield e t al. 14have made it worthwhile to follow this line of research to try to determine the factors which dictate the frictional properties of proteins. II. Geometry of Oligomeric Proteins By combination of Stokes' law [Eq. (3)] with the Svedberg equation [Eq. (2)], the following relation is obtained for a series of smooth spherical molecules of differing molecular weight. 17

s~1,3 (?)1,3 ( 6 7 r T ~ ° ) - l N - 2 1 3 M 2 1 3 Up -

1 -

(8)

In water at 20°, the term preceding M 2/3 on the right-hand side of Eq. (8) has the value 0.012 Svedberg cm g-1 m012/3. This equation is not accu15j. Rotne and S. Prager, J. C h e m . Phys. 50, 4831 (1969). la H. Yamakawa, J. Chem. Phys. 53, 436 (1970). 17 K. E. Van Holde, Proteins, 3rd Ed. 1, 226 (1975).

[8]

FRICTION C O E F F I C I E N T OF PROTEINS

107

rate for proteins because of the rugosity of their surface. For monomeric proteins we found empirically TM (Fig. 1): s v l / 3 / ( 1 -- VO) -----0 . 0 1 0 M 2/3

(9)

The factor 0.010 is believed to account for the average rugosity of the surface of grossly globular proteins. In order to compute the sedimentation coefficient of an oligomeric protein one may assume that the subunits behave as if they possess the hydrodynamic behavior of the monomer, but pack together as spheres.19 Van Holde 17adapted the bead on a string model of Kirkwood 13'2°to calculate the ratio of the sedimentation coefficient of the n-mer to that of the monomer, s n / s l = n~/f,, (10) where ~ is the friction coefficient of the monomer, f , that of the n-mer, and n is the number of subunits. Following Teller and de Hahn, TM substitution of Eq. (9) into Eq. (10) yields Sn~) 1f3

1 - ~p

O'OlOMn2t3nl/3(~/fn)

(11)

where M, is the molecular weight of the n-mer. One can define Fn = nl/S(~/fn)

(12)

where F , is purely a geometric factor. The geometric factor F , can be calculated from either the Kirkwood approximation (Section III) or from the more rigorous theories of hydrodynamic interaction as presented in Sections IV and V. Table I presents the numerical values o f F , for a variety of subunit arrangements calculated from the rigorous theory? 1 With knowledge of s, ~, and M,, the subunit geometry is determined by calculating F , and choosing the closest value in Table I. Alternately, a graph such as Fig. 2 can be constructed and the geometry determined from the closest line. For those oligomeric proteins that can be separated into subunits without denaturation of the polypeptide chains (e.g., in high concentrations of CaClz), Eqs. (11) and (12) serve as a reasonably accurate means of the determination of subunit number or geometry of the protein. From the 18 D. C. Teller and C. de Hahn, Fed. Proc., Fed. Am. Soc. Exp. Biol. 34, 598 (Abstr. 2143) (1975). 19 D. C. Teller, Nature (London) 260, 729 (1976). ~o j. G. Kirkwood, J. Polym. Sci. 12, 1 (1954). ~1 D. C. Teller, E. Swanson, and C. de Hahn, Fed. Proc., Fed. Am. Soc. Exp. Biol. 36, 668 (Abstr. 2092) (1977).

108

INTERACTIONS

[8]

TABLE I GEOMETRIC FACTORS (F.) FOR VARIOUS ARRANGEMENTS OF SUBUNITS

n

Arrangement

F."

Dimer Trimer Trimer Tetramer Tetramer Tetramer

Linear Equilateral triangle Linear Square Tetrahedron

0.9449 0.8639 0.9555 0.8064 0.9259 0.9772

" F, is defined by Eq. (12) in the text and is computed using the rigorous theory of Section IV.

0

25 50 I

Moleculor Weight (xlO -3) I00 150 2 0 0 Z00

I

I

r

f

400

I

500

I

60

I

B-

/

_

/,~ ,X / / _.,-

.I "Phos. a

,~.

40

i/i.//. / Z.//.."

-', I~%

GSPDH GAI . 2 ~ , , , , ' " P h ° s "

b

,," Avid.

- "/~- Loct. 0(~

I

I

2

i

I

4

i

I

6

MZ/3(xlO-3) FIG. 2. Relationship of sedimentation coefficient and molecular weight of oligomeric proteins, for which there exists independent evidence for the geometrical arrangement of subunits. O, dimers; [~, square planar tetramers; O, tetrahedral tetramers; Theoretical lines computed from the values in Table I are , tetrahedral tetramers; - - - - - - , dimers; ..... , square planar tetramers; . . . . , linear tetramers, fl-Lact., fl-lactoglobulin; Avid, avidin, hen egg white; MDH, malic dehydrogenase; Hb, hemoglobin; Y Aid., yeast aldolase; LDH, lactic dehydrogenase; GAl.2, asparaginase from Acinetobacter; G3PDH, glyceraldehyde-3-phosphate dehydrogenase; Phos. b, phosphorylase b; Phos. a, phosphorylase a; B-Gal., ~galactosidase. (From D. C. Teller, E. Swanson, and C. de Hahn, unpublished results.)

[8]

FRICTION C O E F F I C I E N T O F PROTEINS

109

sedimentation coefficient and the Svedberg equation [Eq. (2)], the values of ~ and f~ are obtained for the monomer and oligomer. The closest value of F~ of Table I together with Eq. (12) and test integers suffices to determine n. If n is known by other experiments, then the geometry is determined by Eq. (12) and Table I. For geometries not included in Table I, the value of F~ can be calculated approximately by the formula =

--

Vij -1

(13)

n i=1 J = l

derived from the Kirkwood approximation. Here, r~j is the distance between subunits of equal size and radius a. The radius r.may be calculated from the subunit molecular weights and e as in Eq. (3). We presently do not know the general accuracy of this procedure but it should be good. Figure 2 shows the results for some proteins of known subunit arrangement using the values of Table I for the geometric factor Fn. The data in this figure are seen to agree with the geometric arrangements found by independent methods. III. Friction Properties of Proteins--Approximate Theory Kirkwood 13'z° derived a very simple formula for calculation of the translational friction coefficient of an array of beads exposed to solvent which was based on the Oseen tensor. Later Bloomfield et al. 14 extended this approach to shells of beads describing solid objects, such as viruses and proteins, that can be modeled as being composed of subunits, e.g., serum albumin at low pH. 22 Kirkwood's result for an assembly of beads of radius a was f K = 6zr~ona/

1 +

rij -1 "=.

(14)

=

where f~ is the calculated translational friction coefficient, "00 is the solvent viscosity, n is the number of beads of radius a, and rij is the distance between beads i and j. This formula was soon recognized to contain an error. 23-25 Later it was realized that it actually constituted an approximation z6 although the expected error could not be estimated. The worst discrepancy between results from Eq. (14) and a more rigorous solution in ~2 V. Bloomfield, Biochemistry 5, 684 (1966). 2z y . Ikeda, K o b a y a s h i Rigaku K e n k y u s h o Hokoku 6, 44 (1956). ~4 j. j. Erpenbeck and J. G. Kirkwood, J. Chem. Phys. 29, 909 (1958). 25 j. j. Erpenbeck and J. G. Kirkwood, J. Chem. Phys. 38, 1023 (1963). 26 H. Yamakawa and J. Yamaki, J. Chem. Phys. 58, 2049 (1973).

110

INTERACTIONS

[8]

the case of proteins is 7%. 27 For an infinitely large rigid ring of infinitesimal beads, it is known to be in error by 8.3%. zsa9 The errors always operate to underestimate f when calculated by Eq. (14). Equation (14) is an approximation to the true value of f and is quite easy to calculate. In contrast, rigorous calculations are usually prohibitively time consuming, even on fast computers. For this reason, it is useful to find empirical ways to correct the discrepancy between values observed and those predicted by Eq. (14) for known objects. Bloomfield et al. 14used a shell model of tiny beads to describe the surface of rigid impenetrable objects, such as spheres, ellipsoids, cylinders, etc., for which other and more acurate hydrodynamic calculations were available. The correspondence of the results was sufficiently accurate to encourage further investigation into the limits of applicability of Kirkwood's formula [Eq. (14)], both theoretically 1~'16and empirically. Indeed, we have found z1"27that a shell model of proteins together with Eq. (14) can be used to accurately describe the friction properties of proteins. The procedure involves the following calculations: 1. Atomic coordinates of known protein structures are used to calculate the coordinates of nonoverlapping test spheres of radius 1.4/~, that coat the protein surface. 2. The friction coefficient of this test sphere shell is computed using Eq. (14). The results of this procedure are in very good agreement with the observed friction coefficients of these globular proteins (Table II). A. Description o f the Test Sphere Shell

For the purpose of this article protein atoms shall be defined as all atoms of proteins except hydrogens. Van der Waals radii of these protein atoms are adjusted so as to include the hydrogens in their van der Waals radius. The shell of test spheres with radius 1.4 • is created by arranging around each protein atom 12 test spheres in a hexagonal close packing arrangement. The distance from the center of the protein atom to that of the test sphere is 1.4 ]~ plus the van der Waals radius of the protein atom. Test spheres that overlap with protein atoms are then eliminated. There now remains a set of test spheres some of which may overlap. The rigorous calculations described later can deal with overlapping test spheres, but the Kirkwood formula [Eq. (14)] cannot. Reduction of the set 27D. C. Teller, E. Swanson, and C. de Hahn, in preparation. z8R. Zwanzig,J. Chem. Phys. 45, 1859 (1966). 29E. Paul and R. M. Mazo,J. Chem. Phys. 51, 1102 (1969).

[8]

111

FRICTION COEFFICIENT OF PROTEINS TABLE II FRICTION COEFFICIENTS a OF PROTEINS COMPARED TO CALCULATED VALUES

Protein Ferredoxin Ribonuclease S Lysozyme Trypsin Subtilisin BPN' Carboxypeptidase A Thermolysin Hemoglobin (deoxy)

fD 3.79 3.61 4.18 4.48

f~

fa

2.62 3.80 3.71 4.36 4.48 4.91 4.57 6.06

2.626 3.623 3.592 4.176 4.382 4.679 4.77 5.913

fa/fD 0.956 0.995 0.999 0.978

fa/fs 1.002 0.953 0.%8 0.958 0.978 0.953 1.044 0.976

afo, experimental friction coefficient by diffusion; fs, experimental friction coefficient by sedimentation; fK, friction coefficient computed by Kirkwood approximation [Eq. (14)]; (From D. C. Teller, E. Swanson, and C. de Hahn, unpublished results.)

to nonoverlapping test spheres is done in three passes, in order to retain the maximum number of test spheres. During the first pass, any test sphere in contact with three or more others is eliminated, updating the count of overlap contacts after each elimination. During pass two, test spheres in contact with two others are eliminated, and, in the final pass, test spheres contacting one other are eliminated. Occasionally test spheres are placed by this procedure into cavities in the interior of the protein where they cannot contribute to the frictional resistance. Such test spheres are identified with the help of a program which displays a slice of the shell of arbitrary thickness on an oscilloscope, and allows the observer to eliminate such test spheres. This yields the final set of test sphere coordinates. B. Calculations Calculations of friction coefficients are made on test sphere shells using Eq. (14) and the coordinates of the final test sphere shell. Table II compares such calculated friction coefficients with those observed in sedimentation and diffusion experiments. Despite the limitations of this approach, it can definitely be concluded that the rugosity of the protein surface plays a major role in determining the friction coefficient. This is a determinant that is overlooked if axial ratios are derived by the procedures of Oncley 1 or later refinements, s The test spheres were chosen to have the radius ascribed to water molecules by Pauling, a° because water molecules are the smallest units to 30 L. Pauling, "The Nature of the Chemical Bond." Cornell Univ. Press, Ithaca, New York, 1960.

112

INTERACTIONS

[8]

p r o b e the protein surface. H o w e v e r , the test spheres should not be identified with w a t e r o f hydration, b e c a u s e this would result in a specific hydration of a b o u t 0.5 g w a t e r per gram o f protein, larger than is reasonable. 31 T h e y are an artifice to e x p a n d the frictional surface in a m a n n e r that compensates for the low values o f friction coefficients obtained with Kirkw o o d ' s formula applied to protein surface atoms directly. In early studies, a similar result was obtained by isomorphous expansion o f all protein coordinates. A best fit of calculated and m e a s u r e d friction coefficients was obtained, when the surface a t o m s are m o v e d on the average by 3.6/~ out f r o m the center of m a s s of the proteins, is After considering the rigorous theory, we will return to the question o f hydration. IV. Friction Properties of Proteins-Rigorous T h e o r y The basis f r o m which K i r k w o o d ' s formula [Eq. (14)] was derived is the h y d r o d y n a m i c interaction tensor for the pairwise interaction b e t w e e n point sources o f friction d e v e l o p e d b y C. W. Oseen ~2 in 1927 and applied by Burgers 33 to objects o f various geometries. Z w a n z i g et al. 34 pointed out that e v e n if the error (or approximation) introduced by K i r k w o o d 13'2° (vide s u p r a ) is avoided, for some objects negative friction coefficients will be calculated. The Oseen tensor applied to b e a d models treats the b e a d s as point sources of friction so the matrix to be inverted is not necessarily positive-definite, leading to nonphysical results. Rotne and Prager '5 and Y a m a k a w a ' 6 derived a m o d i f i e d O s e e n tensor which is always positivedefinite p r o v i d e d that S t o k e s ' law is used for the friction coefficient o f the beads. Friction coefficients calculated f r o m this modified Oseen tensor are not necessarily exact, but we k n o w from the w o r k o f Rotne and Prager 15 that f * < f t , where f * is the friction coefficient calculated from the tensor a n d f t is the exact (true) value. By true value we m e a n one that would result from an a p p r o x i m a t i o n free solution of the steady state N a v i e r - S t o k e s equation. 15 We have m a d e calculations of ellipsoids for which an e x a c t theory is known. 34a Results of the c o m p u t a t i o n s extrapolated to an infinite n u m b e r of b e a d s of infinitesimal size placed on the inside or outside surface of rotational ellipsoids find f * = f t within the precision of the calculation (---0.1%). F o r cylinders, the values o f f * agree with experimental values 35 within experimental error (---1%). Thus, at 31I. D. Kuntz, Jr. and W. Kauzmann, Adv. Protein Chem. 28, 239 (1974). 32C. W. Oseen, Nevere Methoden und Ergebnisse in der Hydrodynamik, in "Mathernatik und ihre Anwendungen in Monographien und Lehrbiichern" (E. Hilb, Ed.). Akad. Verlagsges., Leipzig, 1927. 3aj. M. Burgers, Verh. K. Ned. Akad. Wet., Afd. Natuurkd., Reeks 1 16 (4), In "Second Report on Viscosity and Plasticity," Chapt. 3, p. 113 (1938). 34R. Zwanzig, J. Kiefer, and G. H. Weiss, Proc. Natl. Acad. Sci. U.S.A. 60, 381 (1968). 34aE. Swanson, D. C. Teller, and C. de Hai~n,J. Chem. Phys. 68, 5097 (1978). 35j. F. Heiss and J. Coull, Chem. Eng. Prog. 48, 133 (1952).

[8]

FRICTION COEFFICIENT OF PROTEINS

1 13

present we have good confidence in the accuracy of friction coefficients calculated using the modified Oseen tensor. According to Oseen a2 (as cited, e.g., by M c C a m m o n and Deutch 36) for a rigid object made of an aggregate of n beads moving in a solvent that is at rest at infinite distance from the object, the force, Fi, exerted on the ith bead is given by /-t r

F~ + ~

TuF i = ~ ( U i - Vi°)

(15)

j=l

where ~ is the friction coefficient of a bead of radius a, and ~ = 6rr'o0a. 37 Ui is the velocity of the ith bead, Vi ° is the velocity which the solvent would have at the position of the ith bead if the assembly were absent, n is the number o f beads, and the prime on the summation indicates that j = i is not included. For all future considerations we take I/i° = 0. The modified Oseen tensor 1~ multiplied by the friction coefficient of a bead is given by

~Tokl= 6rr~?0a [ 1 + r°kr'J------~t+ 2a~ (~1 8 7r~qoro r ij 2 -r' ' ij T

r°h'rJ )]

(16)

r ij 2

r~jk and ro ~are the projections of vector r~ onto the cartesian axes k and l, where k and l each in turn are cartesian x, y, and z axes of the coordinate system c o m m o n to all beads (i.e., x, y, z coordinate system of atomic coordinates o f protein). In this equation rij is the vector between the center of bead i and bead j , r u is the distance between nonoverlapping beads i and j , and I is the unit tensor. When the beads overlap, then ~T~y=

1-

9ru~

32a]

I+

3rijkriJ 32ar u

(17)

The transition from Eq. (16) to Eq. (17) is continuous and smooth. 3 4 a Note that the original Oseen tensor 32 only contained the first two terms in the bracket of Eq. (16). The translation of these equations into computational forms will be presented later. Following McCammon and Deutch, z6 we now write Eq. (15) as an algebraic set o f simultaneous linear equations, = ~

(18)

where ~ i s a 3n x 3n " s u p e r m a t r i x " consisting of 3 x 3 blocks M u, where ~6 j. A. M c C a m m o n and J. M. Deutch, Biopolymers 15, 1397 (1976). 37 Note the following conventions or symbols: scalars and scalar e l e m e n t s of m a t r i c e s : f , ~, T~j; vectors a n d 3 × 3 matrices of vectors: r is, F~; tensors and 3 × 3 matrices o f scalars: T, I, M~s, f; " s u p e r m a t r i c e s " and " s u p e r v e c t o r s " c o m p o s e d of 3 × 3 matrices of vectors as scalars:

114

[8]

INTERACTIONS ~j M~j= f gT~j,i I , i =j

(19)

Subscripts i and j indicate the ith and jth bead, respectively. and ~ / a r e matrices ( " s u p e r v e c t o r s " ) with dimensions 3n × 3. ~ is given by F1 x~ FI~ FiX ~ F 1 ux FlU u F l uz F a zx F 1 zu F1 zz Fzxx .

F2xY

F2X~ . I

F , zx

Fn~U

Fn ~ /

J =

(20)

/ where the subscript indicates the bead index, the first superscript the direction of the flow and the second the direction o f the force. ~/is given by , o 0 1 0 0 0 1 1 0 0 0 1 0 ~/= (21) 0 0 1

t

~0

0

i;

Note that we have used unit velocities. The unknown terms o f Eq. (18) are the individual forces on each bead as given in Eq. (20). These forces can be c o m p u t e d either by inversion of d~, or by iterative solution o f the system of equations. Once ~ is obtained, the scalar elements are added to produce a 3 x 3 total force matrix Ft =

( FXX FUX FZX

FX~ F ~ F~

FXZ ) F~Z F~

(22)

where Fxx = F,XX + F2 xx + F3 xx + • . . + F , xx Fx~ =

FiX~

+

F z x~

+

F a x~

+

• . .

+

F,~ ~v

Fzz = F1 zz + F2 zz + F3 zz + • . . + Fn ~

[8]

FRICTION COEFFICIENT OF PROTEINS

115

In matrix algebra notation this operation is equivalent to the multiplication of ~ b y a matrix, 5~t of dimension 3 x 3n, corresponding to the transpose of Eq. (21). We now have the equality ~5~t~-1 e//= ~ t ~ = Ft

(23)

where Ft is a 3 x 3 total force matrix. In this equation, the first term assumes that we have solved the equations by inversion of ~ . The middle term assumes that the force matrix ~ is known by iteration methods (see below). The total force on a rigid object is given by

Ft

=

fO

(24)

where f is the desired friction tensor and U is the velocity matrix. Consequently,

f

=

VtU -1

(25)

Since U -1 is an identity matrix, it only converts the units. Thus numerically

f

=

Ft

(26)

The 3 × 3 friction matrix f consists of the nine scalar friction coefficients f ~ , f~", f~z . . . . , fzz. f is symmetric and positive definite. 10 All friction matrices f are similar regardless of particle orientation. To obtain the average scalar friction coefficient f * , it is necessary to average the reciprocal values of the coefficients. 2'1° Thus, we obtain the value of f * from the trace of inverse f (trace equals sum of diagonal entries). I/f* = ktr f-1

(27)

It is of interest to note, in passing, that one inverse o f the nonsquare matrix °//is ~-1 = (1/n)oCt. Conversely one inverse of the nonsquare matrix ~t is (It) -1 = (1/n)U. By virtue of these equalities and Eq. (26), we may now write fK- ' =

[ ~octJ/(-l°-//] - 1 = (1/~)°-//-lJ//(t') - 1 = (1/rt20 5~t,/~c//

(28)

Equation (28) yields an expression whose identity with Kirkwood's formula [Eq. (14)] can be demonstrated. The error (or approximation) in K i r k w o o d ' s formula appears in this derivation in the guise of a lack of uniqueness o f the inverses of 07/and 5~t. V. Calculation M e t h o d for the Rigorous T h e o r y In this section, we illustrate the mechanics of the calculation of the friction coefficient of a three bead asymmetric object with beads o f unit radius a by the theory o f Section IV. The coordinates of this object are

116

INTERACTIONS

[8]

given in Table III. Bead 3 has been raised out of the x , y plane in order to illustrate some aspects o f the calculation. By Eq. (19), the 3 × 3 block Mi~ for i = 1,j -- 1 is simply Mll =

1

0 N o w consider i = 1 , j = 3, r13 = 2V2. The entries for M13 are obtained through Eq. (16), because beads do not overlap. With k = x and l = x, ~T13 xx

= 30 [ (Xl-X3) 2 20 (1 4rza 1 + + - rz3 - 2 rz3 2 _

3

[1+4

( x z - 3)2 --; ~] rz3 / J

2

= 0.3867 and with k = x a n d l = y,

[

3a (Xl - x3)(yl ~Tla~ = _ _ 4r03 0 + rz32 8x/-f

'

-

Y3) +

2a( XxX3, yly )] 0

-r132 -

-

r132

"

+8

= 0.0703 In analogous manner all six different entries o f M13 (not 9, because Mz3 is symmetric) may be obtained. //0.3867 M13-- |0.0703 \0.0703

0.0703 0.3370 0.0497

0.0703~ 0.0497~ 0.3370/

T A B L E II1 COORDINATES OF A TRIANGULAR OBJECT OF THREE NONOVERLAPPING SPHERICAL BEADS OF RADIUS a = l Coordinates (alternative ames a) number

x

y

z

(Ix)

(2x)

(3x)

0 2

0 0

0 0

2

X/f

x/f

The alternative names have been introduced in Eq. (29) for computational ease.

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FRICTION COEFFICIENT OF PROTEINS

1 17

Equation (16) was given in this form, because if only the first two terms in the brackets are considered, one has the original Oseen tensor. 32 H o w e v e r , for computational purposes a more convenient form can be written.3e Let the x, y, and z directions be denoted ~x, Zx, and 3x, or in general kxor Zx, where k a n d / a s s u m e values 1 to 3. We then get from Eq. (16) and (17) 3a ( (kxi - kXJ)(-----tX'- IX~)) ~T~y = ~ B 8 kt q- C rift

(29)

~Ti=~~ t = 8 kl

(29a)

For i = j,

where 8kt = 1 if k = l, and 8kt = 0 otherwise. For i ¢ j and nonoverlapping beads, 2a 2 B -- 1 + 3r~----7

and

2a 2 C = 1 ---rij2

(29b)

3rij C = 32---~

(29c)

and for i ~ j, with overlapping beads, 9rij B = 1 - 32--~

and

Finally (kX i -- kXj)2

rij =

(29d)

Obviously it is indicated to compute all in 3 × 3 blocks of interacting beads i and j. For small problems such as the 3-bead right triangle, for which the entire matrix ~t can be stored in the computer m e m o r y and inverted, we use Eqs. (23) and (26) as f

=

Ft

:

~.~t~-lo~

As an example, Table IV gives the 9 × 9 matrix for the 3-bead right triangle described above. Note that the left and right upper corner 3 × 3 submatrices of the matrix M are the matrices Mll and MI3 given above. In this case, matrix inversion by Gauss elimination is used and M-1 is given in Table IVb. Table IVc gives the forces on each subunit [Eq. (20] in each direction. Note that the largest force is directed along the velocity direction but there are finite components in other directions. Table IVd is the friction matrix ! [Eq. (26)]. Inversion of this matrix gives the nine entries of f-1 (Table IVe), where f-1 is necessarily symmetric. We will later show

118

[8]

INTERACTIONS

¢~ ¢$ ¢:/

V-

.J

.¢ z ©

¢:/ ~

¢/

V-

i

I

i

I

I

I

I

I

I

~2 i

b~

f

z~ I

.¢

o z

I

I.J

7

I

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FRICTION COEFFICIENT OF PROTEINS

I

I

II

T 6-

1-

gh

C ~ -r"

E t. O r~

E

o"

~o,.E

119

120

INTERACTIONS

I8]

that the off-diagonal elements can be eliminated by rotation of the object. The average friction coefficient is calculated from Eq. (23) as l/f* = 31(3.4288 + 3.2719 + 3.2719) or

f* = 0.30082 For large problems, inversion of M may be impractical, and an iterative solution to the system of linear equations corresponding to Eq. (18) is the better method. The Seidel method z8 is well suited since convergence is relatively rapid and computer memory requirements are minimal. For each step of the iteration, the necessary IVIu are recomputed (rather than storing M), and only the progressively refined entries of ~ are stored. Required total storage for arrays is 12n, where n is the number of beads (9n for storage of ~ , 3n for coordinates). The largest problem we have solved is the test sphere set of hemoglobin, which involves about 3300 simultaneous equations. Each iteration by the Seidel method required eight hours of time on a PDP-12 computer with Floating Point Processor. Fifteen iterations were required to estimate f* to 0.16% uncertainty. In order to begin the iteration, we use the following procedure: The 3 x 3 matrices Muare summed across the rows o f ~ to yield a 3n x 3 matrix of n blocks of 3 x 3 submatrices. These 3 x 3 submatrices are individually inverted to yield an initial estimate of the 3 x 3 blocks of the force vector ~. As an example consider the interaction of bead 1 with all the others: 2

j=l

MIj

:

[ F I g/g

F1 ~/y

F1//x /

\FxZ.~

FlZU

FI= /

(30)

While a Seidel iteration applied to the system of equations represented by Eq. (18) will converge, regardless of the starting point for ~ , provided the modified Oseen tensor [Eqs. (16) and (17)] is used, it may take an excessively long time. The above procedure [Eq. (30)] gives a much better starting point for the iteration, than choosing (for example) all entries for equal to unity. VI. Location of the Principal Axes of Translation Because the orientation of particles in an external Cartesian coordinate system is arbitrary, in general, the off-diagonal elements in f will freaa j. Westlake, "A Handbook of Numerical Matrix Inversion and Solution of Linear Equations," p. 55ff. Robert E. Krieger, Huntington, 1975.

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121

FRICTION COEFFICIENT OF PROTEINS

quently be finite. If an object, such as the 3-bead right triangle, is dropped into a solution parallel to any of the three coordinate axes used, it will move with an angle to the axis because of the off-diagonal resistances. According to Happel and Brenner 1° (p. 167), because of the symmetry of t, such a particle must possess at least three mutually perpendicular axes, fixed in the particle, such that if it is translating without rotation parallel to one of them, it will experience a force only in this direction; that is, if we can find these axes, there will be no lateral forces and the off-diagonal elements of f should be zero. Happel and BrenneP ° call these axes the "principal axes of translation." If we know the scalar friction coefficients fk, (k, l = x , y , z ) in our external Cartesian coordinates, then the principal axes of translation can be determined from the eigenvectors. The fkt coefficients corresponding to this special coordinate system are termed the "principal friction coefficients of translation. ''1° Here we present the method used when we know the matrix t in the arbitrary coordinate system. First we solve the cubic equation corresponding to if== _ f det | f ' = \

if=

fx~ f"-f fz.

F= p=

= 0

(31)

if=_

which has three real, positive roots, corresponding to the eigenvaluesf~. For the friction matrix t, of Table IVd, for example, the values are f= = 0.27995, fu = 0.30437, and f f = 0.32102 as presented in Table V. The average of the reciprocals of these values is the same as I/f* previously obtained. In order to find the principal translation axes, we must find the eigenvectors. To find these, we must solve three sets of three simultaneous equations for the unknown eigenvector values e k=, e k~, and e k=, ek:~(fx= _ f k ) ekXfyx ekXf ~

+ ekUfx~, + ek~,(f~,y _ f k ) + ek~f ~

+ e k ~ f xz + ek~fu= + e k ~ ( f ~ -- f ~ )

= 0 = 0 = O,

(32)

where k = x,y,z indicates the three directions of motion. Table V gives the eigenvectors associated with t of Table IVd, normalized to unit length. Solution of Eq. (32) does not identify which of the eigenvectors correspond to the directions x , y , and z. The correspondence is obtained by inspection of the components of the eigenvectors, the largest positive component indicating the direction of the new axis. In order to align the principal axes of translation, rigidly fixed in the particle, with the external coordinate system of the laboratory the object with coordinates x, y, and z is rotated through appropriately determined

122

INTERACTIONS

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TABLE V EIGENVALUES AND EIGENVECTORS OF THE FRICTION MATRIX OF THE THREE-BEAD OBJECT OF TABLE l I I '~

Eigenvector k

Eigenvalue

ek~

ek~

ekz

x y z

0.27995 0.30437 0.32102

0.70711 -0.70711 0.00000

0.50000 0.50000 -0.70711

0.50000 0.50000 0.70711

Eulerian angles of the principal axes of translation: tb = 135°, 0 = 45°, ~ = 180°. E u l e r i a n angles ~, 0, a n d 0, yielding n e w c o o r d i n a t e s x ' , y ' , and z'.39 T h e E u l e r i a n angles r e p r e s e n t (1) a rotation b y ~b a b o u t z axis, (2) a r o t a t i o n b y 0 a b o u t the n e w x axis, and (3) a r o t a t i o n b y ~ a b o u t the n e w z axis c r e a t e d in step 2. T h e t w o s y s t e m s are c o n n e c t e d b y the p r o d u c t o f the three rotation matrices, x' y' z'

=

/COStb si 0~

-- sin~b cos(b 0

0 1

0 0

0 cos0 sin0

- n0 cos0 /

tsi0~

in0 °t(x ) 0y cos~0

1

Z

(33)

or x' |cosg~sint~ + cos0cos~sin~b i = \ sin0sin~

-sint~sin~ + cos0costbcos~ sin0cos~

-sin0costbJ|y J cos0 / \ z /

(34)

B y e q u a t i n g the unit e i g e n v e c t o r s with the e l e m e n t s o f the r o t a t i o n m a t r i x R o f Eq. (34), the v a l u e s o f 0, (b, a n d ~ m a y b e d e d u c e d , since e zz = cos0, e z~ = sin0cosd/, e uz = - s i n 0 c o s t b , etc., as illustrated in T a b l e V. T o rotate the particle, it is not n e c e s s a r y to d e d u c e the angles. T h e c o o r d i n a t e s o f the particle r o t a t e d s u c h t h a t the f o r m e r x, y, and z a x e s are parallel to the principal translation a x e s are o b t a i n e d b y multiplying the r o t a t i o n m a t r i x R t i m e s the particle c o o r d i n a t e s (x, y , z) as i n d i c a t e d b y E q . (34) a n d Table VI. R o u n d i n g e r r o r s in c o m p u t i n g e i g e n v a l u e s a n d e i g e n v e c t o r s m a y bec o m e significant, so the e i g e n v e c t o r m a t r i x (Table V) should be c h e c k e d to e n s u r e that it is a p r o p e r r o t a t i o n m a t r i x . A n o r t h o g o n a l r o t a t i o n m a t r i x a9 We have chosen to rotate the particle coordinates in a right-handed manner, rather than to rotate the external coordinate system in a right-handed manner.

[8]

123

F R I C T I O N C O E F F I C I E N T OF P R O T E I N S

TABLE VI COORDINATES OF THE THREE-BEAD OBJECT OF TABLE III W1TH THE AXES OF THE NEW COORDINATE SYSTEM PARALLEL TO THE PRINCIPAL TRANSLATION AXES OF THE PARTICLE a

rdinates number

x'

y'

Z'

0

0

0

,~

-~/~

o 0

2X/2

0

T h e s e coordinates were obtained by rotating the object according to Eq. (34) through the angles of 4, 0, and ~b from Table V.

has a determinant equal to + l, and the product of the transpose, R t, and the matrix R is an identity matrix: RtR

= I

or

R = ( R t ) -1

(35)

The second equality of Eq. (35) can be used to smooth out any calculation inaccuracies by averaging the elements of R and (R0 -1. Table VI gives the coordinates of the reoriented three-bead particle that we have been considering. In the initial external orientation that we considered (Table III), it would tend to move off-axis and possibly tumble if dropped into a fluid along the axis x, y, and z since off-diagonal elements appear in the friction matrix f (Table IVd). If dropped into a fluid parallel to x', y', or z' coordinates of Table VI, however, it would follow a stable translational motion along the chosen axis with no tumbling, provided it is not perturbed by Brownian motion. In considering model studies of asymmetric molecules, these considerations may become important. VII. Results, Conclusions, and Future Directions Using the rigorous theory on protein atoms alone yields friction coefficients which are less than the observed values. Calculations based on the complete test sphere shell (section III) gives values greater than observed for proteins. Placing test spheres only on charged groups and computing the friction coefficients based on these test spheres and surface protein atoms yields values essentially in agreement with experimental values. Considering that a rigorous theory has been used, and that charged groups are well recognized to be hydrated, it appears legitimate to equate test spheres in this case with water of hydration. Thus, the frictional behavior of proteins is determined by several factors: overall di-

124

INTERACTIONS

[8]

mensions, rugosity of the surface, and hydration of charged (and perhaps some polar) groups on the surface. If the rugosity of globular proteins would on the average be similar, one could expect a relationship between the friction coefficient (or sedimentation or diffusion coefficient) and the surface area of proteins accessible to solvent. Indeed, for those monomeric proteins whose accessible surface area has been calculated 4°-~ and for which hydrodynamic data are available, the following empirical equation holds quite well. s~1/3/(1 - Up) = kA~

(36)

As is the accessible surface area in ~z, and k = 8.9 -+ 0.7 × 10-4 Svedberg cm g-l/3/~-2. Thus, hydrodynamic data can be used to estimate accessible surface area. One problem area that has not been approached to date is the coupling of translation and rotation of asymmetric objects such as proteins. Consider an object such as a propeller translating through a solvent. It should spin, even if it moves along the principal axes of translation, one of which in this case is the rotational symmetry axis. Translation and rotation are coupled. As required by the properties of principal axis of translation, the object will, however, not tend to leave the direction of translation. We presently do not know the magnitude of translation-rotation coupling for real molecules. Another area which should be explored is the rotational frictional behavior of proteins. Are the conclusions we have reached for translational friction valid for rotation as well? Hopefully, the answers to these questions and related ones will become available in the near future. A ck n o wled g men t s This work was supported by United States Public Health Service Grant GM-13401, and in part by AM-02456.

40 B. 41 A. 42 C. 43 C.

Lee and F. M. Richards, J. Mol. Biol. 55, 379 (1971). Shrake and J. A. Rupley, J. Mol. Biol. 79, 351 (1973). Chothia, Nature (London) 254, 304 (1975). Chothia, J. Mol. Biol. 105, 1 (1976).