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Radius of gyration of multisubunit macromolecules: application to myosin heads, myosin rod and whole myosin
Radius of gyration of multis.ubunit macromolecules: apphcatmn to myosin heads, myosin rod and whole myosin Juan A. Soivez, Angel Iniesta and Jose Garcifi de la Torre* Departamento de Quimica Fisica, Facultad de Ciencias, Quimicas y Matematicas, Universidad de Murcia, 30001 Murcia, Spain (Received 30 October 1986; revised 3 April 1987) The radius of gyration, R, of macromolecules composed by subunits of arbitrary shape can be easily obtained flora the radii of gyration of the subunits and the position of their centres. The procedure can be applied also to obtain R for particles having cavities, as illustrated for a model of myosin head. B'hen the macromolecule presents segmental flexibility, R must be calculated as a conformational average. As an example of the procedure, we analyse the experimental values of R for whole myosin and myosin rod. Our analysis supports strongly the existence of a small but appreciable flexibility in the myosin rod. Keywords: Radius of gyration; myosin; conformation; flexibility
Introduction In the low-angle region, the techniques of electromagnetic scattering (light, X-rays or neutrons) by macromolecules in solution are particularly easy to interpret. The dependence of scattered intensity on scattering angle is in part determined by the radius of gyration, R, which is a measure of the global size of the macromolecule. When combined with other physical data like molecular weight andpartial specific volume, it can give information about the macromolecular shape. Well-known formulas for R are available for simple, symmetrical shapes such as ellipsoids or cylinders. However, in the field of biological macromolecules, one is often facing rather complex, irregular structures that cannot be described adequately by such simple models. This situation has motivated much theoretical work aimed at the interpretation of hydrodynamic properties (sedimentation, diffusion, viscosity) of complex biological macromolecules 1. R is a practical alternative to hydrodynamic measurements; indeed, its sensitivity to size and shape should be comparable to that of the sedimentation or translational diffusion coefficients. For macromolecules of arbitrary shape, R can be calculated from a model in which point-like or in which small, spherical elements are distributed throughout. Such is the kind of model needed to compute the angular variation of the scattering intensity using Debye's formula. This approach can be nearly exact if the model contains a very high number of scattering elements, so that the shape of the macromolecule is accurately described, but at the cost of much computational effort. Models composed of spherical elements have the advantage of allowing the joint interpretation of hydrodynamic and X-ray results; interesting illustrations of joint modelling are available in recent literature 2,3. * To whom correspondence should be sent. 0141-8130/88/010039-05503.00 ~) 1988 Butterworth & Co. (Publishers) Ltd
In this paper, we explore the possibility of modelling
the macromolecule for the calculation of R, using a moderate number of building blocks of known radius of gyration and various shapes, like spheres, cylinders or ellipsoids. In some instances, these blocks may actually correspond to functionally significant subunits. Then, from the experimental R of the whole macromolecule, and the R (or dimensions) of the subunits, one can infer plausible structures for the spatial arrangements of the subunits. Concretely, we consider two alternative formulae for R of multisubunit structures. The formulae can be also applied to particles having cavities, as illustrated here for a model of myosin head. Another particular situation is that of segmentally flexible macromolecules, for which R is a conformational average. A typical example, considered in this paper, is whole myosin.
Theory The particle under study is assumed to be composed of n subunits whose radii of gyration are Ri, i = 1. . . . . n. If fi is the mass fraction of subunit i and ei is the distance from the centre of mass of the subunit to the centre of mass of the particle, R can be calculated as R 2= ~ fi(c~+R 2)
(1)
i=l
There is an alternative formula which does not require the previous calculation of the centre of mass of the particle, but only the distances r o between the centres of all the i,i-pairs of subunits:
R 2=
f, R2+E 2 f , flr2j i=l
(2)
i
Equations (1) and (2) are not reported in a recent monograph on X-ray scattering 4. After deriving these
Int. J. Biol. Macromol., 1988, Vol 10, F e b r u a r y
39
Radius of gyration of multisubunit macromolecules: J. A. Solvez et al. Z
Applications to myosin proteins Myosin is a typical example of a macromolecule in which rigid subunits can be clearly differentiated. Furthermore, the joints at which the myosin subunits are connected are usually supposed to be flexible. The segmental flexibility of myosin plays an essential role in muscle contraction 9. A schematic model for the myosin molecule is depicted in Figure 1. Two myosin heads (subfragment S1) are connected to the myosin rod. The rod may not be straight; instead, it is assumed to contain a small, 'soft' region which can act as a flexible joint (joint J1 in Figure 1), and divides the rod into two subfragments, $2 and LMM. The heads are also flexibly joined to the $2 subfragment of the rod by means of the joint denoted J2 in
s1
~y x r"
$2
Figure 1.
Figure 1 Model for whole myosin in an instantaneous conformation. Subfragments S1, $2 and light meromyosin (LMM) are indicated. The orientation of the heads are specified by polar angles 01, ~'l (shown) and 02, ~2 (not shown), referred to a system of Cartesian coordinates such that the rod ($2 plus LMM) lies on the y, z plane. The conformation of the rod is described by angle ~t equations we were kindly informed by a referee that they were presented in a review article on ribosome structure by Damaschun et al. 5, quoting previous, lessknown works of their group 6-s. In the Appendix of this paper we give the derivation of equations (1) and (2). The procedure based on equation (1) would be as follows. From the geometry of the particle, the position vectors of the centres of the subunits with respect to an arbitrary origin are evaluated first. Next, the centre of the particle, C, is calculated using the subunit volumes and densities. Then, the position vectors of the subunits are referred to C. Finally, these positions and the individual radii are substituted into equation (1) to find R. The alternative procedure based on equation (2) is more appealing because the intercentre distances can be evaluated directly from the geometry, and the centre of the whole particle is not needed. However, the calculation from equation (2) requires a number of operations proportional to n 2. In special cases in which n is very large, computing time may be appreciably longer than in the other procedure, for which the number of operations is proportional to n. For a particle composed by only two subunits, equation (2) takes a simple form:
R 2 = f l R2 q-f2R 2 +flf2r22
Int. J. Biol. Macromol., 1988, Vol. 10, February
Myosin heads Garrigos et al. 11 have proposed that chymotryptic, light-chain-2(LC2)-free S1 has the shape of a prolate ellipsoid with a hole near one of its extremities. This hole would be the place where LC2 is located. For simplicity, it was assumed that the hole is also ellipsoidal, and has the same axial ratio as the main ellipsoid. The model is depicted in Figure 2, where its geometrical parameters are defined. With p = 2 . 3 , a * = 6 . 0 n m , d = 1.5 nm and ~ = 4 . 0 n m , the theoretical results for a number of hydrodynamic properties and the angular dependence of X-ray scattering were in excellent agreement with experimental data 11. Equation (3) can be used to obtain the radius of gyration of a particle having a hole. If RF, R and RH are the radii of gyration of the filled particle, hollow particle and hole, respectively, and VF, V and VH are their volumes, with V= VF--VH, it follows from equation (3) that R2
2 2 = RF/f-R~LfH/f-rl2 zfH
(4)
a ~
,=
r
- -i
\
--
-
(3)
which is known as the Parallel Axes Theorem of solution scattering. Equations (1), (2) and (3) tell us that R depends on the dimensions of the subunits and the position of their centres but, if the latter are fixed, R does not depend on their orientation. Let us suppose, for instance, a dimer composed of two rods of length L and diameter d joined side-to-side with their axes parallel. For this model, r12=d and RE=R2=L2/12 +d2/8 so that RE=L2/12+ 3d2/8. Now, if one of the rods is arbitrarily rotated around its centre, so that the dimer takes a cross-like shape, R is unchanged because r12 remains constant.
40
Hydrodynamic properties of myosin fragments and whole myosin have been extensively used to learn about myosin structure and flexibility (see for instance, Ref. 10). In this paper, we shown how data on the radii of gyration can be useful for the same purpose.
,8
2
Figure 2 Model for LC2-free Sl. a* and b* are the semimajor and semiminor axes of the ellipsoid, respectively. The axial ratio is p = a*/b*. The ellipsoidal hole is centred at a distance d from the centre of the ellipsoid. Its major and minor semiaxes are ct and fl, respectively, with the same axial ratio, p=ct/fl, as the main ellipsoid
Radius of gyration of multisubunit macromolecules: J. A. Solvez et al.
50
40
ely-
30
20 0
I 30
I 60
I 90
I 120
I 150
180
0 (A), a (B, C, D) (degrees)
Figure 3
Radius of gyration, R, of whole myosin. Curve A: R
versus 0t=02 for Ls2=LLMM=72nm, Ct=0 and $t=~b2=0. Curve B: R versus ot for LS2=LLMM=72nm, 01=02=30 °, ~b~=~k2=0. Curve C: variation of the value of R, conformationally averaged over the orientation of the heads, with the fixed angle ~t. Ls2 = LLMM= 72 rim. Curve D: The same as for curve C with Ls2 =43 nm, LLMi = 113 nm
where f = V/Vv and fu = Vu/VF are the volume fractions (equal to mass fractions since we are assuming uniform density), and the distance between the centres of the hole and hollow particle is r12 = d/f. Using expressions for the volume and radius of gyration of ellipsoids, we obtain, in the notation of Figure 2, R 2 = [(p2 + 2)/5](b.S _ fls)/(b.a _flu) _ d2b ,afla/(b,a _ fl3)2
(5)
Due to an arithmetic error, equation (8) in Ref. 11 differs slightly from the correct expression, equation (5). Repeating the calculations we have seen that the effect from this error on the previous analysis of the shape of LC2-free S1 is very small. For the parameters given above, equation (5) leads to R = 3.3 nm, which coincides with the experimental value ~2. It must be pointed out that the size and shape of myosin heads are objects of much recent controversy t°-16. Our purpose here was only to illustrate the applicability of calculations of R in the case of a specific model.
We note that the actual shape of S 1 is not relevant in the calculations for the whole molecule. The length of rodlike subfragments $2 and L M M are not known accurately. Two rather extreme choices are Ls2 =/-~MM = 72 nm and Ls2 = 43 nm, and LLMM= 113 nm. The diameter of $2 and L M M is taken as 1.7 nm, which is compatible with experimental data for the molecular weight of the rod, and slightly smaller than the hydrodynamic radius 7. An instantaneous conformation of myosin is specified by the polar angle of the heads, 01, ~k1, 02, ~k2, and the angle in the rod, ct. These angles are shown in Figure 1. The experimental radius of gyration of the myosin molecule, R~xp= 45-47 nm (Refs 17 and 18), corresponds to a conformational average over these angles, Thus it seems possible to obtain information on myosin flexibility by comparing R~xpwith calculated values. However, the dependence of R on the conformation of the rod is much more pronounced, as shown by curve B in Figure 3. It is instructive to calculate R firstly for instantaneous conformations of the model, i.e. for fixed angle. Some results are presented in Figure 3. Curve A is for a straight rod (ct=0) with Ls2=LLMM=72nm and varying orientation of the heads. It shows that R depends very weakly on the conformation of the heads. In fact, the change in R between a fully extended conformation (0 t = 02'-'30 °) and a retracted one ( 0 1 = 0 2 2 1 5 0 °) is about
10%. According to current views of myosin flexibility, the mobility of the heads is rather extensive, so that J1 is nearly a universal joint. Therefore, the average over 0~, ~bl, 02, ~k2 can be a simple, unweighted one. We note that if the flexibility of joint J1 were somewhat restricted, the error introduced by the small influence of the heads' conformation on R would be small. Then, we decided to carry out the average by numerical simulation to avoid cumbersome algebra. For fixed angle, ~, in the rod, about 400 allowed conformations of the heads were generated picking at random 01 and 02 with a sine distribution, and ff~ and ~k2 with a uniform distribution. A conformation is allowed if there is no overlap between the four subunits. This procedure was repeated for varying values of ct, and the results are presented in curves C and D of Figure 3. In a simple approach 1° the flexibility of the rod is characterized by a certain equivalent angle, Ct~q,such that the value of a given radius calculated for ~q, and averaged over the random orientation of the heads, coincides with the experimental datum. Thus, Ct~qcan be regarded as some mean bending angle. The result would be Ct,q- 0 ° if the rod was rigid and straight. However, values of ~,q of up to 60 ° have been found in the analysis of hydrodynamic properties 1°. In the present study, ~q is estimated interpolating R~xpin curves C and D. We find atoq---20°-45 ° with curve C for LS2= L L M M = 72 nm, and ~ q - 600-70 ° with curve D for Ls2 = 43 nm and LLMM= 113nm. These values, as well as those from hydrodynamic properties, are significantly different from 0 ° and therefore confirm the existence of a moderate flexibility of the myosin rod.
Whole myosin In the calculations for whole myosin, subfragment S1 is considered as a particle with a volume of 142nm a, and a radius of gyration of 3.3 nm, as discussed in the previous section. Its centre of mass is located at a distance of 8 nm from joint J1. This distance is obtained adding an excess of 2 nm to the long semiaxis of S1, which is about 6 nm.
Myosin rod The flexibility of the myosin rod can be analysed with more detail in the rod itself (headless myosin) since it has only two subunits and a single joint. The proper model is a broken, or hinged rod having two rodlike subunits of length L1 ( = Ls2) and L 2 (-----L L M M ) . The radius of gyration
Int. J. Biol. Macromol., 1988, Vol 10, February
41
Radius of oyration of multisubunit macromolecules: J. A. Solvez et al. of the broken rod can be obtained from equations (3) with
fl=LJL=7, where L = L t + L 2 is the total length, f 2 = 1 - 7 , R2=L~/12, (d2/8,~L2/12), R2=L2/12, and r22 = L2/4 + L2/4 + L 1L2 c o s a/2. The result is (R2)=(L2/2)[1/6+72(1-7)2((cos~¢-l)] (6) where ( - - - - ) denote conformational average. We note that only conformation-dependent quantity is cos ~. A formula equivalent to equation (6) was derived by Garcia Molina and Garcia de la Torre 19 though in a more indirect way. Hvidt et al. 2° have measured the radius of gyration of the myosin rod, obtaining values in the range 37-41 nm with a negligible temperature dependence. From equation (6) we find (cos a) = 0.45-0.93 if Lsz = LLMM = 72nm and (cos a) < 0.29 if L s 2 = 4 3 n m and LLMM= 113 nm. These values deviate to a lesser or greater extent from the rigid limit (cos a) = 1, and is again a proof of the flexibility of myosin rod. For the first choice, the rod is quite stiff (Ct~q=20°~50°) while for the second one its behaviour is almost completely flexible (~q> 75°). We recall that these two choices are extremes. As R increases with increasing L and decreasing 7, the first choice corresponds to lowest R, while for the second one has the highest value. Thus, the parameters obtained for the two choices of length should be regarded as lower and upper bounds for the true values of myosin. Flexibility of the joint in the rod can be quantified by means of an elastic potential V(a) =½ k~ 2
(7)
where k¢ is the elastic constant. Then, the conformational average needed for R is (Ref. 19)
Acknowledgement This work was supported by grant 561/84 from the Comision Asesora de Investigacion Cienfifica Tbcnica.
References 1 2 3 4 5 6 7 8 9 I0 11 12 13 14 15 16 17 18 19 20
=
fo o
d ~ cos a sin ~ e - q "
/fo o
dct sin ct e-°~2
(8)
Appendix
with
Q=k./2kbT
(9)
where kbTis Boltzmann factor. In Ref. 19, the upper limits of the integrals in equation (8) where rt instead of ~ . However, the influence of this difference in the resulting values of (cos ~) is negligible except for very low Q. The plot of (cos ~) versus Q is practically the same as that in Fioure 1 of Ref. 19. Interpolating the (cos ct) values for myosin rod obtained from equation (6), we find Q = 0.5-8 if Ls2 = LLMM= 72 nm and Q < 0.3 if Ls2 = 43 nm and LLMM= 113 nm. The agreement between the values of the flexibility parameter, ~q, of myosin and myosin rod is better for L S 2 = LLM M ~---72 nm than for Ls2 = 43 nm, LLMM= 113 rim. For the first set of lengths the myosin rod is found to be rather stiff, but not straight, while for the other set the flexibility is perhaps too high. We should note that our analysis is based on a single property, namely, the radius of gyration, which depends on several geometrical and conformational parameters of the molecule, and therefore we cannot ignore the possibility of over-interpretation of our model. Nonetheless, different choices of those parameters lead to coincident qualitative conclusions: our analysis of the radius of gyration of myosin and myosin rod confirms the moderate but appreciable flexibility of the rod.
42
Garci/l de la Torre, J. and Bloomfield, V. A. Q. Rev. Biophys. 1981, 14, 81 Perkins, S. J. Biochem. J. 1985, 228, 13 Perkins, S. J. and Sire, R. B. Eur. J. Biochem. 1986, 157, 155 Glatter, O. and Kratky, O. (Eds) 'Small Angle X-ray Scattering', Academic, New York, 1982 Damaschun, G., Muller, J. J. and Bielka, H. Meth. Enzymol. 1979, 59, 706 Damaschun, G., Muller, J. J. and Purschel, H. V. Acta Biol. Med. Ger. 1968, 20, 379 Damaschun, G., Fichtner, P., Purschel, H. V. and Reich, J. G. Acta Biol. Med. Get. 1968, 21,308 Damaschun, G. and Purschel, H. V. Acta Biol. Med. Ger. 1970, 24, 59 Harvey, S. C. and Cheung, H. in 'Muscle and Nonmuscle Motility', (Eds R. M. Dowben and H. Cheung), Plenum, New York, 1982, p. 279 Garcia de la Torre, J. and Bloomfield, V. A. Biochemistry 1980, 19, 5118 Garrigos, M., Morel, J. E. and Garcia de la Torre, J. Biochemistry 1983, 22, 4961 Mendelson, R. Nature 1982, 298, 665 Mendelson, R. Nature 1985, 318, 20 Highsmith, S. and Eden, D. Biochemistry 1986, 25, 2237 Bachouchi, N., Gulik, A., Garrigos, M. and Morel, J. E. Biochemistry 1985, 24, 6305 Craig, R., Trinick, J. and Knight, P. Nature 1986, 320, 688 Holtzer, A. and Lowey, S. J. Am. Chem. Soc. 1959, 81, 1370 Herbert, T. J. and Carlson, F. D. Biopolymers 1971, 10, 2231 Garcia Molina, J. J. and Garci/i de la Torre, J. Int. J. Biol. Macromol. 1984, 6, 170 Hvidt, S., Chang, T. and Yu, H. Biopolymers 1984, 23, 1283
Int. J. Biol. Macromol., 1988, Vol. 10, February
The radius of gyration, R, of a particle which occupies a volume V is given by
R2= fv r2p(r) d~ / fv p(r) d~
(A1)
where p(r) is some density at a point P whose position vector with respect to the particle's centre, C is r. This centre has the property
vrp(r) dz = 0
(A2)
Ifr' and e' are the position vectors of the general point P and the centre C, respectively, with respect to an arbitrary origin 0, so that r' = e ' + r, we have from equation (A2)
c'=fvr'p(r')d~/fj(r')d~
(A3)
Now we assume that the particle is composed of n subunits. The volume, centre and radius of gyration of subunit i are denoted as V, C; and R~, respectively. For simplicity, we restrict ourselves to subunits of uniform density. If pi is the density of subunit i, equations (A1) and (A2) lead to R 2=
(tlfv )/(tl gi) Pi
i
r 2 dz
~
i
i
(A4)
Radius of gyration of muhisubunit macromolecules:J. A. Solvez et al. and
and p
C-----
i
I
r d'r
i
fvridr = 0
i i
The centre of subunit i, C~, is placed at e [ = f r' dx/Vi
(A6)
,d V~
so that e ' = ~. f,c~
(A7)
i=1
(A8)
For the particular case in which all the subunits have the same density, f~isjust the volume fraction of subunit i. Particularizing equations (A1) and (A2) for subunit i we have
g~=~,fv,r~d,
where ri is the vector from Ci to P. Now, if ei is the position vector of C i with respect to C, we have r = c~+ r~. Substituting this into equation (A4) and using equations (A9) and (A10) we finally find equation (1). To derive equation (2) we note that it follows from equations (A2) and (A10) that
Z f,ei=O
(All)
i
and therefore
where
f i= piVi/ (i~=l piVi)
(A10)
(A9)
~. f.ff~cj = 0 i j Now, if
(A12)
r0=c i-ci
(A13)
the law of cosines gives
e~cj-_ (ci2 + cj2 _
r~)/2
(A14)
and substituting equation (A14) into equation (A12) and equation (1) we obtain equation (2).
Int. J. Biol. Macromol., 1988, Vol 10, February
43
Introduction In the low-angle region, the techniques of electromagnetic scattering (light, X-rays or neutrons) by macromolecules in solution are particularly easy to interpret. The dependence of scattered intensity on scattering angle is in part determined by the radius of gyration, R, which is a measure of the global size of the macromolecule. When combined with other physical data like molecular weight andpartial specific volume, it can give information about the macromolecular shape. Well-known formulas for R are available for simple, symmetrical shapes such as ellipsoids or cylinders. However, in the field of biological macromolecules, one is often facing rather complex, irregular structures that cannot be described adequately by such simple models. This situation has motivated much theoretical work aimed at the interpretation of hydrodynamic properties (sedimentation, diffusion, viscosity) of complex biological macromolecules 1. R is a practical alternative to hydrodynamic measurements; indeed, its sensitivity to size and shape should be comparable to that of the sedimentation or translational diffusion coefficients. For macromolecules of arbitrary shape, R can be calculated from a model in which point-like or in which small, spherical elements are distributed throughout. Such is the kind of model needed to compute the angular variation of the scattering intensity using Debye's formula. This approach can be nearly exact if the model contains a very high number of scattering elements, so that the shape of the macromolecule is accurately described, but at the cost of much computational effort. Models composed of spherical elements have the advantage of allowing the joint interpretation of hydrodynamic and X-ray results; interesting illustrations of joint modelling are available in recent literature 2,3. * To whom correspondence should be sent. 0141-8130/88/010039-05503.00 ~) 1988 Butterworth & Co. (Publishers) Ltd
In this paper, we explore the possibility of modelling
the macromolecule for the calculation of R, using a moderate number of building blocks of known radius of gyration and various shapes, like spheres, cylinders or ellipsoids. In some instances, these blocks may actually correspond to functionally significant subunits. Then, from the experimental R of the whole macromolecule, and the R (or dimensions) of the subunits, one can infer plausible structures for the spatial arrangements of the subunits. Concretely, we consider two alternative formulae for R of multisubunit structures. The formulae can be also applied to particles having cavities, as illustrated here for a model of myosin head. Another particular situation is that of segmentally flexible macromolecules, for which R is a conformational average. A typical example, considered in this paper, is whole myosin.
Theory The particle under study is assumed to be composed of n subunits whose radii of gyration are Ri, i = 1. . . . . n. If fi is the mass fraction of subunit i and ei is the distance from the centre of mass of the subunit to the centre of mass of the particle, R can be calculated as R 2= ~ fi(c~+R 2)
(1)
i=l
There is an alternative formula which does not require the previous calculation of the centre of mass of the particle, but only the distances r o between the centres of all the i,i-pairs of subunits:
R 2=
f, R2+E 2 f , flr2j i=l
(2)
i
Equations (1) and (2) are not reported in a recent monograph on X-ray scattering 4. After deriving these
Int. J. Biol. Macromol., 1988, Vol 10, F e b r u a r y
39
Radius of gyration of multisubunit macromolecules: J. A. Solvez et al. Z
Applications to myosin proteins Myosin is a typical example of a macromolecule in which rigid subunits can be clearly differentiated. Furthermore, the joints at which the myosin subunits are connected are usually supposed to be flexible. The segmental flexibility of myosin plays an essential role in muscle contraction 9. A schematic model for the myosin molecule is depicted in Figure 1. Two myosin heads (subfragment S1) are connected to the myosin rod. The rod may not be straight; instead, it is assumed to contain a small, 'soft' region which can act as a flexible joint (joint J1 in Figure 1), and divides the rod into two subfragments, $2 and LMM. The heads are also flexibly joined to the $2 subfragment of the rod by means of the joint denoted J2 in
s1
~y x r"
$2
Figure 1.
Figure 1 Model for whole myosin in an instantaneous conformation. Subfragments S1, $2 and light meromyosin (LMM) are indicated. The orientation of the heads are specified by polar angles 01, ~'l (shown) and 02, ~2 (not shown), referred to a system of Cartesian coordinates such that the rod ($2 plus LMM) lies on the y, z plane. The conformation of the rod is described by angle ~t equations we were kindly informed by a referee that they were presented in a review article on ribosome structure by Damaschun et al. 5, quoting previous, lessknown works of their group 6-s. In the Appendix of this paper we give the derivation of equations (1) and (2). The procedure based on equation (1) would be as follows. From the geometry of the particle, the position vectors of the centres of the subunits with respect to an arbitrary origin are evaluated first. Next, the centre of the particle, C, is calculated using the subunit volumes and densities. Then, the position vectors of the subunits are referred to C. Finally, these positions and the individual radii are substituted into equation (1) to find R. The alternative procedure based on equation (2) is more appealing because the intercentre distances can be evaluated directly from the geometry, and the centre of the whole particle is not needed. However, the calculation from equation (2) requires a number of operations proportional to n 2. In special cases in which n is very large, computing time may be appreciably longer than in the other procedure, for which the number of operations is proportional to n. For a particle composed by only two subunits, equation (2) takes a simple form:
R 2 = f l R2 q-f2R 2 +flf2r22
Int. J. Biol. Macromol., 1988, Vol. 10, February
Myosin heads Garrigos et al. 11 have proposed that chymotryptic, light-chain-2(LC2)-free S1 has the shape of a prolate ellipsoid with a hole near one of its extremities. This hole would be the place where LC2 is located. For simplicity, it was assumed that the hole is also ellipsoidal, and has the same axial ratio as the main ellipsoid. The model is depicted in Figure 2, where its geometrical parameters are defined. With p = 2 . 3 , a * = 6 . 0 n m , d = 1.5 nm and ~ = 4 . 0 n m , the theoretical results for a number of hydrodynamic properties and the angular dependence of X-ray scattering were in excellent agreement with experimental data 11. Equation (3) can be used to obtain the radius of gyration of a particle having a hole. If RF, R and RH are the radii of gyration of the filled particle, hollow particle and hole, respectively, and VF, V and VH are their volumes, with V= VF--VH, it follows from equation (3) that R2
2 2 = RF/f-R~LfH/f-rl2 zfH
(4)
a ~
,=
r
- -i
\
--
-
(3)
which is known as the Parallel Axes Theorem of solution scattering. Equations (1), (2) and (3) tell us that R depends on the dimensions of the subunits and the position of their centres but, if the latter are fixed, R does not depend on their orientation. Let us suppose, for instance, a dimer composed of two rods of length L and diameter d joined side-to-side with their axes parallel. For this model, r12=d and RE=R2=L2/12 +d2/8 so that RE=L2/12+ 3d2/8. Now, if one of the rods is arbitrarily rotated around its centre, so that the dimer takes a cross-like shape, R is unchanged because r12 remains constant.
40
Hydrodynamic properties of myosin fragments and whole myosin have been extensively used to learn about myosin structure and flexibility (see for instance, Ref. 10). In this paper, we shown how data on the radii of gyration can be useful for the same purpose.
,8
2
Figure 2 Model for LC2-free Sl. a* and b* are the semimajor and semiminor axes of the ellipsoid, respectively. The axial ratio is p = a*/b*. The ellipsoidal hole is centred at a distance d from the centre of the ellipsoid. Its major and minor semiaxes are ct and fl, respectively, with the same axial ratio, p=ct/fl, as the main ellipsoid
Radius of gyration of multisubunit macromolecules: J. A. Solvez et al.
50
40
ely-
30
20 0
I 30
I 60
I 90
I 120
I 150
180
0 (A), a (B, C, D) (degrees)
Figure 3
Radius of gyration, R, of whole myosin. Curve A: R
versus 0t=02 for Ls2=LLMM=72nm, Ct=0 and $t=~b2=0. Curve B: R versus ot for LS2=LLMM=72nm, 01=02=30 °, ~b~=~k2=0. Curve C: variation of the value of R, conformationally averaged over the orientation of the heads, with the fixed angle ~t. Ls2 = LLMM= 72 rim. Curve D: The same as for curve C with Ls2 =43 nm, LLMi = 113 nm
where f = V/Vv and fu = Vu/VF are the volume fractions (equal to mass fractions since we are assuming uniform density), and the distance between the centres of the hole and hollow particle is r12 = d/f. Using expressions for the volume and radius of gyration of ellipsoids, we obtain, in the notation of Figure 2, R 2 = [(p2 + 2)/5](b.S _ fls)/(b.a _flu) _ d2b ,afla/(b,a _ fl3)2
(5)
Due to an arithmetic error, equation (8) in Ref. 11 differs slightly from the correct expression, equation (5). Repeating the calculations we have seen that the effect from this error on the previous analysis of the shape of LC2-free S1 is very small. For the parameters given above, equation (5) leads to R = 3.3 nm, which coincides with the experimental value ~2. It must be pointed out that the size and shape of myosin heads are objects of much recent controversy t°-16. Our purpose here was only to illustrate the applicability of calculations of R in the case of a specific model.
We note that the actual shape of S 1 is not relevant in the calculations for the whole molecule. The length of rodlike subfragments $2 and L M M are not known accurately. Two rather extreme choices are Ls2 =/-~MM = 72 nm and Ls2 = 43 nm, and LLMM= 113 nm. The diameter of $2 and L M M is taken as 1.7 nm, which is compatible with experimental data for the molecular weight of the rod, and slightly smaller than the hydrodynamic radius 7. An instantaneous conformation of myosin is specified by the polar angle of the heads, 01, ~k1, 02, ~k2, and the angle in the rod, ct. These angles are shown in Figure 1. The experimental radius of gyration of the myosin molecule, R~xp= 45-47 nm (Refs 17 and 18), corresponds to a conformational average over these angles, Thus it seems possible to obtain information on myosin flexibility by comparing R~xpwith calculated values. However, the dependence of R on the conformation of the rod is much more pronounced, as shown by curve B in Figure 3. It is instructive to calculate R firstly for instantaneous conformations of the model, i.e. for fixed angle. Some results are presented in Figure 3. Curve A is for a straight rod (ct=0) with Ls2=LLMM=72nm and varying orientation of the heads. It shows that R depends very weakly on the conformation of the heads. In fact, the change in R between a fully extended conformation (0 t = 02'-'30 °) and a retracted one ( 0 1 = 0 2 2 1 5 0 °) is about
10%. According to current views of myosin flexibility, the mobility of the heads is rather extensive, so that J1 is nearly a universal joint. Therefore, the average over 0~, ~bl, 02, ~k2 can be a simple, unweighted one. We note that if the flexibility of joint J1 were somewhat restricted, the error introduced by the small influence of the heads' conformation on R would be small. Then, we decided to carry out the average by numerical simulation to avoid cumbersome algebra. For fixed angle, ~, in the rod, about 400 allowed conformations of the heads were generated picking at random 01 and 02 with a sine distribution, and ff~ and ~k2 with a uniform distribution. A conformation is allowed if there is no overlap between the four subunits. This procedure was repeated for varying values of ct, and the results are presented in curves C and D of Figure 3. In a simple approach 1° the flexibility of the rod is characterized by a certain equivalent angle, Ct~q,such that the value of a given radius calculated for ~q, and averaged over the random orientation of the heads, coincides with the experimental datum. Thus, Ct~qcan be regarded as some mean bending angle. The result would be Ct,q- 0 ° if the rod was rigid and straight. However, values of ~,q of up to 60 ° have been found in the analysis of hydrodynamic properties 1°. In the present study, ~q is estimated interpolating R~xpin curves C and D. We find atoq---20°-45 ° with curve C for LS2= L L M M = 72 nm, and ~ q - 600-70 ° with curve D for Ls2 = 43 nm and LLMM= 113nm. These values, as well as those from hydrodynamic properties, are significantly different from 0 ° and therefore confirm the existence of a moderate flexibility of the myosin rod.
Whole myosin In the calculations for whole myosin, subfragment S1 is considered as a particle with a volume of 142nm a, and a radius of gyration of 3.3 nm, as discussed in the previous section. Its centre of mass is located at a distance of 8 nm from joint J1. This distance is obtained adding an excess of 2 nm to the long semiaxis of S1, which is about 6 nm.
Myosin rod The flexibility of the myosin rod can be analysed with more detail in the rod itself (headless myosin) since it has only two subunits and a single joint. The proper model is a broken, or hinged rod having two rodlike subunits of length L1 ( = Ls2) and L 2 (-----L L M M ) . The radius of gyration
Int. J. Biol. Macromol., 1988, Vol 10, February
41
Radius of oyration of multisubunit macromolecules: J. A. Solvez et al. of the broken rod can be obtained from equations (3) with
fl=LJL=7, where L = L t + L 2 is the total length, f 2 = 1 - 7 , R2=L~/12, (d2/8,~L2/12), R2=L2/12, and r22 = L2/4 + L2/4 + L 1L2 c o s a/2. The result is (R2)=(L2/2)[1/6+72(1-7)2((cos~¢-l)] (6) where ( - - - - ) denote conformational average. We note that only conformation-dependent quantity is cos ~. A formula equivalent to equation (6) was derived by Garcia Molina and Garcia de la Torre 19 though in a more indirect way. Hvidt et al. 2° have measured the radius of gyration of the myosin rod, obtaining values in the range 37-41 nm with a negligible temperature dependence. From equation (6) we find (cos a) = 0.45-0.93 if Lsz = LLMM = 72nm and (cos a) < 0.29 if L s 2 = 4 3 n m and LLMM= 113 nm. These values deviate to a lesser or greater extent from the rigid limit (cos a) = 1, and is again a proof of the flexibility of myosin rod. For the first choice, the rod is quite stiff (Ct~q=20°~50°) while for the second one its behaviour is almost completely flexible (~q> 75°). We recall that these two choices are extremes. As R increases with increasing L and decreasing 7, the first choice corresponds to lowest R, while for the second one has the highest value. Thus, the parameters obtained for the two choices of length should be regarded as lower and upper bounds for the true values of myosin. Flexibility of the joint in the rod can be quantified by means of an elastic potential V(a) =½ k~ 2
(7)
where k¢ is the elastic constant. Then, the conformational average needed for R is (Ref. 19)
Acknowledgement This work was supported by grant 561/84 from the Comision Asesora de Investigacion Cienfifica Tbcnica.
References 1 2 3 4 5 6 7 8 9 I0 11 12 13 14 15 16 17 18 19 20
fo o
d ~ cos a sin ~ e - q "
/fo o
dct sin ct e-°~2
(8)
Appendix
with
Q=k./2kbT
(9)
where kbTis Boltzmann factor. In Ref. 19, the upper limits of the integrals in equation (8) where rt instead of ~ . However, the influence of this difference in the resulting values of (cos ~) is negligible except for very low Q. The plot of (cos ~) versus Q is practically the same as that in Fioure 1 of Ref. 19. Interpolating the (cos ct) values for myosin rod obtained from equation (6), we find Q = 0.5-8 if Ls2 = LLMM= 72 nm and Q < 0.3 if Ls2 = 43 nm and LLMM= 113 nm. The agreement between the values of the flexibility parameter, ~q, of myosin and myosin rod is better for L S 2 = LLM M ~---72 nm than for Ls2 = 43 nm, LLMM= 113 rim. For the first set of lengths the myosin rod is found to be rather stiff, but not straight, while for the other set the flexibility is perhaps too high. We should note that our analysis is based on a single property, namely, the radius of gyration, which depends on several geometrical and conformational parameters of the molecule, and therefore we cannot ignore the possibility of over-interpretation of our model. Nonetheless, different choices of those parameters lead to coincident qualitative conclusions: our analysis of the radius of gyration of myosin and myosin rod confirms the moderate but appreciable flexibility of the rod.
42
Garci/l de la Torre, J. and Bloomfield, V. A. Q. Rev. Biophys. 1981, 14, 81 Perkins, S. J. Biochem. J. 1985, 228, 13 Perkins, S. J. and Sire, R. B. Eur. J. Biochem. 1986, 157, 155 Glatter, O. and Kratky, O. (Eds) 'Small Angle X-ray Scattering', Academic, New York, 1982 Damaschun, G., Muller, J. J. and Bielka, H. Meth. Enzymol. 1979, 59, 706 Damaschun, G., Muller, J. J. and Purschel, H. V. Acta Biol. Med. Ger. 1968, 20, 379 Damaschun, G., Fichtner, P., Purschel, H. V. and Reich, J. G. Acta Biol. Med. Get. 1968, 21,308 Damaschun, G. and Purschel, H. V. Acta Biol. Med. Ger. 1970, 24, 59 Harvey, S. C. and Cheung, H. in 'Muscle and Nonmuscle Motility', (Eds R. M. Dowben and H. Cheung), Plenum, New York, 1982, p. 279 Garcia de la Torre, J. and Bloomfield, V. A. Biochemistry 1980, 19, 5118 Garrigos, M., Morel, J. E. and Garcia de la Torre, J. Biochemistry 1983, 22, 4961 Mendelson, R. Nature 1982, 298, 665 Mendelson, R. Nature 1985, 318, 20 Highsmith, S. and Eden, D. Biochemistry 1986, 25, 2237 Bachouchi, N., Gulik, A., Garrigos, M. and Morel, J. E. Biochemistry 1985, 24, 6305 Craig, R., Trinick, J. and Knight, P. Nature 1986, 320, 688 Holtzer, A. and Lowey, S. J. Am. Chem. Soc. 1959, 81, 1370 Herbert, T. J. and Carlson, F. D. Biopolymers 1971, 10, 2231 Garcia Molina, J. J. and Garci/i de la Torre, J. Int. J. Biol. Macromol. 1984, 6, 170 Hvidt, S., Chang, T. and Yu, H. Biopolymers 1984, 23, 1283
Int. J. Biol. Macromol., 1988, Vol. 10, February
The radius of gyration, R, of a particle which occupies a volume V is given by
R2= fv r2p(r) d~ / fv p(r) d~
(A1)
where p(r) is some density at a point P whose position vector with respect to the particle's centre, C is r. This centre has the property
vrp(r) dz = 0
(A2)
Ifr' and e' are the position vectors of the general point P and the centre C, respectively, with respect to an arbitrary origin 0, so that r' = e ' + r, we have from equation (A2)
c'=fvr'p(r')d~/fj(r')d~
(A3)
Now we assume that the particle is composed of n subunits. The volume, centre and radius of gyration of subunit i are denoted as V, C; and R~, respectively. For simplicity, we restrict ourselves to subunits of uniform density. If pi is the density of subunit i, equations (A1) and (A2) lead to R 2=
(tlfv )/(tl gi) Pi
i
r 2 dz
~
i
i
(A4)
Radius of gyration of muhisubunit macromolecules:J. A. Solvez et al. and
and p
C-----
i
I
r d'r
i
fvridr = 0
i i
The centre of subunit i, C~, is placed at e [ = f r' dx/Vi
(A6)
,d V~
so that e ' = ~. f,c~
(A7)
i=1
(A8)
For the particular case in which all the subunits have the same density, f~isjust the volume fraction of subunit i. Particularizing equations (A1) and (A2) for subunit i we have
g~=~,fv,r~d,
where ri is the vector from Ci to P. Now, if ei is the position vector of C i with respect to C, we have r = c~+ r~. Substituting this into equation (A4) and using equations (A9) and (A10) we finally find equation (1). To derive equation (2) we note that it follows from equations (A2) and (A10) that
Z f,ei=O
(All)
i
and therefore
where
f i= piVi/ (i~=l piVi)
(A10)
(A9)
~. f.ff~cj = 0 i j Now, if
(A12)
r0=c i-ci
(A13)
the law of cosines gives
e~cj-_ (ci2 + cj2 _
r~)/2
(A14)
and substituting equation (A14) into equation (A12) and equation (1) we obtain equation (2).
Int. J. Biol. Macromol., 1988, Vol 10, February
43