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Int. J. Peptide Protein Res. 19,1982,549-555 Internal fluctuations in globular proteins P.K. PONNUSWAMY and R. BHASKARAN Department of Physics, Autonomous Postgraduate Centre, University of Madras, Tamilnadu, India Received 20 April, accepted for publication 25 November 198 1 An attempt was made to study the dynamics and hence the fluctuations in globular proteins treating them as prolate and oblate spheroidal bodies. The fluctuations were obtained by solving the dynamical differential equation of motion derived from the elastic stress-strain relations. The results provide information on the nature of variation of the displacement of the atoms due to fluctuations in the distance from the centroid of the protein. Key words: dynamical displacement; elastic oblate spheroids. It has been convincingly demonstrated that the native structure of a globular protein exists as a well-defined densely packed matrix. However, in its functional form, the piotein molecule is under the influence of a variety of physical factors: temperature, pH, inter- and intra- molecular interactions, and other environ- mental conditions including the solvent. These factors cause the globular protein to undergo observable structural and positional fluctuations. Such fluctuations have been predicted from both experimental and theoretical studies (1-9). Recently, a number of reports have appeared on this aspect of protein structure. The conformational fluctuations in the protein were investigated by means of the statistical mechanical treatment, incorporating various kinds of interactions (10-13). Molecular dynamics was applied to study the fluctuations in the atom positions and in the bond angles in specific proteins (14-1s). The structural fluc- tuations have also been studied through com- puter simulation technique (16-18). All these studies have added valuable information about fluctuations; globular proteins; prolate and the dynamical behaviour of globular protein molecules in their native state. As a continu- ation of this laboratory’s study of the protein conformation (19, 20), we have recently con- sidered the problem of small amplitude fluctu- ations in globular proteins. The results are pre- sented in this article. We treat the proteins in prolate and oblate spheroidal shapes and set up dynamical differential equations of motion, the solutions of which under appropriate boundary conditions provide valuable information as to the relationship between the positional fluctu- ations of atoms and their distance from the centroid of the protein molecule. THEORY In our study, we treat the globular protein molecule as a continuous, isotropic material in accordance with the treatment of Suezaki & Go (7). In their study, Suezaki & Go assumed a spherical shape for the protein and the low frequency fluctuations in it were taken to arise from the action of a force which is purely 0367-8377/82/050549-07 $02.00/0 0 1982 Munksgaard, Copenhagen 549

Internal fluctuations in globular proteins

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Int. J. Peptide Protein Res. 19,1982,549-555

Internal fluctuations in globular proteins

P.K. PONNUSWAMY and R. BHASKARAN

Department of Physics, Autonomous Postgraduate Centre, University of Madras, Tamilnadu, India

Received 20 April, accepted for publication 25 November 198 1

An attempt was made to study the dynamics and hence the fluctuations in globular proteins treating them as prolate and oblate spheroidal bodies. The fluctuations were obtained by solving the dynamical differential equation of motion derived from the elastic stress-strain relations. The results provide information on the nature of variation of the displacement of the atoms due to fluctuations in the distance from the centroid of the protein.

Key words: dynamical displacement; elastic oblate spheroids.

It has been convincingly demonstrated that the native structure of a globular protein exists as a well-defined densely packed matrix. However, in its functional form, the piotein molecule is under the influence of a variety of physical factors: temperature, pH, inter- and intra- molecular interactions, and other environ- mental conditions including the solvent. These factors cause the globular protein to undergo observable structural and positional fluctuations. Such fluctuations have been predicted from both experimental and theoretical studies (1-9). Recently, a number of reports have appeared on this aspect of protein structure. The conformational fluctuations in the protein were investigated by means of the statistical mechanical treatment, incorporating various kinds of interactions (10-13). Molecular dynamics was applied to study the fluctuations in the atom positions and in the bond angles in specific proteins (14-1s). The structural fluc- tuations have also been studied through com- puter simulation technique (16-18). All these studies have added valuable information about

fluctuations; globular proteins; prolate and

the dynamical behaviour of globular protein molecules in their native state. As a continu- ation of this laboratory’s study of the protein conformation (19, 20), we have recently con- sidered the problem of small amplitude fluctu- ations in globular proteins. The results are pre- sented in this article. We treat the proteins in prolate and oblate spheroidal shapes and set up dynamical differential equations of motion, the solutions of which under appropriate boundary conditions provide valuable information as to the relationship between the positional fluctu- ations of atoms and their distance from the centroid of the protein molecule.

THEORY

In our study, we treat the globular protein molecule as a continuous, isotropic material in accordance with the treatment of Suezaki & Go (7). In their study, Suezaki & Go assumed a spherical shape for the protein and the low frequency fluctuations in it were taken to arise from the action of a force which is purely

0367-8377/82/050549-07 $02.00/0 0 1982 Munksgaard, Copenhagen 549

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P.K. Ponnuswamy and R. Bhaskaran

elastic in nature. This elastic model for the protein is, indeed, a good approximation since it gives due consideration to the extensional/ contractional/deformational behaviour of the protein. In considering the globular protein as a single elastic body, we encounter, using a simple harmonic oscillator, fluctuations of an oscillatory nature. For low frequency fluc- tuations, whose wavelengths are larger than the size of the individual residues, the protein could be taken to be a continuous body and if we assume that the elastic constants are the same in all the directions, the protein can be con- sidered isotropic. Suezaki & Go (7) calculated the angular frequency in the fundamental mode of vibration of the isotropic, continuous, elastic spherical shaped protein to be around 26 cm-’ and this theoretical value was in good agreement with the Laser Raman spectroscopic result of 30cm-’ for the proteins, pepsin and a-chymotrypsin. While Suezaki &Go considered the protein in a spherical shape, we consider it in prolate and oblate spheroidal shapes and characterise its low frequency fluctuations.

Differential equation of motion for the fluctuations Prolate. The relations between the Cartesian and prolate spheroidal coordinates are,

x = a sinh [ sin 7) cos @

y = a sinh C; sin 77 sin @

z = acoshC;cosq (1)

To relate the general equation of motion for the fluctuations in an elastic, spheroidal, iso- tropic body with a protein molecule, we have to define the parameters appropriately. Due to the existence of fluctuations in the positions of atoms and in the protein as a whole, there arises a resultant oscillatoryfvibrational force and hence according to the energy conservation law, the dynamical differential equation of motion can be obtained from the equation of equilibrium by adding this oscillatory force (of inertia) to the component of the body force ‘R’ along the radial direction.”

The normal stresses and the shearing stresses could be obtained from eqn. 1 and substituting these stresses in the general formula of the dynamical differential equation, we obtain the

550

equation of motion for the fluctuations along the radial direction (77 = 90e), assuming that the scalar quantity a = 1, as

(X + 2p) sinh g cosh [ (a2 Ut/i3t2)

+ ( h + 2 ~ ) cosh2[(aU,/a[)

+ [A sinh 6 cosh C; - (X + 2p) coth g cosh2 [

- 2(X + p) tanh [ s i n h 2 t ] U ~ + p * R

= p - (azut/atz) (2) where X and p are the Lame’s constants, U, is the radial component of the displacement of the internal units (atoms) in the elastic material, R is the internally existing force and p is mass per unit volume. The body force is, however, small and could be left out in the case of globular proteins as a first approximation. p could be taken as the mass density of the globular protein which is here assumed to be unity.

Considering the fluctuations occurring in a protein to be of simple harmonic nature with an angular frequency o, a particular solution UE for the net displacement of the final second order partial differential equation with variable coefficients can be obtained. In defining o, we retain the spherical shape for the protein and use the formula mentioned in the article by Suezaki & Go (7) and consider only the case of fundamental mode. The solution Ut may be divided into two parts: one, the time part specified by a cosine term, and the other, the space part specified by a function u(t) which depends only on the radial component C; of the prolate spheroidal protein. Let the assumed solution be

UE = A cos (wt + e,,)u(C;) (3) This solution implies the net displacement of the atom in the radial direction, and is a com- bination of the variation of the atom’s dis- placement with respect to time and its position from the centroid of the protein molecule. As such, it is evident from the solution that the time-displacement relation is harmonic in nature. Therefore, to find out the space- displacement relation in one-dimension (along the radial direction), this particular solution (eqn. 3) has to be applied in the final dynamical partial differential equation of second order

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Internal fluctuations in globular proteins

ponding positions (distances from the centroid) and the differential coefficients of the displace- ments with respect to their corresponding positions. Thus, the displacements of the atoms situated at different points from the centroid can be determined. If the function F2 becomes non-defined for any given initial condition, an alternative value of t in the close neighbour- hood of zero is chosen for the initial condition such that it will not affect the system. For different values of the slope p, different sets of g, u and p values will be obtained, but the nature of the variation of the displacement of the atom with respect to its position will not be affected. Thus, the numerical solution can be obtained for the displacements of the atoms within and on the surface of the prolate spheroidal protein.

(eqn. 2) so that an ordinary differential equa- tion of second order will be obtained, from which an analysis on the motion of the atoms in the interior as well as on the surface of the protein with respect to its position from the centroid can be carried out. So the final differ- ential equation (wholly a function of t ) , which has to be solved to get the space-displacement variation, is given by,

d2 u/dt2 + coth t(du/dt)

+ [B - cothZC;- C tanh' r; + D sech 5' cosech E ] u

where, = o (4)

B = h/(h + 2 p )

c = 2 ( p + h)/(h + 2 p )

D = p o 2 / ( h + 2 p )

The above differential equation has been solved numerically by the Runge-Kutta method for two simultaneous first order equations. This has been achieved by splitting the above differential equation as,

dp/df: = - p coth 4 - [B - coth2t - C tanh2C;

+ D sech f cosech t ] u

= FZ(t,U,P) (6) Eqn. 5 implies the differential coefficient of the displacement of an atom with respect to its position from the centroid of the protein and is considered to be equal to p (a function of t , u and p), which is the slope of the space- displacement relation. Eqn. 6 implies the differential coefficient of the slope of the space- displacement relation of an atom in a protein with respect to its corresponding position from the centroid, which is taken to be equal to a function Fz , of C;, u and p.

In the Runge-Kutta method, the functions F1 and F2 are calculated for an initial condition defined by a set of constants. From these func- tions, the consecutive values o f t , u and p are calculated. This procedure will, actually, give the displacements of the atoms at their corres-

Oblate. When the protein is considered in the oblate spheroidal shape, the corresponding dynamical differential equation of motion with the conditions q = 90" and a = 1 is given by

(A + 2 p ) sinh C; cosh C;(a'Ut/a$z)

+ ( A + 2 p ) sinh2t(aut/at)

+ [B - tanh t sinh' .$ - C coth 5 cosh2t] Ut

+ p R = p *(a2Ug/atZ) (7)

which when we proceed as for the prolate case, is split into

and,

dp/d,$ = - p tanh C; - [B - tanh2C; - C coth2C;

d d d t = P = Fi(S,u,p) (8)

- D sech cosech 5 ) u

= FZ(C;,U,P) (9)

The functions F1 and Fz have the same meaning as in the prolate case.

RESULTS AND DISCUSSION

Using the initial values of the variables C;, u and p (initial value of p is fixed by the boundary conditions), the subsequent variation in the dis- placement (u) of the atoms with respect to their positions from the centroid is obtained.

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P.K. Ponnuswamy and R. Bhaskaran

Radius Vector, 1 4 “ h I - FIGURE 1 Space-plane plots for the fluctuations in a protein when treated as a prolate spheroid in shape. Initial values of the slope, p, are indicated.

The boundary conditions are chosen such that the nature and magnitude of the numerical solution suits our specification for a globular protein .

It is well known that the residues which are on the surface part of the protein molecule are free to move about with fewer interactions than those in the interior of the protein, which are restricted to move as per the interactions among themselves and with that of the water medium: hence it is expected that the surface part of the protein experiences larger displace- ments whereas the displacements of parts in the close neighbourhood of the centroid will be very small. For the sake of simplicity we assume that the atom (or part) situated at the centroid experiences nil fluctuation (assignment of any kind of fluctuation in the neighbour- hood of the minimum value in magnitude to the centroid will not alter the nature of our results).

As a maximum limit of the boundary condition, the surface part of the protein is assumed to have a displacement of 10% of the

55 2

I 0 25 0 50 0 75

Displacement, (10 d l - FIGURE 2 Phase-plane plot for the fluctuations in a protein when treated as a prolate spheroid in shape. Initial value of the slope, p, is indicated.

respective protein’s maximum radius vector value. Thus, at = 0, u = 0, and at ,$ = r, u = r/lO were the boundary conditions. With these boundary conditions, the kinds of fluctuation the atoms undergo have to be studied as we proceed from the centroid to the surface along the radial path. The sets of 5, u and p values were then obtained from the numerical solu- tions of eqns. 5,6,8 and 9 by varying the radial component (C;) in steps of 0.28,, from the initial zero value up to the 208, limit. For different values of the slope p, different sets of values of C;, u and p are obtained. Plots relating ,$ vs u, and u vs p, the former being the space- plane type and the latter being the phase-plane type, were drawn and discussed. In general, if the fluctuation at a site exists in harmonic form (u = sin x), the plot in the space-plane will be sinusoidal and that in the phase-plane will be a circle, and if the fluctuation possesses a non- linear behaviour, the phase-plane plot will be a closed curve such as an ellipse/spiral/limit cycle (22).

The space-plane (C; vs u) plots corresponding

Page 5: Internal fluctuations in globular proteins

Internal fluctuations in globular proteins

O ' d

I

Tun. I," 0.1-

FIGURE 3 The variation of the netdisplacement (Ut) with respect to time at different positions (g = 1 A and 2 A ) from the centroid of the protein for a slope value p = - 0.005. The values of Ug are given in units of A.

to a prolate spheroidal protein are shown in Fig. 1 for two initial values of the slope p = duldf. From these plots, we note that as the distance of the atom (part of the protein) from the centroid increases in the radial direc- tion (one-dimensional), the displacement increases in a nonlinear form.

The phase-plane plot for an initial slope value is shown in Fig. 2. We note from this plot that as the displacement increases, its differ- ential coefficient with respect to the radius vector increases, specifying a linear relationship between the displacement and the radial vector (if the phase-plane plot were a closed curve, it would imply an oscillatory relationship between the variables).

Once the space part solution u(f) is found out (displacement which depends on .$ only), the net displacement U&, t) along the radial

FIGURE 4 The variation of the netdisplacement with respect to the radius-vector at three different instants t = 0, 1 , 2 ps for an initial value of the slope p = - 0.005.

direction can be found out by combining the time part solution cos(wt) with that of the space part solution u(f) for any particular value of w. The variation of this net displacement with time (keeping the radius vector constant) and with the radius vector (keeping time constant) are shown in Figs. 3 and 4, respec- tively, for an initial value of slope p = - 0.005 and w = 26 cm-' . The net displacement Ut possesses a nonlinear oscillatory behaviour with respect to time (Fig. 3). Fig. 4 implies a more or less linear relationship between the variables f and Ug for small values o f f , and for higher values, UE increases steeply in a non- linear fashion. As time increases, the displace- ment becomes both positive and negative (a positive value indicates a movement away from the centroid and the negative value indicates a movement towards the centroid with refer- ence to the observing point) (Fig. 4).

In Figs. 5 and 6, the space-plane plots and the phase-plane plot corresponding to the case of an oblate spheroidal protein are shown. The space-plane plots for low .$ values indicate a

553

Page 6: Internal fluctuations in globular proteins

P.K. Ponnuswamy and R. Bhaskaran

Radius Vector, ( i n A 1 - FIGURE 5 Space-plane plots for the fluctuations in a protein when treated as an oblate spheroid in shape. Initial slope values are indicated.

Dmpiocement , u (in I ) - FIGURE 6 Phase-plane plot for the fluctuations in a protein when treated as an oblate spheroid in shape. Initial slope value is indicated.

554

- 1 a l a ia 15 20

J Tim. 1 8 0 w l -

FIGURE 7 The time us netdisplacement (Ut) plots at the positions = 1 A and 2 A from the centroid of an oblate spheroidal protein for a value of the slope p = 0.001. The values of UE are given in units of 10-3 A.

RDdtU. v.c,or, [I" t - FIGURE 8 The radius vector us netdisplacement plots at the instants t = 0, 1, 2ps. in oblate spheroidal protein for a value of the slope p = 0.001.

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Internal fluctuations in globular proteins

7. Suezaki, Y. & Go, N. (1975) Int. J . Peptide Protein Res. 7,333-334

8. Udea, Y. & Go, N. (1976)Int. J. Peptide Protein Res. 8,55 1-558

9. Yguerabide, J., Epstein, H.F. & Stryer, L. (1970) J. Mol. Biol. 51,573-590

10. Go, N. (1976)Adv. Biophys. 9,65-113 11. Go, N. & Taketomi, H. (1979) Int. J. Peptide

Protein Res. 13,235-252 12. Go, N. & Taketomi, H. (1979) Inf . J. Peptide

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Peptide Protein Res, 7,445-449 14. McCammon, J.A., Gelin, B.R. & Karplus, M.

(1977) Nuture 267,585-590 15. Northrup, S.H., Pear, M.R., McCammon, J.A. &

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(1979) Biochemistry 18,927-942 18. Northrup, S.H. & McCammon, J.A. (1980)

Biopolymers 19,1001-1017 19. Ponnuswamy, P.K., Prabhakaran,M. &Manavalan,

P. (1980) Biochim. Biophys. Acfa 623,300-316 20. Prabhakaran, M. & Ponnuswamy, P.K. (1980)

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more or less linear relationship between u and 4 and those for higher 4 values indicate a non- linear relationship between them. The phase- plane plot also indicates a linear relationship between u and du/de.

The corresponding time-net displacement, and space-net displacement plots for the oblate case (Figs. 7 and 8) show similar behaviour to those for the prolate case.

Thus, in general, for both prolate and oblate spheroidal shaped proteins, for the funda- mental mode, though their time-displacement relation is oscillatory in nature, their space (radius-vector)-displacement relation is not. We are now working on the characteristic fluctu- ations of individual proteins and also on higher modes, the results of which will be reported elsewhere.

ACKNOWLEDGEMENT

This work was supported by a research grant to P.K.P. from the Department of Science and Technology, Government of India.

REFERENCES

1. Brown, K.C., Erfurth, S.C., Small, E.W. & Peticolos, W.L. (1972) Proc. Natl. Acad. Sci. US

2. Cooper, A. (1976) Proc. Nutl. Acad. Sci. US 75,

3. Hull, W.E. & Sykes, B.D. (1975) J. Mol. Biol.

4. Lakowicz, J.R. & Weber, G. (1975) Biochemistry

5 . McCammon, J.A., Gelin, B.R., Karplus, M. & Wolynes, P.G. (1976) Nature 262,325-326

6. Saviotti, M.L. & Galley, W.C. (1974) Proc. Natl. Acad. Sci. US 71,4154-4158

69,1467-1469

2740-2741

98,121-153

12,4171-4179

Address:

P.K. Ponnuswamy Department of Physics Autonomous Postgraduate Centre University of Madras Tiruchirapalli - 620 020 Tamilnadu India

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