Energetic approach to the packing of .alpha.-helixes. 1. Equivalent

kcal/mol, are nearly antiparallel with orientational torsion angles of-154,170, and 146°, .... sin 0. COS 0. (2) f. Figure 1. Helical coordinate syst...
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J. Phys. Chem. 1983, 87, 2869-2881

2869

Energetic Approach to the Packing of a-Helices. 1. Equivalent Helices Kuo-Chen Chou,+ George NOmethy, and Harold A. Scheraga’ Baker Laboratory of Chemistry, Cornell University, Ithaca, New York 14853 (Received: November 4, 1982)

A mathematical framework has been developed for generating interacting a-helices and for calculating and minimizing the interaction energy between a-helices. Computationswere carried out on a system of two equivalent right-handed a-helices, each having the structure CH3CO-(L-Ala)lo-NHCH3. Ten low-energy packing arrangements were found. The three lowest-energy arrangements, occurring within an energy interval of 1.8 kcal/mol, are nearly antiparallel with orientational torsion angles of -154, 170, and 146’, in order of increasing energy. The other seven arrangements,with energies of 3.3-5.3 kcal/mol above that of the most stable structure, have a variety of orientational torsion angles. The major contribution to the intermolecular energy arises from nonbonded interactions, with a minor contribution from the electrostatic interactions. The latter tend to favor an antiparallel arrangement of two a-helices that are located side by side. This is one of the reasons why the packing with an orientational torsion angle of -154’ has the lowest energy among the computed packings. This packing arrangement is observed frequently in globular proteins. In all minimum-energy structures, the two helices are closely packed, with the side-chain CHB groups of one helix intercalated between those of the other helix. This is indicated by the small distance of closest approach (6.7-7.7 A) of the helix axes. This distance is about 4 A less than twice the distance of the van der Waals surface of the methyl groups from the helix axis. While most of the computed packing arrangements are similar to some packings derived in other studies on the basis of geometrical considerations of packing, the latter by itself cannot determine the order of stability of the various arrangements. Some geometrically feasible packing arrangements are of high energy; on the other hand, some low-energy arrangements could not be found from geometrical considerations alone. These results show the importance of energy calculations for studying packing.

I. Introduction A frequently occurring feature of the folding pattern of globular proteins is the close packing of regular structural elements, such as a-helices and/or P-sheets., In particular, two or more a-helices often pack tightly against each other. It is important, therefore, to elucidate the geometrical and energetic features of helices that determine their manner of packing, and to derive the optimal packing arrangements. This problem has been treated in the past in terms of the geometrical fitting of the surfaces of two a-helices, without taking the energetics of interhelical interactions into consideration. Geometrical models generally emphasize the unevenness of the surface of the a-helix, caused by the presence of regularly space side chains. Such models have been named “knobs into holes”,2“string and sausagen,3“ridges into groove~”,4,~ “closepacked sphere”,6 and “polar and apolar ~ p h e r e ” . ~A, ~ requirement for hydrophobic interactions also is included in some of the models.6-8 Each of these models has some advantageous features for describing a given packing arrangement.l Because of the great variation of size and geometry of side chains in actual a-helices, it is possible to describe contacts by quite different idealized geometrical models. Actually, however, the energetics of interaction between the helices determines p a ~ k i n gthe ; ~ geometrical feature of surface complementarity describes only part of the total interaction energy. It is therefore necessary to consider helix packing in terms of interaction energies. This paper is devoted to such an analysis for two equivalent poly(Lalanine) a-helices. A preliminary report on this work has been presented elsewhere.’O Poly(L-alanine) has been chosen in order to investigate the general effectagJOarising from the interactions of the backbones and the simplest side chain that is common to all chiral L-amino acids. In this paper, we consider only a-helices with a rigid backbone, Le., we keep d backbone dihedral angles fixed. The ‘Visiting Professor from Shanghai Institute of Biochemistry, Chinese Academy of Sciences. 0022-3654/83/2087-2069$0 1.50/0

interactions between nonequivalent a-helices containing other amino acid residues are being studied and will be reported elsewhere.

11. Mathematical Formulation A. Coordinate Systems. In discussing the interactions between two a-helices, it is convenient to use a helical coordinate system (fg,h),as illustrated in Figure 1. The axis of helix 1 coincides with the h axis, and the origin 0, is the projection of a reference atom (taken here arbitrarily as the peptide nitrogen atom of the fourth residue) of the a-helix onto the h axis. The perpendicular line from O1 to this atom defines the f axis, and the g axis is perpendicular to both f and h so that (f.g,h)forms a right-handed coordinate system; ef, eg,and ehare unit vectors along these axes. In Appendix A, we show how to transform a set of coordinates from any general reference system ( x , y , z ) to the helical coordinate system (f,g,h).We now consider the geometrical relationship between the two helices in the helical reference frame of Figure 1. Suppose the second a-helix is generated by a rotational operation followed by a translational operation on the first a-helix, i.e. r2 = T + Rr, (1) where rl and r2are the coordinates of corresponding atoms (1) Richardson, J. S. Adu. Protein Chem. 1981, 34, 167. (2) Crick, F. H. C. Acta Crystallogr. 1953, 6, 689. (3) Ptitsyn, 0. B.; Rashin, A. A. Dokl. Akad. Nauk. SSSR (Biochem. Ser.) 1973,-213, 413. (4)Chothia, C.; Levitt, M.; Richardson, D. Proc. Natl. Acad. Sci. U S A . 1977, 74, 4130. (5) Chothia, C.; Levitt, M.; Richardson, D. J . Mol. Biol. 1981,145, 215. ( 6 ) Richmond, T. J.; Richards, F. M. J . Mol. Biol. 1978, 119, 537. (7) Efimov, A. V. Dokl. Acad. Nauk. SSSR. (Biochem. Ser.) 1977,235, fi99.

(8) Efimov, A. V. J . Mol. Biol. 1979, 134, 23. (9) Chou, K. C.; Pottle, M. S.; NBmethy, G.; Ueda, Y.; Scheraga, H. A. J.Mol. Biol. 1982, 162, 89. (10) Scheraga, H. A.; Chou, K. C.; NBmethy, G. In ‘Conformation in Biolod: Srinivasan, R.; Sarma. R. H., Ed.; Adenine Press: Guilderland, NY; 1982, p 1.

0 1983 American Chemical Society

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Chou et al.

The Journal of Physical Chemistry, Vol. 87, No. 15, 1983 y - sin a 0:

cos y

-cos cr sin

cos p sin 7

+ cos a cos 9 sin y

~

sin cr sin y

sin 9 cos y

-

sin 0: cos p cos y

sin 01 sin 0

+

cos cr cos 0 cos y

-

cos a sin p

cos P

I

(2)

f

t

h

Figure 1. Helical Coordinate system ( f , g , h )in which the axis of helix 1 coincides with the h axis. The origin O1 is the projection of a specified (reference) atom of the a-helix on the h axis. The perpendicular line from point 0,to this atom defines the f axis. The g axis is perpendicular to both f and h so that ( f , g , h )forms a right-handed rectangular coordinate system. Helix 2 Is shown at an arbitrary orientation and distance with respect to helix 1. The point on its axis that is equivalent to 0,of the first helix is denoted 02.Therefore, 0,02 = T = (tf,tg,th).A and B represent, respectively, the projections of the first (N terminal) and last (C terminal) heavy atoms (Le., non-hydrogen atoms) of each helix onto its axis. The length of each helix is defined as the distance AB.

in helices 1 and 2, respectively, T = (tf,tg,th)is a translational vector which transforms point O1 of helix 1 into point O2of helix 2, as shown in Figure 1,and R is the Euler rotational operatorg given by eq 2, where a, 0,and y are the Euler angles. The two identical helices are superposed on each other when a = 8 = y = t f = t , = th = 0. The six parameters a, P, y , tf, t,, and th completely define the relative position and orientation of the two helices (Figure 2). If one is concerned only with the relative position of the axes of the two helices, this position can be specified (except for the displacement of each helix along its own axis) by using only two (instead of six) parameters, to be defined in sections IIB to IID: the distance D of closest approach between the two helix axes5 and an angle describing the relative orientation of the two helix axes. The latter can be expressed either as Ro, the angle of orientation between the axes of the two helices, or as Qp, the angle between the projections of the axes of the two helices onto the “contact plane” between them5 (to be defined below). B. The Orientation Angle Qo. To obtain an expression for R,, we introduce two unit vectors eH1 and eH2which are parallel to the axes of helices 1 and 2, respectively, and pointing in the direction from the N terminus of the helix toward its C terminus (i.e., from A toward B in Figure 1). By definition (3)

Hence, using eq 2

Figure 2. Definition of the parameters that describe the relative position and orientation of two helices in an ( f , g , h )coordinate system. The h axis of the reference coordinate system coincides with the axis of helix 1. Parts a and b define the rotation of helix 2 with respect to helix 1, expressed in terms of Euler angles. (a) Definition of the axis f’ around which the axis of helix 2 is rotated (by the angle p). This axis is located in the ( f , g )plane, and it forms an angle a with the f axis, corresponding to a rotation by a around the h axis. The g’ axis also lies in the (f.g) plane, and it forms an angle CY with the g axis. When P = 0, the axes of helices 1 and 2 coincide. (b) Rotation of the axis of helix 2 by an angle 0 around axis f’and rotation of helix 2 around its own axis by the angle y. The plane of this drawing contains the h axis and it is perpendicular to the f’axis. I n an alternative definition of the angle a , it can also be considered as the negative of the rotation of helix 1 around its own axis, with respect to f’. (c) Translation of the origin O2 of helix 2 (marked by a heavy dot) with respect to helix 1. The displacement in the ( f , g )plane (from O1 to 0,) is described by the vector t,, positioned at an angle tmwith respect to the f axis. Equivalently, this vector can be described by its f and g components, tf and tg . The displacement along the h axis is described by the vector th .

When these two vectors are used, the magnitude of the angle between them can be expressed in the usual sense of vector algebra as follows: IQOl = c0s-I (eH1’eH*) = IPI (5) Equation 5, however, does not give the sign of R,,. The following derivation is necessary to define the sign. Construct two unit vectors, e, and e,, as follows: e, = (cos a)ei + (sin a)eg (6) (7)

where e , is a unit vector, lying in the (f,g) plane. It specifies the orientation of the f f axis of a new coordinate system v’,gf,h)that is obtained from the coordinate system

Energetic Approach to the Packing of a-Helices

The Journal of Physical Chemistry, Vol. 87, No. 15, 1983 2871

where

ts tf e,* = r e f - ;eg r

is a unit vector, in the (fg)plane, perpendicular to e, (defined in eq 7), and the vector e, is the projection of eH2 onto the S, plane. The quantity eH;er* is the projection of eH2(and of e,) onto e,*, and eH;eH1 is the projection of eH2 (and of e,) onto eH1,as shown by the dotted lines in Figure 3. Substituting eq 3, 4, 8, and 11 into eq 10, we obtain Q, = tan-l (p tan 8) for -90” Ip I90’ = tan-l (p tan p)

P + -180’

IPI P = tan-’ (p tan 0) - -180’ lPl

f

J

Figure 3. Definition of the projected torsion angle Q,, lying in the “contact plane” S,. The unit vector e,, defined in eq 7, lies in the ( f , g ) plane of the ( f , g , h )coordinate system. The plane S, is perpendicular to e,, and hence it is parallel to the h axis. The unit vector e,* is perpendicular to e, and lies in the intersection of the ( f , g ) and S, planes. The unit vectors eH1and e,, are parallel to the axes of helix 1 and 2, respectively;eH1lies in the plane, while eH2generally does not lie in S,. The vector e, denotes the projection of eH2onto the S, plane. Then Qp is defined as the angle between e,,, and ep. A positive !Jpcorresponds to a clockwise rotation when going from eH1toward e,, as seen from the origin 0,.Therefore, !Jp is negative for the example shown in the figure.

6,

(f,g,h) by a rotation by an angle a! around the h axis (see Figure 2a). The vector t, = t#f + tge,, lying in the (fg) plane, specifies the position of the origin 0, of helix 2 after the application of the particular translational operation T,= (tf,tg,O)(see Figure 2 4 ; e, is a unit vector specifying the direction of the vector t,, i.e., t, = t,e, where t , = ( t r t,2)1/2. A projection factor p may be defined as the projection of e, on e,, viz. as

+

As shown in Appendix B, the magnitude and sign of Qo can thus be given by Qo = p f o r p > 0 = -p f o r p < 0 (9) With this choice of the definition the orientation angle Qo is positive if the far helix is rotated in a clockwise sense with respect to the near one (stated eq~ivalently?~ if the near helix is rotated counterclockwise with respect to the far one). When p = 0, the two helix axes are coplanar. In this case, the magnitude of Qo is given by eq 5, but its sign cannot be defined in a unique manner. C. The Projected Torsion Angle Q,,. This quantity (which was used in ref 5) can be used as an alternative to Qo for comparisons with earlier work.5 It is the angle between the projections of the two helix axes onto the ”contact planen5 S, (see Figure 3). The “contact plane” is a useful formal mathematical construct. It is defined as any plane perpendicular to e, and hence parallel to the h axis. Then, Q, is defined as the angle between the projections of the two helix axes, i.e., of the two vectors eH1and eH2,onto this plane (Figure 3), and it is given by

for 90’ I@ I180’ for -180’ 5 p 5 -90’

(12) A comparison of eq 9 and 12 shows that the difference between Qo and Q, lies in the projection factor p . When p = 0, the two helix axes are contained in the plane, defined by h and e,, that is perpendicular to the “contact plane”; hence, Q, = 0. When the projection factor p = k l , we have $2, = Qo = kp,as expected, because in this case e, and e , are colinear, and therefore both eH1and eH2 are perpendicular to e, and hence parallel to the “contact plane”. When 0 < lpl < 1, Qo z Q,. The above definitions of the “contact plane” and of Q, are valid for any pair of helices, whether they are actually in contact or not. In the special case of two smooth cylinders in actual contact, however, an intuitive geometrical picture can be associated with the “contact plane”: a particular position of the “contact plane” can be chosen along t, such that this plane actually contains the point of contact (in the case of nonparallel helices) or the line of contact (in the case of parallel helices) between the two smooth cylinders. The torsion angle Q used in ref 5 corresponds to Q, as defined here. Hence, the present results can best be compared with those in ref 5 in terms of Q,. D . The Distance of Closest Approach, D . Along the two helix axes, the ends of the helices are given by pairs of points Al,Bl and A2,B2,respectively (Figure l),where A and B are the projections of the first and last heavy atoms (i.e., non-hydrogen atoms) of each helix on its own axis. Then the length of these t,wo iden!ical heljces is L = lAiB1l = 101A1-01Bll = IA2B21 = 101A2 - 01B21 (13) where 03;is the vector from the origin 0 to point Al, etc. Then D is defined as the distance of closest approach between the line segments AIBl and A & . For two infinite nonparallel lines, D is the length of a segment that connects the two lines and is perpendicular to both. Its derivation is given in eq 14-24. This definition also is valid for certain relative positions of finite helices [section IID(l)], but it must be modified for parallel or antiparallel helices [section IID(2)l and for certain relative orientations of nonparallel helices of finite length, as described ‘in sections IID(3) and IID(4). In order to specify concisely and unambiguously whether a given point on the helix axis (considered as an infinite line) lies within the helix (i.e., on the line segment AIBl or A a 2 )or outside it, we found it helpful5 to describe the location of this point in terms of a numerical parameter S1 or S2. The position of any point on the axis of the first helix cankexpressed, by vector addition, in parametric form as OIAl + SILeHland that of any point on the axis

-

__)+

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-+

The Journal of Physical Chemistry, Vol. 87, No. 75, 7983

of the second helix as OIAz &LeH, (Figure l),where S1 and S2are variable parameter^.^ If 0 IS1I1, the point defined by S1is located on the line segment bounded by points A, and B1,and it is defined to fall within the helix, but if S1 < 0 or S1 > 1, the point defined by S1 is located on the extension of this segment along the helix axis. An analogous relation holds for Sz. The distance between any two points on the first and second helix axes can be expressed as a function of S1and S2,viz. f(S,,S2) = + SlLeH,) - ( z 2 + S&eH,)I (14)

Chou et al.

(a)

Hellx 2 P

2

((OX

The distance of closest approach, D, between the axes of the two helices is D = min f(S,,S,) = f(Slo,SZo) (15) To obtain D we set -.+ (df/ds,)s, = Ls1 - L(eH;eH,)S, + (0.4,- OA2).eHl = 0 (16) - + (df/dS2),, = L(eH;eH,)S, - LS2 + (OAl - OA,)*eH, = O (17) Solving eq 16 and 17 for S1 and S2 gives ( ~ A -I OA2)'[(eH1*eH2)eH2 -e ~ ~ ] SI0 = (18) L[1 - (eHl.eHJ21

(b)

Helix

2

--

- +

S*' =

( O A z - OAi)'[(eH1'eH2)eH1- e ~ , ]

If we let

=[

r

r

r

z,=E]

then, according to eq 1 --+ OA2=T-.[H]

(19)

L[1 - (eHl*eHJ21

Flgure 4. Definition of the distance of closest approach D between two helices with parallel @ = 0")or antiparallel = 180') axes. Only the case of parallel axes is shown here; the definition is analogous for antiparallel axes. Helix 2 is displaced along its own axis (parallel to h ) by t,, = L I- where is defined in eq 25. L is the projection of the vector M ,connecting the midpoints of the two helices onto the axis of helix 1. For positive values of (Le., t,, > 0), the displacement is in the positive direction of the h axis (to the right in the figure). For negative values of the displacement is in the negative direction (to the left). (a) Displacement corresponding to -1 I I 1, Le., by lthl 5 L . I n this case, the projection of segment A 262onto the h axis overlaps with the segment A 16 and the closest distance between these two line segments is given by the perpendicular distance D = t , between them, as given in the first lines of eq 26 and 27. (b) Displacement corresponding to > 1 or < -1, Le., by lthl > L . I n this case, the projection of A 2B2 onto the h axis does not overlap with the segment A @,, and the closest distance D between the two helices equals the distance between their two nearest end points, Le., the distance B,A2 for the example in the figure, as given in the second and subsequent lines of eq 26 and 27.

r,

,,

1

h , sin a sin 6 + t f - h , C O S s~ i n 0 + t, h , cos3 + t h

(21)

Substituting eq 3 , 4 , 20, and 21 into eq 14,18, and 19, we obtain f(S,,S2) = ( [ t f+ (h, + S2L) sin a sin 01' + [t, - (h, + S2L) cos cy sin PI2 + [h, + S1L - t h - (h, + S2L) cos /3]2)"2 (22) (tgcos a - tf sin a) cos /3 + (th- h,) sin p SI0 = (23) L sin p t, cos a - tf sin a - hl sin p s20 = (24) L sin /3 The numerical values of Sl0 and SZodetermine the manner in which the distance of closest approach of the two helices is computed. Four cases must be distinguished. (1) When both 0 ISlo5 1 and 0 5 S2" 5 1,the distance of closest approach is given by eq 15, i.e., the value of D is obtained by substituting into eq 22 the values of S,' and SZoobtained from eq 23 and 24. (2) When p = 0" or f18Oo, the axes of the helices 1 and 2 are parallel or antiparallel, respectively, and eq 23 and 24 cannot be applied. In this case, we define a scalar criterion r as follows, in order to determine whether the relative positions of the two helices are as in part a or b, respectively, of Figure 4

r

r

-

where r is the projection of the vector MIMzconnecting the midpoints of the two helices onto the axis of helix 1 (expressed in dimensionless units). When -1 I r I 1, the projection of the line segment AzBz onto helix axis 1 overlaps with at least some part of the line segment A,&. In this case, D is given by the perpendicular distance t , between the two helix axes (Figure 4a). When r > 1 or r < -1, the projection of AzBz on helix axis 1 does not overlap with the line segment A$,, because the longitudinal displacement thof helix 2 along its own axis (relative to helix 1) is larger than the length L of each helix. In this case, the distance of the two helices equals the distance between their nearest ends, i.e., it is the distance B1A2(as shown in Figure 4b) or A,& for parallel helices, BIB2or AlA2 for antiparallel helices, depending on the direction of the displacement described by the sign of tk Therefore, the closest approach between the axes of helices 1 and 2 can be expressed as 4 D = (OAz - OA,) X eH1 for -1 I r 5 1 and /3 = 0' or f180° for r > 1 and p = Oo = f(1,O) for r < -1 and /3 = 0" = f(0,l) for F > 1 and p = &180° = f(1,l) for r < -1 and ,6 = f180' = f(0,O) (26)

The Journal of Physical Chemistry, Vol. 87,

Energetic Approach to the Packing of &-Helices

Helix

1

(C)

,, ,, Htlii 1

Flgwe 5. Illustration of closest approach of the axes of helices 1 and 2 when (a) S Z ois outside the range (0,l)and S l 0 is inside the range, (b) both S l 0and S20 are outside the range (O,l), and (c) not only S l 0 and S20, but also P,and P, are outslde the range (0,l). For simplicity, the axes of the two helices ( A ,,8 1) and ( A 2 , 6 2 ) are assumed to be coplanar, Le., the two helix axes are shown to intersect at the point X. This point is defined in terms of S l 0and S20, given by eq 23 and 24.

or, substituting eq 3,4,20, and 21 into eq 25 and 26, we have D = t, for -L I t h I L and = Oo or 2hl I th I 2(h1 + L) and = f180° for t h > L and /3 = Oo = f(1,O) for t h < -L and p = Oo = f(0,l) for t h > 2(hl L) and /3 = f180° = f(1,l)

+

for t h < 2hl and p = f180° (27) where the function f is given by eq 22. (3) When the values of Sl0and/or S20, obtained from eq 23 and 24, do not fall into the range (O,l), the points defined by Sl0and/or S20 lie outside the line segments AIBland/or AzB2,i.e., a t least one of them is located outside the line segment defining the length of the helix. Therefore, the distance between the points defined by Sl0 and S , O does not correspond to the distance between the two finite-length helices. Examples of this case are shown in Figure 5. The figure is drawn for the sake of simplicity with the axes of the two helices coplanar, but coplanarity represents a special case, and it is not required for the arguments that follow. In parts b and c of Figure 5, Sl0 < 0 and SZo< 0, while in part a of Figure 5,O < Sl0< 1 but S20 < 0. The two helix axes intersect in these examples a t the points marked X, and Sl0 and S20, obtained by solving eq 23 and 24, define this intersection. In the examples of parts a and b of Figure 5, the distance of closest approach between the two line segments defining the where P1A2is perhelices is given by the distance pendicular to AIB1.This distance must be obtained by a special consideration, as indicated below, instead of applying eq 22-24. In Figure 5a, it would be incorrect to

1983

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substitute in eq 22 S1 = Sl0(as obtained from eq 18) and S2 = 0 (as was done in ref 5), because this would give the incorrect distance AzX instead of the correct distance A$',. Similarly, in Figure 5b, it would be incorrect to substitute S1= S2= 0 in eq 22 because this would give the distance A,A2instead of A,P,. In both examples shown in parts a and b of Figure 5, the value of S1 a t the point P1 is obtained directly by solving eq 16 for S1 after setting S, = 0, and it is donated as 6,. Similarly, when the distance of closest approach is given by B2P1(instead of A2P1), the value of S1a t the point P1is obtained by solving eq 16 for S1after setting S2= 1, and it is denoted as A,. Should P, fall outside the line segment A$, (Le., when 6, or A, < 0, or 6, or A, > 11, an alternate point P2 must be located as the perpendicular projection of A, (or B,, as the case may be) onto the axis of helix 2. In this case, the value of Sz at the point P2is obtained directly by solving eq 17 after setting either S1 = 0 or S1 = 1, respectively, and it is given by Sz = a2 (if 0 I 82 I 1) or Sz = A, (if 0 I A, 5 1). (4) When 6,, 62, A,, and A2, obtained as described above, all fall outside the range (O,l),P1is located outside the line segment AIBland P2 is located outside A2B2, as shown in the example of Figure 5c. It is only in this fourth case that the distance between the two helices is equal to the smallest distance between the ends of the two line segments describing the helices. For the example shown in Figure 5c, D = f(0,O) of eq 22, i.e., D is the distance between points A, and A2,with analogous expressions for other relative positions of the two helices. On the basis of the above analysis, the distance of closest approach in cases 3 and 4 is given by D = f(0,O) for 6, < 0, < 0 = f(0,6,)

for S1 < 0, 0 -< 6, -< 1

< 0, bz > 1 = f(6,,0) for S, < 0, 0 -< 6, -< 1 for S2> 1, 0 IA, I1 = f(Al,l) for 6, > 1, A, < 0 = f(1,O) for S, > 1,0 -< A, I 1 = f(l,Az) = f(1,l) for A, > 1, A, > 1 (28) = f(0,l)

where

= f(0,O)

lm,

No. 15,

for A,

--

(OA2 - OAl).eH, 6, = L

62

--

+

~ ~ C p O S th

- hl

L

(29)

= (OAl - O A ~ ) * ~ H ,=/ L

--

+ (ts cos a - tf sin a) sin p - h,]/L (31) A2 = [L(eH;eH,) + (OAl - OA~)*C?H,]/L = [ ( L+ hl - t h ) cos + (tgcos a - tf sin a)sin 0 - h,]/L [ ( h ,- t h ) cos p

(32) The above equations are derived directly from eq 16 and 17 by setting Sz = 0 or 1 or S1= 0 or 1, respectively. 111. Energy Minimization

For brevity, and a-helix is symbolized here by aW-(R),-Yl where X and Y denote end groups, R represents any amino acid residue, and n, is the number of residues in the helix.

The Journal of Physical Chemistry, Vol. 87, No. 15, 1983

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Chou et al.

TABLE I: Initial Positions for t h e Minimization of t h e Interaction Energy between Two CH,CO-(L-Ala),,-NHCH, a-Helicesa Euler angles, deg a

B

b 0 0 0 0 0 0 0

45 90f 135 180g -135 -90f -. 45

5

oe

0 b b b b b b b

translational displacements, A

tf 11.15 11.15 11.15 11.15 11.15 11.15 11.15 11.15

t, 0.00 -2.21 -3.12 -2.21

0.00 2.21 3.12 2.21

thd 0.00 0.91 3.12 5.33 6.24 5.33 3.12 0.91

a All starting values for the Ca-C$ dihedral angles (XI) were 60". This angle was varied in 30 steps from 0 This dist o 360" t o obtain the various starting points. tance is two times the maximal peripheral radius of the a-helix considered here. The peripheral radius r p of any atom j of an @-helixis defined as rp, = Rj + r j , where Rj is the distance of atom j from the he ix axis and rj is its van der Waals radius (given in Table 11). The maximal peripheral radius of an o-helix is defined as the highest value The starting values of tg and th were taken by any r p . chosen so as to keep the distance between the midpoints M I and M 2of the t w o helices equal t o t f for any initial choice o f 3 , Le.. the two helices are rotated around an axis connecting their midpoint in the starting positions with various values of p . e Parallel orientation. f Perpendicular orientation. Antiparallel orientation.

We calculated and minimized the energy of interaction between two identical right-handed a(CH3CO-(L-Ala)loNHCH3)molecules, i.e., two equivalent a-helices each of which consists of ten alanine residues with CH,CO- and -NHCH3 end groups. The atomic coordinates within the first a-helix were computed from the dihedral angles by means of the ECEPP algorithm'l (empirical conformational energy program for peptides). This procedure generates the coordinates in a coordinate system (x,y,z) that is based on the orientation of the first residue of the polypeptide chain.'l For the purposes of the present study, it is advantageous to express the coordinates of the a-helix in terms of the helical coordinate system (f,g,h)defined in section IIA and in Figure 1. Coordinates expressed in the (x,y,z) system can be transformed into the (f,g,h)system by using the relations given in Appendix A. In the latter coordinate system, the coordinates of the second helix (taken here to be identical in sequence and conformation with the first helix) are obtained as a function of six external variables, i.e., the three Euler angles (a,p, y) and the three translational displacements (tf,tg,t h ) , as described in section 11. During energy minimization (with the ECEPP algorithm), the first helix was fixed and the second helix was allowed to move. In the study reported here, interactions of only regular poly(L-alanine) a-helices are considered. All backbone dihedral angles were kept fixed a t the reference values" (@,+,u) = (-57",-47°,1800). Thus, the intramolecular energy of each helix is constant, except for very small variations due to small changes of the side-chain dihedral angles x. The total energy of the two a-helices (including both intra- and intermolecular energies) was minimized with respect to all side-chain dihedral angles (x's) and the six external variables (a,P,y,tf,tg,th). In transforming the coordinates of an a-helix from the ( x , y , z ) to the (f,g,h)system, the values of L and hl are (11) Momany, F.A.;McGuire, R. F.; Burgess, A. W.; Scheraga, H. A. J. Phys. Chem. 1975, 79, 2361. (12) IUPAC-IUB Commission on Biochemical Nomenclature, Biochemistry 1970, 9, 3471

I Flgure 6. Schematic illustration of the nonbonded (van der Waals) interaction energy U ( r ) between two atoms a s a function of their separation r . The energy is a minimum at point M ( r = d o ) and it is zero at point M I(r = CT). Point M , (with r = d ' ) is chosen so that the distances M,M and MM, are equal.

TABLE 11: van der Waals Radii of Atoms in Polypeptidesa atom typeb

HI H, C, C,

N,, 0,-

description

r o . ."i

aliphatic hydrogen primary and secondary amine or amide hydrogen aliphatic carbon carbonyl, carboxylic acid, o r peptide bond carbon primary or secondary amide nitrogen carbonyl or carboxylic acid oxygen

1.46 1.34 2.06 1.87 1.99 1.56

Derived from column 5 of Table IV in ref 15. The minimum of the pairwise nonbonded contact energy between t w o like atoms occurs a t a distance 2r0. This disUsed in ref 11 and tance was denoted ( r g k h ) in ref 15. 15.

obtained directly (see eq 13 and A-13). Substitution of L, h,, and the energy-minimized values of a, 0,y, tf, tg,and th into eq 9, 12, and 22-32 yields Q,, Q,, and D, the parameters that characterize the optimal packing of two a-helices. Ninetysix different relative positions of the two helices were selected as starting points for energy minimization, as described in Table I. The choices of the initial values of /3 cover the entire range of relative orientations of the helix axes, and the choices of y (or a ) cover all relative rotations of the helices. The initial distance between the two axes ( t f )was chosen as the smallest separation that assures the absence of repulsive overlaps between any two atoms of the two helices. The value of t f is twice the maximal peripheral radius of an a-helix as defined in footnote c of Table I. The choice of the initial t, and t h is explained in Table I. Energy minimization was carried out by using the POWELL algorithmI3 on a Prime 350 minicomputer with an attached Floating Point Systems array processor.l*

IV. Physical Picture of Packing In order to analyze the packing of helices in detail, we must have a physical picture of the extent of contact between residues, or better yet, between atoms. First, we define what is meant here by contact between two atoms. Figure 6 shows the nonbonded (van der Waals) potential Um(r) as a function of the distance between two (13) Powell, M.J. D. Comput. J. 1964, 7, 155. (14) Pottle, C.;Pottle, M. S.; Tuttle, R. W.; Kinch, R. J.; Scheraga, H. A. J. Comput. Chem. 1980, 1, 46.

The Journal of Physical Chemistry, Vol. 87, No. 15, 1983 2875

Energetic Approach to the Packing of a-Helices

t

A

l

Figure 7. Stereoscopic picture of two CH3CO-(L-Ala),o-NHCH, ahelices in the lowest-energy packing state, with ! l =o -153.5' or Qp = -155.0'. The helix axes are indicated by arrows, with the head of the arrow pointing in the direction of the C terminus of each helix. The helix in front is indicated with shaded atoms and bonds. Hydrogens of the alanyl and terminal methyl groups have been omitted. Only the centers of atoms are indicated, not their van der Waals surfaces. The intercalation of the two helices can be seen in the region of contact: Each turn of one helix fis into the space between the turns of the other helix.

atoms. According to the expression for the Lennard-Jones 6-12 potential15 used here

d" = 21/6g (33) where do is the minimum of the potential curve, and g is its intersection with the abscissa. It is considered here to be the sum of the van der Waals radii of the two atoms concerned (see Table 11). In practice, it is necessary to consider a range of distances around d o , within which atoms are defined to be in contact. UNBis negative (i.e., favorable) a t distances r > CT. In order to allow for equal fluctuations for r above and below r = do, we define point M 2 so that the distances MIM and M M 2 are equal (Figure 6). Thus, we define a cutoff distance d* d* = do + (do - g) (2 - 2-1/6)d' (34) The following two definitions of contact are based on the application of eq 34. Definition 1. Two atoms, from the first and second helix, respectively, are defined to be in contact if the distance r between their centers is less than d*, i.e. T < d* = (2 - 2-l/9(ri0 rj") (35)

+

where ri' and rjo are the respective van der Waals radii (Table 11). Definition 2. Two residues, from the first and second helix, respectively, are defined to be in contact if they have at least one pair of atoms in contact, as given by definition 1.

V. Results and Discussion A. Results of the Energy Minimization. Starting from the 96 initial positions of the two helices (Table I), we obtained 18 minimum-energy packing arrangements. The three lowest-energy structures have nearly antiparallel helices (with Qo = -153.5', 170.0°, and 145.7", respectively), (15) Momany, F. A.; Carruthers, L. M.; McGuire, R. F.; Scheraga, H. A. J. Phys. Chem. 1974, 78, 1595.

Figure 8. Stereoscopic picture of the packing of two a-helices with Qo = 170.0' or Q,, = 172.0'. See Figure 7 for further explanation.

and their energies are within 1.8 kcal/mol of the energetically most favorable arrangement. Seven other structures have relative energies ranging from 3.3 to 5.3 kcal/mol, while the relative energies of all other structures are a t least 6.0 kcal/mol. The energies and geometrical parameters characterizing the ten lowenergy packing arrangements are listed in Table 111. Stereoscopic drawings of representative examples are shown in Figures 7-12. In addition to the 18 packing arrangements, some other energy minima were obtained, but these do not represent different structures. They correspond to the movement of one helix along the other along a screw line into equivalent positions, i.e., with the same orientational torsion angles and similar sets of residues in contact, but with different values of the Euler angles and translational displacements. Such structures were not considered further. The packings described in Table I11 are essentially the resultants of the optimization of the total interchain energy, which in turn is the sum of interchain nonbonded and electrostatic energies. The intrachain energy is nearly invariable, the only source of its change being the variation of the xi's. The interchain energy (and hence the total energy) is most favorable in the three packing arrangements with large absolute values of Qo, i.e., when the two a-helices are nearly antiparallel. This reflects to a considerable extent the important role played in the packing of a-helices by electrostatic interactions of the net dipoles of the two a-helices. In an a-helix, the individual peptide dipoles are oriented nearly parallel to the axis of the helix. Their alignment gives rise to a dipole of considerable strength.16-18 As a result, the electrostatically most favorable orientation of two a-helices would be one with an antiparallel orientation of the helix axes. In fact, two of the nearly antiparallel packings (lines 1and 2 of Table 111) have much lower electrostatic energy than the others, confirming the above conclusion. In general, the electrostatic energy increases with decreasing IQol, as expected for two dipoles located side by side. The parameters used (16) Wada, A. Adv. Biophys. 1976, 9, 1. (17) Hol, W. G. J.; Halie, L. M.; Sander, C. Nature (London) 1981,294, 532. (18) Sheridan, R. P.; Levy, R. M.; Salemme, F. R. Proc. Natl. Acad. Sci. U.S.A. 1982, 79,4545.

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The Journal of Physical Chemistry, Vol. 87, No. 15, 1983

Chou et al.

t r

\

\

Figure 9. Stereoscopic picture of the packing of two a-helices with Qo = -35.7'

or

Qp

= -28.7'.

I Figure 10. Stereoscopic picture of the packing of two a-helices with Q,

I = -135.9' or !Jp= -144.5'.

See Figure 7 for further explanation.

= 29.5'.

See Figure 7 for further explanation.

Figure 11. Stereoscopic picture of the packing of two a-helices with Qo = 30.1' or

in the ECEPP algorithmll for the representation of electrostatic interactions correspond to a macroscopic dielectric constant of about 4-8, i.e., to a polar organic medium.l'

See Figure 7 for further explanation.

Qp

Therefore, the electrostatic interactions computed here are the appropriate ones for the energetics of a-helices inside a globular protein molecule.

Energetic Approach to the Packing of a-Helices

The Journal of Physical Chemistry, Vol. 87,

No. 15, 1983 2077

The role of electrostatic interactions was pointed out in several recent studies,l6-l8but it was already indicated in an earlier, niore limited investigation, in which it was shown that a strictly antiparallel arrangement of two long (24-residue)linked poly(L-alanine)a-helices is energetically more favorable than a conformation of the same polypeptide that consists of a single long helix.lg An approximately antiparallel packing of two linked short ahelices was also obtained in the energy minimization of a simplified model of a polypeptide chainsz0 The electrostatic energy is, however, not the only determinant of the favored torsion angle of packing (ao).The total interchain energy of the stable packings contains a major contribution from nonbonded interactions. Such interactions between atoms on the surface of the two helices tend to favor orientations in which the atoms pack most compactly. The a-helix is not a smooth cylinder, and its surface exhibits regions of appreciable convexity and concavity, described in the various geometrical models as "knobs and holesnZor "ridges and groove^".^^^ Optimization of the nonbonded interactions requires good geometrical fitting of the surfaces because this maximizes the number of atoms in close contact; hence some kind of intercalation between the convex and concave regions of the two helices is favored. The presence of such intercalation is indicated by the values of D, the distance of closest approach of the two helix axes, listed in Table 111. They are in the range 6.7-7.7 A, Le., they are considerably smaller than 11.1A, twice the size of the maximal peripheral radius of the a-helix (as defined in footnote c of Table I). This indicates that the minimum-energy conformations correspond to closer packing than what would result from peripheral contacts between two "knob" regions. The values of D also are smaller than twice what has been called the "radius of an a - h e l i ~ (-4.7 " ~ A). While the lowest-energy packing (i.e., the one with Oo = -153.5') also possesses the largest number of pairwise atomic contacts (Table 111), it must be remembered that the optimal fit is characterized not by the purely geometrical quantity of number of contacts but by the optimization of the total energy summed over all pairs of atoms. In summary, the packing of the helices in the minimum-energy structures listed in Table I11 is influenced by both nonbonded and electrostatic interactions. In terms of total values, the interchain nonbonded energy is much larger in magnitude than the electrostatic energy (Table 1111, and thus it dominates the overall stability of packed a-helical arrangements. The contributions of the two energy components to the energy differences between the various structures listed in Table I11 generally are, however, comparable in magnitude. This indicates that both components are equally important in determining preferences of packing. This result underscores the need to consider the total intermolecular energy rather than only geometric factors in the analysis of the packing of structural elements in folded globular proteins. It is seen from Table I11 that the total energy bears no simple relationship to any single one of the geometrical parameters (such as D, Qo, Qp, or the number of residues or atoms in contact), i.e., the energy depends on the details of the particular packing arrangements rather than on any individual feature of the packing. The dependence of the packing energy on the length of the interacting a-helices was not investigated in this study, but some general conclusions can be drawn from the re(19) Silverman, D.N.;Scheraga, H. A. Arch. Biochem. Biophys. 1972, 153, 449. (20) Rackovsky, S.; Scheraga, H. A. Macromolecules 1978, 11, 1.

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The Journal of Physical Chemistry, Vol. 87, No. 15, 1983

t

s

'I Figure 12. Stereoscopic picture of the packing of two a-helices with

sults. Lengthening of the a-helices is likely to increase the energy difference between the low-energy nearly antiparallel packings and the others listed in Table 111, for two reasons. (a) The electrostatic energy of the nearly antiparallel packings is lowered because of the linear dependence of the dipole moment on the length of the helix, while there is less of an effect on the other packings. (b) The total nonbonded interchain energy of the nearly antiparallel (as well as parallel) packings is lowered, because of the increase in the number of contacts, while that of other packings remains essentially constant.21 Thus, the first three packings listed in Table I11 should be stabilized even more when the helices become longer. B. Comparison with Observed Protein Structures. A frequently observed arrangement in proteins consists of a bundle of four interconnected a-helices.1J8922*B Adjacent helices in such a bundle are nearly antiparallel to each other, with the mean angle between adjacent helix p a i d s near -162'. This structure is in excellent agreement with the lowest-energy packing computed here. The need to pack four instead of two helices and the presence of larger side chains can cause small deviations of the minimumenergy four-helix structure from the two-helix minimumenergy structure. Various other orientations of adjacent helices have been observed in proteins, in addition to the antiparallel arrangement.1*5 A detailed comparison of observed and computed a-helix packings will be carried out in a later paper dealing with nonequivalent helices containing other residues besides L-alanine, because the presence of larger side chains can affect the packing and result in some changes of the angles of orientation. It should be noted, however, that a frequently observed packing5occurs with Qp ranging from about -30 to -50'. This packing is very close to the lowest-energy computed conformation (line 4 of Table 111) among those that are not antiparallel. C. Comparison with a Geometrical Model of a-Helix Packing. The packing arrangements obtained in this study can be compared with those derived from a geometrical model of "ridges into grooves" introduced by Chothia et al.5 This model has been described5 as follows: "On the helix surface rows of residues form ridges separated by shallow grooves, and helices pack with the ridges of one packing into the grooves of the other and vice versa. The ridges and grooves are formed by residues whose separation

I

a,, = -86.5'

or

a,

= -85.5'.

in the sequence is usually four, occasionally three, and, rarely On the basis of this assumption,the projected torsion angle was calculated as Qij = -(q+ U j ) (i, j = 1, 3, 4) (36) where the subscripts i and j refer to the type of intercalation. For example, in the packing with the torsion angle QM, which we shall abbreviate as packing 34, ridges formed by residues whose separation in the sequence is three residues and the grooves located between these ridges on one helix are intercalated with ridges whose separation is four residues and grooves located between these ridges in the second helix. In eq 36, aiis the angle between the helix axis and the line that joins residues separated by i in the ~ e q u e n c e .The ~ numerical values of ai,viz.

w3 =

tan-l

o4 = tan-l

TR [ b( 7 )] [ $(TR n)] 3-n

= -49.0'

4-n

= 25.8"

(38) (39)

correspond to a particular geometrical model: wth R = 4.7

A the "radius of a helix" (not defined clearly in ref 5), d = 1.51 A the rise per residue along the helix axis, and n

= 3.6 the number of residues in each turn of helix. They change when the parameters are altered.5 When comparing the present results with those calculated in ref 5, it must be noted that the direction of the individual helices was ignored in the geometrical m0de1,~ i.e., no distinction can be made between packings with Qij and Qijf 180'. This is a consequence of the use of a purely geometrical packing scheme, and it is a shortcomingof the geometrical model, because the actual difference in energy between a parallel and an antiparallel pair of helices is considerable,16 as discussed above in section VA. As a result, each type of intercalation i j can have two values of Qij associated with it, as summarized in eq 40 (calculated from eq 36-39). The i j = 11 packing was not considered Q44 = -51.6" or 128.4' Q34 = -156.9" or 23.1" Q14 = -105.3' or 74.7" Q33

(21) Ngmethy, G. Biopolymers 1983,22, 33. (22) Argos, P.; Rossmann, M. G.; Johnson, J. Biochem. Biophys. Res. Commun. 1977, 75, 83. (23) Weber, P. C.; Salemme, F. R. Nature (London) 1980, 287, 82.

See Figure 7 for further explanation.

= -82.2' or 97.8'

Q13

= -30.6' or 149.4'

Qll

= -159.0" or 21.0'

The Journal of Physical Chemistv, Vol. 87, No. 15, 1983 2879

Energetic Approach to the Packing of a-Helices

i

TABLE IV: Relationshipa between t h e Torsion Angles (in degrees) Obtained from Energy Minimizationb and Those Calculated from a Geometrical ModelC energy-minimized structuresb

A

E

0.00 0.76 1.74 3.35 3.86 4.35 4.57 4.92 5.17 5.30 6.31 9.18 10.53

projected torsion classification ~ angle ~ a , ~ of packinga ~ ij - 155.0 172.0 149.9 - 28.7 128.6 29.5 76.8 - 144.5 - 159.3 -85.5 -52.3 - 120.1 84.9

/””

geometrical (“ridges into grooves”) modelC

34(-) nonef 13(+) 13(-) 44(+ ) 34(+)

torsion angle ai, - 156.ge 149.4 - 30.6

128.4 23.1e 74.7

14(+)

none 34(-) 33(-) 44(-) 14(-) 33(+)

156.ge -82.2 -51.6 -105.3 97.8

-

The value of ij and the sign associated with each computed minimum-energy packing has been selected as described in the t e x t , Present work. Reference 5. Total energy, &Etot,from Table 111. Some high-energy packings, not shown in Table 111, are included here. e As discussed in the t e x t , packings with ij = 34 and 11 have nearly identical values of a ii and are therefore not distinguishable. f There is no “ridges into grooves” arrangement corresponding t o this computed packing. a

in ref 5. In fact, the values of all are very close to those of 334,so that the two packings are not distinguishable within the precision of the geometrical model. In order to distinguish the two orientations of the helices corresponding to a given ij classification, we introduce the nomenclature ij(+) and ij(-), referring to the orientations with positive and negative values of aij,respectively, in eq 40. The packings of the geometrical model5 are compared in Table IV with the minimum-energy packings computed in this work. Each computed packing was associated with an ij class by using two criteria: (a) the values of 3, and 9 , had to be similar, and (b) the residues in actual contact (as defined in section IV) in each computed structure had to be those defined by the values of i and j. Energy-minimized structures can be found that correspond to every one of the ten classes of packing derived from the geometrical model, but some of these structures have a high energy. This result indicates that good geometry of packing does not necessarily mean that the packing is favorable in terms of energetics. On the other hand, two low-energy structures (with AI3 < 5.0 kcal/mol) were found which could not be obtained by means of the geometrical “ridges into grooves” model, Le., there are some energetically favorable ways of packing two a-helices that do not correspond to the idealized “ridges into grooves” model. Furthermore, there are two minimum-energy packings with greatly different energies (AI3 = 0.00 and 5.17 kcal/mol, respectively) but similar values of Q,, which correspond to the ij = 34(-) packing. Thus, for a given class ij and the same direction of the helices, it is possible to have two slightly different packings with different energies, i.e., the ij classification is not unique. For comparable energy-minimized and geometrical structures in Table IV, the respective values of 3, and 3i; are closely similar, except in some structures with very high energies. This result indicates that the geometrical model predicts the relative orientation of two packed poly(Lalanine) a-helices well, but it cannot compare the relative stability of the various arrangements of packing, and it

x

f Flgure 13. A general Coordinate system (x ,y , z ) and the helical coordinate system ( f , g , h ) . N,, ca,and C’, (i = 1, 2 , ...) are the atoms of the backbone chain of the ahellx. The dashed-line circles indicate the path of each atom when the helix is rotated about its own axis (Le., the coordinate axis h ) .

Figure 14. The projection of the nitrogen atoms of an a-helix onto a plane perpendicular to the h axis. The projections are denoted by asterisks. See the text of Appendix A for the explanation of the other symbols.

misses some low-energy structures.

VI. Conclusions Energy minimization with respect to relative position and orientation was carried out for two equivalent rigid a-helices, each consisting of ten L-alanine residues with CH3CO- and -NHCH3 end groups. Ten low-energy packing arrangements were found. The orientation torsional angle between the helix axes in these minimumenergy structures depends mainly on the nonbonded interaction energy between residues in contact, but it is also influenced by the electrostatic energy. The latter arises largely from the interaction between the helix dipoles, and it tends to align the helices in an antiparallel fashion. Because of favorable electrostatic and nonbonded interactions, the intermolecular energy of three nearly antiparallel packings (no= -153.5’, 170.0°,and 145.7’) is much lower than the energy of other stable packings of two helices. The first arrangement is observed frequently in

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Chou et al.

The Journal of Physical Chemistry, Vol. 87,No. 15, 1983 -

proteins that contain a bundle of four closely packed helices.

CY-

Note Added in Proof. The computations reported in this paper are based on a reference a-helix with dihedral angles (4, J/, w ) = (-57O, -47O, M O O ) , as described in section 111. Energy minimization of an a-helix with respect to the dihedral angles gives a minimum-energy structure with somewhat different dihedral angles, viz. (4, J/,w ) = (-68O, -38", 180'). The packing of two a-helices with these dihedral angles gives essentially the same minimum-energy structures as shown in Table 111, with Ehkrchanging less than 0.2 kcal/mol and nochanging less than 3O. Thus, the results reported here are valid for both sets of dihedral angles. Acknowledgment. We thank M. S. Pottle and S. Rumsey for help with the computations and for preparation of the ORTEP diagrams. A valuable discussion with Professor R. Connelly is also acknowledged. This work was supported by grants from the National Science Foundation (PCM79-20279 and PCM79-18336), from the National Institute of General Medical Sciences (GM-14312 and GM-25138) and the National Institute on Aging (AG00322) of the National Institutes of Health, US.Public Health Service, from the Mobil Foundation, and from the National Foundation for Cancer Research. Appendix A. Transformation of Coordinates t o a Helical Frame The coordinates of an a-helix are expressed in a common reference frame (x,y,z)when they are generated by means of the ECEPP algorithm1' and then transformed to a helical coordinate system (f,g,h).The method described by Sugeta and MiyazawaZ4and used by McGuire et a1.26has been applied in this work. Their derivations were re-formulated here, in order to adapt the formalism to the special case of a-helices and to simplify the definitions of various quantities used in this paper. Figure 13 shows a helical coordinate system in which h is the axis of a regular CYhelix, and N,, C,* and C,' are the atoms along the backbone chain of the a-helix, where the subscript i (i = 1, 2, ...) refers to the ith amino-acid residue. Let the perpendicular distances from the atoms N,, Cia,and C,' to the helix axis be pN,,pc, and pcz, respectively. Then, for a regular a-helix PN1 = P N 2 PCp

*e*

- PC2n --

PC,/ =

Let

--

Pc,

--

--

A = N,-lN, - N,N,+'

*e.

-- PN, = P N -- PC,n = Pc, -- Pc,, = Pc,

(i = 2, 3, ..., n - 1) (A-2)

... = lAn-ll = A

_

f

,

ii

I

.

(A-5)

A'

Since the triangles ON*i-lN*i and N*i-lN*iQiare similar, we have

where (A-7) Combining eq A-6 and A-7, we obtain PN =

A

4 sin2 (8/2) Since N*i-lN*iNi is a right triangle, and by locating N*;-l at Ni-l (A-9) As indicated in Figure 13, the origin 0 of the helical coordinate system is defined as the projection of the atom Ni onto the h axis, and the line from 0 to Ni defines the direction of axis f. The axis g is chosen so that (f,g,h)forms a right-handed coordinate system. Thus, according to Figures 13 and 14, the unit vectors along the helical coordinate axes can be expressed as ei = Ai/A (A-10)

eg = e,, X ef

(i = 2, 3, ..., n - 1)

(A-11) (A-12)

On the other hand, the vector from 0', the origin of the (x,y,z)system, to 0, the origin of the (f,g,h)system (see Figure 13), is equal to the vector from 0' to N, minus the vector from 0 to N,, i.e. 4 0 3 = O'N, - pNef (A-13) From eq A-10 to A-13, we immediately obtain the transformation relationship from the (x,y,z) system to the (f,g,h) system as follows:

(A-1)

where N X is the vector from the (i- 1)th nitrogen atom to the ith one. Clearly [A21 = 1A3) =

_

vectors N*iN*i+land N*,-,Qi are parallel and equal to each other. Then. bv vector addition, A; can be written as + = QiN*i = N * ~ - ~ N *N*,_,Q*) ~= N*i-lN*! - N*iN*i+: (A-4) and is directly radially. The helical parameters, viz. 8, the angle of twist per residue, p N , the helical radius (i.e., the distance from the helix axis), and d, the rise per residue, are obtained as follows:

where (ef), is the x component of ef, etc; XN,, Y N , ,ZN, are the coordinates of the N, atom in the (x,y,z)system; and (ef)x

(er),

(ef)z

(eh)x

(eh)\

(eh)z

(A-3)

The projections of points N,-', N,, N,+' and N,+2onto a plane perpendicular to the h axis are N*,-', N*,, N*,+', and N*r+2,respectively. For a regular helix, these projections are spaced equidistantly on the circumference of a circle (Figure 14). Point Q, is defined such that the

The reverse transformation is given by

(24) Sugeta, H.; Miyazawa, T. Biopolymers 1967, 5, 673. (25) McGuire, R. F.; Vanderkooi, G.; Momany, F. A.; Ingwall, R. T.;

Crippen, G. M.; Lotan, N.; Tuttle, R. W.; Kashuba, K. L.; Scheraga, H. A. Macromolecules 1971, 4 , 112.

where

J. Phys. Chem. 1083, 87, 2881-2889

H-’=

[

]

(ef)x

(egIx ( e h I x

(ef)s

(eg)s ( e h I y

(ef)Z

(eg)s

(A-17)

(eh)r

Appendix B. Definition of the Projection Factor P

The projection factor p , as defined in eq 8, permits a decision concerning the sign of no,by correlating the direction of translation of helix 2 in the (fg)plane, given by t, = t,ef tgeg (Figure 2c) and the direction of the axis f ’ (Figure 2a) around which the axis of helix 2 is rotated with respect to the axis of helix 1 by the angle /3 (Figure 2b). According to the definition of the angles a,/3,y in the Euler rotational operation, the axis f ’lies in the (fg)plane (Figure 2a). The angle between axes f and f ’is a. Looking along f’ toward the positive direction of the f’ axis, /3 is defined to be positive if the axis of helix 2 is rotated in the clockwise direction relative to helix 1 (Figure 2b). In order to define the sign of Q,, we consider it as a dihedral angle, formed by three vectors, eH and eHp(eq 3 and 4), pointing in the positive direction dong the axes of helices 1 and 2, respectively, and D which is perpendicular to both eH, and eH2(defined here to be directed from HI to Hz).According to the usual definition of the sign of dihedral angles,12this sign is taken to be positive if the unit vector that is farther from the observer is ro-

+

2881

tated clockwise with respect to the closer unit vector, or conversely, if the nearer one of the two unit vectors is rotated counterclockwise with respect to the farther one. The sign is negative for opposite rotations. In order to relate !doto 0, it is necessary to decide which helix is farther for an observer looking along the f ’axis in the positive direction. For this purpose, we consider the direction of the translational displacement t, relative to the direction of the f ‘axis. If e, is a unit vector along the f ’axis and e, is a unit vector pointing in the direction of the translational vector t,, as defined in eq 6 and 7, respectively, then the projection of e, on e, is given by eq 8. If p is positive, then the absolute value of the angle between f’and t, is 90° and helix 2 is moved toward the observer, so that it becomes the nearer helix. In this case, /3 and nohave opposite signs. The relationships described result in eq 9. If p = 0, t, is perpendicular to t,, i.e., the two helices are a t the same distance from the observer, measured along the f ’axis. Rotation by /3 around f’ takes place in the plane defined by eH and t,, i.e., the two axes of the two helices are coplanar. fn this case, the sign of Oo cannot be defined uniquely. Registry No. Ac-(L-Ala)lo-NHMe, 85251-49-6.

Time-Correlation Functions from Computer Simulations of Polymers Thomas A. Weber’ and Eugene HeHand Bell Laboratories, Murray Hi//, New Jersey 07974 (Received: December 14, 1982; I n Final Form: March 2, 1983)

Many of the rapid relaxation processes in polymers are related to conformational transitions of bonds from one state to another. Earlier research, based on Brownian motion computer simulation and kinetic theory, revealed that conformational transitions occur as single events and also as cooperative pairs. That research focused on kinetic rate parameters, but the experiments generally measure time-correlation functions of the system’s properties. This paper is a report on the results of computer simulations of polymer time-correlation functions. The decay with time of both initial conformational states and initial vector orientations has been studied. The correlation functions fit well to a two-factor functional form, suggested by recent theoretical research. The first factor is an exponential decay arising from single transitions. The second is a diffusional term related to the correlated pair transitions. The total transition rate when compared with the earlier kinetic studies is in reasonable agreement. Since the functional form has been shown here to fit the simulations,it is suggested that it will be of value in interpreting relaxation experiments.

I. Introduction Fundamental to the rapid relaxation processes which occur in polymers are the conformational transitions from one rotational isomeric state to another. NMR, dielectric relaxation, dynamical light scattering, and fluorescence depolarization are a few of the experimental techniques which may be used to study such relaxation processes. The maximum amount of information which can be extracted 0022-3654/83/2087-288 1$01.50/0

from these experiments would be a time-correlation function of some orientational quantity (generally a vector or a second-rank tensor). This orientation might be defined, for instance, by an electric dipole, a polarizability anisotropy, or the direction of a proton relative to a 13C nucleus. Because the direction is fairly rigidly defined relative to the polymer backbone, and reorients when bonds undergo conformational transitions, one could hope 0 1983 American Chemical Society