Utilization of shape data in molecular mechanics using a potential based on spherical harmonic...

Utilization of Shape Data in Molecular Mechanics Using a Potential Based on Spherical Harmonic Surfaces

ARUN MALHOTRA, ROBERT K.-Z. TAN, and STEPHEN C. HARVEY* Department of Biochemistry and Molecular Genetics, University of Alabama at Birmingham, Birmingham, Alabama 35294

Received 25 May 1993; accepted 30 August 1993

ABSTRACT This article introduces a novel potential function that allows the use of topographical information in molecular modeling. Quantitative shape data are provided by techniques such as electron-microscopy-based three-dimensional image reconstruction for large macromolecular assemblies. Such data can provide important constraints for molecular mechanics. We represent topogra hical data by spherical harmonic surfaces, first used by Max and Getzoff used to constrain atoms within these spherical harmonic surfaces. This potential was implemented in the yainmp molecular mechanics package.27 Implementation details and results of several test cases are discussed here.

for displaying molecular surfaces. A simple harmonic potential is

0 1994 by John Wiley & Sons, Inc.

magnetic resonance (NMR) or X-ray crystallography data, or by molecular modeling.

In the absence of a high-resolution atomic structure, knowledge about the shape of a macromolecule can provide crucial structural details. Bio- physical techniques such as low-angle X-ray or neutron scattering and hydrodynamic studies provide only gross shape data for macromolecules, and are thus of limited use in modeling. However, other techniques, such as three-dimensional reconstruction of electron-microscopy (EM) images, provide topographical data that are detailed enough to direct molecular modeling. Three-dimensional

Introduction

hree-dimensional shape is an important struc- T tural attribute of a macromolecule and of mac- rOmolecular aggregates. Several methods have been used to visualize and represent md~cular shapes. Such n ~ t h o d s are usually based on the positions of atoms in a molecule as determined by high-resolution structural refinement with nuclear

‘Author to whom all correspondence should be addressed.

Journal of Computational Chemistry, Vol. 15, No. 2, 190-199 (1994) 0 1994 by John Wiley & Sons, lnc. CCC 0192-8651 /94/020190-10

SHAPE DATA IN MOLECULAR MECHANICS

reconstruction techniques have been developed for single particles and two-dimensional Electron crystallography of 2-D crystals, which combines 3-D image reconstructions with electron diffraction patterns, has been extended into high resolution.*-'j

Three-dimensional reconstruction techniques have yielded detailed topographies for a number of macromolecules-a few recent examples include yeast RNA polymerase 11,7 the nicotinic acetylcho- line receptor,* and the thermosome.' The ribosomal subunits have been studied by 3-D reconstruction of electron microscopy images in both two-dimensional crystals'" and single

Topography information from electron microscopy has been used to guide to overall shape of several physical and computer graphics models of the 16s RNA in the Escherichiu coli 30s sub-

These efforts have used 30s shape models based on visual interpretation of electron microscopy images, and the EM data was used to direct the RNA modeling process qualitatively.

In molecular mechanics, atoms in a molecule are represented by point masses, and potentials are used to mimic covalent and non-covalent interactions between these atom^.'^,'^ This article presents a novel method to use three-dimensional shape data quantitatively during molecular mechanics. Our approach is inspired by the representation of molecular surfaces by dot surfaces20 and spherical harmonics,",'2 and we use spherical harmonics to approximate experimentally derived molecular topography. A simple harmonic potential is used to impose shape data as a constraint for molecular mechanics. The techniques introduced in this article are being used for quantitative modeling of the E. coli 30s ~ubunit . '~

Spherical harmonics originate from the solution of Laplace's equation in spherical coordinates by separation of variables, and they satisfy the or- thonormality relation:

where df2 represents integration over a unit sphere and is equal to sin 0 d8 dcp, 8 and cp are the polar and azimuthal coordinates, and n and kare integers

where - n 5 k 5 n. The spherical harmonics Y,,k(8, cp) are related to the associated Legendre polyno- mials:

and can be evaluated for any set of n and k using recurrence relations.24 Since spherical harmonics form an orthonormal basis set on a unit sphere, they are useful for representing any function that can be mapped on a spherical surface. The distance along a direction vector can be used to map simple closed three-dimensional surfaces onto a unit sphere when the origin is placed within the surface. A molecular surface can thus be represented as an expansion of spherical harmonic function terms as:21

N t i i

r<(8t 'P) = ciiAyiik(8, cp) (3) 11=0 k = - 1 1

where ~ ~ ( 8 , cp) is the distance to the surface from the center of the unit sphere for the angular coordinates (0, cp), YllA are the spherical harmonics of order n, CllA are the corresponding expansion coefficients, and N is the order of the expansion. Such a function is only suitable when ~ , ( 0 , cp) is single valued for any given value of 8 and cp (i.e., the surface has no overhangs or cavities and is starlike25), though transforms can be used to represent non-starlike surfaces by spherical harmonics as well.26 The expansion coefficients are determined by the surface integral:

CIA = I rs(8t (P)YIIL(~T 9)dfl (4)

computed over the unit sphere.21 A simple shape potential to constrain an atom i

within a surface can be constructed as a harmonic term:

where E,, is the surface topography energy for atom i, k, is the surface topography force constant, Y, is the distance between atom i and the center of the surface, and 8, and cpI are the polar and azimuthal coordinates of atom i with respect to the center of the surface.

For energy minimization and other molecular mechanics calculations, it is also necessary to be able to compute a force or gradient for a potential

JOURNAL OF COMPUTATIONAL CHEMISTRY 191

MALHOTRA, TAN, AND HARVEY

function. Since it is computationally expensive to evaluate the analytical derivative of eq. (5), we use a simplifying assumption where rS(8;, q;) is taken as a constant r, and thus the gradient along the Cartesian x-axis is

where xi and x, are the x-coordinates of the atom i and the center of the surface, respectively. Similar expressions can be written along the y- and z-axes.

Our surface potential thus exerts a centripetal force on any atom i outside the surface, which pulls it toward the center of the surface, irrespective of the topography of the surface around atom i. This simplistic formulation of the potential gradient is adequate when the local surface is smooth and nearly spherical, but can lead to artifacts when the surface is undulated. For example, with our defi- nition of the potential gradient, an atom in a surface invagination will be pulled toward the center of the unit sphere used to define the surface rather than toward the part of the surface closest to it.

Since the spherical harmonic approximation in eq. (3) can only be used with starlike surfaces, complex molecular surfaces with overhangs or cavities have to be divided into several closed, starlike sub- surfaces. Combinations of surfaces can be used, and surfaces can be made inclusive or exclusive to mimic complex shapes. Surfaces are defined to be inclusive (where an atom outside the surface has a potential and experiences a gradient which tries to force the atom into the surface), or exclusive (where an atom inside the surface has a potential and experiences a radial gradient which tries to force it out of the surface). The expressions for potential energy and gradients for exclusive surfaces are similar to those in eqs. (5) and (6). To facilitate the use of groups of surfaces, we define AND and OR surfaces based on the Boolean AND and OR operators. AND surfaces form a part of a group where an atom i is required to be within all the surfaces in the group. Surface potentials and gradients on an atom within a set of AND surfaces are computed individually for each surface and are then added. OR surfaces can form groups (and several groups can be defined) where an atom is required to be within any one of the surfaces in the group. For OR surfaces, the energy E,; and the associated gradient are computed only to the surface closest to the atom i.

Each surface has an associated atom list to specify the atoms which are constrained by it. Ob- viously, atom lists for a group of OR surfaces are identical. Figure 1 shows the format used to specify the surface constraints in our implementation of this potential function.

The surface potential was implemented in C as a new potential type surf in the molecular modeling package y~rrzrrzp.~~ For each atom in a surface atom list, the angles 8 and cp are computed with respect to the surface center, and the corresponding value of r, is computed using a spherical harmonic expansion of order N (specified by the user). Depending on the surface type, the energy and gradient for the atom are computed using eqs. (5) and (6). Since the evaluation of r, is computationally intensive, a list of r,, 8, and cp is maintained for all the atoms in each surface list, and a new value of r, is computed only if the angles 8 and cp change by more than a user-specified limit, typically 0.01 radians. The use of an angular cutoff requires additional memory, but decreases computational time for the surface potential severalfold. Other schemes can be used to reduce the repeated evaluation of the spherical harmonics for r,, such as the use of look- up tables for r, for sets of [8, cp] angles for each surface, though these also involve the trade-off of additional memory usage. Also, unlike the evaluation of nonbond interactions where cutoff list sizes are proportional to N 2 (where N equals the number of atoms in the system), the use of angular cutoffs in the surf potential uses lists whose size increases linearly with N.

Table I shows a comparison of normalized cen- tral processing unit (CPU) times and memory usage for Monte Carlo and energy minimization runs with and without the surf potential. As can be seen, with the use of an angular change cutoff, the evaluation time for surface terms is comparable to the evaluation time of van der Waals terms. Compu- tational time increases with the order of spherical harmonics used to represent the shape data, as can be seen in Table 11.

Examples of the Use of the Surf Potential

The surface potential introduced above was used to impose shape constraints during modeling of large RNA chains. Several approaches are being explored for modeling large RNA molecules2*-- our protocol uses reduced representations where

VOL. 15, NO. 2 192


# s u r f sum-surface> <exponent> <angular cutoff> # # <surface-i> <k-surf> <surf-thickness> <surf-order> <surf-type> <num-j> # <surface-center-x> <surface-center-y> <surface-center-z> # # <n> <k> <spherical harmonics coefficient Cnk> # # # . # . # . # # <atom-j 1> <atom-j2> <atom-j3> ... <atom-num-j> # . # . # . # <num-surface> sets #

<n = 0 to surface-order> <k = 1 to 2n+l corresponding to the range -n to +n>

surf 2 2 0.01 Surf potential w i th two surfaces

1 100.0 0.00 0 1 10 Surface 1 of order zero. This i s an inclusive surface. 0.00 0.00 0.00 Surface 1 i s centered at the or ig in

0 1 177.245385 Spherical harmonic coefficients for surface 1

1 2 3 4 5 6 7 8 9 10 List o f atoms t o be enclosed w i th in surface 1

2 100.0 0.00 0 -1 10 Surface 2 of order zero. This i s an exclusive surface. 0.00 0.00 0.00 Surface 2 i s centered at the origin.

0 1 70.898154 Spherical harmonic coeficents f o r surface 2

1 2 3 4 5 6 7 8 9 10 List of atoms t o be excluded f rom surface 2

FIGURE 1. File format for specification of the surf constraints in ~arnrnp.~’ The set of comment lines at the top of the file (starting with # in column 1) define the file format we use to specify surfaces with the surf potential function. The rest of the file describes a surf potential which constrains a group of 10 atoms within a shell formed by two concentric spheres of radius 50 A and 20 A. The term surface thickness, added to the surface radius r,, is used to correct for errors introduced due to the probe size in solvent-accessible surfaces. Each surface is also assigned an integer “type“ to distinguish between AND and OR surfaces. The type is set negative (or positive) to specify that a surface is exclusive (or inclusive). Groups of surfaces with type 1 or - 1 are used to define a set of inclusive or exclusive AND surfaces. Surfaces with types other than 1 or - 1 define set of OR surfaces. For example, when three surfaces with the type 2 are specified, atoms are confined within any of the three.

every nucleotide is represented by a single pseudo- Secondary structure and tertiary inter-

actions are imposed on these pseudoatoms using bonds, angles, improper torsions, and other potential terms. The examples described below use such chains of pseudoatoms.

Figure 2 shows a simple case where the surf potential is used to constrain a random chain of atoms

within a spherical surface (Fig. 2b), outside a spherical surface (Fig. 2c), and within a shell created by two concentric spherical surfaces (Fig. 2d). An initially extended chain of atoms connected by simple bonds and traversing the sphere (Fig. 2a) was subjected to energy minimization with a simple spherical surface constraint (a spherical harmonic surface of order 0). When an inclusive surface potential is



TABLE 1. Comparison of Normalized CPU Times for Energy Minimization and Monte Carlo, with and without Surface Constraints and Nonbond Interactions.

System with: Monte Carlo Steepest descent

Nonbond Number of Memory usage for (200 Cycles) (1 000 cycles) interaction surface Monte Carlo (Normalized CPU (Normalized CPU included? constraints (Normalized) time) time)

No Yes Yes Yes Yes No

1 .oo 1.22 1.26 1.29 1.33 1.12

1 .oo 2.12 2.88 3.1 7 3.71 2.52

1 .oo 4.04 5.22 4.80 5.41 2.1 1

System consists of an RNA chain with 1912 atoms and bond, angle, and improper torsion constraints. CPU times for two identical runs on two systems with different initial conformations were averaged. The nonbond list was updated every 10 cycles with a cutoff of 6 A. All surfaces were of order 6, and an angular change cutoff of 0.01 radians was used for the surface potential. All runs were on a Silicon Graphics 4D 40-MHz R-3000 processor.

used (Fig. 2b), parts of the random chain that are outside of the sphere rapidly crumple into the sphere during energy minimization. Figure 2c shows the same random chain minimized with an exclusive surface, where the atoms within the sphere are pushed to the periphery by the surface potential. In Figure 2d, a shell was created by using a set of two spherical surfaces, with the inner sphere being an exclusive surface and the outer sphere being an inclusive surface. The potential thus created forces the atoms into the shell. In ad- dition, two bonds were used to tether the ends of the extended chain to two opposite ends of the

TABLE II. Comparison of Normalized CPU Times for a Surf Potential with a Surface Approximated by Spherical Harmonics of Order N.

Monte Carlo Steepest descent (200 cycles) (1 000 cycles)

N Normalized CPU times Normalized CPU times

0 1 .oo 1 .oo 2 1.02 0.99 4 1.09 1.02 6 1.16 1.02 8 1.26 1.03

System consists of an RNA chain with 1912 atoms and bond, angle, and improper torsion constraints. CPU times for two identical runs on two systems with different initial conformations were averaged. The nonbond lists are updated every 10 cycles with a cutoff of 6 A, and angular change cutoff for the surface potential was set at 0.01 radians. All runs were on a Silicon Graphics 4D 40-MHz R-3000 processor.

shell. After energy minimization, though the ends of the chain are pulled to the two poles of the shell by the tethers, the radial nature of the surf gradient pushes the random chain out on the side where the chain was initially positioned.

Figure 3 illustrates the use of molecular surfaces to impose shape constraints during energy refinement. Figure 3a shows the solvent accessible surface32 for the tRNAPh' crystal structure33 computed using the ms program20 with all atoms assigned a radius of 5 A to smooth the surface invaginations. The L shape of tRNA is not starlike and cannot be represented by a single spherical harmonic expansion as defined by eq. (3). We thus compute the solvent accessible surface of the two arms of tRNA separately, as shown in Figure 3a. The surface defined by the points in the Connolly "dot" surface can be approximated by a spherical harmonics surface of any order N using the sphinx program,21 which computes C,,k coefficients for the spherical harmonic expansion around a center computed by averaging the coordinates of all the points on the Connolly surface [eq. (4)]. Figures 3b-e show the tRNA solvent accessible surface represented by spherical harmonic expansions of several orders. As can be seen, the tRNA shape can be well approximated by two spherical harmonic surfaces of order 8.

Figure 3e shows energy refinement of a simpli- fied representation of the tRNA chain with secondary structure, helix stacking, and a few tertiary constraints,% where several random chains are subjected to a surf potential requiring all the 76 pseudoatoms in the chain to be within one of the

194 VOL. 15. NO. 2

FIGURE 2. Example of the use of the surf potential to constrain an extended chain of atoms with spherical surfaces by energy minimization. (a) Initial random-walk chain of atoms. (b) Chain of atoms shown in (a) refined with an inclusive spherical surface constraint. (c) Chain of atoms shown in (a) refined with an exclusive spherical surface constraint. (d) Chain of atoms shown in (a) constrained to a shell by the use of two AND spherical surfaces. The two ends of the chain are tethered to the poles of the shell using simple harmonic constraints. Panels (b), (c), and (d) show the chain of atoms after simple energy refinement.

two surfaces (specified as OR surfaces). Three positional constraints are used on nucleotides 1, 31, and 51 to position them correctly with respect to the two surfaces. Figure 3f shows three random chains refined in the absence of these positional constraints, and as can be seen the surf potential stuffs the RNA chains into the surfaces in arbitrary orientations. The three tethers used in Figure 3e are thus necessary to orient correctly the RNA chain within the L shape of tRNA.

The use of tethers (positional constraints on parts of the molecular chain) becomes important


when shape constraints are imposed during molecular modeling, as the surface potential described here has no inherent orientation information. Thus, unless the molecule being modeled is initially close to the correct conformation with respect to the imposed surface, the surf potential will stuff it into the surface in an arbitrary orientation. Teth- ers can help direct parts of a molecule toward the correct regions within a surface, if such information is available.

Reduced representation models of RNA chains based only on secondary structure and a few tertiary contacts are severely underdetermined. The use of shape information, as shown here, can help impose the global geometry for a model where in- teratomic distances are by themselves not enough to specify a unique conformation. We present these RNA models primarily to illustrate the use of surface constraints; details on the general modeling protocol and results are presented e l s e ~ h e r e . ~ ~ - ~ ]

The use of electron microscopy data to impose shape constraints is illustrated in Figure 4. We have used topographical data for the E . c d i 30s ribosomal subunit, derived from 3-D image reconstruction of electron microscopic images,35 to develop quantitative models of the 16s RNA in the 30s subunit.23 Three-dimensional reconstructions of EM images are usually available as density or contour data, and data on the 30s subunit was provided as contour data.36 This data, in MOVIE BYU format,37 consists of points along contours on the 30s subunit surface as shown in Figure 4a.

The conversion of such contour data into spherical harmonic surfaces requires several transfor- mations. Each surface should be starlike, as ex- plained earlier. The 305 contour data, shown in Figure 4a, was manually divided into three inde- pendent starlike surfaces corresponding to the head, the body, and the platform of the 30s subunit. These are shown in Figure 4b.

To generate coefficients accurately for spherical harmonic approximations, a large number (tens of thousands) of closely spaced points is required on each of these surfaces. The points along the contours are too sparse and ill spaced to meet this criteria. A convenient way around this utilizes the wzs program,20 which peppers the solvent accessible surface of a molecule with ”dots.” By placing additional points on contour faces and positioning large pseudoatoms on these points, the contour data can be converted into a surface covered with atoms. The program ms is then used to compute the solvent accessible surface for this “molecular”



FIGURE 3. Use of spherical harmonic surfaces with succinct models of tRNA to impose global geometry. (a) Solvent-accessible surface of the two arms of the tRNAPhe crystal structure.33 All atoms in the crystal structure were assigned radii of 5 A to get a relatively smooth surface. (b), (c), (d), and (e): Approximation of these two surfaces by two spherical harmonics functions of orders 0, 2, 4, and 8, respectively. Panel (e) also shows three random RNA chains energy refined using the secondary and tertiary constraints for tRNAPhe (ref. 31), along with a surf potential that imposes the surfaces shown on the RNA chains as two OR surfaces. Nucleotides 1, 31, and 51 are tethered by positional constraints at the acceptor terminus, anticodon loop, and elbow region, respectively. (f) Three random RNA chains refined with the same constraints as in panel (e), but without the three positional tethers.

shell. Both the outside and the inside surfaces of this shell are computed by ms, and we use a simple procedure to remove the inside surface points. Fig- ure 4c shows the full 305 subunit surface, trans- formed into three "dot" surfaces using this procedure. Coefficients for the spherical harmonic approximation of these surfaces were computed using the sphinx program,21 and the resulting surfaces are shown in Figure 4d.

EM data provide important shape information for the modeling of the E . coli 16s RNA in the 30s subunit. Additional data on the orientation of the RNA chain within the 30s subunit are provided by immunoelectron microscopy experiment^,^'-^' which have localized parts of the ribosomal RNA and proteins to specific regions on the 30s surface.

These and other data used in our models of the 16s RNA in the 30s ribosomal subunit, along with modeling details and results, are presented else- where.23 Figure 4e presents a typical final model and shows the packing of the RNA chain within the EM reconstructed 30s shape by the use of the surf potential during energy refinement and simulated annealing. For proper imposition of the surface constraints, the force constants for the surf potential must be comparable (in order of magni- tude) to the other terms in the potential function. Thus, instead of using very low force constants to reflect experimental uncertainty in the EM-derived shapes, a surface thickness of 10 A (which is added to r,) was used. This mimics experimental uncertainty by allowing atoms to be placed as much as

196 VOL. 15, NO. 2


10 A beyond the EM surface before the surf potential is activated. The force constants for the surf potential were given a value comparable to the smallest force constants in the potential function. The surface thickness and the soft surf force constants allow parts of the RNA chain in Figure 4e to venture out of the imposed surfaces. Figure 4f shows the final conformation of the same RNA chain after an identical cycle of energy refinement and simulated annealing without any surface constraints.

Conclusions

Three-dimensional reconstructions of EM images, from both single particles and 2-D crystals,

are providing quantitative topographical data on many large macromolecular assemblies. These shape data can provide important constraints for molecular modeling of such systems.

This article introduces a potential for the use of shape data in molecular mechanics, based on the approximation of molecular surfaces by spherical harmonic expansions. As can be seen from Table I and 11, this potential does not impose an undue computational burden during Monte Carlo, molecular dynamics, or energy minimization, and the computational load of introducing shape constraints is comparable to that of nonbond interactions. Though the use of spherical harmonics requires starlike surfaces, complex shapes can be constructed easily using multiple surfaces.

Simplifying assumptions have been made in our

FIGURE 4. Use of 3-D image reconstructed electron microscopy data in modeling of the 16s RNA in the 30s ribosomal subunit. (a) Electron microscopic image reconstruction of the shape of .€. coli 30s subunit35 displayed as contour data.% (b) Division of the 30s shape data into three starlike surfaces. Additional points have been added to compute a solvent-accessible surface. (c) Complete 30s subunit surface, represented by three solvent-accessible surfaces with the internal surface “dots” removed. (d) Spherical harmonic approximation of the surfaces in panel (c). The head and the platform are approximated by spherical harmonics of order 6; order 8 is used to approximate the 30s body. (e) A 16s RNA model, energy refined with the surfaces shown in panel (d) imposed as OR surface constraints using the surf potential.23 (f) Conformation of the 16s RNA chain, energy refined without any surface constraints and starting from the same initial conformation as the model in panel (e). All views are from the solvent side of the 30s subunit with the 505 interface at the back.



FIGURE 4. (Continued)

formulation of the gradient of the surface potential [eq. (6)], which can cause problems in regions with deep invaginations or sharp inflections on the imposed surfaces, or at the junctions of surfaces. The potential is thus more suited for relatively smooth surfaces.

In models where the initial atomic chain is not well oriented with respect to the surface being imposed, position or orientation data are usually necessary to place the molecule properly within the surface. This can be done by either positional constraints or the use of additional smaller surfaces to which a subgroup of atoms is constrained.

Acknowledgments

Specific contributions to this work are as follows: S.C.H. for inception of the surface potential and overall guidance: R.K.Z.T. for development of the yummp molecular modeling package and program- ming assistance; A.M. for development, implementation, and testing of the surface potential and the writing of this article. The authors thank Dr.

Elizabeth Getzoff for the sphinx program, and Dr. Joachim Frank for providing data on 3-D image reconstructions of E . coli ribosomal particles. This work was supported by a grant from the National Science Foundation (DMB-90-05767). Additional grants for the purchase of Silicon Graphics work- stations were provided by the National Science Foundation (NSF) and the state of Alabama. Some computer time was provided by the Alabama Su- percomputer Network.

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