Computing device for calculating energy and pairwise central forces of particle interactions

This invention provides a computing device which calculates the total energy and total pairwise, central forces between selected pairs of interacting particles within a system. The device comprises (a) a means for generating both (i) a retrievable identifier for the spatial coordinates (x, y, z) of a particle i and (ii) retrievable identifiers of the type of pairwise, central interaction between i and a second particle j; (b) a means for determining r.sup.2, the square of the radius vector between i and j; (c) a means for determining r, the square root of r.sup.2 ; (d) a means for calculating energy and force/r values for each ij pair; (e) means for determining the force on each particle; and (f) means for accumulating energy and force/r values for each ij particle pair.

Throughout this application various publications are referenced by number 
within parentheses. Full citations for these publications may be found at 
the end of the specifications immediately preceding the claims. The 
disclosures of these publications in their entireties are hereby 
incorporated by reference into this application in order to more fully 
describe the state of the art to which this invention pertains. 
The computational aspect of molecular modelling has received increasing 
attention from many investigators during the past several years (1). This 
is due not only to the great progress made since the 1950's in 
conventional protein crystallography but is also due to the promise of 
protein structure determination using synchrotron radiation and to the 
availability of new kinds of information from the use of neutron and 
electron diffraction and 2-D nuclear magnetic resonance (2). The 
application of improved computational techniques for molecular dynamics 
and energy minimization to proteins, to liquids and gases, to foreign 
atoms in solids, and to the interactions between molecules have all 
contributed to the use of very large amounts of computer time (3). In 
these calculations the time required is used primarily to determine the 
energy and force on each atom due to its interaction with all of the 
others. Much less time is needed to calculate the displacements which each 
atom should undergo because of the forces and the past history of the 
energy, forces and velocities of the particles. 
It is obvious that increasing by a thousand fold the speed of computer 
programs which require large amounts of time will have significant effect 
on the nature of the problems which can be dealt with successfully. To 
make this statement more explicit, a description of some of the problems 
which are now being pursued and the impact that this invention would have 
on them is described herein. Further, some of the other problems which are 
now using large amounts of computer time and for which this invention will 
have a significant impact are discussed herein. 
Modelling of the combining sites of antibody molecules has become a 
particularly important problem in recent years because it is now possible 
to obtain the amino acid sequence of monoclonal antibodies relatively 
easily by sequencing the cDNA which encodes the protein (15). Further, it 
is now possible to alter a portion of the peptide or complemetarity 
determining region by manipulating cloned cDNA. Thus, it is in principal 
possible to change the amino acid sequence in such a way that the molecule 
has an altered combining affinity. The questions being asked are whether 
one can understand the structure-function relationships well enough to 
predict the conformation of the combining site from the amino acid 
sequence alone, and whether one can predict changes in the binding 
affinity and selectivity when specific mutations are introduced by in 
vitro mutagenesis. 
A second problem which is currently being investigated is the 3-D structure 
of the portion of the colicin El protein which produces an ion channel 
through a biological membrane (16). The work involves extensive protein 
model building and efforts to predict changes in the electro-physiological 
properties of the protein when the amino acid sequence is changed by in 
vitro mutagenesis. 
Investigations in which features of the 3-D structure and the function of 
proteins from the amino acid sequence or changes in the sequence are 
studied also require large amounts of computer time (7,18 ). It is 
practical with the VAX to find a minimum energy conformation which is at 
the bottom of whatever local energy valley one happens to start in. But, 
even for a small portion of a complete protein, enormous amounts of 
computer time are needed if one wishes to explore a significant region of 
conformation space. For example, in order to find a reasonable packing 
arrangement for the antibody combining sites, it is necessary to start 
with many different conformations and for each to carry out a process of 
minimizing the packing energy. The starting conformations can be obtained 
either by a random or a systematic alteration of the structure to be 
minimized. Alternatively, one can use a molecular dynamics program to 
explore a space and use the results to aid in the search for a global 
minimum. 
Even with the device presented herein, the time required to find the global 
minimum for an entire protein from the amino acid sequence alone would be 
prohibitive. However, for the antibody combining site, for the colicin 
channel, or for enzymes whose structures have been solved 
crystallographically, it should be possible to explore the changes induced 
by mutation. In the case of antibodies it is reasonable to explore the 
possible combining site conformations, given the structure of the 
framework region in which the complimentary-determining regions are 
embedded, since they are the same for all antibody molecules. 
Another class of problems which has not been approachable until now 
involves understanding the details of the passage of ions through 
channels, the specificity of channels and the interaction of portions of a 
protein with water molecules on the exterior (19). In the case of the 
colicin El channel, for example, neither the observed value of its 
conductance or the ion specificity is understood in detail. It is our 
expectation that applying molecular dynamics to the water, the ions, and 
the amino acid side-chains facing into the channel will contribute greatly 
to the understanding of how the channel functions. 
There are several disciplines in addition to protein physical chemistry 
which should benefit from the device of this invention. This device is 
applicable to any many-body problem subject to central forces. Such 
problems arise in many fields of science and engineering, ranging from 
astronomy through solid-state and plasma physics to biology and chemistry. 
The design of pharmaceutical drugs would be enhanced by the type of 
computer simulations possible using this device. In the past, drug 
companies have proceeded with their goal of developing new drugs by the 
method of "blind" analog development. Given a successful drug, new drugs 
have been tried by manually synthesizing hundreds or thousands of analogs 
for testing by injection into test animals. The energies of the 
computational capabilities of molecular dynamics; Monte Carlo, energy 
minimization; and template forces searching etc., have enhanced the 
possibility for a new approach to the entire problem, called 
"rationalized" drug design. Briefly, and ideally, the approach involves 
the design and screening of drugs by computer simulation at the atomic 
level with a description of the interaction of a drug with its receptor 
site. This can be done if the structure of the receptor site is known by, 
for example, x-ray crystallography. The predicted change in the binding 
constant of a new analog to the site is approachable by using free energy, 
simultations, and slowly mutating the structure of the old drug whose 
binding is known into the structure of a new and better drug. Other kinds 
of energetic manipulations on both the analogue and the protein can 
provide extremely valuable guidance not only for computer screening of 
analogs but also for suggesting new directions for drug development which 
might not have been obvious without the simulational tools. 
Similarly, energy minimization calculation can be used to tackle issues in 
protein engineering. To take full advantage of the emerging technology of 
recombinant DNA requires the development of predictive capability: e.g., 
if one modifies the gene which codes for the light chain of an antibody 
molecule, can one predict what change this will have on the function of 
that antibody? If one modifies the gene for a nerve channel protein, can 
one design organisms with modified nervous system response? If one 
modifies the gene for photoreceptors, can one turn the sensitivity to new 
ranges of wavelengths, extending vision into the near infrared? All of 
these problems, as well as others, can be, and are, under active 
investigation in various laboratories, both in industry and at 
universities. However, the computations involved in tackling energy 
minimization calculations for biologically interesting molecules can be 
prohibitive. 
SUMMARY OF THE INVENTION 
The invention concerns a computing device which calculates the energy and 
pairwise, central forces between pairs of discrete interacting particles. 
The components include a means for generating a retrievable identifier for 
the spatical coordinates (x, y, z) of a first particle (i), e.g. an oxygen 
atom with coordinates (2,4,2), and a means for identifying the type of 
pairwise, central relationship between i and a second particle or group 
(j), e.g., when j is a carbon atom the relationship is an oxygen-carbon 
interaction. These means are responsive to the generation of digital words 
from an external word generator. 
The components also include a means for determining the square of the 
radius vector (r.sup.2), e.g., for any particle with coordinates (a,b,c) 
and a second particle with coordinates (a',b',c') the equation for r.sup.2 
=(a-a').sup.2 +(b-b').sup.2 +(c-c').sup.2. This calculation is performed 
with a system of subtracters and multipliers responsive to digital words 
describing the spatial coordinates of any i and j, and the solution, 
r.sup.2, l is sent in digital word form to a means for calculating r. 
A means for calculating the square root of r.sup.2 is also a component of 
the computing device. This means is responsive to a digital word for the 
value of r.sup.2 and comprises a quadratic look-up table which gives a 
value of r for the corresponding r.sup.2, and outputs the r so obtained as 
a digital word. 
A means for calculating the energy and force/r for an ij pair is also a 
component of the device. This means contains quadratically-interpolated 
tables which are responsive to digital words describing the ij pair type 
and the value of r. These digital words are entered into the formula: 
E=Cf1(pt) * Table 1(pt,r)+Cf2(pt) * Table 2(pt,r) 
A solution for the total energy of the ij pair is sent as a digital word to 
an energy accumulating memory. A solution for force/r is sent as a digital 
word to a force/r accumulating memory. 
A means for accumulating energy values is also a component of the device. 
This means sums energy for each ij formed by a given i, and comprises an 
adder responsive to digital word input obtained as solutions of the energy 
formula. 
A means for accumulating force/r for each i and j of each and all ij pairs 
is also a component of the device. This means is responsive to digital 
words for solutions to the force/r equation, and forms a product of each 
solution and the difference of each dimensional coordinate of the ij pair, 
e.g. (force/r)(a-a') and (force/r)(b-b') and (force/r)(c-c') when the 
coordinates of i and j are (a,b,c) and (a',b',c'), respectively. These 
products are then summed to the existing force/r for i and added as 
negative values to the total force/r for j. 
Finally, a molecular mechanics calculating device for molecular 
conformation studies is described. This device comprises a means for 
generating a retrievable identifier for the spatial coordinates (x, y, z) 
of a first atom; and for identifying the type of pairwise, central 
relationship between i and a second defined particle j, thereby defining 
an ij pair, a means for determining the square of the radius vector 
r.sup.2 between atoms i and j, a means for determining the square root of 
r.sup.2, a means for calculating the energy and magnitude of force/r for 
any defined ij pair when r is known, and a means for accumulating energy 
and force/r for each i and j.

DETAILED DESCRIPTION OF THE INVENTION 
The processor can functionally be broken into five units, as shown in FIG. 
2: 
Unit I: The Neighbor List. This contains the addresses necessary to look up 
the coordinates of the individual atoms; an ij bit which identifies an 
atom as an i or a j atom; and a pair type, which indicates the type of 
pair a atom forms with the most recently accessed i atom. Each clock 
cycle, these three atom oriented pieces of information are delivered to 
the other units. 
Unit II: The r Squared Calculation. This unit contains the coordinate 
memories, and the associated hardware to calculate 
EQU r.sup.2 =(x.sub.i -x.sub.j).sup.2 +(y.sub.i -y.sub.j).sup.2 +(z.sub.i 
-z.sub.j).sup.2 
Unit III: The r Calculation. This unit performs a quadratic table look-up 
to obtain the value of r, given the value of r squared. 
Unit IV: The Energy and Force/r Tables. This unit contains the 
quadratically-interpolated tables used to look up the energy and the 
magnitude of the force/r between a j atom and the most recently considered 
i atom. The Energy and the Force/r sections are identical, and consist of 
weighted sums of two contributions: for the Energy, 
E =ACoeff*(Table A)+BCoeff*(Table B). 
The coefficient tables A, B, C, and D contain information relating to the 
products of the partial charges (q.sub.1 .times.q.sub.2) on the atoms and 
information relating to the van der Waals forces, as is well known. 
Unit V: The Energy and Force Accumulators. This unit contains sum 
accumulation memories which accumulate the total energies associated with 
a few selected classes of atoms and the total force components associated 
with each atom. The components are calculated by multiplying F/r by 
(x.sub.i -x.sub.j) to obtain F.sub.x etc. 
Throughout the diagram, various delays, either labelled "Delay" or "D", 
have been indicated. These are either file register delays (where the 
delays are long) or shift register delays (where the delays are short). 
These delays guarantee the arrival of information pertaining to a given 
pair at the appropriate input to a unit in synch. 
The quadratic interlopers, shown schematically in FIG. 3, are used to look 
up the values of the energy, the force/r, and the square root. The central 
memories, y.sub.0, y', and y", contain tables of the functional values, 
slopes, and 1/2 times the second derivatives of the desired function. The 
beginning of the appropriate table for a given pair type is stored in an 
offset memory. This offset is added to the high order bits of the 
incoming(integer) independent parameter, and the result is used to address 
the central memories. This value is represented by the letter x in the 
figure. The adders and multipliers form the combination 
EQU y =y +y.sub.O +(d.sub.x) (y'+d.sub.x (Y")) 
where d consists of the low order bits of the independent parameter. It is 
understood that the Notation dx represents delta x; which is, for example, 
the output of the x-coordinate subtractor in FIG. 2. There are also two 
memories, Max and Min, which store the maximum and minimum addresses of 
the tables associate with any given pair type. If the generated address 
falls outside of this range, an "invalid bit" is set. 
The invention described herein is designed to be extremely flexible and 
will serve for any currently addressed problem in molecular mechanics. The 
flexibility arises from the fact that all function evaluations are done as 
table look-ups in which the functions and their first and second 
derivatives are stored in tables which represent a local quadratic fit for 
each function needed. Since the tables are loaded from the host computer 
and can be changed to suit a given problem, new force fields can always be 
accommodated. In addition, the invention is designed to calculate only the 
force of each atom due to the pairwise interactions, leaving all other 
calculations to be done in a programmable array processor. The overall 
system, consisting of the device of this invention, the array processor, a 
host computer, mass storage, and communications and display hardware, will 
yield speeds roughly an order of magnitude greater than those available on 
Cray-1-lS or Cyber 205 for molecular mechanics calculations. 
On any general purpose computer, the most time intensive part of the 
calculation is the evaluation of the interaction energy and the vector 
forces on each atom. Once the 3N component force vector is formed, a much 
smaller computation consists of updating the 3N coordinates of the atoms 
given the forces and total energy. To limit the size of the calculation, 
most investigators carrying out simulations or minimizations generate a 
list of atom-pairs which are close enough to each other to be included in 
the force and energy calculation. The size of these pair-lists or 
"neighbor" lists depend on the distance cutoff used in the force and 
energy calculations and may vary for different energy types. The pair-list 
must be regenerated as the atoms move and the number of interactions, or 
time steps, between the regenerations will depend on the number of 
neighbors we assume for each atom. The larger the pair-list, the less 
often it must be regenerated, and in operation the system will have to be 
optimized for each problem with respect to pair-list size. 
The overall hardware architecture of the present invention, which mirrors 
the software architecture described above, is shown in FIG. 1. There are 
two central modules, one of which is home built and the other of which is 
a commercially available array processor. The home built module, the 
device of this invention, calculates the total energy and vector force on 
each atom due to pairwise forces only. The commercial array processor 
calculates those forces involving 3 or 4 atoms (torsional terms, angle 
bending terms, cross terms), and uses these results combined with those 
from the device of this invention to update the coordinates to a new 
molecular conformation. Roughly 95% of the time spent in a calculation on 
a conventional computer is spent evaluating the pairwise forces; as such, 
this device has been designed to give speeds roughly ten times greater 
than those available on supercomputers of the Cray-lS class. By 
concentrating on the pairwise forces, only those portions of the inner 
loop which are the most time intensive and stable are frozen in hardware. 
The implementation of the coordinate update in a programmable array 
processor allows flexibility in tailoring the integration or minimization 
technique to a specific problem. 
The overall schematic of this device is shown in FIG. 2. The architecture 
adopted is a parallel, synchronous pipeline. Each element in the machine 
is a multiplier, an adder, or a table look-up. Most arithmetic is done in 
32 bit floating point, and is accomplished by utilizing commercially 
available 32 bit floating point chips recently introduced by the Weitek 
Corporation. Accumulation of forces and energies is done to higher bit 
precision, and the pipeline cycle time is 100ns. Each clock cycle, an 
entry corresponding to an atom is etched from a Pair List Memory. The 
structure of the pair-list is such that every i atom entry (slow moving 
index of a j pair) is followed by all j i atom entries (fast moving index) 
in the list. Each entry (32 bit total) consists of a coordinate address, 
used to look up the coordinates of that atom; a pair type (j atoms only), 
which identifies the type of interaction which that j atom forms with the 
most recent i atom; and an ij bit, which identifies the atom as an i or j 
atom. The coordinate addresses are fed simultaneously to three coordinate 
memories (x, y, and z). These coordinates are passed to one of the inputs 
of three subtractors. If the atom is an i atom, nothing happens except 
that an empty cycle is generated. If the atom is a j atom, the subtraction 
is done to form delta-x, delta-y, and delta-z. These are squared and 
summed to form delta-r, squared, which is used as input to a table look-up 
to obtain the value of an interpolation function, which in the present 
invention is a square root table yielding r. This r, along with the 
pairtype which describes what kind of pair of atoms this ij pair is, is 
then used to look up the value of the energy and the value of F/r. The 
Cartesian components of the force vector are obtained by multiplying F/r 
by delta-x, delta-y, and delta-z, and the total energy and the force 
vector memories are updated. 
In order to limit the total size of memory, the energy and the value of F/r 
are both calculated as a sum of two contributions, i.e. as E 
=a(pt)*Ea(pt,r) +b(pt)*Eb(pt,r), where pt is the pair type and r is the 
scalar distance between the atoms comprising the pair. Since all function 
evaluations, including the square root, are handled by table lookups, this 
device is memory intensive. More specifically, as shown in FIG. 2, the 
coefficient tables and interpolated tables are accessed or addressed by 
the pair-type (pt) information acting as retrievable identifiers, so that 
the proper coefficient corresponding to the particular pair-type under 
scrutiny is read out from the table. Overall, there are about 5-10 Mbytes 
of memory distributed, throughout the system when it is equipped to handle 
16,000 atoms and 1,000,000 interacting pairs. Note that if all hydrogens 
are included in the calculations, 16,000 atoms might require of the order 
of 1.5 million interacting pairs. The system has been designed so that 
additional pair list memory can be added at any time. 
Each of the table look-ups is done by quadratic interpolation, as shown in 
FIG. 3. In each bin, the value of the function, the first derivative, and 
half of the second derivative are stored. The incoming value of the 
independent variable is scaled, integerized, and split into high and low 
order bits. The high order bits are added to an offset which is looked up 
as a function of the pair type and which gives the starting address of the 
table associated with that pair type. This sum is used to fetch the value, 
slope, and half of the second derivative from the tables, and these are 
used to compute: 
EQU y =y .sub.0 +(dx)(y'+dx(y")) 
The energy and force calculator described above has been designed to 
evaluate pairwise, central forces between objects in three dimensional 
space only. In the empirical Hamiltonian which is usually manipulated in 
molecular mechanics calculations, there are additional terms associated 
with bond angle bending, torsional excursions, and out-of-plane excursions 
which are not pairwise and central. The angle bending terms have been 
centralized by placing a non-Hooks law spring between atoms 1 and 3 in a 
1-2-3 atom triplet which maps out the angular dependence normally used. 
This introduces a maximum error of 10% at 10 kt. The torsional terms can 
be handled in one of two ways: first, by positioning a point in space away 
from the 1-2-3-4 quadruplet using atoms 1,2 and 3 and rigid springs and 
subsequently dropping a non-Hooks law spring to mimic the torsional 
potential for excursions of the 4th atom; and second, by calculating these 
torsional contributions in the array processor which would otherwise be 
idle during the time the force and energy accumulator is working. 
The length of the pipe is several hundred cycles, since each Weitek chip is 
9 cycles deep internally. This effectively limits the minimum size the 
problem which is worth running of on this device, since a problem 
involving several hundred pairs would incur an effective overhead of a 
factor of two. However, this overhead is insignificant for typical 
problems involving proteins or DNA, since the number of pairs for these 
problems is typically two orders of magnitude greater than the length of 
the pipe. For any problem, once the pipe is full, a pair is evaluated 
every 100ns. This speed can be compared to that which is available on 
several general purpose machines; on a VAX 780 with a floating point 
accelerator, a pair can be evaluated every 150 .mu.s using assembler code. 
On a Cyber 205, a benchmark provided by Osguthorpe et al (Osguthorpe, 
D.G., P. Dauber-Osguthorpe, Kitson, D., Wolff, J. and Hagler, A.T., A 
State of the art Calculation of a Biological System using a Cyber 205 
Supercomputer), achieves the evaluation of a pair every 600ns but only if 
there is no pair list and all pairs are evaluated. On a Star ST100 array 
processor, it is possible to evaluate a pair every 1-2 .mu.s using 
microcoded routines with a pair- list. 
The device of this invention has been designed around the idea of easy 
maintainability. All tables are connected to a slow bus (Q bus) and can be 
down loaded either from the host VAX or through a resident Motorola 68010 
microprocessor. The system is designed so that a single step mode is 
available for debugging, with selected pipeline registers readable from 
the slow bus. The hardware is being implemented in 6 layer DEC boards 
(13".times.18"), with interboard connections implemented through the back 
plane and via ribbon cable. There are approximately 20 different board 
types. 
The array processor chosen to complement this device is the Star ST100. The 
Star ST100 is presently the fastest commercially available array 
processor, with a theoretical rating of 100 Mflop. One feature of this 
machine which is crucial to this application is its ability to communicate 
with this device at a high rate, along with its ability to overlap 
communications and calculations. A crucial performance number for the 
array processor is the time required to evaluate a torsional couplet and 
resolve the forces into Cartesian components on each of the four atoms. 
This involves four square roots or divides, along with about 35 multiplies 
or adds. The ST100 is capable of evaluating a torsional couplet as a 
pipelined routine in 1.6 us, which is roughly ten times faster than its 
nearest presently available competitor. It is sufficiently fast to allow 
evaluation of torsional and angle bending terms during the time this 
device is evaluating the non-bonded pairwise central terms for most 
systems of biological interest. 
Another crucial task assigned to the array processor is the updating of the 
pair-list. The pair-list is separated into two different parts containing 
fixed and variable pairs. The fixed pairs reflect the chemical 
connectivity of the atoms and thus do not change as the positions of the 
atoms move during the calculations. The variable pair-list is the one 
which is recalculated in the array processor after a number of iterations 
of the coordinate updates. The fixed pairlist is calculated once in the 
host and only changed by it if the connectivity changes. The structure of 
the Star ST100 permits rapid generation of the variable pair-list. 
Although the details of the algorithm are flexible and change as, for 
example, boundary conditions change, a pair of atoms can be considered and 
accepted a lying within as cutoff in 3 equivalent cycles (3 pairs in 9 
cycles; each cycle in the ST100 is 40 ns). 
The speed of this device as measured in megaflops/sec (m flops) is simple 
to determine. The Weitek computing chips perform one floating point 
operation every clock cycle (100ns); that is, they operate at the rate of 
10 m flops each. There are about sixty Weitek chips in FASTRUN; thus, it 
operates at a speed of 600 m flops once the pipe is full. Note that, as 
discussed above, it takes several hundred cycles to fill the pipe. For a 
typical problem involving protein and solvent, this represents an overhead 
of about one percent. There is also an overhead associated with the 
introduction of each i atom into the pipe, since this generates an empty 
cycle. Since each i atom is followed in practice by 50 to 100 ]atoms 
(depending on the cutoff used), this represents a one to two percent 
overhead to the operation of the device of this invention. The total 
fraction of running time (i.e., duty-cycle for this device) varies 
depending on what algorithm is being used in the STAR to update the 
coordinates. However, it would be above 70% for molecular dynamics using 
Verlet integration, conjugate gradient energy minimization, and most Monte 
Carlo schemes. 
There are several ways in which future systems using a similar architecture 
can be made faster. The most obvious is to use faster calculating chips 
along with faster memory, and this will certainly happen as the technology 
develops. In addition, one can consider ways in which some aspects of the 
calculations can be done in parallel without duplicating all of the 
memories. However, at this time, it is not obvious that it would be 
practical to install more than four memory shared parallel pipes with the 
present architecture. Thus, one can see ways of getting to several 
gigaflops in computing speed but much faster systems would probably 
require a very high order of parallelism in a very different architecture. 
An orthogonal future development might involve the installation of 64 bit 
multipliers and adders as they become cost competitive at 100 ns speeds. 
There are applications which would gain significantly by this increase in 
precision. The development of a 64 bit machine, while not trivial since 
all pathways and memories double in size, is feasible and would be 
strongly aided by the prior development of a 32 bit/64 bit hybrid; that 
part of the calculation which is most prone to accumulated error, the 
summing of the forces and energies, is done to 64 bit accuracy. 
REFERENCES 
1. Go, W., Ann. Rev. Bioph. Bioeng. 12, 183 (1983). 
2. Physics Today, 35, 6 (June 1983) 
3. Wumthrich, K., Billeter, M., Braun, W.J., Mol. Biol. 196, 13, 42 (1983). 
4. Schoenborn, B. and Kunes, A., Ann. Rev. Biophys. Bioengineer. 1:52.9 
(1972) 
5. Schoenborn, B. Brookhaven Symp. Biol. 27: II 3-II II (1976). 
6. Schoenborn, B., Trends in Biochemical Sciences, 2, 206 (1977). 
7. Schoenborn, B., Schneider, D., and Wise, D., ACA Transaction 19 (1983). 
8. Mc Cammon, J., Wolynes, P., and Karplus, M., Biochemistry 18:6, 927 
(1979). 
9. Mc Cammon, J., Gelin, D.R., Karplus, M. Nature, 267,587 (1977). 
10. Stillinger, F.N. Rahman, A.J., Chem., Phys., 60, 1545 (1974). 
11. Pern, B., Ed. Modern Theoretical Chemistry, Volume 6, Plenum Press, New 
York (1980). 
12. Wipff, G., Dearing, A., Weiner, P., Blaney, J., Kollman, P., J. Am. 
Chem. Soc. 105, 997 (1983). 
13. Pincus, M., Burgess, A., Scheraga, H., Biopolymers 15, 2485 (1976). 
14. Wodak, S.N. Janin, J., J. Mol. Biol., 124, 323 (1978). 
15. Kaaartinen, M., Griffiths, G., Markham, G., and Milstein, C., Nature 
304, 320 (1983). 
16. Cleveland, M., Slatin, S., Finkelstein, A., and Levinthal, C., Proc. 
Natl. Acad. Sci. 80, 3706 (1983). 
17. Berens, P. and Wilson, K, J. Comp. Chem, July, 82. 
18. Brooks, B. Brucollori, R., Olafson, B., States, D., Fwaminathan, S., 
Karplus, M., J. Comp. Chem., 4, 187 (1983). 
19. Hille, B., Ann, Rec. Physiol. 38:139, 52 (1976).