Minimizing the Lennard-Jones Potential Function on a Massively Parallel Computer
G.L. Xue R.S. Maier J.B. Rosen University of Minnesota, Minneapolis, MN
E-mail: {xue,maier,rosen} Qcs.umn.edu
Abstract
The Lennard-Jones potential energy function arises in the study of low-energy states of proteins and in the study of cluster statics. This paper presents a math- ematical treatment of the potential function, dleriving lower bounds as a function of the cluster size, in both two and three dimensional configurations. Thlese re- sults are applied to the minimization of a linear chain, or polymer, in two-dimensional space to illustrate the relationship between energy and cluster size. An alg- rithm is presented for finding the minimumenergy lat- tice structure in two dimensions. Computational results obtained on the CM-5, a massive] y parallel processor, support a mathematical proof showing an esalentially linear relationship between minimum potential energy and the number of atoms in a cluster. Computational results for as many as 50000 atoms are presented. This largest case was solved on the CM-5 in approximi~tely 40 minutes at an approximate rate of 1.1 32-bit gigaflops.
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1 Introduction
The minimum-energy configuration of a small cluster of atoms, molecules, or ions is known as the molecular con- formation problem [1,2,4,5, 7,8,9, 13, 14, 17, 18]. It is a central problem in the study of cluster statics, or the topography of a potential energy function in an internal configuration space. From the viewpoint of mathemat- ical optimization, it is a difficult global optimization problem which doea not yield e=ily either to discrete or continuous optimization methods [1, 2, 10, 15, 16, 17].
The most successful efforts in identifying minimum en- ergy conformations have been achieved by a combina- tion of discrete methods and numerical optimization.
To illustrate the difficulty of the problem, Hoare [8] has developed an empirical function relating the number of local minima of a certain pote:tial energy function to the number n of atoms as O(e” ). Nonetheless, by care- ful study of minimum-energy conformations, suitably good starting points for numerical optimization have been identified, permitting researchers to locate puta- tive global minima for as many as n = 46 atoms [16].
The study of such conformations draws from analysis of crystalline structures, as well as results from sphere- packing [3].
While the real problem requires a 3-dimensional model, efficient methods for finding the minimum en- ergy conformation can best be developed using a 2- dimensional model [2]. Therefore in this initial com- putational work we have limited consideration to the special case of a linear chain of atoms confined to a plane. The results include a lower bound on the po- tential function, a method for choosing starting points for local minimization procedures, and the application of parallel computation to the discrete and continuous minimization problems. More specifically, we have de- veloped a method for this problem which permits the solution in a reasonable time, for values of n as large aa n = 50,000. This method is very well suited for the CM- 5 architecture, and solved this largest case (n = 50, 000) in approximately 40 minutes on the CM-5, at an approx- imate rate of 1.1 32-bit gigaflops. A theoretical lower bound is also obtained for the potential energy as a func- tion of n, which is shown to depend linearly on n. This is confirmed by the computational results.
2 Molecular Conformation Prob- lem
Suppose the cluster configuration of n atoms is given by the Cartesian coordinates
Pi={ Zi, Vi, ~iJ, ~=1, ”.”, n. (1) The simplest assumption about interaction energy is that of tw~body forces between component atoms, lead- ing to the general two-body, n-atom potential
n j-1
Vn(P) = ~~v(llPj ‘PillZ), (2)
j=2 i=l
where v(r) is a pair potential function satisfying the conditions [8]:
v(r)~O-asr~cm (3)
v(r) + ~ for r < t’min, l’mi~ >0 (4) v’(ro) = O for some r~in ~ ro < m (5) v“(ro) >0 and v(ro) <0 (6) While this model is clearly a greatly simplified rep- There are a number of pair potential functions of inter- resentation of the correct molecular configurate ion, the est, however we focus exclusively on the Lennard-Jones techniques developed should be applicable to more re- pair potential (Figure 1),
alistic models. This simplified model also serves as an
excellent scalable test problem for parallel computation v(r) = r-12 – 2r-6 (7)
on the CM-5. It is believed that the method described
here can be successfully extended to more realistic 3- in which r. = 1 and v(ro) = – 1, giving the two-body,
dimensional molecular models. n-atom energy potential function
The outline of this paper is as follows. In section 2, relevant background on the molecular conformation problem is presented, and a new lower bound on the Lennard-Jones potential function is derived. Then the linear chain problem is described, a special case involv- ing constraints on certain atomic pair potentials. A de- tailed formulation of the linear chain problem is given in which the constraints are enforced implicitly, leading to an unconstrained minimization problem. In section 3, a heuristic method for obtaining low-energy configura- tions on a t we-dimensional lattice is described, along wit h computational results obtained on the CM-5, a massively parallel computer with 512 SPARC process- ing elements. It is shown that the method takes full ad- vantage of the MIMD architecture in solving problems with up to 105 atoms. In section 4, local minimization of the potential function by a parallel implementation of the Limited Memory BFGS algorithm is described, in- cluding computational results obtained by applying the method to heuristic starting points. The resulting func- tion values are stable configurations with lower binding energy than the previously best known solutions.
n ]-1
Vn(p) = ~ ~(r,:j2 – 2r,;~) (8)
j=2 i=l
where ri,j = Ilpj – pi IIz, with gradient
n
VjV~(p) = ’12 ~ (7’,~~4-r~~)(Pj-pi), j = 1, ....n.
i=l, i#j
(9) Note that for p such that VjVn(p) = (), j = 1,..., ~ (a stationary point), the configuration is stable if p is a lo- cal minimum and metastable otherwise. An important characteristic of the function (8) is invariance under ro- tation and permutation. Clearly, any permutation and translation of the ordering of {pj } results in an equiv- alent function value, since Vn (p) depends only on the pair distances.
2.1 Lower Bounds
In this subsection, lower bounds on the molecular po- tential function are derived, First, linear lower bounds as a function of n are proved for the two and three- dimensional cases, under the assumption that the mini- mum distance between any pair of molecules is bounded
1
II
-1 ~ v I
0 1 2 34 s 6 7 8
* —,
Figure 1: Lennard-Jones pair potential function
away from zero. Second, we derive a lower bound for the minimum distance between any pair of mcJecules at a stationary point which leads to lower bounds for the Lennard-Jones potential function which are nearly linear in n and much sharper than earlier results.
For any positive number a and integer n ~ 2, define En(a, p) = En(~, Pl, P2, .--, Pn) =
= x((;)” - (:)6)7 (w
@3 ‘
where ri,j is the Euclidean distance between pi and pj.
Note that En(a,.) is more general function than Vn(o) defined in the previous section, for En(a, ●) = ~Vn(0) for a = 2- 1/6, Since En (a,.) is bounded from bellow for any fixed n and a, and is continuous on {p[En(a, p) ~ O}, it achieves its global minimum. Define ~(a, n) = rein= En (a, p). We will prove several useful prc)perties of 4(cJ, n) in the following.
Theorem 2.1 #(a, ●) is a monotonically decreasing function of n.
Proof, Let qt(a, n) = l?~(a, PI, . . . . p~). Now choose Pn+l such that ri,~+l > 2a for i = 1, ..., n. Then
@(a,n+ 1) s En+l(a,pl,””,pn,pn+l) =
= 44a,~)+ jj(~)”-(:)6]<
i=l
< #(a, n). (11)
This completes the proof.
Lemma 2.1 Let {ak}, {bk} and {ck} be sequences of nonnegative real numbers such that
~a~<&b~,cm— ~cm+~,m=l,2,3,---- (12)
k-l k=l
Then
m m
~akCk~~bkCk,m=l, 2,3, . . . . (13)
For any two positive real numbers R and r, define T(R, r, d) to be the maximum number of d-dimensional balls with radius r that can be packed in a d-dimensional ball with radius R + r. We have
Lemma 2.2
Proof The right hand sides follow from the comparison of volumes. The left hand sides follow from the hexago- nal lattice packing and the face-centered cubic packing, respectively [3] .
Theorem 2.2 (Linear Lower Bound) Assume that there exists a positive constant c s ~a such that the distance between any pair of molecules is no less than 2( at a global minimizer. Then there exists a negative constant a such that
cj(a, n) > cm (16)
holds for every integer n >2.
Proof. We need only to prove the theorem for 3- dimensional space, because a lower bound for the 3-d Lennard-Jones potential is always a lower bound for the 2-d Lennard-Jones potential.
In order to prove (16) we just need to prove that En(a, pl, . . ..p~)>cm (17) holds for all pi c R3.
Define Rk = {jli < j < n,ri,j > a,ka – 2C < ri,j ~ (k+ l)a - 2c} for any fixed i c {1,2,...,73} and any integer k. We have
It follows from Lemma 2.1 that
E [(:)12 - (:)61
j=i+l ‘
~- = [-]3 - [~]’ 0.728(&)3
x
k=l (~ - $)6 -
(;)6
~ .$ * [(k+ l)a]’ – [ka – ~a]” 0.728(:)3
x (k - $)6 - (+)’
k=l
- 2(k–$)2+$(~ –+)+%
> -$[0.278 X 36 + ~
(k–:)” 1
k=l
k=l k=l = const. (18)
Let a be the constant in (18). Then it follows that
~(a, n) ~ an for all n ~ 2. This completes the proof.
Note that the sum in (18) is the product of (:)3 and an absolute constant. For the 2-dimensional case, we can similarly prove that ~~=i+l [(%) lZ - ( ~)6] is greater than or equal to (+ )2 times an absolute constant,
Theorem 2.3 (Minimum Bond Length)
At any stationay status, the minimum distance be- tween any pair of molecules is greater than or equal to min{n-+, *}a.
Proof. Let c be the minimum distance between the molecules in a stationary status. Assume that the con- clusion is not true. We will show that this leads to a contradict ion. Within the proof, we will use the follow- ing notations. ~={1, . . ..k}. ~={k+l, n},l~=, l~=
{(i,j)\i e I,j E J})H= {(i,j) C IJl(i, j) # (k, k+l)}
and ~ = {(i, j) E ~lri,j > a2*}. We will also assume that ~ ~ a.
Assume that we have reordered the points pl, . . . . pn such that rk,k+~ = c, and that xk = O, $k+l = c, xi ~ O,i= 1,2,... ,k, ~j >0, i=k+l,..., n, where pi=
(~i, yi, ~i). Also, we will write En(a,p) M f(z, ~, z).
For any k, the partial derivative of ~ with respect to
~ $(fi)%k - zj)[-2(&)’ + 1]
j#k ‘
~ $(_&)’(zk - ~i)[-z(fi)’ + 1]
iGI,i#k
Therefore
xkeI Gaf = (I,J)GIJ~ :(:)% ‘ - %)[-X:)* + 11
= :(& )8(Zk - Zk+, )[-2( *)’ +11
+ x :(:)8(” – “)[-2(:)’ +11
(i,j)cfi
—— ;(:)8(-4[-2(3’ + 1]
+ ~ $(~)8(~i – $j)[–z(fi)” +1]
(I!j)m ‘
> $(:)8 (-4[-(:)61
+ ~ ;(:)%* -2,)[–2( ~)’a +11
(:,j)~fi ‘
= :(:)”+
2 :(:)13-:+(y)’
= :[(:)’3-(:)W .
= > 0 if if c < a2~n-i$.
This shows that Ek~I a.k
Therefore c ~ a2~ n- *. This completes the proof, By Theorem 2.3 and the proof of Theorem 2.2, we have the following lower bounds on the Lennard-Jones potential function.
Theorem 2.4 (Lower Bound on 3-d Function) For the %d problem, there exists a negative constant a such that
@(a, n) ~ Crnfi (19)
holds for every integer n >2.
Theorem 2.5 (Lower bound for 2-d function) For the 2-d problem, there ezists a negative constant Q such that
~(a, n) ~ CYn~ (20)
holds for evey integer n >2.
2.2 The Linear Chain
A simple protein structure can be modeled as a sp~
cial case of the n-atom conformation problem. Define a chain of n molecules connected by bonds of fixed length. Suppose, for simplicity, that the bonds are all unit length. Then the problem may be viewed as a con- strained problem in the original configuration space
(21)
subject to
ri,i+l = l,i= 1,..., n— 1. (22) Since the bond lengths are given, it is possible to ex- press the chain configuration as a set of angle pairs. In this formulation, the location of molecule i + 1 is speci- fied by two angular offsets from the i-th molecule, This
formulation leads to an unconstrained problem in the configuration space of angles.
Formulation by angles implies that the positions of at first two molecules are fixed, relative to the rest of the chain. Thus, without loss of generality, fix th,e loca- tion of the first two molecules pl at (1, O, O) and pz at (2, O, O). Then for any k c {3,..., n}, the position of the kth molecule is determined by a sequence of anglle pairs {~i, /3i}, i = 1,..., n – 2 such that pk can be represented by the triple
k-2
(z+ ~ Cocos, ~ shoos, ~sh(/3i))
1=1 *=1 i=l
(23) The computational complexity of the potential func- tion and gradient are 0(n2) and 0(n3), respectively,
3 Discrete Optimization
By studying the grid points and the optimal topolo- gies of small problems, a heuristic algorithm has been derived for minimizing the two-dimensional Lennard- Jones potential function. It was observed that in op timal topologies for the two dimensional problem, the molecules are positioned at certain regular grid, points on the plane which correspond to a hexagonal lattice packing. This is analogous to the observation that reg- ular icosahedra correspond to low-energy configurations in the three-dimensional problem [8].
3.1 A Heuristic Algorithm
The discrete algorithm folds the linear chain of molecules into a hexagonal lattice packing. The dif- ficulty of the problem arises from two points. First, the placement of molecules on the plane must satisfy the constraints of a linear chain, i.e. must describe a Hamiltonian path among the molecules with bonds of unit length. For computational purposes, it, is also important to express the placement in terms of angles between adjacent molecules, rather than coordinates in the plane. Second, the number n of molecules in the chain does not always permit a symmetric lattice pack- ing, so it is necessary to fold the chain at a linlk which leads to the densest possible packing. A nonsymmetric configuration is illustrated in Figure 2,
The discrete algorithm is presented aa two modules.
Algorithm 3.2 is the “inner” module which, for a chain of n + 2 molecules and a fold point at molecule p, finds the sequence of angles {61, .... On} corresponding to a low-energy hexagonal lattice packing. Algorithm 3.1 is the “outer” module, which determines the range of fold points p which includes the minimum-energy conforma- tion.
0
-1- .2- -3 -
4 -
-5<
4 -
.7 -
-8 -
-9 -
.“
-2-1012345 678
Figure 2: Optimal configuration for a 100-cluster
Algorithm 3.1 (Discrete Minimization). Given a number n of molecules, and the Lennard-Jones potential function (8), the discrete optimization algorithm finds the lattice structure with minimum energy.
forp=f, ...!. +#l@+3
1.
2.
generate a chain configuration for n molecules with fold point p using Algorithm 3.2.
evaluate Lennard Jones potential (21) for the cur- rent configuration and compare with the previous minimum, replacing the incumbent minimum with the new minimum.
Algorithm 3.2 (Chain). Given a number n of molecules and a fold point at molecule p, the chain al- gorithm generates a sequence of angles {81, .... On} de- scribing a linearly connected chain of molecules forming a tw~dimensional hexagonal lattice.
k=p; L=O;
while k >0 do
forj=lto 3dofori =lto Ldo{O~=O; k= k-l}
endfor; 6k = -$; k = k - 1; endfor;
L= L+l;
endwhile;
ep+l = -:7r; OP+2 =$r; k=p+3; L=O;
while k < n do
for j = lto3dofori= lto Ldo{O~=O; k=k–1}
endfor; ok = $ r; k = k – 1; endfor;
L= L+l;
endwhile;
fori=2to ndo Oi=Oi+Oi_1.
3.2 Computational Results for Discrete Optimization
The discrete algorithm was implemented on a CRAY-2 and a Thinking Machines CM-5 operated by the Min- nesota Supercomputer Center for the AHPCRC. The CRAY-2 implementation was written in Fortran 77 and compiled wit h the cjl 7’7 version 4.0 compiler in single- precision (64-bit). The CM-5 version was written in Fortran 77 with message-passing extensions and com- piled under the crn~ version 1.0 compiler. All the com- putation on the CM-5 were in single-precision (32-bit) except the reevaluation of the minimum function which was computed in double-precision (64-bit). Results for the CM-5 were obtained with a 512-processor partition.
It should be noted that the CM-5 processors have not yet been equipped with vector processing units.
Results for the discrete algorithm are presented in Ta- ble 1. For the CM-5, the number of seconds is the total elapsed (wall clock) front end process run time. For the CRAY-2, the number of seconds is total CPU time.
The megaflop rate is calculated using the approximation f~OPS = $((11/2)332 + 872) for both machines.
Note that the minimum energy is monotonically de- creasing as a function of the number of molecules n in the chain. The time requirement increases as a func- tion in the form of O(n25): the potential function is evaluated approximately ~ times in Algorithm 3.1, and each evaluation requires O(n2) calculations. The megaflop rate increases monotonically with the problem size, as the proportion of local computation to com- munication increases. The peak rate of 1.1 32-bit gi- gaflops on the CM-5 was obtained for the largest prob- lem, n = 50000 (Figure 3). Figure 4 compares CRAY-2 and CM-5 execution time. Note that the CM-5 is 8 times faster than the Cray-2 (in single processor mode) for n = 10000.
The computational results for V.(p) provide an em- pirical basis for estimating coefficients of the theoretical model of section 2. In section 2, a lower bound on ~(n) was derived of the general form
where ~ was shown to be ~ = 17/13, Given the values of Vn (p) obtained by discrete optimization in Table 1, the parameters a, /3,7 of this model were estimated by a nonlinear least-squares procedure (subroutine LMDER in MINPACK). The resulting estimates were
V~(p) ~ –11.63 – 3.209 n0”9999. (25)
Although the residuals of the fitted equations tend to increase with n, the effect is minor and the conclusion is that the potential function is essentially linear in n for the range n <105 (Figure 5).
n
1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 20000 30000 40000 50000
Vn(p)
-3231,048 -6544.523 -9870.816 -13203.476 -16542.495 -19882.841 -23226.991 -26572.420 -29919.131 -33267.115 -66794.213 -100363.262 -133954.326 -167559.440
Cray-2 results CM-5 results second mflop second mflop
1.56 112 0.54 297
8.34 118 1!81 471
22.00 123 3.81 617
45.07 124 6.78 711
77.24 126 10.59 795
121.27 127 15.52 856
176.61 128 21.39 913
245.91 128 28.56 955
333.36 127 37.19 984
436.07 126 47.57 1001
— — 250.23 1077
— — 674.94 1100
— — 1374.70 1109
— — 2425.03 1098
Table 1: Results for Discrete Optimization. Re- sults for a chain of n molecules include best known potential energy function value, execution time, and megaflops on the Cray-2 (single processor mode) and the CM-5.
1300 , ,
, 1200.
1100.
1000.
900.
800.
700-
600-
500-
400-
300 4 $ i
o Sooo 100001500020000 25000 3000035000 400004500050000 cluiter size
Figure 3: Mega flops sustained on the CM-5
4 Local Minimization
400 .
350-
300-
250.u : 200-
150-
100.
50 -
“
450I , , 1 , t
“may. tim ‘ +-
“Ca-s.ti!ze’
/
I
“o 1000 2000 3000 4000 5000 6000 7000 8000 90(}010000 cluster size
Figure 4: Execution time On CRAY-2 and CM-5.
for discrete algorithm
Xl&
o
-2-
4 -
-6-
-8-
-lo -
-12.
-14-
-16-
-180 0.5 1 M 2 ‘2s 3 3s 4 45 5
Chl$mSk Xlw
Figure 5: Minimum energy as a function of num- ber of molecules. Computed values of Vn (p) obtained by discrete optimization.
Local minimization procedures are designed to locate stationary points of the potential function, Due to the exponential number of such stationary points, it is impractical to apply local minimization from random starting points, since there is little hope that the global minimum will be located from among the many station- ary points. A more typical use of local minimization in the molecular conformation problem is to find a sta- ble configuration using as a starting point some unsta- ble, low-energy configuration obtained by other meth- ods, often based on physical andlor geometrical consid- erations.
The Limited Memory BFGS (LM-BFGS) algorithm is a modification of the standard BFGS quasi-Newton algorithm for general purpose minimization. The stan- dard BFGS algorithm is one of the most effective and widely used minimization algorithms employed where second-derivative information is not available. For a complete description, see [6].
The LM-BFGS method was applied as a local mini- mization procedure for the unconstrained problem, us- ing a code developed by Liu and Nocedal [12] and adapted to the CM-5. See [11] for a description of the implementation of LM-BFGS on the CM-2 using For- tran 90. The method was applied using the heuristic starting point described in section 3. Recall that for a given n, this starting point is an unstable, low-energy configuration obtained by discrete optimization. The code was implemented on the Minnesota Supercomputer Center CM-5 and compiled under the cnzfFortran com- piler version 1.0 in double precision. The code was run for values of n ranging from 10 to 10,000. The results are presented in Table 2.
Table 2 shows the results of the local minimization procedure for various values of n. The table includes the starting and ending function value, gradient norm, and the number of iterations.
Note in Table 2 that the starting and ending func- tion values are not very different, although a substan- tial number of iterations are required to move from the starting point to the stable solution. Recall that the time required by the discrete minimization method is less than a single gradient evaluation. Therefore it is a very efficient starting point generator.
5 Acknowledgement
This research was supported in part by the United States Army Contract Number DAAL03-89-C-O038, the Air Force Office of Scientific Research grant AFOSR91- 0147, and the Minnesota Supercomputer Institute.
Thanks are due to Don Aust& for his support and encouragement, to Jorge Mor4 and Panes Pardalos
n fo f 11911 s’= itn
10 -20.085 -20.102 0.1E06 0.6 15
20 30 40 50 60 70 80 90 100 1000 10000 -
-47.519 -77.248 -107.122 -137.178 -168.239 -199.303 -230.457 -261.701 -292.946 -3231.048 33267.115
-47.595 -77.392 -107.343 -137.488 -168.631 -199.776 -231.019 -262.358 -293.697 -3240.966 -33374.617
0.5E-06 0.6E06 0.213-05 0.1E05 0.lE-05 0.2E05 0.2E05 0.2E05 0.2E05 0.3E04 0.3E03
2.6 2.’7 3.4 3.4 4.5 4.9 5.3 6.7 6.6 126.7 4783.1
55 50 48 50 63 62 63 73 68 172 454
Table 2: Results of Local Minimization. Results for clusters of n atoms include initial and final func- tion value, initial and final gradient L2-norm, execution time, LM-BFGS iterations, and number of function- gradient evaluations.
for providing many useful references, to Juan Meza, Jill Mesirov, George Wilcox, David Ferguson, Keith Kastella and David Garrett for very helpful discussions.
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