A Parallel Algorithm for Calculating the Potential Energy in DNA

Transcription

1 Proceedngs of the 28th Annual Hawa Internatonal Conference on System Scences 995 A Parallel Algorthm for Calculatng the Potental Energy n DNA John S. Conery, * Warner L. Petcolas, Thomas Rush III, Kesavan Shanmugam, * and Jose Domnguez * * Department of Computer and Informaton Scence Department of Chemstry Unversty of Oregon Eugene, OR emal: conery@cs.uoregon.edu Abstract The Dredng force feld s a seven-term equaton that descrbes the potental energy n a molecule as a functon of the relatve postons of bonded atoms and electrostatc nteractons between atoms that do not share a bond. For large molecules such as DNA, wth several thousand atoms, the O( n 2 ) nonbonded terms can requre a sgnfcant amount of computaton. In ths paper we present a data-parallel algorthm that takes tme O( n) on n processors. We compare the executon tme of our algorthm on a MasPar MP- wth an effcent sequental program runnng on an SGI workstaton. Introducton The long polymerc structure of deoxyrbonuclec acd (DNA) may change sgnfcantly as t performs ts vtal functons n lvng cellular organsms. The two man functons of DNA nclude replcaton, whch s needed for growth and the passng along of heredtary nformaton, and the transcrpton of the rbonuclec acds whch make the protens that regulate most of our bologcal processes. To help understand DNA s role n such events, researchers have been expendng consderable effort n determnng the three-dmensonal structure, or conformaton, of DNA under a great varety of condtons. In the past, X-ray crystallography has been a tremendous help n determnng some of the conformatons that DNA may adopt. It has the advantage that t expermentally provdes the average Cartesan coordnates for every atom n a molecule, and therefore the overall conformaton, but t also has several dsadvantages when t comes to studyng bologcal molecules n ther natural envronment. Two maor dsadvantages are that good crystallne DNA samples are not always easy to obtan, and that DNA s not naturally found as a sold. A promsng approach to ths problem s to combne computer modelng wth expermental methods to provde a better descrpton of the structure of DNA. Ths approach uses an expermental technque such as Nuclear Magnetc Resonance (NMR) or Raman spectroscopy to provde an ntal conformaton and then uses a mnmzaton algorthm to fnd the lowest energy conformaton n the neghborhood of the startng pont (e.g [2]). In ths paper we descrbe a new data-parallel algorthm for calculatng the energy of a sngle conformaton of a DNA molecule. Our algorthm wll compute the potental energy n a molecule wth n atoms, takng nto account the O( n 2 ) nteractons between all the atoms. The parallel algorthm runs n tme O( n) on n processors, whch s an mprovement over the O( n 2 ) tme complexty of a seral algorthm runnng on one processor. In order to avod calculatng all n 2 nteractons, t s common to set a cutoff dstance beyond whch energes are not calculated; for example, by settng the cutoff to 0 angstroms (Å) the program wll not compute the coulombc attracton of any par of atoms further than 0Å apart. Snce our algorthm s lnear n the number of atoms t s easer to calculate all nteractons and thus acheve a more accurate value of the total energy. The dfference n the calculated energy s nontrval 5% n one example molecule contanng 264 atoms and thus t s worthwhle to compute all nteractons when possble. In Secton 6 we gve a table that shows the energy values wth and wthout cutoffs. We also compare the executon tmes of a seral algorthm runnng on a Slcon Graphcs Indgo workstaton (part of a commercal package named BIOGRAF []) and our parallel algorthm runnng on a MasPar MP /95 $ IEEE 23

2 2 Background: Raman Spectroscopy Over the past twenty years, Raman spectroscopy has been developed to dentfy partcular conformatons of olgonucleotdes (short synthetc strands of DNA) n soluton, crystal and fber forms [7]. Only recently has ths nformaton been able to be used to determne a reasonable set of Cartesan coordnates for a whole olgomer of DNA n soluton. To date, only a few soluton structures have been solved because wthout other supportng expermental data (such as NMR data) there are a large number of calculatons whch need to be performed. Raman spectroscopy provdes nformaton about the energy (frequency) of the vbratons wthn a molecule [5]. Data are obtaned by shnng a monochromatc laser on the sample and detectng the lght beng scattered. A monochromator separates the lght nto ts component frequences, and a photomultpler tube detects the ntensty of lght of the dfferent frequences. Dfferent vbratons wthn a molecule wll produce dfferent peaks at dfferent frequences. The most lkely structure for the olgomer n soluton s the one whch wll reproduce the Raman spectrum and have the lowest energy. It s also true, and very mportant for ths applcaton, that a partcular vbraton wll have a slghtly dfferent frequency dependng on the geometry of the molecule. An example of ths s the Guanne breathng vbraton, whch appears at 625 cm, 665 cm, and 680 cm dependng on three dfferent conformatons of the DNA molecule (the unt cm - s a wavenumber, whch s drectly proportonal to frequency). Over the past twenty years, researchers have compared Raman spectra of crystals wth known Cartesan coordnates to make a lst of Raman conformatonal markers (some theoretcal calculatons were also performed). Usng these markers, a Raman experment can determne most of the structural nformaton of a molecule wth unknown geometry. Ths rough guess at a geometry may then be entered nto an energy mnmzaton program where the expermental geometry s to be determned. The Raman data can potentally dctate a structure whch has torsonal angles (see Secton 3.3) whch fall wthn ±0 degrees of the average soluton conformaton of the molecule. Wth ths ntal guess at the structure and a well defned, constraned regon of the molecule s multdmensonal conformatonal space, t s possble to more easly search for the locally global mnma. The long range goal of our research s to provde a more accurate descrpton of the average conformaton of large bologcal molecules by combnng expermental and computatonal methods. Nether class of methods alone s accurate enough. X-ray crystallography analyzes DNA and other molecules n crystallne form, whch has nherent naccuraces, and Raman and NMR methods can gve only partal nformaton at well-known markers. Computatonal technques alone are also nadequate: mnmzaton algorthms are prone to becomng trapped n local mnma and are very expensve for larger molecules snce there are so many dmensons n the search space. A combned expermental and computatonal method may be able to overcome these dffcultes and provde a more accurate pcture than ether method alone. The combned Raman/mnmzaton technque wll consst of. usng Raman conformatonal markers to obtan an ntal guess at a structure; 2. mnmzng ths structure to obtan a refned ntal structure; 3. systematcally mnmzng many ntal conformatons wthn the conformatonal space descrbed by the vbratonal spectra; and fnally 4. analyzng these mnmzed structures to obtan a fnal average soluton structure. 3 Energy Calculatons The energy calculatons are performed usng the Dredng force feld [6], whch s the sum of seven ndependent components: bond energy: -- k 2 b ( R R 0 ) 2 b angle energy: --k 2 θ ( θ θ 0 ) 2 θ torson energy: p --k 2 τ, n [ d cos( nφ) ] τ n = nverson energy: --C ( cosω cosω 2 0 ) 2 Coulombc attracton: R Q Q C ε 0 R 24

3 van der Waals nteracton: hydrogen bond nteracton: The frst four terms n ths energy feld are the bonded terms. They represent contrbutons to the total energy arsng from atoms whch are drectly bonded to each other. The four stuatons whch are taken nto account here are:. the energy due to a bond beng stretched or compressed; 2. the energy due to an angle formed by three atoms beng dsplaced from ts equlbrum poston; 3. the energy due to the torsonal angle formed by four sequentally bonded atoms beng dsplaced from ts equlbrum poston; and 4. the energy due to an atom beng dsplaced from ts equlbrum poston above the plane of three nearby atoms (known as an nverson). The remanng nonbonded terms are the electrostatc nteracton energes of an atom wth another atom n the molecule or the surroundng envronment. In these calculatons we do not compute nonbonded energy terms for atoms that are bonded,.e. the electrostatc nteractons between two bonded atoms s accounted for n the bond energy. In a large molecule, wth a thousand or more atoms, calculatng the nonbonded terms s the most tme consumng part of the calculaton snce there are on the order of n 2 nteractons n a molecule of n atoms. The overall goal of the computaton s to mnmze ths energy term by varyng the conformaton usng an optmzaton algorthm such as a conugate gradent or Newton method. In ths paper we present an effcent parallel algorthm for computng the energy of a sngle conformaton; at the end of the paper we descrbe how our algorthm can be embedded n an optmzaton algorthm. 3. Bonds R 0 D R R0 6 R R R D 0 5 R R R0 0 4 cos θ R H In the four bonded nteractons bond, angle, torson, and nverson energes a contrbuton to the total energy arses when atoms are dsplaced from ther equlbrum postons. In the bond term, R 0 s the equlbrum bond dstance specfc for the par of atoms n the bond, R s the current bond dstance, whch s calculated from ther Cartesan coordnates, and k b s a constant whch s also specfc for the par of atoms. When R s dfferent from R o (ether a postve or negatve dfference) ths term contrbutes to the total energy. Ths s analogous to, and actually modelled after, two balls connected by a sprng (Fgure a). When the sprng s stretched out past ts natural length, t reacts by tryng to return to ts equlbrum poston because ts energy s too hgh. The same s true when you compress the sprng,.e. the bond s too short. 3.2 Angles The term for the energy due to the angle formed by three atoms s dentcal to the bond term, except nstead of current and equlbrum bond length we compare the equlbrum angle θ 0 to the current angle θ (Fgure b). k θ s agan a constant that s specfc for the three atoms n the angle. As wth the bond term, only dsplacements from the equlbrum angle cause an energy contrbuton. 3.3 Torsons Torson refers to the force requred to twst a bond about ts horzontal axs. A torsonal term wll arse when there are four atoms,, k, and l lnked by bonds -, -k, and k-l. The current torsonal angle φ s the angle between the kl plane and the k plane (Fgure c). The torsonal term has the same concept as the bond and angle terms but s wrtten slghtly dfferently because there may be several equlbrum torsonal angles at perodc ntervals. The parameters that are specfc to the four-atom combnaton makng up the torsonal angle are the perodcty p of the potental, the force constant k τ, n, and the locatons of the p mnma, whch are accounted for by the constant d shown n the formula. The conventon s to defne a postve angle as clockwse lookng from atom toward atom k. In the Dredng force feld, t s also customary to rescale the force constant by dvdng t by the number of torsons defned around that same bond n the molecule. 3.4 Inversons --k 2 b ( R R o ) 2 An nverson occurs when an atom l that s at equlbrum wth respect to three other atoms,, and k s pushed to the other sde of the k plane (Fgure d). The nverson 25

4 (a) R (c) (d) ω l k (b) θ k φ k l l (e) θ H Fgure. (a) A bond between atoms and that are R angstroms apart. (b) The angle θ formed by atoms,, and k; atom s the central atom. (c) Torsonal energy s the result of twstng the bond between atoms and k. (d) The nverson energy term, from movng atom l to the other sde of the k plane. (e) Nonbonded hydrogen nteracton between donor and acceptor ; s bonded wth a hydrogen, θ s the angle formed by H. term has the same form as the bond and angle terms, where the nverson angle ω s the angle between the l axs and the k plane. The force constant here s labeled C, the current nverson angle s ω, and the equlbrum nverson angle s ω Nonbonded Coulombc Interactons In the coulombc term, the nteracton s due to the charges Q and Q on atoms and, where lke charges repel and unlke charges attract. Ths energy depends nversely on the dstance between the atoms, whch s agan calculated from ther Cartesan coordnates and scaled by ε 0 R. The scalng factor ε 0 s usually taken as, and the overall coulombc scalng factor, C 0 = puts the term n the proper unts to be added n the total energy summaton. As n all of the nonbonded nteractons, the nteractons between the atom beng summed over and any atoms t s drectly bonded to are not ncluded. In the coulombc term we also gnore -3 nteractons, or the nteracton between an atom and ts second nearest bonded neghbor. 3.6 Nonbonded van der Waals Interactons The van der Waals nteracton s another type of electrostatc nteracton. It has almost the same shape potental as the bond energy. The mnmum energy s at the equlbrum van der Waals dstance, R 0, and the potental energy depends on the magntude of the dsplacement from equlbrum. The constant D 0 s agan specfc to the atom par nvolved. Due to lmted expermental data, the R 0 and D 0 values between unlke atoms are calculated from the values for the nteracton between lke atoms. For example, R 0 for a C-H par can be estmated from the known R 0 values for C-C and H-H nteractons usng the formula R 0 CH R 0 CC R0 HH The values for D 0 are computed smlarly, replacng R by D n the above formula. 3.7 Hydrogen Bond Interactons = The hydrogen bond nteracton s calculated n the specal case where a donor atom (ether oxygen, ntrogen, or sulphur) has a bond to a hydrogen atom. The hydrogen s sngle electron s weakly attached, and the donor has room n ts outer shell for another electron, so the electron s pulled toward the donor. Ths alters the electrostatc charge between the donor and other atoms (known as acceptors). 26

5 H 3 H 5 C H 4 The hydrogen bond term s a functon of the angle H where s the donor and s the acceptor (Fgure e). It s almost dentcal to the form of the van der Waals term 4 except for the cos term. 4 Data Dstrbuton In order to calculate the energy of a confguraton we need to dstrbute the data both the Cartesan coordnates of the atoms and the relevant energy term constants to the processng elements (PEs) that need the data. In our data-parallel approach we dstrbute the atoms among PEs so that each PE wll hold the descrpton of one atom and the bonds t partcpates n. A PE wll perform all the calculatons concernng ts atom, whch we call the local atom. In the descrpton of the algorthm we wll assume there are more PEs than atoms; the mappng of atoms to PEs when there are fewer actual PEs s straghtforward and C 2 H 6 H 8 Bonds Angles Torsons Fgure 2. Ethane Molecule and ts Bonds, Angles, and Torsons H can often be done by a compler. At the end of the data dstrbuton, each PE wll contan:. An Atom structure ntalzed wth nformaton about one atom 2. An array of bond nformaton (one entry for each atom the local atom s connected to n the molecule) 3. Informaton about angles n whch the local atom s the central atom 4. An array of torsons and nversons, wth one entry for each possble torson where the local atom s one of the two central atoms The reason angles and torsons are lmted to those n whch the local atom s a central atom s to reduce the amount of redundant nformaton. If we kept all the angles n whch an atom partcpates every angle would be stored on three PEs, one for each atom n the angle. Once the data has been dstrbuted, the energy calculaton problem becomes one of fndng all the bond, angle, torson and non-bonded terms n whch ths atom partcpates and then calculatng the force due to each term. The total energy of the molecule s the sum of all these forces over all atoms, beng careful not to nclude duplcate energy terms. For example, f there s a bond between atoms A and B, we want the processor for ether A or B, but not both, to calculate the bond energy and contrbute t to the fnal sum. As an example, Fgure 2 shows an ethane molecule and the bond, angle and torson terms present n the molecule. The energy calculaton algorthm should generate these terms and calculate the forces due to them. In addton, t should calculate the force due to non-bonded terms. The forces wll be calculated n parallel on each PE and then combned (also n parallel) and forwarded to a front-end processor or user node for output. 4. The Vstor Algorthm Our method for computng bond, angles, torsons, and nversons s based on the Fox rng algorthm used to compute the nteractons n an n-body problem [8]. Our algorthm, whch we dubbed the vstor algorthm, s smlar to the Fox method n that descrptons of atoms are sent around a vrtual rng to PEs where other atoms resde, but t has the extra complcaton that we need to compute angles, torsons, and other terms that nvolve more than two bodes. If each PE has an atom, the vstaton operaton can be accomplshed by shftng the atoms around a vrtual rng, as llustrated n Fgure 3. Almost every physcal processor 27

6 topology has an embedded rng structure, so ths communcaton pattern s lkely to be very effcent [4][8]. It wll take n steps to send an atom around the entre rng, where n s the number of atoms n the molecule. We assume every PE mantans a local atom, whch s named a n the followng descrpton. In addton, each PE holds the descrpton of a second atom, named x. The step send x to the rght means the PE should send the descrpton of x on the communcaton channel that connects t to ts neghbor on the rght n the vrtual rng. Smlarly, read x from the left means the PE should read nformaton from the communcaton channel connectng t to ts left neghbor and store ths descrpton n x. Thus durng the algorthm x s contnually updated, and at any one tme x holds the descrpton of a vstng atom. Vstor Algorthm (Bonds Only) do n parallel on each PE that contans an atom a: x = a repeat n tmes: send x to the rght read x from the left f a s connected to x: calculate the bond between a and x else: calculate the nonbonded nteracton between a and x After n cycles an atom that starts at PE wll have passed completely around the rng and wll now be vstng the PE drectly to the left of PE. Durng ts ourney atom wll be the vstng atom on every other PE n the rng, and each of these PEs wll have had a chance to compute the energy term represented by ther local atoms and the travelng atom. Note also that durng each cycle n PEs wll smultaneously compute n dfferent energy terms, correspondng to the nteractons between ther local atom and the atom that s currently vstng that PE. Snce there are n cycles n the outer loop of the vstors algorthm, the tme requred to compute all energy terms wll be O( n) on n processors, whch s an mprovement over the O( n 2 ) that would be requred on a sngle processor. The above algorthm does not address the problem of duplcate calculatons, and t omts the steps that do the calculatons for angles and torsons. As we wll see, communcaton wll be more effcent f we go ahead and duplcate the bond calculatons, but we wll avod duplcatng angles and torsons. 4.2 Angles and Torsons n the Vstor Algorthm allow PE a to compute all the angles that contan atom a. Ths means that PE a wll generate angles of the form a-x-x, x-a-x, and x-x-a, where x s a nonlocal atom, and that every angle a-b-c wll be duplcated n PE a, PE b and PE c. A problem wth ths approach s that t requres a second pass around the rng because we do not know the order n whch atoms wll vst each other. For example, assume the molecule contans an angle a-b-c. Wth the straghtforward approach the angle should be generated by PE a. But f c vsts PE a before b does, ths angle wll not be generated because bond a-b wll not be present when c vsts PE a. Untl b vsts PE a, PE a wll not know t has to look for c n order to compute the angle a-b-c. We can solve ths problem by usng two communcaton cycles, one for bonds and another for angles. On the frst pass all the bonds are calculated, and by the second pass each PE wll be able to detect when the thrd atom of an angle passes by. A better approach smultaneously solves the problems of duplcate angle calculatons and usng an extra pass. Instead of generatng all the angles that contan a, PE a wll generate only those angles n whch a s the central atom. That s, PE a wll generate only angles of the form x-a-y. Now each angle occurs only once n the system on the PE that s the home of the center atom and two connectng atoms can be recognzed and used on the frst pass around the vrtual rng. Compute Angles [done when atom x vsts PE a :] f x s connected to a: calculate and store bond x-a for all bonds a-b created so far: calculate and store angle x-a-b else: calculate nonbonded nteractons between a and x We can use a smlar strategy for torsons and nversons. PE a wll compute torsons n whch atom a occurs as ether the second or thrd atom,.e. torsons of the form x-a-x-x or x-x-a-x. However, the prevous problems have returned: each torson s computed on two PEs, and t wll take multple passes to make sure a PE has all the nformaton t needs. Once all the angles have been computed, however, t wll take only one addtonal pass to compute the torsons. The complete vstor algorthm thus uses the frst pass to compute bonds and angles and a second pass to compute torsons and nversons. We wll correct for the fact that bonds, torsons, and nversons are computed twce durng the summng phase of the algorthm, whch s descrbed next. The straghtforward opton for computng angles s to 28

7 5 Summng the Energy Terms When the vstor algorthm termnates, the bond, angle, torson, and nonbonded energy terms are dstrbuted across all the PEs. The fnal step s to collect and sum all the terms. Every parallel processor has a method for applyng a bnary assocatve operator such as addton to elements that are dstrbuted over all the PEs. Ths operaton, often called a reducton, can be done n O( log 2 n) steps on n processors. For example, f a vector of 000 elements s allocated such that x s stored on PE, we can compute n 0 steps ( log ). In the followng descrpton, bonds means the sum of all the bond energes for the atom a,.e. all the bond forces computed at PE a n the prevous phases. bonds and torsons are angle and torson forces computed at PE a for atom a. The expresson reduce(x) means compute the sum of all the local varables x on each of the PEs, returnng a sngle global sum. Sum Energy Terms do n parallel on each PE that contans an atom a: f = bonds f2 = angles f3 = torsons f4 = nversons f5 = nonbonded terms x 000 bond-force = reduce(f)/2 angle-force = reduce(f2) torson-force = reduce(f3)/2 nverson-force = reduce(f4)/2 nonbonded-force = reduce(f5)/2 E = bond-force + angle-force + torson-force + nverson-force + nonbonded-force Snce the reductons are O( logn) the total tme to compute the energy s O( logn + n) = O( n). 6 Implementaton and Evaluaton We mplemented he vstor algorthm on a MasPar 04, an SIMD machne wth 4096 processng elements. The code was wrtten n MPL, a data-parallel language based on ANSI C. The vrtual rng was mplemented va the X-net, a nearest neghbor communcaton channel wth torodal wrap. To make sure our program was producng the correct results and to compare our parallel algorthm wth a very good seral algorthm we used BIOGRAF [] to compute the free energy n seven molecules, rangng n sze from 3 atoms to 2544 atoms. We ran BIOGRAF on a Slcon Graphcs Indgo workstaton. We then computed the energy of the same seven molecules on the MasPar. The tmng results are lsted n Table. The table shows two columns for BIOGRAF. The frst column s the executon tme when the nonbonded nteractons were cut off at 0Å,.e. when two atoms were farther PE PE 2 PE n Atom: Label: Type: C Charge: X: Y: Z: Connectons: vdw: bonds: angles: torsons: Atom: Label: 2 Type: H Charge: X: Y: Z: Connectons: vdw: bonds: angles: torsons: Atom: Label: n Type: C Charge: X: Y: Z: Connectons: vdw: bonds: angles: torsons: Fgure 3. A Rng of Processors 29

8 Table : Sequental Executon on SGI vs. Parallel Executon on MasPar Executon Tme (seconds) Molecule Sze (#Atoms) Number of Nonbonded Terms BIOGRAF on SGI cutoff no cutoff bonds, angles, torsons MasPar calculate energy Cutoff Error (%) <. 66 2, <. 35 8, , , , , ,544 3,273, than 0Å apart the program dd not compute any nonbonded energy terms. The second column shows the executon tme when all nonbonded terms were calculated, whch was acheved by settng the cutoff dstance to a very large value. The last column n the table shows the dfference n energy (as a percentage) when t was calculated wth and wthout all nonbonded nteractons. From ths small set of data t appears that the error grows wth the sze of the molecule, and that t wll be mportant to use all nteractons when searchng for the mnmal energy conformaton of large molecules. The executon tmes shown for our MasPar program are the tme to construct the bonds, angles, and torsons (two passes around the vrtual rng) and the tme to calculate the energy of a sngle conformaton (one pass, calculatng all seven terms at each pont). It does not nclude the tme to load the descrptons of the atoms to ndvdual PEs, whch s substantal on our machne wth a low bandwdth connecton between the PEs and the front-end workstaton (the tme to load the descrptons of every atom n a molecule s comparable to the tme to calculate the energy). The executon tmes for BIOGRAF measure only the tme to compute the energy. BIOGRAF allows users to specfy whch of the seven energy terms to use n a calculaton. We frst ran the program and measure the executon tme t 0 when no energy terms were ncluded. We then ran the program agan, wth and wthout nonbonded nteractons, and subtracted t 0 from these tmes to get the tmes shown n the table. Wth 2544 atoms there were too many hydrogen bond nteractons and BIOGRAF was unable to produce a result. Fgure 4 s a plot that compares the energy calculaton tmes for BIOGRAF, usng no cutoffs, wth the tme to do the same calculaton on the MasPar. As expected, for small atoms, the SGI s much faster: t has a 50MHz processor and 32-bt nternal data path, whereas the MasPar MP- has a 2.5MHz processor and 4-bt data path n the PEs. The plot (whch has a log scale on the vertcal axs) also shows the expected lnear growth of the executon tme of the MasPar program as a functon of the number of atoms. For ths set of data the BIOGRAF executon s worse than expected; t should grow quadratcally but nstead s steeper. Most of ths tme s accounted for by the hydrogen bond nteracton term; when ths term s removed, the BIOGRAF plot s closer to quadratc and t does not exceed the MasPar tme untl the number of atoms s greater than Summary and Future Work Ths paper descrbes the desgn and mplementaton of a data-parallel algorthm for computng the potental energy n a bologcal molecule usng the Dredng force feld. The tme to compute the energy n a molecule wth n atoms s O( n) on n processors. The parallel algorthm allows one to compute all nonbonded nteractons, whch would requre tme O( n 2 ) on a sngle processor. Ignorng nonbonded terms for atoms that are beyond a small (0Å) cutoff dstance can lead to errors as large as 5%, so t would appear to be worthwhle to compute all nteractons. The parallel algorthm, when mplemented on a MasPar MP-, begns to be more effectve (faster and more accurate) than a sequental algorthm runnng on an SGI Indgo workstaton on molecules wth about 750 atoms. 30

9 bograf maspar Tme (sec) to Calculate Energy n a Sngle Conformaton Fgure 4. Comparson of the executon tme of the vstor algorthm runnng on a MasPar MP-04 and BIOGRAF on an SGI Indgo. The gray bars represent the error when not all nonbonded terms are calculated (scale on the rght) % 0% 5% Number of Atoms Our future plans are to optmze the algorthm so t can be used as part of an energy mnmzaton procedure. In a mnmzaton program the structure of the molecule (ts bonds, angles, and torsons) needs to be computed only once; after that a sngle pass around the vrtual rng wll suffce to calculate the energy of each new conformaton. A straghtforward method for usng the vstor algorthm on the MasPar wll be to run a standard sequental optmzaton code on the DECstaton front-end and have t nvoke the parallel algorthm to compute the energy n each new conformaton. A longer range proect wll be to nvestgate a parallel mnmzaton technque. The vstor algorthm should be easy to mplement on a general parallel machne. It wll be smple to code t for SPMD executon n a language such as Dataparallel C [3] or Hgh Performance Fortran and let complers for those languages allocate atoms to processors. Wth a lttle extra work, for example fgurng out how to ppelne a batch of atoms from one processor to the next whle processors are workng on a dfferent set of atoms, t should be possble to mplement the method n a message passng style and get t to run effcently on a dstrbuted memory parallel processor or heterogeneous network of workstatons. 8 References [] BIOGRAF ver. 3.0, Molecular Smulatons, Inc., Waltham, MA. [2] Da, Z., Dauchez, M., Thomas, G., and Petcolas, W.L. J. Bomolecular Structure and Dynamcs 9(6), pp , 992. [3] Hatcher, P. J. and Qunn, M. J. Dataparallel Programmng. MIT Press, 99. [4] Leghton, F. T. Introducton Parallel Algorthms and Archtectures: Arrays, Trees, and Hypercubes. Morgan Kauffman Publshers, San Mateo, 992. [5] Long, D.A. Raman Spectroscopy. McGraw-Hll, 977. [6] Mayo, S.L., Olafson, B.D., and Goddard, W.A. III. J. Phys. Chem. 94, pp , 990. [7] Petcolas, W.L. and Evertsz, E. Methods n Enzymology, vol. 2, pp , 992. [8] Setz, C. L. The Cosmc Cube. Communcatons of the ACM 28(), Jan