in Molecular Simulations

Transcription

1 A Fast and Accuate Analytcal Method fo the Computaton of Solvent Effects n Molecula Smulatons Thess by Geogos Zamanaos In Patal Fulfllment of the Requements fo the Degee of Docto of Phlosophy Calfona Insttute of Technology Pasadena, Calfona 2002 (Defended Decembe 14, 2001)

2 2002 Geogos Zamanaos All Rghts Reseved

3 Abstact The solvent envonment of molecules plays a vey mpotant ole n the stuctue and functon. In bologcal systems t s well nown that wate has pofound effects n the functons of potens. Smulatons assst us n mcoscopc studes of chemcal and bologcal phenomena. It s mpotant then to nclude solvaton effects accuately and effcently n molecula smulatons. In ths wo we pesent a novel appoxmate analytcal method fo calculatng the solvaton enegy fo evey atom of a molecula system and the foces that act on each atom because of the solvent. The solvaton enegy s pattoned nto long-ange and shot-ange contbutons. The longange contbutons ae due to pola nteactons between the solvent and the solute and the shot-ange ae due to van de Waals and entopc effects. We show how the calculaton of these effects, unde cetan appoxmatons, can be educed to the calculaton of the volume and exposed aea of each atom, assumng a fused-sphee model fo the solute. We demonstate a fast method fo the exact, analytcal calculaton of the volume and aea of each atom n the fused-sphee model and the gadents wth espect to the atom s poston. We ncopoate the fast geometc algothms nto the appoxmate fomulas we deved fo the calculaton of the solvaton enegy, to get ou solvaton model, the Analytcal Volume Genealzed Bon - Solvent Accessble Suface (AVGB- SAS) model. The pedctons of the pola pat of the method (AVGB) ae vey good as compaed to numecal solutons of the undelyng physcal model, the Posson-Boltzman equaton, fo small and lage molecula systems. AVGB does not depend on any fttng

4 v paametes, whch s common n the lteatue fo such appoxmate methods. It s vey fast compaed to numecal solutons of the PB equaton o othe Genealzed Bon methods. Also, the method s paallelzable whch allows us to study much lage systems. The AVGB-SAS method has been mplemented n a paallel molecula dynamcs softwae pacage and a molecula docng softwae pacage. We have demonstated the qualty of the esults of the AVGB-SAS model n the dynamcs of DNA and n atonal dug desgn applcatons.

5 v Contents Abstact Contents Fgues Tables v v xv 1 Solvaton n Molecula Smulatons Molecula Smulatons Solvent Effects Electostatc Effects Shot Range Effects Total Fee Enegy of Solvaton Solvaton Models Suface Tenson Models Contnuum Delectc Model: The Posson-Boltzman Equaton Appoxmate Solutons fo the Contnuum Delectc Model Includng Fst Solvaton Shell Effects 19 2 The Genealzed Bon (GB) Model The Bon Model The Genealzed Bon Appoxmaton Bon Rad and the Coulombc Appoxmaton Calculaton of the Bon Rad Impovements on the Genealzed Bon Model 41 3 Geometc Algothms fo the Fused-Sphee Model 44

6 v 3.1 Volume Calculaton Aea Calculaton Topologcal Analyss Intesecton of Half-Spaces (IHS) Geometc Dualty and the Convex Hull (CH) Lnea Pogammng Constucton of the Convex Hull Detemnng the GB-paths Implementaton of the Geometc Algothms Robustness Scalng and Pefomance 94 4 The AVGB-SAS Solvaton Model Valdaton of the AVGB model Small Molecules Lage Molecules Intemolecula Pola Solvaton Eneges The Shot-Range Tem Implementaton of the AVGB-SAS Solvaton Model Paallel Molecula Dynamcs Molecula Docng Applcatons of the AVGB-SAS Solvaton Model B-DNA Molecula Dynamcs Vtual Lgand Sceenng (VLS) 133

7 v 6 Bblogaphy 141 Appendx 148 Small Molecule lst 148

8 v Fgues Fgue 1. Reoganzaton of the solvent aound a solute chage and delectc sceenng of ntamolecula nteactons 4 Fgue 2. The Solvent Accessble Suface (tanspaent) of 3 cabon atoms wth adus 1.7 Å (gay), as s taced by a pobe of adus 1.4 Å (yellow). 8 Fgue 3. Sgmod pemttvty pofle fo the dstance dependent delectc model 16 Fgue 4. Effectve delectc pemttvty fo poten A calculated usng equaton (10), whee the pa eneges whee calculated fom numecal solutons of the PB equaton [22]. The sgmod pofle s only qualtatvely descbed by an equaton of the fom (11). The phenomenon s moe complex. (Fgue fom efeence [32].) 17 Fgue 5. The functonal fom of equaton 1 / f (fom equaton (19)) wth Bon ad α α = 2, compaed to the coulombc behavo 1 /. 25 = j Fgue 6. Compason between numecal solutons of the PB equaton and the pedctons of the GB model wth PBF-deved Bon ad, fo 376 small molecules. Lnea egesson ft and coelaton coeffcent shown. 27 Fgue 7. Compason between PB and GB pedctons of the salt contbuton to the solvaton enegy fo a B-DNA stuctue, as a functon of the squae oot of the concentaton of added monovalent salt. (Fgue fom efeence [32].) 42 Fgue 8. An example of the fused-sphee model: The cental atom (whte) s suounded by a numbe of neghbos that defne ts exposed suface aea and volume. 45 Fgue 9. Two ntesectng sphees and the ccle of ntesecton (COI) 47 Fgue 10. Thee ntesectng sphees. The COI s ntesect wth each othe. 47

9 x Fgue 11. Two ntesectng sphees, and, sepaated by the sepaatng plane. The dstance between the sepaatng plane and the cente of sphee s g. 47 Fgue 12. Thee ntesectng sphees and the coespondng sepaatng planes fo the cental sphee (ed). 47 Fgue 13. Decomposton of the fused-sphee model nto the buldng blocs that coespond to each atom. 48 Fgue 14. The weghted Voono dagam (o powe dagam) fo a set of sphees, n two dmensons (Fgue fom efeence [64].) 49 Fgue 15. The buldng bloc and the plana sectons fomed by the neghbos. 50 Fgue 16. Decomposton of the buldng bloc nto cone-pyamds and a sphecal secto. 51 Fgue 17. Decomposton of a plana secton nto tangles and ac-sectos. 52 Fgue 18. Applcaton of the Gauss-Bonnet theoem on the suface of sphee, ntesected by neghbos j,, l. (Fgue adapted fom efeence [73].) 55 Fgue 19. Paametezaton of the Gauss-Bonnet acs. In ths example, the cental atom s ntesected by thee neghbos, j,, l. The j and COI s ntesect each othe, as the and l do also. See text fo explanaton of the vecto quanttes. (Fgue adapted fom efeence [73].) 58 Fgue 20. Patton of the smulaton space nto cells. Fo each atom we seach the cell the atom belongs to (da gay) and the 26 neghbong cells (lght gay). 62 Fgue 21. The cental atom (ed) s ntesected by the neghbos A, B and C (geen). Neghbo B s occluded by A and C. 63

10 x Fgue 22. Same as Fgue 21, wth also showng the ntesectng planes of each neghbo. 63 Fgue 23. The ntesectng plane between the cental atom (left) and a neghbo. 64 Fgue 24. The two half-spaces H 1 and H 2 defned by the ntesectng plane. The exposed aea and excluded volume of the cental atom s on H Fgue 25. Example of a swallowe: the neghbo (geen) swallows the cental atom (whte) but not completely. 65 Fgue 26. The half-spaces defned n the case of a swallowe neghbo (ght). The exposed aea and excluded volume of the cental atom (left) s on half-space H Fgue 27. The IHS fo the example of Fgue 21. The half-spaces of neghbos A, B and C that nclude the cental atom s exposed aea and excluded volume ae coloed blue, ed and yellow espectvely. The ovelap of the half-spaces s coloed by the coespondng ovelappng colo,.e. yellow+ed=oange, blue+ed=puple, blue+yellow=geen. The common nteo of the constants s the IHS (geen) and t s only due to the A and C half-spaces. Neghbo B s occluded. 67 Fgue 28. Geometc dualzaton of a tetahedon. The vetces ae mapped to faces and the faces to vetces. The topology (e.g. faces connected by a common edges) s peseved. 68 Fgue 29. The convex hull of a set of ponts n two dmensons. 69 Fgue 30. The lnea constants. 71 Fgue 31. The nomals to the constant planes. 71 Fgue 32. Dual ponts of the constant planes. 71 Fgue 33. The convex hull of the dual ponts. 71

11 x Fgue 34. The nomal vectos to the faces of the CH. 71 Fgue 35. The dual ponts of the faces of the CH. 71 Fgue 36. Connectng the dual ponts of the CH. 71 Fgue 37. The IHS. 71 Fgue 38. Constants and the half-spaces, n 2D. 73 Fgue 39. Pojecton of the 2D poblem n 3D. The constant lnes on the hypeplane x 3 = +1 become planes that pass though the 3D ogn. The poblem s mapped on a 3D unt sphee. (Fgue adapted fom [65].) 73 Fgue 40. The vetex of the IHS of the shned constants (dotted lnes) s an nteo pont of the ognal IHS. 75 Fgue 41. Tanslatng the constant plane p by ε, towads the half-space H 1, fo the case of a non-swallowe neghbo. 76 Fgue 42. Tanslatng the constant plane p by ε, towads the half-space H 1, fo the case of a swallowe neghbo. 76 Fgue 43. The constants and the objectve functon. 78 Fgue 44. Choosng a andom pont on the decton of the objectve functon. 78 Fgue 45. Pojectng the pont to a andomly pced constant. 78 Fgue 46. Addng anothe constant. The pont satsfes ths constant. 78 Fgue 47. Addng the last constant. Solvng the poblem n one dmenson (on the constant ed-1) 78 Fgue 48. Optmzng the pont wth espect to the last constant n one dmenson and lftng the soluton on the two-dmensonal space. 78 Fgue 49. Set of ponts. 81

12 x Fgue 50. Intal smplex. 81 Fgue 51. Vsblty chec. 81 Fgue 52. Add new faces. 81 Fgue 53. Remove vsble faces. 81 Fgue 54. Vsblty chec. 81 Fgue 55. Add new faces. 81 Fgue 56. Remove vsble faces. 81 Fgue 57. Vsblty chec. 81 Fgue 58. Add new faces. 81 Fgue 59. Convex hull. 81 Fgue 60. The IHS polyhedon fomed by the neghbos of the cental atom (whte) fo the example n Fgue Fgue 61. The IHS polyhedon of Fgue 60 as t cuts though the cental atom. 84 Fgue 62. The IHS polyhedon of Fgue Fgue 63. Bued-bued edge. 86 Fgue 64. Bued-exposed edge. 86 Fgue 65. Exposed-exposed ntesectng edge. 86 Fgue 66. Exposed-exposed non-ntesectng edge. 86 Fgue 67. A bued IHS vetex coesponds to thee connected neghbos and thee GBponts. 87 Fgue 68. Two-dmensonal epesentaton of a bued IHS vetex. 87 Fgue 69. An exposed IHS vetex coesponds to thee dsjont neghbos and thee GBponts. 87

13 x Fgue 70. Two-dmensonal epesentaton of an exposed IHS vetex. 87 Fgue 71. Example of topology of neghbos on the cental atom. 88 Fgue 72. The connectvty gaph fo the example of Fgue 71. The oented edges coespond to GB-ponts. 88 Fgue 73. The connectvty table fo the example of Fgue Fgue 74. Tavesng the connectvty gaph of Fgue 71: statng fom the GB-pont A on neghbo 5, the next GB-pont has to be on neghbo 1. Out of the thee possbltes B, C, D, the GB-pont B s the coect choce. 90 Fgue 75. The GB-paths fo the example of Fgue Fgue 76. The selected cycles (GB-paths) of the connectvty gaph fo the example of Fgue 71. The two GB-paths ae: and Fgue 77. Relaton of the plana sectons of the buldng bloc of Fgue 15 wth the IHS and the GB-paths. 92 Fgue 78. Lnea scalng of the aea/volume calculaton wth espect to the numbe of atoms n the system. 96 Fgue 79. Compason of the pola solvaton eneges between Delph and UHBD fo the molecule set of Table 6. The RMS dffeence s 0.62 Kcal/Mol. 99 Fgue 80. Compason of the pola solvaton eneges between PBF and UHBD fo the molecule set of Table 6. The RMS dffeence s 0.41 Kcal/Mol. 99 Fgue 81. Compason of the pola solvaton eneges between PBF and Delph fo the molecule set of Table 6. The RMS dffeence s 0.73 Kcal/Mol. 100 Fgue 82. Compason of the pola solvaton eneges between AVGB and UHBD fo the molecule set of Table 6. The RMS dffeence s 1.79 Kcal/Mol. 101

14 xv Fgue 83. Compason of the pola solvaton eneges between SGB and UHBD fo the molecule set of Table 6. The RMS dffeence s 1.93 Kcal/Mol. 101 Fgue 84. Compason of AVGB wth ε = 1. 3 to UHBD wth ε = 1. 0 fo the n molecule lst of Table 6. The RMS dffeence s 0.46 Kcal/Mol. 104 Fgue 85. Compason of AVGB and UHBD wth ε = 1. 0, fo the potens of Table 1. The RMS dffeence s 1169 Kcal/Mol. 106 Fgue 86. Compason of AVGB wth ε = 1. 3 and UHBD wth ε = 1. 0, fo the n potens of Table 1. The RMS dffeence s 303 Kcal/Mol. 106 Fgue 87. THF dme wth the pola pats facng each othe. 109 Fgue 88. Pola solvaton enegy fo the system of Fgue 87 fom AVGB as a functon of the dstance between the two THF molecules. The ed lne shows the enegy when the molecules ae nfntely sepaated fom each othe. 109 Fgue 89. THF dme wth the pola pats away fom each othe. 110 Fgue 90. Pola solvaton enegy fo the system of Fgue 89 fom AVGB as a functon of the dstance between the two THF molecules. The ed lne shows the enegy when the molecules ae nfntely sepaated fom each othe. 110 Fgue 91. Compason between the expemental and pedcted wate solvaton eneges fo the AVGB-SAS solvaton model, usng solvaton types by element. The dffeent chemcal goups ae shown. 114 Fgue 92. Compason between the expemental and pedcted wate solvaton eneges fo the AVGB-SAS solvaton model, usng the solvaton types of Table 3. The dffeent chemcal goups ae shown. 117 n n n

15 xv Fgue 93. Total tmes fo the AVGB-SAS model fo a system of 3401 atoms, fo dffeent platfoms. The contbutons of the dffeent pats of the calculaton ae shown. 119 Fgue 94. Scalng of the AVGB-SAS method as a functon of the sze (numbe of atoms) of the molecula system. The calculatons wee pefomed on an Intel Pentum III 866MHz. 120 Fgue 95. Compason of CPU tme spent fo calculatng the solvaton enegy of 1mcp between SGB and AVGB, fo thee dffeent platfoms. 121 Fgue 96. Paallel scalng of AVGB-SAS on a shaed memoy symmetc 4-pocesso Intel Pentum III Xeon 550MHz, fo a 3401 atom poten (1mcp). 123 Fgue 97. Senstvty of the AVGB-SAS enegy wth the update fequency of the Bon ad. The test was pefomed on a small poten (4pt, 454 atoms) fo 200 steps. 124 Fgue 98. Themodynamc cycle fo the calculaton of the bndng enegy of a eceptolgand complex n soluton. 125 Fgue 99. The ntal stuctue: canoncal B-DNA. 131 Fgue 100. B-DNA afte 80ps n vacuum wth fee tps. 131 Fgue 101. B-DNA afte 80ps n mplct solvent wth fee tps. 131 Fgue 102. B-DNA afte 80ps n vacuum wth fxed tps. 131 Fgue 103. B-DNA afte 80ps n mplct solvent wth fxed tps. 131 Fgue 104. RMSD of non-hydogen atoms between smulaton snapshots and the canoncal B-DNA, fo vacuum and solvent smulatons usng AVGB-SAS. 132

16 xv Tables Table 1. AVGB and UHBD pola solvaton eneges fo 11 potens. 105 Table 2. Suface tensons fo wate n (Kcal/Mol Å 2 ) pe element, fo the AVGB-SAS solvaton model. 113 Table 3. Solvaton types defntons and suface tenson values. 115 Table 4. Lst of potens and co-cystal complexes examned. 134 Table 5. Compason of VLS esults fom seachng the lgand database of [105], usng the potocol descbed, wth and wthout solvaton, along wth the esults fom Doc, FlexX and ICM. The shaded entes dentfy the co-cystal lgands that an n the top 2%. 138 Table 6. Lst of small oganc compounds wth the expemental solvaton eneges n wate fom efeence [43] and the pedcted solvaton eneges fom the AVGB-SAS model, n Kcal/Mol. 148

17 1 1 Solvaton n Molecula Smulatons 1.1 Molecula Smulatons Smulatons play a ey ole n the detemnaton of vaous physcal and chemcal popetes of molecula systems. In scence, they ae the ln between expement and theoy ethe by valdatng and challengng new theoes o by pobng expemental esults on the atomc scale. In ndusty, they can seve as a geat cost-cuttng tool, fo example by pe-sceenng molecula tagets fo a cetan popety and thus educng the amount of expemental wo needed to dentfy the appopate compounds. Fo that eason, a geat amount of wo has taen place n the feld of molecula smulatons and geat pogess has been acheved, due to both algothmc mpovements and ncease n computatonal powe. The ecent dscovey of the sequence of the human genome [1] has opened the way to undestandng, on the molecula scale, many human dseases ncludng possble methods fo peventon and teatment. Fo ths goal to be acheved, t s mpeatve to undestand vaous popetes of macomolecules, such as stuctue, bndng and pocesses. Molecula smulatons can addess the above questons n dffeent ways. Homology modelng and enegy mnmzaton can be used to detemne the stuctue of a poten n wate. Molecula docng s used to sceen and dentfy lgands, potental dugs, whch fom enegetcally favoable complexes wth potens n wate. Molecula

18 2 dynamcs can povde atomstc detal on bologcal and chemcal eactons and pocesses. Thee ae many ssues that have to be esolved n ode to accuately smulate a molecula system. These ssues can ange fom the stuctue and geomety of the system, the natue of the ntemolecula and ntamolecula foces, the level of accuacy (quantum-mechancal o classcal), the paametes used such as chages and atomc ad, the effect of extenal factos such as tempeatue and pessue and vaous othe elements that can affect the qualty of the smulaton [2]. Also, computatonal effcency s of paamount mpotance snce typcal studes of chemcal and bologcal systems eque the computaton of many (on the ode of hundeds of thousands o moe) consecutve calculaton steps n ode to acheve the accuacy needed. Thus, fast methods, paallel algothms, and hadwae mpovements ae anothe aea of focus fo molecula smulatons. 1.2 Solvent Effects The defnton of solvent s a substance that s lqud unde the condtons of applcaton, n whch othe substances can be dssolved and fom whch they can be ecoveed unchanged on emoval of the solvent [3]. Wate n patcula s the envonment n whch all bologcal pocesses tae place. Bologcal macomolecules le potens pefom complex functons, such as tanspot of substances, bndng of lgands and catalyzng chemcal eactons, n wate. The effect of the wate envonment on those pocesses s pofound: the solvent nfluences electonc popetes, nuclea dstbuton,

19 3 spectoscopc functons, acdty/bascty, eactve pocesses and molecula assocaton [4]. It s cucal to undestand and accuately calculate solvaton effects n molecula smulatons. The solvaton pocess s defned as the pocess n whch a patcle of the solute s tansfeed fom a fxed poston n the gas phase nto a fxed poston n soluton at constant tempeatue [5]. The ey paamete to descbe the effects of the solvent s the fee enegy of solvaton, Gsol and s defned as the evesble wo spent n the tansfe of the solute unde the afoementoned condtons at equal numbe denstes n the gas phase and n soluton [4]. Mcoscopcally, the solvaton effect s due to ntemolecula nteactons between the solute and the solvent, as well as a change n the ntamolecula nteactons of the solute and a eoganzaton of the solvent because of the solute. In geneal, the calculaton of the solvent effects s pattoned nto thee sepaate pats: electostatcs, shot-ange effects and cavtaton (see [6] and efeences theen) Electostatc Effects Electostatc foces domnate the nteactons of molecules due to the stength and long ange. Electon dstbutons aound nucle n molecules ceate an electostatc feld that nteacts wth that of othe nucle. The chage dstbutons of the solute and solvent play a fundamental ole n the solvaton pocess. The pola contbuton to the solvaton enegy, G pola, ncludes the wo necessay to ceate the solute s gas-phase chage dstbuton n soluton and the wo equed to polaze the solute chage dstbuton. The solute chage dstbuton polazes the solvent, whch n tun nduces an

20 4 electc feld on the solute. Ths s called the eacton feld and t changes the self-enegy of the solute atoms. Also, the ntamolecula coulomb nteactons of the solute ae sceened because of the pesence of the solvent (Fgue 1). Fgue 1. Reoganzaton of the solvent aound a solute chage and delectc sceenng of ntamolecula nteactons In addton, the pesence of salt n the solvent affects the electostatc enegy of solvaton and has a sgnfcant effect on confomatonal changes and bndng. Fo example, DNA s nown to go though a stuctual tanston, fom B to Z fom as the salt concentaton changes [7] Shot Range Effects Besdes the pola nteacton, thee s also the dspeson-epulson (o van de Waals) nteacton between the solvent and the solute that affects the solvaton enegy. These stec foces ae of shot-ange natue and they ae due to an effectve dpole-dpole

21 5 nteacton between solute and solvent [8]. Usually they ae favoable to solvaton snce the dspeson foces ae stonge than the epulsve foces aound the solute cavty. Othe contbutons that tae place ae the hydogen bondng between the solvent and the solute and chage tansfe to o fom the solute. Ths s patculaly tue fo wate. All these effects occu n a shot ange aound the solute, the fst solvaton shell. Cavtaton, o hydophobc effect s defned as the enegetc cost of ceatng a cavty n the solvent fo the solute to ft n. Ths tem s entopc n natue. It accounts fo the loweng of entopy due to the eoganzaton of wate aound non-pola solutes. Fo wate n patcula, t attbutes the decease n the numbe of ways that favoable hydogen bondng can be acheved by solvent wate because of the pesence of a nonhydogen bondng solute. Cavtaton ncludes changes n solvent-solvent dspesonepulson due to the mssng solvent n the cavty and changes n the local solvent stuctue. It s unfavoable to solvaton because the entopy deceases Total Fee Enegy of Solvaton Fom the above, t s clea that the fee enegy of solvaton has to tae nto account all sots of effects: long ange, shot ange and entopc. It s fomally gven by the fomula: G = G + G + G (1) solv pola vdw cav Obvously, the dffeent tems n equaton (1) wll contbute n dffeent ways fo vaous combnatons of solute and solvent. Fo example, fo a pola solvent le wate the electostatc tem wll domnate and the shot-ange stec nteactons wll be

22 6 modeate. On the othe hand, fo non-pola solvents the cavtaton penalty and the electostatc tems should be smalle and the stec nteacton should domnate due to weae nteactons among the solvent molecules. Fo pola solutes n pola solvents the electostatc tem should domnate, wheeas fo non-pola solutes n non-pola solvents the stec nteactons should domnate. Popely teatng the solvaton effect n smulatons of bologcal systems s ctcal to obtanng accuate nfomaton. Because the effects of solvaton ae so complex, a numbe of assumptons and appoxmatons need to be made n ode to mae such smulatons computatonally tactable. The ey assumpton, s that we can patton the solvaton enegy nto the dffeent contbutons, the shot-ange van de Waals and entopc effects and the long-ange pola effects. Vaous methods that exst to calculate these contbutons wll be pesented n the followng. 1.3 Solvaton Models The exstng models fo the calculaton of solvaton n molecula smulatons can be sepaated nto two classes: explct and mplct. The most obvous way of tang account of the solvent s by explctly ncludng solvent molecules n the smulaton. Ths method has many dawbacs, the fst of whch s computatonal effcency. Evey atom that s explctly ncluded n a smulaton adds 3 degees of feedom. Fo 200 wate molecules we add 1800 degees of feedom, wheeas fo octanol molecules we add degees of feedom. In ode to measue stuctual and dynamcal popetes of the system we ll need fst to equlbate the system and then aveage ove those addtonal

23 7 degees of feedom. Ths mples that we need to pefom the smulaton fo a lage numbe of steps whee each addtonal step costs moe CPU tme. One mght thn that ths addtonal cost comes at the beneft of a moe accuate smulaton, but ths s not necessaly the case. Explct solvent models ae only as good as the smulaton method and paametes used. Solute electonc polazaton s usually not taen nto account and the electostatc nteactons between solvent and solute ae dependent on the chages of the focefeld paamete set used. Fo the above easons, the focus of most eseach has been on mplct solvaton models. In such models, the solvent s mplctly ncluded by assumng t s a contnuous medum suoundng the solute. That way the effect of the solvent on the solute s aleady aveaged and the solute s always n statstcal equlbum. The challenge then s to descbe the effects of the solvent accuately unde the contnuum appoxmaton. In the followng we wll pesent dffeent ways that deal wth ths poblem Suface Tenson Models Suface tenson models wee fst ntoduced by Esenbeg n 1986 [9], [10]. In such models, the fee enegy of solvaton s gven as a poduct of the solvent accessble suface aea and an empcally detemned suface tenson paamete, fo each atom: N G solv = σ A (2) = 1 fo a molecule of N atoms, whee A s the solvent accessble suface aea and σ s the suface tenson of atom.the solvent accessble suface aea (SASA) s defned as the

24 8 suface taced by the cente of a sphee of cetan pobe adus, as t olls ove a fusedsphee model of the solute [11]. An example of the SAS of 3 cabon atoms and the pobe sphee that taces t s shown n Fgue 2. Teatng the solvent molecules as sphees s easonable fo molecules of sphecal symmety. The appopate pobe adus fo solvents wth dffeent popetes s addessed n [12]. Fo example, n a non-pola solvent le hexadecane, we expect the solvent effects to tae place n the fst solvaton shell snce dspeson nteactons should domnate. Thus, fo hexadecane a solvent pobe adus of appoxmately 1.5 Å s moe appopate than a much lage adus that would esult f one taes nto account the sze and shape of the solvent molecule. Fgue 2. The Solvent Accessble Suface (tanspaent) of 3 cabon atoms wth adus 1.7 Å (gay), as s taced by a pobe of adus 1.4 Å (yellow).

25 9 Suface tenson models ae conceptually smple and thus not vey elable. The shotcomngs ae: Accuate and effcent calculaton of the SASA and ts gadent wth espect to atomc coodnates s necessay n ode fo ths model to be pactcally useful n molecula smulatons. The suface tenson paametes ae empcally obtaned fom a molecule data set. It s obvous that the accuacy of the model s heavly dependent on the tanng set used and the extenson of t onto molecules out of that set s questonable. Snce the model s puely based on the exposed suface, only atoms on the suface of the solute feel the effect of the solvent. It s thus not capable of calculatng long-ange effects such as delectc sceenng whch n pola solvents domnate the solvaton pocess. Thee have been attempts to coect fo the above defcences, by modfyng slghtly the fom of equaton (2), [13], o by ncopoatng the occuped volume of the solute [14], [15] n the calculaton. Howeve, the poblem of not teatng the electostatc contbuton wth any sold theoetcal foundaton emans n those theoes and that s whee we wll shft ou focus n the followng Contnuum Delectc Model: The Posson-Boltzman Equaton Electostatc nteactons n macomolecules have been studed extensvely due to the pofound effects on the macomolecules functons [16], [17]. Most mpotant

26 10 bologcal phenomena nvolve changes n the nteacton of vaous goups wth the suoundng wate envonment and snce pola effects ae domnatng those nteactons, methods fo the calculaton of those effects have polfeated [18]. Contnuum delectc methods teat the solvent as a delectc contnuum suoundng the solute molecule. The calculaton space s chaactezed by the delectc pemttvty ε ( ), and the solvent s sepaated fom the solute cavty by a bounday suface. The solute s teated as a chage dstbuton, ρ ( ) (whch could also be a set of pont chages). The electostatc potental, Φ ( ) at evey pont n space would descbe the nteacton between the solute chage dstbuton and the solvent delectc. It s clea then, that n ths model and n the absence of salt, the potental Φ ( ) would be gven by the Posson equaton: (3) ( ε ( ) Φ( )) = 4πρ( ) In the pesence of salt, we can employ the Debye- Hucel theoy [19] to ncopoate salt effects. In ths theoy, we assume that the ato of the concentaton of on type aound the solute to ts concentaton fa away fom the solute s gven by the v Boltzman dstbuton, exp( W ( ) / BT), whee B s Boltzman s constant, T the absolute tempeatue and () W the wo equed to move the on of type fom nfnty (whee Φ ( ) = 0 ) to the pont. We assume that we have only two types of ons, negatve and postve (such that the total system s electcally neutal). Then, f e c s the absolute chage of one electon, we must have fo each onc speces: W ) = + e Φ( ) W ( ) = e Φ( ) (4) 1( c 2 c

27 11 and f we assume that the concentaton of each speces M +, M at nfnty s M, the Boltzman dstbuton law gves: M + = M exp( e Φ( ) / T ) M = M exp( + e Φ( ) / T ) (5) c and thus, the chage densty of ons aound the solute should be: B ecφ( ) ρ = = on ( ) ec ( M + M ) 2Mec snh (6) BT By applyng equaton (6) on (3) and usng the onc stength, I, nstead of the salt c B concentaton M, we get (snce N I = 0.5 c z 2 = 1000M / N = 1 A, whee N A s Avogado s numbe, c s the mola concentaton and z the chage n electons of on speces ): 2 ecφ( ) ( ε ( ) Φ( )) + κ ( ) snh = 4πρ( ) BT (7) whee the constant κ s called the Debye-Hucel sceenng paamete, whch s zeo nsde the solute and gven by the fomula 2 8πN Aec I κ ( ) = (8) 1000 ε( ) T outsde of the solute. If we assume a constant delectc pemttvty value, ε out, fo the egon outsde of the solute, then the Debye-Hucel paamete s also a constant. Equaton (7) s called the Posson-Boltzman equaton (PB) and t s a nonlnea, ellptc, second ode, patal dffeental equaton. In cases of low onc stength, one could use the fst tem of a Taylo expanson of the exponental sgn tem to get the lneazed PB equaton. It s a vey dffcult equaton to solve fo abtay systems but t descbes vey B

28 12 accuately the pola effects of the solvent on the solute n the contnuum delectc appoxmaton. The most common assumpton fo solvng the PB equaton s that the delectc pemttvty taes two values, ε n n the solute cavty and ε out outsde. Fo wate envonment, ε out s The electostatc fee enegy of solvaton G pola can be obtaned by solvng ths equaton twce, once wth the solute nsde the solvent delectc and once wth the solute n vacuum ( ε = 1). The pola solvaton enegy, assumng that the solute chage densty s a set of N pont chages q at postons s: out G pola = 1 solv vac ( Φ ( ) Φ ( )) = 1 N q 2 (9) Exact analytcal solutons of the PB o n the absence of salt, the Posson equaton, ae not possble except fo vey few smple cases. In ode to get analytcal solutons we must mae cude smplfcatons n the shape of the solute. Fo example, small molecules and globula potens ae teated as sphecal cavtes, wheeas DNA s modeled as a chaged cylnde. Kwood [20] ntoduced an analytcal soluton fo equaton (3) fo a set of pont chages nsde a sphecal cavty and Jayaam [21] has gven the analytcal soluton fo the ntemolecula poblem of two ons embedded n a delectc contnuum. Although analytcal, these solutons ae of lttle value fo any pactcal pupose because the smplfcatons needed n the shape of the solute lmt seveely the applcablty of the model to ealstc systems. Fo that eason, numecal solutons of the PB and Posson equatons have been developed nstead.

29 13 A numbe of numecal methods have been developed fo the numecal soluton of the PB equaton ethe usng fnte dffeences o bounday element methods. The best nown n the lteatue ae the DelPh pogam [22], UHBD [23] and PBF [24]. A majo poblem wth numecal methods s that they do not calculate the solvaton electostatc foces along wth the eneges. These foces can be computed but only at a geat computatonal expense snce they would have to be calculated fom numecal dffeences. Ths maes numecal methods useless fo molecula mechancs smulatons. At the same tme, snce a spatal gd s used n ode to solve the PB equaton, the accuacy of the soluton and the CPU tme needed fo ts calculaton ae hghly dependent on the densty of the gd. Too spase a gd would esult n fast calculatons wth naccuate solutons. Too fne a gd would esult n accuate solutons but slow calculatons. Fo example, DelPh would tae about 25 mnutes on a 195MHz SGI pocesso to solve poblems on a 185x185x185 gd fo a system of 600 atoms. Also, snce the soluton s based on a gd, the algothm used wll scale wth the sze of the 3 solute as O ( N ), mang t mpactcal fo studyng vey lage systems. Fnally, paallelzaton of such algothms has met wth lmted success. Regadless of the computatonal effcency shotcomngs, numecal solutons of the PB equaton have been successful at dffeent applcatons [18]. Ths s a poof that the delectc contnuum appoxmaton, despte ts conceptual smplcty (the PB equaton gnoes the molecula natue of the solvent, the fnte sze of the ons and on-on coelaton effects), s a vald appoxmaton. In ode to get an accuate descpton of the

30 14 pola solvaton effects n molecula systems, but wth a cetan computatonal effcency, appoxmate analytcal solutons to equatons (3) and (7) have to be found. The goal s to fnd solutons that coelate well to numecal solutons of the PB equaton qualtatvely and quanttatvely, wth analytcal fomulas that ae deved usng appoxmatons that captue the physcs of the PB equatons. Such theoes wll be descbed n secton Appoxmate Solutons fo the Contnuum Delectc Model Multpole Expansons In the multpole expanson appoach the electostatc potental s detemned by assumng vey smple shapes fo the solute cavty and usng lmted multpole expansons to epesent the solute chage dstbuton. The electostatc potental can be wtten as a sees of sphecal hamonc tems. Ths method was fst ntoduced by Kwood [20] as an analytcal soluton to the Posson equaton but has snce then been extended fo moe complex cavty shapes [25]. Although faste than the numecal soluton of the PB equaton, t stll s a slow method snce the sees must convege fo the esults to be valuable, whch means many tems have to be ncluded. Thee have been attempts fo a faste calculaton, [26], [27], but the nheent poblems of naccuate descpton of the molecula cavty and the need to tuncate the sees at some pont lmt the sutablty of these methods to smple systems o qualtatve studes. Dstance Dependent Delectc (DDD) methods As was aleady descbed above, the solvent molecules suoundng the solute ae polazed due to the solute chage dstbuton. Ths geneates a eacton feld, whch n

31 15 tun polazes the solute. The ntamolecula coulombc nteactons ae sceened because of the suoundng solvent molecules (Fgue 1). Ths effect of delectc sceenng on the pola enegy of two atoms, E j pol, can be epesented quanttatvely by the delectc pemttvty ε : E j pol qq j = (10) ε j Howeve, ths fomula cannot be accuate fo small dstances j snce when two atoms ae close togethe thee s not enough space fo the solvent to sceen the nteacton. Fo lage dstances though, we expect that thee wll be enough solvent and the sceenng wll be sgnfcant. Ths motvates us to assume that the delectc pemttvty should be dependent on the dstance by a sgmod pofle (Fgue 3). Such sgmod pofle can be descbed mathematcally by an equaton of the fom: ε ( j B ) = A + (11) 1+ exp( λb ) In pactce, one would calbate the esults obtaned fom such model by assgnng dffeent paametes n (11), accodng to the type of atoms nvolved n the nteacton, [28], [29]. Compasons wth expemental solvaton eneges o numecal solutons of the PB equaton wll detemne the exact values of the paametes. j

32 16 Fgue 3. Sgmod pemttvty pofle fo the dstance dependent delectc model Othe fomulas moe o less complex have been poposed, howeve all these models ae ad-hoc n natue. Sgmod pemttvty pofles ae pedcted by the Loentz- Debye-Sac (LDS) theoy of pola solvaton [30], wth eacton feld coectons ncluded [31]. Thus, although qualtatvely equaton (11) should be able to descbe delectc sceenng, thee s lttle fomal justfcaton fo t. On the othe hand, studes on the ntemolecula sceenng of the pola enegy due to the solvent have been done wth the PB equaton [32]. It s shown thee that the delectc sceenng only qualtatvely s descbed by a sgmod behavo. The phenomenon s just too complex to be descbed by such a smple fomula (Fgue 4). Ths method s also heavly dependent on the molecule set used to tan the paametes that descbe the exact fom of the sgmod pemttvty, fo each atom type. Ths would mae the extenson of the method n dffeent systems

33 17 questonable. Nevetheless, DDD models ae stll extensvely used because they ae easy to mplement and computatonally vey effcent. DDD models howeve should always be used wth cauton n applcatons, snce compasons wth othe solvaton electostatc models have shown that the esults fom these models can be qualtatvely eoneous [33], [34]. Fgue 4. Effectve delectc pemttvty fo poten A calculated usng equaton (10), whee the pa eneges whee calculated fom numecal solutons of the PB equaton [22]. The sgmod pofle s only qualtatvely descbed by an equaton of the fom (11). The phenomenon s moe complex. (Fgue fom efeence [32].)

34 18 Genealzed Bon Model In the Genealzed Bon (GB) model the goal s to fnd an expesson fo the pola solvaton enegy of the fom G pola 1 1 = 2 εn 1 ε out N N = 1 j= 1 q q γ j j (12) whee γ j s an ad-hoc functon that somehow descbes the effects of polazaton and delectc sceenng. The GB model has been poven to be vey successful n pedctng pola solvaton eneges and has eceved consdeable attenton n the lteatue [33], [34], [35]. It comes n dffeent vaatons that dffe n the functonal fom of γ j. A cetan vaaton of the GB model s employed n ths wo and the theoetcal foundatons and appoxmatons behnd the GB theoy wll be dscussed n detal n chapte 2. Othe methods An ssue that ases wth the applcaton of the PB equaton on the solvaton electostatcs s the value of the delectc pemttvty nsde the solute, ε n. A value of 1 s appopate only fo small molecules but s pobably not ght fo macomolecules. Electonc polazaton and feld-nduced nuclea eoentaton effects affect the value of ε n. Typcally, values that ange fom 2 to 8 have been used, but even then the assumpton that the solute delectc pemttvty should be sotopc s questonable. In geneal, the defnton of the delectc pemttvty n potens can ambguous [17]. Fo

35 19 that eason, the Langevn Dpoles (LD) model was developed [36] n ode to avod the use of ε n. In ths model, the solute s placed n the cente of a cubc gd. Langevn dpoles ae placed on the gd ponts and the polazaton of the solvent wth the solute s accounted fo by eoentng the solvent dpoles, whch geneates the eacton feld. The shotcomngs of ths model ae manly due to the assumpton that the electostatc potental s epesented by a dpole tem only, the accuacy and speed depend on the esoluton of the gd used and the esults may not be otatonally nvaant. Howeve, t s an nteestng dea and an altenatve to the contnuum delectc methods. Anothe method s the conductng sceenng model (COSMO) that assumes that the suoundng medum s well modeled as a conducto [37]. The delectc behavo s deved usng analytcal fomulas that allow fo the calculaton of gadents, whch ae necessay fo molecula dynamcs smulatons Includng Fst Solvaton Shell Effects Due to the bologcal mpotance of the effect of wate on macomolecules, the focus of the calculaton of solvaton effects n the lteatue has been on the calculaton of pola effects. Ths s because wate s a hghly pola solvent and potens have pola goups. In such stuatons the pola effect domnates dspeson-epulson and cavtaton effects. Shot-ange effects though can play a sgnfcant ole n non-pola solvents and should be ncluded n ode to have a complete solvaton model. The natue of van de Waals and cavty effects was dscussed n secton

36 20 The most obvous way to nclude these effects n solvaton calculatons would be by explctly ncludng solvent molecules n a shot ange aound the solute. Howeve ths would lead to the same poblems of the explct solvent calculatons, namely the aveagng n tme of many moe degees of feedom and the lac of polazaton effects between solute-solvent. Instead, the suface tenson models (1.3.1) ae an attactve altenatve. In ths model, the fee enegy of solvaton s detemned by equaton (2) and a set of suface tenson paametes that ae empcally detemned. Although ths model cannot descbe accuately long-ange pola effects, t should be able to epoduce shotange effects, le dspeson-epulson and entopc effects le cavtaton. Thus, the fst solvaton shell effects wll be descbed by the fomula: G vdw + G cav = N = 1 σ A (13) whee A s the solvent accessble suface aea [11] and σ the suface tenson paamete fo atom of the solute. The justfcaton behnd ths appoxmaton s that the magntude of the fee enegy of solvaton due to fst solvaton shell effects can be consdeed popotonal to the numbe of solvent molecules n the fst solvaton shell. One expects that entopc tems le cavtaton, along wth the aveagng of shot-ange stec nteactons between solute and solvent should be a functon of the geomety of the solute and coelate statstcally wth the exposed aea. The exposed aea can be thought of as a non-ntege aveage (ensemble o tme aveage) numbe of solvent molecules n the fst solvaton shell. The assumpton n equaton (13) s that the enegy s popotonal to the solvent accessble suface aea (SASA), as defned by Rchads [11] and weghted by paametes

37 21 that ae empcally detemned. The qualty of the fttng to expemental esults and ts pedctve ablty wll be the ultmate judge of the success of the model. The fact that suface tenson models, despte the smplcty, have yelded qualtatvely accuate esults pompts us to accept the valdty fo shot-ange effects. Thus, a model that has an accuate method fo pedctng electostatc contbutons to the solvaton enegy accompaned by a suface tenson model fo ncludng fst solvaton shell effects should be capable of pedctng solvaton eneges fo a multtude of solvents and solutes. Of couse, the qualty of the fst solvaton shell contbuton wll depend on the suface tenson paametes used.

38 22 2 The Genealzed Bon (GB) Model 2.1 The Bon Model The electostatc enegy G pol of a chaged conductng sphee of adus R and chage q, embedded n a delectc of pemttvtyε, can be easly calculated fom Gauss s law ( E ds 4πq ) = by use of sphecal symmety and the elaton between the electostatc enegy and the electc feld E, 1 = E dv. It s gven by the 4π G pol 2 fomula: G pol 2 = q 2ε R (14) Thus, f we assume that the sphee has nteo delectc of 1, and t s evesbly tansfeed fom the vacuum to a medum of delectc pemttvty ε, the change n the electostatc fee enegy change should be: 1 1 = 1 2 ε q R 2 G pol (15) In geneal, f the nteo delectc of the sphee s ε n and the suoundng medum has pemttvty ε out then the electostatc fee enegy change s: G pol 1 1 = 2 ε n 1 ε out 2 q R (16) If we assume now that the sphee s actually an on and the exteo delectc s the solvent envonment, we can see that equaton (16) actually descbes the solvaton

39 23 enegy of an on of adus R. Ths model was fst ntoduced by Bon [38] and has been used successfully fo the calculaton of the solvaton effects on ons [39], [40] and on pas [41]. The success of the Bon model on ons has gven the mpetus to genealze the Bon equaton (16) and ceate an analytcal, appoxmate model fo the descpton of electostatc solvaton effects on mult-atom systems such as macomolecules, wth abtay sze and shape. In the followng, the fomalsm of the GB model wll be deved and the valdty of the appoxmatons made wll be dscussed. 2.2 The Genealzed Bon Appoxmaton We wll now ty to genealze the Bon equaton (16) fo a system of N atoms. As n secton 2.1, we assume that evey atom s a conductng sphee of adus α and chage q. If the sphees ae at a vey lage dstance away fom each othe, then t s a safe appoxmaton that the nteacton enegy between evey atom wll follow Coulomb s law. Ths s because the sphees ae vey fa away and thus the fnte sze of them has no effect on the enegy. Effectvely the sphees nteact as pont chages, as long as the sepaaton dstances ae much lage than the sphees adus. At the same tme we have to nclude the electostatc self-enegy of solvaton fo evey atom, whch s gven by equaton (16). Thus, fo the system of N atoms vey fa away fom each othe the solvaton enegy s:

40 24 G pol 1 1 = 2 ε n 1 ε out N = 1 2 q + α N N j= 1j= 1 q q j j (17) whee j s the dstance between atoms and j. We would le to genealze ths fomula to be applcable to molecula systems of abtay shape and sze. Fo ths, we have to coect equaton (17) fo when the sphees get close to and possbly ntesect each othe. We see fo an analytcal fomula that has the fom of equaton (12) o, ewtten to esemble moe of coulomb s law, G pol 1 1 = 2 ε n 1 ε out N N = 1 j= 1 q q f j j (18) The functonal fom of f can be detemned only by an ad-hoc way, as long as t maes j physcal sense and satsfes appopate bounday condtons. The most common fom used s [42]: f j 2 j j 2 ( α α ) = + α α exp 4 (19) j j The paametes α and α j ae called the Bon ad and they epesent an effectve adus fo the espectve atoms. Ths functonal fom s chosen because t epoduces the ght solvaton eneges at the two lmts: when thee s only one atom, N = 1 and j = 0, then f = α, whch means that the solvaton polazaton enegy s: q G pol = 2 ε n ε (20) out α whch s what s expected fom the Bon model.

41 25 At vey lage nteatomc dstances compaed to the ad,, the j exponental n equaton (19) falls fast to zeo, and f. Then, the j j ntamolecula solvaton enegy between atoms, j, becomes G j pol ε n 1 ε out q q j j (21) whch s, as expected, Coulomb s law. Equaton (19) s bascally an ntepolaton fomula between the Bon and Coulomb lmts, as s shown n Fgue 5. Fgue 5. The functonal fom of equaton 1 / f (fom equaton (19)) wth Bon ad α = 2 = j α, compaed to the coulombc behavo 1 /.

42 26 The ey paametes n ths model ae the Bon ad α. The physcal meanng of the Bon ad becomes obvous when one sets all chages q = 0 except fo atom. Then, the solvaton enegy of the system becomes: q G pol = 2 ε n ε (22) out α Fo a molecule that only atom s chaged, the solvaton enegy s effectvely the selfenegy of polazaton fo atom. Fom equaton (16) we can ntepet the Bon adus paamete α as the effectve adus of an on of chage q, whose solvaton enegy s equal to the self-enegy of polazaton of atom n the molecule. Ths means that n ode to use the GB model we wll have to aleady now the self-enegy of polazaton, snce that s the only way to now the values of the Bon ad. Obvously ths pocedue would be of no pactcal use snce we would need to now the answe n ode to solve the poblem. We wll have to ntoduce some appoxmatons n ode to pedct the values of the Bon ad, and these wll be dscussed n secton 2.3. Nevetheless, t s nstuctve to examne the accuacy of the appoxmatons made aleady at ths pont. The ntepolaton fomula (19) along wth equaton (18) povdes an analytcal fomula to get the solvaton enegy fo an abtay molecula system. Although we do not have a fomal way to calculate the Bon ad yet, we can stll use numecal solutons of the PB equaton n ode to pedct these paametes. Specfcally, fo a molecule of N atoms, we do N numecal calculatons of the solvaton enegy whee each tme all chages ae set to zeo except fo one atom,. The numecal answe can be plugged nto equaton (22) and thus calculate the Bon adus α. Then, we epeat

43 27 the numecal soluton fo the eal, fully chaged molecule and compae the solvaton enegy to the pedcted enegy fom the GB model wth the numecally deved Bon ad. We pefomed these calculatons on a set of 376 small molecules, not lage than 40 atoms each (see efeence [43] and the Appendx fo the lst of molecules). The numecal calculatons wee pefomed usng the PBF solvaton code [24] and the esults ae shown n Fgue y = x R 2 = GB (Kcal/Mol) PBF (Kcal/Mol) Fgue 6. Compason between numecal solutons of the PB equaton and the pedctons of the GB model wth PBF-deved Bon ad, fo 376 small molecules. Lnea egesson ft and coelaton coeffcent shown. The esults show excellent coelaton (coelaton coeffcent 0.97) between the GB model and the numecal solutons of the PB equaton. The lnea egesson ft shows

44 28 that the esults fom the two methods ae vey close to each othe although thee s a small systematc eo; the elaton s not exactly of the fom y = x, but nstead the slope s Ths s poof that as long as we have an accuate descpton of the Bon ad paametes fo the system, the ntepolaton fomula (19) n the GB model (18) descbes emaably well electostatc solvaton eneges, at least fo small molecules. Despte ts success, thee have been attempts to modfy equaton (19). Dffeent fomulas have been poposed that satsfy the same lmtng condtons aleady dscussed, but pefom bette fo specfc applcatons [44]. Howeve, thee s no systematc way to ntoduce an ntepolaton fomula. It s always an appoxmaton that s put to the test by dect compasons wth numecal solutons to the PB equaton. 2.3 Bon Rad and the Coulombc Appoxmaton In ode to calculate the Bon ad fo the GB model, we need to come up wth an analytcal appoxmate soluton to the polazaton self-enegy of an atom n the molecule. Such a soluton was fst gven n [45] by use of the electostatc enegy densty. Instead, we wll pesent a novel, fomal poof, nsped by [35], that clealy shows the physcal meanng of the assumptons made n ths calculaton. As was aleady descbed n 1.2.1, the electostatc self-enegy of solvaton of an atom s due to the nteacton of the solute chage dstbuton, ρ ( ) wth the nduced dpoles of the solvent. Ths s called the eacton feld Φ ( ) and t s esponsble fo polazng eac

45 29 the solute atoms. The polazaton s descbed by the nduced suface chage, σ (), on the suface of the solute atoms. Then accodng to electostatc theoy [46], the pol eacton feld s gven by: Φ eac σ pol ( R) ( ) = d R S 2 R (23) whee S s the solvent accessble suface of the solute. The pola fee enegy of solvaton s a functonal of the eacton feld: G pol = 1 3 ρ ( ) Φ eac ( ) d (24) 2 Assumng that the solute chage dstbuton s a set of N pont chages q located at ponts, ρ ( ) = N = 1 q δ ( ), the pola fee enegy of solvaton becomes: G pol = 1 2 σ N pol q = 1 S ( ) d 2 (25) By applyng Gauss s law on an nfntesmal pllbox of suface S on the bounday suface that sepaates the two delectcs, ε n and ε out, we can calculate the dscontnuty of the eacton feld on the suface of the solute: E ds = 4πq ( E E ) nˆ = 4πσ S out S n o, pol (26) whee nˆ s the nomal to the bounday suface and E s the electc feld due to the local polazaton chage densty. Accodng to the bounday condton fo the delectc

46 30 dsplacement D( ) on the nteface between the two delectcs, the nomal component of the delectc dsplacement has a dscontnuty that s popotonal to the bae suface chage densty σ (whch does not nclude the polazaton chage) [46]. ( D D ) nˆ = 4πσ out n (27) But the bae chage densty σ s zeo snce we assume pont chages and the delectc dsplacement s popotonal to the delectc constant, D = ε E. Thus, fom equatons (26) and (27) we have an expesson fo the polazaton chage densty σ () as a functon of the nomal component of the electostatc feld on the suface of the molecule, E n ( ) nˆ : 1 ε n σ pol ( ) = 1 En ( ) nˆ 4 (28) π ε out By pluggng equaton (28) nto equaton (25) and settng all chages equal to zeo except fo atom, whch s located at poston, we get an expesson fo the self-enegy of pol polazaton fo atom, G pol,, n the molecula cavty: G pol, = 1 ε n q 8π ε out 1 S En ( ).ˆ n d 2 (29) Equaton (29) s exact, but not vey useful snce the functonal fom of the electc feld s not nown. We wll need to ntoduce an appoxmaton fo the electc feld n ode to get a fomula fo the Bon ad. We now fom Gauss s law that fo a pont chage electc feld s: q located at pont n a sphecal cavty wth delectc constant ε n, the

47 31 E n 1 ( ) = q (30) ε 3 n whch s Coulomb s law. If we use equaton (30) n equaton (29) we can have an expesson fo the self-enegy of polazaton fo atom, only n tems of the geomety of the system: G pol, 1 1 = 8π ε n 1 ε out q 2 S nˆ d 4 ( ) 2 (31) The use of Coulomb s law fo the electc feld s exact only n the case of a sngle chage q n the cente of a sphecally symmetc cavty. We can expect that ths appoxmaton fo the local electostatc feld wll be vald fo cases that the molecule s suface s locally convex. Ths appoxmaton s nown as the Coulombc appoxmaton and t s valdty has been examned n [45] and [47]. Fo cases that the suface s not locally convex we cannot be sue how well ths appoxmaton wll hold. Howeve, ou tests fom chapte 4 showed that the Coulombc appoxmaton wos vey well fo a dvese set of molecules. We wll attempt to efomulate equaton (31) fom a suface to a volume ntegal snce ths wll allow us an analytcal calculaton of the Bon adus. Fo ths, we employ Geen s theoem fo a vecto feld A, fom vecto analyss: V = S A ds = V 3 Ad (32)

48 32 We wll set the vecto feld A equal to the expesson nsde the suface ntegal n (31), 4 A =, and n ode to avod the sngulaty at the cente, =, we bea the ntegaton ove the solute volume V nto two egons: the volume of a sphee of adus R centeed at, R, and the volume of the solute excludng the sphee of adus R, { } R V > R = Ω : 3, whee R s an abtay adus. Then we get: Ω = + = = R V S V d d d ds (33) The ntegal ove the volume of the sphee of adus R can be ewtten, as a suface ntegal ove the suface of the sphee and t yelds R π 4 : R R R R R R ds d π π = = = = (34) Fo the ntegal ove the volume Ω we need to calculate the dvegence of the vecto feld 4. If we use patal dffeentaton and the dentty ( ) 3 = we get: = (35)

49 33 Hence, by applyng equatons (34) and (35) n (33) we get an expesson fo the suface ntegal: V = S 4π 1 ds = R 4π Ω d (36) If we apply equaton (36) nto (31), we can get an expesson fo the self-enegy of polazaton of atom, G pol,. By compang ths expesson to equaton (22) we get an analytcal fomula fo the Bon adus of atom, α : 1 α = 1 R 1 4π Ω 1 4 d 3 (37) The value of the adus R can be detemned f we examne the case of the solute beng just one on. Then, the GB fomula (18) becomes the well-nown Bon expesson (16) and hence the adus R wll be the onc adus of that atom. In geneal, we wll tae ths to be the van de Waals adus of the atom. The van de Waals ad ae atomc paametes that ae usually dependent on the focefeld paamete set used. The concluson dawn fom equaton (37) s that the Bon adus, n the Coulombc appoxmaton at least, s dependent only on the geomety of the solute. Howeve, the volume ntegal n (37) s stll vey dffcult to be calculated analytcally fo cases of abtay molecula geomety. Clealy, we wll need to ntoduce addtonal appoxmatons n ode to ave to an analytcal fomula fo the calculaton of the pola pat of the solvaton enegy.

50 Calculaton of the Bon Rad The ntegal n equaton (37) cannot be calculated analytcally fo all but the smplest case of a sphecal solute. Fo that eason thee has been consdeable nteest n the lteatue fo the calculaton of ths ntegal and we wll pesent n the followng dffeent methods that have been poposed. Numecal ntegaton The most obvous way to calculate such ntegal fo abtay geometes s by numecal ntegaton. The ntegaton doman s dvded by a cubc gd, elements of whch ae assgned as beng nsde o outsde the solute. Then each gd element (of volume V and cente coodnate ) contbutes V to the ntegal of the Bon adus fo atom. Compasons of the esults of ths method wth numecal solutons of the PB equaton on a set of small molecules and molecula complexes show that the GB esults coelate vey well to the PB answes although thee s a systematc eo [48]. Ths s encouagng snce t poves that the Coulombc appoxmaton that was ntoduced n 2.3 s vald, at least fo the molecula systems tested. Howeve, the numecal ntegaton s not pactcal fo molecula smulatons snce t lacs devatves (whch ae necessay fo the calculaton of foces), t s vey slow and the accuacy and speed depend on the esoluton of the gd used. The asymptotc model

51 35 Snce the numecal soluton s not pactcal and we need an analytcal fomula to calculate gadents, the asymptotc model attempts to povde an ad-hoc analytcal soluton. The model assumes that the Bon adus of atom s gven by [49]: 1 α = 1 P2V j P3 V j R + φ + P1 j stetch j j bend j j nonbond P 4V jc 4 j (38) The paametes φ, P 1, P 2, P 3, P 4, ae scalng factos that ae detemned by fttng the pedcted solvaton eneges to the numecal solutons of the PB equaton of a set of small molecules. CCF s a close contact functon that adjusts ad fo nonbonded atoms that ae too close to the cental atom and atom j. V j s the van de Waals volume of The smlaty of equaton (38) to equaton (37) s obvous. Equaton (38) s an adhoc fomula wth not much fomal justfcaton besdes the fact that the tem V 4 j / j coesponds to the enegy loss of a classcal chage-nduced dpole nteacton between the chage of atom and the delectc medum that s dsplaced by atom j [49]. One can thn ths as a fst ode appoxmaton to the exact ntegal of equaton (37). In fact, the V elatonshp fo the contbuton of atom j to the self-enegy of atom wll hold 4 j / j only asymptotcally, fo lage dstances j. At the same tme, the tue volume of atom j n the solute s dffeent than the van de Waals volume V j snce the atoms ntesect each othe. Ths s the eason that the paametes of the model need to be ftted to numecal solutons of the PB equaton and ths maes questonable the applcaton of (38) to lage molecules. Fo example, n ode to use ths model fo potens and nuclec acds a e-

52 36 paametezaton was necessay [50]. Agan, those paametes wll be applcable only fo the molecule set that wee taned on and the focefeld paametes used. The pawse desceenng appoxmaton (PDA) In ths model the pola solvaton enegy s calculated usng a slghtly dffeent but equvalent fomula fo the Bon ad [47], [51]: ( ) whee { }, ', R ' all ' (,{ ', R '} ) all ' d A 1 α = (39) 4 4π R A s the exposed suface aea of a sphee of adus centeed at atom and s ntesected by all the othe sphees ' of adus locatons of all the othe atoms ' at dstance R ', centeed at the ' fom atom [51]. Agan, the ntegal n equaton (39) s not possble to be calculated analytcally because the atomc sphees ' ovelap wth each othe. In ode to account fo ths eo, the adus of each atom ' s scaled by a facto S '. These factos ae less than one so that each sphee s educed to an effectve volume. The fnal fomula fo the Bon ad n ths model has the fom: 1 = 1 α R H ( ', S ' R ) (40) ' whee H ( ', S ' R ) s a complex expesson [51]. The scalng factos S ' ae detemned fom fttng the solvaton eneges pedcted by the PDA model to the numecal solutons of the PB equaton, fo a set of small molecules. Although ths method povdes fo an analytcal appoxmaton to the Bon ad, the dependence on scalng factos and fttngs to numecal solutons cause t to be not

53 37 easly appled to othe molecula systems. Dffeent paametezatons of the model have to be developed fo t to be applcable to dffeent systems [52], [53]. Thee have been attempts to mpove on the esults of the PDA model [54] but the man dawbac of the dependence of the esults on empcally detemned scalng factos emans. The suface genealzed Bon model (SGB) In the SGB model [55] the Bon ad ae calculated usng the suface ntegal fomula fo the polazaton self-enegy of solvaton, equaton (31), nstead of the volume ntegal (37). The two expessons ae fomally equvalent but the suface ntegal has the advantage of beng faste to calculate numecally than the volume ntegal. By ceatng a tangulaton of the suface of the solute we can calculate the contbuton to the Bon adus at each suface element and add up to get the value of the ntegal ove the solute. The advantage of ths method s that the CPU tme fo the suface ntegaton scales bette than the volume numecal ntegaton, as a functon of the sze of the solute, and thee s no need fo any paametezaton o scalng factos. In pactce howeve, empcal shot-ange (nvolvng atom-pas whose sphees ovelap) and long-ange coectons (based on the amount of nvagnaton of the molecula suface) ae added to mpove ageement wth numecal solutons of the PB equaton. Also, the accuacy of the method depends on the esoluton of the gd used and devatves of the Bon ad ae not eadly avalable. The ovelappng sphees appoxmaton and the analytcal volume model

54 38 The ntegaton egon Ω n the volume ntegal of equaton (37) s the volume of the solute that s naccessble to the solvent, V, mnus the volume of a sphee of adus R centeed at. The solute volume s defned as the nteo of the solvent accessble suface, whch n tun s defned by the van de Waals sphees of each atom extended by the pobe adus of the solvent, p, as s shown n Fgue 2 [11]. We can patton the ntegaton egon nto the set of sub-volumes V ' that each neghbong atom ' occupes and then ewte the ntegal as a sum of the ntegals ove each sub-volume: Ω d = d 4 (41) 4 ' V ' Equaton (41) s exact as long as the patton of the ntegaton egon nto sub-volumes s consstent.e. ' V ' = Ω. Howeve, the shape of the sub-volumes s hghly egula and the volume ntegals n (41) cannot be solved analytcally except n the smple case that V ' s a sphee (t s not ntesected by the neghbong atoms). In patcula, the ntegal of the quantty 4 1 ove the volume of a sphee of adus R ', centeed at, mnus a possble ovelap wth the sphee ( R ) ',, can be solved analytcally [45]:

55 39 ( ) ( ) V V V R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R d ' 2 ' 2 2 ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' 2 ' 2 ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' 2 ' 2 ' ' whee f 0, 0 f, 4 4 f, ln 4 2 f, ln 2 f, ln 2 1 ' ' + = < = < = = θ π π π π π π π θ π π π (42) The fve cases esult fom the dffeent topologes that ase between the two sphees, ( ) R V, = and ( ) ' ' ', R V =. They coespond, espectvely, to the cases of no ovelap, patal ovelap, ' completely swallows but the sphees ae not concentc, ' completely swallows and the sphees ae concentc, and completely swallows '. Equaton (42) would allow fo an analytcal calculaton of the Bon ad, f the solute wee a set of non-ovelappng sphees. In ealty though the atoms ntesect each othe so we cannot use (42) wthout ntoducng some appoxmatons. In the ovelappng sphees appoxmaton, we attempt to epesent the sub-volumes ' V as sphees and use equaton (42) fo the calculaton of the Bon adus of atom. We cannot smply use the ad ' R (whch ae equal to the van de Waals adus of atom ', vdw R ' plus the solvent s pobe adus p, p vdw R R + = ' ' ) fo the sub-volumes because we would oveestmate the solute volume due to the nteatomc ovelap. An attempt to use

56 40 effectve ad that depend on atom types [45], deved fom studes of cystallogaphc poten stuctues [56], lead to modeate coelaton between the calculated atomc selfeneges of polazaton and the esults of numecal calculatons [57]. Ths s pobably due to an nadequate descpton of the molecula geomety. Standad effectve volumes that ae dependent only on atom types cannot dstngush between atoms on the suface of the solute and atoms that ae deeply bued nsde the molecula cavty. Secondly, these effectve volumes wee deved fom studes on a fnte numbe of potens and the applcaton of those values to othe systems s not obvous. If we had howeve a fast and accuate way of calculatng the tue sub-volume of each atom n the molecula cavty by usng a fused-sphee model of the solute, V, we could defne fo each atom an effectve adus adus eff R s equal to the tue sub-volume V : eff R such that the volume of a sphee of 3V eff 3 R = (43) Then, we can pefom the volume ntegaton n equaton (41) analytcally by use of (42), 4π whee each ntegaton egon s a sphee of adus eff R '. Effectvely, n ths appoxmaton we assume that the atoms ae not ovelappng but the sphecal volumes that we assgn to each atom have been coected fo the ovelap. If the method fo the calculaton of the tue sub-volumes V s analytcal then the fnal fomula fo the Bon ad wll be also analytcal and we wll have the capablty to calculate the full gadent of the pola solvaton enegy. Obvously, fo the method to be

57 41 pactcally applcable n molecula smulatons, the volume calculatons have to be computatonally vey effcent. In the Analytcal Volume Genealzed Bon model (AVGB) that we popose hee, the volumes ae calculated accuately and effcently by means of analytcal algothms. The algothms fo the volume calculatons ae descbed n detal n chapte Impovements on the Genealzed Bon Model The GB model as descbed hee has two lmtatons: the effect of salt on the solvaton enegy s not taen nto account and the chage densty of the solute s assumed to be a set of pont chages. Howeve, thee ae mpovements that can be done on the GB fomalsm that addess these poblems. Incluson of salt effects The GB model was deved as an appoxmaton between two lmts, the case of two wdely sepaated sphees and the case of a sphecal on. When we nclude salt effects we can get analytcal solutons fo these two specal cases, n the lmt of low salt concentaton fom the lneazed fom of the PB equaton (7). If κ s the Debye-Hucel delectc sceenng paamete, then fo two wdely sepaated sphees and j, the fee enegy of polazaton s [32]: κj e qq G 1 pol = ε n ε out j and fo the case of a sphecal on of adus α t s [19]: j (44)

58 42 G pol 1 1 = 2 ε n e ε κ out j 2 2 q q κ α 2ε out ( 1+ κ ) p (45) whee p s the pobe adus of the solvent. To a close extend, these two lmts can be obtaned by the smple substtuton [32]: 1 ε n 1 ε out 1 ε n e ε κf out j (46) whee f j s the ntepolaton fomula (19). Although ths s a vey smple appoxmaton, ths model can epoduce well the salt contbuton to the solvaton enegy, as s shown n Fgue 7. Fgue 7. Compason between PB and GB pedctons of the salt contbuton to the solvaton enegy fo a B-DNA stuctue, as a functon of the squae oot of the concentaton of added monovalent salt. (Fgue fom efeence [32].)

59 43 Gaussan chage dstbutons In the devaton of the Bon ad of atom, equaton (37), we made the assumpton that the solute chage dstbuton was a set of pont chages. If nstead we assume that the chage densty fo evey atom has a gaussan shape, 2 2 ( ( ) ) ( π ) a ; a ρ ( ) = q πa = exp R 1 2 (47) we can e-deve the fomalsm of secton 2.3 and get a modfed expesson fo the Bon ad. Ths theoy was descbed n [57] and [58], along wth a modfcaton fo the patton of the ntegal n the ovelappng sphees appoxmaton by assumng patal atomc denstes of also gaussan shape. The esultng fomulaton s moe complcated than the ognal GB theoy (and potentally moe CPU ntensve) but ths s an nteestng addton to the GB model.

60 44 3 Geometc Algothms fo the Fused-Sphee Model In chapte 1 we showed how the shot-ange contbutons to the solvaton enegy ae dependent on the exposed aea of the solute. In chapte 2, usng the Genealzed Bon appoxmaton, we educed the calculaton of the pola contbuton to the solvaton enegy to the computaton of the occuped volume of the solute. In patcula, fo evey atom n the solute (consstng of N atoms total), the exposed aea A and the occuped volume V of that atom need to be calculated, along wth the gadents wth espect to the atomc coodnates. The sub-volumes V ae needed fo the pola pat of the calculaton and the aeas A fo the shot-ange cavty-van de Waals tem. The defnton of the exposed aea and the solvent excluded volume fo the solute s, as defned by Rchads, the solvent accessble suface aea (SASA) and solvent excluded volume [11]. We assume a fused-sphee model fo the solute whee each sphee has adus equal to the van de Waals adus of the atom t epesents, vdw extended by the pobe adus of the solute, p.e. = +. We ae extendng the ad of the atoms by the pobe vdw p adus, accodng to the defnton of the SASA as the suface taced by the cente of a sphecal solvent pobe, as t olls aound the van de Waals sphees of the solute (Fgue 2). Snce many atoms ntesect each othe, the sphees ae fused nto each othe, as s shown n Fgue 8.

61 45 The calculaton of the volumes and aeas has to be analytcal and fast, n ode fo ths model to be pactcal fo molecula smulatons. At the same tme the algothms used wll have to be applcable to all dffeent topologes that may ase between an atom and ts neghbos. It s clea fom Fgue 8 that these topologes can vay wldly fom atom to atom. Theefoe, t s cucal to have vey obust algothms. Fgue 8. An example of the fused-sphee model: The cental atom (whte) s suounded by a numbe of neghbos that defne ts exposed suface aea and volume.

62 Volume Calculaton Numecal technques wee the fst to be used fo the calculaton of molecula volumes, ethe by gd o Monte Calo methods. Gd ponts o andomly geneated ponts ae dentfed as beng n o out of the fused-sphee model. The facton of the ponts nsde detemnes the volume of the solute. Such methods besdes not beng able to calculate devatves ae vey computatonally neffcent and cannot povde the ndvdual sub-volumes contbuted by each sphee. Anothe method poposed was the ncluson-excluson method by Katy [59] whee usng the ncluson-excluson fomula of set theoy we can calculate the unon of N sphees as N summatons ove combnaton of ntesectons of the ndvdual sphees: N ( N ) = V ( ) V ( j) + V ( j )... N V (48) > j N > j> Obvously ths algothm can be vey complex fo complcated topologes of multple ntesectons and fo systems wth lage numbe of sphees. Thee have been attempts to smplfy these expessons [60], [61] but the mplementaton of these methods s stll patculaly cumbesome and t s not possble to nclude all topologes. Also, the method does not povde the ndvdual sub-volumes of each atom. In ode to calculate the ndvdual sub-volumes we need to unambguously defne a way to patton the fused-sphee model to the ndvdual contbutons of each sphee. The smplest case of two ovelappng sphees can gve us the pncple of the decomposton: the two sphees ntesect each othe and fom a ccle on the bounday, the ccle of ntesecton (COI) (Fgue 9). The COI defnes a sepaatng plane that cuts

63 47 Fgue 9. Two ntesectng sphees and the ccle of ntesecton (COI) Fgue 10. Thee ntesectng sphees. The COI s ntesect wth each othe. Fgue 11. Two ntesectng sphees, and, sepaated by the sepaatng plane. The dstance between the sepaatng plane and the cente of sphee s g. Fgue 12. Thee ntesectng sphees and the coespondng sepaatng planes fo the cental sphee (ed). though the sphees (Fgue 11). The ntesecton of the sepaatng plane wth the suface of each sphee s the COI. If thee ae moe than two sphees, the sepaatng planes mght

64 48 ntesect each othe, fomng a moe complcated topology. The ccles of ntesecton ntesect each othe (Fgue 10), as the sepaatng planes do also (Fgue 12). The ntesectng planes that cut though the sphees sepaate the fused-sphee model nto buldng blocs. Each buldng bloc s a complcated geometcal shape that s made of a sphee that has been cut by the coespondng ntesectng planes fom each neghbo. Fgue 13 llustates the decomposton of the fused-sphee model nto the buldng blocs. The buldng blocs consst of plana faces and egons of the sphee suface that s left uncut. Fgue 13. Decomposton of the fused-sphee model nto the buldng blocs that coespond to each atom. The decomposton pocedue that we ntoduce hee s vey smla to the concept of the Voono dagams [62] and the weghted Voono dagams o powe dagams [63], whch ae fundamental stuctues n computatonal geomety and have found many applcatons n dffeent felds n scence. In shot, gven a set of ponts n space the

65 49 Voono dagam dvdes the space nto egons accodng to the neaest-neghbo ule: each pont s assocated wth the egon of space closest to t. Thus, fo evey pont nsde a egon ts dstance to the geneatng pont s less than (o equal to) ts dstance to any othe pont n the set. Accodng to ths defnton, the egons ae defned by sepaatng planes, whch ae the bsecto planes between two neghbong ponts. The weghted Voono dagam s a genealzaton of the Voono dagam; the sepaatng plane s not the bsecto plane but t s paallel to t. If we assgn sphees of dffeent ad to evey pont n the set, the sepaatng plane s defned as the plane fo whch evey pont n t has equally long tangent lne segments to both of the sphees. If the sphees ntesect, ths plane s the sepaatng plane of the two sphees. An example of a weghted Voono dagam s gven n Fgue 14. Fgue 14. The weghted Voono dagam (o powe dagam) fo a set of sphees, n two dmensons (Fgue fom efeence [64].)

66 50 The dffeence between the weghted Voono dagam and the decomposton descbed hee s that we do not tae nto account sepaatng planes between sphees that do not ntesect, as s shown n Fgue 14. Howeve, both methods poduce exactly the same decomposton of the fused-sphee model. We pefe the method poposed hee, whch s essentally a smplfed veson of the weghted Voono dagam because t facltates the calculaton of each atom ndependently of the othes, as t wll be shown n the followng. Instead, methods fo the calculaton of the Voono dagams ae global n chaacte [65] whch has dsadvantages n the paallelzablty and obustness of the mplementaton of the calculaton. A fomal study of the applcatons of advanced computatonal geomety constucts on the fused-sphee model s gven n [66]. Fgue 15. The buldng bloc and the plana sectons fomed by the neghbos. In ode to calculate the volume of the buldng blocs we need to contnue the decomposton pocess heachcally, untl we ae able to descbe the objects fom well-

67 51 defned geometcal shape. The buldng bloc s made fom the atom s sphee cut by the ntesectng planes between the atom and ts neghbos. These cuts ae plana sectons on the suface of the bloc and they coespond to the neghbo that fomed them (Fgue 15). If we connect each pont on a plana secton to the cente of the sphee we fom a sold that has the shape of a cone-pyamd wth the plana secton as ts base. If we cave out all the cone-pyamds fom the buldng bloc, we ae left wth a sphecal secto. The sphecal secto s the sold that esults by connectng all ponts of the exposed suface of the atom (the sphecal pat of the buldng bloc) to the cente of the sphee. The conepyamd decomposton was poposed n [67] and t s llustated n Fgue 16. Buldng Bloc Sphecal Secto Cone-pyamds Fgue 16. Decomposton of the buldng bloc nto cone-pyamds and a sphecal secto. The advantage of ths decomposton s that the volume of cone-pyamds and sphecal sectos can be calculated analytcally. If a cone-pyamd has a base of aea A and dstance fom the tp d, then the volume s V con py 1 = 3 Ad. Smlaly, the volume of

68 52 the sphecal secto of adus s the sum of the volumes of nfntesmal pyamds of base aea ds : V 1 1 dv ds S sph sec = = = 3 3 (49) Thus, the volume of the sphecal secto s popotonal to the exposed aea of the atom. In geneal, fo the buldng bloc of an atom wth adus and exposed suface aea S, fomed by j { 1,..., M } exp = neghbos that ae sepaated fom by planes of dstance g j fom the cente of, the volume V s gven by: V 1 = S 3 exp + M j= 1 1 g 3 j A j (50) whee A j s the aea of the plana secton that s fomed on atom fom neghbo j. Note that n equaton (50) the dstance g j of the sepaatng plane of neghbo j fom atom can be negatve f the neghbo swallows the atom, that s, the cente of s bued by j (see Fgue 11). The algebac sum subtacts coectly the ovelaps that may appea between the cone-pyamds n the case of a swallowe neghbo [68]. Plana Secton Tangles Ac-Sectos Lne Segment Ac (pat of a COI) Fgue 17. Decomposton of a plana secton nto tangles and ac-sectos.

69 53 The aea of a plana secton, A j, between atom and neghbo j can be calculated n a smla fashon by decomposng t nto tangles and ac-sectos (Fgue 17). In geneal, the plana secton s bounded by a set of acs and lne segments. The acs ae pats of the COI fomed between and neghbo j. The lne segments coespond to ntesectons of the atom and neghbo wth othe neghbos of the atom. They ae fomed by the ntesectons of the atom-neghbo plane wth the othe neghbos ntesecton planes. Fo the decomposton, we need to pc a efeence pont on the suface of the j ntesectng plane that wll defne the tangles and ac-sectos. Ths pont s defned to be the ntesecton between the lne that connects the centes of and j, and the ntesecton plane. Obvously, the connectng lne s nomal to the ntesecton plane. Thus, f we have acs M j acs on the plana secton and seg M j lne segments, then the plana secton s aea s: A j = acs M M j j seg 1 1 a Sλ + hµ tµ (51) 1 12 j λ = 2 µ = whee th S λ s the length of the λ ac on the plana secton, a j s the adus of the j th COI, t µ s the length of the µ lne segment and h µ the dstance between the efeence pont and the lne segment. Equatons (50) and (51) allow us to calculate the volume of the buldng blocs and thus the volume of each atom accuately, as long as we can calculate the exposed aea exp S and all the othe quanttes, g j, a j, t µ, h µ. We wll pesent a method to calculate exposed aeas n secton 3.2.

70 Aea Calculaton The calculaton of the suface aea of the fused-sphee model has attacted consdeable attenton n the lteatue because of ts mpotance n the descpton of the solvaton enegy. The descpton of solvaton n tems of the solvent accessble suface aea (SASA) -see equaton (2)- made the calculaton of the SASA and ts gadent wth espect to the atomc coodnates necessay. Numecal methods ae chaactezed by the way of appoxmatng the suface and they ae too slow to be used n molecula smulatons (see [69] and efeences theen). Analytcal methods wee fst poposed by Connolly [70] and Rchmond [71] and they use the Gauss-Bonnet theoem of dffeental geomety [72]. The Gauss-Bonnet (GB) theoem s the most fundamental theoem n dffeental geomety and topology and n ts smplest fom t assets that the excess ove π of the sum of the nteo angles ϕ 1, ϕ 2, ϕ 3 of a geodesc tangle T s equal to the ntegal of the gaussan cuvatue K ove T, o fomally: 3 () s ϕ π = K ds (52) = 1 T The geneal fom of the GB theoem (global GB theoem) s: Let R be a egula egon of an oented suface and let C 1, C 2,, C n be the closed, smple, pecewse egula cuves whch fom the bounday R of R. If each C s postvely oented and Ω 1, Ω 2,, Ω m ae all the extenal angles of the cuves n = 1 C g C, then: () l dl + K() s ds + Ω = ( R) R m = 1 2πχ (53)

71 55 whee l denotes the ac length of C, g () l s the geodesc cuvatue of the ac, K () s s the gaussan cuvatue of the suface and χ ( R) the Eule-Poncaé chaactestc of the suface R. The Eule-Poncaé chaactestc s a topologcal constant of the suface and n geneal, f a two-dmensonal suface has q holes and h handles, then ( h + q) χ = 2 2 [74]. The global GB theoem can be appled n the case of an atom n the fused sphee model. We wll attempt to calculate the aea of an atom that s bued by the neghbong atoms (Fgue 18). Fgue 18. Applcaton of the Gauss-Bonnet theoem on the suface of sphee, ntesected by neghbos j,, l. (Fgue adapted fom efeence [73].) The ntesectng neghbos fom ccles on the sphee of atom, the COI s, whch ntesect each othe. The bued suface of s bounded by acs whch ae peces of the

72 56 COIs. We name these acs the Gauss-Bonnet acs (GB-acs) and the closed oented path th that bounds the bued suface the Gauss-Bonnet path (GB-path). Fo the λ ac, the ac length s Φ λ and the adus of the coespondng COI s a λ and the pola angle of the COI s Θ. The exteo angle between ac λ and λ + 1 s λ Ω λ. The gaussan cuvatue of the sphee of adus s K = 2 1. In ode to calculate the geodesc cuvatue of the ac we apply the GB theoem (53) on a sphecal cap of heght the base. Snce fo a sphecal cap the aea s π ( d ) λ d and adus λ 2 and χ = 1 we have: a λ on a λ g ( d ) λ g 2π λ + 2π λ 2 hence d = a λ λ 1 = 2π (54) Then, snce cos Θ λ = d λ, the geodesc cuvatue of the λ ac must be: λ Θ λ g aλ = cos (55) By applyng the GB theoem (53) on the suface bued by the neghbos of atom and usng (55), we get the bued aea: P λ= 1 S cos Θ a Bued λ λ Φ = 2 λ a λ + S 2πχ Bued hence + Ω = 2πχ P ( Ωλ + Φ λ cosθλ ) λ= P λ = 1 λ (56) The Eule-Poncaé chaactestc descbes the topology of the bued suface. The bued suface s topologcally equvalent (homomophc) to a sphee wth n holes, as many as the closed paths fomed by the neghbos, hence χ = 2 n. In geneal, thee can be moe

73 57 than one dsconnected pece of the suface of the atom that s bued. The exposed aea of the atom s the total aea 2 4 π mnus the sum of the bued aeas of each dsconnected pece, gven fom equaton (56): P exp 2 S = 2π ( 2 χ ) ( Ω λ + Φ λ cosθλ ) (57) λ= 1 whee now χ s the sum of the ndvdual χ ' s fo each dsconnected pece of the bued suface and P s the total numbe of GB-acs on the suface of the atom. Usng the GB theoem, the calculaton of the exposed aea s educed to the calculaton of the acs of the COI s of the neghbos on the suface of the atom and the angles between them. In ode to calculate these quanttes we need to paameteze the geometcal poblem. Intal attempts used a Catesan system that made the fnal fomulas extemely cumbesome [71]. Instead, we wll use a paametezaton that s equvalent fo each atom and was ntoduced n ts basc fom n [70] and fomalzed n [73]. We wll use the cente of the cental atom as the cente of the coodnate system (Fgue 19). If atom of adus s located at x and ts neghbo of adus at x, then the dstance fom the cente of to the COI that s fomed by s g µˆ, whee, g ˆ µ = x x = x and 2 x 2 x x + x x 2 2 (58) and the adus of the COI, a s: a ( g ) 2 = (59) 2

74 58 The pola angle Θ of the COI s: cos Θ = g (60) Fgue 19. Paametezaton of the Gauss-Bonnet acs. In ths example, the cental atom s ntesected by thee neghbos, j,, l. The j and COI s ntesect each othe, as the and l do also. See text fo explanaton of the vecto quanttes. (Fgue adapted fom efeence [73].) Now, f the neghbo s ntesected by two othe neghbos, j and l, the COI s ntesected by the j and l COIs (located at g µˆ and j j g µˆ espectvely) l l and the COI becomes a GB-ac. The oentaton of the GB-Path s vey mpotant because dependng on ths oentaton the calculaton of the bued aea, equaton (57), wll yeld the aea on the one o the othe sde of the GB-path. The ght oentaton fo

75 59 the GB-acs n ou poblem s CCW, loong fom above (outsde the cental atom). Then, the GB-ac s bounded by two ponts, ponts between the P j and Q l, that ae the ntesecton and j COIs and and l COIs espectvely. We call these ponts GB-ponts. If the mdpont of the segment defned by the ntesecton of the and j COIs s ηj, vecto on the decton of the segment, then we have: 2 γ j s the total length of the segment and ωˆ j the unt γ j cosφj = ˆ µ ˆ µ j η = τ ˆ µ + τ ˆ µ τ j = j = ˆ ω j g 2 j g sn j 2 g τ φ j j cosφ j ˆ µ ˆ µ j = snφ j j j g τ j j (61) whee φ j s the angle between the and j COIs. We ae now able to calculate the GB-ponts: P Q j l = ηj + γ j ˆ ωj = η γ ˆ ω l l l (62) and then the tangent unt vectos that defne the exteo angles Ω between consecutve GB-acs, nˆ j mˆ Ω l j ˆ µ P = a ˆ µ Q = a j l = accos j ( nˆ mˆ ) j (63)

76 60 snce the exteo angles ae negatvely oented [72]. The angula ac length of the GB ac can be calculated fom the nne poduct of the vectos n j and m l. The ac length s ethe the ac-cosne of the nne poduct o the complmentay angle. In compact fom, we have: Φ S jl jl = ( 1 S ) π + S accos( n m ) whee = sgn jl ( µ ( n m ) ˆ j jl l j l (64) S jl s the sgn of the elatve oentaton of the vecto µˆ and the tangent vectos n j and m l. Thus, the ac-length of the COI s GB-ac bounded by neghbos j and l s: S jl = a Φ (65) jl Ths s the ac-length of each ac-secto n equaton (51). Fnally, fo the plana secton fomed by neghbo, snce the efeence pont fo the plana secton s g µˆ, the base of a tangle n the decomposton coesponds to the lne segment fomed by the ntesecton of the COIs of the neghbos and j that defne t. Then, the heght of the tangle s (see Fgue 19): h j g g j cosφj = (66) snφ j and the length of the base, t j, can be detemned by the postons of the vetces that defne t. These vetces ae the ntesecton ponts of thee sphees, whch mght o mght not nclude the cental atom (see Fgue 15). If the cental atom s ncluded then the vetex s the coespondng GB-pont, othewse t can be calculated analytcally by employng the same vecto paametezaton that we have used n ths secton.

77 61 We now have analytcal expessons fo the aeas and volumes of each atom n the fused-sphee model. We can dffeentate these expessons wth espect to each neghbo s coodnates to get patal gadents of these quanttes. Snce movng the cental atom n one decton s equvalent to movng all the neghbos n the opposte decton, we can compute the gadent wth espect to the cental atom by summng the patal gadents wth espect to ts neghbos postons. In patcula, fo the exposed suface aea of atom, exp S, f the cental atom has M neghbos then: exp S S = (67) x M exp = 1 x The devaton of the patal gadents s tedous but staghtfowad and has been pesented n [73]. The total gadent s the sum of the patal gadent wth espect to the cental atom and the patal gadent of the neghbos wth espect to the cental atom: S exp exp S S = + (68) x M exp j j= 1 x 3.3 Topologcal Analyss The analyss n sectons 3.1 and 3.2 llustates the fomulas needed fo the analytcal calculaton of aeas and volumes wth gadents fo evey atom n the fusedsphee model. Ths assumes that we now fo evey atom n the model whch neghbos ntesect t and n whch ode. In patcula, the applcaton of the Gauss-Bonnet theoem mples that we now whch neghbong atoms ceate the GB-acs on the suface of the cental atom and also the odeng of the GB-acs as they fom the closed GB-paths. The

78 62 topologes that may ase n a molecula smulaton can vay geatly and can be extemely complex (see Fgue 8). It s mpeatve that we have an algothm that can deal wth all possble topologes and also be computatonally vey effcent. Thee ae two poblems that we have to solve: Fst, we need to dentfy whch neghbos that ntesect the cental atom ae tuly contbutng to the exposed aea and volume of the atom. Second, we need to ode the tue neghbos as they fom the GB-paths on the suface of the cental atom Intesecton of Half-Spaces (IHS) In ode to dentfy whch neghbos ntesect the cental atom n an effcent way we dvde the smulaton space nto cells (Fgue 20). We assgn each atom nto the cell that contans t and then seach whch atoms ntesect the cental atom only fo the cell t belongs to and the 26 neghbong cells. Cental atom s cell Neghbo cells Fgue 20. Patton of the smulaton space nto cells. Fo each atom we seach the cell the atom belongs to (da gay) and the 26 neghbong cells (lght gay).

79 63 If the sphee of adus + 2max centeed on the cental atom of adus and a neghbong cell do not ntesect, no atom n that cell can ntesect the cental atom. Thus, only f the neghbong cell and the + 2max sphee ntesect we seach fo ntesectng atoms n that cell. In ode fo the seach to be complete when we estct ouselves to seachng only the neaest neghbo cells, the length of each cell has to be at least twce the maxmum adus n the system, max. Fgue 21. The cental atom (ed) s ntesected by the neghbos A, B and C (geen). Neghbo B s occluded by A and C. Fgue 22. Same as Fgue 21, wth also showng the ntesectng planes of each neghbo. Howeve, fndng the atoms that ntesect the cental atom s not enough. Thee can be many sphees that ntesect the cental sphee that do not eally contbute to the exposed aea and volume accodng to the decomposton descbed n secton 3.1. Ths s because most of the ntesectng sphees ae actually occluded by the tuly ntesectng neghbos. In the example n Fgue 21 and Fgue 22 neghbo B s occluded by the othe

80 64 two neghbos A and C and does not contbute to the aea and volume calculaton. In ode to dentfy the tuly ntesectng neghbos we wll tansfom the poblem nto a well-nown poblem n computatonal geomety, the poblem of the ntesecton of N half-spaces [65], [73], [75]. Evey neghbong atom that ntesects the cental atom defnes an ntesecton plane that cuts though the two sphees, as shown n Fgue 23. Ths plane dvdes the space nto two half-spaces, H 1 and H 2 (Fgue 24). gµˆ Fgue 23. The ntesectng plane between the cental atom (left) and a neghbo. Fgue 24. The two half-spaces H 1 and H 2 defned by the ntesectng plane. The exposed aea and excluded volume of the cental atom s on H 1. The half-spaces can be fomally descbed as follows: A plane n space that s at dstance l fom the ogn can be defned by a vecto p that s nomal to that plane and such that p = l. All ponts n space that belong to ths plane obey the plane equaton: p p 2 = 0 (69)

81 65 The plane s the bounday between the two half-spaces t defnes: p p p p 2 2 < 0 > 0 fo fo H H 1 2 (70) Accodng to the vecto paametezaton defned n secton 3.2, the vecto that defnes the ntesectng plane between atom and neghbo s g µˆ, whee the unt vecto µˆ ponts towads the cente of the neghbo. Fo the case shown n Fgue 24 the solvent excluded volume and solvent accessble suface of the cental atom ae on the H 1 halfspace that ncludes the ogn, whch s defned as: 2 ( g ) 0 g ˆ µ (71) whee we also nclude the bounday. A slghtly dffeent case ases when the neghbo swallows the cental atom, but not completely (Fgue 25 and Fgue 26). µˆ Fgue 25. Example of a swallowe: the neghbo (geen) swallows the cental atom (whte) but not completely. Fgue 26. The half-spaces defned n the case of a swallowe neghbo (ght). The exposed aea and excluded volume of the cental atom (left) s on halfspace H 1.

82 66 In such case, the ntesectng plane s stll defned by the vecto g µˆ although now g s negatve. The half-space H 1 that ncludes the excluded volume and accessble aea s the one that does not nclude the ogn, thus: g ˆ µ g o, ˆ µ + ( g ) ( g ) 0 (72) In geneal, thee can be both non-swallowe and swallowe neghbos. The soluton to the set of equatons (71) and (72) s the ntesecton of half-spaces (IHS) and t s the egon of space that satsfes all the lnea constants mposed by the ntesectng planes of all the neghbos. Clealy, the IHS wll be ethe an empty set o a convex set (nfnte o fnte) bounded by the planes that tuly contbute to the exposed aea and excluded volume. Thus, the faces of the convex polyhedon that bound the IHS belong to the ntesecton planes of the neghbos that tuly ntesect the cental atom (Fgue 27). By dentfyng the IHS we dentfy the tue neghbos. The IHS s an empty set f the constants posed by the ntesectng planes ae nconsstent. Ths occus f the cental atom s completely swallowed by ts neghbos. The exposed aea and excluded volume fo such case s zeo. The poblem of fndng the feasble ponts fo a set of N lnea constants has many applcatons n computatonal geomety and mathematcal optmzaton and ts effcent soluton has attacted consdeable attenton. We wll pesent hee a smplfed veson of the algothm pesented n [75] that s sutable fo ou needs. But fst we wll

83 67 have to ntoduce the concepts of geometc dualty and the convex hull (CH) of a set of ponts. Fgue 27. The IHS fo the example of Fgue 21. The half-spaces of neghbos A, B and C that nclude the cental atom s exposed aea and excluded volume ae coloed blue, ed and yellow espectvely. The ovelap of the half-spaces s coloed by the coespondng ovelappng colo,.e. yellow+ed=oange, blue+ed=puple, blue+yellow=geen. The common nteo of the constants s the IHS (geen) and t s only due to the A and C half-spaces. Neghbo B s occluded Geometc Dualty and the Convex Hull (CH) As was shown n 3.3.1, a plane n space can be descbed by the vecto p nomal to t. The dstance between the plane and the ogn s p. At the same tme, any vecto can defne a pont n space. In geneal, we defne the geometc nveson n d R as a

84 68 pont-to-pont tansfomaton of d R whch maps a vecto p appled to the ogn to the vecto ' p = p p 2, appled to the ogn [65]. Usng geometc nveson we can defne the dual of a plane p as the pont p ' 2 =, and vce vesa, the dual of a pont ' p p p as the plane ' ' 2 p = p p. The geometc dualzaton maps ponts to planes and planes to ponts. Thus, f t s appled to a convex polyhedon, ts vetces ae mapped to faces n the dual space and the faces ae mapped to vetces. In the example n Fgue 28 we see how a tetahedon s mapped to anothe tetahedon wth equvalent topology. Fgue 28. Geometc dualzaton of a tetahedon. The vetces ae mapped to faces and the faces to vetces. The topology (e.g. faces connected by a common edges) s peseved. Vetex 1, whee faces A, C and D meet, s mapped to face 1 n dual space, whch s defned by the vetces A, C and D, the duals of the espectve faces. Smlaly, face A,

85 69 defned by vetces 1, 2 and 4 s mapped to ts dual vetex A, whee dual faces 1, 2 and 4 meet. Edge 1-2, whch connects faces A and C s mapped to edge A-C, whch connects the dual faces 1 and 2. The oentaton of each face s peseved n the dual space. Ths topologcal equvalency between a convex polyhedon and ts dual wll pove to be cucal n ou algothm. The convex hull (CH) of a set of ponts S n convex doman n d R s the bounday of the smallest d R contanng S [65]. We can ntutvely thn of the convex hull of a set of ponts n space as the geometc fgue that would ase f we wee to tghtly wap the outsde set of ponts wth an elastc band. It s obvously a convex polyhedon and fo any set of ponts thee exsts a convex hull. A two-dmensonal example of a convex hull s shown n Fgue 29. The CH s defned by the most outwads ponts of the set of ponts. Fgue 29. The convex hull of a set of ponts n two dmensons. Now we have all the tools needed to solve the IHS poblem. The cucal popety of the IHS s that f we dualze the ntesecton planes, whch defne the IHS, to ponts, and constuct the CH of that set of ponts, the dual of the CH s the IHS. In othe wods,

86 70 the dual of the IHS s the CH of the dual of the constants. A poof of ths theoem usng pojectve geomety s gven n [65]. Ths gves us the ecpe fo computng the IHS: Each ntesectng plane of the cental atom s a lnea constant descbed by the vecto g µˆ. The dual of each ntesectng plane s the pont µˆ g. Then we calculate the convex hull of all the dual ponts, ncludng the ogn (whch s the cente of the atom ). Fo each face of the CH, we fnd ts dual pont. Usng the topologcal equvalence between the CH and ts dual, as shown n Fgue 28, we connect the duals of the faces of the CH. The esultng polyhedon s the IHS. Ths pocedue s shown schematcally n Fgue 30 - Fgue 37, fo a two-dmensonal example. Effectvely ths pocedue s able to emove the occluded constants because of the natue of the geometc nveson. Ths mappng bngs ponts close to the cente fa away fom t, and vce vesa. The occluded constants coespond to fa-away ponts. By nvetng them, we bng the duals close to the cente and then the convex hull of the duals selects the ones fathest away fom the cente, n the dual space, (closest to the cente n eal space), thus emovng the occluded constants. We nclude the cente of the atoms n the set of ponts fo whch we constuct the CH because t coesponds to a constant at nfnty. We call the cente n the dual space the zeo pont. If the IHS s open, le a convex cone nstead of a convex polyhedon, as n the example n Fgue 27, the cente wll be a vetex of the CH. Afte dualzng the CH, all faces that nclude the zeo pont ae emoved fom the IHS polyhedon, thus ceatng the open IHS. These faces ae called zeo faces.

87 71 Fgue 30. The lnea constants. Fgue 31. The nomals to the constant planes. Fgue 32. Dual ponts of the constant planes. Fgue 33. The convex hull of the dual ponts. Fgue 34. The nomal vectos to the faces of the CH. Fgue 35. The dual ponts of the faces of the CH. Fgue 36. Connectng the dual ponts of the CH. Fgue 37. The IHS.

88 72 A ey popety fo ths pocedue s the exstence of the coodnate cente nsde the IHS. In ode to dualze the ntesecton planes, we must have a pont wth espect to whch we do the geometc nveson. Ths pont was taen as the cente of the atom, because t s usually nsde the IHS. Howeve, f at least one of the neghbos s a swallowe, the cente of the atom s no longe n the IHS, as shown n Fgue 25 and Fgue 26. Thus, befoe we can apply the afoementoned pocedue, we need to fnd an nteo pont of the IHS. Ths s a non-tval ssue snce we ae seeng fo a pont nsde a egon fo whch we do not yet now ts boundaes, and fndng these boundaes s the actual poblem we have to solve and we need the nteo pont fo. We can fnd though a vetex of the IHS, wthout nowng anythng else but the constants, by usng lnea pogammng, as descbed n secton An addtonal complcaton that ases when thee ae swallowe neghbos s the exstence of both a common nteo (the IHS) as well as a common exteo, the egon of space fo whch the constants (71) and (72) ae nveted (f the constants ae not nconsstent). As descbed above, the duals of the faces of the CH coespond to vetces of the common nteo (IHS). In the pesence of swallowes howeve, the duals of cetan faces of the CH coespond to vetces of the common exteo. These faces have to be dentfed and emoved fom the CH n ode to buld coectly the IHS. The pocedue then s slghtly modfed as follows: The zeo pont s not ncluded n the constucton of the CH. Afte the CH s calculated, we dentfy whch faces of t ae vsble fom the zeo pont (the concept of vsblty of a face fom a pont s explaned n detal n secton 3.3.4). The duals of those faces ae vetces of the common exteo. We name these faces

89 73 negatve faces because f we efomulate the poblem usng homogeneous coodnates n fou dmensons, then these vetces ae ponts of the hypeplane x = 1, wheeas the 4 thee-dmensonal space we e wong on s the hypeplane x 4 = 1. (See Fgue 38 fo a two-dmensonal example and Fgue 39 fo the pojecton n thee dmensons and [65] fo detals on homogeneous coodnates and the ntepetaton of the poblem n hghe dmensons). Negatve faces actually coespond to constants at nfnty, much le the zeo pont coesponds to a constant at nfnty, thus ceatng an open IHS. Common Inteo x3 = +1 x 3 x 2 x3 = 1 Common Exteo Fgue 38. Constants and the half-spaces, n 2D. Fgue 39. Pojecton of the 2D poblem n 3D. The constant lnes on the hypeplane x = 1 become planes that pass though the 3D ogn. The 3 + x 1 poblem s mapped on a 3D unt sphee. (Fgue adapted fom [65].) Lnea Pogammng Lnea pogammng (LP) s a fundamental optmzaton poblem that has found applcatons n vaous felds, fom compute scence to economcs to busness admnstaton, and t has attacted consdeable attenton n the lteatue. In ts geneal

90 74 fom t s fomulated as follows [76]: Fo N ndependent vaables x,..., 1, x2 xn we see a vecto n the N dmensonal space that maxmzes the lnea functon f N ( x1 x2,..., xn ) =, c x (called the objectve functon) subject to M lnea constants, = 1 N j= 1 a j x j a N + 1, whee = 1,..., M (some of the nequaltes could also be equaltes). Any vecto that satsfes all the constants s called a feasble vecto. The feasble vecto that optmzes the objectve functon s called the optmal feasble vecto. If the ndependent vaables ae estcted to be postve, the LP poblem s sad to be n ts nomal fom. An optmal feasble vecto can fal to exst fo two possble easons: the constants ae ncompatble o thee s a decton n the N - dmensonal space fo whch one o moe of the vaables can be taen to nfnty whle stll satsfyng all the constants, gvng an unbounded value fo the objectve functon. The lnea constants effectvely educe the seach space nto a convex polyhedon, whch could be open. If thee s an optmal feasble vecto, snce the objectve functon s lnea, t wll have to be a vetex of that polyhedon, whch s the pont at whch some N of the constants meet. Thus, we can apply LP on the thee-dmensonal lnea constants (71) and (72), the halfspace nequaltes, wth an abtay objectve functon, to fnd a vetex of the IHS. Howeve, a vetex of the IHS s nappopate to use as the coodnate cente fo dualzaton, because the planes that meet at that vetex wll be dualzed to nfnty. We must fnd a tuly nteo pont of the IHS nstead.

91 75 In ode to fnd a pont n the nteo of the IHS we have to shn the constant planes by a postve constantε, towads the half-spaces of nteest. Then, f we apply LP fo the shned constants the optmal feasble vecto wll be a vetex of the shned constaned polyhedon and thus an nteo pont of the IHS (Fgue 40). Fgue 40. The vetex of the IHS of the shned constants (dotted lnes) s an nteo pont of the ognal IHS. fom The constant planes ae descbed by equatons (71) and (72), whch have the p p p p (73) espectvely, whee p = gµˆ s the vecto that defnes each plane. The constants can be ewtten as: pˆ pˆ p p 0 0 (74)

92 76 The fst equaton coesponds to the half-space that ncludes the ogn and the second that excludes the ogn, whch s the case of a swallowe neghbo as shown n Fgue 26. We want to tanslate the constants towads the half-spaces of nteest by a constant postve value ε, so the constant nequaltes become: pˆ pˆ ( p ε ) 0 ( p + ε ) 0 (75) as s shown n Fgue 41 and Fgue 42. p ε p ε Fgue 41. Tanslatng the constant plane p by ε, towads the half-space H 1, fo the case of a non-swallowe neghbo. Fgue 42. Tanslatng the constant plane p by ε, towads the half-space H 1, fo the case of a swallowe neghbo. The value of ε descbes by how much the constants ae shfted. We want an optmal value fo ε to ensue we fnd a pont way n the nteo of the IHS. We can acheve ths by settng t as an ndependent vaable and optmzng ts value usng LP. So, nstead of solvng the thee-dmensonal LP poblem of the M constants (71) and (72), whee

93 77 M s the total numbe of neghbos ntesectng the cental atom, we solve the LP poblem n the fou-dmensonal space (,ε ) g ˆ µ + g ε g ˆ µ g ε + ' ' ' that conssts of the constants: ( g ) ( g ' ) 2 2 0, 0, ε 0 whch esult fom equaton (75) by pluggng p = gµˆ. We want to optmze the value of ε to ensue we have an optmal nteo pont, so the objectve functon to be maxmzed f f g g ' > 0 < 0 (76) s: f (,ε ) = ε (77) If thee s a soluton to the fou-dmensonal LP poblem of equatons (76) and (77), the optmal feasble vecto (, ε ) opt opt gves the nteo pont of the IHS, opt. If the constants ae ncompatble, thee s no soluton and the atom s completely swallowed by ts neghbos, so the exposed aea and excluded volume ae zeo. The case of an unbounded soluton has to be consdeed wth cae, snce t can stll povde us wth an nteo pont as long as we can dentfy the decton n the N dmensonal space that gves an unbounded value fo the objectve functon. The standad algothms used to solve LP poblems [76], [77], assume the poblem n ts nomal fom and cannot deal wth unbounded solutons. Instead, we mplemented the algothm ntoduced by Sedel [78], whch can deal effectvely wth these ssues. In Sedel s algothm, the optmum vetex s detemned by a ecusve pocedue. Intally, a constant s pced n andom, and a guess fo the optmal vetex s made

94 78 Fgue 43. The constants and the objectve functon. Fgue 44. Choosng a andom pont on the decton of the objectve functon. Fgue 45. Pojectng the pont to a andomly pced constant. Fgue 46. Addng anothe constant. The pont satsfes ths constant. Fgue 47. Addng the last constant. Solvng the poblem n one dmenson (on the constant ed-1) Fgue 48. Optmzng the pont wth espect to the last constant n one dmenson and lftng the soluton on the two-dmensonal space.

95 79 along the decton of the objectve functon. Then we eep addng the est of the constants and chec f the tal vetex satsfes them. If thee s a constant that s not satsfed by the tal vetex, we poject the vetex and all the othe constants on the hypeplane of that constant and ecusvely solve the poblem of M 1 constants n N 1 dmensons. The ecuson wll eep gong nto lowe dmensons untl we each a one-dmensonal poblem whose soluton s tval. That soluton s then lfted onto the hghe dmensons. Addtonal constants ae added dynamcally n the algothm that bound the optmal vetex n case of an unbounded soluton. The pseudocode fo ths algothm s descbed n detal n [78]. A two dmensonal example of ths pocedue s shown n Fgue 43-Fgue 48. Thee, the lnes n ed ae the boundaes of half-spaces that do not nclude the ogn and the lnes n blue bound the half-spaces that nclude the ogn Constucton of the Convex Hull As was descbed n 3.3.2, afte we dualze the ntesecton planes to ponts, we calculate the convex hull of the duals n ode to elmnate the edundant constants. If thee ae no swallowes the dualzaton s wth espect to the cente of the atom, othewse we apply LP to fnd a pont n the nteo of the IHS and use ths pont as the cente. The calculaton of the convex hull s done by means of a andomzed ncemental algothm that has optmal expected pefomance [79]. We stat the calculaton by choosng fou ponts n andom to fom a tetahedon. Ths s the statng pont fo the constucton of the CH. At evey consecutve step we

96 80 wll ncementally add and emove faces to ths polyhedon untl we ave at the CH. To do that, we pc a pont n andom fom the emanng set of ponts and chec all the faces of the cuent polyhedon to detemne whch faces ae vsble fom that pont. The faces of the polyhedon ae oentated n a CCW fashon, loong fom the outsde. Ths oentaton defnes a vecto nomal to the face pontng outwads. A face s vsble fom a pont f the pont s on the half-space that the nomal vecto s pontng towads. Mathematcally, f the vetces of the face ae the vectos a, b and c, and the pont s d, then the sgn of the detemnant D detemnes f the face s vsble, whee a c x x x d d x x x a c y y y d d y y y a c z D = b d b d b d (78) z z = d d z z z If D < 0 then the face s vsble, f D > 0 the face s not vsble and f D = 0 then the face and the pont ae coplana. Ths s because the sgned volume of the tetahedon s D 6 and the sgn has to do wth the oentaton of the tplet a d, b d, c d, centeed at the vetex d. Afte we detemne the vsble faces of the cuent polyhedon fo the pced pont, we delete the vsble faces that shae an edge, along wth the common edge. If vsble faces shae a common vetex t s also deleted. We then ceate new edges fom the pont to the undeleted vetces of the deleted faces and fom new faces, mang sue we peseve the CCW oentaton. The polyhedon that ases wth the new faces ncludes the deleted vetces n ts nteo. The pocedue s contnued by pcng anothe pont n andom fom the emanng set and addng and deletng faces to the polyhedon afte the vsblty checs. At the end of each addton the polyhedon ceated s the convex hull of the subset of ponts that have been utlzed up to that step.

97 81 Fgue 49. Set of ponts. Fgue 50. Intal smplex. Fgue 51. Vsblty chec. Fgue 52. Add new faces. Fgue 53. Remove vsble faces. Fgue 54. Vsblty chec. Fgue 55. Add new faces. Fgue 56. Remove vsble faces. Fgue 57. Vsblty chec. Fgue 58. Add new faces. Fgue 59. Convex hull.

98 82 When thee ae no moe ponts emanng, the algothm guaantees the fnal polyhedon s the CH of the ntal set of ponts. Ths pocedue s shown schematcally n Fgue 49- Fgue 59, fo a two dmensonal case. The vsble faces to be emoved at evey step ae maed wth ed. It s clea fom the above that the actual speed of the algothm depends a lot on the nput set of ponts and the numbe of ponts elmnated at evey step. The wost case s f all ponts ae on the suface of a sphee, then all ponts ae ncluded n the CH and we must examne evey pont. Also, due to the andom natue of the algothm, the pefomance wll depend on ode of the ponts pced. The futhe they ae at the begnnng of the pocess, the moe ponts ae excluded and thus fewe ponts have to be examned. Ths mples that f we somehow bas the selecton pocess towads ponts futhe away, we should ncease the computatonal speed. Ths s the dea behnd the QucHull vaaton of the andomzed ncemental algothm, whch was poposed n [80] and was utlzed n ths mplementaton. The only change on the ncemental algothm because of QucHull s that at the begnnng, we loop ove all the ponts and fo each pont we fnd the fst face that s vsble to t. Fo each face we ceate an outsde lst of ponts that t s vsble to, and the lst s soted accodng to the dstance of the pont fom the face. Ths way we patton the set of ponts to the outsde lsts of the faces. Then, nstead of selectng n andom a pont fo the ncemental algothm, we select the pont wth the lagest dstance fom ts face. At each successve step, afte some faces ae deleted and othes ceated, the ponts

99 83 n the outsde lsts of the deleted faces ae epattoned to the outsde lsts of the new faces. In addton, the ntal tetahedon s ceated by fou ponts that have maxmum coodnates. These addtons to the QucHull algothm mae the CH constucton much moe effcent Detemnng the GB-paths The pocedue descbed n sectons allows us to detemne whch neghbos ae tuly ntesectng the cental atom, o equvalently, whch neghbos fom GB-acs on the suface of the cental atom. The next tas s to dentfy the exact topology of the tue neghbos on the suface of the atom. If the COIs of these neghbos ae ntesectng each othe then they ntesect at the GB-ponts (see Fgue 18 and vectos P v j, Q l n Fgue 19). Each tue neghbo contbutes at least one GB-ac (except n the tval case of an solated neghbo, as n Fgue 9) and two neghbos that ntesect each othe may contbute at most two GB-ponts. Each GB-ac s a secton of the COI of a patcula neghbo, bounded by two GB-ponts at the begnnng and the end. The oentaton of the GB-acs s CCW, loong fom top, accodng to the conventon we set n secton 3.2. The goal then s to dentfy whch neghbos COIs ntesect each othe thus fomng a GB-pont, goup the GB-ponts nto subsets that belong to each GB-path and then ode the GB-ponts of each goup as they fom the GB-path n a CCW fashon. When the GB-paths have been detemned we can use equaton (57) to calculate the exposed aea and equaton (50) fo the excluded volume. The ey popety that wll allow us to solve ths poblem s the fact that the edges of the IHS pece the suface of the atom

100 84 Fgue 60. The IHS polyhedon fomed by the neghbos of the cental atom (whte) fo the example n Fgue 8. Fgue 61. The IHS polyhedon of Fgue 60 as t cuts Fgue 62. The IHS polyhedon of Fgue 60. though the cental atom.

101 85 at exactly the GB-ponts. Ths s clealy shown n Fgue 60, Fgue 61 and Fgue 62, whee the IHS polyhedon fo the cental atom n the example of Fgue 8 was calculated. Each face of the IHS s a polygon-shaped secton of the plana bounday of the half-space mposed by the coespondng neghbo. The COI of that neghbo les on the plane of the face. If the whole COI les nsde the face, the neghbo s solated (Fgue 9). If the whole COI les outsde the face (but on the bounday plane) then the neghbo does not contbute any GB-acs. If the COI les patally nsde and outsde the face, then each pont on the COI that s on an edge of the IHS s a GB pont (see Fgue 19). An edge of the IHS s common to two of ts faces. If an edge peces the cental atom, then the COIs of the coespondng atoms ntesect at the GB ponts. In the case that all the vetces of the IHS ae completely bued nsde the atom, no edges ntesect the suface of the atom and thus thee ae no GB-ponts. In such case, the atom has no exposed suface because t s bued unde all ts neghbos. Howeve, the volume s not zeo and t s the volume of the IHS. The edges of the IHS pece the cental atom zeo, one o two tmes. Ths depends on the locaton of the vetces of the IHS that defne the edges, wth espect to the suface of the cental atom. If both vetces ae bued nsde the atom then the edge does not ntesect the suface of the cental atom. If both vetces ae exposed the edge can ntesect n two ponts o none at all. Ths wll have to be detemned explctly by the followng sphee-lne ntesecton test: If a lne goes though a pont 0 and s paallel to the decton û then evey pont on the lne obeys the lne equaton = 0 + λ uˆ, fo any

102 86 eal value of λ. The unt vecto û s detemned by the postons of the two vetces. All ponts on a sphee of adus R, centeed at pont C obey the equaton C = R. The system of the two equatons has a soluton f the quantty 4 = ( a uˆ ) ( a R ) s postve, whee a = 0. Fnally, f one vetex of the edge s bued and the othe C exposed then the edge peces the atom at exactly one pont. (See Fgue 63-Fgue 66.) Fgue 63. Bued-bued edge. Fgue 64. Bued-exposed edge. û a û a Fgue 65. Exposed-exposed ntesectng edge. Fgue 66. Exposed-exposed non-ntesectng edge. The vetces of the IHS can gve us even moe nfomaton. By the vey natue of the IHS, a vetex of t s the pont n space whee thee bounday planes meet. If a vetex s bued then the thee neghbos that coespond to each plane ae connected on the suface of the atom. The thee edges that emege fom that vetex wll necessaly pece the cental atom once each, ceatng thee GB-ponts (see Fgue 67 and Fgue 68). If the vetex s exposed then the neghbos ae not connected and the edges pece twce,

103 87 ceatng sx GB-ponts (see Fgue 69 and Fgue 70). We call these edges double peces. Fgue 67. A bued IHS vetex coesponds to thee connected neghbos and thee GB-ponts. Fgue 68. Two-dmensonal epesentaton of a bued IHS vetex. Fgue 69. An exposed IHS vetex coesponds to thee dsjont neghbos and thee GB-ponts. Fgue 70. Two-dmensonal epesentaton of an exposed IHS vetex. In the case of an open IHS, cetan vetces ae at nfnty. These vetces ae the duals of the zeo faces of the CH n the case that thee ae no swallowes. If thee ae swallowes thee ae no zeo faces snce we do not nclude the cente but thee ae negatve faces, as explaned n secton The duals of those faces ae taen to be at nfnty, n ode to ceate the tue IHS, and clealy, they ae exposed vetces (outsde the

104 88 cental atom). The edges that have such vetces wll fall ethe n the exposed-exposed o exposed-bued categoy, dependng on the popetes of the othe vetex. The analyss fo these cases s the same as befoe, wth the excepton that the decton vecto û of the edge (fo the ntesecton chec) s taen towads the open face of the IHS, whch n the case of a negatve face s opposte to the dual of the face. Usng the above deas, we can geneate the lst of GB-ponts that ae on the suface of the cental atom. Afte the IHS s constucted we loop ove each edge and analyze ts vetces to ecognze how many GB-ponts t ceates. Each edge s connectng two neghbos. The GB-ponts ae then assgned to one of the two neghbos, accodng to the CCW oentaton that we have chosen. Fo example, n Fgue 19, the GB-pont P j would be assgned to neghbo j and the GB-pont Q l to neghbo. Ths way we ceate a lst of connectvtes between the tue neghbos on the suface of the atom. The Fgue 72. The connectvty Fgue 71. Example of topology of neghbos on the cental atom. gaph fo the example of Fgue 71. The oented edges coespond to GB-ponts. Fgue 73. The connectvty table fo the example of Fgue 71

105 89 connectvty lst s an oented gaph, whee each oented edge s a GB-pont and each node s a neghbo. In the example of Fgue 71 the COIs of the neghbos on the suface of an atom ae shown unfolded onto a plane. The esultng connectvty gaph and connectvty table of the GB-ponts s shown n Fgue 72 and Fgue 73. It s possble that the connectvty gaph can have moe than one connected components. Each connected component coesponds to a dsconnected pece of the bued suface of the atom, fo whch the Eule-Poncaé chaactestc s 2 n whee n s the numbe of GB-paths on that pece, as was explaned n secton 3.2. In ode to dentfy the connected components of the connectvty gaph we use the Depth-Fst- Seach (DFS) algothm [81], a ecusve gaph-seachng algothm. Afte fndng the connected components we dvde the gaph to the espectve sub-gaphs. We then need to sot the vetces of each sub-gaph, thus odeng the GB-ponts as they fom the GBpaths. The GB-paths coespond to cycles of the connectvty gaph. In the example of Fgue 71 thee ae two GB-paths. Howeve, the elated connectvty gaph of Fgue 72 has moe than two cycles. The eason fo ths nconsstency s that as we tavese the gaph, thee can be moe than one optons to select the next node, as shown n Fgue 74. In ths example, we stat tavesng the connectvty gaph fom node 5, so the next node can only be node 1. Howeve, fom that node we have thee possbltes fo the next step, nodes 2, 3 and 5, as s shown n Fgue 72 and Fgue 74, because thee ae 3 GB-ponts on the COI of neghbo 1. To pc the ght one, we use the paametezaton shown n Fgue 19. The tangent vecto on the COI nˆ j, at the GB-pont P j fom

106 90 neghbos j, ponts to the decton of the GB-ac fo that GB-pont. The next GBpont has to be the end pont of that GB-ac and thus the fst pont we encounte as we tavese the COI n a CCW fashon. Hence, the next GB-pont has to be the left-most pont elatve to the decton nˆ j, whch s easly detemned by the dot poduct of the decton vecto to the vecto that connects the cuent GB-pont to the canddate next GB-ponts, as s shown n Fgue ˆn 5 Fgue 74. Tavesng the connectvty gaph of Fgue 71: statng fom the GB-pont A on neghbo 5, the next GB-pont has to be on neghbo 1. Out of the thee possbltes B, C, D, the GB-pont B s the coect choce. The detemnaton of the cycles (the GB-paths), fo each connected component of the connectvty gaph, poceeds as follows: we pc a node n andom, and pc an edge fo that node n andom. Ths node s the head of the cuent GB-path. The next node s chosen accodng to the left-most cteon descbed above. When the next node to be chosen s the head, we fom a cycle, whch s the GB-path. If thee ae oented edges n the gaph that have not been tavesed, we pc one n andom and contnue ths

107 91 pocedue untl all edges have been tavesed. When we have fnshed ths analyss fo all connected components, we have detemned all the GB-paths fo that atom (see Fgue 75 and Fgue 76 fo the GB-paths and cycles of the example of Fgue 71.) Fgue 76. The selected cycles (GB-paths) of the Fgue 75. The GB-paths fo the example of Fgue 71. connectvty gaph fo the example of Fgue 71. The two GB-paths ae: and Wth the topology of the tue neghbos on the suface of the cental atom detemned, we ae fee to poceed wth the aea and calculaton as shown n secton 3.3. Also, the nowledge of the IHS and the GB-paths allow us to undestand bette how the plana sectons of the buldng blocs ae fomed, as shown n Fgue 77. The plana sectons ae bounded by ac-segments and lne segments, as was explaned n Fgue 17. The ac-segments ae the GB-acs and the lne-segments ae pats of the edges of the IHS. The vetces of the lne-segments ae ethe bued vetces of the IHS o GB-ponts. The dentfcaton of these vetces s necessay fo the calculaton of the volumes usng equatons (50) and (51). In patcula, the nowledge of whch neghbos ntesect to

108 92 ceate the bued IHS vetces of the GB-ponts s needed fo the calculaton of the length of the lne-segments, t µ n equaton (51). The calculaton of the IHS and ts analyss, as explaned above, povdes ths topologcal nfomaton. Fgue 77. Relaton of the plana sectons of the buldng bloc of Fgue 15 wth the IHS and the GBpaths. 3.4 Implementaton of the Geometc Algothms Robustness The geometc algothms pesented n sectons wee mplemented n a compute pogam n the C pogammng language [82]. It s mpotant fo the mplementaton to be obust and effcent n ode fo t to be used n lage molecula systems of complcated topology. The fnte pecson athmetc used n computes can cause ound-off eos that may lead to eoneous esults, poduce numecal nfntes o even cause the pogam to cash. Ths s because of the dualzaton pocedue descbed n secton If the dstance of a plane fom the cente, g, s too small (the plane s

109 93 vey close to the cente), then the dual pont of that plane wll be vey fa away fom the cente n the dual space, and vce vesa. Also, f fou atoms come vey close to ntesectng at the same pont n space, some GB-acs can be vey close to zeo length and ths can cause vey small faces of the CH and numecal nstabltes fo the gadent calculaton (see [83] fo an analyss of poblematc atom topologes). The geometc pedcates used to calculate the CH, the vsblty of a face fom a pont can gve wong answes f the ound-off eos n such ccumstances. If the constucton of the CH fals then the esult of the calculaton can poduce unphyscal esults, le negatve suface aea o volume, o nfnte gadents. In molecula dynamcs whee the atoms ae popagated n space by nfntesmal dstances at evey step, t s nevtable that we wll come acoss such cases n the couse of a long smulaton fo a lage system. It s cucal that we can deal wth these poblems effectvely. These numecal pecson poblems wee aleady nown n the computatonal geomety communty. The calculaton of geometc constucts (convex hulls, Voono dagams, Delaunay tangulatons etc) faces smla ssues due to fnte pecson. The solutons poposed fall nto two categoes: petubaton schemes and abtay pecson athmetc technques. In petubaton schemes, the esults ae checed fo unphyscal condtons and f eos ae detected the sphees n the fused-sphee model ae petubed about the centes by a vey small dstance that ovecomes the degeneaces and pecson poblems [84], [85], [86]. Abtay pecson technques attempt to ceate geometc pedcates that poduce esults of abtay pecson on fnte pecson machnes. These technques ae conceptually bette but computatonally neffcent. New

110 94 adaptve methods howeve pomse effcent calculatons fo the geometc pedcates [87], [88]. In ths mplementaton we used an adaptve abtay pecson softwae lbay fo the calculaton of the geometc pedcates, whch s n the publc doman [88]. Fnally, anothe topology that can poduce numecal nstablty s coplana (o almost coplana) molecules, le benzene. The poblems ae smla n natue as descbed above but the soluton s smple: f the centes of the atoms ae almost coplana, nstead of calculatng the CH n thee dmensons we poject the cente of evey atom on a plane and solve the two-dmensonal CH poblem Scalng and Pefomance Fo a molecula system of N atoms we have to pefom N calculatons fo the aea and volume of each atom. Each calculaton n tun nvolves the calculaton of the IHS and ts analyss. The tme fo the calculaton of the IHS depends on the numbe of neghbo atoms, M, that ntesect each atom. The calculaton of M half-spaces scales as O( M log M ) [65] thus the calculaton of the whole system taes tme ( NM M ) O log. It s mpotant then to detemne f and how the numbe of neghbos M depends on the total sze of the system, N [89]. Let us defne max and mn the maxmum and mnmum adus of any sphee n the system and κ = max mn the ato. Also, f d mn s the mnmum dstance between any two atoms (physcally coespondng to the mnmum valence bond length) then we

111 95 defne the ato λ = d mn max. Let M be the neghbos of any atom, o adus. Then, all M neghbos sphees wll be completely engulfed n a sphee centeed at wth adus +. Fo each neghbo we defne a sphee of adus λ 2. Then 2max mn these sphees do not ntesect each othe because the centes ae at a dstance smalle than d mn : λ d mn mn 2 mn = λmn = mn = < d mn 2 max κ d (79) snce κ > 1. Hence, the sum of the volumes has to be less than the volume of the engulfng sphee: M 4 λ M π ( + 2 ) ( 3 ) λ 2 max 3 mn mn 3 4 π 3 max λ mn 2 Thus : 3 κ M 6 λ ( + 2 ) 3 3 max 3 (80) So, the numbe of neghbos M s bounded by a constant that depends on the ad and dstances of the sphees of the fused-sphee model. In pactce, fo the paametes we used, we notced that the maxmum numbe of ntesectng neghbos neve exceeded 150, wheeas the aveage value was aound 80. Hence, the calculaton of the aeas and volumes scales lnealy wth the sze of the system, O ( N ) (See Fgue 78 fo scalng on an Intel Pentum Xeon 866 MHz).

112 96 The algothms pesented wee mplemented n thee dffeent platfoms: Lnux (Intel), Ix (Slcon Gaphcs) and AIX (IBM) opeatng systems. The esults ae extemely accuate as compaed to numecal calculatons of the volumes of test molecules and vey fast. On an Intel Pentum III 866 MHz the aveage tme spent fo the aea and volume calculaton of one atom s aound 0.8ms (Fgue 78). Aeas of potens of typcal sze, atoms can be calculated analytcally, wth gadents, n just a couple of seconds. To ou nowledge, ths s the fastest mplementaton of the analytcal calculaton of aeas and volumes pe atom fo a fused-sphee model to date. Scalng wth system sze y=0.0008x CPU Tme (s) Numbe of atoms Fgue 78. Lnea scalng of the aea/volume calculaton wth espect to the numbe of atoms n the system.

113 97 4 The AVGB-SAS Solvaton Model The algothms n chapte 3 allow us to calculate accuately and effcently the solvent accessble suface aea A and solvent excluded volume V, wth the gadents, fo evey atom n a molecula system, assumng a fused-sphee model fo the system whee each atom s epesented by a sphee of adus + p, whee s the van de Waals adus of atom and p the pobe adus of the solvent, as t olls aound the solute. The volumes of each atom ae necessay fo the calculaton of pola solvaton effects accodng to the contnuum delectc theoy, n the Genealzed Bon appoxmaton, as was descbed n chapte 2. In patcula, unde cetan appoxmatons that wee llustated n sectons 2.3 and 2.4, the Bon ad ae calculated by equatons (37) and (41) fo whch the solvent excluded volumes V of each atom ae needed. Snce these volumes ae calculated analytcally, we name ths veson of Genealzed Bon the Analytcal Volume Genealzed Bon (AVGB) method. At the same tme, as was explaned n secton 1.3.4, shot ange solvaton effects ae lnealy dependent on the solvent accessble suface (SAS) aeas of each atom n the system. The volume calculatons have as a peequste the exposed aea, so the shot-ange tem s eadly avalable n ths calculaton. The full solvaton model ncludes both shot-ange and longange (pola) effects and we call ths the AVGB-SAS solvaton model. In the followng we wll nvestgate the pefomance of the model as fa as physcal pedctons and computatonal effcency.

114 Valdaton of the AVGB model The AVGB model s an appoxmaton to the solutons of the PB equaton (7). In ode to assess the qualty of the AVGB model we must compae ou pedctons to the esults of numecal solutons of the PB equaton. The best-nown mplementatons ae the DelPh pogam [22], UHBD [23] and PBF [24]. We wll also compae ou esults to the SGB veson [55] of the GB appoxmaton snce t s the only GB mplementaton that does not depend on fttng paametes, as AVGB does nethe. The compason wll be pefomed ove dffeent sets of molecules: small oganc molecules, amnoacds, lage potens and dmes Small Molecules The 376-molecule set fo the compasons s fom efeence [43] and each molecule has at most 40 atoms. The complete lst s gven n the Appendx. In ode fo the compasons to be unbased we must use the same paametes fo each test molecule n all methods. The van de Waals ad ae taen fom the DREIDING focefeld paamete set [90]. The chages ae calculated fom electostatc potental (ESP) fttng [91] of the quantum-mechancal wave functons of each atom n the molecule, whch n tun wee estmated by Hatee-Foc ab-nto electonc stuctue calculatons usng the JAGUAR pogam [92] and the 6-31G** bass set. The pobe adus of wate was taen to be 1.4Å, the had-sphee adus of wate n the lqud state [12]. The delectc constant of wate s 78.2 and fo the nteo delectc constant we chose a value of 1.

115 99 Delph vs UHBD 0-4 y = x R 2 = Delph (Kcal/Mol) UHBD (Kcal/Mol) Fgue 79. Compason of the pola solvaton eneges between Delph and UHBD fo the molecule set of Table 6. The RMS dffeence s 0.62 Kcal/Mol. PBF vs UHBD 5 0 y = x R 2 = PBF (Kcal/Mol) UHBD (Kcal/Mol) Fgue 80. Compason of the pola solvaton eneges between PBF and UHBD fo the molecule set of Table 6. The RMS dffeence s 0.41 Kcal/Mol.

116 100 PBF vs Delph 5 0 y = x R 2 = PBF (Kcal/Mol) Delph (Kcal/Mol) Fgue 81. Compason of the pola solvaton eneges between PBF and Delph fo the molecule set of Table 6. The RMS dffeence s 0.73 Kcal/Mol. In Fgue 79, Fgue 80 and Fgue 81 we compae the dffeent methods that calculate numecally the PB equaton, UHBD, Delph and PBF. All methods coelate stongly wth each othe, as s shown by the lnea egesson coeffcent 2 R. The lnea egesson ft fo evey compason s vey close to the deal y = x lne. Howeve, loong moe closely we see that UHBD behaves bette oveall, as s shown by the RMS dffeences between the esults of the dffeent methods. Delph poduces eoneous esults fo a few molecules and PBF does not coelate as stongly as the othe two methods wth each othe. Thus, fo the est of the compasons we wll focus on UHBD as the method of choce fo compang GB to PB esults.

117 101 AVGB vs UHBD 0-5 y = x R 2 = AVGB (Kcal/Mol) UHBD (Kcal/Mol) Fgue 82. Compason of the pola solvaton eneges between AVGB and UHBD fo the molecule set of Table 6. The RMS dffeence s 1.79 Kcal/Mol. SGB vs UHBD y = 1.356x R 2 = SGB (Kcal/Mol) UHBD (Kcal/Mol) Fgue 83. Compason of the pola solvaton eneges between SGB and UHBD fo the molecule set of Table 6. The RMS dffeence s 1.93 Kcal/Mol.

118 102 In Fgue 82 we compae the esults between AVGB and UHBD and n Fgue 83 between SGB and UHBD. In both cases we see that the GB esults ae vey well coelated to the PB numecal solutons, as shown by the lnea egesson coeffcent, wth AVGB coelatng slghtly bette to the PB esults than SGB does ( R 2 = fo AVGB and R 2 = fo SGB). Howeve, n both methods we obseve a systematc devaton fom the PB esults, as s shown by the egesson fts to the lne. The slope a s dffeent than unty and almost the same fo both methods, 1.36, whch means that both GB methods oveestmate the pola solvaton enegy as the system gets moe solvated. We obseved such systematc devaton n Fgue 6, whee we used PB-deved Bon ad and the ntepolaton fomula (19). At the same tme, the Coulombc appoxmaton ntoduced n the Bon adus calculaton n secton 2.3, along wth the pawse appoxmaton, equaton (41), may lead to addtonal eos that esult n the systematc devaton fom the coect PB esults. The fact that both GB methods have almost the same systematc eo hnts to the fact that the devaton s pobably not due to the volume o aea calculaton o the appoxmatons used fo the calculaton of equaton (37). It s the ntepolaton fomula (19) and the coulombc appoxmaton that leads to the fomula fo the calculaton of the Bon ad, equaton (37), that ae nducng the systematc eo n the calculaton of the pola solvaton eneges. The fact that the eo s systematc allows us to calbate ou paametes such that the pedctons of AVGB match exactly the PB solutons. Of couse, the success of such appoach wll depend on the numbe of paametes that need calbaton, and the applcablty of the calbated esults to molecula systems outsde the set used fo

119 103 calbaton. Snce the devaton s systematc and lnea n natue, we can tae advantage of the dependence on the nteo delectc constant ε n of equaton (18) to calbate the AVGB esults. In patcula f U PB s the PB pola solvaton enegy and U GB the GB pola solvaton enegy, then, accodng to Fgue 82 t s U = au. We can ewte GB PB equaton (18) as: U GB 1 = ε n 1 ε out S (81) whee S = 1 2 N N = 1 j= 1 q q f j j '. We see fo a value ε n of the nteo delectc constant such ' that U = U GB PB. Then, we must have: = ' ε n ε out a ε n o, = + ' ε n ε out a ε n 1 ε out 1 ε out (82) ' Usng equaton (82) we can pedct the value ε n that woul d gve esults that ae vey close to the PB solutons. In ou case, usng a = 1. 36, ε = and ε = 1. 0 we get ' ε 1.3. In Fgue 84 we compae the esults of AVGB wth ε = 1. 3 to UHBD wth n ε = 1.0. The coelaton coeffcent R 2 = obvously s the same as n Fgue 82 but n the slope of the lnea ft s now much close to unty, a = The RMS dffeence between the two methods s 0.46 Kcal/Mol. The ageement of the AVGB esults to the numecal solutons of the PB equaton s excellent and the dffeences ae not lage than out n n

120 104 the ones between the dffeent mplementatons of numecal solutons of the PB equaton, as shown n Fgue 79, Fgue 80 and Fgue 81. AVGB vs UHBD 5 0 y = x R 2 = AVGB (Kcal/Mol) UHBD (Kcal/Mol) Fgue 84. Compason of AVGB wth = The RMS dffeence s 0.46 Kcal/Mol. ε to UHBD wth = 1. 0 n ε fo the molecule lst of Table n Lage Molecules The success of AVGB wth small molecules s an encouagng step but n ode fo the method to be applcable to ealstc systems, t must be equally successful n pedctng the pola solvaton eneges of lage molecula systems. Also, the calbaton of the nteo delectc constant that was descbed n secton should not have to be

121 105 edone fo the lage systems. We compaed the pola solvaton eneges fo 11 potens, shown n Table 1.The ad used wee fom the DREIDING focefeld [90], chages fom the CHARMM22 focefeld [93] and the poten stuctues fom the PDB poten databan. The solvent delectc constant was taen ε = fo wate and we tested AVGB wth ε = 1. 0 and ε = 1. 3, and compaed to UHBD wth ε = The solvent pobe adus was 1.4Å. n n out n Table 1. AVGB and UHBD pola solvaton eneges fo 11 potens. Sze Poten (ncname) (Atoms) AVGB ε = 1 n AVGB ε = 1. 3 (Kcal/Mol) (Kcal/Mol) (Kcal/Mol) n UHBD L-Aabnose (aa) Cabonc Anhydase II (cah) Caboxypeptdase A (ca) Cytochome P-450cam (cyt) Intestnal FABP (fab) Neuamndase (nad) Pencllopepsn (pep) ε-thombn (et) Rbonuclease T1(b) Themolysn (tmn) Typsn (tp)

122 106 AVGB vs UHBD y = x R 2 = b fab AVGB (Kcal/Mol) et tmn nad aa tp cah ca pep cyt UHBD (Kcal/Mol) Fgue 85. Compason of AVGB and UHBD wth ε = 1. 0, fo the potens of Table 1. The RMS dffeence s 1169 Kcal/Mol. n AVGB vs UHBD y = x R 2 = b fab AVGB (Kcal/Mol) pep et tmn nad aa cah ca tp cyt UHBD (Kcal/Mol) Fgue 86. Compason of AVGB wth = 1. 3 The RMS dffeence s 303 Kcal/Mol. ε and UHBD wth = 1. 0 n ε, fo the potens of Table 1. n

123 107 In Fgue 85 we see how AVGB compaes to the numecal PB soluton fo the 11 potens when we use the same value fo the nteo delectc, ε = We see that the two methods coelate vey well to each othe, R 2 = 0. 95, but as wth the small molecules, thee s a systematc eo snce the slope of the lnea ft s In Fgue 86 we compae AVGB wth the calbated value of the nteo delectc, ε = The AVGB pedctons ae vey close to the PB esults as shown by the lnea ft slope The RMS dffeence s 303Kcal/Mol, whch s about 10% of the pola solvaton enegy of these systems. The dffeences between AVGB and UHBD ae on the same ode as between UHBD and Delph, whch ae dffeent mplementatons of the numecal soluton of the PB equaton. n n The fact that the calbated value of the nteo delectc that we obtaned fom the small molecule set wos so well wth these lage systems mples that we can use ε = 1.3 fo any molecula system n ode to get the pola solvaton enegy of that n system, as pedcted by the PB equaton wth ε = Obvously, f we want the pola solvaton enegy fo ε 1. 0 we need to edo the calbaton pocedue explaned n n secton Fom ths analyss t s clea that the calbated AVGB esults ae guaanteed to pedct the pola solvaton enegy fo molecules of any sze and any solvent and solute delectc constants. n

124 Intemolecula Pola Solvaton Eneges In sectons and we poved that AVGB can successfully epoduce the pola solvaton eneges fo small and lage molecules. Fo the method to be applcable to any system, t must be able to descbe accuately complexes and mult-molecula systems. Ths poblem s moe dffcult than a sngle molecule calculaton because of the complexty of the geomety of the solute-solvent bounday and the sceened ntemolecula nteactons that have to be accounted fo. To test the behavo of AVGB n such cases, we examned the pola solvaton enegy of a THF dme n dffeent oentatons, as a functon of the dstance between the two molecules. Qualtatvely, we expect that the enegy of the dme at nfnte dstance should be equal to the sum of the ndvdual molecules eneges. At vey close dstances, the pola solvaton enegy should ncease as the dstance deceases, othewse solvaton would favo the collapse of the dme. At ntemedate dstances we expect to have a mnmum fo whch the confguaton s optmally favoable. In Fgue 88 we see the pola solvaton enegy fo the THF dme of Fgue 87, whee the pola oxygen atoms face each othe. The solvaton enegy coectly epoduces the nfnty and zeo dstance lmts and t s a smooth functon of the ntemolecula dstance. Smlaly, n Fgue 90 the pola solvaton enegy of the THF dmme of Fgue 89 obeys smla behavo. In ths case, the pola atoms ae away of each othe. AVGB s able to coectly epoduce the ntemolecula pola solvaton enegy. We note that fo the same test cases, all othe methods (UHBD, Delph, PBF and SGB) dd not pedct the ght eneges at the nfnty lmts and the pola solvaton enegy was not a smooth

125 109 Fgue 87. THF dme wth the pola pats facng each othe. Pola Solvaton Enegy vs Dstance 0-1 AVGB (Kcal/Mol) Dstance (A) Fgue 88. Pola solvaton enegy fo the system of Fgue 87 fom AVGB as a functon of the dstance between the two THF molecules. The ed lne shows the enegy when the molecules ae nfntely sepaated fom each othe.

126 110 Fgue 89. THF dme wth the pola pats away fom each othe. AVGB (Kcal/Mol) Pola Solvaton Enegy vs Dstance Dstance (A) Fgue 90. Pola solvaton enegy fo the system of Fgue 89 fom AVGB as a functon of the dstance between the two THF molecules. The ed lne shows the enegy when the molecules ae nfntely sepaated fom each othe.

127 111 functon of the dstance. It s not clea why these methods fal to calculate the pola solvaton enegy coectly fo the ntemolecula poblem, although they can calculate the eneges coectly fo ndvdual molecules. Snce the man dffeence between AVGB and all othe methods s the analytcal calculaton of the geomety of the molecules, we beleve that the numecal calculaton of the boundaes of the molecules n multmolecula systems used s wong fo such cases n these methods. At the same tme, t s possble that snce dmes wth lage ntemolecula dstances wee examned, a much hghe gd esoluton would be needed to solve accuately the PB equaton, but that would mae the calculaton pactcally nfeasble. 4.2 The Shot-Range Tem In secton 4.1 we showed that the AVGB model fo the calculaton of the pola solvaton enegy and foces gves esults wth geat accuacy, as compaed to numecal solutons of the PB equaton. In ode fo ou model to ncopoate all solvaton effects, as explaned n secton 1.2, we must have a tem that accounts fo shot-ange effects such as van de Waals and entopc effects. In secton we showed that such effects can be descbed by a tem that s lnea dependent on the solvent accessble suface aea A of evey atom n an N - atom system: N GvdW + Gcav = σ A (83) The paametes σ have unts of suface tenson, Enegy/Aea, and n geneal could be dffeent fo evey atom n the system. Howeve, n ode to educe the numbe of paametes n the model and to have a method that can be applcable to any system, thee

128 112 have to be some ules on whch suface tenson would coespond to each atom n the system. Such ules have been pevously developed fo suface tenson models (secton 1.3.1) and the complexty can vay a lot. The suface tenson can be a constant fo all atoms, o depend on the atom element only, o depend on the element, the hybdzaton and connectvty wth ts valence-bond neghbos. In ode to detemne the suface tenson paametes fo wate, accodng to the AVGB-SAS model, we subtact the pola eneges as pedcted fom AVGB fom the expemental solvaton eneges of a lage set of molecules, to get the pedcted shotange contbuton to the solvaton eneges. We then ft the suface tensons fo each atom so that the dffeences n the enegy pedcted fom equaton (83) compaed to the shot-ange tem, fo each molecule, ae mnmal. Clealy, the suface tensons that we get fom ths pocedue depend on the molecule set and the paametes used n the AVGB calculaton. It s mpotant then to have expemental solvaton fee eneges fo a lage set of molecules wth dvese chemcal goups. We use a set of 376 oganc compounds fo whch the expemental solvaton eneges n wate ae gven n [43] and the efeences theen. The paametes fo AVGB ae the atomc ad, the pobe adus of the solvent, the chages of the atoms, and the thee-dmensonal stuctue of each molecule. Fo the atomc ad we used the values fom efeence [94], whch wee optmzed to epoduce wate solvaton eneges of a small numbe of oganc compounds by means of Hatee- Foc and Posson-Bolzman calculatons. The pobe adus of wate was taen to be 1.4Å,

129 113 accodng to [12] and the chages wee deved fom quantum-mechancal calculatons descbed n secton The thee-dmensonal stuctues of the molecules wee deved fom enegy mnmzaton usng the DREIDING focefeld [90]. Fnally, the solute delectc constant was 78.2 and fo the nteo we used the calbated value of 1.3, accodng to secton Intally we attempted to epoduce the expemental wate solvaton eneges fo ou molecule lst by usng solvaton types that depend only on the element of each atom. Wth the above-mentoned paametes the lnea optmzaton pocedue gave the suface tensons shown n Table 2 fo each element epesented n the molecule lst. Table 2. Suface tensons fo wate n (Kcal/Mol Å 2 ) pe element, fo the AVGB-SAS solvaton model. Suface Tenson Element (Kcal/MolÅ 2 ) H C O N F S Cl B I P

130 114 The compason between the expemental and the pedcted wate solvaton eneges fo the 376 small oganc compounds s shown n Fgue 91. The name, expemental and pedcted solvaton eneges ae shown n the Appendx. AVGB-SAS vs Expement 5 0 y = x R 2 = AVGB-SAS (Kcal/Mol) H/C O N F B, Cl, I S, P Expement (Kcal/Mol) Fgue 91. Compason between the expemental and pedcted wate solvaton eneges fo the AVGB- SAS solvaton model, usng solvaton types by element. The dffeent chemcal goups ae shown. The coelaton between expemental and pedcted values s qute good, R 2 = The mean unsgned eo s 1.23Kcal/Mol and the RMS devaton s 1.61Kcal/Mol. The slope of the lnea egesson s These esults show that the use of equaton (83) fo the shot-ange tem s a vald appoxmaton because the pedcted eneges coelate qute well. Howeve, the use of only element-dependent solvaton

131 115 types s pobably an ovesmplfcaton. The local envonment of each atom, as descbed by ts hybdzaton and valence bond connectvty cetanly plays a ole n the shotange solvaton enegy. Thus we wll attempt to edo the shot-ange tem paametezaton fo a moe complcated set of solvaton types. The solvaton types we used ae fom [95] and they ae shown, along wth the coespondng suface tensons n Table 3. The suface tensons wee deved n a smla fashon as befoe. The statng pont fo the lnea optmzaton pocedue was the suface tensons pe element that we aleady deved. Then, the lnea optmzaton was done sepaately fo each set of molecules belongng to the same chemcal goup. Each tme we examned a new chemcal goup, all the pevously detemned suface tensons emaned constant and only the new solvaton types that ae ntoduced by the new goup wee optmzed. Table 3. Solvaton types defntons and suface tenson values. Descpton Solvaton Type Suface Tenson H bonded to sp3 C H_MET H bonded to sp2 C H_BZN H bonded to sp1 C H_ACE H bonded to sp3 C bonded to OH goup H_ALC H bonded to sp3 N n pmay amne H_N H bonded to sp2 N bonded to sngle C H_N H bonded to sp2 N n seconday amne H_N H bonded to sp2 N bonded to two C H_N H bonded to S H_S sp3 C n a dol C_DOL Any C bonded to sp1 N C_N

132 116 Any C bonded to sp2 N C_N sp3 C bonded to sp3 N C_N Any othe C C_ sp3 O bonded to two C O_ Ntous O O_2N Cabonyl O n peptde bond O_PEP Caboxylate O O_2CM Pmay alcohol O o any othe O O_ sp3 N n tetay amne N_ sp2 N bonded to sp2 C and H N_ Ntous N N_O sp2 N n peptde bond N_PEP sp2 pmay N bonded to sp2 C bonded to thee sp2 N N_NNH Any othe N N_ F bonded to sp3 C bonded to thee F F_MET Any othe F F_ Any B B_ Any Cl Cl_ Any I I_ Any P P_ Any S S_ The compason between the expemental and pedcted solvaton eneges usng the suface tensons of Table 3 s shown n Fgue 92. The aveage unsgned eo s 0.72Kcal/Mol and the RMS devaton 0.98Kcal/Mol. The slope of the lnea egesson ft s 0.95 and the coelaton coeffcent s 0.9. Oveall, the qualty of the pedctons of AVGB-SAS wth the solvaton types of Table 3 s vey good. It s clea that the addtonal solvaton types mpove damatcally the pedctons of the AVGB-SAS model. Ths fact could pompt us to contnue efnng ou esults by addng moe solvaton types. Howeve, ths would mae the model moe dependent on fee paametes, whch s not desed. At the same tme, t s not clea by how much the

133 117 deved suface tensons ae applcable to molecules outsde the set we used fo optmzaton. We cetanly expect that these esults would be at least qualtatvely ght, but pedctng the expemental values fo all possble molecules s a athe futle goal. In any case, the method pesented n ths secton can be used wth any solvaton type set. AVGB-SAS vs Expement 5 0 y = x R 2 = AVGB-SAS (Kcal/Mol) H/C O N F B, Cl, I S, P Expement (Kcal/Mol) Fgue 92. Compason between the expemental and pedcted wate solvaton eneges fo the AVGB- SAS solvaton model, usng the solvaton types of Table 3. The dffeent chemcal goups ae shown. The suface tensons we deved ae stctly applcable fo solvaton n wate. In ode to use the AVGB-SAS model fo othe solvents one must epeat the optmzaton pocedue pesented hee. The only lmtaton s the need fo expemental solvaton enegy data fo a suffcently lage and dvese molecule set, fo the solvent of nteest.

134 118 Afte the atomc ad, the solvent pobe adus and the atomc chages ae detemned, the pola eneges can be calculated wth AVGB wth the espectve nteo/exteo delectc constants fo the new solvent. Then the pocedue fo calculatng the suface tensons s dentcal to the one we used fo wate. The qualty of the pedctons, as compaed to the expemental data, wll detemne f new solvaton types wll need to be defned. 4.3 Implementaton of the AVGB-SAS Solvaton Model The AVGB-SAS solvaton model was mplemented and poted nto two dffeent molecula smulaton pacages, MPSm [96] and Doc [97]. MPSm s a paallel molecula dynamcs pogam that uses the POSIX-theads multpocessng standad [98] fo paallelzaton. It s capable of pefomng molecula dynamcs and enegy mnmzaton fo vey lage systems unde vaous themodynamc ensembles. It can do catesan, tosonal o gd-body dynamcs and uses the cell-multpole method fo fast calculaton of the non-bond contbutons to the atomc eneges and foces [99]. Doc s a molecula database seachng pogam that ans lgands by the ablty to bnd n a specfed ste of a ecepto. Fo evey lgand, t geneates a numbe of confomatons nto the taget ste of the ecepto that ae scoed and aned by the bndng enegy. The pupose of Doc s to dentfy the lgands that bnd best to a specfc ecepto as pat of atonal dug desgn. The eason fo developng the AVGB-SAS model was to be able to ncopoate solvaton effects n an accuate and effcent way n such methods. The accuacy of the AVGB-SAS model was establshed n sectons 4.1 and 4.2. Hee we wll examne the pefomance of the model n these methods.

135 Paallel Molecula Dynamcs The AVGB-SAS model was mplemented n MPSm [96]. The solvaton enegy and the solvaton foce wee added nto the total enegy and foce of evey atom. MPSm s poted n thee dffeent platfoms/opeatng Systems: SGI/Ix, IBM/AIX, Intel/Lnux. The AVGB-SAS method was mplemented n all thee platfoms and the tmngs fo a 3401 atom poten (PDB code: 1mcp) fo each platfom ae shown n Fgue 93. The tests wee pefomed on an SGI Ogn R MHz, IBM Powe3-II 375MHz and Intel Pentum III 866MHz espectvely. Contbutons to Total Tmes Test system: 1mcp, 3401 atoms Tme (seconds) Enegy/Foces Bon Rad Aea/Volume Neghbo Seach 5 0 SGI-Ix IBM-AIX Intel-Lnux Platfom Fgue 93. Total tmes fo the AVGB-SAS model fo a system of 3401 atoms, fo dffeent platfoms. The contbutons of the dffeent pats of the calculaton ae shown.

136 120 Each enegy/foce calculaton ncludes the followng steps: neghbo seach fo each atom, dentfcaton of the tue neghbos and the aea and volume calculaton, Bon ad calculaton usng equatons (37) and (41) and pola solvaton enegy calculaton fom equaton (18). The shot-ange tem s easly calculated afte the aeas fo each atom have been detemned. It s clea that the bul of the CPU tme spent fo the calculaton s on the Bon ad and secondaly on the pola solvaton enegy/foce calculaton. Ths s 2 because of the ( N ) O natue of these calculatons whee N s the numbe of atoms n the molecula system. The aea and volume calculatons scale as O ( N ), as was shown n secton The oveall scalng of the AVGB-SAS method s shown n Fgue 94. Scalng of AVGB-SAS Tme (seconds) Enegy/Foces Bon Rad Aea/Volume Neghbo Seach (4pt) 1001 (6lyz) 1629 (2ptn) 3401 (1mcp) Numbe of atoms (system) Fgue 94. Scalng of the AVGB-SAS method as a functon of the sze (numbe of atoms) of the molecula system. The calculatons wee pefomed on an Intel Pentum III 866MHz.

137 121 The CPU tme equed fo the solvaton enegy calculaton of the same test system, usng numecal solutons to the PB equaton, depends on the softwae and the gd esoluton used. In any case, the mnmum tme spent was about a mnute fo Delph and about 8 mnutes fo UHBD whch s obvously vey neffcent fo use n molecula dynamcs whee a typcal un nvolves at least calculatons fo a 1000 steps. The CPU tme compason of AVGB to SGB s shown n Fgue 95. AVGB s 3-5 tmes faste than SGB, dependng on the platfom. Ths s a sgnfcant mpovement ove othe vesons of the Genealzed Bon method. AVGB vs SGB Computaton Tmes Test system: 1mcp, 3401 atoms 70 Tme (seconds) SGB AVGB 10 0 SGI-Ix IBM-AIX Intel-Lnux Platfom Fgue 95. Compason of CPU tme spent fo calculatng the solvaton enegy of 1mcp between SGB and AVGB, fo thee dffeent platfoms.

138 122 We can boost the pefomance of AVGB-SAS even moe by usng paallel computes. The AVGB-SAS model s staghtfowadly paallelzable due to the natue of the calculaton. All the atoms n the system ae unfomly dstbuted to all avalable pocessos. The solvent accessble suface and solvent excluded volume s calculated fo evey atom, ndependently of all othe atoms. Afte the aea and volume calculatons ae done, the Bon ad and pola solvaton eneges can agan be calculated ndependently fo each atom usng equatons (41) and (18). The fact that the aea, volume, Bon ad and pola solvaton enegy fo each atom ae calculated ndependently of all othe atoms means that the nfomaton necessay to be passed between dffeent pocessos s mnmal and ths maes the algothm natually paallel. The scalng of the CPU tme spent as a functon of the numbe of pocessos used s shown n Fgue 96. Ths test was pefomed on a 4-pocesso SMP (Symmetc Mult Pocesso) shaed memoy machne, fo the poten1mcp (3401 atoms). The scalng s vey good whch means that the ovehead due to pocesso communcaton s ndeed mnmal and the paallel mplementaton s vey effcent. Fo 4 pocessos we get a boost of about 3.6 (o 360%) n the computaton tme. The tme spent fo the calculaton of the same system usng SGB (whch s not paallelzable due to the global aea calculaton of the suface of the solute) s about 4 tmes moe than AVGB fo one pocesso and 14 fo 4 pocessos. The excellent paallel pefomance of AVGB and the smplcty of the paallel mplementaton open the way fo the smulaton, ncludng solvaton effects, of vey lage systems wth the use of massvely paallel computes. In contast, ths s vey dffcult to do fo methods that calculate numecally solutons of the PB equaton.

139 123 AVGB Paallel Scalng Test system: 1mcp, 3401 atoms SGB: seconds Tme (seconds) Enegy/Foces Bon Rad Aea/Volume Neghbo Seach Pocessos Fgue 96. Paallel scalng of AVGB-SAS on a shaed memoy symmetc 4-pocesso Intel Pentum III Xeon 550MHz, fo a 3401 atom poten (1mcp). Fom the beadown of the total CPU tme to the ndvdual contbutons of the dffeent pats of the AVGB-SAS calculaton, Fgue 93, t s clea that the most expensve pat of the calculaton s the computaton of the Bon ad. In molecula dynamcs, the calculaton s done n successve steps that usually coespond to a tme of 1fs. Afte evey step, the stuctue of the molecules changes vey lttle. Only afte a athe lage numbe of steps (usually of the ode of 100 o moe) does the stuctue change notceably. The exposed aeas, volumes, Bon ad and thus the pola solvaton enegy and shot-ange tem ae dependent exclusvely on the stuctue of the solute. We expect then that the Bon ad wll not change by any sgnfcant amount n a few steps and

140 124 accodngly the solvaton enegy. It s pobably a good appoxmaton then to update the Bon ad only evey few steps, thus savng a lot of computaton tme n the meantme. To test ths we un dynamcs fo a small poten fo 200 steps wth dffeent Bon ad update fequences: 1, 2, 5, 10, 20 and 50 steps. The solvaton enegy fo each un s shown n Fgue 97. Senstvty on Bon Rad Update Fequency AVGB-SAS Enegy (Kcal/Mol) step 2 steps 5 steps 10 steps 20 steps 50 steps Dynamcs Steps Fgue 97. Senstvty of the AVGB-SAS enegy wth the update fequency of the Bon ad. The test was pefomed on a small poten (4pt, 454 atoms) fo 200 steps. We notce that the dffeences n the solvaton enegy ae vey small, just a few Kcal/Mol, and the tajectoy of the dynamcs s stable dung the couse of the un, even when updatng the Bon ad evey 50 steps. By peodcally updatng the Bon ad we

141 125 gan sgnfcant speedup n the molecula dynamcs, whch ae necessay fo long tme smulatons Molecula Docng In molecula docng the goal s to an a lage numbe of lgands by the ablty to bnd n a specfed ste of a ecepto as pat of atonal dug desgn. The patcula confomatons of each lgand n the bndng ste ae scoed accodng to the bndng enegy. It s mpotant to nclude solvaton effects n such calculatons so that the selected lgands bnd best wth the ecepto n the natual aqueous envonment. The bndng enegy n solvent s calculated by usng the themodynamc cycle of Fgue 98. Fgue 98. Themodynamc cycle fo the calculaton of the bndng enegy of a ecepto-lgand complex n soluton.

142 126 Let us set as U L, R U, C U the ntenal enegy and L G, R G, C G the solvaton enegy of the lgand, ecepto and lgand-ecepto complex espectvely. State 1 n Fgue 98 coesponds to the lgand and ecepto n vacuum, sepaated by an nfnte dstance. State 2 s the complex n vacuum. State 3 s the lgand and ecepto n solvent, sepaated by an nfnte dstance, and state4 s the complex n the solvent. The enegy equed to get fom state1 to state 2 s: W C L R 1 2 = U U U (84) The enegy equed to get fom 2 to 4 s the solvaton enegy of the complex, W G C 2 4 = and the enegy equed to get fom 1 to 3 s the sum of the ndvdual solvaton eneges of the lgand and the ecepto, W L R 1 3 = G + G. The bndng enegy of the lgand-ecepto complex n wate, B.E., s then: B. E. = W = W = W 2 4 W 1 3 C C L L R R ( U + G ) ( U + G ) ( U + G ) (85) So, fo evey lgand confomaton geneated, to calculate the bndng enegy we must calculate the solvaton enegy of the lgand, the ecepto and the complex, along wth the ntenal eneges of each system. In ode to save CPU tme fo the calculaton, t s vey common n docng smulatons to assume the ecepto to be fxed n space and only the lgand to be flexble as we seach aound the ecepto s bndng ste to fnd an optmal confguaton. Ths s a easonable appoxmaton because of the dffeence n the szes of the two systems. A typcal ecepto has at least a few thousand atoms, whee a lgand not moe than a

143 127 hunded o so. Ths way, only the enegy of the lgand and the nteacton enegy between lgand and ecepto need to be calculated. The ntamolecula nteactons n the ecepto can be neglected snce they ae a constant fo all possble lgand confomatons. We can use ths fact to save addtonal CPU tme n the calculaton of the solvaton eneges. In patcula, fo most of the fxed atoms n the complex we need not calculate the aeas and volumes snce the ntesectng neghbos ae also fxed and thus the aea and volume that they buy do not change. The fxed (ecepto) atoms that need to be ecalculated ae the neghbong atoms of the movable (lgand) atoms. Also, an addtonal tmesavng appoxmaton that can be done s at the Bon adus calculaton: the Bon adus of an atom s dependent on the volumes of all othe atoms n the system. The contbuton of each atom to the Bon adus of atom falls appoxmately as the nvese of the dstance between the two atoms (see equaton (42)). Thus, fo atoms that ae fxed and fa away fom the bndng ste (whee the lgand s) the Bon adus should not change by much because of the pesence of the lgand. Fo these atoms we use the ntal Bon adus value, wthout the lgand s contbuton (ecepto only). The Bon ad ae updated only fo ecepto atoms close to the lgand. Wth these addtonal tmesavng schemes, we have been able to educe the solvaton bndng enegy of a atom complex fom 14s down to 2.5s, whch s a sgnfcant mpovement.

144 128 5 Applcatons of the AVGB-SAS Solvaton Model In chapte 4 we examned the pefomance of the AVGB-SAS model as fa as the qualty of ts pedctons and the CPU tme effcency. The AVGB model has many advantages wth espect to all pevous methods fo calculatng the electostatc solvaton effects. In patcula, the qualty of the esults s as good as fo numecal solutons of the PB equaton, but AVGB s odes of magntude faste than vaous mplementatons of the PB model. At the same tme the method s capable of calculatng foces, analytcally, whch s necessay fo the model to be useful n molecula dynamcs. Unle most othe flavos of the Genealzed Bon theoy, AVGB does not depend on any fttng paametes. In fact, we beleve that the eason that we acheve such good esults s ou ablty to calculate accuately the solute geomety, though the volume and aea calculatons descbed n chapte 3. Compang AVGB to SGB, the only othe GB method that does not depend on any ad-hoc paametezatons, AVGB s fou tmes faste and the esults ae bette. Also, the way the aea and volume calculatons ae done allow fo dstbuted computaton of the solvaton effects and as s shown n Fgue 96 the ovehead of the paallel calculaton s mnmal, thus achevng vey good scalng and excellent CPU tme pefomance. Fnally, n the pocess of calculatng the pola effects we have to calculate the solvent accessble aeas fo each atom, whch s needed fo the ncluson of shot-ange effects n the solvaton enegy. In the AVGB-SAS model, unle othe solvaton methods, the long-ange and shot-ange tems of the solvaton enegy and the devatves ae analytcally calculated at the same tme, whch esults n the excellent pefomance of the AVGB-SAS model.

145 129 At the same tme, we should be awae of the lmtatons of the AVGB-SAS model. As any mplct solvaton model t s unable to explctly tae nto account chage tansfe and hydogen bondng effects between solute and solvent. Also, although salt effects ae ncopoated nto the model n an aveage way, t s only n an appoxmaton and the esults ae expected to be vald only fo low salt concentatons. Fnally, ph effects ae not ncluded at all n ths model. Oveall, the qualty of the AVGB-SAS pedctons can be only as good as the delectc model s an applcable appoxmaton fo the system studed, whch s often the case. Keepng the afoementoned successes and lmtatons n mnd, we mplemented and ncopoated the AVGB-SAS solvaton model n MPSm [96], a paallel molecula dynamcs pogam, and Doc [97] a lgand database seachng pogam. In sectons 5.1 and 5.2 we wll apply the AVGB-SAS model n the smulaton of the dynamcs of nuclec acds and n vtual lgand sceenng (VLS). 5.1 B-DNA Molecula Dynamcs Nuclec acds play a fundamental ole n bologcal functons and the accuate smulaton s of pmay mpotance. Because of DNA s mpotance and complexty thee s a stong focus n the scentfc communty to smulate accuately ts dynamcs (see [100] fo a evew of molecula dynamcs smulatons of nuclec acds). The natual envonment of DNA s wate and DNA n soluton exhbts complex dynamcal behavo such as bendng and stetchng.

146 130 DNA has tadtonally been a dffcult system to smulate. It s a hghly chaged molecula system and long-ange nteactons have to be calculated accuately n ode to mantan a stable double-helx stuctue. Also, t s essental that we nclude the effects of the wate solvent n the smulaton f we want to accuately smulate the behavo of DNA n ts natual envonment. Usually the wate effects wee ncopoated by explctly ncludng wate molecules n the smulaton, wth geat expense n CPU tme. In ths secton we show that by ncludng solvaton effects usng the AVGB-SAS model, n a molecula dynamcs smulaton of DNA, we ae able to acheve a stable double-helx stuctue and obseve some dynamcal effects. Fo the smulatons we stated fom the canoncal fom of B-DNA [101] (Fgue 99) of the dodecame d(cgcatatatgcg) 2 and used the molecula dynamcs softwae pacage MPSm [96] wth the AMBER focefeld [102]. We pefomed 80 ps of dynamcs n vacuum and n solvent, usng AVGB-SAS fo the calculaton of eneges and foces fom the solvent to the DNA. We un the smulaton wth two dffeent bounday condtons: fee tps and fxed tps. The latte was done so we could pevent the double helx fom bendng, and allow us to compae the fnal stuctues to the canoncal B-DNA stuctue. The fnal stuctues fo the unconstant smulatons ae shown n Fgue 100 n vacuum and Fgue 101 n solvent. We notce that when solvent s ncluded, the double helx s bended, whch s a phenomenon that has been obseved expementally [103]. In contast, the vacuum smulaton does not clealy show such effect.

147 131 Fgue 99. The ntal stuctue: canoncal B-DNA. Fgue 100. B-DNA afte 80ps n vacuum wth fee tps. Fgue 101. B-DNA afte 80ps n mplct solvent wth fee tps. Fgue 102. B-DNA afte 80ps n vacuum wth fxed tps. Fgue 103. B-DNA afte 80ps n mplct solvent wth fxed tps.