Canonical dual theory applied to a Lennard-Jones potential minimization problem

Canonical dual theory app lied to a L ennard-Jones p oten tial min imiza- tion problem Jiapu Z hang Cen tre for Informatics and Applied Optimization, & Graduate Sc ho ol of Sciences, Information T ec hnology and Engineering, The Univ ersit y of Balla rat, Mount Helen, VIC 33 50, Australia. Emails: j.zhang@ballarat.edu.au, jiapu zhang@hotmail.com T elephones: 61-3-5327 9809 (office), 61-4 2348 7360 (mobile) Abstract The simplified Lennard-Jones (LJ) p oten tial minimization problem is minimize f ( x ) = 4 N X i =1 N X j =1 ,j 0 } . Let Ξ( x, ς ) ′ = 0, we get three critical p oin ts of Ξ ( x, ς ): ( ¯ x 1 , ¯ ς 1 ) = (2 . 11491 , 0 . 236417) , ( ¯ x 2 , ¯ ς 2 ) = ( − 1 . 86081 , − 0 . 26870 1) , ( ¯ x 3 , ¯ ς 3 ) = 4 -4 -2 2 4 -4 -2 2 4 6 Figure 2: The prime and du al double-we ll fun ctio ns (Prime: b lue, Dual: red). ( − 0 . 2541 02 , − 1 . 96772). By Theorem 1, w e kno w ¯ x 1 = 2 . 11491 is th e global mini- mizer of (13), ¯ ς 1 = 0 . 236 417 is th e global maximizer of (14) o v er S + a , and P ( ¯ ς 1 ) = Ξ( ¯ x 1 , ¯ ς 1 ) = P d ( ¯ ς 1 ) = − 1 . 02951. By Theorem 2, w e kno w that the lo cal minimizers: ¯ x 2 = − 1 . 86081 , ¯ ς 2 = − 0 . 268701 (o ver S − a ), P ( ¯ ς 2 ) = Ξ ( ¯ x 2 , ¯ ς 2 ) = P d ( ¯ ς 2 ) = 0 . 9665031 and the lo cal maximizers: ¯ x 3 = − 0 . 254 102 , ¯ ς 3 ) = − 1 . 96772 (o v er S − a ), P ( ¯ ς 3 ) = Ξ( ¯ x 3 , ¯ ς 3 ) = P d ( ¯ ς 3 ) = 2 . 063. 3 Applications to a L-J p oten tial optimization problem In 20 07, Sa w a ya et al. got a breakthrough fin ding: the atomic structures of all am yloid fibrils reve aled steric zipp ers, with strong vdw interact ions (LJ) b et w een β -sh eets and h ydrogen b ond s (HBs) to main tain the β -strands [1 2]. Similarly as (1), i.e. the p oten tial energy for the vdw interactio ns (Fig. 1 ) b et w een β -sheets: V LJ ( r ) = A r 12 − B r 6 , (15) the p oten tial en er gy for the HBs b et w een the β -strands has a similar form ula V H B ( r ) = C r 12 − D r 10 , (16) where A, B , C, D are constan ts giv en. Thus, the amyloi d fi bril molecular mo del build- ing p roblem is reduced to w ell solv e the optimization problem (3) or (6) (in this pap er w e ap p ly the CDT introdu ced in Section 2 to solv e (6)). In this section, w e will use suitable templates 3nvf.pd b, 3nvg.pd b and 3n vh.p db from the Protein Data Ba nk (h ttp://www.rcsb.org/) to build some amyl oid fibril mod - els. 5 3.1 3NVF Constructions of the A GAAAA GA am yloid fibr il m olecular str uctures of prion 113– 120 region are based on the most r ece n tly r elea sed exp erimental molecular structur es of I IHF GS segmen t 13 8–143 from human prion (PDB en try 3NV F released into Prote in Data Bank (http:// www.rcsb.org) on 2011-03 -02) [1 ]. The atomic-resolution structure of th is p eptide is a steric zipp er, with strong vdw in teractions b etw een β -sheets and HBs to main tain the β -strands (Fig. 3). In Fig. 3 w e see that H chain (i.e. β -sheet 2) Figure 3: Protein fibril structure of I IHF GS segmen t 138–143 from h uman prion. The purp le d ashed lines denote the hydrogen b onds. A, B, ..., I, J denote the 10 chains of the fibrils. of 3NVF.p db can b e obtained from A c h ain (i.e. β -sheet 1) b y H =   − 1 0 0 0 − 1 0 0 0 1   A +   27 . 546 0 0   , (17) and other c hains can b e got b y C ( G ) = A ( H ) +   0 0 4 . 8   , B ( F ) = A ( H ) + 2   0 0 4 . 8   , (18) 6 D ( I ) = A ( H ) −   0 0 4 . 8   , E ( J ) = A ( H ) − 2   0 0 4 . 8   . (19) Basing on the template 3NVF.pd b from the Protein Data Bank, three pr ion A GAAAA GA palindrome am yloid fibril mo dels - an AGAA AA mo del (3n vf-Mod el 1), a GAAAA G mo del (3nvf-Model 2), and an AAAA GA mo del (3n vf-Mod el 3) - will b e successfully constructed in th is pap er. Chain A of 3nvf-M o dels 1-3 were got from A Chain of 3NVF.p db us in g the muta te mo dule of the free pack age S wiss-PdbView er (SPDBV V ersion 4.01) ( ht tp://sp db v.vital-it.ch ). It is pleasant to see that almost all th e hydro- gen b onds are still kept after the m utations; thus we just need to consider th e vd w con tacts only . Making m utations f or H Ch ain of 3NVF.p db, we can get H C hain of 3nvf-Models 1-3. Ho w ev er, w e find that the vdw con tacts b et wee n A Chain and H Ch ain are to o far at this moment. W e kno w that for 3n vf-Mo del 1 at least the vdw in teract ion b et w een A.GL Y2.CA-H.GL Y2.CA, A.ALA4.CB-H.GL Y2.CA should b e maintained, for 3nvf-M o del 2 at least th ree vdw inte ractions b et w een A.ALA4.CB- H.ALA2.CB, A.ALA2.CB-H.ALA2 .CB, A.ALA2 .CB-H.ALA4.CB should b e main tained, and for 3nvf-Model 3 at least three vdw in teractions b et ween A.ALA2.CB-H.ALA2.CB, A.ALA2.CB-H.ALA4 .CB, A.ALA4.CB-H.ALA2.CB should b e maintained. Fixing the co ordinates of A.GL Y2.CA and A.ALA4.CB (tw o anc hors) ((-10.919, -3.862 ,-1.487), (6.357 ,1.461 ,-1.905)) for 3nvf-Model 1, fixing the co ordinates of A.ALA2.CB and A.ALA4.CB (t wo anc hors) ((11.959 ,-2.844 ,-1.977), (6.357 ,1.461 ,-1.905)) for 3nvf-Models 2-3, letting d equal to the t wice of the vdw radius of Carb on atom (i.e. d = 3 . 4 angstroms), and let- ting the coord inates of H.GL Y2.CA of 3nvf-Model 1 (t w o sensors) and the co ordinates of H.ALA2.CB and H.ALA4.CB of 3n v f -Models 2-3 (t wo sensors) b e v ariables, we may get a simple MDGP with 3/6 v ariables and its dual with 2/3 v ariables for 3n vf-Mod el 1: P ǫ ( x 1 ) = 1 2  ( x 11 + 10 . 91 9) 2 + ( x 12 + 3 . 862) 2 + ( x 13 + 1 . 487 ) 2 − 3 . 4 2  2 + 1 2  ( x 11 − 6 . 357 ) 2 + ( x 12 − 1 . 461 ) 2 + ( x 13 + 1 . 905 ) 2 − 3 . 4 2  2 − (0 . 05 x 11 + 0 . 05 x 12 + 0 . 05 x 13 ) , P d ǫ ( ς 1 , ς 2 ) = 124 . 79 08 ς 1 − 1 2 ς 2 1 + 34 . 615 ς 2 − 1 2 ς 2 2 − 1 2   0 . 05 − 21 . 83 8 ς 1 + 12 . 71 4 ς 2 0 . 05 − 7 . 724 ς 1 + 2 . 922 ς 2 0 . 05 − 2 . 974 ς 1 − 3 . 81 ς 2   T   1 2 ς 1 +2 ς 2 0 0 0 1 2 ς 1 +2 ς 2 0 0 0 1 2 ς 1 +2 ς 2     0 . 05 − 21 . 83 8 ς 1 + 12 . 71 4 ς 2 0 . 05 − 7 . 724 ς 1 + 2 . 922 ς 2 0 . 05 − 2 . 974 ς 1 − 3 . 81 ς 2   . W e can get a global maximal solution (70.1 836,70 .1812) for P d ǫ ( ς 1 , ς 2 ) and its corre- sp onding lo cal maximal solution to P ǫ ( x 1 ): ¯ x = ( − 2 . 28 097 , − 1 . 20037 , − 1 . 69582) . 7 By Theorem 1 w e kn o w that ¯ x is a global minimal solution of P ǫ ( x 1 ). Th us we get H =   − 1 0 0 0 − 1 0 0 0 1   A +   − 4 . 5619 − 2 . 4009 0 . 0004   (20) for 3nvf-Model 1, whose other c hains can b e got b y (18)-(19) (Fig. 4). F or 3n vf-Mo d els 2-3, similarly we may get a simp le MDGP with 6 v ariables and its dual with 3 v ariables: P ǫ ( x 1 , x 2 ) = 1 2  ( x 11 − 11 . 95 9) 2 + ( x 12 + 2 . 844 ) 2 + ( x 13 + 1 . 977 ) 2 − 3 . 4 2  2 + 1 2  ( x 21 − 11 . 95 9) 2 + ( x 22 + 2 . 844 ) 2 + ( x 23 + 1 . 977 ) 2 − 3 . 4 2  2 + 1 2  ( x 11 − 6 . 357 ) 2 + ( x 12 − 1 . 461 ) 2 + ( x 13 + 1 . 905 ) 2 − 3 . 4 2  2 − (0 . 05 x 11 + 0 . 05 x 12 + 0 . 05 x 13 + 0 . 05 x 21 + 0 . 05 x 22 + 0 . 05 x 23 ) , P d ǫ ( ς 1 , ς 2 , ς 3 ) = 143 . 4545 ς 1 − 1 2 ς 2 1 + 143 . 4 545 ς 2 − 1 2 ς 2 2 + 34 . 61 50 ς 3 − 1 2 ς 2 3 − 1 2         0 . 05 + 23 . 91 80 ς 1 + 12 . 71 40 ς 3 0 . 05 − 5 . 688 0 ς 1 + 2 . 922 0 ς 3 0 . 05 − 3 . 954 0 ς 1 − 3 . 810 0 ς 3 0 . 05 + 23 . 9180 ς 2 0 . 05 − 5 . 688 0 ς 2 0 . 05 − 3 . 954 0 ς 2         T          1 2 ς 1 +2 ς 3 0 0 0 0 0 0 1 2 ς 1 +2 ς 3 0 0 0 0 0 0 1 2 ς 1 +2 ς 3 0 0 0 0 0 0 1 2 ς 2 0 0 0 0 0 0 1 2 ς 2 0 0 0 0 0 0 1 2 ς 2                  0 . 05 + 23 . 91 80 ς 1 + 12 . 71 40 ς 3 0 . 05 − 5 . 688 0 ς 1 + 2 . 922 0 ς 3 0 . 05 − 3 . 954 0 ς 1 − 3 . 810 0 ς 3 0 . 05 + 23 . 91 80 ς 2 0 . 05 − 5 . 688 0 ς 2 0 . 05 − 3 . 954 0 ς 2         . W e can get a global maximal solution (0.920088,0 .0127 286,0.921273) for P d ǫ ( ς 1 , ς 2 , ς 3 ) and its corresp onding lo cal maximal solution to P ǫ ( x 1 , x 2 ): ¯ x = (9 . 1 6977 , − 0 . 676538 , − 1 . 9274 , 13 . 9231 , − 0 . 879925 , − 0 . 0129248) . By Theorem 1 w e kn o w that ¯ x is a global minimal solution of P ǫ ( x 1 , x 2 ). Th us we get H =   − 1 0 0 0 − 1 0 0 0 1   A +   20 . 845 9 − 2 . 1533 0 . 6638   . (21) for 3nvg-Models 2-3, whose other c hains can b e got by (18)-(19) (Fig. 4). W e fin d 3nvf-Model 1 has some atoms w ith bad/close con tacts, 3nvf-Model 2 h as 6 b ad/clo se con tacts, and 3nvf-Model 3 has no bad/close con tact. This m eans it not necessary at all to fur ther refin e 3n vf-Mo del 3. W e remo v e these b ad conta cts b y p er- forming energy minimization using Am b er 11 [2]. Ev en if there are no obvious bad 8 Figure 4: Protein fibril structure of 3nvf-Models 1-3 (from left to righ t resp ectiv ely) for prion AG AAAA GA segmen t 113-120 . The pur ple dashed lines d enote the h ydrogen b onds. A, B, ..., I, J den ote the 10 c h ains of the fi b rils. con tacts, it is still a go o d idea to ru n a short energy m in imizati on to r elax the str u c- tures a bit. W e will p erf orm the energy minim ization in 2 stages. In the first stage, w e’ll only m inimize the w ater molecules and hold the protein fixed for 500 steps of steep est descen t metho d and then 500 steps of conju gate gradien t metho d. Our goal is just to r emo ve b ad cont acts, there is no need to go o verb oard with min imizat ion. In the second stage we pro ceed directly to min imizing the en tire system as a whole for 1500 steps of steep est descent m ethod and then 1000 steps of conju gate gradien t metho d. RMSD (ro ot mean s q u are deviation) is an ind icat or for stru ctural c hanges in a protein. It is us ed to measure the scalar spatial d istance b et w een atoms of the same t y p e (for example the C α atoms) f or tw o str u ctures in differen t time. The RMS Ds b et ween the last snapshot after the r efinemen t and the s n apshot illuminated in Fig. 4 are 2.157 96, 1.3089 087, 1.04531 8 angstroms for these three 3n vf-Models resp ective ly . The v ery small v alues of RMSD are very go o d measure of precision of CDT for our mo del building. T h is shows us that CDT p erforms w ell. 3.2 3NV G In this su bsection, the constructions of th e AG AAAA GA am yloid fi bril molecular structures of prion 113–1 20 region are based on the most recent ly released exp er i- men tal molecular str uctures of MIHF GN segment 137–142 fr om m ou s e pr ion (PDB en try 3NV G released into Protein Data Bank (http:/ /www.rcsb.org) on 2011-03-02 ) [1]. The atomic-resolutio n structur e of this p eptide is a steric zipp er, with s trong vdw in teractio ns b et wee n β -sheets and HBs to main tain th e β -strands (Fig. 5). In Fig. 5 w e s ee that H Chain (i.e. β -sh eet 2) of 3NV G.p db can b e obtained from A Ch ain (i.e. β -sheet 1) b y H =   − 1 0 0 0 1 0 0 0 − 1   A +   − 27 . 28 2 . 385 15 . 738   , (22) 9 Figure 5: Protein fib ril structure of MIHFG N segmen t 137–14 2 from mouse pr ion. The purp le d ashed lines denote the hydrogen b onds. A, B, ..., I, J denote the 10 chains of the fibril. and other c hains can b e got b y C ( G ) = A ( H ) +   0 4 . 77 0   , B ( F ) = A ( H ) + 2   0 4 . 77 0   , (23) D ( I ) = A ( H ) −   0 4 . 77 0   , E ( J ) = A ( H ) − 2   0 4 . 77 0   . (24) Basing on the template 3NV G.p db fr om the Protein Data Bank, three prion A GAAAA GA palindrome am yloid fib ril mo dels - an AGAAA A mo del (3n vg-Mod el 1), a GAAAA G mo del (3n vg-Mod el 2), and an AAAA GA mo del (3nvg-M o del 3) - will b e su cce ssfully constructed in th is pap er. Ch ain A of 3nvg- Mo dels 1-3 were got from A Chain of 3NV G.p db using the mutate mo dule of the free pac k age Sw iss-PdbView er (S PDBV V ersion 4.01) ( h ttp://sp db v.vital-it.ch ). It is p leasa n t to see that almost all the hydrogen b onds are still k ept after the mutatio ns; thus we just need to consider th e vdw conta cts 10 only . Making muta tions for H Chain of 3NV G.p d b , we can get the H Chains of 3n vg- Mo dels 1-3. Ho wev er, the vd w con tacts b et w een A Chain and H C h ain are to o far at this momen t. W e know that f or 3nvg- Mo del 1 at least the thr ee vdw int eraction b e- t ween A.GL Y2.CA-H.GL Y2.CA, A.GL Y2.CA-H .ALA4.CB, A.ALA4.CB-H.G L Y2.CA should b e mainta ined, for 3nvg-Model 2 at least the three vdw inte ractions b et w een A.ALA2.CB-H.ALA2 .CB, A.ALA2.CB-H.ALA4.CB, A.ALA4.CB-H.ALA2.CB should b e main tained, and for 3nvg -Mo d el 3 at least th e three vdw in teractio ns b et w een A.ALA2.CB-H.ALA2 .CB, A.ALA2.CB-H.ALA4.CB, A.ALA4.CB-H.ALA2.CB should b e maint ained. Fixing the co ordinates of A.GL Y2.CA and A.ALA4.CB (t w o an- c h ors) ((-11.15 9,-2.24 1,4.126), (-5.865 ,-2.618 ,8.696)) for 3n vg-Mo del 1, fixing the co ordi- nates of A.ALA2.CB an d A.ALA4.CB (t w o anchors) ((-12.04 0,-2.6 75,5.307), (-5.86 5,- 2.618, 8.696) ) f or 3n vg-Models 1-2, letting d equal to the twice of the vdw radius of Carb on atom (i.e. d = 3 . 4 angstroms), and letting the co ordinates of H.GL Y2.CA and H.ALA4.CB of 3nvg- Mo del 1 (t wo sensors ) and the co ord inates of H.ALA2.CB and H.ALA4.CB of 3nvg-Models 2-3 (tw o s en sors) b e v ariables, w e may get a sim p le MDGP with 6 v ariables and its d ual with 3 v ariables for 3n vg-Model 1: P ǫ ( x 1 , x 2 ) = 1 2  ( x 11 + 11 . 15 9) 2 + ( x 12 + 2 . 241 ) 2 + ( x 13 − 4 . 126 ) 2 − 3 . 4 2  2 + 1 2  ( x 21 + 11 . 15 9) 2 + ( x 22 + 2 . 241 ) 2 + ( x 23 − 4 . 126 ) 2 − 3 . 4 2  2 + 1 2  ( x 11 + 5 . 865 ) 2 + ( x 12 + 2 . 618 ) 2 + ( x 13 − 8 . 696 ) 2 − 3 . 4 2  2 − (0 . 05 x 11 + 0 . 05 x 12 + 0 . 05 x 13 + 0 . 05 x 21 + 0 . 05 x 22 + 0 . 05 x 23 ) , P d ǫ ( ς 1 , ς 2 , ς 3 ) = 135 . 009238 ς 1 − 1 2 ς 2 1 + 135 . 0 09238 ς 2 − 1 2 ς 2 2 + 105 . 3 125 ς 3 − 1 2 ς 2 3 − 1 2         0 . 05 − 22 . 31 8 ς 1 − 11 . 73 ς 3 0 . 05 − 4 . 482 ς 1 − 5 . 236 ς 3 0 . 05 + 8 . 252 ς 1 + 17 . 39 2 ς 3 0 . 05 − 22 . 31 8 ς 2 0 . 05 − 4 . 482 ς 2 0 . 05 + 8 . 252 ς 2         T          1 2 ς 1 +2 ς 3 0 0 0 0 0 0 1 2 ς 1 +2 ς 3 0 0 0 0 0 0 1 2 ς 1 +2 ς 3 0 0 0 0 0 0 1 2 ς 2 0 0 0 0 0 0 1 2 ς 2 0 0 0 0 0 0 1 2 ς 2                  0 . 05 − 22 . 31 8 ς 1 − 11 . 73 ς 3 0 . 05 − 4 . 482 ς 1 − 5 . 236 ς 3 0 . 05 + 8 . 252 ς 1 + 17 . 39 2 ς 3 0 . 05 − 22 . 318 ς 2 0 . 05 − 4 . 482 ς 2 0 . 05 + 8 . 252 ς 2         . W e can get a global maximal solution (0.708403,0 .0127 287,0.699001) for P d ǫ ( ς 1 , ς 2 , ς 3 ) and its corresp onding lo cal maximal solution to P ǫ ( x 1 , x 2 ): ¯ x = ( − 8 . 51 192 , − 2 . 41048 , 6 . 4135 , − 9 . 19493 , − 0 . 276929 , 6 . 09007) . By Theorem 1 w e kn o w that ¯ x is a global minimal solution of P ǫ ( x 1 , x 2 ). Th us we get H =   − 1 0 0 0 1 0 0 0 − 1   A +   − 18 . 133 923 0 . 6673 703 11 . 955 023   (25) 11 for 3n vg-Mo del 1, whose ot her c h ains can b e g ot b y (2 3)-(24) (Fig . 6). F or 3nvg- Mo dels 2-3, similarly we may get a simp le MDGP with 6 v ariables and its dual with 3 v ariables: P ǫ ( x 1 , x 2 ) = 1 2  ( x 11 + 12 . 04 0) 2 + ( x 12 + 2 . 675 ) 2 + ( x 13 − 5 . 307 ) 2 − 3 . 4 2  2 + 1 2  ( x 21 + 12 . 04 0) 2 + ( x 22 + 2 . 675 ) 2 + ( x 23 − 5 . 307 ) 2 − 3 . 4 2  2 + 1 2  ( x 11 + 5 . 865 ) 2 + ( x 12 + 2 . 618 ) 2 + ( x 13 − 8 . 696 ) 2 − 3 . 4 2  2 − (0 . 05 x 11 + 0 . 05 x 12 + 0 . 05 x 13 + 0 . 05 x 21 + 0 . 05 x 22 + 0 . 05 x 23 ) , P d ǫ ( ς 1 , ς 2 , ς 3 ) = 168 . 7214 74 ς 1 − 1 2 ς 2 1 + 168 . 7 21474 ς 2 − 1 2 ς 2 2 + 105 . 3 12565 ς 3 − 1 2 ς 2 3 − 1 2         0 . 05 − 24 . 08 0 ς 1 − 11 . 73 ς 3 0 . 05 − 5 . 35 ς 1 − 5 . 236 ς 3 0 . 05 + 10 . 61 4 ς 1 + 17 . 39 2 ς 3 0 . 05 − 24 . 08 0 ς 2 0 . 05 − 5 . 35 ς 2 0 . 05 + 10 . 61 4 ς 2         T          1 2 ς 1 +2 ς 3 0 0 0 0 0 0 1 2 ς 1 +2 ς 3 0 0 0 0 0 0 1 2 ς 1 +2 ς 3 0 0 0 0 0 0 1 2 ς 2 0 0 0 0 0 0 1 2 ς 2 0 0 0 0 0 0 1 2 ς 2                  0 . 05 − 24 . 08 0 ς 1 − 11 . 73 ς 3 0 . 05 − 5 . 35 ς 1 − 5 . 236 ς 3 0 . 05 + 10 . 614 ς 1 + 17 . 39 2 ς 3 0 . 05 − 24 . 080 ς 2 0 . 05 − 5 . 35 ς 2 0 . 05 + 10 . 614 ς 2         . W e can get a global maximal solution (0.8497 35,0.01 27287,0.84036) for P d ǫ ( ς 1 , ς 2 , ς 3 ) and its corresp onding lo cal maximal solution to P ǫ ( x 1 , x 2 ): ¯ x = ( − 8 . 95 484 , − 2 . 63187 , 7 . 00689 , − 1 0 . 0759 , − 0 . 710929 , 7 . 27107) . By Theorem 1 w e kn o w that ¯ x is a global minimal solution of P ǫ ( x 1 , x 2 ). Th us we get H =   − 1 0 0 0 1 0 0 0 − 1   A +   − 19 . 310 2 0 . 6644 13 . 531 6   . (26) for 3nvg-Models 2-3, whose other c hains can b e got by (23)-(24) (Fig. 6). W e did same r efinemen ts for 3n vg-Models 1-3 as for 3n vf-Mo dels 1-3. The RMSDs b et ween the last snapshot after the r efinemen t and the s n apshot illuminated in Fig. 6 are 1.572438 , 1.404648, 1.46476 7 angstroms for these three 3n vg-Models resp ectiv ely . The v ery small v alues of RMSD sh o w us that CDT p erforms we ll and pr ecisel y for 3n vg-Model building. 3.3 3NVH Similar as the a b o v e tw o sub sectio ns, th is subsection constructs the A GAAAA GA amy- loid fib ril molecular stru ctures of p r ion 113–1 20 region basing on the most recen tly re- leased exp eriment al molecular structures of MIHFG ND segmen t 137–143 f r om mouse 12 Figure 6: Protein fibr il s tructure of 3n vg-Mo dels 1-3 (fr om left to righ t r esp ectiv ely) for prion AG AAAA GA segmen t 113-120 . The pur ple dashed lines d enote the h ydrogen b onds. A, B, ..., I, J den ote the 10 c h ains of the fi b rils. prion (PDB en try 3NVH r eleased into Pr otei n Data Bank (h ttp://www.rcsb.org) on 2011- 03-02) [1]. The atomic-resolution structure of this p eptide is a steric zipp er, with strong vd w in teractions b et w een β -sh eets and HBs to mainta in th e β -strands (Fig. 7 ). In Fig. 7 we see that H c hain (i.e. β -sheet 2) of 3NVH.p db can b e obtained f r om A c h ain (i.e. β -sheet 1) by H =   − 1 0 0 0 1 0 0 0 − 1   A +   0 2 . 437 − 15 . 553   , (27) and other c hains can b e got b y C ( G ) = A ( H ) +   0 4 . 87 0   , B ( F ) = A ( H ) + 2   0 4 . 87 0   , (28) D ( I ) = A ( H ) −   0 4 . 87 0   , E ( J ) = A ( H ) − 2   0 4 . 87 0   . (29) Basing on the template 3NVH.pd b from the Protein Data Bank, three prion A GAAAA GA palindrome am yloid fibril mo dels - an AG AAAA G mod el (3n vh-Mo del 1), a GAAAA GA mo del (3n vh-Mo del 2) - will b e su ccessfully constructed in this pap er. A chain of 3nvh- Mo dels 1-2 were got from A chain of 3NVH.p db u sing the mutate mo dule of th e free pac k age Swiss-Pdb View er (SPDBV V ersion 4.01) ( http://spdbv.vital-it.ch ). I t is pleas- an t to s ee that almost all th e hydrogen b onds are still k ept after the muta tions; thus w e ju st need to consider the vdw con tacts only . Making m utations for H c hain of 3NVH.p db, w e can get the H chains of 3nvh-Models 1-2. Ho wev er, the vd w contac ts b et ween A c hain an d H c h ain are to o far at this moment ( ≥ 4.25 Angstroms). w e ma y kno w that for 3n vh-Mo dels 1-2 at least one vdw in teract ion b et ween A.ALA4 .CB- H.ALA4.CB shou ld b e mainta ined. Fixing the co ordinates of A.ALA4.CB (the anc hor) 13 Figure 7: Protein fibril structure of MIHF GND segmen t 137–14 3 from mouse p r ion. The p urple d ashed lines denote the h ydrogen bon d s. A, B, ..., I , J denote the 10 c hains of the fi bril. ((1.73 1,-1.51 4,-7.980)), letting d equal to th e t wice of the v d w radius of Carb on atom (i.e. d = 3 . 4 angstroms), and letting the co ordinate of H.ALA4.CB (one sensor) b e v ariables, we ma y get a simple MDGP w ith 3 v ariables and its d ual with 1 v ariable: P ǫ ( x 1 ) = 1 2  ( x 11 − 1 . 731 ) 2 + ( x 12 + 1 . 514) 2 + ( x 13 + 7 . 980 ) 2 − 3 . 4 2  2 − 0 . 05 x 11 − 0 . 05 x 12 − 0 . 05 x 13 , P d ǫ ( ς 1 ) = 57 . 409 ς 1 − 1 2 ς 2 1 (0 . 05 + 3 . 46 2 ς 1 ) 2 + (0 . 05 − 3 . 028 ς 1 ) 2 + (0 . 05 − 15 . 96 ς 1 ) 2 4 ς 1 . W e can easily get the global maximal solution 0 . 012728 7 ∈ { ς ∈ R 1 | ς i > 0 , i = 1 } for P d ǫ ( ς 1 ). Then, we get its corresp onding solution for P ǫ ( x 1 ): ¯ x = (3 . 6 9507 , 0 . 450071 , − 6 . 01593) . By Theorem 1 w e know th at ¯ x is a global minimal solution of P ǫ ( x 1 ), i.e. for H.ALA4.CB. 14 Th us w e get H =   − 1 0 0 0 1 0 0 0 − 1   A +   5 . 4260 7 1 . 9640 71 − 13 . 995 93   (30) for 3nvh-Mod els 1-2, wh ose other c hains can b e got by (28)-(29) (Fig. 8). Figure 8: Protein fibril structure of 3n vh -Mo dels 1-2 (fr om left to right resp ecti v ely) for prion AG AAAA GA segmen t 113-120 . The pur ple dashed lines d enote the h ydrogen b onds. A, B, ..., I, J, K, L denote the 12 c h ains of the fi brils. W e carried on the same refinements for 3n vh-Mo dels 1-2 as for 3n vf-Mod els 1-3 and 3nvg-Models 1-3. T he RMSDs b et w een the last s napshot after the refinement and th e s n apshot illuminated in Fig. 8 are 1.534417, 1.57 2836 angstroms for th e tw o 3n vh-Mod els resp ectiv ely . The ve ry small v alues of RMSD again show to us that CDT p erforms well and pr ecisel y for 3n vg-Model building. 3.4 Refined 3n vf-Mo dels 1-3, 3n vg-Mo dels 1-3, 3n vh-Models 1-2 The amyloid fib ril mo dels of prion AGAA AA GA segmen t refined b y Am b er 11 are illu- minated in Fig.s 9 -11. All these m o dels are w ithout any b ad conta ct n o w (chec k ed by pac k age Sw iss-PdbView er), and th e v d w inte ractions b et w een the t w o β -sheets are in a v ery p erfect w a y now. All the initial stru ctures b efore dealt b y CDT approac h ha v e v ery f ar vdw con tact s b et wee n the t w o β -sheets. CDT easily made the vd w conta cts come closer and reac h a state with the low est p oten tial energy , w h ic h has p erf ect vdw con tacts as shown in Fig.s 9-11. 4 Conclusion Global optimization of Lennard -Jones clusters is a challe nging problem f or researchers in the fi eld of b iolo gy , physics, c h emistry , computer science, m aterials science, and es- 15 Figure 9: P erfect 3n vf-Mo dels 1-3 (from left to r igh t resp ectiv ely) for prion A GAAAA GA segment 113-12 0 . Th e pur ple dashed lines denote the hydrogen b ond s. A, B, ..., I , J denote the 10 c hains of the fibrils. Figure 10: P erfect 3n vg-Models 1-3 (fr om left to righ t resp ectiv ely) for pr ion A GAAAA GA segment 113-12 0 . Th e pur ple dashed lines denote the hydrogen b ond s. A, B, ..., I , J denote the 10 c hains of the fibrils. Figure 11: P erfect 3nvh-Mod els 1-2 (from left to righ t resp ectiv ely) for prion A GAAAA GA segment 113-12 0 . Th e pur ple dashed lines denote the hydrogen b ond s. A, B, ..., I , J, K, L denote the 12 c hains of the fibrils. 16 p ecially for experts in mathematical optimizatio n researc h field b ecause of the noncon- v exit y of the L-J p oten tial en er gy function and enorm ou s lo cal minima on the p otent ial energy sur face. In Marc h 2008, American Mathematical Programming So ciet y sp e- cially prod uced one whole issue, No. 76, to discuss this problem. In this pap er through clev er u se of global optimization techniques of Gao’s canonical du al theory (CDT), we successfully tac kle this challengi ng pr oblem illuminated by the amylo id fi bril m olec u- lar mo del build ing. Clearly , this pap er sho ws to readers that C DT is v ery usefu l and p o w erf u l to tac kle challe nging pr oblems in optimization area and man y other areas. Ac knowledgmen ts: This researc h w as supp orted b y US Air F orce Office of Scien- tific Researc h und er the gran t AF OSR F A9 550-1 0-1-04 87, b y a Victorian Life Sciences Computation Initiativ e (VLSCI) gran t n um b er VR00 63 on its P eak Computing F acilit y at th e Univ ersit y of Melb ourne, an initiativ e of th e Victorian Go vernmen t, and by the Head and Colleagues of Gradu ate School of Ballarat Univ ersit y . References [1] Ap ostol, M.I., Wiltzius, J.J., Sa w a ya, M.R., Cascio, D., E isenb erg, D.: A tomic structures suggest determinant s of transmission barriers in mammalian pr ion dis- ease. 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