EFFECTS OF SECONDARY STRUCTURE ON ELASTIC BEHAVIOR OF PROTEIN-LIKE CHAINS *

Chns Journal of Polymr Scnc Vol. 27, No. 2, (2009), 253 265 Chns Journal of Polymr Scnc 2009 World Scntfc EFFECTS OF SECONDARY STRUCTURE ON ELASTIC BEHAVIOR OF PROTEIN-LIKE CHAINS * Tng-tng Sun a**, Ha-zhu Ma a, Zhou-tng Jang b and Yu Shn c a Collg of Informaton and Elctronc Engnrng, Zhjang Gongshang Unvrsty, Hangzhou 310018, Chna b Dpartmnt of Physcs, Chna Jlang Unvrsty, Hangzhou 310018, Chna c Dpartmnt of Physcs, Zhjang Unvrsty of Scnc and Tchnology, Hangzhou 310023, Chna Abstract Th lastc bhavor of protn-lk chans was nvstgatd by usng th Prund-Enrchd-Rosnbluth Mthod (PERM). Thr typcal protn-lk chans such as all-α, all-β, and α + β(α/β) protns wr studd n our modfd orntaton dpndnt monomr-monomr ntracton (ODI) modl. W calculatd th rato of <R 2 >/N and shap factor <δ * > of protn-lk chans n th procss of longaton. In th mantm, w dscussd th avrag nrgy pr bond <U>/N, avrag contact nrgy pr bond <U> c /N, avrag hlcal nrgy pr bond <U> h /N and avrag sht nrgy pr bond <U> b /N. Thr maps of contact formaton, α-hlx formaton, β-sht formaton wr dpctd. All th rsults duc a vw that th hlx structur s th most stabl structur, whl th othr two structurs ar asy to b dstroyd. Bsds, th avrag Hlmholtz fr nrgy pr bond <A>/Ns was prsntd. Th forc f obtand from th fr nrgy was also dscussd. It was shown that th chan xtndd tslf spontanously frst. Th forc was studd n th procss of longaton. Lastly, th nrgy contrbuton to lastc forc f u was calculatd too. It was notd that f u for all-β chans ncrasd frst, and thn dcrasd wth x 0 ncrasng. Kywords: Protn-lk chan; Elastc bhavor; Scondary structur; PERM. INTRODUCTION Elastc bhavor of protn molculs s of grat sgnfcant n undrstandng th mchancal charactrstcs of protns. Thr ar a srs of sngl-molcul xprmnts, n whch protns such as botn and avdn ar strtchd by mchancal forcs. Lots of nformaton about th mchancal proprts of structural protns has bn rvald. Ths xprmnts also provd dtald nsghts nto ntrmolcular and ntramolcular forcs [1 15]. Mchancal xprmnts of atomc forc mcroscop (AFM) can xtract nformaton about fr nrgy landscaps of protns and study drctly th mchancal charactrstcs of molcular motors and mchancal functon n lvng organsms [3 5, 16 19]. On th othr hand, how to us th thortcal mthod to study th mchancal proprts of protns s also an ntrstng topc. In fact, th lastc bhavor of protn molculs s almost th sam as that of gnral polymr chans. For xampl, th changs of Hlmholtz fr nrgy nclud nrgy and ntropy componnts for both protns and gnral polymrs. Th lastcty s an mportant phnomnon n polymr physcs. Many rsarchrs hav provdd a non- Gaussan thory of rubbr-lk lastcty basd on rotatonal somrc stat smulatons of ntwork chan confguratons and nvstgatd rubbr-lk lastcty on th bass of dstrbuton functons for nd-to-nd sparaton r of th chans usng th Mont Carlo mthod [20 23]. Howvr, th thrmodynamc proprts of rubbr-lk lastcty hav not bn nvstgatd n dtal. Thn Dmtr and Zhsong nvstgatd th forc * Ths work was fnancally supportd by th Natonal Natural Scnc Foundaton of Chna (Nos. 20174036, 20274040, 10747160) and th Natural Scnc Foundaton of Zhjang Provnc (No. R404047). ** Corrspondng author: Tng-tng Sun ( 孙婷婷 ), E-mal: Tngtngsun@mal.zjgsu.du.cn Rcvd Dcmbr 28, 2007; Rvsd Fbruary 25, 2008; Accptd March 7, 2008

254 T.T. Sun t al. xtnson curvs of sngl protn molculs from th probablty dstrbuton P(r) [24]. Howvr, th ntractons btwn rsdus (or atoms) n protn molculs ar gnord n thr calculatons. In fact, th ntractons btwn rsdus n protn molculs ar vry mportant. It has bn mprovd n our arly work whr contact ntractons ar contand [25 27]. It has bn found that spcal bhavors of lastc forcs n protn molculs dpnd manly on rsdu ntractons. Th ntractons n protns hav many typs, and ths ntractons play an mportant rol n protn foldng and th stablty of a protn molcul [28 33]. Th ntractons also nflunc dply th structurs of protns [30 33]. Globular protns ar groupd nto four structural classs namd all-α, all-β, α + β and α/β protns. Th classs ar basd on th contnt and topology of α-hlcs and β-strands whch hav a dffrnt xhbton n structur. Th structurs of α-hlx and β-strand ar th most common scondary structurs n natv protns. Mayb dffrnt scondary structur wll xhbt dffrnt lastc bhavor. Thrfor, th man am of ths papr s to study th ffct of scondary structur on th lastc bhavor of protns. As a thr-dmnsonal smpl cubc lattc modl s adoptd hr, protns ar calld as protn-lk chans. Frst of all, th mthod of calculaton s prsntd. Thn th changs of chan dmnsons and shaps n th procss of longaton ar dscussd. Som thrmodynamc proprts such as avrag nrgy, avrag Hlmholtz fr nrgy ar also prsntd. Lastly, th lastc forc and nrgy contrbuton to lastc forc both ar nvstgatd. METHOD OF CALCULATION In ordr to dscrb th scondary structur of protns, w us a modfd ODI modl whch w hav prsntd bfor [34, 35], whch s basd on th ODI Modl [36] (Orntaton-Dpndnt Monomr-Monomr Intractons). Th mrt of our modl s th α-hlcal and β-sht structurs can b takn nto account smultanously. Thrfor, th modl s much closr to th ral protn. Howvr, ths modl s also a lattc approxmaton. That s to say a protn-lk chan s schmatcally vwd as a lnar squnc, and th chans ar constrand by thr narst nghbors on a thr-dmnsonal cubc lattc, and ach lattc st can only b occupd by on monomr. In th modl, th nrgy of a protn-lk chan s assumd to contan four trms: E N 1 = c δ rj a) + ε bδ ( rj a) + ω( ss + 1) j 3 j 3 = 1 ε ( + ε n whr r j s th dstanc btwn monomrs and j. a s th lattc spacng ( n fact, a = 1 n our calculaton). Hr δ(x) = 1 for x = 0, and δ(x) = 0 for x 0. Th unt vctor s rprsnts th orntaton of monomr, and ( s s + 1) s a scalar product of th orntaton vctors. ε c s th nrgy of on contact. Monomr and j wll form a contact f th two monomrs ar locatd at adjonng lattc sts, but not adjacnt along th chan. Th frst trm of Eq. (1) rprsnts th total contact nrgy of th chan. ε b corrsponds to th hydrogn bonds n topologcal contacts. Barng n mnd th condtons ncssarly for formaton of hydrogn bonds, w consdr that th valus of ε b s ngatv f () on of th monomrs s drctd toward th othr monomr (.g., r + s = rj ), and () smultanously th vctors s and s j ar paralll. In all th othr cass, ε b = 0. In fact, w consdr th scond trm of Eq. (1) to b th β-sht nrgy. Th thrd trm (wth ω > 0), constructd n analogy wth th Isng antfrromagntc spn-spn ntracton, dscrbs th orntatonal dpndnc of th ntracton btwn narst-nghbor monomrs lnkd va th pptd bonds. Its goal s to rproduc an antparalll orntaton of narst-nghbor monomrs. In th last trm, ε h s th nrgy of on hlx, n h s th numbr of hlx, ε h n h rprsnts th total nrgy of hlx. Thrfor, th α-hlcal and β-sht nrgs ar both consdrd n th conformatonal nrgy. In our smulaton, w lt ε c = 0.5, and ω = 0.2 (n unts of k B T ). In th mantm, w chang th valu of ε b and ε h. Hr w calculatd sx groups of ε b and ε h : ε b = 0, ε h = 0.5; ε b = 0, ε h = 1.0 (thy rprsnt all-α protns); ε b = 0.5, ε h = 0; ε b = 1.0, ε h = 0 (thy ar smulatd as all-β protns); ε b = 0.5, ε h = 0.5; ε b = 1.0, h h (1)

Effcts of Scondary Structur on Elastc Bhavor of Protn-Lk Chans 255 ε h = 1.0 (thy rprsnt α + β or α/β protns). In th llumnaton of th rsults, w us th nams of all-α, allβ and α + β (α/β) protn-lk chans nstad of th groups of ε b and ε h. Mont Carlo smulaton mthod s wdly usd n polymr scnc to study th thrmodynamc proprts, kntcs, phas bhavors of polymrs and so on [37 39]. Howvr, n ths papr, a nw smulaton mthod s usd whch s calld th prund-nrchd-rosnbluth mthod (PERM) [40, 41]. Grassbrgr had usd ths algorthm for smulatng flxbl chan polymrs, and thr rsults can llumnat that ths mthod s th most ffcnt for thr-dmnsonal polymrs on th smpl-cubc lattc. Also PERM can b usd nstad of th numraton calculaton mthod, and th partton functon can b calculatd. Now w wll consdr th lastc bhavor of protn-lk chans. Th frst monomr of th chan s fxd. Th forc acts at th nd of th chan s n th drcton of th x-axs. Howvr, th forc s not a ral forc whl t can b obtand from fr nrgy. Fgur 1 shows th procss of longaton. Thn w count th total conformatons wth dffrnt x valus. Th partton functon of th systm s Z ( x) xp( E ) (2) = whr s th sum of all conformatons at dffrnt dstanc x. Th man-squar nd-to-nd dstanc <R 2 (x)> and th man-squar radus of gyraton <S 2 (x)> ar dfnd as < R < S 2 2 2 R ( x) >= (3) 2 S ( x) >= (4) Fg. 1 Th procss of longaton Th strtchng coordnat s th componnt of th nd-to-nd dstanc vctor n th x-drcton; Th strtchng forc s along th x-axs; Th dffrnt longatd stats ar dscrbd n (a) and (b). W can also gt avrag nrgy <U(x)>, avrag contact nrgy <U(x)> c, hlcal nrgy <U(x)> h and sht nrgy <U(x)> b. E >= (5) E c > c = (6)

256 T.T. Sun t al. E h > h = (7) E E b > b = (8) Hlmholtz fr nrgy A(x) can b drvd from th partton functon: A( x) = kt ln Z( x) (9) At th sam tm, th lastc forc can b obtand from th dpndnc of A on th longatd dstanc along th forc drcton [42, 43]. A f = (10) x Accordng to Nwton s thrd law, th forc f s th lastc forc stord n th protn-lk chans. In th mantm, nrgy contrbuton to th lastc forc f u s dfnd as: < U > fu = (11) x RESULTS AND DISCUSSION Chan Sz and Shap Th man-squar nd-to-nd dstanc <R 2 > s vry mportant n nvstgatng th chan dmnsons. Fgur 2 prsnts th plots of th charactrstc rato <R 2 >/N vrsus x 0 for protn-lk chans wth dffrnt valus of th α-hlcal nrgy ε h and β-sht nrgy ε b. Hr, N = 50 s consdrd, and x 0 s dfnd as x x 0 = (12) N Hr x s th x-drcton dstanc btwn th last monomr (th pont of forc actng) and th frst monomr. Th ntroducton of paramtr x 0 s to compar som lastc proprts of protn-lk chans wth dffrnt lngths. It s found that <R 2 >/N ncrass wth x 0 for th sx nrgy groups whch hav bn prsntd n part 2 of th papr. It frst ncrass slowly, thn ncrass quckly. Thr xst som dffrncs btwn th Fg. 2 Charactrstc rato <R 2 >/N vrsus x 0 for protn-lk chans wth dffrnt valus of th α-hlcal nrgy ε h and β-sht nrgy ε b (N = 50)

Effcts of Scondary Structur on Elastc Bhavor of Protn-Lk Chans 257 sx nrgy groups n th arly stag of longaton. Th valu of <R 2 >/N for all-α protn-lk chans s largr than that of all-β or α + β(α/β) chans. Howvr, thr s bascally no obvous dffrnc xstng for dffrnt hlcal nrgs ε h = 0.5 and ε h = 1.0. W prsum that th hlcal nrgy has lttl nflunc on <R 2 >/N. For all-β chans, t s also notd that th valu of <R 2 >/N for largr sht nrgy ε b = 1.0 s smallr than that for ε b = 0.5. Th sam phnomnon for α + β(α/β) chans s found. It s du to that th largr sht nrgy wll form mor compact structur. It s also obsrvd that th valus of <R 2 >/N for ε h = 0, ε b = 0.5 and for ε h = 0.5, ε b = 0.5 ar vry nar. In th mantm, <R 2 >/N valus for ε h = 0, ε b = 1.0 and for ε h = 1.0, ε b = 1.0 ar also clos. That provs agan that th hlcal nrgy dos hav lttl nflunc on th rato <R 2 >/N. In ordr to xplor th chang of th shap of protn-lk chans, a shap paramtr <δ * > s dpctd. Th nstantanous shap of an ndvdual confguraton may b dscrbd by svral ratos basd on th prncpal 2 2 2 2 2 2 2 componnts L1 L2 L3 of S = L1 + L2 + L3,.., th orthogonal componnts of th squard radus of gyraton takn along th prncpal axs of nrta [44, 45]. <δ * > s obtand by combnng th rducd componnts of S 2 to a sngl quantty that vars btwn 0 (sphr) and 1 (rod) [46, 47] : 2 2 1 L2 2 ( L1 2 2 2L3 2 2 + 2 2 L3L1 2 2 3 ) * L + L + < δ >= 1 3 (13) + L L Th plots of <δ * > vrsus x 0 for chans wth lngth N = 30 and N = 50 ar shown n Fg. 3(a) and Fg. 3(b) rspctvly. All-α, all-β and α + β(α/β) protn-lk chans wth dffrnt nrgs ar consdrd. In th two fgurs, thr xsts a common trnd. <δ * > ncrass wth x 0 slowly frst, thn thr s a larg ncrasng, and thn t ncrass slowly agan, untl t rachs to a crtan valu. For small valu of x 0, th valu of <δ * > for allα structur s largr than that for all-β and α + β(α/β) structurs. W not that <δ * > for ε h = 0.5 and ε h = 1.0 ar clos. Howvr, for th lattr two structurs, th valu of <δ * > for largr nrgs s smallr. It shows that th chans wth largr sht nrgs ar mor symmtrc than thos wth small nrgy. As x 0 bcoms largr, th dffrncs btwn th sx nrgy groups dsappar gntly. Comparson of <δ * > for dffrnt chan lngths N s also nvstgatd. For chans wth ε h = 1.0, t s notd that at x 0 = 0, th valu of <δ * > s 0.22 for N = 30 and 0.18 for N = 50. Undr th sam nrgy group, th shap of th chans wth longr lngth s mor lk a globul wthout longaton. In th procss of longaton, th valu of <δ * > for N = 30 ncrass from 0.22 to 0.26 n th rang of x 0 = 0 0.1. It ncrass by 18.18%. For N = 50, th valu changs from 0.18 to 0.25, ncrasng by 38.89%. In th ara of x 0 = 0.1 0.2, th valu for N = 30 s from 0.26 to 0.46, ncrasng by 76.92%, whl for N = 50, t changs from 0.25 to 0.58, ncrasng by 132.00%. Th rason of th grat dffrncs btwn th ncrasng xtnts for dffrnt chan lngths may b that thr ar mor numbr contacts, α-hlcs and β-shts Fg. 3 Shap factor <δ * > vrsus x 0 for protn-lk chans wth dffrnt valus of th α-hlcal nrgy ε h and β-sht nrgy ε b a) N = 30; b) N = 50

258 T.T. Sun t al. formd for N = 50. Howvr, at x 0 = 0.2 0.3, th valu of <δ * > for N = 30 changs from 0.46 to 0.67, almost ncrasng by 45.65%, whl th valu for N = 50 s from 0.58 to 0.80, th ncrasng xtnt s 37.93%. Th ncrasng xtnt for N = 50 s lss than that for N = 30 n ths ara. Thrmodynamc Proprts In Fg. 4, th valus of avrag nrgy pr bond <U>/N ar plottd vrsus x 0 for protn-lk chans wth sx nrgy groups undr study. Fgur 4(a) s for chan lngth N = 30, and Fg. 4(b) s for N = 50. It s found that th trnds of th curvs ar almost th sam. Thus, w manly dscuss th avrag nrgy of N = 50. It s shown that <U>/N dos not chang much for all-α protn-lk chans. W fnd that th curvs of <U>/N for all-β chans and α + β(α/β) chans wth th sam sht nrgy hav th smlar trnds. On can s that for largr sht nrgs, <U>/N xhbts mor ncras. Fg. 4 Avrag nrgy pr bond <U>/N vrsus x 0 for protn-lk chans a) N = 30; b) N = 50; ε h = 0.5, ε b = 0; ε h = 1.0, ε b = 0; ε h = 0, ε b = 0.5; ε h = 0, ε b = 1.0; ε h = 0.5, ε b = 0.5; ε h = 1.0, ε b = 1.0 In ordr to s th chang of nrgy n th procss of longaton n dtal, w also plot th avrag contact nrgy pr bond <U> c /N, avrag hlcal nrgy <U> h /N and avrag sht nrgy <U> s /N vrsus x 0 for N = 50 n Fgs. 5 7 rspctvly. It s nvstgatd that th contact nrgy frst ncrass slowly and thn obvously wth x 0 ncrasng. Th ncrasng xtnt of all-α chans s lss than that of all-β and α + β(α/β) chans. Fgur 5(b) shows th contact formaton btwn rsdus and j n th procss of longaton for ε h = 1.0, ε b = 1.0. Four contact maps whch ar for x 0 = 0, 0.1, 0.2, 0.3 ar prsntd. Ths s asy to obsrv that th numbr of contact drops whn x 0 ncrass. Now lt us dscuss th hlcal nrgy pr bond <U> h /N n th procss of longaton for dffrnt valu of ε h and ε b. Th rsults ar shown n Fg. 6(a). It can b found that for all-α and α +β(α/β) chans, th hlcal nrgy changs vry lttl. It can b concludd that α-hlx structur n th chan s dffcult to b dstroyd. Only whn x 0 s larg, th α-hlx structur s to b dstroyd, and th hlcal nrgy ncrass. At th sam tm, th hlx formaton btwn rsdus and j for x 0 = 0, 0.1, 0.2, 0.3 s prsntd n Fg. 6(b). From th map, on can also obsrv that th α-hlx structur s vry stabl. W plot sht nrgy pr bond <U> b /N for dffrnt valus of α-hlcal nrgy ε h and β-sht nrgy ε b n Fg. 7(a). Th valu of <U> b /N s sn to ncras wth x 0. Howvr, t dos not ncras much at frst. Fnally, at larg x 0, th sht nrgy pr bond <U> b /N rachs zro. That s to say thr s almost no β-sht at th nd of longaton. Fgur 7(b) gvs th β-sht formaton btwn rsdus and j for x 0 = 0, 0.1, 0.2 and 0.3, rspctvly. It can b obsrvd that th numbr of β-sht drops whn x 0 ncrass. W conclud that wth th ncrasng of x 0, th β-sht structur s dstroyd.

Effcts of Scondary Structur on Elastc Bhavor of Protn-Lk Chans 259 a) b) Fg. 5 (a) Avrag contact nrgy pr bond <U> c /N vrsus x 0 for protn-lk chans wth dffrnt valus of th α-hlcal nrgy ε h and β-sht nrgy ε b (N = 50); (b) Th contact formaton btwn rsdus and j n th procss of longaton for ε h = 1.0 and ε b = 1.0 To furthr confrm th changs of contact, hlx and sht n th procss of longaton, w ntroduc thr paramtrs P c, P h and P b. Th dfntons of th thr paramtrs ar gvn blow: > P c( x) = (14) < U (0) > c c

260 T.T. Sun t al. > h P h ( x) = (15) < U (0) > h a) > b P b( x) = (16) < U (0) > b b) Fg. 6 (a) Avrag hlcal nrgy pr bond <U> h /N vrsus x 0 for protn-lk chans wth dffrnt valus of th α-hlcal nrgy ε h and β-sht nrgy ε b (N = 50); (b) Th α-hlx formaton btwn rsdus and j n th procss of longaton for ε h = 1.0 and ε b = 1.0

Effcts of Scondary Structur on Elastc Bhavor of Protn-Lk Chans 261 a) b) Fg. 7 (a) Avrag sht nrgy pr bond <U> b /N vrsus x 0 for protn-lk chans wth dffrnt valus of th α-hlcal nrgy ε h and β-sht nrgy ε b (N = 50); (b) Th β-sht formaton btwn rsdus and j n th procss of longaton for ε h = 1.0 and ε b = 1.0 Fgur 8 gvs th thr paramtrs vrsus x 0 for protn-lk chans wth ε h = 1.0, ε b = 1.0. Hr N = 30 and 50 ar both consdrd. It s found that th trnds of contact, hlx and sht changs for N = 30 and 50 ar almost th sam. Thrfor, w manly dscuss N = 50. It s found that P h changs lttl. It s n good agrmnt wth th rsults gvn n Fg. 6. At x 0 = 0.38, th valu of P h only drops to 96.32%. It can b obtand from th fgur that th valu of P c s 58.53% at x 0 = 0.38. Howvr, th droppng of β-sht numbr s vry sharp. Th valu of P b narly drops to zro at x 0 = 0.38.

262 T.T. Sun t al. Fg. 8 Th changs of contact prcntag P c (, ), hlx prcntag P h (, ), and sht prcntag P b (, ) vrsus x 0 for protn-lk chans wth th α-hlcal nrgy ε h = 1.0 and β-sht nrgy ε b = 1.0 Opn symbols rprsnt N = 30 and th sold symbols ar N = 50. In Fg. 9, th avrag Hlmholtz fr nrgy pr bond A/N vrsus x 0 for protn-lk chans wth dffrnt valus of α-hlcal nrgy ε h and β-sht nrgy ε b s xamnd, chan lngths N = 30 and 50 ar consdrd. As a rsult, th valu of A/N frst dcrass n th rang of x 0 = 0 to 0.02. Whn x 0 > 0.02, A/N ncrass wth x 0. From th shap of A/N, th forc longatng th chan s rflctd. Thrfor, t s most sgnfcant to compar th forc for dffrnt typs of protn-lk chans. Fg. 9 Avrag Hlmholtz fr nrgy pr bond A/N vrsus x 0 for protn-lk chans wth dffrnt valus of th α-hlcal nrgy ε h and β-sht nrgy ε b a) N = 30; b) N = 50 W calculat th lastc forc stord n th protn-lk chans accordng to th Eq. (10). Sx nrgy groups ar consdrd. Th forc for N = 30 and 50 s prsntd n Fg. 10. For N = 50, at x 0 = 0.02, th forcs ar all ngatv. Th valus ar almost nar 0.65. It mans thr s no forc actng on th chan. Th chan can xtnd tslf spontanously at th bgnnng of th longaton. For x 0 > 0.02, t nds forc to longat th chans. Frst, w dscuss th forcs for all-α chans wth ε h = 0.5 and ε h = 1.0. Th forc valus ar almost th sam for x 0 < 0.1. It s bcaus that n ths ara, th α-hlx structur s narly not dstroyd. Whl for x 0 > 0.1, th forc of ε h = 1.0 s largr than that of ε h = 0.5. It rflcts th α-hlx structur bcoms to b dstroyd. Thn, w compar th forcs of all-β chans wth ε b = 0.5 and ε b = 1.0. Th β-sht structur s vry asy to b dstroyd n th procss of longaton. It causs that th forc valus of ε b = 0.5 and ε b = 1.0 ar dffrnt

Effcts of Scondary Structur on Elastc Bhavor of Protn-Lk Chans 263 from bgnnng. W not that th forc always ncrass abruptly frst, thn th chang of forc s vry small. It s bcaus that th β-sht structur s dstroyd wth x 0 ncrasng. For larg x 0, th β-sht structurs almost dsappar, so th forc dos not chang much. For α + β(α/β)chans, th forcs ncras wth x 0 whn x 0 > 0.02. Howvr, for ε h = 0.5 and ε b = 0.5, th forc s smallr than that for ε h = 1.0 and ε b = 1.0. It s found that for th nrgy groups ε h = 0, ε b = 0.5 and ε h = 0.5, ε b = 0.5, th forcs ar almost th sam at small x 0. Howvr, th dffrncs btwn thm appar at hghr x 0 valus. Th phnomnon s also obsrvd for th nrgy groups ε h = 0, ε b = 1.0 and ε h = 1.0, ε b = 1.0. It can b prsumd that at th bgnnng of th longaton, th α-hlx structur s vry hard to b dstroyd, and only th contact and β-sht numbrs dcras. At hghr longatons, th α-hlx structur also bgns to b dstroyd. It may b th rason of th phnomna. Fg. 10 Elastc forc f vrsus x 0 for protn-lk chans wth dffrnt valus of th α- hlcal nrgy ε h and β-sht nrgy ε b a) N = 30; b) N = 50 Th nrgy contrbuton to th lastc forc f u s shown n Fg. 11. Th dffrnt chan lngths N = 30 and N = 50 ar consdrd rspctvly. It can b sn that for all-α chans, th valu of f u frst dcrass, and thn ncrass wth x 0. Smlar to th plots of f for small x 0, th dffrncs of f u for ε h = 0.5 and ε h = 1.0 ar not obvous. Whl at larg x 0, f u for ε h = 1.0 s largr than that for ε h = 0.5. For all-β chans, th valu of f u for Fg. 11 Enrgy contrbuton to lastc forc f u vrsus x 0 for protn-lk chans wth dffrnt valus of th α-hlcal nrgy ε h and β-sht nrgy ε b a) N = 30; b) N = 50

264 T.T. Sun t al. ε b = 1.0 s largr than that for ε b = 0.5. Alk th curvs of f, th valu of f u for ε h = 0.5, ε b = 0.5 s smallr than that for ε h = 1.0, ε b = 1.0. At th bgnnng of longaton, snc th hlx structur s narly not dstroyd, th curvs of all-β chans and α + β(α/β) chans wth sam sht nrgy ar smlar. Howvr, wth x 0 ncrasng, th dffrncs btwn f u for all-β chans and α + β(α/β) chans ar obvous. It s also du to th chang of α-hlx structur. It s qut mportant to b obsrvd that th valu of f u for all-β chans frst ncrass thn dcrass. Whl thr s no such trnd for all-α chans and α + β(α/β) chans. That s bcaus, at frst, th numbr of vanshng shts ncrass, and n som rgon, th numbr bcoms th largst, thrfor thr xsts a maxmum of f u. Howvr, wth th ncrasng of x 0, th numbr of vanshng shts bcoms small. That s why f u drops. Howvr, for th curvs of all-α chans and α + β(α/β) chans, snc contacts and hlcs ar mor dffcult to b dstroyd, thr trnds of f u valus ar dffrnt from thos of th all-β chans. It s also concludd that α-hlx scondary structur s mor stabl than β-sht and contact structur. All ths studs basd on th lastc bhavor of protn-lk chans wll provd som nsghts nto th lastc bhavor of protn chans. REFERENCES 1 Zhuang, X. and Rf, M., Curr. Opn. Struct. Bol., 2003, 13: 88 2 Znobr, R.C., Brockwll, D.J., Bddard, G.S., Blak, A.W., Olmstd, P.D., Radford, S.E. and Smth, D.A., Protn Sc., 2002, 11: 2759 3 Rf, M., Gautl, M., Ostrhlt, F., Frnandz, J.M. and Gaub, H.E., Scnc, 1997, 276: 1109 4 Rf, M., Frnandz, J.M. and Gaub, H.E., Phys. Rv. Ltt., 1998, 81: 4764 5 Rf, M., Pascual, J., Sarast, M. and Gaub, H.E., J. Mol. Bol., 1999, 286: 553 6 Thompson, J.B., Hansma, H.G., Hansma, P.K. and Plaxco, K.W., J. Mol. Bol., 2002, 322: 645 7 Pnns, M.E., Scnc, 1999, 283: 168 8 Obrhausr, A.F., Hansma, P.K., Carron-Vazquz, M. and Frnandz, J.M., Proc. Natl. Acad. Sc. U.S.A., 2001, 98: 468 9 L, H., Obrhausr, A.F., Fowlr, S.B., Clark, J. and Frnandz, J.M., Proc. Natl. Acad. Sc. U.S.A., 2000, 97: 6527 10 Erckson, H.P., Scnc, 1997, 276: 1090 11 Brockwll, D.J., Bddard, G.S., Clarkson, J., Znobr, R.C., Blak, A., Trnck, J., Olmstd, P.D., Smth, D.A. and Radford, S.E., Bophys. J., 2002, 83: 458 12 Bst, R.B., L, B., Stward, A., Daggtt, V. and Clark, J., Bophys. J., 2001, 81: 2344 13 L, H., Carron-Vazquz, M., Obrhausr, A.F., Marsamant, P.E. and Frnandz, J.M., Nat. Struct. Bol., 2000, 7: 1117 14 Obrhausr, A.F., Badlla-Frnandz, C., Carron-Vazquz, M. and Frnandz, J.M., J. Mol. Bol., 2002, 319: 433 15 Marsalk, P.E., Lu, H., L, H., Carron-Vazquz, M., Obrhausr, A.F., Schultn, K. and Frnandz, J., Natur (London), 1999, 402: 100 16 Smth, B.L., Schaffr, T.E., Van, M., Thompson, J.B., Frdrck, N.A., Kndt, J., Blchr, A., Stucky, G.D., Mors, D.E. and Hansma, P.K., Natur (London), 1999, 399: 761 17 Kllrmayr, M.S.Z., Smth, S.B., Granzr, H.L. and Bustamant, C., Scnc, 1997, 276: 1122 18 Obrhausr, A.F., Marszalk, P.E., Erckson, H. and Frnandz, J.M., Natur (London), 1998, 393: 181 19 Tskhovrbova, L., Trnc, J.A., Slp, J.A. and Smmons, R.M., Natur (London), 1997, 387: 308 20 Mark, J.E. and Curro, J.G., J. Chm. Phys., 1983, 79: 5705 21 Curro, J.G. and Mark, J.E., J. Chm. Phys., 1984, 80: 4521 22 Yang, X.Z. and L, X.F., Chns J. Polym. Sc., 1998, 16: 279 23 Yang, X.Z., Scnc n Chna B, 2001, 44: 154 24 Dmtr, M.E. and Zhsong, W., J. Chm. Phys., 2002, 116: 7760 25 Sun, T.T., Zhang, L.X., Chn, J. and Shn, Y., J. Chm. Phys., 2004, 120: 5469 26 Zhang, L.X. and Sun, T.T., Polymr, 2004, 45: 3547 27 Y, W.Q. and Zhang, L.X., Polymr, 2004, 45: 6735 28 Anfnsn, C.B., Scnc, 1973, 181: 223

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