Exact solution of a linear molecular motor model driven by two-step fluctuations and subject to protein friction

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Duane Powers
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1 Exact olution of a linea molecula moto model dien by two-tep fluctuation and ubject to potein fiction PADOVA, MAY 13, 2005 Han Foedby, Aahu Ralf Metzle, Nodita Axel Sane, Aahu

2 Motion in Bioloy Bioloical moto: Wok like man-made machine Tanpot of bioloical mateial Copy and tanlate enetic code Inteact with othe cell Inteact with uoundin Effectuate cell diiion and motion Collectiely enable bacteia to wim Collectiely enable mucle to contact

3 Moto potein: Myoin and Kinein Linea moto potein KINESIN STRUCTURE MYOSIN STRUCTURE

4 Moto walk alon tack Myoin walk on actin tack Kinein walk on micotubule tack Moto conet chemical eney to motion ATP-ADT hydolyi poide fuel Actin tack in e.. mucle Micotubule in e.. cell Motion in fluctuatin themal enionment

5 Myoin on the Actin Tack

6 Kinein on the Micotubule Tack

7 Manipulation of inle biomolecule Optical tweeze (1) Manetic tweeze (2) AFM cantilee (3) Gla micofibe (4) 2) 1) 3) 4)

8 Bioloical moto model Linea molecula moto: Kinein moin alon micotubule Myoin moin alon actin filament Moto potein moe unidiectionally alon pola tack Cental quetion: Chaacteize mechanochemical mechanim eneatin diected foce and moto moement by the coneion of chemical eney to mechanical eney Themal atchet model: Rectifyin Bownian motion by peiodic potential aymmetic in pace Powe toke model: Motion due to confomational chane in moto potein Chemical eney ouce diin moto potein Cyclical hydolyi ATP -> ADP phophate

9 The myoin moto at wok

10 The powe toke of kinein Ditance (nm) Foce (pn) Time () Time ()

11 Expeimental fact Size of moto ~ 5 20 nm. Kinein moe alon a micotubule with tep ize of ~ 8 nm. One ATP molecule hydolyed on aeae pe tep. Maximal elocitie of moto ane fom nm/ to µm/. Maximal foce to halt motion (tall foce) ~ pn Chemical cycle time ~ m. Fee eney pe ATP i ~ kt. Linea moto Rotay moto

12 Fundamental equiement fo theoetical model Vicoity detemine motion of moto potein ATP-ADP hydolyi die the moto potein Chemical eney lae compaed to themal eney Moto potein ae non equilibium ytem Motion of moto i tochatic due to chemical and themal fluctuation Themal adient anih much fate than cycle time of ATP hydolyi Moto potein ae iothemal machine on the nanomete cale

13 Linea moto model Powe toke model popoed by Moilne Moto dien by cyclical ATP hydolyi Moto with two head: a wokin head and an idle head Potein fiction when wokin head in contact with tack ATP hydolyi bin about confomational chane between elaxed and tained tate Confomational chane modelled by time dependent ize and elatic modulu fo moto potein Moto wok in ambient enionment at oom tempeatue

14 Remak on model Model i intended to chaacteize featue of wokin moto potein (myoin, kinein) Model i baed on two coupled oedamped ocillato Model i dien chemically, not by Bownian motion Model i a enuine non-equilibium ytem No detailed balance Confomational chane chaacteized by two-tep Mako poce Model acceible to analyi

15 How the moto wok State altenate between elaxed (R) and tained (S) confomation In elaxed tate wokin head attached to tack - potein fiction In tained tate wokin head lip - no potein fiction - moto moe (fom Moilne)

16 Detail of model Wokin head coodinate: xt ( ) Idle head coodinate: yt ( ) Elatic module: kt ( ) Lenth: Lt ( ) Modulu in tained tate: k Modulu in elaxed tate: k Lenth in tained tate: L Lenth in elaxed tate: L Pobability in elaxed tate: P Pobability in tained tate: P Rate of hydolyi: Rate of elaxation: Potein fiction in elaxed tate: Potein fiction in tained tate: () t = p ( t) = Vicou fiction in medium: Ambient noie: N ( t) Load foce: f ν xy,

17 Mate equation Two-tep Mako poce Moto chaed up by ATP-ADP hydolyi Pobability in chaed tate: P (t) Pobability in elaxed tate: P (t) Rate of hydolyi: Rate of elaxation: dp () t dt dp () t dt = P() t P() t = P() t P() t

18 Lanein equation Coupled oe damped ocillato Actie head: x(t), Idle head: y(t) Lenth: L(t), module: k(t), load: f Potein fiction: (t) dx () t = f k()( t y() t x() t L()) t Nx() t dt dy = kt ()( yt () xt () Lt ()) Ny() t dt Noie coelation < N x, y( t) N x, y( t' ) >= 2kT p, δ ( t t' )

19 Bioloical paamete Rate of hydolyi... Rate of elaxation... Elatic modulu in elaxed tate pn/nm Elatic modulu in tained tate.. k 0. 5 pn/nm Ret lenth in elaxed tate... L Ret lenth in tained tate... L Themal eney... kt k 40 nm 20 nm Room Fee eney of hydolyi... F 25 kt 4.1 pn nm Vicoity of wate... η 10 pn /nm Size of potein head nm Stoke law... 6πη Vicou fiction coefficient pn /nm Potein fiction coefficient... p Load foce... f 1 pn pn /nm

20 Method of olution Sole Lanein equation fo x(t) and y(t) Themal noie independent of two-tep poce Aeae oe themal noie Aeae oe two-tep Mako poce Show that moto head moe toethe Show eodic behaio: time-aeae equal two-tep enemble aeae

21 Fee moto olution ) )( ( 2 ) )( ( Mean moto elocity without load, 0 p p p k k k k k L L f = =

22 Dicuion of fee moto Moto moe in abence of load No detailed balance Moto conet chemical eney to motion Moto only need ooe to wok (no pola o peiodic tack equied) No motion fo L = L (no confomational chane) No motion fo ν = p (no potein fiction, no attachment to tack) Moto opeate in bioloical eime

23 Mean elocity a function of ate of hydolyi Bioloical

24 Mean elocity a function of potein fiction Bioloical

25 Moto olution with load f = μ μ 0 Define mobility p p p k k P P k k k k P P P k P k γ γ γ γ γ γ γ γ γ γ γ γ γ γ μ = = = = =, 1 1 2,, ) 1/ )(1/ ) / ( ) / )(( ( ) ( 2 / ( ) ( ) 2 / ( 2 Mobility:

26 Dicuion of moto with load Velocity eu load foce Mobility μ, Stall foce : f 0 = μ f 0 = μ

27 Mobility a function of ate of hydolyi Bioloical

28 Mobility and tall foce a function of ate of elaxation

29 Mobility and tall foce a function of potein fiction Bioloical

30 Summay and Concluion Molecula moto abundant in bioloy Moto ae enuine nonequilibium ytem Moto conet chemical eney to diected motion Moto dien by ATP-ADP hydolyi Peent wok: Moto modeled by coupled oedamped ocillato Moto opeate in themal enionment ATP-ADP hydolyi induce moto confomational chane Relaxation of confomational chane yield motion on tack Deciption in tem of Lanein equation and Mate equation Moto model amenable to analyi Moto model opeate in bioloical eime

31 FINE

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