NANOPHYSIQUE INTRODUCTION PHYSIQUE AUX NANOSCIENCES Pierre GASPARD 2011-2012 6. MOTEURS MOLECULAIRES.

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1 NANOPHYSIQUE INTRODUCTION PHYSIQUE AUX NANOSCIENCES Pierre GASPARD 2011-2012 6. MOTEURS MOLECULAIRES

2 BIOMOLECULES -ACATGTAATTCATTTACACGC- -GTACATTAAGTAAATGTGCGT- A: adénineT: thymine C: cytosineG: guanine 1 paire de bases = 2 bits d’information ~ 64 atomes adénosine monophosphate adénosine triphosphate (stockage d’énergie) ADN: Acide Désoxyribo-Nucléique (stockage d’information) Watson & Crick, Franklin, Wilkins (1953)

3 En plongée dans la cellule L’évolution biologique a transformé de simples vésicules en cellules munies de nombreuses organites. taille ~10-30  m Chr. de Duve, Une visite guidée de la cellule vivante (De Boeck Université, Bruxelles, 1987).

4 Dans une mitochondrie Chr. de Duve, Une visite guidée de la cellule vivante (De Boeck Université, Bruxelles, 1987). taille ~ 2-3  m Centrale énergétique de la cellule: production de l’ATP (carburant cellulaire) membrane double, membrane interne avec: (1) pompes à protons (2) ATP synthases

5 Moteur moléculaire F o F 1 -ATPase ATP synthase F : turbine à protons F : génératrice d’ATP mitochondrie: centrale énergétique de la cellule protéine: polymère d’acides aminés F : moteur à ATP F : pompe à protons o o 1 1

6 combustible: adénosine triphosphate (ATP) Moteur moléculaire F 1 -ATPase puissance = 10 Watt  18 H. Noji, R. Yasuda, M. Yoshida, & K. Kinosita Jr., Nature 386 (1997) 299  m 1,3 tour / sec 0,5 tour / sec actine Moteur F 1

7 H. Wang & G. Oster, Nature 396 (1998) 279 F : moteur à ATP 1 Moteur moléculaire F 1 -ATPase puissance = 10 W (Watt)  18 H. Noji, R. Yasuda, M. Yoshida, & K. Kinosita Jr., Nature 386 (1997) 299  m 1,3 tour / sec 0,5 tour / sec

8 Locomotive à vapeur puissance = 5 10 W (Watt) 

9 F o F 1 -ATPase INSIDE MITOCHONDRIA Chr. de Duve, Une visite guidée de la cellule vivante (De Boeck Université, Bruxelles, 1987). size ~ 2-3  m Power plant of the cell: synthesis of ATP Internal membrane with: F o = proton turbine F 1 = ATP synthase (23500 atoms) H. Wang & G. Oster, Nature 396 (1998) 279

10 ACTIN-MYOSIN MOLECULAR MOTOR myosin II: protein of about 6700 atoms cross-bridge mechanism

11 ROTARY AND LINEAR MOLECULAR MOTORS Linear motors: actin-myosin II (muscles) kinesin-microtubule (anterograde transport cargo) dynein-microtubule (retrograde transport cargo) Rotary motors: F 1 -ATPase + actin filament or bead Powered by chemical energy: ATP hydrolysis ATP ADP + P i difference of free energy:  G 0 =  30.5 kJ/mole =  7.3 kcal/mole =  50 pN nm =  k B T nonequilibrium thermodynamicsk B T = 4 pN nm = 0.026 eV (300 K) importance of the chirality of the molecular structure for the directionality of motion under specific nonequilibrium conditions ATP

12 F 1 -ATPase NANOMOTOR H. Noji, R. Yasuda, M. Yoshida, & K. Kinosita Jr., Nature 386 (1997) 299 R. Yasuda, H. Noji, M. Yoshida, K. Kinosita Jr. & H. Itoh, Nature 410 (2001) 898 power = 10  Watt chemical fuel of F 1 : ATP chiral molecules F 1 = (  ) 3  cycle:

13 DISCRETE-STATE STOCHASTIC PROCESSES FOR MOLECULAR MOTORS Markovian jump process between the discrete states  : master equation A. B. Kolomeisky & M. E. Fisher, Ann. Rev. Phys. Chem. 58 (2007) 675 R. Lipowsky & S. Liepelt, J. Stat. Phys. 130 (2008) 39 A. Garai, D. Chowdhury & M. P. Betterton, Phys. Rev. E 77 (2008) 061910 Fluctuation theorems: U. Seifert, EPL 70 (2005) 36 (rotary motor, 3 states) D. Andrieux & P. Gaspard, Phys. Rev. E 74 (2006) 011906 (rotary motor, F 1 -ATPase, 6 states) D. Lacoste, A. W. C. Lau & K. Mallick, Phys. Rev. E 78 (2008) 011915 (linear motor)

14 CONTINUOUS STOCHASTIC PROCESSES coupled Fokker-Planck equations for the probability densities: Chemical part: transition rates of the reactions Arrhenius’ law of chemical kinetics potentials for the wells: U i (  ) potentials for the transition states: U i * (  ) diffusion coefficient:Mechanical part: probability currents: friction coefficient P. Gaspard & E. Gerritsma, J. Theor. Biol. 247 (2007) 672 F. Jülicher, A. Adjari & J. Prost, Rev. Mod. Phys. 69 (1997) 1269

15 FREE-ENTHALPY POTENTIALS P. Gaspard & E. Gerritsma, J. Theor. Biol. 247 (2007) 672 Potential wells obtained by inverting the experimental probability distributions: R. Yasuda, H. Noji, M. Yoshida, K. Kinosita Jr. & H. Itoh, Nature 410 (2001) 898 potentials for the wells U i (  ) potentials for the transition states U i * (  ) three-fold rotation symmetry: group C 3 absence of parity symmetry (chirality)

16 RANDOM TRAJECTORIES OF THE F 1 -ATPase MOTOR Random trajectories observed in experiments R. Yasuda, H. Noji, M. Yoshida, K. Kinosita Jr. & H. Itoh, Nature 410 (2001) 898 Random trajectories simulated by a model: P. Gaspard & E. Gerritsma, J. Theor. Biol. 247 (2007) 672 Michaelis-Menten kinetics

17 F 1 -ATPase ROTATION RATE VERSUS FRICTION Crossover from reaction-limited regime to friction-limited regime P. Gaspard & E. Gerritsma, J. Theor. Biol. 247 (2007) 672

18 F 1 -ATPase UNDER AN EXTERNAL TORQUE (e.g. from F o ) stall torque ATP synthesis ATP consumption H. Itoh, A. Takahashi, K. Adachi, H. Noji, R. Yasuda, M. Yoshida, K. Kinosita Jr., Mechanically driven ATP synthesis by F 1 -ATPase, Nature 427 (2004) 465

19 EFFICIENCIES OF F 1 -ATPase F 1 -ATPase under an external torque (e.g. from F o ) number of ATP synthesized or consumed per revolution: chemical efficiency in ATP synthesis: tight chemomechanical coupling for |  | < 27 pN nm mechanical efficiency in energy transduction: F. Jülicher, A. Adjari & J. Prost, Rev. Mod. Phys. 69 (1997) 1269

20 tight coupling condition: entropy production: TIGHT/LOOSE CHEMOMECHANICAL COUPLING chemomechanical affinity: E. Gerritsma & P. Gaspard, unpublished shift of effective equilibrium by the external torque

21 The angle jump at each reactive event: the tight coupling condition is always fulfilled. DISCRETE-STATE MODEL FOR THE F 1 -ATPase MOTOR 1 E. Gerritsma & P. Gaspard, unpublished Markovian jump process between the discrete states  : dependence on friction  and torque  : transition rates: fitted to the continuous model

22 DISCRETE-STATE MODEL FOR THE F 1 -ATPase MOTOR 2 D. Andrieux & P. Gaspard, Phys. Rev. E 74 (2006) 011906 E. Gerritsma & P. Gaspard, unpublished mean rotation rate (rev/sec): master equation : stationary solution:

23 mean rotation rate: highly nonlinear dependence on A linear regime around equilibrium: nonlinear regime far from equilibrium: The F 1 molecular motor typically works in a highly nonlinear regime far from equilibrium. dimensionless affinity or thermodynamic force: equilibrium: F 1 -ATPase ROTATION RATE VERSUS AFFINITY A = 1: 3.1 days/rev ! E. Gerritsma & P. Gaspard, unpublished

24 discrete-state model: 1st cumulant: mean rate chemomechanical affinity: FULL COUNTING STATISTICS & FLUCTUATION THEOREM D. Andrieux & P. Gaspard, Phys. Rev. E 74 (2006) 011906 E. Gerritsma & P. Gaspard, unpublished generating function of the statistical cumulants of the number N t of reactive events during the time t : 2nd cumulant: diffusivity fluctuation theorem: A. B. Kolomeisky & M. E. Fisher, Ann. Rev. Phys. Chem. 58 (2007) 675

25 affinity or thermodynamic force: FLUCTUATION THEOREM FOR THE F 1 -ATPase MOTOR: NO EXTERNAL TORQUE t = 10 4 s D. Andrieux & P. Gaspard, Phys. Rev. E 74 (2006) 011906 very long time interval: P(S t =  s) exp(sA/2) Fluctuation theorem for the number S t of substeps:

26 chemomechanical affinity: Fluctuation theorem for the number S t of substeps: FLUCTUATION THEOREM FOR THE F 1 -ATPase MOTOR: WITH EXTERNAL TORQUE E. Gerritsma & P. Gaspard, unpublished x P(S t =  s) exp(sA/2) o P(S t = s) shorter time interval:

27 Loose coupling: independent mechanical & chemical fluctuating currents FLUCTUATION THEOREM & TIGHT CHEMOMECHANICAL COUPLING Tight coupling: chemomechanical affinity: D. Andrieux & P. Gaspard, J. Chem. Phys. 121 (2004) 6167 D. Andrieux & P. Gaspard, Phys. Rev. E 74 (2006) 011906 D. Lacoste et al., Phys. Rev. E 80 (2009) 021923 E. Gerritsma & P. Gaspard, unpublished U. Seifert, EPL 70 (2005) 36 (rotary motor, 3 states) D. Andrieux & P. Gaspard, Phys. Rev. E 74 (2006) 011906 (rotary motor, F 1 -ATPase, 6 states) D. Lacoste, A. W. C. Lau & K. Mallick, Phys. Rev. E 78 (2008) 011915 (linear motors)

28 OUT-OF-EQUILIBRIUM DIRECTIONALITY IN THE F 1 -ATPase NANOMOTOR at equilibrium: detailed balance between …212132131223132… forward and backward rotations, (random) zero currents out of equilibrium: directionality of motion: …123123123123123… non-zero currents, (more regular) dynamical order   