NANOPHYSIQUE INTRODUCTION PHYSIQUE AUX NANOSCIENCES Pierre GASPARD MOTEURS MOLECULAIRES

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)

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).

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

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

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

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

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

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

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

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 = eV (300 K) importance of the chirality of the molecular structure for the directionality of motion under specific nonequilibrium conditions ATP

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:

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) Fluctuation theorems: U. Seifert, EPL 70 (2005) 36 (rotary motor, 3 states) D. Andrieux & P. Gaspard, Phys. Rev. E 74 (2006) (rotary motor, F 1 -ATPase, 6 states) D. Lacoste, A. W. C. Lau & K. Mallick, Phys. Rev. E 78 (2008) (linear motor)

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

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)

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

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

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

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

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entropy production: TIGHT/LOOSE CHEMOMECHANICAL COUPLING chemomechanical affinity: E. Gerritsma & P. Gaspard, unpublished shift of effective equilibrium by the external torque

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

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 E 74 (2006) E. Gerritsma & P. Gaspard, unpublished mean rotation rate (rev/sec): master equation : stationary solution:

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

discrete-state model: 1st cumulant: mean rate chemomechanical affinity: FULL COUNTING STATISTICS & FLUCTUATION THEOREM D. Andrieux & P. Gaspard, Phys. Rev. E 74 (2006) 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

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) very long time interval: P(S t =  s) exp(sA/2) Fluctuation theorem for the number S t of substeps:

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:

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) D. Lacoste et al., Phys. Rev. E 80 (2009) E. Gerritsma & P. Gaspard, unpublished U. Seifert, EPL 70 (2005) 36 (rotary motor, 3 states) D. Andrieux & P. Gaspard, Phys. Rev. E 74 (2006) (rotary motor, F 1 -ATPase, 6 states) D. Lacoste, A. W. C. Lau & K. Mallick, Phys. Rev. E 78 (2008) (linear motors)

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