Introductory slide

1. Introductory slide

2. Motor: eg. myosin Chemical Chemical flux potential Torque rotation eg. F1 ATP-synthase Free energy transduction ForceConjugate flux Chemical Chemical flux potential Force Displacement

3. “food” e.g. glucose electrons reproduction pmf ATP photons growth transport movement A molecular motor: acto-myosin R.D. Vale & R. Milligan

4. The mechanism of muscle contraction organization of proteins within a sarcomere insect flight muscle hierarchical organization

5. Actin-myosin in vitro motility assay 10μm J.E. Molloy

6. ATP binding completes the cycle Once bound, Phosphate release allows myosin head to relax. This “powerstroke” pulls the thick filament even against an external force, doing Work = Force x distance ATP hydrolysis transfers free energy to “strained” straight form of myosin head. This form also binds actin. ATP binding makes myosin release actin Actin-myosin crossbridge cycle

7. 4 5 Conformational states of the myosin motor revealed by X-ray crystallography structural model of power stroke Houdusse, A. & Sweeney, H.L. Curr. Opin. Struc. Biol. 11, 182 (2001)

8. P r x Free energy landscape: phosphate release with power stroke Reaction coordinate r 5 4 Mechanical coordinate x

9. 4 5 Free energy Mechanical coordinate x Free energy landscape: phosphate release with power stroke Reaction coordinate r 5 4 Mechanical coordinate x

10. kXY pX pX, pY are probabilities that states A, B are occupied; kXY, kYX are rate constants    probability fluxes kYX pY Free energy G* ΔG X Y Reaction coordinate r Rate constants often obey Arrhenius equation: G* is an activation barrier; reaction rate depends on probability that system has sufficient energy to cross it. Consistent with detailed balance:  and, in equilibrium, Barrier-crossing rates Reaction coordinate r Y X Mechanical coordinate x

11. Free energy lowest barrier a b c Reaction coordinate r Free energy landscape: position-gated transition Reaction coordinate r a c b Mechanical coordinate x transition is most probable at b, where the activation barrier is lowest

12. 1 4 6 1’ 5 2 3 Rates and free energies in crossbridge cycle Key AM actin-myosin T ATP M free myosin DP ADP+Pi D ADP - empty

13. 3 → 4actin binding 4 → 5 phosphate release 2→3 ATP hydrolysis ΔGATP -25kBT 5→6 ADP release 6→1 ATP binding Free energy 1→2´ unbinding from actin position power-stroke Rate constant -5nm 0 position x0 x0+Δx 0 Crossbridge model for muscle acto-myosin AMDP (4) AMD (5) AM (6) AMT (1) MT (2) 0 MDP (3) -10 kT -20 kT MT (2’) • Each cycle: • hydrolyzes one molecule of ATP, • releasing ~25 kT of free-energy, • produces one ~5 nm power stroke • (about half of the available free • energy can be converted to work) binding un-binding

14. where Fokker-Planck Equation mechanical: motor movement while in state i chemical:transitions into and out of state i at position x Reaction-Diffusion Equation Reaction-Diffusion Equation Let P(x)dx be the probability of finding a motor in the range x→x+dx. Consider diffusion in the presence of an external force: “Probability flux”: & D,  are diffusion, drag coefficients Continuity equation: Including the possibility of transitions between states:

15. actin binding and phosphate release power stroke unbinding -5nm 0 Dx Minimal model for muscle acto-myosin o o If we assume that the muscle contracts at a constant speed, so x = vt, we can use the minimal model to predict the relationship between force and speed…

16. Reaction-Diffusion Equation: 2 states: P ≡ Pbound = 1-Punbound rates: kon, xo < x < xo + Δx koff, x > 0 constant velocity: F/γ = v = constant x = vt & D = 0 (motion determined by ensemble of motors) steady state: P/ t = 0 -5nm 0 o o Special case: constant velocity, two states

17. Model vs data • Solve in 3 pieces: • Binding zone around x = xo (= -5 nm) (assume initial condition P(xo)=0; binding zones are widely separated) • power stroke zone xo < x < 0 • drag-stroke zone x > 0 • If stiffness of crossbridge is κ, average force • exerted per crossbridge is: N.B. please use d, not D as in problem set, for distance between binding sites

18. kij kij´ j´,x j,x i,x Pj Random number here, no transition Random number in this interval, transition to j 0 P(any) 1 Monte Carlo: chemical transitions The probability of making a transition from state i to state j during a time step Δt is At time t, motor is in state i at position x. State i has chemical transitions to states j ( j´ ) For sufficiently small Δt: A random number between 0 and 1 decides whether any transition should occur in time Dt and, if so, which: Rate constants for these transitions ensure detailed balance: (Non-zero rate constants define positions where the motor can switch states.)

19. At time t, motor is in state i at position x, moving with instantaneous velocityv. “Thermal force” due to collisions with water molecules “Langevin Equation” Inertial forces are negligible (Re = 10-8 ~10-6): The second term is the distance the particle would have travelled by free diffusion in time Dt : substitute a random distance drawn from a Gaussian distribution with mean square 2DDt. The distance moved in time Dt is… Brownian motion (stochastic) normally distributed: mean 0, variance 2Dt motion due to potential (deterministic) Monte Carlo: mechanical motion