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26 3 Self-organization and cooperativity of weakly coupled molecular motors asymmetric potential, Fig. 3.1. In this model we assume only a conformational change which leads to the power stroke of the motor. Any other conformational change is considered instantaneous when compared to chemical reaction times of the order of milliseconds. (a) (b) ATP ATP ADP P P ADP ADP Unbinding Diffussion Binding Power stroke (c) α1 β2 α2 Fig. 3.1. Equivalence between a molecular motor cycle (a) and the two state model (b-c) explained in the text. In Fig. 3.1 we schematically show the equivalence between a motor cycle and the two state model. Initially, the motor is attached to the filament in what is called ”rigor state” until a molecule of ATP binds to the motor, which detaches from the filament and hydrolyzes the ATP molecule, β1 ATP ⇋ ADP + P, (3.3) this reaction represents the transition from the bound state (1) to the unbound state (2). Associated to this reaction there is a chemical potential difference ∆µ ≡ µAT P − µADP − µP which measures the free-energy change per ATP consumed. At the unbound state, when a phosphate molecule P is released the motor attaches to the filament and performs the power stroke after the molecule of ADP is released. The transition from the unbound state (2) to the bound state (1) is thermal (passive), M − ADP − P ⇋ M + ADP + P. (3.4)
Transition rates 3.1 Introduction 27 There are two transition rates associated to each of the two different reactions (Eq 3.3 and 3.4), Fig. 3.1c. α1,2 correspond to the hydrolysis of ATP, and their ratio depends on the difference of energy between the two states, ∆U ≡ U1 −U2, and the difference of chemical energy ∆µ. β1,2 correspond to the thermally induced release of phosphate and ADP, and their ratio satisfies detailed balance, α1(x) −∆U(x)+∆µ = exp (3.5) α2(x) kBT β1(x) −∆U(x) = exp , (3.6) β2(x) kBT The global transition rates between the two states w1,2, include both α and β rates: wi ≡ αi + βi, which in terms of α2 ≡ α, and β2 ≡ β, −∆U w1(x) = (α exp∆µ + β)exp (3.7) kBT w2(x)=α + β, (3.8) which do not satisfy detailed balance unless ∆µ = 0. When a cell has excess of ATP (∆µ > 0), the motors are driven out of thermodynamic equilibrium and perform a directional movement along its associated polar filaments. In physiological conditions, ∆µ ∼ 10kBT , the system is far from equilibrium, and α exp∆µ ≫ β. If in addition ∆U ≫ kBT , which is equivalent to neglect thermally excited transitions with respect to conformational changes, w1(x)=α exp∆µ (3.9) w2(x)=α + β. (3.10) For simplicity we can assume U2 to be a flat potential, and de-exitations (corresponding to the rate w2) to be spatially delocalized. Moreover, it is reasonable to assume that the transition from the bound to the unbound state, which depends on ATP binding, is localized at the minimum of U1, corresponding to the fact that ATP is not likely to bind to a motor during the ”power stroke”. In Fig. 3.2, we schematically represent the potentials and the transition rates between the two states.
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Page 5 and 6: Acknowledgements This thesis has be
Page 7 and 8: Contents 1 General introduction . .
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Page 12 and 13: 2 1 General introduction gies. In t
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Page 16 and 17: 6 2 Probing elastic anisotropy from
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5 General conclusions In this work
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5 General conclusions 115 number of
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A Resum en català El present treba
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A.1 Auto-organització i cooperativ
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A.2 Mesura de l’anisotropia elàs
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A.3 Interaccions de membrana i còr
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A.3 Interaccions de membrana i còr
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B Résumé en Français Ce travail
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B.1 Auto-organisation et coopérati
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B.2 Mesure de anisotropie élastiqu
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B.3 Interactions de membrane et cor
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B.4 Conclusions 135 une perturbatio
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References Alberts, B., Johnson, A.
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REFERENCES 139 Evans, E. (2001). Pr
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REFERENCES 141 Kruse, K., Joanny, J
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REFERENCES 143 Sit, P., Spector, A.
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