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32 3 Self-organization and cooperativity of weakly coupled molecular motors (a) (b) (c) Fig. 3.5. Schematics of two motors connected by a rigid rod. When both motors are in the diffusive or bound state (a) and (b), they behave as single a motor (with half the total force). (b) In the crossed configurations, the advancing probability of the motor in the diffusive state do not rely on noise since the bound motor is pushing (pulling), and the motor advances deterministically. damped), with its engine that has an asymmetric combustion cycle (the cycle of the motor) but moves at a constant velocity. The two motor case would correspond to two cars that are pushing together against a force, sharing exactly the same force, and hence moving at the velocity they would move if they were alone pushing against half of the applied force (Fif. 3.6b). This softly enough interacting potential results in the intuitive result that a force applied to a cluster of N motors is equally shared and their velocity VN is related to the velocity of a single motor, VN(F)=V1(F/N). (a) (b) F F/2 F/2 〈 ˙x〉 = v v Fig. 3.6. Schematics of the interaction of two motors satisfying mean field. (a) the spring connecting the two motors represents a soft connection that is insensible over a period ℓ. In (b), the car analogy, where an over damped car is moving at a constant velocity although its internal motor has asymmetric combustion cycles. More rigorously, we define the mean field approximation for a cluster of N motors as satisfying the relation: 〈−W ′ (r)〉 −W ′ (〈r〉), (3.15) so that correlations between positional and internal degrees of freedom of different motors are neglected. The steady state solution of Eq. 3.11 for the mean field approximation implies that the force between two motors is F
3.2 Self-organization and cooperativity of weakly coupled molecular motors 33 constant, and, consequently, each motor behaves as an isolated motor subject to an equal share F/N of the external force. The collective dynamics is then reduced to the single-motor problem with VN(F) =V1(F/N) and FN(V )=NF1(V ). The physical picture corresponds to N ballistic motors defined solely by V1(F). It is illustrative to obtain the exact velocity-force curve for N motors in mean field approximation, or equivalently, for a single motor at a force F/N. We use these values later as a reference to compare with interacting potentials that not satisfy the conditions for mean field approximation. The mean velocity over a time t corresponding to m cycles is given by 〈 ˙x〉 = ∆x t ≡ ∑m i=0 δxi ∑ m i=0 δti = 〈δx〉 , (3.16) 〈δt〉 where ∆x is the total distance traveled by the motor during the time t. This distance can be decomposed in steps of distances δxi, which correspond to the advanced distance per cycle. Similarly, the time t is the sum of intervals δti corresponding to the time lapse of the i-th cycle. Finally, the mean velocity is the ratio of the mean displacement and the mean time lapse per cycle. We compute the mean velocity in two steps: first, the mean displacement and second the corresponding mean time per cycle, Mean displacement per cycle We consider the initial position of the motor as an arbitrary minimum of the asymmetric potential U1, Fig. 3.7. The motor instantaneously changes its state to U2 and starts diffusing for a time τ. During this diffusive state, the motor is drifted by the external force at a velocity F/λ. After a time τ the motor deexitate to the state U2 at a position x with a gaussian probability distribution centered at x0 ≡−Fτ/λ, P(x)= 1 (x + Fτ/λ)2 √ exp − 2πa 2a2 , (3.17) where a2 ≡ Dτ. At the state U2, the motor diffuses and slides over the ramp of the potential towards the minimum. The probability of going forward towards the minimum of the potential (and not going backwards) is given by (Parmeggiani et al., 1999) (F −U1/ℓ)z P→ 1 − exp − . (3.18) kBT
Page 1: Studies of dynamical phenomena in s
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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82 4 Membrane-cortex interactions:
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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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