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38 3 Self-organization and cooperativity of weakly coupled molecular motors tor will generically push against the first motor. In particular, due to symmetry, the maximum performance will correspond to mod(σ,ℓ) ∼ 1/2, Fig. 3.11. The amount of deviation from mean-mean field, depends on the correlations between motors. If entropic repulsion between motors dominates, D (V1(0)−V1(F))ℓ ≫ 1, correlations between motors are fully suppressed, and we recover the mean field approximation results. For very weak noise, the increased motor performance becomes dramatic, Fig. 3.10. (a) (b) (c) (d) v − F/(2λ) −F/(2λ) v − F/(2λ) v − F/λ Fig. 3.9. Schematics of cooperativity in the case of two motors with hard-core repulsion. (a) the second motor pushes the foremost motor which is at the diffusive state. This configuration represents a net gain with respect to mean field, since the advancing mechanism for the foremost motor do not rely on thermal fluctuations as in the mean field case. (b) the two motors slide over the U1 potential, sharing the external load as in mean field. (c) the foremost motor takes over all the load when sliding over the U1 potential whilst the second motor diffuses. This configuration corresponds to a loss with respect to mean field since a larger force is applied on the foremost motor. (d) the two motors diffuse on the U2 state sharing the external load as in mean field approximation. N motors For large N, we find that VN( f ) converges to a limiting curve VN → V∞( f ), that corresponds to a force per motor f that tends to a constant. One would be tempted to state that the force per motor tends the corresponding mean field approximation, that is, the force per motor is equally distributed and f = F/N. As we will see in section 3.3.1, the force distribution is not homogeneous throughout the motors, although its rescaled value saturates to a constant. In this sense, we recover the extensive scaling as in mean field, but with a V∞( f ) = V1( f ), Fig. 3.12. This is an important difference with respect to the prediction of the lattice model in (Campas et al., 2006), where VN(F) (the non rescaled velocity-force curve) was found to saturate with N to a limiting curve VN(F) → ˜V (F), i.e. they obtained a non-extensive scaling of force, except near stall. Consequently even with an unlimited supply of motors, the cluster could not advance against an arbitrarily large force, contrarily to our
3.2 Self-organization and cooperativity of weakly coupled molecular motors 39 VN/V 0 1 (0) 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 F/(NF 0 s ) Fig. 3.10. Velocity-force relationship for N = 1, 2 motors, α = 0.15 and hard-core repulsion (repulsive part of a Lennard-Jones potential truncated at r = 2 1/6 σ, with ε = 1 and σ = 0.2ℓ). Circles correspond to β = 10; empty circles, N = 1 and N = 2 with mean field; solid circles, N = 2. Squares correspond to β = 44.7; empty squares, N = 1 and N = 2 with mean field; solid squares, N = 2. Axes are normalized by D = 0 values of the 1-motor stall force and the 1-motor free velocity (see text). V2(2F0)/V1(F0) 1.8 1.6 1.4 1.2 1 0.8 0.6 0 0.5 1 1.5 2 2.5 3 Fig. 3.11. Dependence on the hard-core size σ (in units of ℓ) of the gain with respect to mean fied for an external force F0 = 1 3 F0 s . result which states that any arbitrary force can be overcome if enough motors are supplied. 3.2.4 Long range interaction - transition to mean field Early in section 3.2.2, we described the criteria for mean field approximation. In the last section, we showed that increasing noise leads to mean field σ
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Page 5 and 6: Acknowledgements This thesis has be
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Page 12 and 13: 2 1 General introduction gies. In t
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88 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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