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40 3 Self-organization and cooperativity of weakly coupled molecular motors
VN/V 0
1 (0)
1
0.8
0.6
0.4
0.2
0.5
0 5 10 15
N
20 25 30
0
0 0.2 0.4 0.6 0.8 1
F/(NF 0 s )
Fig. 3.12. Convergence with the number of motors N, of velocity vs force per motor, for
α = 0.15 and β = 14.1. Diamond, N = 1; inverted triangle, N = 2; triangle, N = 3; empty
square, N = 5; empty circle, N = 10; solid square, N = 15; solid circle, N = 30. In the inset,
the convergence for a fixed force F0 = 1/3F 0
s
(see Fig. 3.10). Another route to mean field is to impose a soft interacting potential
such that the variation of the force over a ratchet period is negligible,
ℓW ′′ (r)/W ′ (r), which ensures the decoupling of internal and spatial degrees
of freedom between motors. This condition is easily achieved for a (longranged)
interaction of the form W(r) ∼ exp(−r/λ), with λ ≫ ℓ. Adding to
this potential a hard-core part to prevent motor-motor crossing, we end up
with a long range interaction (see Fig. 3.3),
σ 6 σ 12
4ε −
W(r)= r6 r12
+ Aλ exp − r
r ≤ 2
λ
1/6 σ
Aλ exp − r
r > 2
λ
1/6 (3.25)
σ,
where the parameter A, sets the intensity force of the long-range potential.
We expect a crossover force between hardcore repulsion and long-range interaction,
if the force provided by the long-range part is not enough to overcome
the force per motor: −W ′ (2 1/6 σ) < F/N. In Fig. 3.13, we show several
plots of the 2-motor velocity-force curves for different long-range intensities
intensity. The transition region from mean field to cooperative behavior
corresponds qualitatively to a cross-over force F ∗ 2A/λ exp(−2 1/6 σ/λ).
Above the cross-over force, the velocity-force curve approaches the hardcore
curve (region in between the top and bottom curve in Fig. 3.13). The gain
in velocity from respect to mean field in this region grows linear with the force
until it reaches the hard-core region, inset of Fig. 3.13, with a slope that is
VN (NF0)/V1(F0)
2.5
2
1.5
1