Thesis (pdf) - Espci

34 3 Self-organization and cooperativity of weakly coupled molecular motors
When the external force approaches the force due to the power stroke, U1/ℓ,
the advancing probability is considerably reduced. For the sake of simplicity,
we will neglect here the noise in the U1 state, and the motor always move
forward towards the minimum, advancing a multiple of the potential period,
nℓ, each cycle.
Expressing the transition position from the state U2 to U1, x ≡ (n−1)ℓ+z
(see Fig. 3.7), the mean displacement per cycle is 〈δx〉 = ℓ∑ ∞ n=−∞ np(n), with
p(n) the probability of falling at any position between n(ℓ − 1) and nℓ,
〈δx〉 = ℓ
∞
∑
n=−∞
ℓ
n
0
1 (ℓ(n − 1)+z + Fτ/λ)2
dz√
exp − . (3.19)
2πDτ 2Dτ
For an external force F = λℓ/(2τ), the motor at state U1 after the transition
time τ is positioned at half the period of the potential from the initial position.
In this case, the probability of advancing forward or backwards (+nℓ or
−nℓ) is exactly the same and the mean displacement is δx = 0. This force
corresponds to the stall force of the motor if λℓ/(2τ) < U/ℓ, otherwise, the
stall force is U/ℓ. Defining the two dimensionless parameters α = vτ/ℓ, the
ratio of the lifetime of the state U2 and the sliding time on the state U1, and
β ≡ ℓ/(4Dτ) 1/2 , the ratio of the potential period and the diffusive displacement
in U2, we can express the stall force as,
F 0
s = λv min
1, 1
2α
where the super-index 0 stands for mean field.
Mean time lapse per cycle
, (3.20)
The time lapse corresponding to advancing a distance nℓ is the sum of the U2
life-time τ and the sliding time on the U1 state, δt = τ +ts, with ts =(ℓ−z)/v,
see Fig. 3.7. The mean time lapse is then 〈δt〉 = ∑ ∞ n=−∞ δt(n)p(n), with p(n)
the probability of falling at any position between n(ℓ − 1) and nℓ,
〈δt〉 =
∞
∑
n=−∞
ℓ
0
dz τ +
ℓ − z
v
1 (ℓ(n − 1)+z + Fτ/λ)2
√ exp −
2πDτ 2Dτ
(3.21)
Finally the mean velocity V 0 ≡〈˙x〉 is the ratio of Eq. 3.19 and Eq. 3.21,
which in terms of the dimensionless parameters α, β and the dimensionless
force f ≡ F/(λv), can be expressed as