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30 3 Self-organization and cooperativity of weakly coupled molecular motors
Ui
U
ℓ
v
√ Dτ
Fig. 3.4. Two state potential of a molecular motor. The state U1 is a totally asymmetric potential
with period ℓ with localized transitions at each minima. The unbound state U2 is modeled
as a flat potential with delocalized transitions after a deterministic time τ.
xi( j + 1)=xi( j)+
U2
U1
x
δki1v −∑ W
k=i
′ (xi( j) − xk( j)) − Fδi1
∆t (3.12)
+(−2Dlog(χ)∆t) 1/2 cos(2πχ), (3.13)
where the last term is a discretization of the thermal noise ξi(t), using a uniform
distributed stochastic variable, χ ∈ (0,1) (Sancho et al., 1982). For the
transitions between states U1 and U2, we define the following rules: From U1
to U2,
If(ℓ + δ >| mod(xi,ℓ) |>ℓ − δ), ki = 2, (3.14)
which corresponds to a transition localized in a small interval δ from the
minimum of the potential (see Fig.3.2). For the transition from U2 to U1, the
motor remains at U2 for a deterministic interval of time τ, and then changes
to U1.
Simulation parameters
The fixed simulation parameters are
• The time step ∆t is fixed to 5 × 10 −5 s.
• The periodicity of the potential U1 is fixed to 8 nm which is roughly the
length of a heterodimer of α-β tubulin (a microtubule monomer).
• The interval around the potential minimum for motor transition is fixed to
δ = 0.002ℓ.
• The de-excitation time of the state U2, τ = 1/w2 is fixed to 50 ms.