6.4 Brownian Dynamics with Hydrodynamic Interactions
89
Here L is a typical length scale, in our case the length of a single ratchet element,
v is the average stationary velocity of the particle, in our case we used ||v cx |. The
larger the Péclet number, the more net drift predominates over diffusion.
We performed the simulation in dimensionless units. Energy is in units of kT .
Distance is in units of the separation distance l 0 and time is in units of l 2
0 /D 0 , We
used in the simulations the following parameters: l 0 = 1, L = 1, κ = δ −2 = 100,
radius = 0.25, D 0 = 1., A = 7.0, ω = 5., F 0 = V 0 /L = 4.0, t = 0.00125,
the simulation time was t = 1250, which corresponds to 1 × 10 6 steps. The
corresponding dimension units are: l 0 = 8.nm, L = 8.nm, κ = 6.43pN/nm,
radius = 2.0nm, D 0 = kT /(6πηa) = 1.226 × 10 −9 m 2 /s, A = 3.60pN,
t = 52.2ns, t = 65.2ps, ω = 0.6GH z. F 0 = 2.057pN.
In Brownian dynamics simulations with hydrodynamic interactions the size of
the physically meaningful time step is restricted to values which are sufficiently
long t m i D 0 /kT . In our case m i D 0 /kT = 3.4fs where we have used m i =
138 × 10 −24 Kg for the protein dimer unit of 2.5Å of radius.
The average velocity of a molecular motor is a function of the load force resisting
the motor’s advancement. One of the characteristic of a molecular motor is the load
force-velocity curve.
In Fig. 6.4 we show v cx as a function of the load Force F load . At the stationary
state, the ratio SE (v cx ) / v cx ≤ 10 −6 , where SE (v cx ) is the standard error of
the mean velocity v cx .
In the range −4 ≤ F load ≤ 0 we observe, the motor continue with a
positive velocity in spite of the negative load force (motor effect). We also observe
a substantially increase of the motor velocity in the case with hydrodynamic
interaction as compared without it.
6.4.1 Efficiency
The motor efficiency is defined as the ratio of the output work to input energy
η =
˙
W
˙
E in
(6.27)
where ˙
W = |F load v cx | is the average work done against the load per unit of time
and ˙
E in is the average input power, both quantities averaged with respect to all
random processes and time, see Chap. 6, 100. of Reference [29]. We can estimate
the efficiency in our case the periodic force F p = A|sin(ωt)|, then the average input
power per cycle, ˙
E in , will be
˙
E in = =F p v cx (t) + |F load v|
(6.28)
89
Here L is a typical length scale, in our case the length of a single ratchet element,
v is the average stationary velocity of the particle, in our case we used ||v cx |. The
larger the Péclet number, the more net drift predominates over diffusion.
We performed the simulation in dimensionless units. Energy is in units of kT .
Distance is in units of the separation distance l 0 and time is in units of l 2
0 /D 0 , We
used in the simulations the following parameters: l 0 = 1, L = 1, κ = δ −2 = 100,
radius = 0.25, D 0 = 1., A = 7.0, ω = 5., F 0 = V 0 /L = 4.0, t = 0.00125,
the simulation time was t = 1250, which corresponds to 1 × 10 6 steps. The
corresponding dimension units are: l 0 = 8.nm, L = 8.nm, κ = 6.43pN/nm,
radius = 2.0nm, D 0 = kT /(6πηa) = 1.226 × 10 −9 m 2 /s, A = 3.60pN,
t = 52.2ns, t = 65.2ps, ω = 0.6GH z. F 0 = 2.057pN.
In Brownian dynamics simulations with hydrodynamic interactions the size of
the physically meaningful time step is restricted to values which are sufficiently
long t m i D 0 /kT . In our case m i D 0 /kT = 3.4fs where we have used m i =
138 × 10 −24 Kg for the protein dimer unit of 2.5Å of radius.
The average velocity of a molecular motor is a function of the load force resisting
the motor’s advancement. One of the characteristic of a molecular motor is the load
force-velocity curve.
In Fig. 6.4 we show v cx as a function of the load Force F load . At the stationary
state, the ratio SE (v cx ) / v cx ≤ 10 −6 , where SE (v cx ) is the standard error of
the mean velocity v cx .
In the range −4 ≤ F load ≤ 0 we observe, the motor continue with a
positive velocity in spite of the negative load force (motor effect). We also observe
a substantially increase of the motor velocity in the case with hydrodynamic
interaction as compared without it.
6.4.1 Efficiency
The motor efficiency is defined as the ratio of the output work to input energy
η =
˙
W
˙
E in
(6.27)
where ˙
W = |F load v cx | is the average work done against the load per unit of time
and ˙
E in is the average input power, both quantities averaged with respect to all
random processes and time, see Chap. 6, 100. of Reference [29]. We can estimate
the efficiency in our case the periodic force F p = A|sin(ωt)|, then the average input
power per cycle, ˙
E in , will be
˙
E in = =F p v cx (t) + |F load v|
(6.28)