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8.7 Bio or Molecular Motors 187

problem of the generation of directional motion without temperature or force ﬁeld

gradients.

Thus, let us consider [207] the molecular motor as a Brownian particle in local

thermal equilibrium with ﬁxed temperature T . The action of the motor is generated

by generalized forces of two types. The ﬁrst type f

ext

is connected to the mechan-

ical interaction of the motor with the surrounding environment and, in particular,

involves the forces of viscous friction. The second type Δμ is generated by the

difference of chemical potential, which is equal to free-energy per consumed “fuel”

molecule. The chemical potential difference Δμ for the process AT P → ADP+P

is given by

Δμ = μ(AT P) −μ(ADP) − μ(P) . (8.82)

At chemical equilibrium Δμ = 0, whereas it is positive when ATP is in excess and

negative when ADP is in excess.

The action of generalized forces leads to fuel consumption and the motion of

the motor. It is useful to introduce generalized currents to describe these effects:

the average rate of consumption of fuel molecules r (i.e., the average number of

ATP molecules hydrolyzed per unit time, per motor) and the average velocity of

the motor’s displacement v. The dependencies v( f

ext

, Δμ) and r( f

ext

, Δμ)arein

general nonlinear, since bio-motors often function far from thermal equilibrium

(Δμ  10 kT ). First let us consider the linear regime

(

Δμ  kT

)

.Inthisregime

linear response theory [207, 209] allows us to write:

v = λ

ext

+λ

Δμ

r = λ

ext

+λ

Δμ. (8.83)

Here λ

is the mobility coefﬁcient, λ

and λ

are the coefﬁcients of mechano-

chemical coupling , λ

is a generalized mobility relating ATP consumption and

chemical potential difference. According to the Onsager principle of the symme-

try of kinetic coefﬁcients λ

= λ

. Whenever f

ext

v<0, some work is per-

formed by the motor; whenever rΔμ<0, chemical energy is generated. A given

motor/ﬁlament system can work in eight different regimes. However, only four

regimes are of interest: when mechanical work and chemical energy have different

signs. If both f

ext

v and rΔμ are positive, there is no energy output from the system;

all work performed at the system is simply dissipated in the thermal bath. Cases for

which both f

ext

v and rΔμ are negative are forbidden by the second law of thermo-

dynamics. We shall now discuss a concrete model for the motion of bio-motors. We

restrict ourselves to the consideration of the so-called two-state model [210, 211].

In this model, energy consumption by a bio-motor leads to coordinated transitions

between states 1 and 2. These transitions could be described in terms of chemi-

cal kinetics. For each of the states the one-dimensional potential V

(x)(i = 1, 2)

is introduced (for systems with a variable number of particles it is more exact to

talk about free-energy), where x is a coordinate of the motor’s center of mass. It

188 8 The Appearance of Regular Fluxes Without Gradients

Fig. 8.13 The basic working

mechanism of a pulsating

ratchet [207]

is supposed that potential is periodic and spatially asymmetric (the symmetry of

the potential reﬂects the symmetry of the ﬁlament). Although the exact form of the

potential is unknown, experiments indicate that its period is about 10 nm [212].

According to the classiﬁcation represented in Sect. 8.4, this is a pulsating ratchet

(see Fig. 8.13). The mechanism of current generation within it is quite simple. Let

us consider a case when V

(x) has a saw-tooth potential with spatial asymmetry, and

(x) = const. When k

T << ΔV (ΔV is high in potential), the dynamics of the

motor splits into two stages: diffusion in potential V

(x), leading to the narrowing

of the diffusion bell, and transition to an exited state due to the breaking of the

phosphate bond, when dynamics of the diffusion bell are reduced to free symmetric

broadening. Because of the spatial asymmetry of the potential, the bio-motor could

transit to the neighboring right minimum with the probability proportional to the

stroked area of the probability density P. The value of the induced current depends

on the relation between times τ

and τ

of work of potentials V

and V

. The optimal

time could be deﬁned as a narrow diffusive peak formation time, which is approxi-

mately equal to the time of the rolling down from the top of the inclined plane to the

potential minimum. This time is deﬁned by parameters of the potential and mass of

the bio-motor. We can also evaluate the upper boundary of the time τ

: the diffusive

bell must not become so blurred that the considerable reverse current appears. More

complicated models of molecular motors are discussed in reviews [183, 207].

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Index

active control, 91

adenosine diphosphate, 186

adenosine triphosphate, 186

algorithm, 8

alternation effect, 150

amplitude, 22

amplitude spectral density, 23

amplitude spectrum, 22

analog realization, 130

angular momentum conservation, 98

aperiodic orbit, 89

aperiodic orbits control, 89

aperiodic signal, 90

attractor, 32, 36

attractor dimension, 46

autocorrelation function, 24, 144, 145

Belousoff-Zhabotinski reaction, 101

billiard, 96

binary calculus system, 15

bio-motor, 186

biological membrane, 185

biological motor, 185

bistable potential, 136

black-box, 35

breaking of spatial reﬂective symmetry, 163

broken symmetry, 160

Brownian motor, 163

Brownian particle, 138, 163

calculus system with the radix b,14

Cantor set, 43

capacity, 42

capture region, 76

Carno cycle, 159

cascade, 25

chaos, 14

chaotic behavior, 19

chaotic orbit, 19

chaotic oscillation synchronization, 101

chaotic scattering, 71

chaotic sea, 97

chaotic synchronization, 101

characteristic equation, 28

chemical cycle, 186

chemical potential, 187

Church’s thesis, 11

climate change, 135

climates, 155

colored noise, 136

command, 9

complete replacement, 103

complex system, 75

complexity, 8, 11

conditional Lyapunov exponent, 104

conﬁguration space, 96

continuous control, 81

continuous control with external force, 81

continuum, 13

control setup time, 65

controlling chaos, 51

controlling perturbation, 87

coordinate, 28

correlation function, 23

correlation tensor, 23

correlational dimension, 46

countable set, 14

criterion, 24

critical index, 30

Curie principle, 171

current direction, 184

decay of correlations, 24

delayed synchronization, 106

195

196 Index

deterministic ratchet, 182

diffusive ratchet, 172

diffusive transport, 159

dimension, 38

directed motion, 159

discrete parametric control, 53

disorder, 7

divergence, 25

driven subsystem, 108

driving subsystem, 108

Dufﬁng oscillator, 79

dynamic tunneling, 95

eigenvalue, 28

eigenvector, 28

Einstein-Brillouin-Keller tori, 95

elliptic ﬁxed point, 74

embedding dimension, 40

empty cell, 9

energy barrier, 92

energy spectrum, 24

energy splitting, 96, 98

equilibrium state, 33

ergodisity, 34

Euler constant, 15

Euler-Lagrange equations, 138

exponential divergence, 17, 54

exponential sensibility, 25

exponential sensitivity, 17

external alphabet, 9

external force, 81

external monochromatic ﬁeld, 92

external noise, 51

extrusion, 181

feedback control, 81

ﬂuctuating force ratchet, 173

ﬂuctuating potential ratchet, 172

focus, 32

Fokker-Planck equation, 164

fractal dimension, 45

free-energy, 187

Frobenius-Peron equation, 33

δ-function, 33

gedanken experiment, 36

generalized synchronization, 106

global cluster dimension, 45

global control, 77

global stochasticity region, 76

good approximation of irrational number, 15

gravitational perturbation, 135

enon mapping, 59, 66

enon-Heiles Hamiltonian, 20

Hamiltonian, 91

Hamiltonian system, 32

Hausdorff dimension, 41

Hausdorff measure, 41

Hermitian operator, 91

high-period orbit, 73

higher harmonics, 148

hyperbolic scattering, 71

hypersurface, 39

ice ages, 135

impact parameter, 98

individual random sequences, 13

information transmission, 129

informational dimension, 45

inorganic phosphate, 186

internal alphabet, 9

invariant curves, 21

invariant density, 32

invariant distribution function, 34

ionization, 92

irrational number, 14

irregular wave, 78

Jacobi matrix, 53

Jacobian matrix, 28

Klausius principle, 160

Klein bottle, 39

Kramers formula, 137

Langevin equation, 142

Langevin force, 141

limit cycle, 32, 36, 37

linearized dynamics, 54

local cellular dimension, 45

local control, 77

local stability, 84

logistic mapping, 54

Lorenz system, 105

Lyapunov exponent, 25

Lyapunov functions method, 115

manifold, 20, 38