Rolf Luchsinger Oct. 98



10. Molecular Motors



Experimental facts:

• nano-microscale
  ( $\Rightarrow$ thermal noise relevant)
• motors move along linear, periodic structure
  (Kinesin and Dynein on Microtubule)
  (Myosin on Actin)
• energy (ATP) is needed
• two states (bound $\Leftrightarrow$ unbound)
• isothermal ( $\eta_c = 1-T_2/T_1 = 0$)

FORCED THERMAL RACHET
(Magnasco,1993)

LANGEVIN Eq. for an over damped particle



where

f(x) = f(x+l)

$$F_{eff}=\frac{1}{\tau} \int_0^{\tau} F(t) dt = 0$$

$$\langle \xi(t)\xi(t')\rangle =2 k_B T \delta(t-t')$$

Corresponding FOKKER-PLANCK Eq.
(Smoluchoswki equation)

$$\frac{\partial P(x,t)}{\partial t}= \mu \left[-\frac{\partial... ...f(x)+F(t)\Bigg) P+ k_B T \frac{\partial}{\partial x} P \right]$$

with $\mu = 1/\alpha$ (mobility)

Rewrite the Smoluchowski Eq.

$$\frac{\partial P(x,t)}{\partial t}+ \frac{\partial}{\partial x} J(x,t) = 0$$

$$J(x,t) = \mu \left[- k_B T \frac{\partial}{\partial x}P +\Bigg(f(x)+F(t)\Bigg)P\right]$$

Stationary solution:

$$\frac{\partial P(x,t)}{\partial t} = 0\quad \Rightarrow J = const.$$

Simplest case: Piecewise linear potential:
$f^- = \;\;\;Q/a$
f⁺ = -Q/b

$$P(x) = \left\{\begin{array}{r@{\; ;\;}l } c_1 \exp(\frac{(F... ...{J}{\mu(F+f^+)} & \;\;\;0 \le x \le b \end{array} \right.$$

1) P(-0) = P(0) (continuous)
2) P(-a) = P(b) (periodic)
3) $\int_{-a}^b P(x) dx = 1$ (normalized)

$$1) \Rightarrow c_1-c_2 = \frac{J}{\mu}\left( \frac{1}{F+f^+}-\frac{1}{F+f^-}\right)$$

$$1) + 2)\quad \Rightarrow \quad \frac{c_1}{c_2} = \frac{\ex... ...rac{F b - Q}{k_B T}) - 1} {\exp(\frac{-F a - Q}{k_B T}) - 1}$$

Investigate different cases:

a.) F = 0:

$$\frac{c_1}{c_2} = \frac{\exp(\frac{- Q}{k_B T}) - 1} {\exp(\frac{- Q}{k_B T}) - 1} = 1 \quad \Rightarrow J = 0$$

b.) $F \ne 0,\;\; a = b \Leftrightarrow V ={\rm sym}$:

$$\frac{c_1}{c_2} = \frac{\exp(\frac{Fa- Q}{k_B T}) - 1} {\exp(\frac{-Fa-Q}{k_B T}) - 1} \ne 1$$

$$\tilde{F} := -F\; \Rightarrow \;\frac{\tilde{c_1}}{\tilde{c_2... ...\; \Rightarrow \;\tilde{c_1} = c_2, \tilde{c_2} = c_1 \quad 3)$$

$$\tilde{c_1}-\tilde{c_2} = \frac{J}{\tilde{J}}(c_1-c_2) \; \Rightarrow \; \tilde{J} = -J$$

$$J_{av} = \frac{1}{\tau}\int_0^{\tau} J(F(t)) dt = \frac{1}{2} (J(F)+J(-F)) = 0$$

c.) $F \ne 0,\;\; a \ne b \Leftrightarrow V ={\rm asym}$:

$$J(F) = \frac{P_2^2 \sinh(lF/2k_B T)}{k_B T (l/Q)^2 \{\cosh[... ...F/2)/k_B T]-\cosh(lF/2k_B T)\} -(l/Q)P_1P_2\sinh(lF/2k_B T)}$$

where $P_1 = \Delta +\frac{l^2-\Delta^2}{4}\frac{F}{Q},\;\; P_2 = \left(1-\frac{\Delta F}{2Q}\right)^2 - \left(\frac{lF} {2Q}\right)^2$
$\;\;\;\quad \quad l = a + b, \;\; \Delta = b-a$


$\quad J(-F) \ne -J(F)\quad \Rightarrow \quad J_{av} \ne 0$

The average probability current J_av as a function of the magnitude of the external forcing.

The average probability current J_av as a function of the temperature.





A net probability current J_av results for:

• Asymmetric Potential
• Finite Temperature
• Forcing with time correlations

There are physical applications of the Forced Thermal Ratchet (Molecular Pumps, Filters) but the relevance for biological molecular motors is not clear.

TWO-STATE MODEL
(Prost et al, 1994)


Instead of the time-correlated forcing, this model assumes a two-state system (bound/unbound) for the motor. The external action (ATP hydrolysis) drives the transition rates between the two states away from their equilibrium values.

$$\frac{\partial P_1}{\partial t}+ \frac{\partial}{\partial x} J_1 = -w_1(x)P_1+w_2(x)P_2$$

$$\frac{\partial P_2}{\partial t}+ \frac{\partial}{\partial x} J_2 = w_1(x)P_1-w_2(x)P_2$$

$$J_i = \mu_i \left[- k_B T \frac{\partial}{\partial x}P_i - P_i\frac{\partial}{\partial x} V_i \right]$$

This set of equations can be transformed to an equation for the total probability current J=J₁+J₂ as a function of P=P₁+P₂

$$J = \mu_{eff} \left[- k_B T \frac{\partial}{\partial x}P - P\frac{\partial}{\partial x} V_{eff} \right]$$

$$\mu_{eff} = \mu_1 \lambda +(1-\lambda)\mu_2,\;\;\; \lambda= \frac{P_1}{P}$$

$$V_{eff}(x)-V_{eff}(0) = \int\limits_0^x dx' \frac{\mu_1 \lamb... ...\partial}{\partial x'}V_2}{\mu_1 \lambda + \mu_2 (1-\lambda)}$$

$$\quad \quad \;\; + k_B T \ln (\mu_{eff})\vert _0^x$$

If the effective force
F_eff = (V_eff(x+l)-V_eff(x))/l
vanishes, then J vanishes.


One can show that

• V₁, V₂ symmetric $\Rightarrow F_{eff} = 0$
• $w_1(x) = w_2(x)\exp((V_1(x)-V_2(x))/k_B T)\\ \Rightarrow F_{eff} = 0$ (detailed balance)
• detailed balance broken by energy input
  through ATP $\Rightarrow F_{eff} \ne 0 \Rightarrow J\ne 0$

An effective force can be obtained by breaking detailed balance.

The basic mechanism for the Two-State System can be understood from the Figure. A particle makes a transition from 1 to 2. It diffuses in state 2. After a certain time there is a finite probability that it will fall into the next well during the transition from 2 to 1.

Matching of transition rates, diffusion rate and potential form is important.

SUMMARY

• Basic understanding of molecular motors
  emerges
• Theories based on Fokker-Planck equations
• Three important ingredients:
  -broken symmetry
  -noise
  -correlated forcing/broken detailed balance
• Stochastic resonance behavior
• So far the importance of noise in biological motors not proven, but if models are correct, then:

Noise keeps us going!

REFERENCES:


M.O. Magnasco, Phys.Rev.Lett. 71, 1477 (93)
J. Prost et al, Phys.Rev.Lett. 72, 2652 (94)
F. Jülicher et al, Rev.Mod.Phys. 69, 1269 (97)
R.D. Astumian, Science 276, 917 (97)


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