Thermodynamics and Kinetics of a Brownian Motor

R Dean Astumian

Shreya Ray 20091069

Prashant Beeraka 20091032

Langevin’s Thermal Noise

Brownian Motion of Particles in solvent has two components: •Fluctuating force that changes direction and magnitude very fast, averages to zero over time = THERMAL NOISE •Viscous drag force damps the motion induced by these fluctuations

The amplitude of thermal noise depends on: •Viscosity (which again dampens it) •Temperature

Feynman’s Ratchet and Pawl Allows motion only in one direction (sawtooth potential) Diffusion acts in both directions, but ratchet moves only in one direction: So can we have a perpetual motion machine that violates the Second Law of Thermodynamics? (Can anisotropy drive a motor?)

NO. Despite the anisotropy, without an energy supply, probabilities of moving in either direction are exactly EQUAL. (counterintuitive?) We need to couple diffusion to thermal gradient/gravity/electrostatic or any other force field in order to extract work.

Noise harms, Noise heals

Electrophoresis, centrifugation, chromatography: Long-range gradients used. Thermal noise causes band-broadening. Must be turned off each time a new batch of particles is added.

Coupling short-range non-equilibrium fluctuations in an anisotropic medium with diffusive Brownian motion can bias the direction of motion. In fact, the thermal noise provides part of the energy required to for transport across energy barrier! So, small voltages are required; can be used continuously. Fluctuation-driven transport requires: •Thermal Noise to cause Brownian motion •Anisotropy arising from the structure of the medium •Energy supplied either by external variation of constraints or by a chemical reaction far from equilibrium (the fluctuation)

The Anisotropic Periodic Potential

•Interdigitated electrodes deposited on glass using photolithography •Linear array of dipoles aligned head-to-tail, via aggregation/polymerisation •Alpha=anisotropy parameter

Flashing Ratchet : Fluctuating Potential

•Uphill transport against sufficiently small Fext •Sawtooth potential is periodically turned on and off •On-state favours trapping inside well- going back is difficult •Off-state favours thermal diffusion to the right side too, although gaussian probability distribution drifts downhill. •Toff should be sufficient for this diffusion to occur. It should however not be high enough for velocity drift. •Larger particles diffuse slowly- principle of separation? Too broad •Assisted by tilting (gravity) or constant electric field.

Rocking Ratchet: Fluctuating Force

•Square wave modulation of sawtooth potential •Overall, favours motion in the direction of sharper edge of sawtooth because of appearance of traps in the other direction. •Mean velocity increases with the push given by the square wave, but this shouldn’t be so large as to mask the effects of sawtooth. •Mean velocity increases when this process is assisted by thermal noise upto an optimum T, beyond which again noise becomes unwanted. (too shallow wells)

Chemical Modulation

•Our particle catalyses hydrolysis of SH •Local conc. Of S and H vary with position because of dipole array, hence a sawtooth potential •Away from eqm transition probability is same in all directions •Near eqm we have trabsitions from charged to uncharged form and vice-versa •Charged state is affected by dipole, neutral state is not: an on-off switch •Mean velocity increases away from eqm, as delG produces the sawtooth. Larger barriers, however, disfavour the process.

Molecular Pumps and Motors

•External oscillating fields have been shown to drive transport by the sodium-potassium triphosphatase ion pump •Externally imposed electric oscillations can substitute for energy from ATP-hydrolysis to power uphill transport of ions •It is not clear whether molecular motors too use this mechanism