1Department of Chemistry, University of South Alabama,
Mobile, AL 36688,
2Department of Pharmacology, University of California
at San Diego, La Jolla, CA 92093-0365,
3EMBL Meyerhofstr 1, 69012, Heidelberg, Germany.
Abbreviations
BD = Brownian Dynamics; EI =; Electrostatic Inleracdons; HI = HydrodynamicInteractions;
MC = Monte-Carlo; MD = molecular dynamics; PB = Poisson-Boltzmann; TIM=
triose phosphate isomerase; WENDS = Weighted Ensemble Brownian DynamicsSimulations.
Introduction
In 1827 Robert Brown, a British botanist, observed pollen grains moving erratically in water. It was not until 1905 that Albert Einstein and Marian von Smoluchowski, independently, explained this phenomenon. What Brown had observed was the effect of the water molecules randomly colliding with the large pollen particle. These random collisions lead to an erratic displacement, i. e. diffusion, of the pollen particle in water. This chaotic motion is commonly referred to as Brownian motion. Therefore Brownian motion is really a diffusion process. Using the concept of a one-dimensional random walk, Einstein found that the average displacement of a particle in the x direction is
where t is time and D is the diffusion coefficient which is
where kB, is Boltzmann's constant, T is the absolute temperature, h is the solvent viscosity, and r is the solute's hydrodynamic radius.
The dynamics of the diffusion process can be simulated using Newton's second law of motion. The form of Newton's equation of motion used in simulating Brownian motion is
where m is the mass of the diffusing particle, dv/dt is the particle's acceleration, F(x) is the systematic interaction force, z is the friction coefficient and is equal to kBT/D, n is the velocity and f(t) is a fluctuating, statistical force that represents the random component of the particle's interactions with the solvent. The first term on the right hand side of equation 3 is the force due to intra- and intermolecular interactions while the second term represents a dynamical friction experienced by the particle. The third term is characteristic of the Brownian motion which is continuously fluctuating. Two principal assumptions are made for f(t). These are that f(t) is independent of the particle velocity and that it varies extremely rapidly compared to the variations of the velocity. Equation 3 is known as Langevin's equation.
The diffusive behavior of a panicle can be modeled using the Langevin equation. For example, in certain reactions, the two reactants must come in contact in order for a reaction to take place. In these types of reactions the rate determining step may be due to the diffusion of the two reactants in the solvent. Computer simulations, commonly known as Brownian dynamics simulations, are employed at the molecular level to study these diffusion-controlled processes and is aiding in the design of systems with prescribed transport or kinetic properties.
This type of bimolecular diffusion-limited reaction is prevalent in various enzyme-substrate encounters. Brownian dynamics simulations are being used to study the encounter rate of an enzyme with a substrate, protein-protein association. and nucleic acid-protein association which are a few of the many biochemical processes that occur on characteristic time scales of diffusion-controlled systems.
In ``Encyclopedia of Computational Chemistry''
Eds. Schleyer,P.v.R., Allinger,N.L., Clark,T., Gasteiger,J., Kollman,P.A. and Schaefer,H.F., Schreiner,P.R.
John Wiley & Sons: Chichester, UK, (1998) 1, 141-154.