CHE3935. Lecture 10 Brownian Dynamics

Transcription

1 CHE3935 Lecture 10 Brownian Dynamics 1

2 What Is Brownian Dynamics? Brownian dynamics is based on the idea of the Brownian motion of particles So what is Brownian motion? Named after botanist Robert Brown Random drifting of particles in a fluid due to collisions of fluid molecules with the particle. First identified by Lucretius in 60 BC for dust in air and used a proof of the existence of atoms 2

3 Brownian Motion Lucretius, On the Nature of Things, c. 60 BC: "Observe what happens when sunbeams are admitted into a building and shed light on its shadowy places. You will see a multitude of tiny particles mingling in a multitude of ways... their dancing is an actual indication of underlying movements of matter that are hidden from our sight... It originates with the atoms which move of themselves [i.e., spontaneously]. Then those small compound bodies that are least removed from the impetus of the atoms are set in motion by the impact of their invisible blows and in turn cannon against slightly larger bodies. So the movement mounts up from the atoms and gradually emerges to the level of our senses, so that those bodies are in motion that we see in sunbeams, moved by blows that remain invisible. Although Lucretius published first, it was Robert Brown who got his name associated with the effect. Brown was a Scottish botanist. In 1827 Brown was using a microscope to study pollen suspended in water and observed the jittery motion of small particles (amyloplasts). The effect had previously been reported for charcoal particles by Jan Ingenhousz in 1784 and Lucky for us, the effect is not named after him. 3

4 Brownian Motion the Einstein Connection One of Einstein s four Annus Mirabilis papers was Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen English: On the movement of small particles suspended in a stationary liquid demanded by the molecular-kinetic theory of heat. This paper dealt with Brownian motion. Einstein s paper provided a way to determine the diffusion coefficient for a particle in a fluid, the diameter of atoms, and the value of Avogadro s constant. 4

5 What Is Brownian Dynamics? Describes the motion of particles in solution without explicitly accounting for the motion of the solvent atoms i.e., replace the explicit solvent with an implicit solvent. Why would we want to do this? (think ) Time scale issues, system size issues, accounting for disparate length scales or time scales in a single system (multi-scale). In other words, BD seeks to reduce the number of degrees of freedom in a problem by averaging out many less important degrees of freedom, while preserving the essential physics (thermodynamics) of the important degrees of freedom. This goal is not exclusive to BD there are many methods used to reduce the degrees of freedom. 5

6 What Is Brownian Dynamics? It is a simplified version of Langevin dynamics. So, what is Langevin dynamics? Method for describing the evolution of a subset of the degrees of freedom in a system. m i r i = F i (t) γ i m i r i + R i (t) where F i are the internal forces, γ i is the friction coefficient, and R i denotes forces due to random fluctuations caused by interactions with the solvent molecules. 6

7 What Is Brownian Dynamics? Relation between Brownian and Langevin dynamics: Start with the Langevin equation: m i r i = F i (t) γ i m i r i + 2γ i m i k B TR i (t). Assume that over long times that the acceleration is zero (or negligible): 0 = F i (t) γ i m i r i + 2γ i m i k B TR i (t). Using the Einstein relation (Annus Mirabilis!) we get r i = F i t ζ i + 2D i R i t, where ζ i = γ i m i = k B T/D i. Note that BD includes friction and stochastic noise. 7

8 Brownian Dynamics Equations of Motion r i t + t = r i t D i k B T F i t t + R i t The stochastic (noise) force has the properties: R i t = 0 R i t 2 =2D i t General observations F i t is assumed to be constant over t. The longer t the smaller the noise. For slow processes over macroscopic times the noise goes to zero. 8

9 Example: Reaction-Diffusion Problem Reactions on cell membranes: Comparison of continuum theory and Brownian dynamics simulations, J. Chem. Phys. 123, (2005). Biochemical transduction of signals received by living cells typically involves molecular interactions and enzyme-mediated reactions at the cell membrane, a problem that is analogous to reacting species on a catalyst surface or interface. We have developed an efficient Brownian dynamics algorithm that is especially suited for such systems and have compared the simulation results with various continuum theories through prediction of effective enzymatic rate constant values. We specifically consider reaction versus diffusion limitation, the effect of increasing enzyme density, and the spontaneous membrane association/dissociation of enzyme molecules. In all cases, we find the theory and simulations to be in quantitative agreement. This algorithm may be readily adapted for the stochastic simulation of more complex cell signaling systems. 9

10 Reactions on Cell Membranes Problem: Activation of substrate (reactant) by membrane enzymes (E). Substrate is converted from inactive (open circles) to active (filled circles) with rate k act [S] 2. Substrate inactivation occurs with rate k i [S]. Dilute enzyme limit, one enzyme. Density of activated substrate falls to zero away from enzyme with effective rate coefficient. High density limit, multiple enzymes, mean-field theory. Reaction rate may be controlled by diffusion of S to the enzyme. 10

11 Reactions on Cell Membranes Problem: Activation of substrate (reactant) by membrane enzymes (E). The reaction rate is modeled as 2ππ n i(r,t) = r=s eff k act n i s, t = k act n i Goal: Find α = k eff act /D, the dimensionless effective enzymatic rate constant. 11

12 BD Code for Cell Membrane Reactions Run the code:./membr inputfile outputfile Input file contents: eta_rec: dimensionless receptor density. N_Rec: number of receptors per simulated area. S: effective radius of receptor-enzyme complex. N_Ras: number of Ras molecules per simulated area. enzyme_on_off: 0 or 1. In simulations with stable (non-dissociating) enzymes, enzyme_on_off=0. For enzymes with final lifetime, enzyme_on_off=1. k_r_on: rate constant of cytosolic enzyme binding (pseudo-first-order reaction). k_r_off: enzyme dissociation rate constant. Da: Damkohler number. Diff: Ras diffusion coefficient. kappa: dimensionless rate constant of Ras activation (second-order reaction). TSTEP: timestep, if a particle reaches the activation layer surrounding an enzyme-receptor complex. ds: thickness of the activation layer. t_out: time interval for output values. 12

13 Running the Code Login to Puccini: ssh puccini.che.pitt.edu Copy the BD.tar file: cp ~karlj/web-docs/classes/che3935/bd.tar. Untar the file: tar xvf BD.tar Go to the Brownian directory: cd Brownian Look at the two input files: membr_case1.data and membr_case2.data Try running each of them:./membr membr_case1.data case1.out./membr membr_case2.data case2.out Play with the parameters and see what happens 13