Lectures in Theoretical Biophysics K. Schulten and I. Kosztin Department of Physics and Beckman Institute University of Illinois at Urbana–Champaign 405 N. Mathews Street, Urbana, IL 61801 USA (April 23, 2000) Contents 1 Introduction 1 2 Dynamics under the Influence of Stochastic Forces 3 2.1 Newton’s Equation and Langevin’s Equation . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Stochastic Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 How to Describe Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 Ito calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.5 Fokker-Planck Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.6 Stratonovich Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.7 Appendix: Normal Distribution Approximation . . . . . . . . . . . . . . . . . . . . . 34 2.7.1 Stirling’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.7.2 Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3 Einstein Diffusion Equation 37 3.1 Derivation and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.2 Free Diffusion in One-dimensional Half-Space . . . . . . . . . . . . . . . . . . . . . . 40 3.3 Fluorescence Microphotolysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.4 Free Diffusion around a Spherical Object. . . . . . . . . . . . . . . . . . . . . . . . . 48 3.5 Free Diffusion in a Finite Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.6 Rotational Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4 Smoluchowski Diffusion Equation 63 4.1 Derivation of the Smoluchoswki Diffusion Equation for Potential Fields . . . . . . . 64 4.2 One-Dimensional Diffuson in a Linear Potential . . . . . . . . . . . . . . . . . . . . . 67 4.2.1 Diffusion in an infinite space Ω∞ = ]−∞,∞[ . . . . . . . . . . . . . . . . . . 67 4.2.2 Diffusion in a Half-Space Ω∞ = [0,∞[ . . . . . . . . . . . . . . . . . . . . . 70 4.3 Diffusion in a One-Dimensional Harmonic Potential . . . . . . . . . . . . . . . . . . . 74 5 Random Numbers 79 5.1 Randomness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.2 Random Number Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.2.1 Homogeneous Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.2.2 Gaussian Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.3 Monte Carlo integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 i ii CONTENTS 6 Brownian Dynamics 91 6.1 Discretization of Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 6.2 Monte Carlo Integration of Stochastic Processes . . . . . . . . . . . . . . . . . . . . . 93 6.3 Ito Calculus and Brownian Dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6.4 Free Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.5 Reflective Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 7 The Brownian Dynamics Method Applied 103 7.1 Diffusion in a Linear Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 7.2 Diffusion in a Harmonic Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 7.3 Harmonic Potential with a Reactive Center . . . . . . . . . . . . . . . . . . . . . . . 107 7.4 Free Diffusion in a Finite Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 7.5 Hysteresis in a Harmonic Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 7.6 Hysteresis in a Bistable Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 8 Noise-Induced Limit Cycles 119 8.1 The Bonhoeffer–van der Pol Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 119 8.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 8.2.1 Derivation of Canonical Model . . . . . . . . . . . . . . . . . . . . . . . . . . 121 8.2.2 Linear Analysis of Canonical Model . . . . . . . . . . . . . . . . . . . . . . . 122 8.2.3 Hopf Bifurcation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 8.2.4 Systems of Coupled Bonhoeffer–van der Pol Neurons . . . . . . . . . . . . . . 126 8.3 Alternative Neuron Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 8.3.1 Standard Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 8.3.2 Active Rotators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 8.3.3 Integrate-and-Fire Neurons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 8.3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 9 Adjoint Smoluchowski Equation 131 9.1 The Adjoint Smoluchowski Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 9.2 Correlation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 10 Rates of Diffusion-Controlled Reactions 137 10.1 Relative Diffusion of two Free Particles . . . . . . . . . . . . . . . . . . . . . . . . . . 137 10.2 Diffusion-Controlled Reactions under Stationary Conditions . . . . . . . . . . . . . . 139 10.2.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 11 Ohmic Resistance and Irreversible Work 143 12 Smoluchowski Equation for Potentials: Extremum Principle and Spectral Ex- pansion 145 12.1 Minimum Principle for the Smoluchowski Equation . . . . . . . . . . . . . . . . . . . 146 12.2 Similarity to Self-Adjoint Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 12.3 Eigenfunctions and Eigenvalues of the Smoluchowski Operator . . . . . . . . . . . . 151 12.4 Brownian Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 13 The Brownian Oscillator 161 13.1 One-Dimensional Diffusion in a Harmonic Potential . . . . . . . . . . . . . . . . . . . 162 April 23, 2000 Preliminary version CONTENTS iii 14 Fokker-Planck Equation in x and v for Harmonic Oscillator 167 15 Velocity Replacement Echoes 169 16 Rate Equations for Discrete Processes 171 17 Generalized Moment Expansion 173 18 Curve Crossing in a Protein: Coupling of the Elementary Quantum Process to Motions of the Protein 175 18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 18.2 The Generic Model: Two-State Quantum System Coupled to an Oscillator . . . . . 177 18.3 Two-State System Coupled to a Classical Medium . . . . . . . . . . . . . . . . . . . 179 18.4 Two State System Coupled to a Stochastic Medium . . . . . . . . . . . . . . . . . . 182 18.5 Two State System Coupled to a Single Quantum Mechanical Oscillator . . . . . . . 184 18.6 Two State System Coupled to a Multi-Modal Bath of Quantum Mechanical Oscillators189 18.7 From the Energy Gap Correlation Function ∆E[R(t)] to the Spectral Density J(ω) . 192 18.8 Evaluating the Transfer Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 18.9 Appendix: Numerical Evaluation of the Line Shape Function . . . . . . . . . . . . . 200 Bibliography 203 Preliminary version April 23, 2000 iv CONTENTS April 23, 2000 Preliminary version Chapter 1 Introduction 1 2 CHAPTER 1. INTRODUCTION April 23, 2000 Preliminary version Chapter 2 Dynamics under the Influence of Stochastic Forces Contents 2.1 Newton’s Equation and Langevin’s Equation . . . . . . . . . . . . . . . . 3 2.2 Stochastic Differential Equations . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 How to Describe Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 Ito calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.5 Fokker-Planck Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.6 Stratonovich Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.7 Appendix: Normal Distribution Approximation . . . . . . . . . . . . . . 34 2.7.1 Stirling’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.7.2 Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.1 Newton’s Equation and Langevin’s Equation In this section we assume that the constituents of matter can be described classically. We are interested in reaction processes occuring in the bulk, either in physiological liquids, membranes or proteins. The atomic motion of these materials is described by the Newtonian equation of motion d2 ∂ m r = − V(r ,... ,r ) (2.1) i dt2 i ∂r 1 N i where r (i = 1,2,... ,N) describes the position of the i-th atom. The number N of atoms is, of i course, so large that solutions of Eq. (2.1) for macroscopic systems are impossible. In microscopic systems like proteins the number of atoms ranges between 103 to 105, i.e., even in this case the solution is extremely time consuming. However, most often only a few of the degrees of freedom are involved in a particular biochemical reaction and warrant an explicit theoretical description or observation. For example, in the case of transportoneissolelyinterestedinthepositionofthecenterofmassofamolecule. Itiswellknown thatmoleculartransportincondensedmediacanbedescribedbyphenomenologicalequationsmuch simpler than Eq. (2.1), e.g., by the Einstein diffusion equation. The same holds true for reaction 3 4 Dynamics and Stochastic Forces processes in condensed media. In this case one likes to focus onto the reaction coordinate, e.g., on a torsional angle. In fact, there exist successful descriptions of a small subset of degrees of freedom by means of Newtonian equations of motion with effective force fields and added frictional as well as (time dependent) fluctuating forces. Let us assume we like to consider motion along a small subset of the whole coordinate space defined by the coordinates q ,... ,q for M (cid:28) N. The equations which 1 M model the dynamics in this subspace are then (j = 1,2,... ,M) d2 ∂ d µ q = − W(q ,... ,q ) − γ q + σ ξ (t). (2.2) j dt2 j ∂q 1 M j dt j j j j The first term on the r.h.s. of this equation describes the force field derived from an effective potential W(q ,... ,q ), the second term describes the velocity (dq ) dependent frictional forces, 1 M dt j and the third term the fluctuating forces ξ (t) with coupling constants σ . W(q ,... ,q ) includes j j 1 M the effect of the thermal motion of the remaining n−M degrees of freedom on the motion along the coordinates q ,... ,q . 1 M Equationsoftype(2.2)willbestudiedindetailfurtherbelow. Wewillnot“derive”theseequations from the Newtonian equations (2.1) of the bulk material, but rather show by comparision of the predictions of Eq. (2.1) and Eq. (2.2) to what extent the suggested phenomenological descriptions apply. To do so and also to study further the consequences of Eq. (2.2) we need to investigate systematically the solutions of stochastic differential equations. 2.2 Stochastic Differential Equations We consider stochastic differential equations in the form of a first order differential equation ∂ x(t) = A[x(t),t] + B[x(t),t]·η(t) (2.3) t subject to the initial condition x(0) = x . (2.4) 0 In this equation A[x(t),t] represents the so-called drift term and B[x(t),t]·η(t) the noise term which will be properly characterized further below. Without the noise term, the resulting equation ∂ x(t) = A[x(t),t]. (2.5) t describes a deterministic drift of particles along the field A[x(t),t]. Equations like (2.5) can actually describe a wide variety of phenomena, like chemical kinetics or the firing of neurons. Since such systems are often subject to random perturbations, noise is added to the deterministic equations to yield associated stochastic differential equations. In such cases as well as in the case of classical Brownian particles, the noise term B[x(t),t]·η(t) needs to be specified on the basis of the underlying origins of noise. We will introduce further below several mathematical models of noise and will consider the issue of constructing suitable noise terms throughout this book. For this purpose, one often adopts a heuristic approach, analysing the noise from observation or from a numerical simulation and selecting a noise model with matching characteristics. These characteristics are introduced below. Before we consider characteristics of the noise term η(t) in (2.3) we like to demonstrate that the one-dimensional Langevin equation (2.2) of a classical particle, written here in the form µq¨ = f(q) − γq˙ + σξ(t) (2.6) April 23, 2000 Preliminary version 2.3. HOW TO DESCRIBE NOISE 5 is a special case of (2.3). In fact, defining x ∈ R2 with components x = m, and q˙, x = mq 1 2 reproduces Eq. (2.3) if one defines f(x /m)−γx /m σ 0 ξ(t) A[x(t),t] = (cid:18) 2 1 (cid:19), B[x(t),t] = (cid:18) (cid:19), and η(t) = (cid:18) (cid:19) . (2.7) x 0 0 0 1 The noise term represents a stochastic process. We consider only the factor η(t) which describes the essential time dependence of the noise source in the different degrees of freedom. The matrix B[x(t),t] describes the amplitude and the correlation of noise between the different degrees of freedom. 2.3 How to Describe Noise We are now embarking on an essential aspect of our description, namely, how stochastic aspects of noise η(t) are properly accounted for. Obviously, a particular realization of the time-dependent process η(t) does not provide much information. Rather, one needs to consider the probability of observingacertainsequenceofnoisevaluesη ,η ,... attimest ,t ,.... Theessentialinformation 1 2 1 2 is entailed in the conditional probabilities p(η1,t1;η2,t2;...|η0,t0;η−1,t−1;...) (2.8) when the process is assumed to generate noise at fixed times t , t < t for i < j. Here p( | ) is i i j the probability that the random variable η(t) assumes the values η ,η ,... at times t ,t ,..., if 1 2 1 2 it had previously assumed the values η0,η−1,... at times t0,t−1,.... An important class of random processes are so-called Markov processes for which the conditional probabilities depend only on η and t and not on earlier occurrences of noise values. In this case 0 0 holds p(η1,t1;η2,t2;...|η0,t0;η−1,t−1;...) = p(η1,t1;η2,t2;...|η0,t0) . (2.9) This property allows one to factorize p( | ) into a sequence of consecutive conditional probabilities. p(η ,t ;η ,t ;...|η ,t ) = p(η ,t ;η ,t ;...|η ,t ) p(η ,t |η ,t ) 1 1 2 2 0 0 2 2 3 3 1 1 1 1 0 0 = p(η ,t ;η ,t ;...|η ,t ) p(η ,t |η ,t ) p(η ,t |η ,t ) 3 3 4 4 2 2 2 2 1 1 1 1 0 0 . . . (2.10) The unconditional probability for the realization of η ,η ,... at times t ,t ,... is 1 2 1 2 p(η ,t ;η ,t ;...) = p(η ,t ) p(η ,t |η ,t ) p(η ,t |η ,t ) ··· (2.11) 1 1 2 2 (cid:88) 0 0 1 1 0 0 2 2 1 1 η 0 where p(η ,t ) is the unconditional probability for the appearence of η at time t . One can 0 0 0 0 conclude from Eq. (2.11) that a knowledge of p(η0,t0) and p(ηi,ti|ηi−1,ti−1) is sufficient for a complete characterization of a Markov process. Before we proceed with three important examples of Markov processes we will take a short detour and give a quick introduction on mathematical tools that will be useful in handling probability distributions like p(η0,t0) and p(ηi,ti|ηi−1,ti−1). Preliminary version April 23, 2000