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Unformatted text preview: Langevin and Brownian Dynamics Overview Langevin Dynamics of a Single Particle We consider a spherical particle of radius r immersed in a viscous fluid and suppose that the dynamics of the particle depend on two forces: A drag force arising from friction between the particle and the viscous fluid: Fdrag = -v where v is the velocity of the particle and is a friction constant. For a spherical particle, Stokes' law gives = 6r , where is the viscosity. A random force R(t) arising from the collisions of the solvent molecules with the particle. This is a phenomenological model in the sense that the solvent molecules are not modeled explicitly. Jay Taylor (ASU) APM 530 - Lecture 10 Fall 2010 1 / 29 Langevin and Brownian Dynamics Overview To make further progress, we need to specify R(t). Our first assumption is that on average the solvent fluctuations are non-directional: E R(t) = 0. Our second assumption concerns the covariance function of the solvent fluctuations: Cov R(t), R(t ) = 2 (t - t )I, where is a parameter and I is the 3 3 identity matrix. This is motivated by two assumptions: Solvent fluctuations are isotropic and therefore uncorrelated along orthogonal directions. The particle is much heavier than the individual solvent molecules and so R(t) fluctuates much more rapidly than v. Jay Taylor (ASU) APM 530 - Lecture 10 Fall 2010 2 / 29 Langevin and Brownian Dynamics Overview Together, these assumptions imply that R(t) is a white noise process on R3 , i.e., R(t) = Wt where Wt is a three-dimensional Brownian motion. Substituting the drag force and the white noise process into Newton's laws of motions leads to the following Langevin equation: mv = -v + Wt , which can be rewritten as a SDE: dvt = - vt dt + dWt m m -vt + dWt . Jay Taylor (ASU) APM 530 - Lecture 10 Fall 2010 3 / 29 Langevin and Brownian Dynamics Overview Furthermore, if we write vt = vt , vt , v t (1) (2) (1) (2) (3) (3) (i) W = Wt , Wt , Wt , then each component of v satisfies a SDE of the form dvt (i) = -vt dt + dWt , (i) which we recognize as the Ornstein-Uhlenbeck equation. From the previous lecture, we know that the solution is a diffusion process given by the stochastic integral t -t vt = e v0 + e -(t-s) dWt . 0 Jay Taylor (ASU) APM 530 - Lecture 10 Fall 2010 4 / 29 Langevin and Brownian Dynamics Overview Furthermore, if the initial velocity v0 is prescribed, then the distribution of vt at time t is a three-dimensional Gaussian distribution with the following mean vector and covariance matrix: m(t) = v0 e -t 2 (t) = 1 - e -2t I. 2 In particular, notice that as t , the limits m(t) 0 and (t) ( 2 /2)I, are independent of the initial velocity. Since limits of sequences of Gaussian variables are also Gaussian, it follows that the stationary distribution of the velocity of the particle is just N(0, ( 2 /2)I). Jay Taylor (ASU) APM 530 - Lecture 10 Fall 2010 5 / 29 Langevin and Brownian Dynamics Overview This last result can be used to identify . If the solvent is held at constant temperature T , then in the limit t , the distribution of vt will tend to the Maxwellian distribution N(0, (kB T /m)I). Thus 2 kB T = 2 m which along with = /m implies that 2 = 2 2kB T 2kB T = m m2 2 2 = m = 2kB T . This is an example of the fluctuation-dissipation theorem: the energy imparted to the particle by solvent fluctuations is on average balanced by the energy lost to friction. Jay Taylor (ASU) APM 530 - Lecture 10 Fall 2010 6 / 29 Langevin and Brownian Dynamics Overview We can also solve for the trajectory of the particle by integrating the velocity process: Xt = X0 + vs ds t = X0 + e -s v0 + 0 e -(s-u) dWu ds 0 0 t t 1 -t u = X0 + 1-e v0 + e dWu e -s ds 0 u 1 t -t -(t-u) = X0 + 1-e v0 + 1-e dWu , 0 s t where we have interchanged the order of integration in passing to the third line. Jay Taylor (ASU) APM 530 - Lecture 10 Fall 2010 7 / 29 Langevin and Brownian Dynamics Overview This shows that if the initial position X0 = x0 is fixed, then Xt is Gaussian with mean vector and covariance matrix 1 m(t) = x0 + 1 - e -t v0 2 2 t t = 1 - e -(t-s) ds I 0 2 = 2t - 3 + 4e -t - e -2t I 2 3 where I is the 3 3 identity matrix. Jay Taylor (ASU) APM 530 - Lecture 10 Fall 2010 8 / 29 Langevin and Brownian Dynamics Overview This last result can be used to calculate the mean squared displacement of the particle in time t: E (Xt - x0 ) 2 = 2 v0 2 1 - e -t + 3 2 -t -2t 2t - 3 + 4e -e . 2 3 When t is small, this is approximately 2 E (Xt - x0 )2 = v0 t 2 + O(t 3 ) which shows that the particle moves linearly over very short time intervals. Jay Taylor (ASU) APM 530 - Lecture 10 Fall 2010 9 / 29 Langevin and Brownian Dynamics Overview In contrast, for large t >> -1 , the mean squared displacement is approximately 3 2 3kB T E (Xt - x0 )2 = t = t 2 which shows that the particle moves diffusively over long time intervals, with diffusion coefficient D= kB T kB T = 6r (Einstein-Stokes relation). Jay Taylor (ASU) APM 530 - Lecture 10 Fall 2010 10 / 29 Langevin and Brownian Dynamics Overview Brownian Dynamics of a Single Particle If the friction constant 1 is large, then the motion of the particle is approximately Brownian: 1 = X0 + 1 - e -t v0 + 1/2 kB T X0 + Wt . t 0 Xt 1 - e -(t-u) dWu This reflects the dominance of the stochastic forces acting on the particle over the inertial forces that lead to short-range linear motion. Jay Taylor (ASU) APM 530 - Lecture 10 Fall 2010 11 / 29 Langevin and Brownian Dynamics Langevin Dynamics Langevin Dynamics for Molecules Langevin dynamics have been used to address several needs in molecular dynamics simulations: as phenomenological models of solvent-macromolecular interactions; to enhance sampling of molecular conformations; to stabilize simulations using multiple timestep methods. Jay Taylor (ASU) APM 530 - Lecture 10 Fall 2010 12 / 29 Langevin and Brownian Dynamics Langevin Dynamics The Langevin equation for a molecule with potential energy function U is: MXt = -U(Xt ) - MXt + R(t) where M is the diagonal matrix of molecular masses; is a damping constant; R is a stationary Gaussian process with E Rt = 0 and Cov Rt , Rs = 2kB T M(t - s). Remark: The variance of the white noise process has been chosen to satisfy the fluctuation-dissipation theorem. Jay Taylor (ASU) APM 530 - Lecture 10 Fall 2010 13 / 29 Langevin and Brownian Dynamics Langevin Dynamics To put this on solid mathematical footing, we rewrite the Langevin Eqn. as a system of It^ SDEs: o MdVt dXt where Wt is a 3N-dimensional Brownian motion; M1/2 is diagonal with elements Mii . Remark: Because of the white noise and drag forces, this system is neither Hamiltonian nor symplectic. 1/2 = -U(Xt )dt - MVt dt + (2kB T M)1/2 dWt = Vt dt Jay Taylor (ASU) APM 530 - Lecture 10 Fall 2010 14 / 29 Langevin and Brownian Dynamics Langevin Dynamics One algorithm that has been used to numerically solve the Langevin equation for MD simulations is the Brooks-Brnger-Karplus (BBK) u discretization: Vn+1/2 = Vn + M-1 t [-U(Xn ) - MXn + Rn ] 2 Here Xn+1 = Xn + tVn+1/2 t Vn+1 = Vn+1/2 + M-1 -U(Xn+1 ) - MXn+1 + Rn+1 . 2 R1 , R2 , are IID Gaussian RVs with mean 0 and covariance matrix 2kB T M/t. This scheme reduces to the velocity Verlet method when = 0. Jay Taylor (ASU) APM 530 - Lecture 10 Fall 2010 15 / 29 Langevin and Brownian Dynamics Langevin Dynamics The Damping Constant The choice of determines the relative strengths of the inertial forces and the external stochastic forces. The choice of depends on the purposes of the simulation. When noise is added to stabilize MTS methods, is taken as small as possible, e.g., = 20 ps-1 . When Langevin dynamics are used to model solvent effects, can be chosen according to Stokes' law. For protein atoms exposed to water at room temperature, this gives 50 ps-1 . In the diffusive limit, can be chosen to reproduce the measurable translation diffusion coefficient Dt of the molecule: Dt = kB T M (M = molecular mass). Jay Taylor (ASU) APM 530 - Lecture 10 Fall 2010 16 / 29 Langevin and Brownian Dynamics Langevin Dynamics Brownian Dynamics Langevin dynamics can be simplified if is so large that inertial forces are negligible. In this case the momentum derivatives can be dropped: Xt = -U(Xt ) + Rt which can be expressed as the following SDE: dXt = - D U(Xt )dt + (2D)1/2 dWt kB T where D = kB T -1 is the diffusion coefficient. In this case, the molecular configuration Xt is itself a diffusion process and the dynamics are known as Brownian dynamics. Jay Taylor (ASU) APM 530 - Lecture 10 Fall 2010 17 / 29 Langevin and Brownian Dynamics Langevin Dynamics More sophisticated BD models also account for the hydrodynamic interactions between the solvent and the macromolecule: dXt = T(Xt )U(Xt )dt + D(Xt )dt + SdWt where T = (Tij ) is the hydrodynamic tensor, which accounts for the transmission of frictional forces through the molecule. D = (kB T )T = SST is the diffusion tensor. S can be calculated using Cholesky decomposition or Chebyshev approximations. Jay Taylor (ASU) APM 530 - Lecture 10 Fall 2010 18 / 29 Langevin and Brownian Dynamics Transcription Bubble Kinetics Alexandrov et al. (2009) Toward a Detailed Description of the Thermally Induced Dynamics of the Core Promoter. PLoS Comp. Biol. 5: e1000313. Background Eukaryotic protein-coding genes are transcribed from DNA to mRNA by RNA polymerase II. Transcription initiation involves several steps: Basal transcription factors bind upstream of the gene; The basal apparatus recruits pol II to the promoter. pol II binds to the transcription start site (TSS). pol II requires ssDNA at the TSS to initiate transcription. Thermal noise can cause spontaneous separation of dsDNA. This study used Langevin dynamics to study the kinetics of transcription bubble formation in core promoter sequences. Jay Taylor (ASU) APM 530 - Lecture 10 Fall 2010 19 / 29 Langevin and Brownian Dynamics Transcription Bubble Kinetics Eukaryotic Transcription Complexes Jay Taylor (ASU) APM 530 - Lecture 10 Fall 2010 20 / 29 Langevin and Brownian Dynamics Transcription Bubble Kinetics Peyrard-Bishop-Dauxois (PBD) Model The PBD model is a one-dimensional Hamiltonian model of the transverse opening of dsDNA with the following potential energy: U= where N is the number of base pairs (bp's); yn is the transverse displacement of the complementary bases of the n'th bp; The first term is a Morse potential for each displacement The second term is a harmonic potential with a nonlinear coupling constant to account for stacking interactions between bp's. Jay Taylor (ASU) APM 530 - Lecture 10 Fall 2010 21 / 29 N n=1 Dn e -an yn 2 k -(yn +yn-1 ) 2 -1 + 1 + e (yn - yn-1 ) 2 Langevin and Brownian Dynamics Transcription Bubble Kinetics Cooperative interactions between neighboring base pairs promotes bubble formation. Alexandrov et al., Fig. 2. Jay Taylor (ASU) APM 530 - Lecture 10 Fall 2010 22 / 29 Langevin and Brownian Dynamics Transcription Bubble Kinetics A Langevin PBD Model The effects of thermal noise on bubble formation can be investigated by incorporating the PBD potential energy function into the Langevin equation: mdvt dyt = -U(y)dt - mVt dt + = vt dt, mdWt where is the friction constant and Wt is an N-dimensional Brownian motion. According to Alexandrov et al. (2006), = 0.05 ps-1 gives good agreement between the model and experiment (e.g., melting transitions and nuclease digestion). Jay Taylor (ASU) APM 530 - Lecture 10 Fall 2010 23 / 29 Langevin and Brownian Dynamics Transcription Bubble Kinetics For this study, 1000 independent 1 ns LD simulations were carried out for eight different mammalian pol II core promoters. Jay Taylor (ASU) APM 530 - Lecture 10 Fall 2010 24 / 29 Langevin and Brownian Dynamics Transcription Bubble Kinetics Fig. 6: Bubble formation in collagen promoter and intron This suggests that bubble formation occurs more readily in promoters than in nonpromoter sequences. Jay Taylor (ASU) APM 530 - Lecture 10 Fall 2010 25 / 29 Langevin and Brownian Dynamics Transcription Bubble Kinetics Fig. 3: Bubble probabilities by length and amplitude Jay Taylor (ASU) APM 530 - Lecture 10 Fall 2010 26 / 29 Langevin and Brownian Dynamics Transcription Bubble Kinetics Fig. 4: Bubble lifetimes (ps) by length Jay Taylor (ASU) APM 530 - Lecture 10 Fall 2010 27 / 29 Langevin and Brownian Dynamics Transcription Bubble Kinetics Conclusions pol II promoters are prone to bubble formation near the TSS. TSS bubbles are around 10 bp long and have mean lifetimes of 5-10 ps. Bubble formation is less prevalent in non-promoter sequences. Bubble formation is less concentrated along non-classical G/C rich promoters such as HSV-1 UL11 and snRNA. However, this could be due to model inadequacy. Jay Taylor (ASU) APM 530 - Lecture 10 Fall 2010 28 / 29 Langevin and Brownian Dynamics References References Alexandrov, B.S., Wille, L.T., Rasmussen, K.O., Bishop, A.R. and Blagoev, K.B. (2006) Bubble statistics and dynamics in double-stranded DNA. Phys. Rev. E 74: 050901. Chaikin, P. M. and Lubensky, T. C. (1997) Principles of condensed matter physics. Cambridge University Press. Schlick, T. (2006) Molecular Modeling and Simulation. Springer. Jay Taylor (ASU) APM 530 - Lecture 10 Fall 2010 29 / 29 ...
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