Analysis of MD Simulations Data - Statistical Mechanics of Proteins - NAMD Tutorial (Part 2)

Transcription

1 Phys 45: Computational iological Physics Winter 5 Lecture 9- Analysis of MD Simulations Data - Statistical Mechanics of Proteins -. Structural properties.equilibrium properties 3. on-equilibrium properties Today: - Theoretical background followed by AMD tutorial (part Analysis AMD Tutorial (Part. Equilibrium properties.. RMSD for individual residues.. Maxwell-oltzmann Distribution..3 Energies..4 Temperature distribution..5 Specific Heat. on-equilibrium properties of protein.. Heat Diffusion.. Temperature echoes

2 Equilibrium (Thermodynamic Properties MD simulation microscopic information Statistical Mechanics macroscopic properties Phase space trajectory Γ[r(t,p(t] Ensemble average over probability density ρ(γ Statistical Ensemble Collection of large number of replicas (on a macroscopic level of the system Each replica is characterized by the same macroscopic parameters (e.g., T, PT = # of particles, =volume, T=temperature, P=pressure The microscopic state of each replica (at a given time is determined by Γ in phase space

3 Time vs Ensemble Average For t, Γ(t generates an ensemble with phase space density: ρ( Γ dγ = limdτ / t t Ergodic Hypothesis -Timeand Ensemble averages are equivalent, i.e., for any physical quantity A : A( r, p = A( Γ t ρ Time average: Ensemble average: A t = T T dt A[ r( t, p( t] A = dγρ( Γ A( Γ Thermodynamic Properties from MD Simulations Thermodynamic (equilibrium averages can be calculated via time averaging of MD simulation time series A i = A( t i Thermodynamic average MD simulation time series Finite simulation time means incomplete sampling!

4 Common Statistical Ensembles. Microcanonical (,,E: ρ ( Γ δ[ H ( Γ E] E ewton s eq. of motion. Canonical (,,T: ρ ( Γ = exp{[ F H ( Γ]/ k T} T 3. Isothermal-isobaric (,p,t ρ ( Γ = exp{[ G H ( Γ]/ k T} PT Langevin dynamics ose-hoover method Different simulation protocols [Γ(t Γ(t+δt ] sample different statistical ensembles Examples of Thermodynamic Observables Energies (kinetic, potential, internal, Temperature [equipartition theorem] Pressure [virial theorem] Thermodynamic derivatives are related to mean square fluctuations of thermodynamic quantities Specific heat capacity C v and C P Thermal expansion coefficient α P Isothermal compressibility β T Thermal pressure coefficient γ

5 Total (internal energy: Kinetic energy: Potential energy: Mean Energies E = E( i= t i M p j K = i= j= U = E K ( t m i j TOTAL KIETIC OD AGLE DIHED IMPRP ELECT DW ote:you can conveniently use namdplot to graph the time evolution of different energy terms (as well as T, P, during simulation Temperature From the equipartition theorem T = K 3k mv / = kt/ i ix Instantaneous kinetic temperature T = K 3k namdplot TEMP vs TS ote: in the TP ensemble - c, with c =3

6 From the virial theorem P The virial is defined as with W = 3 M j= Pressure rk H/ rk = = k T + W r j f w ( r = r dv ( r / dr j = 3 i, j> i w( r Instantaneous pressure function (not unique! P = ρkt+ W/ ij k T pairwise interaction Thermodynamic Fluctuations (TF δa [ A( t i A ] i = Mean Square Fluctuations (MSF [also called variance ] δa = ( A A = A A According to Statistical Mechanics, the probability distribution of thermodynamic fluctuations is ρ fluct δp δ δt exp kt δs

7 TF in T Ensemble In MD simulations distinction must be made between properly defined mechanical quantities (e.g., energy E, kinetic temperature T, instantaneous pressure P and thermodynamic quantities, e.g., T, P, For example: ut: Other useful formulas: C = ( E / T γ = ( P / T δ E = ktc = δp δp kt / βt δk δu = = 3 k T ( k ( C T 3k δuδp = k T ( γ ρk How to Calculate C?. From definition C = ( E / T / Perform multiple simulations to determine E E as a function of T, then calculate the derivative of E(T with respect to T. From the MSF of the total energy E C = δe / k T with δe = E E see AMD Tutorial:..5 Specific Heat

8 Analysis AMD Tutorial (Part. Equilibrium properties.. RMSD for individual residues.. Maxwell-oltzmann Distribution..3 Energies..4 Temperature distribution..5 Specific Heat. on-equilibrium properties of protein.. Heat Diffusion.. Temperature echoes Organization of AMD Tutorial Files

9 ... RMSD for individual residues Objective: Find the average RMSD over time of each residue in the protein using MD. Display the protein with the residues colored according to this value... Maxwell-oltzmann Distribution Objective: Confirm that the kinetic energy distribution of the atoms in a system corresponds to the Maxwell distribution for a given temperature. ( ( 3/ ε k p εk = kt εk exp π kt ( p ε k ε / kt k normalization condition: dε p( ε =

10 ..3 Energies Objective: Plot the various energies (kinetic and the different internal energies as a function of temperature. sample:..4 Temperature Fluctuations Objective: Simulate ubiquitin in an E ensemble, and analyze the temperature distribution. According to the central limit theorem T(t follows (approximately a normal (Gaussian distribution Gaussian (normal distribution: ( x x p( x = πσ exp, σ = x x σ probability density variance mean

11 Central Limit Theorem the sum of many independent identically distributed random variable is approximately normally distributed x = x i= i note that x i may not be normally distributed! x = x, σ = σ x = x, σ = σ / i i ( x / x px ( = ( πσ / exp σ..4 Temperature Fluctuations Temperature time series: Tt Kt mv t ( = ( = i i( 3k 3k i= mv ( t ε = = i i i Xi(, t Xi( t i= 3k 3k According to the central limit theorem: mv i i 3 T = X = = kt = T 3k 3k T T σ = X X = σ = σ / = 3 3 ( T T 4π T / 3 px ( = exp 3 4T thermodynamic temperature

12 For next class: : Continue the AMD tutorial. on-equilibrium properties of protein.. Heat Diffusion