Organization of NAMD Tutorial FilesOrganization of NAMD Tutorial Files

2.1.1. RMSD for individual 2.1.1. RMSD for individual residuesresiduesObjective: Find the average RMSD over time of each residue in the protein using VMD. Display the protein with the residues colored according to this value.

2.1.2 Maxwell2.1.2 Maxwell--Boltzmann DistributionBoltzmann DistributionObjective: Confirm that the kinetic energy distribution of the atoms in a system corresponds to the Maxwell distribution for a given temperature.

( ) 3/ 22( ) exp kk B k

B

p k Tk Tε

ε επ

− = −

0 1 2 3 40

0.1

0.2

0.3

0.4

/k Bk Tε

( )kp ε

0

( ) 1d pε ε∞

=∫normalization condition:

2.1.3 Energies2.1.3 EnergiesObjective: Plot the various energies (kinetic and the different internal energies) as a function of temperature.

sample:

2.1.4 Temperature Fluctuations2.1.4 Temperature Fluctuations2

1

2

1

2 1( ) ( ) ( )3 3

( ) 2( ) , ( )3 3

N

i iiB B

Ni i i

i ii B B

T t K t m v tk Nk

m v tX t X tNk Nk

ε=

=

= =

= ≡ =

∑

∑

Temperature time series:

According to the central limit theorem:

2

0 02 2 33 2 3 2

i iB

B B

m vT N X k T Tk k

= = = = thermodynamic temperature

0 0

2 222 2 2 20 0

2

2 2/3 3T TX X NN N

σ σ σ= − = ⇒ = =

( )1/ 2 2200

20

34( ) exp3 4

T TTp xN Tπ

− − = −

Analysis of MD DataAnalysis of MD Data

1. Structural properties2. Equilibrium properties3.Non-equilibrium properties

Can be studied via both equilibriumand/or non-equilibrium MD simulations

Time Correlation FunctionsTime Correlation Functions⟩−⟨=⟩⟨=− )0()'()'()()'( BttAtBtAttCAB

since ρeq is t independent !

BA ≠BA =

cross-auto- correlation function

Correlation time: ∫∞

=τ0

)0(/)( AAAAc CtCdt

Estimates how long the “memory” of the system lasts

In many cases (but not always): )/exp()0()( ctCtC τ−=

Free Diffusion (Brownian Motion) of ProteinsFree Diffusion (Brownian Motion) of Proteinsin living organisms proteins exist and function in a viscous environment, subject to stochastic (random) thermal forcesthe motion of a globular protein in a viscous aqueous solution is diffusive

2 ~ 3.2R nm

e.g., ubiquitin can be modeled as a spherical particle of radius R~1.6nm and mass M=6.4kDa=1.1x10-23 kg

Diffusion can be Studied by MD Simulations!Diffusion can be Studied by MD Simulations!

solvate

ubiquitin in water

1UBQPDB entry:

total # of atoms: 7051 = 1231 (protein) + 5820 (water) simulation conditions: NpT ensemble (T=310K, p=1atm), periodic BC, full electrostatics, time-step 2fs (SHAKE) simulation output: Cartesian coordinates and velocities of all atoms saved at every other time-step (10,000 frames = 40 ps) in separate DCD files

Goal: calculate D and Goal: calculate D and ττ

theory:

by fitting the theoretically calculated center of mass (COM) velocity autocorrelation function to the one obtained from the simulation

2 /0

20

( ) ( ) (0) tvv

B

C t v t v v ek T DvM

τ

τ

−= =

= =

simulation: consider only the x-componentreplace ensemble average by time average

( )xv v→

1

1( )N i

vv i n i nn

C t C v vN i

−

+=

≈ =− ∑

( ), , #i n nt t i t v v t N≡ = ∆ = = of frames in vel.DCD

(equipartition theorem)

Velocity Autocorrelation FunctionVelocity Autocorrelation Function

0.1 psτ ≈

2 11 2 13.3 10B

x

D k Tv m s

γτ − −

= / == ≈ ×

0 200 400 600 800 1000

050100150200250300

100 200 300 400

50100150200250300

100 200 300 400

50100150200250300

0 2 4 6 8 10-40-2002040

time [ps]

V x(t

) [m

/s]

( )vvC t

[ ]time fs

[ ]time fs

/( ) tvv

DC t e τ

τ−=

Fit

Probability distribution of Probability distribution of

( ) ( )( )

1/ 22 2 2

2

( ) 2 exp

exp

p v v v v

D v D

π

τ π τ

−= − /2

= /2 − /2

, ,x y zv

, ,x y zv v≡with

Maxwell distribution of Maxwell distribution of vvCOMCOM

COM velocity [m/s]

( ) ( )3/ 2 2 2

3/ 2 22

( )

exp 4

2 exp

( ) )

2

( ) (x y z x y z

B B

P v

D v D v

M Mvvk T k T

dv p v p v p v dv dv dv

dv

dv

τ π τ π

π

−

−

/2 − /2

−

=

=

=