CHEM 498Q / 630QMolecular modeling of proteins

Winter 2010 Term

Instructor:Guillaume LamoureuxConcordia University, Montréal, Canada

1

NONBONDED nbxmod 5 atom cdiel shift vatom vdistance vswitch -

cutnb 14.0 ctofnb 12.0 ctonnb 10.0 eps 1.0 e14fac 1.0 wmin 1.5

!adm jr., 5/08/91, suggested cutoff scheme

!

!V(Lennard-Jones) = Eps,i,j[(Rmin,i,j/ri,j)**12 - 2(Rmin,i,j/ri,j)**6]

!

!epsilon: kcal/mole, Eps,i,j = sqrt(eps,i * eps,j)

!Rmin/2: A, Rmin,i,j = Rmin/2,i + Rmin/2,j

!

!atom ignored epsilon Rmin/2 ignored eps,1-4 Rmin/2,1-4

!

!carbons

C 0.000000 -0.110000 2.000000 ! ALLOW PEP POL ARO

! NMA pure solvent, adm jr., 3/3/93

CA 0.000000 -0.070000 1.992400 ! ALLOW ARO

! benzene (JES)

CC 0.000000 -0.070000 2.000000 ! ALLOW PEP POL ARO

! adm jr. 3/3/92, acetic acid heat of solvation

CD 0.000000 -0.070000 2.000000 ! ALLOW POL

! adm jr. 3/19/92, acetate a.i. and dH of solvation

CE1 0.000000 -0.068000 2.090000 !

! ! ! for propene, yin/adm jr., 12/95

CE2 0.000000 -0.064000 2.080000 !

! ! ! for ethene, yin/adm jr., 12/95

CM 0.000000 -0.110000 2.100000 ! ALLOW HEM

! Heme (6-liganded): CO ligand carbon (KK 05/13/91)

CP1 0.000000 -0.020000 2.275000 0.000000 -0.010000 1.900000 ! ALLOW

! alkane update, adm jr., 3/2/92

CP2 0.000000 -0.055000 2.175000 0.000000 -0.010000 1.900000 ! ALLOW

! alkane update, adm jr., 3/2/92

CP3 0.000000 -0.055000 2.175000 0.000000 -0.010000 1.900000 ! ALLOW

! alkane update, adm jr., 3/2/92

CPA 0.000000 -0.090000 1.800000 ! ALLOW HEM

! Heme (6-liganded): porphyrin macrocycle (KK 05/13/91)

CPB 0.000000 -0.090000 1.800000 ! ALLOW HEM

! Heme (6-liganded): porphyrin macrocycle (KK 05/13/91)

CPH1 0.000000 -0.050000 1.800000 ! ALLOW ARO

! adm jr., 10/23/91, imidazole solvation and sublimation

CPH2 0.000000 -0.050000 1.800000 ! ALLOW ARO

! adm jr., 10/23/91, imidazole solvation and sublimation

par_all22_prot.inp

2

NONBONDED nbxmod 5 atom cdiel shift vatom vdistance vswitch -

cutnb 14.0 ctofnb 12.0 ctonnb 10.0 eps 1.0 e14fac 1.0 wmin 1.5

!adm jr., 5/08/91, suggested cutoff scheme

ctonnb ctofnb

r

r

“vswitch”(for Lennard-Jones)

“shift”(for electrostatics)

ctofnb

ctonnb ctofnb

r“fswitch”

Cutoff schemes

Uelec(r) =qiqjr

(1− r2

ctofnb2

)2

3

Periodic boundary conditions

Lattice size should be large enough to accommodate tumbling motions of the solute and to preserve a layer of solvent of at least 2 Debye lengths between the images.

4

Numerical integration

Non-integrable system of differential equations

Finite-difference approximations :

We can calculate ri(t+!t) from ri(t) and ri(t–!t) :

ri(t+ δt) = 2ri(t)− ri(t− δt) + δt2Fi(t)

mi

(Verlet formula, 1967)

miai(t) = Fi(t)

vi(t+12δt) !

ri(t+ δt)− ri(t)

δt

vi(t− 12δt) "

ri(t)− ri(t− δt)

δt

ai(t) !vi(t+

12δt)− vi(t− 1

2δt)

δt

Finite-difference version of Newton’s equations :

miri(t+ δt)− 2ri(t) + ri(t− δt)

δt2= Fi(t)

5

Choice of time step (!t)

Rule of thumb :

!t should be at least one tenth of the shortest period of oscillation of the system, but ideally one twentieth.

O–H stretch : ~3500 cm–1 ! ! ~10 fs ! !t ~ 0.5 fs

C=O stretch : ~1700 cm–1

N–O stretch : ~1600 cm–1

C=C stretch : ~1600 cm–1 ! !t ~ 1.0 fs

To avoid using a 0.5 fs time step, the X–H bonds can be kept rigid (using the SHAKE algorithm).

time step

time step

period

(too large!)

(OK)

6

How long should the simulation be ?

Typical time for crossing an energy barrier :

τ = τo e∆G‡/kBT

∆G‡

The problem gets worse if the system has multiple barriers…

1 kcal mol–1 ! 1.2 ps–1 (“1.2 crossings per ps”)

5 kcal mol–1 ! 1.4 ns–1

10 kcal mol–1 ! < 1 ms–1

Using :τ0 = 1 ps

7

“Thermostatted” Newton’s equations(Nosé–Hoover)

miri = Fi −miriη

Qη = 2K −NfkBT 2K =∑

j

mj r2jwith

2 times the total instantaneous kinetic energy of the systemnumber of degrees

of freedom

12kBT

According to the equipartition theorem, each degree of freedom of the system should have a kinetic energy of .

Nf = 3N −Nc

inertia factor

variable friction coefficient

Boltzmann constant

8