Molecular Dynamics: Review

Molecular Simulations

• NMR or X-ray structure refinements

• Protein structure prediction• Protein folding kinetics and

mechanics• Conformational dynamics• Global optimization• DNA/RNA simulations• Membrane proteins/lipid

layers simulations

Molecular Dynamics

From Lecture 6 (Robert):• MD is our approximation to how molecules explore theirpotential energy surface in the real world• – The atoms are “heated” by giving them a distribution ofvelocities corresponding to temperature we wish to simulate• – The wiggling and jiggling of the atoms is then obtained byintegrating the Newtonian laws of motion• – This gives us the Ei's of all states “i” occupied at thattemperature as long as we simulate long enough

I. Force Fields

Force Fields: Typical Energy Functions

20

20

12 6

1( )

2

1( )

2

[1 cos( )]2

( )

[ ]

rbonds

angles

n

torsions

improper

i j

elec ij

ij ij

LJ ij ij

U k r r

k

Vn

V improper torsion

q q

r

A B

r r

Bond stretches

Angle bending

Torsional rotation

Improper torsion (sp2)

Electrostatic interaction

Lennard-Jones interaction

Bonding Terms: bond stretch

• Most often Harmonic

20 )(

2

1rrkV r

bondsbond

Harmonic Potential

bond length

Vbond

r0

Bonding Terms: angle bending

• Most often Harmonic

• CHARMM force field’s Urey-Bradley angle term:

20 )(

2

1 kVangles

angle

Harmonic Potential

angle

Vangle

0

20 )(

2

1sskV UB

UBUB

This UB term is only found in CHARMM force field to optimize the fit to vibrational spectra. s: the 1,3-distance.

Mackerell et al. J. Phys. Chem. B 102, 3586, 1998

Bonding Terms: Torsions

• Torsion energy: rotation about a bond (dihedral angles)

)]cos(1[2

nV

Utorsions

ntorsion

Vn: force constant n: periodicity of the angle ( determines how many peaks and wells in the potential, often from 1-6 ) : phase of the angle (often 0º or 180º)

i

lj

k

i-j-k-l

Bonding Terms: Improper Torsions

• Improper torsion is not a regular torsion angle. It is used to describe the energy of out-of-plane motions. It is often necessary for planar groups, such as sp2 hybridized carbons in carbonyl groups and in aromatic rings, because the normal torsion terms described above is not sufficient to maintain the planarity (w~0).

)]1802cos(1[22

improperimproper

VU

20 )(

2

improper

wimproper

kU

or

j

li

k

i-j-k-l

Non-bonded Terms

• Electrostatic interactions (Coulomb’s Law)

• Lennard-Jones interactions

ji ij

jielec r

qqV

41

ji ij

ij

ij

ijijLJ

rrV 6

6

12

12

4

Coulomb Potential

pair distance

Vele

c

LJ Potential

pair distance r/sigma

VL

J

~1/r

II. Solvation Models

Solvation Models

• Explicit solvent models– Fixed charge models: SPC, SPC/E, TIP3P,

TIP4P, TIP5P, ST2,…– Polarizable water models: TIP4P/FQ, POL5,

MCDHO,…

• Implicit Solvent models– Poisson-Boltzman solver (Delphi, Honig)– Generalized Born Model (Still)– Karplus’ EEF1 model – Benoit Roux’s Spherical Solvent Boundary

Potential (SSBP)

Explicit Water modelsSPC, SPC/E, TIPnP, POL5

Water Model Geometries

Water Model Parameters• SPC, SPC/E (Berendsen)• TIP3P, TIP4P, TIP5P (Jorgensen)• TIP4P/FQ, POL5 (Berne)

Implicit Solvent ModelsPBF, GB

Continuum Solvent Model

continuum solvent=80

=1-4protein

III. Molecular Dynamics

Molecular Dynamics

• Solve Newton’s equation for a molecular system:

amF

Integrator: Verlet Algorithm

)()()()(2)( 42 tOtatttrtrttr

)()(2

1)()()( 32 tOtatttvtrttr

)()(2

1)()()( 32 tOtatttvtrttr

Start with {r(t), v(t)}, integrate it to {r(t+t), v(t+t)}:

{r(t), v(t)}

{r(t+t), v(t+t)}

The new position at t+t:

Similarly, the old position at t-t:

(1)

(2)

Add (1) and (2):

Thus the velocity at t is:

(3)

)())()((2

1)()( 2tOttrttr

ttrtv

(4)

Typical MD Flowchart

Program MYMD simple MD program

call init initialization t = 0 do while (t .lt. tmax) MD loop call force (x, f, en) calculate the force call integrate (x, f, en) integrate equation of motion t = t + delt call sample sample averages enddo stop end

Periodic Boundary ConditionsMinimum Image

Central simulationbox

rc

One MD example

Determining voltage threshold for translocation of dsDNA through Si3N4 pores

To establish the threshold field required to drive dsDNA through a 2.0 nanometer diameter pore. The 3.9 V path caused the partial unzipping of the DNA strands prior to reaching the center of the membrane.

http://www.ks.uiuc.edu/Research/nanopore/

Historical Perspective on MD

The Next Generation in MD

• Current longest MD simulations: microsecond vs. time scale of many biologically interesting phenomena is millisecond

• Anton, Desmond• Scientific advances &

Drug Discovery

Faculty in Computer Science Department at Columbia University, till1986

D. E. Shaw & Co., Inc., founded in 1988

1994, pointed by President Clinton, President's Council of Advisors on Science and Technology

Acknowledgement

• Powerpoint slices from Ruhong Zhou