Molecular Dynamics Simulation:

Understanding DNA melting

Dhananjay Bhattacharyya

Computational Science Division

Saha Institute of Nuclear Physics, Kolkata

E-mail: dhananjay.bhattacharyya@saha.ac.in

A

G

T

CJ.D. Watson (Biologist)

and F. Crick (Physicist)

Netropsin like drugs bind in

the B-DNA narrow and deep

minor groove

Actinomycin D like drugs make

their place in between two

stacked base pairs by distorting

the DNA double helix

Open

Semi-Open

Close

HIV-1 Protease

in action

Interactions between Biological

Molecules

• Coulomb Interaction between unlike

charges F = QiQj/ Rij

• Shape complementarity – van der Waals

interaction

• Hydrogen bond

• Covalent bond Rare cases

D

H

A ABHA

R1

R2

),,(fr

R

r

Rd

r

qq)r(V

bondedHij

ijmin,

ij

ijmin,

ij

ij

ji

1012

van der Waals

Approximate Hydrogen Bond

van der Waals + Coulomb

Approximate H-Bond + Coulomb

Ener

gy

(kca

l/m

ol)

Hydrogen Bond

Quantum Mechanical Calculations for

Molecules

)()()()(8

2

2

2

rErrVrm

h

It is analytically solvable for three/four systems (V)

Molecules have complicated V(r)

Approximate solution of is possible

Approximations are of varied range:

a) Semi-empirical

b) Hartree-Fock

c) Density Functional Theory

d) Molar-Plesset Perturbation Theory, etc.

Quantum Mechanics

)()()*

2(

),(

),(),()],(2

[

222

2

'' 2

1 1

2

22

iii

ijiK

i

JK JK

JK

ij ij

n

i

N

K iK

KiKiiK

rErr

e

r

Ze

m

R

ZZ

r

e

r

ZeRrV

RrERrRrVm

e Z

Born-

Oppenheimer

Approximation

Ethane (three fold symmetry)

Ethylene (two fold symmetry)

)}3cos(1{kE

)}2cos(1{kE

)}1802cos(1{ okE

Electrostatic Interaction between polar atoms

through their atom-centered partial atomic charges

3324

coul

ij

ji

coul

ijo

jiK;

r

qqK

r

qqE

Bonds are also stretchable but at a cost of energy

Bond Breaking

energy

2

2

1)bb(KE obondbond

QM evaluation

Deformation from

equilibrium value costs

energy. Simplest form of

energy penalty is:

E k o

Bond Angle Deformation

Minimum Energy

at: Ro

66

126

5.012r

R

r

R

r

B

r

AE

ooo

Dispersion Interaction / Leanard-Jones

Potential / van der Waals Interaction

Classical Physics-based Force Field

202

b

pot bbkE

ncosVn12

612

0 2r

r

r

r oo

ijr

qqji

202

02

1

2

1ikikik rrFk

Classical Knowledge-based Force Field

0.00.30.40.10H(N)

2.4522.09441.0NC’C

1.2-0.53.20.42O

1.2-0.43.60.36N30.5NCC’N

1.20.53.60.40C’3-1.5C’NCC’

1.2-0.13.60.42C212.0CC’NC

0.10.13.40.05H31.2CCCC

qr*Nonbondedn½KAngle

Nonbonded ParametersTorsional Parameters

2.3902.09448.0HC’O0.980405.0NH

2.4102.09454.0C’NC1.457201.0CN

2.0272.09426.0C’NH1.200595.0C’O

2.5551.91118.0CCC1.278403.0C’N

2.2431.91125.0CCH1.490110.0CC

1.821.91140.0HCN1.099286.0CH

rik0½Fik

b0½KAnglesb0½KbBond

Angle Bending ParametersBond Stretching Parameters

Some Simplified Force Field Parameters for Hydrocarbon and Amides

From Warshel’s book

Force-fields:

a) AMBER

b) CHARMM

c) GROMOS

Softwares:

a) AMBER

b) CHARMM

c) GROMACS

d) NAMD

e) DESMOND …

RESI ALA 0.00

GROUP

ATOM N NH1 -0.47 ! |

ATOM HN H 0.31 ! HN-N

ATOM CA CT1 0.07 ! | HB1

ATOM HA HB 0.09 ! | /

GROUP ! HA-CA--CB-

HB2

ATOM CB CT3 -0.27 ! | \

ATOM HB1 HA 0.09 ! | HB3

ATOM HB2 HA 0.09 ! O=C

ATOM HB3 HA 0.09 ! |

GROUP !

ATOM C C 0.51

ATOM O O -0.51

BOND CB CA N HN N CA

BOND C CA C +N CA HA CB HB1 CB

HB2 CB HB3

DOUBLE O C

IMPR N -C CA HN C CA +N O

DONOR HN N

ACCEPTOR O C

Protein/position/species

independent residue

TOPOLOGY for Alanine

in

CHARMM format

ALANINE

ALA INT 1

CORR OMIT DU BEG

0.00000

1 DUMM DU M 0 -1 -2 0.000 0.000 0.000 0.00000

2 DUMM DU M 1 0 -1 1.449 0.000 0.000 0.00000

3 DUMM DU M 2 1 0 1.522 111.100 0.000 0.00000

4 N N M 3 2 1 1.335 116.600 180.000 -0.404773

5 H H E 4 3 2 1.010 119.800 0.000 0.294276

6 CA CT M 4 3 2 1.449 121.900 180.000 -0.027733

7 HA H1 E 6 4 3 1.090 109.500 300.000 0.120802

8 CB CT 3 6 4 3 1.525 111.100 60.000 -0.229951

9 HB1 HC E 8 6 4 1.090 109.500 60.000 0.077428

10 HB2 HC E 8 6 4 1.090 109.500 180.000 0.077428

11 HB3 HC E 8 6 4 1.090 109.500 300.000 0.077428

12 C C M 6 4 3 1.522 111.100 180.000 0.570224

13 O O E 12 6 4 1.229 120.500 0.000 -0.555129

TOPOLOGY for Alanine

in

AMBER format

[ ALA ]

[ atoms ]

N N -0.28000 0

H H 0.28000 0

CA CH1 0.00000 1

CB CH3 0.00000 1

C C 0.380 2

O O -0.380 2

[ bonds ]

N H gb_2

N CA gb_20

CA C gb_26

C O gb_4

C +N gb_9

CA CB gb_26

[ angles ]

; ai aj ak gromos type

-C N H ga_31

H N CA ga_17

-C N CA ga_30

N CA C ga_12

CA C +N ga_18

CA C O ga_29

O C +N ga_32

TOPOLOGY for Alanine

in

GROMOS/GROMACS format

E( x, y, z)

E( x+1, y, z)

E( x+2, y, z) …..

Energy Landscape

of typical bio-

molecules

Ener

gy

Positional Variables

kTE

kTEi

kTEE

i

i

if

e

eQQ

e~xp

xp withxQxpQ

Property Average

1 Available Methods:

Monte Carlo Simulation

Genetic Algorithm

Molecular Dynamics

Numerical Integration as Area

Covered by a Curve

01,0-1,0

1,1-1,1

Random Number

• Probability of finding its value

between xx+ x == that between

yy+ y

• Should not repeat

• Can be generated from mistake in

calculation, such as

IX = IM * M, where IM is largest Integer a

computer can store

rand=IX/IM

Probabilistic Method: Monte Carlo Metropolis

dxkT

xUexp

kTxU

exp

xP

dxxPxQQ

In a system having two states:

Pa/pb = exp(-(Ua-Ub)/kT) = Probability

of transition from b to a

Probability of transition

=1 if Ub > Ua

= exp(- U/kT)

A Bab

ba

Alw

ays

A

cce

pt

Reject

Accept

Ener

gy

Uniformly generated

Random numbers are used

to accept if

exp(- U/kT) > random no

and reject otherwise

Conformation 0: Calculate energy (Ei)

Alter conformation randomly

Calculate energy (Ei+1)

Calculate ρ = exp(-(Ei+1-Ei)/kT)

If ρ > random no

accept the conformation

Repeat the procedure

Periodic Boundary Condition

Energy Optimization

0dx

dE

x

Force = ─ Gradient of Potential Energy

...!3

)()(

!2

)()()()()(

3

3

32

2

2 xxf

dx

dxxf

dx

dxxf

dx

dxfxxf

ooo xxx

oo

Taylor’s Seriese

x

f(x)

xo

Multi-variable Optimization: NP-hard

Problem

• Systematic Grid Search procedure: Impossible, large no. variables

• Guided Grid Search: Depends on Choice

• Approximate Method based on Taylor series

...xdx

Ed

!xxx

dx

Edxxx

dx

dExdx

dEmmm 03

32002

2

002

10

1

2

2

0220dx

Ed

dx

dEx

dxEd

dxdExxm

...)x(''f!/h)x('hf)x(f)hx(f 22

Newton-Raphson

Method

Further approximations:

2. Steepest Descent

3. Conjugate Gradient

4. Truncated Newton-Raphson

1

2

3

2

32

2

13

232

2

2

2

2

31

231

2

21

2

2

1

2

0

dx

Ed

dxdx

Ed

dxdx

Ed

dxdx

Ed

dx

Ed

dxdx

Ed

dxdx

Ed

dxdx

Ed

dx

Ed

dx

dExxm

Energy Landscape

of typical bio-

molecules

Ener

gy

Positional Variables

kTE

kTEi

kTEE

i

i

if

e

eQQ

e~xp

xp withxQxpQ

Property Average

1 Available Methods:

Monte Carlo Simulation

Genetic Algorithm

Molecular Dynamics

Deterministic Method: Molecular Dynamics

2

2

dt

xdmamFxE ii

...t

dt

xdt

dt

xdtxttx

2

2

2

2

t

ttxttxtv

txFtttxtxttx

tdt

xdtt

dt

xdtt

dt

dxttxttx

tdt

xdtt

dt

xdtt

dt

dxttxttx

2

)()()(

))(()()(2)(

)(!3

)(!2

)()()(

)(!3

)(!2

)()()(

2

3

33

2

22

3

33

2

22

Verlet Algorithm:

t0-1/2 t t0+1/2 t t0+3/2 t t0+5/2 t t0+7/2 t

t0 t0+ t t0+2 t t0+3 t t0+4 t

X X X X XE

E

E

EE

E

E

E

E

E

v v v v

Leap-Frog Verlet Algorithm

NkTtvm

tm

tEttvttv

tttvtxttx

ii2

3

2

1

22

2

2

21

0 11T

Tt

vv

T

oldi

newi

Leapfrog Verlet Algorithm

Temperature and Pressure controlled by:

31

1 )PP(t

cp

xcx

o

oldp

new

Other Integrators:

1. Velocity Verlet

2. Langevin Dynamics

Simple Pendulum

Average Position of

a simple pendulum

12

3

4

5

Period of

measurement of

position : ~2.3 T

Recommended period of measurement

~T /10

Recommended dt for MD ~ 10-15sec

Heating phase

Equilibration

Solvated system

Temperature variation

Energy variation

Simulation in water in presence of counter-ions [Na+] to maintainelectroneutrality Periodic Boundary condition with Particle Mesh Ewald for long-range electrostatics Molecular Dynamics by NAMD in CHARMM27 force-field without anyconstraint using Constant Pressure-Temperature [CPT] Algorithm

Use of no constraint/restraint during the simulation allows us to

study the spontaneous evolution of the structure with time.

Molecular Dynamics Study: System Setup for Simulation

Duration of Simulation

• Protein Folding requires 1 s to 1ms

• Ligand binding/dissociation requires 1 s

• No. of steps = 1ms / t = 10-3s/10-15s = 1012

Need of faster computer

Engaging several computers in parallel

Increasing t by Shake, Rattle or Lincs algorithms

Special Decomposition in

Parallel Distributed Computer

CPU 1

CPU 0 CPU 2

CPU 3

Limitations and Further

Improvements

• Atomic movements Rescheduling tasks

• Network speed limitation

• Simulations using large no. of cores in GPU

• Replica Exchange Molecular Dynamics

• Steered Molecular Dynamics

Typical examples

• Our experiences in Simulation of DNA

double helix

• Understanding Sequence Directed DNA

melting thermodynamics

A

T

G

C

A-DNA

B-DNAZ-DNA

Adapted from: Leontis et al, NAR(2002), 30, 3497

Non-canonical Basepairing

Base Pair Finder: BPFIND

Took a base edge

Identify the H-bonding centers (N3G &

N2G)

Look for H-bond partner through distance

calculation (N6A & N7A)

Calculate pseudo-angles for planarity

Calculate E= i(di-3.0)2 + ½ k( k- i are

for two H-bond distances and k are for four

pseudo angles, for comparisons Gives rise to:

1822 A:U W-W(C);

6056 G:C W-W(C) and

847 G:U W-W(C) base pairs

Das, Mukherjee, Mitra & Bhattacharyya (2006) J Biomol Struct Dynam, 24, 149-

161

Structural Motifs of RNA:

G:C W:W C

A:U W:W C

G:U W:W C

A:G H: S T

A:U H:W T

A:A H:H T

G:A W:W C

G:A S:W T

A:A W:W T

A:U W:W T

A:A H: W T

A:U H:W C

G:G S:S T

G:G H:W T

A:C W:W T

C:U W:W T

A:C H:W T

G:G H:WC

G:C W:W T

A:G s:s T

AA HHT

AG SST

AG HST

AU HWT

AU HWT

Double Helical regions of RNA consists of Mainly Base

Pairs (G:C, A:U or G:U) and others

Containing

WC only

Loop residuesBase

paired

Base triples

Double Helices

Double helical fragments from ribosomal RNA

Sequence: 5’-GCAAACCGG-3’

3’-UGAUGGGCC-5’

Motif: A:A s:hT PDB: 1N32

A:U H:WT

A:G H:ST

Motif: G:A S:HT PDB: 2AW4

A:G H:ST

Sequence: 5’-GUGGGAGCACG-3’

3’-CGUCAGUGUGC-5’

A:A s:hT

A:U H:WT

A:G H:ST

Stability of Non-canonical Motifs: Open-angle

Variation

1N32: A:A s:hT A:U H:WT A:G H:ST

2AW4: G:A S:HT A:G H:ST

CHARMM AMBER

Non-Watson-Crick

Basepairs: R:R, Y:Y &

R:Y MismatchesBasepair PDB Sequence

R:R A:G W:WC 1FJG 5’-CGCCAUGG-3’

3’-GCGGGGUC-5’

Y:Y U:U W:WC 1J5A 5’-GUCUGGC-3’

3’-CGGUUCG-5’

R:Y G:U W:WC 1N33 5’-GGGCUCUACCC-3’

3’-CCCGGGAUGGG-5’

CHARMM AMBER

HD-RNAS: Hierarchical Database of RNA Structures

http://www.saha.ac.in/biop/www/HD-RNAS.html

Crystallographic Ensemble from Classification

1J5A: (35) 23S rRNA5’-GUCUGGC-3’

3’-CGGUUCG-5’

2AW4: (52) 23S rRNA5’-GUGGGAGCACG-3’

3’-CGUCAGUGUGC-5’

1FJG: (102) 16S rRNA5’-CGCCAUGG-3’

3’-GCGGGGUC-5’

1N32: (102) 16S rRNA5’-GCAAACCGG-3’

3’-UGAUGGGCC-5’

Summery

• Molecular Dynamics can break a structure

indicating instability of the conformation

• Molecular Dynamics results are compatible with

collective ensemble from crystal structure

database

• RNA double helices with tandem non-canonical

motifs are important for its structure

Thank YouDr. Debashree Bandyopadhyay

Sukanya Halder

Dr. Shayantani Mukherjee

Sanchita Mukherjee

Dr. Sudipta Samanta

Dr. M.C. Blaise

Dr. Geeta Kant

Dr. Sangeeta Kundu

DBT; CDAC; DAE

MD Simulation

Snapshots of

d(CGATTAATCG)2 at

300K

Mukherjee & Bhattacharyya (2012)

J. Biomol. Struct. Dynam.

Kundu, Mukherjee &

Bhattacharyya (2012) J. Biosci.

1EHV 249D 426D

DNA as base pair stacks

IUPAC-IUB

suggestion

RM

SD

Basepairs near terminal can

easily open up

Fraying effect

Non-cooperative melting

Mukherjee & Bhattacharyya JBSD 2012

Ends of the oligonucleotide

Expected DNA melting curve

indicating of cooperative

effect

Observed DNA melting curve for

Base Paired Double Stranded

oligonucleotides

Application: Prediction of Local

Melting in Polymeric DNA

Biological situation Most solution studies

http://www.saha.ac.in/biop/bioinformatics.html

Simulation of

Longer DNA

Stretch with

periodic

boundary

condition

DNA of sequence

d(CGCGCGCGAATTCGCGCGCG)2

Partial

opening after

~10 ns

Complete melting

after 30 ns at 500K

MD Simulations

at 460K

Base pair vs time plot at 460K

0.00

5.00

10.00

15.00

20.00

25.00

30.00

35.00

C1 G2 C3 G4 C5 G6 C7 G8 A9 A10 T11 T12 C13 G14 C15 G16 C17 G18 C19

Base pair step

Tim

e in

ns

Base pair vs step number at 400K

0

20

40

60

80

100

120

C1 G2 C3 G4 C5 G6 C7 G8 A9 A10 T11 T12 C13 G14 C15 G16 C17 G18 C19

Base pair step

Tim

e in

ns

Attempted Simulation of

Polymeric DNA

Samanta, Mukherjee, Chakrabarti & Bhattacharyya (2009) J. Chem. Phys. 130, 115103 (2009)

. . .|. . .

. . .|. . .

1 61 71 81 92 01 11 21 31 41 51 61 71 81 92 01 11 21 31 41 5

0 50 40 30 20 11 00 90 80 70 60 50 40 30 20 11 00 90 80 70 6

ttttt|T-TTTTTTTTTttttt

aaaaa|AAAAAAAAAAaaaaa

Structural properties in Polymeric

Simulation

Samanta, Mukherjee, Chakrabarti & Bhattacharyya (2009) J. Chem. Phys. 130, 115103 (2009)

Work in progress: Life time analysis at 460K of CG-polyL

ife

tim

e w

. r.

t R

oll

0

5

10

15

20

25

30

35

40

C1 G2 C3 G4 C5 G6 C7 G8 A9 A10 T11 T12 C13 G14 C15 G16 C17 G18 C19

0.00

5.00

10.00

15.00

20.00

25.00

30.00

35.00

C1 G2 C3 G4 C5 G6 C7 G8 A9 A10 T11 T12 C13 G14 C15 G16 C17 G18 C19

oligo

poly

MD Tutorial

1 1 200

db2000.dat

ALANINE

ALA INT 1

CORR NOMI DU BEG

0.00000

1 DUMM DU M 0 -1 -2 0.000 0.000 0.000

2 DUMM DU M 1 0 -1 1.449 0.000 0.000

3 DUMM DU M 2 1 0 1.522 111.100 0.000

4 N N3 M 3 2 1 1.335 116.600 180.000

5 H1 H E 4 3 2 1.010 130.000 0.000

6 H2 H E 4 3 2 1.010 60.000 90.000

7 H3 H E 4 3 2 1.010 60.000 -90.000

8 CA CT M 4 3 2 1.449 121.900 180.000

9 HA HP E 8 4 3 1.090 109.500 300.000

10 CB CT 3 8 4 3 1.525 111.100 60.000

11 HB1 HC E 10 8 4 1.090 109.500 60.000

12 HB2 HC E 10 8 4 1.090 109.500 180.000

13 HB3 HC E 10 8 4 1.090 109.500 300.000

14 C C M 8 4 3 1.522 111.100 180.000

15 O O E 14 8 4 1.229 120.500 0.000

CHARGE cc-pvtz esp iterated

.1902 .2056 .2056 .2056 .0040

.0430 -.0968 .0470 .0470 .0470

.6837 -.5819

IMPROPER

CA +M C O

DONE

GLYCINE

GLY INT 1

CORR NOMI DU BEG

0.00000

1 DUMM DU M 0 -1 -2 0.000 0.000 0.000

2 DUMM DU M 1 0 -1 1.449 0.000 0.000

AMBER Residue

Topology format

RESI ALA 0.00

GROUP

ATOM N NH1 -0.47 ! |

ATOM HN H 0.31 ! HN-N

ATOM CA CT1 0.07 ! | HB1

ATOM HA HB 0.09 ! | /

GROUP ! HA-CA--CB-HB2

ATOM CB CT3 -0.27 ! | \

ATOM HB1 HA 0.09 ! | HB3

ATOM HB2 HA 0.09 ! O=C

ATOM HB3 HA 0.09 ! |

GROUP !

ATOM C C 0.51

ATOM O O -0.51

BOND CB CA N HN N CA

BOND C CA C +N CA HA CB HB1 CB HB2 CB HB3

DOUBLE O C

IMPR N -C CA HN C CA +N O

CMAP -C N CA C N CA C +N

DONOR HN N

ACCEPTOR O C

IC -C CA *N HN 1.3551 126.4900 180.0000 115.4200 0.9996

IC -C N CA C 1.3551 126.4900 180.0000 114.4400 1.5390

IC N CA C +N 1.4592 114.4400 180.0000 116.8400 1.3558

IC +N CA *C O 1.3558 116.8400 180.0000 122.5200 1.2297

IC CA C +N +CA 1.5390 116.8400 180.0000 126.7700 1.4613

IC N C *CA CB 1.4592 114.4400 123.2300 111.0900 1.5461

IC N C *CA HA 1.4592 114.4400 -120.4500 106.3900 1.0840

IC C CA CB HB1 1.5390 111.0900 177.2500 109.6000 1.1109

IC HB1 CA *CB HB2 1.1109 109.6000 119.1300 111.0500 1.1119

IC HB1 CA *CB HB3 1.1109 109.6000 -119.5800 111.6100 1.1114

RESI ARG 1.00

GROUP

ATOM N NH1 -0.47 ! | HH11

ATOM HN H 0.31 ! HN-N |

ATOM CA CT1 0.07 ! | HB1 HG1 HD1 HE NH1-HH12

ATOM HA HB 0.09 ! | | | | | //(+)

GROUP ! HA-CA--CB--CG--CD--NE--CZ

CHARMM Residue

Topology

[ ALA ]

[ atoms ]

N N -0.28000 0

H H 0.28000 0

CA CH1 0.00000 1

CB CH3 0.00000 1

C C 0.380 2

O O -0.380 2

[ bonds ]

N H gb_2

N CA gb_20

CA C gb_26

C O gb_4

C +N gb_9

CA CB gb_26

[ angles ]

; ai aj ak gromos type

-C N H ga_31

H N CA ga_17

-C N CA ga_30

N CA C ga_12

CA C +N ga_18

CA C O ga_29

O C +N ga_32

N CA CB ga_12

C CA CB ga_12

[ impropers ]

; ai aj ak al gromos type

N -C CA H gi_1

C CA +N O gi_1

CA N C CB gi_2

[ dihedrals ]

; ai aj ak al gromos type

-CA -C N CA gd_4

-C N CA C gd_19

N CA C +N gd_20

[ ARG ]

[ atoms ]

N N -0.28000 0

H H 0.28000 0

CA CH1 0.00000 1

CB CH2 0.00000 1

CG CH2 0.00000 1

GROMACS Residue

Topology