Introduction to Biomolecular Introduction to Biomolecular Structure and ModelingStructure and Modeling

Dhananjay BhattacharyyaBiophysics Division

Saha Institute of Nuclear Physics

Kolkata dhananjay.bhattacharyya@saha.ac.in

Biomolecular Structures

These are determined experimentally by

• X-Ray Crystallography

• Nuclear Magnetic Resonance Spectroscopy

• Neutron Diffraction Study

• Raman Spectroscopy

And also by theoretical methods

2d sin=n

Nucleic Acid Backbone is Connected to Nucleic Acid Backbone is Connected to Either of Four Different BasesEither of Four Different Bases

A

G

T

C

A-DNAA-DNA

B-DNAB-DNA Z-DNAZ-DNA

Proteins (polymers) are made up of Proteins (polymers) are made up of Amino Acids (monomer units)Amino Acids (monomer units)

There are Twenty different Amino Acids

with different shape, size and electrostatic

properties.

These amino acids form covalent

bonds to form a linear polypeptide chain.

Alanine Phenylalanine

Serine

Cystine

Glutamic Acid (Negatively charged)

Arginine (Positively charged)

Amino Acids are joined together by covalent bonds, called peptide bond, which is structurally very important

helix: Hydrogen bonding between every i i+4 residues

sheet: Hydrogen bonding between ij, i+1j-1 (Antiparallel), or ij, i+1j+1 (parallel)

Coordinate System:

• External coordinates, such as (x,y,z), (r,,), (r,,z)

• Internal coordinates (BondLength, BondAngle, TorsionAngle)

Bond Length

Bond Angle

Torsion Angle

Internal External Coordinate

Generated coordinates

• H 0.000000 0.000000 0.000000

• C 0.000000 0.000000 1.089000

• C 1.367073 0.000000 1.572333

• C 2.050610 -1.183920 1.089000

• C 3.417683 -1.183920 1.572333

• H -0.513360 0.889165 1.452000

• H -0.513360 -0.889165 1.452000

Theoretical Modeling of Biomolecules:

Quantum Mechanics based Methods

Statistics based Methods

Classical or Molecular Mechanics methods

Peptide modeling initiated in India by G.N. Ramachandran

(1950s)Postulates:

• Impenetrable spherical volumes for each atom

• Radius of the sphere depend on atom type

• No two atomic spheres can overlap if they are not covalently bonded

Between H N O C P S

H 2.0 (1.9) 2.4 (2.2) 2.4 (2.2) 2.4 (2.2) 2.65(2.5) 2.65(2.5)

N   2.7 (2.6) 2.7 (2.6) 2.9 (2.8) 3.2 (3.1) 3.1 (3.0)

O     2.7 (2.6) 2.8 (2.7) 3.2 (3.1) 3.1 (2.9)

C       3.0 (2.9) 3.4 (3.2) 3.3 (3.1)

P         3.5 (3.3)  

S            

Normal and Extreme Limit (within parenthesis) distances (Å) used by Ramachandran co-workers

Original Ramachandran Plot

Fully Allowed Regions

Partially Allowed Regions

Ramachandran plot for 202 proteins at 1.5A or better resolution

Variation of angle by 5o allowed to fit observed phi-psi of protein structures.

Schrodinger Equation: Quantum Mechanics

E)Vdx

d

m

h(

ih)V

m

h(

2

22

22

2

2

28Time dependent

(3 Dimensional)

Time independent

DFT formalism with B3LYP 

Pseudoeigenvalue equation: 

where 

Potential due to exchange-correlation, is defined by 

with a, b and c as parameters obtained from fit with experimental data for sample compounds, Ex are for electron exchange and Ec are for

correlation.

N

iNiN

N

i

ksih

1321321

1

......

xci

nuclei

k ki

kksi V'dr

'rr

)'r(

rr

Zh

1

2

2

1

xcV

xcxc

EV

LYPc

LSDAc

Bx

HFx

LSDAxxc cEE)c(EbaEE)a(E 11

Essentials of Computational Chemistry by C.J. Cramer (2002) John Wiley & Sons Ltd,

Input data (atom coordinates, basis sets)

Generate input guess density (overlap integrals)

Construct the potential andSolve Kohn-Sham equation

Generate output densities fromSolutions to Kohn-Sham equations

Are input and output density

same?

Analyze electronic population

Repeat the cycle using the output density as the

input density

YESNO

FLOW CHART DESCRIBING THE DFT METHODOLOGY

G:C W:W CE = -26 kcal/mol

A:U W:W CE = -14

G:U W:W C E = -15

A:G H:S T E = -10

A:G s:s T E = -6

A:U H:W TE = -13

A:A H:H TE = -10

G:A W:W CE = -15

G:A S:W TE = -11

A:A W:W TE = -12

A:U W:W TE = -13

A:A H:W TE = -11

2=>NH..O

1=>NH..N

1=>NH..O

1=>NH..N 2=>NH..O

2=>NH..N

1=>NH..N

1=>CH..O

1=>NH..O

1=>NH..N2=>NH..N

1=>NH..O

1=>NH..N

2=>NH..N

2=>NH..N 1=>NH..O

1=>NH..N

1=>NH..O

1=>NH..N

Strengths of different H-bonds from 33 non-canonical Base Pairs

Considered Energy components, ENHO, ENHN, etc are additive. Additional stabilities, i may come from van der Waals, dipole-dipole etc interactions.

Least Squares Fit indicates i, errors should be smallest for best Fit

iCHNCHN

iCHOCHO

iOHNOHN

iNHNNHN

iNHONHO

ii EnEnEnEnEnE int

2int2

i

CHNCHNi

CHOCHOi

OHNOHNi

NHNNHNi

NHONHOi

i

ii EnEnEnEnEnE

Type of H-bond E (kcal/mol)

N-H…O -7.82

N-H…N -5.62

O-H…N -6.89

C-H…O -1.33

C-H…N -0.67

A. Roy, M. Bhattacharyya, S. Panigrahi, D. Bhattacharyya, (2008) J. Phys. Chem. B (in press)

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

DNA kinks by 90o at the dyad location while binding to two subunits of Catabolite Activator Protein (CAP)

TATA-box binding protein transforms the interfacing DNA region to A-DNA like structure

DNA Smooth Curvature induced by Histone proteins in Chromatin (Nucleosome)

Definition and Nomenclature of Base Definition and Nomenclature of Base Pair Doublet ParametersPair Doublet Parameters

Calculation of Base Pair Calculation of Base Pair parameters by NUPARMparameters by NUPARM

Local Step Parameters: Mean Local Helix Axis: Zm = Xm Ym, where Xm = Xaxis1 + Xaxis2 and Ym = Yaxis1 + Yaxis2

M is Base Pair Center to Center Vector

Tilt : 2.0 * sin-1 ( -Zm Y1) Roll: 2.0 * sin-1 ( Zm X1)Twist: cos-1 (( X1 Zm) ( X2 Zm)) Shift (Dx) M XmSlide(Dy) M YmRise(Dz) M Zm 

Partial list of DNA crystal Partial list of DNA crystal structures available at structures available at

http://ndbserver.rutgers.eduhttp://ndbserver.rutgers.edubd0001 12: A C C G A C G T C G G T bd0003 12: A C C G G T A C C G G T bd0004 12: C G C G A A T T C G C G bd0006 10: G G C C A A T T G G bd0011 12: C G C A A A T A T G C G bd0014 12: C G C G A A T T C G C G bd0015 10: C C G C C G G C G G bd0017 9: C G C G C G G A G bd0018 11: G C G A A T T C G C G bd0019 12: G G C G A A T T C G C G bd0022 12: A C C G G C G C C A C A bd0023 10: C C A G T A C T G G Bd0024 10: C C G A A T G A G G

Average Structural Parameters from Crystal Structures

Base-Pair Step

Size of Database

Tilt Roll Twist Rise

G:G 37 -0.24 5.80 30.99 3.46

G:C 106 -0.33 -5.37 38.52 3.32

C:G 157 0.66 3.81 36.26 3.46

A:A 116 -0.01 0.67 35.92 3.21

A:T 54 0.20 -0.60 32.76 3.25

T:A 18 -0.02 0.07 40.39 3.30

A:C 20 -0.37 0.97 32.73 3.43

C:A 47 -0.19 2.17 37.75 3.48

A:G 34 0.16 5.34 31.92 3.44

G:A 55 -0.23 0.52 38.40 3.14

DNA Bending: Experimental and DNA Bending: Experimental and TheoryTheory

Sequence Experimental RL

Theoretical bending (d/l)

Random 1.00 0.98

(AAANNNNNNN)n 1.23 0.85

(AAAANNNNNN)n 1.60 0.81

(AAAAANNNNN)n 2.00 0.74

(AAAAAANNNN)n 2.31 0.72

(AAAAAAAANN)n 2.21 0.67

(AAAAAAAAAN)n 1.73 0.82

Curved DNA models built Curved DNA models built from Crystal parametersfrom Crystal parameters(A3G7)n

(A6G4)n

(A10)n

Bond Angle Deformation

Deformation from equilibrium value costs energy. Simplest form of energy penalty is:

Eko

Bonds are also stretchable but at a cost of energy

Bond Breaking energy

2

2

1)bb(KE obondbond

Ethane (three fold symmetry)

Ethiline (two fold symmetry)

)}cos({kE 31

)}cos({kE 21

Normal and Extreme Limiting (within parenthesis) distances (Å) used by Ramachandran co-workers

Minimum Energy position: rij

o

Between H N O C P S

H 2.0 (1.9) 2.4 (2.2) 2.4 (2.2) 2.4 (2.2) 2.65 (2.5) 2.65 (2.5)

N   2.7 (2.6) 2.7 (2.6) 2.9 (2.8) 3.2 (3.1) 3.1 (3.0)

O     2.7 (2.6) 2.8 (2.7) 3.2 (3.1) 3.1 (2.9)

C       3.0 (2.9) 3.4 (3.2) 3.3 (3.1)

P         3.5 (3.3)  

S            

Interaction between Instantaneous Atomic dipoles and Induced Atomic dipoles

Force Field for Biomolecular Simulation

202 b

pot bbkE

ncosVn 12

612

0 2r

r

r

r oo

ijr

qqji

(optional) r

D

r

C

ij

ij

ij

ij

1012

202

0 2

1

2

1ikikik rrFk

E(x, y, z)

E(x+1, y, z)

E(x+2, y, z) …..

Search for Search for Conformation with Conformation with Lowest EnergyLowest Energy

Multivariable Optimization: NP-hard Problem

• Systematic Grid Search procedure: Impossible, large no. variables• Guided Grid Search: Depends on Choice• Approximate Method based on Taylor series

Newton-Rhapson Method:

...xdx

Ed

!xxx

dx

Edxxx

dx

dEx

dx

dEmmm 03

32

002

2

00 2

10

1

2

2

0220

dx

Ed

dx

dEx

dxEd

dxdExxm

Energy Landscape of typical bio-moleculesE

nerg

y

Positional Variables

kTE

kTEi

kTEE

i

i

if

e

eQQ

e~xp

xp withxQxpQ

Property Average

1

Alw

ay

s Acc

ep

t

Reject

Accept

Ene

rgy

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

Deterministic Method

Molecular Dynamics

2

2

dt

xdmamFxE ii

...t

dt

xdt

dt

xdtxttx

2

2

2

2

t

)tt(x)tt(x)t(v

))t(x(Ft)tt(x)t(x)tt(x

)t(dt

xd

!

t)t(

dt

xd

!

t)t(

dt

dxt)t(x)tt(x)tt(x

2

2

322

3

33

2

22

Verlet Algorithm:

NkTtvm

ttattvttv

tttvtxttx

ii 2

3

2

1

22

2

2

Leapfrog-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 XEE

EE

EE

EE

EE

v v v v

Time scale of Vibrational MotionsType Wave no (cm-1) Period

Tp(λ/c) (fs)Tp/π (fs)

O-H, N-H stretch 3200-3600 9.8 3.1

C-H Stretch 3000 11.1 3.5

O-C-O Asymm. Stretch 2400 13.9 4.5

C=C, C=N stretch 2100 15.9 5.1

C=O (carbonyl) stretch 1700 19.6 6.2

C=C stretch

H-O-H bend 1600 20.8 6.4

C-N-H, H-N-H bend 1500 22.2 7.1

C-N stretch (amides) 1250 26.2 8.4

Water Libration (rocking) 700 41.7 13

C=C-H bending

Simple Pendulum

Average Position of a simple pendulum

12

3

45

Period of measurement of position : ~2.3 T

Recommended period of measurement ~T /10

Duration of Simulation

• Protein Folding requires 1s to 1ms• Ligand binding/dissociation requires 1s• 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

Softwares for Molecular Simulation

• Accelrys, MOE, SYBYL, TATA-BioSuite (Composite package, costly)

• CHARMM, AMBER (for Simulation, special Academic Price)

• GROMACS, NAMD (for Simulation, FREE)

• MOLDEN (for molecule Building, FREE)• GAMESS (for QM calculation, FREE)

Heating phase

Equilibration

Dickerson Dodecamer seq: d(CGCGAATTCGCG)2

CURVES calculated values

S replaces O in backbone of substituted DNA.

It yields two chiral conformers of

DNA – PSR and PSS.

S. Mukherjee and D. Bhattacharyya (2004) Biopolymers 73, 269–282

PS-R PS-S

Normal PO

PS-R

PS-S

Students:Dr. Debashree Bandyopadhyay

Dr. Shayantani MukherjeeDr. Kakali Sen

Mr. Sudipta Samanta

Partially Supported by CSIR, DBT and CAMCS (SINP)

Collaborators:Dr. Rabi MajudarDr. Samita Basu

Dr. Sangam BanerjeeDr. Abhijit Mitra (IIIT, Hyderabad)

Dr. N. Pradhan (NIMHANS, Bangalore)