Hydro-pathy/phobicity/philicityHydro-pathy/phobicity/philicity

• One of the most commonly used properties is the suitability of an amino acid for an aqueous environment

• Hydropathy & Hydrophobicity– degree to which something is “water hating” or

“water fearing”

• Hydrophilicity– degree to which something is “water loving”

Hydrophobicity/Hydrophilicity Tables

Hydrophobicity/Hydrophilicity Tables

• Describe the likelihood that each amino acid will be found in an aqueous environment - one value for each amino acid

• Commonly used tables– Kyte-Doolittle hydropathy– Hopp-Woods hydrophilicity– Eisenberg et al. normalized consensus

hydrophobicity

Kyte-Doolittle hydropathyKyte-Doolittle hydropathyAminoAcid

Index AminoAcid

Index

R -4.5 S -0.8K -3.9 T -0.7D -3.5 G -0.4Q -3.5 A 1.8N -3.5 M 1.9E -3.5 C 2.5H -3.2 F 2.8P -1.6 L 3.8Y -1.3 V 4.2W -0.9 I 4.5

Example Hydrophilicity PlotExample Hydrophilicity Plot

This plot is for a tubulin, a soluble cytoplasmic protein. Regions with high hydrophilicity are likely to be exposed to the solvent (cytoplasm), while those with low hydrophilicity are likely to be internal or interacting with other proteins.

Amphiphilicity/AmphipathicityAmphiphilicity/Amphipathicity

• A structural domain of a protein (e.g., an -helix) can be present at an interface between polar and non-polar environments– Example: Domain of a membrane-associated

protein that anchors it to membrane

• Such a domain will ideally be hydrophilic on one side and hydrophobic on the other

• This is termed an amphiphilic or amphipathic sequence or domain

Screenshot of a phospholipid bilayer in the process of its modeling. Shown is a computational cell consisting of 96 PhCh molecules and 2304 water molecules which on the whole make up 20544 atoms.

Average number of hydrogen bonds within the first water shell around an ion

Molecular Dynamics: Introduction

Newton’s second law of motion

We need to know

The motion of the

atoms in a molecule, x(t) and therefore,

the potential energy, V(x)

Molecular Dynamics: Introduction

Molecular Dynamics: IntroductionHow do we describe the potential energy V(x) for amolecule?Potential Energy includes terms for

Bond stretching

Angle Bending

Torsional rotation

Improper dihedrals

Molecular Dynamics: Introduction

Potential energy includes terms for (contd.)

Electrostatic

Interactions

van der Waals

Interactions

Molecular Dynamics: Introduction

In general, given the values x1, v1 and the potential energy V(x), the molecular trajectory x(t) can be calculated, using,

tdx

xdVmvv

tvxx

ixii

iii

1

)(11

11

How a molecule changes during MD

Contributions to Potential Energy

• Total pair energy breaks into a sum of terms( )N

str bend tors cross vdW el polU U U U U U U U r

Intramolecular only

• Ustr stretch

• Ubend bend

• Utors torsion

• Ucross cross

• UvdW van der Waals

• Uel electrostatic

• Upol polarization

Contributions to Potential Energy

• Total pair energy breaks into a sum of terms( )N

str bend tors cross vdW el polU U U U U U U U r

Intramolecular only

• Ustr stretch

• Ubend bend

• Utors torsion

• Ucross cross

• UvdW van der Waals

• Uel electrostatic

• Upol polarization

Contributions to Potential Energy

• Total pair energy breaks into a sum of terms( )N

str bend tors cross vdW el polU U U U U U U U r

Intramolecular only

• Ustr stretch

• Ubend bend

• Utors torsion

• Ucross cross

• UvdW van der Waals

• Uel electrostatic

• Upol polarization

Contributions to Potential Energy

• Total pair energy breaks into a sum of terms( )N

str bend tors cross vdW el polU U U U U U U U r

Intramolecular only

• Ustr stretch

• Ubend bend

• Utors torsion

• Ucross cross

• UvdW van der Waals

• Uel electrostatic

• Upol polarization

Contributions to Potential Energy

• Total pair energy breaks into a sum of terms( )N

str bend tors cross vdW el polU U U U U U U U r

Intramolecular only

• Ustr stretch

• Ubend bend

• Utors torsion

• Ucross cross

• UvdW van der Waals

• Uel electrostatic

• Upol polarization

Contributions to Potential Energy

• Total pair energy breaks into a sum of terms( )N

str bend tors cross vdW el polU U U U U U U U r

Intramolecular only

• Ustr stretch

• Ubend bend

• Utors torsion

• Ucross cross

• UvdW van der Waals

• Uel electrostatic

• Upol polarization

Contributions to Potential Energy

• Total pair energy breaks into a sum of terms( )N

str bend tors cross vdW el polU U U U U U U U r

Intramolecular only

• Ustr stretch

• Ubend bend

• Utors torsion

• Ucross cross

• UvdW van der Waals

• Uel electrostatic

• Upol polarization

Mixed terms

Repulsion

Contributions to Potential Energy

• Total pair energy breaks into a sum of terms( )N

str bend tors cross vdW el polU U U U U U U U r

Intramolecular only

• Ustr stretch

• Ubend bend

• Utors torsion

• Ucross cross

• UvdW van der Waals

• Uel electrostatic

• Upol polarization

Mixed terms

Repulsion

Contributions to Potential Energy

• Total pair energy breaks into a sum of terms( )N

str bend tors cross vdW el polU U U U U U U U r

Intramolecular only

• Ustr stretch

• Ubend bend

• Utors torsion

• Ucross cross

• UvdW van der Waals

• Uel electrostatic

• Upol polarization

Mixed terms

- +- +

Repulsion

Attraction

Contributions to Potential Energy

• Total pair energy breaks into a sum of terms( )N

str bend tors cross vdW el polU U U U U U U U r

Intramolecular only

• Ustr stretch

• Ubend bend

• Utors torsion

• Ucross cross

• UvdW van der Waals

• Uel electrostatic

• Upol polarization

Mixed terms

-+-+

Repulsion

Attraction

Contributions to Potential Energy

• Total pair energy breaks into a sum of terms( )N

str bend tors cross vdW el polU U U U U U U U r

Intramolecular only

• Ustr stretch

• Ubend bend

• Utors torsion

• Ucross cross

• UvdW van der Waals

• Uel electrostatic

• Upol polarization

Mixed terms

-+-+

Repulsion

Attraction

-+

+- +

Contributions to Potential Energy

• Total pair energy breaks into a sum of terms( )N

str bend tors cross vdW el polU U U U U U U U r

Intramolecular only

• Ustr stretch

• Ubend bend

• Utors torsion

• Ucross cross

• UvdW van der Waals

• Uel electrostatic

• Upol polarization

Mixed terms

-+-+

Repulsion

Attraction

+-+

+ -++-+

+-+

u(2)

+- +

u(2)

u(N)

Contributions to Potential Energy

• Total pair energy breaks into a sum of terms( )N

str bend tors cross vdW el polU U U U U U U U r

Intramolecular only

• Ustr stretch

• Ubend bend

• Utors torsion

• Ucross cross

• UvdW van der Waals

• Uel electrostatic

• Upol polarization

Mixed terms

-+-+

Repulsion

Attraction

+-+

+ -+

+-

+-+

u(2)

+- +

u(2)

u(N)

Modeling Potential energy

U(r) U(req ) dUdr rreq

(r req ) 12

d2Udr2

rreq

(r req )2

1

3

d3U

drrreq

(r req )3 ....1

n!

dnU

drn

rreq

(r req )n

Modeling Potential energy

dU

dr rreq

(r req )

U(r) 1

2

d2U

dr2

rreq

(r req )2 1

2kAB (r req )2

U(req )

U(r) 1

2

d2U

dr2

rreq

(r req )2

0 at minimum0

Stretch Energy

• Expand energy about equilibrium position

• Model fails in strained geometries– better model is the Morse potential

22

12 12 12 12 12 122( ) ( ) ( ) ( )

o o

o o o

r r r r

dU d UU r U r r r r r

dr dr

minimumdefine

212 12 12( ) ( )oU r k r r

(neglect)

harmonic

122

12( ) 1 rU r D e

dissociation energy force constant

250

200

150

100

50

0

Ene

rgy

(kca

l/mol

e)

0.80.60.40.20.0-0.2-0.4

Stretch (Angstroms)

Morse

Bending Energy

• Expand energy about equilibrium position

– improvements based on including higher-order terms

• Out-of-plane bending

22

2( ) ( ) ( ) ( )

o o

o o odU d UU U

d d

minimumdefine

2( ) ( )oU k

(neglect)

harmonic

2( ) ( )oU k

u(4)

Torsional Energy

• Two new features– periodic– weak (Taylor expansion in not appropriate)

• Fourier series– terms are included to capture appropriate minima/maxima– depends on substituent atoms

– e.g., ethane has three mimum-energy conformations

» n = 3, 6, 9, etc.

• depends on type of bond– e.g. ethane vs. ethylene

– usually at most n = 1, 2, and/or 3 terms are included

1( ) cos( )nn

U U n

Van der Waals Attraction

• Correlation of electron fluctuations• Stronger for larger, more polarizable molecules

– CCl4 > CH4 ; Kr > Ar > He

• Theoretical formula for long-range behavior• Only attraction present between nonpolar

molecules– reason that Ar, He, CH4, etc. form liquid phases

• a.k.a. “London” or “dispersion” forces

-+-+ - +- +

86

( )attvdW

CU O r

r

Van der Waals Repulsion• Overlap of electron clouds

• Theory provides little guidance on form of model

• Two popular treatmentsinverse power exponential

• typically n ~ 9 - 12 two parameters

• Combine with attraction term– Lennard-Jones model Exp-6

repvdW n

AU

r

rep BrvdWU Ae

12 6

A CU

r r 6

Br CU Ae

r

a.k.a. “Buckingham” or “Hill”

10

8

6

4

2

0

2.01.81.61.41.21.0

LJ Exp-6

Exp-6 repulsion is slightly softer

20

15

10

5

0

x103

8642

Beware of anomalous Exp-6 short-range attraction

Electrostatics 1.• Interaction between charge inhomogeneities

• Modeling approaches– point charges

– point multipoles

• Point charges– assign Coulombic charges to several points in

the molecule

– total charge sums to charge on molecule (usually zero)

– Coulomb potential

• very long ranged

0( )

4i jq q

U rr

1.5

1.0

0.5

0.0

-0.5

-1.0

4321

Lennard-Jones Coulomb

Electrostatics 2.• At larger separations, details of charge distribution are less important• Multipole statistics capture basic features

– Dipole– Quadrupole– Octopole, etc.

• Point multipole models based on long-range behavior– dipole-dipole

– dipole-quadrupole

– quadrupole-quadrupole

i iiq r

i i iiqQ r r

Vector

Tensor

0, 0Q

0, 0Q

Q

Q

1 21 2 1 23

ˆ ˆˆ ˆ ˆ ˆ3( )( ) ( )ddur

r r

21 21 2 1 2 24

3 ˆ ˆˆ ˆ ˆˆ ˆ ˆ( ) 5( ) 1 2( )( )2dQ

Qu Q Q

r

r r r

2 2 2 2 21 21 2 12 1 2 1 2 125

31 5 5 2 35 20

4QQQ Q

u c c c c c c c cr

Axially symmetric quadrupole

Polarization

• Charge redistribution due to influence of surrounding molecules– dipole moment in bulk different

from that in vacuum

• Modeled with polarizable charges or multipoles• Involves an iterative calculation

– evaluate electric field acting on each charge due to other charges– adjust charges according to polarizability and electric field– re-compute electric field and repeat to convergence

• Re-iteration over all molecules required if even one is moved

+ -+

+-

+-+

+ -++-+

+-+

Polarization

ind E

ind ,i Ei

Ei q jrij

rij3

ji

ijrij

3ji

3rij

rij

rij

1

Approximation

Electrostatic field does not include contributions from atom i

Common Approximations in Molecular Models

• Rigid intramolecular degrees of freedom– fast intramolecular motions slow down MD calculations

• Ignore hydrogen atoms– united atom representation

• Ignore polarization– expensive n-body effect

• Ignore electrostatics• Treat whole molecule as one big atom

– maybe anisotropic• Model vdW forces via discontinuous potentials• Ignore all attraction• Model space as a lattice

– especially useful for polymer molecules Qualitative models

Molecular Dynamics: Introduction

Equation for covalent terms in P.E.

)](cos1[)(

)()(

02

0

20

20

nAk

kllkRV

torsions

n

impropers

anglesbonds

lbonded

Molecular Dynamics: Introduction

Equation for non-bonded terms in P.E.

ijr

ji

ij

ij

ij

ij

ji

nonbonded r

qq

r

r

r

rijRV

0

6min

12min

4])(2)[(()(

DNA in a box of water

SNAPSHOTS

Protein dynamics study

• Ion channel / water channel

• Mechanical properties– Protein stretching

– DNA bending

Movie downloaded from theoreticla biophysics group, UIUC

Solvent dielectric models

V QiQ j

rij

Effetive dielectric constant

eff r r r 1

2rS 2 2rS 2 e rS

S 0.15Å 1 ~ 0.3Å 1