hydro-pathy/phobicity/philicity one of the most commonly used properties is the suitability of an...
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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
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