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All-atom Molecular Mechanics
Trent E. BaliusAMS 535 / CHE 535
09/27/2010
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Outline• Molecular models
• Molecular mechanics• Force Fields• Potential energy function
– functional form– parameters and parameterization
• Motivations and applications • Biomolecular Force Fields
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Molecular Models • Quantum Mechanics
– physical, but expensive
– Schrödinger equation: H Ψ = E Ψ – wave functions defines electron density
• Molecular Mechanics– less physical --> empirical parameterization– cheap and accurate
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Connectivity (topology)
Parameters
Functional Form
Force Field
Potential EnergyFunction
Molecular MechanicsForce Field
Coordinates
Potential energy
( )rVr
parm top,Dependent on topology and parameters
Function of coordinates
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Molecular Mechanics • Every atom is a sphere with a radius (Lennard Jones)
• Point charge is located at each atomic center(Coulomb’s law)
• Bonds and angles are held by springs to ideal lengths– eg.– Hooke's Law, Kb: spring constant, r0: ideal length
• Dihedrals are represented by sigmoidal function which has energy wells at favorable angles.
• Improper torsions force atoms to be a defined angle to plane.
( )20rrkV bbond −=
Class I Potential Energy function
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The "Tinker-toy Model"
dihedrals (torsions)
bondsangles
Bonded terms
Through space interactions
( )20rrkV bbond −= ( )2
0θθθ −= kVangle ( )( )δχχ −+= nKVdihidral cos1
−
=
6
,
,
12
,
,,
min2
min
ji
ji
ji
jijiLJ r
R
r
RV ε
ij
jicoul r
qqV
ε=
improper
( )20ϕϕϕ −= kVimproper
jiji εεε *, =2
minminmin ,
jiji
RRR
+=
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Dihedral Term( )( )δχχ −+= nKVdihidral cos1
2*K
δ
π (3.14) radians = 180 degrees
2/n*π
Molecular Modelling Principles and applications, Leach Pearson Prentice hall second edition (chapter 4)
Class I Potential Energy function
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Lennard-Jones Equation
(σ, 0)
(Rmin, ε)
−
=
6
,
,
12
,
,,
min2
min
ji
ji
ji
jijiLJ r
R
r
RV ε
Ene
rgy
(kca
l/mol
distance (Å)
σ*2 6/1min =R
Molecular Modelling Principles and applications, Leach Pearson Prentice hall second edition (chapter 4)
Class I Potential Energy function
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( ) ( ) ( )
( )( )ij
jiN
i
N
ij ji
ji
ji
jiji
dihedrals
impropersanglesbondsb
r
r
R
r
RnK
kkbbkV
εεδχ
ϕϕθθ
χ
ϕθ
+
−
+−++
−+−+−=
∑∑∑
∑∑∑
= +=1 1
6
,
,
12
,
,,
20
20
20
min2
mincos1
Potential Energy function
Different Force-Field• CHARMM• AMBER• GROMOS• OPLS
Mackerell, Vol. 25, No. 13, Journal of Computational Chemistry
Molecular Modelling Principles and applications, Leach Pearson Prentice hall second edition (chapter 4)
Parameterization • Experimental observables• Quantum Mechanical
calculationsInterdependences among
parameters
Class I Potential Energy function
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Applications
• Dynamics of Molecules– how do proteins work
• Energy: affinity, specificity – how do proteins interact with one another and
other molecules (drugs, substrates, DNA or RNA)
• Protein design – can proteins be designed to have a specific
function
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Applications
Angew. Chem. Int. Ed. 2006, 45, 4064 – 4092
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Things to consider
Angew. Chem. Int. Ed. 2006, 45, 4064 – 4092
Energy Function
sampling method
size of the system
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Limitations to Molecular Mechanics ?
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Limitations to MM
MM cannot model the following easily:• Chemical reactions
– Formation and breaking of bonds
• Polarizability
• Protonation states
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• Electronic Polarizability• Combining Rules• 1,4 Interactions• Lone Pairs• All-Atom vs. United Atom Force Fields• Treatment of Solvation• Treatment of Long-Range Interactions
Potential Energy Function
Mackerell, Vol. 25, No. 13, Journal of Computational Chemistry
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• Electronic Polarizability• Combining Rules• 1,4 Interactions• Lone Pairs• All-Atom vs. United Atom Force Fields• Treatment of Solvation• Treatment of Long-Range Interactions
Potential Energy Function
Mackerell, Vol. 25, No. 13, Journal of Computational Chemistry
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+ −
Polarization
+ −
Drude particle
Electronic Polarizability
ii
ipol EU ∑= µ2
1
J. Phys. Chem. B 2007, 111, 2873-2885
µi is the dipole moment of atom iEi is the electrostatic field at atom i
induced dipole models
Mackerell, Vol. 25, No. 13, Journal of Computational Chemistry
Class II Potential Energy function
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Electronic Polarizability
• Common force fields use a fixed charge model for every atom
• Polarizable or non-additive force fields allow for atoms to vary their charges in the context of the electric field
• Drude particle (Drude oscillator method)– every atom has two point charges (atom center
and Drude particle) connected by a spring (hooke's law)
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• Electronic Polarizability• Combining Rules• 1,4 Interactions• Lone Pairs• All-Atom vs. United Atom Force Fields• Treatment of Solvation• Treatment of Long-Range Interactions
Potential Energy Function
Mackerell, Vol. 25, No. 13, Journal of Computational Chemistry
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Combining Rules
• force fields combine parameters differently– arithmetic mean
– geometric mean
−
=
6
,
,
12
,
,,
minmin
ji
ji
ji
jijiLJ r
R
r
RV ε
jiji εεε *, =
2
minminmin ,
jiji
RRR
+=
A1A2
A1 = A2
εi
εj εi,,j
εi,,j
0.1, 0.9
0.5, 0.5
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(0.1 + 0.9) / 2 = 0.5(0.5 + 0.5) / 2 = 0.5
(0.1 * 0.9)^(1 / 2) = 0.3(0.5 * 0.5)^(1 / 2) = 0.5
Rad
ius
j
Radius i
Wel
l dep
th j
Well depth j
Combining RulesArithmetic mean Geometric mean
Rad
ius
i, j
Wel
l dep
th i,
j
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Euclidean relationshipHarmonic mean
(0.1^2 + 0.9^2)^(1 / 2) = 0.9(0.5^2 + 0.5^2)^(1 / 2) = 0.7
2 / (1 / 0.1 + 1 / 0.9) = 0.182 / (1 / 0.5 + 1 / 0.5) = 0.5
Combining Rules
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• Electronic Polarizability• Combining Rules• 1,4 Interactions• Lone Pairs• All-Atom vs. United Atom Force Fields• Treatment of Solvation• Treatment of Long-Range Interactions
Potential Energy Function
Mackerell, Vol. 25, No. 13, Journal of Computational Chemistry
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1,4 (non-bonded) Interactions
3
2
3
1
2
2
3 4
4
4
5
5
• 1,2 and 1,3 are normally ignored by force-fields
• 1,4 are scaled– CHARMM no scaling – AMBER and OPLS scale
0.83
• Bonded terms dominate
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1,5 interactionsNot scaled (s = 1)
3
2
3
1
2
2
3 4
4
4
5
5
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1,4 interactions Maybe scaled (amber: s = 0.83)
Note that 1 / 1.2 = 0.83
3
2
3
1
2
2
3 4
4
4
5
5
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• Electronic Polarizability• Combining Rules• 1,4 Interactions• Lone Pairs• All-Atom vs. United Atom Force Fields• Treatment of Solvation• Treatment of Long-Range Interactions
Potential Energy Function
Mackerell, Vol. 25, No. 13, Journal of Computational Chemistry
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Lone Pairs
• Force field without lone pairs do well
• Adding more parameters • Fitting charges from
quantum mechanical electrostatic potentials
• TIP5P and ST2 Water models
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• Electronic Polarizability• Combining Rules• 1,4 Interactions• Lone Pairs• All-Atom vs. United Atom Force Fields• Treatment of Solvation• Treatment of Long-Range Interactions
Potential Energy Function
Mackerell, Vol. 25, No. 13, Journal of Computational Chemistry
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All-Atom vs. United Atom Force Fields
• Hydrogen atoms are not represented explicitly. incorporated in to the bonded heavy atom
• Methyl group will be represented by only one sphere
• Corse grain models
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• Electronic Polarizability• Combining Rules• 1,4 Interactions• Lone Pairs• All-Atom vs. United Atom Force Fields• Treatment of Solvation• Treatment of Long-Range Interactions
Potential Energy function
Mackerell, Vol. 25, No. 13, Journal of Computational Chemistry
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Treatment of Solvation
• Explicit water models – TIP3P, TIP4P, TIP5P, SPC,ST2
• Implicit water models (Continuum with fixed dielectric)– Poisson-Boltzmann (PB) model
– Generalized Born (GB) model
• This will be discussed in detail in a following class
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Treatment of Solvation
Angew. Chem. Int. Ed. 2006, 45, 4064 – 4092
Explicit waterboundaries
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• Electronic Polarizability• Combining Rules• 1,4 Interactions• Lone Pairs• All-Atom vs. United Atom Force Fields• Treatment of Solvation• Treatment of Long-Range Interactions
Potential Energy Function
Mackerell, Vol. 25, No. 13, Journal of Computational Chemistry
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• Cut-offs
• Partial Mesh Ewald (PME)
• Smoothing functions
• Re-parameterization with different treatments of long-range interactions
Treatment of Long-Range Interactions
Mackerell, Vol. 25, No. 13, Journal of Computational Chemistry
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Solvation and long range electrostatics
Angew. Chem. Int. Ed. 2006, 45, 4064 – 4092
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• Protein• Nucleic Acid• Lipid• Carbohydrates• Drug-Like Molecules • Heterogeneous Biomolecular Systems
(combining different types of biomolecules)
Biomolecular Force Fields
Mackerell, Vol. 25, No. 13, Journal of Computational Chemistry
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Conclusions• Introduced Molecular Mechanics
• Described functional form
• Discussed parameterization
• Considered dependences and other issues
• Discussed applications
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Thank you for your attention.
Questions?
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Mackerell, Vol. 25, No. 13, Journal of Computational Chemistry
Protein Force Field
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Mackerell, Vol. 25, No. 13, Journal of Computational Chemistry
Force Field Optimization
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Mackerell, Vol. 25, No. 13, Journal of Computational Chemistry
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Mackerell, Vol. 25, No. 13, Journal of Computational Chemistry
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Mackerell, Vol. 25, No. 13, Journal of Computational Chemistry
Carbohydrate Force Field