Lecture 11: Potential Energy Functions...• Potential Energy – non-kinetic part of the internal...

Lecture 11: Potential Energy Functions

Dr. Ronald M. Levy

[email protected]

Originally contributed by Lauren Wickstrom (2011)

Statistical Thermodynamics

Statistical ThermodynamicsSpring 2013

Microscopic/Macroscopic Connection

• The connection between microscopic interactions and macroscopic thermodynamic properties is encoded in the potential energy function U(x)

Z=∫dx e−βU (x )

• A model of the system starts with the specification of its potential energy function


Terminology

• Potential Energy – non-kinetic part of the internal energy of a system.

• Molecular Mechanics - Classical mechanics description of the potential energy.

• Function of nuclear positions (electronic motion neglected) – Born-Oppenheimer approximation

• Force field – Parameterized, analytical function of the potential energy

• All-Atom Force Field – Function of the positions of all of the atoms (as opposed to a coarse-grained force field).


Some Common All-atom force fields for biomolecules

• OPLS (Optimized Potential for Liquid Simulations) – Jorgensen/Friesner

• AMBER (Assisted Model Building with Energy Refinement)* – Kollman/Case

• CHARMM (Chemistry at Harvard Molecular Mechanics)* – Karplus

• GROMOS (GROningen MOlecular Simulation package)* –Berendsen and van Gunsteren

• AMOEBA (Atomic Multipole Optimized Energetics for Biomolecular Applications) Ren & Ponder

• ECEPP (Empirical Conformational Energy Program for Peptides) - Scheraga

*These are force fields as well as simulation packages.


Born-Oppenheimer formulation

● Molecular degrees of freedom: nucleii positions (x)● For each configuration of the nucleii we have a series of

electronic states at energies E0, E1, E2, ...

● Assume that electrons readjust instantaneously to new positions of the nucleii.

● System always remains in ground state at energy

E0=E 0(x)

● E0(x) can in princible be evaluated at

each x by QM methods (Gaussian, etc.)● In practice need a simple function that mimics as best as

possible the true E0(x)


Typical formulation of a Non-polarizable Non-dissociative Force Field:

Energy terms Interactions

Vbond 1-2

Vangle 1-3

Vtorsions 1-4

VLJ,Vcoul. Non-bonded

V r =∑bonds

k ij

2d ij−d ij , 0

2 ∑angles

k ijk

2ijk−ijk , 0

2

∑torsions

V ijkl ,n

2[1cos nijkl− ijkl] ∑

non−bonded [4ij ij12

r ij12 −

ij6

r ij6 qi q j

40 r ij ]

Torsion

Bond

Bond Angle

Non-Bonded


Bond stretching

V bond=De [1−e−ad−d 0 ]2

De=depth of potential energy mininum

Morse Potential

a=width of potential welld0=equilibrium bond length

V bond=k2d−d 0

2

Hooke's lawHooke's law

– harmonic approximation – non dissociative– reasonable for small displacements


Bond stretching parameters

• k obtained from vibrational spectra

• d0 obtained from X-ray crystallography

• Hard degree of freedom

• Often constrained in MD

If δd = .2 Å for carbonyl

Vbond = 11.4 kcal/mol

Type k (kcal/mol/ Å2) d0(Å)

CA-N 337 1.44

C=O 570 1.22

CAN CO

V bond=k2d−d 0

2


Angle bending

• Hooke’s Law in anglecoordinates

V angle=k

2−0

2

Type Kθ(kcal/mol/ radian2) θ0(deg)

N-CA-HA 35 109.5

CA-C-O 80 120.4

If δθ= 4° for CA-C-0Vangle = 11.6 kcal/mol

CAC N

O

HA

N


Torsional terms Black: Vn=4; n=2; γ=180Red: Vn=2; n=3;γ=0

• Barriers of rotation

• Ethane – 9 dihedral angles (H-C-C-H)

Butane – 27 dihedral angles (1 C-C-C-C, 10 H-C-C-C,

16 H-C-C-H)

V n=amplituden=periodicityn=phase

V torsion=∑n

V n

2[1cosn−n]

Fourier expansion


• 120 degrees periodicity

• Parameters are obtained by fitting to high level QM data

Torsional parametersTorsion angle Vn

N-CA-CB-HB1 .23

N-CA-CB-HB2 .23

N-CA-CB-HB3 .23

V torsion=V 0V 3

2[1cos 3]


Improper torsions

• Describe out of plane motion

• This is often important to maintain planar structure.

Examples:

Peptide bond

Benzene

• OPLS – adjusts force constants instead of using improper function

V improper=∑n

k i

2ijkl−0

2

V improper=V 2

2[1cos 2ijkl−]


Non-bonded interactions: Electrostatic

• Important “directional” interaction energy term

• Charge distribution calculated from QM calculations

• Electrostatic interaction between two molecules:

V mp=∫dr dr 'mr pr '

∣r−r '∣


Electrostatics• One representation: Multipole Expansion• Charge distribution represented in terms of its moments (charges, dipoles, quadrupoles, octupoles)

r q , , Q ,


Different types of multipole interactions

• Interactions become weaker with higher multipole moments

• Attraction or repulsion -charge and orientation of the dipole

Type of interaction

Distance r Dependence

Charge-charge 1/r

Charge-dipole 1/r2

Dipole/dipole 1/r3

Dipole/induced dipole 1/r6

+ δ+ δ-

Charge Dipole

UNFAVORABLE

- δ+ δ-

Charge Dipole

FAVORABLE


Example:

Benzene – Benzene interaction

• 144 charge-charge interactions

• First term: quadrupole-quadruple calculation (no monopole, no dipole)

• Only valid when the distance between two molecules is much larger than the internal dimensions.

• Multipolar expansions are computationally expensive


More common representation: partial charges

• Charge distribution described by delta functions at “charged sites” (usually atomic sites)

Partial charges from:• Fit to experimental liquid properties (OPLS)• ESP charge fitting to reproduce electrostatic potentials of high level QM• X-ray crystallographic electron density

r ≈∑iqir i

V mp≈∑ij

qi q j

∣r i−r j∣


Polarization

• Charges can induce other charge asymmetries by polarization

• Most common strategy is to employ pre-polarized charge distributions

Example: Dipole of water in vacuum = 1.8 D

Dipole fit to yield liquid properties = 2.2-2.5 D

Sometimes overlooked work for creating polarized charge distribution:

E = electric fieldμ = dipole α=point polarizablity

V self =∫0

iE · d =∫0

i

α· d =i

2/2α


Explicit treatment of polarization

• Induced dipole

= E

● Interactions between permanent charges, between charges and induced dipoles and between induced dipoles

● Induced dipoles and electrostatic field solved to self-consistency

● Polarization energy V pol=−12∑i

i E 0

= Electric field from permanent chargesE0

● Most commonly used “fixed-charge” force fields to not treat explicit polarization


σ

Parameters fit to reproduce experimental liquid properties

σ

ε

ε =Depth of potential wellσ=collision diameter (potential energy = zero)

Repulsive and van der Waals interactions

V LJ=∑i=1

N

∑j=i1

N

4 εij [ σ ij

rij

12−σ ij

r ij

6 ]

εij= εi ε j

σ ij= σ i σ j

Lennard Jones potential


1,4 interactions

• Electrostatics + LJ interactions

• 1-2, 1-3 interactions excluded

• 1-4 interactions scaled

• Inclusion in bonded terms of force field

Force field Electrostatics(factor) van der Waals

OPLS .5 .5

AMBER .83 .5

CHARMM 1 1

1-2

1-3

1-4


Computational timing of different force field terms

• Different parts of the energy scale differently

• Bonded – linear

• Non-bonded interactions – N2

• Requires most computational time

• Cutoffs must be implemented to reduce this cost


Final thoughts

• Force fields rely on experimental data for parameterization

• Validation is also a key aspect of building these models (experimental observables)

• Time scale and sampling are important aspects are essential for force field evaluation

• One must know the limitations of particular models (force field biases)


Solvation: Solvent Potential of Mean Force W:Implicit solvent

• Approximate model

• Averages solvation effect (potential mean force)

• High dielectric = solvent; Low dielectric = solute

• Less friction than water (faster transitions)

• Fewer degrees of freedom

ε=80

ε=1

Lecture 11: Potential Energy Functions...• Potential Energy – non-kinetic part of the internal...

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