SMA5233 SMA5233 Particle Methods and Molecular DynamicsParticle Methods and Molecular Dynamics

Lecture 3: Force FieldsLecture 3: Force Fields

A/P Chen Yu ZongA/P Chen Yu Zong

Tel: 6516-6877Tel: 6516-6877Email: Email: phacyz@nus.edu.sgphacyz@nus.edu.sg

http://http://bidd.nus.edu.sgbidd.nus.edu.sgRoom 08-14, level 8, S16 Room 08-14, level 8, S16

National University of SingaporeNational University of Singapore

22

Force FieldsForce Fields• A force field (also called a forcefield) refers to the functional form

and parameter sets used to describe the potential energy of a system of particles (typically but not necessarily atoms). Force field functions and parameter sets are derived from both experimental work and high-level quantum mechanical calculations. "All-atom" force fields provide parameters for every atom in a system, including hydrogen, while "united-atom" force fields treat the hydrogen and carbon atoms in methyl and methylene groups as a single interaction center. "Coarse-grained" force fields, which are frequently used in long-time simulations of proteins, provide even more abstracted representations for increased computational efficiency.

• The usage of the term "force field" in chemistry and computational biology differs from the standard usage in physics. In chemistry usage a force field is defined as a potential function, while the term is used in physics to denote the negative gradient of a scalar potential.

33

Molecular Mechanics Force Fields: Molecular Mechanics Force Fields: Functional FormsFunctional Forms

The basic functional form of a force field encapsulates both bonded terms relating to atoms that are linked by covalent bonds, and nonbonded (also called "noncovalent") terms describing the long-range electrostatic and van der Waals forces. The specific decomposition of the terms depends on the force field, but a general form for the total energy can be written as

44

Molecular Mechanics Force Fields: Molecular Mechanics Force Fields: Basic Interactions and Their ModelsBasic Interactions and Their Models

The stretching energy equation is based on Hooke's law. The "kb" parameter controls the stiffness of the bond spring, while "ro" defines its equilibrium length.

55

Molecular Mechanics Force Fields: Molecular Mechanics Force Fields: Basic Interactions and Their ModelsBasic Interactions and Their Models

The stretching energy equation is based on Hooke's law. The "kb" parameter controls the stiffness of the bond spring, while "ro" defines its equilibrium length.

66

Molecular Mechanics Force Fields: Molecular Mechanics Force Fields: Basic Interactions and Their ModelsBasic Interactions and Their Models

The bending energy equation is also based on Hooke's law

77

Molecular Mechanics Force Fields: Molecular Mechanics Force Fields: Basic Interactions and Their ModelsBasic Interactions and Their Models

The bending energy equation is also based on Hooke's law

88

Molecular Mechanics Force Fields: Molecular Mechanics Force Fields: Basic Interactions and Their ModelsBasic Interactions and Their Models

The torsion energy is modeled by a simple periodic function

Why?

99

Molecular Mechanics Force Fields: Molecular Mechanics Force Fields: Basic Interactions and Their ModelsBasic Interactions and Their Models

Torsion energy as a function of bond rotation angle.

1010

Molecular Mechanics Force Fields: Molecular Mechanics Force Fields: Basic Interactions and Their ModelsBasic Interactions and Their Models

The non-bonded energy accounts for repulsion, van der Waals attraction, and electrostatic interactions.

1111

Molecular Mechanics Force Fields: Molecular Mechanics Force Fields: Basic Interactions and Their ModelsBasic Interactions and Their Models

• van der Waals attraction occurs at short range, and rapidly dies off as the interacting atoms move apart.

• Repulsion occurs when the distance between interacting atoms becomes even slightly less than the sum of their contact distance.

• Electrostatic energy dies out slowly and it can affect atoms quite far apart.

1212

Molecular Mechanics Force Fields: Molecular Mechanics Force Fields: Basic Interactions and Their ModelsBasic Interactions and Their Models

Types of Hydrogen Bond:

N-H … ON-H … NO-H … NO-H … O

Can be modeled by

• VdW+electrostatic (AMBER)• Modified Linard-Jones (CHARM)• Morse potential (Prohofsky/Chen)

1313

Molecular Mechanics Molecular Mechanics Force Fields: Force Fields: Basic Interactions and Their ModelsBasic Interactions and Their Models

Complete Energy Function:

bondednon ijij

ji

ij

ij

ij

ijrra

bondH

bondS

rra

rotationbond

n

eqbendinganglebond

eqrstretchbondatoms

r

qq

r

B

r

AVeV

VeVn

v

krrkm

pH

][])1([

])1([)]cos(1[

2

)(2

1)(

2

1

2

61202)(

0

02)(

0

222

'0

'0

1414

Molecular Mechanics Force Fields: Molecular Mechanics Force Fields: Basic Interactions and Their ModelsBasic Interactions and Their Models

Concept of energyscale is Important for molecular Modeling

1515

Molecular Mechanics Force Fields: Molecular Mechanics Force Fields: Basic Interactions and Their ModelsBasic Interactions and Their Models

Concept of energy scale is Important for molecular modeling

1616

Molecular Mechanics Force Fields: Molecular Mechanics Force Fields: ParameterizationParameterization

• In addition to the functional form of the potentials, a force field typically defines a set of parameters for each of a number of atom or particle types that correspond to different atoms and bonding patterns in commonly simulated molecules.

• The parameter set includes values for atomic mass and partial charge for individual atoms, and equilibrium bond lengths and angles for pairs, triplets, and quadruplets of bonded atoms.

• Preparation for a molecular dynamics simulation involves assigning an atom or particle type to each atom or particle in the molecules of interest.

• Although many molecular simulations involve biological macromolecules such as proteins, DNA, and RNA, the parameters for given atom types are generally derived from observations on small organic molecules that are more tractable for experimental study and quantum calculation.

1717

Popular molecular mechanics force fields: Popular molecular mechanics force fields:

ClassicalClassical

AMBER (Assisted Model Building and Energy Refinement) - widely used for proteins and DNA

CHARMM - originally developed at Harvard, widely used for both small molecules and macromolecules

CHARMm - commercial version of CHARMM, available through Accelrys CVFF - also broadly used for small molecules and macromolecules GROMACS - The force field optimized for the package of the same name GROMOS - A force field that comes as part of the GROMOS (GROningen

MOlecular Simulation package), a general-purpose molecular dynamics computer simulation package for the study of biomolecular systems. GROMOS force field (A-version) has been developed for application to aqueous or apolar solutions of proteins, nucleotides and sugars. However, a gas phase version (B-version) for simulation of isolated molecules is also available

OPLS-aa, OPLS-ua, OPLS-2001, OPLS-2005 - Members of the OPLS family of force fields developed by William L. Jorgensen at Yale Department of Chemistry.

ECEPP/2 - free energy force field

1818

Popular molecular mechanics force fields: Popular molecular mechanics force fields:

Second-generation Second-generation

CFF - a family of forcefields adapted to a broad variety of organic compounds, includes forcefields for polymers, metals, etc.

MMFF - developed at Merck, for a broad range of chemicals MM2, MM3, MM4 - developed by Norman L. Allinger, for a broad range of

chemicals

Reactive force fields

ReaxFF - reactive force field developed by William Goddard and coworkers. It is fast, transferable and is the computational method of choice for atomistic-scale dynamical simulations of chemical reactions.

1919

Molecular Mechanics Force Fields: Molecular Mechanics Force Fields: Basic Interactions and Their ModelsBasic Interactions and Their Models

Sources of force parameters:

Bonds, VdW, Electrostatic (for amino acids, nucleotides only):• AMBER: J. Am. Chem. Soc. 117, 5179-5197• CHARMM: J. Comp. Chem. 4, 187-217

H-bonds (Morse potential):• Nucleic Acids Res. 20, 415-419.• Biophys. J. 66, 820-826

Electrostatic parameters of organic molecules need to be computed individually by using special software (such as Gaussian)

bondednon ijij

ji

ij

ij

ij

ijrra

bondH

bondS

rra

rotationbond

n

eqbendinganglebond

eqrstretchbondatoms

r

qq

r

B

r

AVeV

VeVn

v

krrkm

pH

][])1([

])1([)]cos(1[

2

)(2

1)(

2

1

2

61202)(

0

02)(

0

222

'0

'0

2020

Molecular Mechanics Force Fields: Molecular Mechanics Force Fields: Atom TypesAtom Types

AMBER: J. Am. Chem. Soc. 117, 5179-5197

2121

Molecular Mechanics Force Fields: Molecular Mechanics Force Fields: Atom Types and Standard ParametersAtom Types and Standard Parameters

AMBER: J. Am. Chem. Soc. 117, 5179-5197

2222

Molecular Mechanics Force Fields: Atom Partial ChargeMolecular Mechanics Force Fields: Atom Partial Charge

AMBER: J. Am. Chem. Soc. 117, 5179-5197

2323

Molecular Mechanics Force Fields: Atom Partial ChargeMolecular Mechanics Force Fields: Atom Partial Charge

2424

Molecular Mechanics Force Fields: Atom Partial ChargeMolecular Mechanics Force Fields: Atom Partial Charge

AMBER: J. Am. Chem. Soc. 117, 5179-5197

2525

Molecular Mechanics Force Fields: Atom Partial ChargeMolecular Mechanics Force Fields: Atom Partial Charge

AMBER: J. Am. Chem. Soc. 117, 5179-5197

2626

Molecular Mechanics Force Fields: Atom Partial ChargeMolecular Mechanics Force Fields: Atom Partial Charge

AMBER: J. Am. Chem. Soc. 117, 5179-5197

2727

Molecular Mechanics Force Fields: Bond ParametersMolecular Mechanics Force Fields: Bond Parameters

AMBER: J. Am. Chem. Soc. 117, 5179-5197

2828

Molecular Mechanics Force Fields: Bond ParametersMolecular Mechanics Force Fields: Bond Parameters

2929

Molecular Mechanics Force Fields: Bond ParametersMolecular Mechanics Force Fields: Bond Parameters

3030

Molecular Mechanics Force Fields: Molecular Mechanics Force Fields: Bond and Non-Bonded ParametersBond and Non-Bonded Parameters

AMBER: J. Am. Chem. Soc. 117, 5179-5197

3131

Water ModelsWater Models

A recent review listed 46 distinct models, so indirectly indicating their lack of success in quantitatively reproducing the properties of real water.

They may, however, offer useful insight into water's behavior.

Some of the more successful simple models are illustrated in the picture

Models types a, b and c are all planar whereas type d is almost tetrahedral

3232

Water ModelsWater Models

Lennard-Jones potential

Buckingham potential

Shown right is the Lennard-Jones potential for the SPC/E model (solid red line). The σ parameter gives the molecular separation for zero interaction energy. The minimum energy (-ε) lies 12% further at σx21/6 Å.

Also shown (dotted blue line) is an equivalent Buckingham potential (σ 3.55 Å, ε 0.65 kJ mol-1, γ 12.75);

3333

Water Water ModelsModels

A recent review listed 46 distinct models, so indirectly indicating their lack of success in quantitatively reproducing the properties of real water.

3434

Water ModelsWater Models

3535

Coarse-Grain Force Fields: Coarse-Grain Force Fields: Basic Interactions and Their ModelsBasic Interactions and Their Models

• A polymer chain is modeled using N+1 beads connected by elastic joints in a three-dimensional space. Each bead represents one or more amino acid or nucleotide. The motion of the ith bead follows the underdamped Langevin equation with a leapfrog algorithm:

m is the mass of a bead, and ui is the vector tangential to the chain. gi(t) is the Gaussian white noise and obeys the fluctuation-dissipation theorem:

0 and n represent friction coefficients for motion parallel and normal to the chain, respectively. We set n = at 1<i<N-1 and n=0 at i=0,N

J. Chem. Phys. 114, 7260-7266 (2001)

3636

Coarse-Grain Force Fields: Coarse-Grain Force Fields: Basic Interactions and Their ModelsBasic Interactions and Their Models

UUintraintra = U = Ubondbond+ U+ Ubendbend+ U+ Unonbondnonbond

UUbondbond is the bonded potential, U is the bonded potential, Ubendbend is the is the

bending potential, and Ubending potential, and Unonbondnonbond

represents the non-bonded interaction represents the non-bonded interaction energy terms. energy terms.

J. Chem. Phys. 114, 7260-7266 (2001)