SMA5233 Particle Methods and Molecular Dynamics Lecture 3: Force Fields A/P Chen Yu Zong Tel: Room 08-14, level 8, S16 National University of Singapore

2 Force 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.

3 Molecular Mechanics Force Fields: Functional 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

4 Molecular Mechanics Force Fields: Basic 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.

5 Molecular Mechanics Force Fields: Basic 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.

6 Molecular Mechanics Force Fields: Basic Interactions and Their Models The bending energy equation is also based on Hooke's law

7 Molecular Mechanics Force Fields: Basic Interactions and Their Models The bending energy equation is also based on Hooke's law

8 Molecular Mechanics Force Fields: Basic Interactions and Their Models The torsion energy is modeled by a simple periodic function Why?

9 Molecular Mechanics Force Fields: Basic Interactions and Their Models Torsion energy as a function of bond rotation angle.

10 Molecular Mechanics Force Fields: Basic Interactions and Their Models The non-bonded energy accounts for repulsion, van der Waals attraction, and electrostatic interactions.

11 Molecular Mechanics Force Fields: Basic 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.

12 Molecular Mechanics Force Fields: Basic Interactions and Their Models Types of Hydrogen Bond: N-H … O N-H … N O-H … N O-H … O Can be modeled by VdW+electrostatic (AMBER) Modified Linard-Jones (CHARM) Morse potential (Prohofsky/Chen)

13 Molecular Mechanics Force Fields: Basic Interactions and Their Models Complete Energy Function:

14 Molecular Mechanics Force Fields: Basic Interactions and Their Models Concept of energy scale is Important for molecular Modeling

15 Molecular Mechanics Force Fields: Basic Interactions and Their Models Concept of energy scale is Important for molecular modeling

16 Molecular Mechanics Force Fields: Parameterization 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.

17 Popular molecular mechanics force fields: Classical AMBERAMBER (Assisted Model Building and Energy Refinement) - widely used for proteins and DNA CHARMMCHARMM - originally developed at Harvard, widely used for both small molecules and macromolecules CHARMmCHARMm - commercial version of CHARMM, available through AccelrysAccelrys CVFFCVFF - also broadly used for small molecules and macromolecules GROMACSGROMACS - The force field optimized for the package of the same name GROMOSGROMOS - 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 availableGROMOS OPLS-aa, OPLS-ua, OPLS-2001, OPLS Members of the OPLS family of force fields developed by William L. Jorgensen at Yale Department of Chemistry.OPLSWilliam L. Jorgensen ECEPP/2ECEPP/2 - free energy force field

18 Popular molecular mechanics force fields: Second-generation CFFCFF - a family of forcefields adapted to a broad variety of organic compounds, includes forcefields for polymers, metals, etc. MMFFMMFF - developed at Merck, for a broad range of chemicals MM2MM2, MM3, MM4 - developed by Norman L. Allinger, for a broad range of chemicalsMM3MM4 Reactive force fields ReaxFFReaxFF - 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.

19 Molecular Mechanics Force Fields: Basic Interactions and Their Models Sources of force parameters: Bonds, VdW, Electrostatic (for amino acids, nucleotides only): AMBER: J. Am. Chem. Soc. 117, CHARMM: J. Comp. Chem. 4, H-bonds (Morse potential): Nucleic Acids Res. 20, Biophys. J. 66, Electrostatic parameters of organic molecules need to be computed individually by using special software (such as Gaussian)

20 Molecular Mechanics Force Fields: Atom Types AMBER: J. Am. Chem. Soc. 117,

21 Molecular Mechanics Force Fields: Atom Types and Standard Parameters AMBER: J. Am. Chem. Soc. 117,

22 Molecular Mechanics Force Fields: Atom Partial Charge AMBER: J. Am. Chem. Soc. 117,

23 Molecular Mechanics Force Fields: Atom Partial Charge

24 Molecular Mechanics Force Fields: Atom Partial Charge AMBER: J. Am. Chem. Soc. 117,

25 Molecular Mechanics Force Fields: Atom Partial Charge AMBER: J. Am. Chem. Soc. 117,

26 Molecular Mechanics Force Fields: Atom Partial Charge AMBER: J. Am. Chem. Soc. 117,

27 Molecular Mechanics Force Fields: Bond Parameters AMBER: J. Am. Chem. Soc. 117,

28 Molecular Mechanics Force Fields: Bond Parameters

29 Molecular Mechanics Force Fields: Bond Parameters

30 Molecular Mechanics Force Fields: Bond and Non-Bonded Parameters AMBER: J. Am. Chem. Soc. 117,

31 Water 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

32 Water 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);

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

34 Water Models

35 Coarse-Grain Force Fields: Basic 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, (2001)

36 Coarse-Grain Force Fields: Basic Interactions and Their Models U intra = U bond + U bend + U nonbond U bond is the bonded potential, U bend is the bending potential, and U nonbond represents the non-bonded interaction energy terms. J. Chem. Phys. 114, (2001)