Introduction to Molecular Simulation

Chih-Hao Lu

China Medical University

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Molecular dynamics (MD) is a form of computer simulation in which atoms and molecules are allowed to interact for a period of time by approximations of known physics, giving a view of the motion of the particles.

This kind of simulation is frequently used in the study of proteins and biomolecules.

It is tempting, though not entirely accurate, to describe the technique as a "virtual microscope" with high temporal and spatial resolution.

Protein Dynamics

Experimental Methods1. X-ray crystallography2. Nuclear magnetic resonance

Computational Methods1. Molecular simulation2. Elastic network model3. Protein-fixed-point model4. Weighted-contact number model

It is possible to take "still snapshots" of X-ray crystal structures and probe features of the motion of molecules through NMR, no conventional experiment allows access to all the time scales of motion with atomic resolution

Protein Dynamics - Experimental Methods

X-ray crystallographyTemperature factor (B factor)

Source: Protein Data Bank

NMROrder parameter (S2)

Source: Literatures

Value

large

small

B factor Order parameter

High fluctuation

Low fluctuation

Low fluctuation

High fluctuation

Experimental Methods

X-ray diffraction NMR

Richard Feynman once said that "If we were to name the most powerful assumption of all, which leads one on and on in an attempt to understand life, it is that all things are made of atoms, and that everything that living things do can be understood in terms of the jigglings and wigglings of atoms."

Richard Phillips Feynman (1918–1988)

Molecular dynamics lets scientists peer into the motion of individual atoms in a way which is not possible in laboratory experiments.

in 1962: "... I took a number of rubber balls and stuck them together with rods of a selection of different lengths ranging from 2.75 to 4 inches. I tried to do this in the first place as casually as possible, working in my own office, being interrupted every five minutes or so and not remembering what I had done before the interruption."

John Desmond Bernal (1901-1971)

Because molecular systems generally consist of a vast number of particles, it is in general impossible to find the properties of such complex systems analytically.

MD simulation circumvents the analytical intractability by using numerical methods.

Babylonian clay tablet YBC 7289 (1800–1600 BC)

?Tip: Sexagesimal

Babylonian clay tablet YBC 7289 (1800–1600 BC)

MD simulation circumvents the analytical intractability by using numerical methods.

long MD simulations are mathematically ill-conditioned, generating cumulative errors in numerical integration that can be minimized with proper selection of algorithms and parameters, but not eliminated entirely.

Current potential functions are, in many cases, not sufficiently accurate to reproduce the dynamics of molecular systems, so the much more computationally demanding ab Initio Molecular Dynamics method must be used.

Design of a molecular dynamics simulation should account for the available computational power. Simulation size (n=number of particles), timestep and total time duration must be selected so that the calculation can finish within a reasonable time period.

However, the simulations should be long enough to be relevant to the time scales of the natural processes being studied.

Most scientific publications about the dynamics of proteins and DNA use data from simulations spanning nanoseconds (1E-9 s) to microseconds (1E-6 s).

A molecular dynamics simulation requires the definition of a potential function, or a description of the terms by which the particles in the simulation will interact.

In chemistry and biology this is usually referred to as a force field.

Most force fields consist of a summation of bonded forces associated with chemical bonds, bond angles, and bond dihedrals, and non-bonded forces associated with van der Waals forces and electrostatic charge.

These potentials contain free parameters such as atomic charge, van der Waals parameters reflecting estimates of atomic radius, and equilibrium bond length, angle, and dihedral

In addition to the functional form of the potentials, a force field defines a set of parameters for each type of atom.

For example, a force field would include distinct parameters for an oxygen atom in a carbonyl functional group and in a hydroxyl group.

34

10 bonds?? angle terms?? torsional terms?? non-bonded interactions

How many types of bonds?

C-C 2C-H 8

35

10 bonds18 angle terms?? torsional terms?? non-bonded interactions

C-C-C 1C-C-H 10H-C-H 7

How many types of angles?

36

10 bonds18 angle terms18 torsional terms?? non-bonded interactions

How many types of torsions?

H-C-C-H 12H-C-C-C 6

37

10 bonds18 angle terms18 torsional terms45 non-bonded interactions

How many types of nonbonded ineractions?

H-H 28C-H 16C-C 1

For simulating molecules in a solvent, a choice should be made between explicit solvent and implicit solvent.

Explicit solvent particles must be calculated expensively by the force field, while implicit solvents use a mean-field approach.

The simplest water models treat the water molecule as rigid and rely only on non-bonded interactions.

The charged sites may be on the atoms or on dummy sites (such as lone pairs). In most water models, the Lennard-Jones term applies only to the interaction between the oxygen atoms.

the general shape of the 3- to 6-site water models

Three-site models have three interaction sites, corresponding to the three atoms of the water molecule.

Each atom gets assigned a point charge, and the oxygen atom also gets the Lennard-Jones parameters. The 3-site models are very popular for molecular dynamics simulations because of their simplicity and computational efficiency.

the parameters for some 3-site models

TIPS SPC TIP3P SPC/Er(OH), Å 0.9572 1.0 0.9572 1.0HOH, deg 104.52 109.47 104.52 109.47A × 10−3, kcal Å12/mol 580.0 629.4 582.0 629.4B, kcal Å6/mol 525.0 625.5 595.0 625.5q(O) −0.80 −0.82 −0.834 −0.8476q(H) +0.40 +0.41 +0.417 +0.4238

The TIP3P model implemented in the CHARMM force field is a slightly modified version of the original.

TIPS SPC TIP3P SPC/Er(OH), Å 0.9572 1.0 0.9572 1.0HOH, deg 104.52 109.47 104.52 109.47A × 10−3, kcal Å12/mol 580.0 629.4 582.0 629.4B, kcal Å6/mol 525.0 625.5 595.0 625.5q(O) −0.80 −0.82 −0.834 −0.8476q(H) +0.40 +0.41 +0.417 +0.4238

The 4-site models place the negative charge on a dummy atom (labeled M in the figure) placed near the oxygen along the bisector of the HOH angle.This improves the electrostatic distribution around the water molecule.

The TIP4P model, first published in 1983, is widely implemented in computational chemistry software packages and often used for the simulation of biomolecular systems.

the parameters for some 4-site models

BF TIPS2 TIP4P TIP4P-Ew TIP4P/Icer(OH), Å 0.96 0.9572 0.9572 0.9572 0.9572HOH, deg 105.7 104.52 104.52 104.52 104.52r(OM), Å 0.15 0.15 0.15 0.125 0.1577A × 10−3, kcal Å12/mol 560.4 695.0 600.0 656.1 857.9B, kcal Å6/mol 837.0 600.0 610.0 653.5 850.5q(M) −0.98 −1.07 −1.04 −1.04844 −1.1794q(H) +0.49 +0.535 +0.52 +0.52422 +0.5897

The 5-site models place the negative charge on dummy atoms (labeled L) representing the lone pairs of the oxygen atom, with a tetrahedral-like geometry.

Mainly due to their higher computational cost, five-site models were not developed much until 2000, when the TIP5P model of Mahoney and Jorgensen was published.

William L. JorgensenDepartment of ChemistryYale University

When compared with earlier models, the TIP5P model results in improvements in the geometry for the water dimer, a more "tetrahedral" water structure.

the parameters for some 5-site modelsBNS ST2 TIP5P TIP5P-E

r(OH), Å 1.0 1.0 0.9572 0.9572HOH, deg 109.47 109.47 104.52 104.52r(OL), Å 1.0 0.8 0.70 0.70LOL, deg 109.47 109.47 109.47 109.47A × 10−3, kcal Å12/mol 77.4 238.7 544.5 590.3B, kcal Å6/mol 153.8 268.9 554.3 628.2q(L) −0.19562 −0.2357 −0.241 −0.241q(H) +0.19562 +0.2357 +0.241 +0.241RL, Å 2.0379 2.0160RU, Å 3.1877 3.1287

MB model. A more abstract model resembling the Mercedes-Benz logo that reproduces some features of water in two-dimensional systems. It is not used as such for simulations of "real" (i.e., three-dimensional) systems, but it is useful for qualitative studies and for educational purposes.

For simulating molecules in a solvent, a choice should be made between explicit solvent and implicit solvent.

Implicit solvation (sometimes known as continuum solvation) is a method of representing solvent as a continuous medium instead of individual “explicit” solvent molecules.

There are two basic types of implicit solvent methods: models based on accessible surface areas (ASA) that were historically the first, and more recent continuum electrostatics models

The accessible surface area (ASA) is the surface area of a biomolecule that is accessible to a solvent.

The free energy of solvation of a solute molecule in the simplest ASA-based method is given by

σi is solvation parameter of atom i, i.e. a contribution to the free energy of solvation of the particular atom i per surface unit area

Hybrid implicit/explicit solvation modelsIt is possible to include a layer or sphere of water molecules around the solute, and model the bulk with an implicit solvent.

Molecular dynamics simulation

• Molecular dynamics simulation of 30 proteins ~50 years CPU (Rueda, PNAS 2007)

? Bonds ? C-H ? C-C

? Angle terms ? C-C-C ? H-C-H ? H-C-C

? Torsional terms ? H-C-C-H ? H-C-C-C

? Non-bonded interactions ? C-H ? C-C ? H-H