Peter J. Steinbach

steinbac@mail.nih.gov

This tutorial is available in PDF format.

**Classical Mechanics Applied to Biology **

The purpose of this tutorial is to introduce several popular numerical techniques used to simulate the structure and dynamics of biomolecules. The discussion is confined to simulation methods that apply classical mechanics to biological systems, although some quantum theory is presented to quantify some shortcomings of classical approximations. Molecular dynamics (MD) simulation, Langevin dynamics (LD) simulation, Monte Carlo (MC) simulation, and normal mode analysis are among the methods surveyed here. There are techniques being developed that treat the bulk of a macromolecule classically while applying quantum mechanics to a subset of atoms, typically the active site. This research frontier will not be addressed here. Completely classical studies remain more common and continue to contribute to our understanding of biological systems.

**When is classical mechanics a reasonable approximation?**

In Newtonian physics, any particle may possess any one of a continuum of energy values.
In quantum physics, the energy is *quantized*, not continuous.
That is, the system can accomodate only certain discrete levels of energy, separated by gaps.
At very low temperatures these gaps are much larger than thermal energy, and the system is confined to one or just a few of
the low-energy states. Here, we expect the `discreteness' of the quantum energy landscape to be evident in the system's behavior.
As the temperature is increased, more and more states become thermally accessible, the `discreteness' becomes less and less
important, and the system approaches classical behavior.

For a harmonic oscillator, the quantized energies are separated by ,
where *h* is Planck's constant and *f* is the frequency of harmonic vibration. Classical
behavior is approached at temperatures for which , where is the Boltzmann constant and
= 0.596 kcal/mol at 300 K. Setting *hf* = 0.596 kcal/mol yields *f* = 6.25/ps, or 209 .
So a classical treatment will suffice for motions with characteristic times of a ps or longer at room temperature.

**Outline - Shades of things to come**

We'll expand on the above argument with a more quantitative analysis of classical and quantum treatments of simple harmonic oscillation. This not-too-mathematical glimpse of quantum mechanical phenomena is included to help simulators estimate how much they can trust various motions that have been simulated with the approximations inherent in classical physics. Then, we'll identify the basic ingredients of a macromolecular simulation: a description of the structure, a set of atomic coordinates, and an empirical energy function. This is followed by a discussion of the most popular simulation techniques: energy minimization, molecular dynamics and Monte Carlo simulation, simulated annealing, and normal-mode analysis. Finally, a few general suggestions are offered to those about to perform their first macromolecular simulation. But first, a little theoretical background is presented to aid the discussion. It's a short summary of the most relevant concepts of classical, quantum, and statistical mechanics, along with a glimpse of classical electrostatics.

- Classical and Quantum Mechanics - in a Nutshell
- Statistical Mechanics - Calculating Equilibrium Averages
- Classical vs. Quantum Mechanics: The Harmonic Oscillator in One Dimension
- Probability(x): Where is the oscillating particle?
- Average Energy
*U*and Heat Capacity - Mean-Square Fluctuation
- Overall Comparison - What does all this mean for simulation?

- Electrostatics and the `Generalized Born' Solvent Model
- Classical Macromolecular Simulation
- A Note on Notation - Some CHARMMing language
- Simulating Biomolecules: The Three Necessary Ingredients

- The Empirical Potential Energy Function

- Classical Simulation and Modeling Techniques
- Questions We Can Ask With a Computer
- Energy Minimization
- Molecular Dynamics (MD) Simulation
- Langevin Dynamics (LD) Simulation
- Monte Carlo (MC) Simulation
- Normal Mode (Harmonic) Analysis
- Simulated Annealing

- What is Unique to Computer Experiments?
- Worth Worrying About:
- Bibliography
- About this document ...