Molecular Mechanics
Overview
Molecular Mechanics Molecular mechanics can be used to study molecules that are too large for quantum mechanical models. Molecules are treated as simple mechanical systems governed by Newtonian mechanics. The potential energy

Quantum Chemistry
Theory
Computational Chemistry
Ab initio methods seek to solve the Schrdinger equation. o
Molecular orbital theory expresses the solution as a linear combination of atomic orbitals. Density functional theory (DFT) attempts to solve for t

Molecular Structure
Background
Valence Each element tends to form a fixed number of bonds that depends on the number of electrons in its outer shell:
Carbon and phosphorus have valence 4 Nitrogen has valence 3 Oxygen and sulphur have valence 2 Hydrogen ha

APM 530 - Mathematical Models of Cell Physiology Jay Taylor
Course web page at http:/math.asu.edu/jtaylor syllabus lecture notes readings
Jay Taylor (ASU)
APM 530 - Lecture 1
Fall 2010
1 / 44
This semester will focus on nucleic acid structure and function

Non-bonded Interactions
Overview
Computation of the Non-Bonded Potential Recall that the non-bonded contribution to the potential function takes the form qi qj Aij Bij Vnb (R) = - 6 + 12 + . rij rij rij
i<j i<j
Since each of these sums contains O(N 2 ) te

APM 530 - Mathematical Models of Cell Physiology Jay Taylor
Course web page at http:/math.asu.edu/jtaylor syllabus lecture notes readings
Jay Taylor (ASU)
APM 530 - Lecture 1
Fall 2010
1 / 44
This semester will focus on nucleic acid structure and function

Rare Event Sampling
Importance Sampling
Statement of Problem Suppose that X is an (E , d)-valued random variable with distribution and that we need to calculate the expectation E [f (X )] = f (x)(dx).
E
Unfortunately, analytical and numerical evaluations

Molecular Mechanics
Overview
Molecular Mechanics Molecular mechanics can be used to study molecules that are too large for quantum mechanical models. Molecules are treated as simple mechanical systems governed by Newtonian mechanics. The potential energy

Symplectic Integration
Introduction
Realistic Objectives for Molecular Dynamics Simulations In general, the aim of a MD simulation is to identify qualitative and statistical properties of molecular motions rather than to reproduce a precise trajectory. La

Quantum Chemistry
Theory
Computational Chemistry
Ab initio methods seek to solve the Schrdinger equation. o
Molecular orbital theory expresses the solution as a linear combination of atomic orbitals. Density functional theory (DFT) attempts to solve for t

Quantum Chemistry
Theory
Computational Chemistry
Ab initio methods seek to solve the Schrdinger equation. o
Molecular orbital theory expresses the solution as a linear combination of atomic orbitals. Density functional theory (DFT) attempts to solve for t

Multiscale model of the lac operon.
Villa et al. (2005) Structural dynamics of the lac repressor-DNA complex revealed by a multiscale simulation. PNAS 102: 6783-6788. Background: The lac operon is a cluster of genes in the E. coli genome that encode prote

3D Geometry of the Human Genome
Background
Lieberman-Aiden et al. (2009) Comprehensive Mapping of Long-Range Interactions Reveals Folding Principles of the Human Genome. Science 326: 289-293. Background: Understanding the 3D conformation of the genome can

Rare Event Sampling
Importance Sampling
Statement of Problem Suppose that X is an (E , d)-valued random variable with distribution and that we need to calculate the expectation E [f (X )] = f (x)(dx).
E
Unfortunately, analytical and numerical evaluations

Langevin and Brownian Dynamics
Overview
Langevin Dynamics of a Single Particle We consider a spherical particle of radius r immersed in a viscous fluid and suppose that the dynamics of the particle depend on two forces: A drag force arising from friction

Stochastic Calculus
The Normal Distribution
Preliminaries: Normal Random Variables Definition: A random variable Z with values in R is said to be normally distributed with mean and variance 2 > 0 if Z has density 1 2 2 p(z) = e -(z-) /2 . 2 In this case,

Symplectic Integration
Introduction
Realistic Objectives for Molecular Dynamics Simulations In general, the aim of a MD simulation is to identify qualitative and statistical properties of molecular motions rather than to reproduce a precise trajectory. La

Molecular Dynamics
Introduction
Motivation Most questions in molecular biology are concerned with dynamic processes in macromolecules. Levinthal's paradox: How does protein folding happen quickly on a high-dimensional energy landscape? How does protein st

Implicit Solvation Models
Overview
Solvation and Macromolecular Structure The structure and dynamics of biological macromolecules are strongly influenced by water: Electrostatic effects: charges are screened by water molecules and counterions. Hydrophobic

Non-bonded Interactions
Overview
Computation of the Non-Bonded Potential Recall that the non-bonded contribution to the potential function takes the form qi qj Aij Bij Vnb (R) = - 6 + 12 + . rij rij rij
i<j i<j
Since each of these sums contains O(N 2 ) te