Simplistic Molecular Mechanics Force Field Van der WaalsCharge - Charge Bond Angle Improper Dihedral  

Electrostatic Energy The electrostatic contribution is modeled using a Coulombic potential. The electrostatic energy is a function of: o (a) charges on the non-bonded atoms; o (b) inter-atomic distance; o (c) molecular dielectric expression that accounts for the attenuation of electrostatic interaction by the molecule itself.

Electrostatic Energy: Dielectrics The molecular dielectric is set to a constant value between 1.0 and 4.0. However, it has to be consistent with how a force field is designed. (not a free parameter) A linearly varying distance-dependent dielectric (i.e. 1/r) is sometimes used to account for the increase in the solvent (aka, water) dielectrics as the separation distance between interacting atoms increases. (This is being abandoned) When it is needed, the Poisson’s equation, or its approximation, has to be used. (This is gaining popularity)

Other Nonbonded Interactions: Hydrogen Bonding Hydrogen bonding term is usually wrapped into the electrostatic term in force fields widely used today. However it does not imply that hydrogen bonding is purely electrostatic in nature. Hydrogen bonding, if explicitly represented, uses a Lennard-Jones potentials. This replaces the 6-12 Lennard-Jones term for atoms involved in hydrogen-bonding.

Other Nonbonded Interactions: Polarization Polarization is important when large environmental changes occur, i.e. from protein interior to water, or from membrane to water. Usually modeled as inducible dipole: μ =  E Note it is not free to induce a dipole: the work done is 1/2  E 2. Finally, electrostatic energy includes charge-charge, charge-dipole, and dipole-dipole; or electrostatic field is from charge and dipole. No stable force fields with polarization available right now!

Scaling of Nonbonded Terms Scaling of electrostatic energy: charge- charge 1/r; charge-dipole 1/r 2, dipole- dipole 1/r 3. Scaling of van der Waals energy: 1/r 6.

maximasaddle point Potential Energy Surface (PES) Force Field v.s. PES Why EPS is so important? Stable structures of a molecule, such as protein folding, protein- ligand binding Vibrational frequencies The molecular basis of thermodynamics and kinetics.

maximasaddle point Potential Energy Surface (PES)

minimum Potential Energy Surface saddle point maxima

Local minimum vs global minimum Many local minima; only ONE global minimum Energy Minimization

Stationary points: points on a PES with all first energy derivatives (gradients) zero. Minima (local and global): stationary points with all eigenvalues of the Hessian matrix (all second derivatives) positive. Saddle points: stationary points with exactly one negative eigenvalue in the Hessian matrix. Classification of Stationary Points on PES

Energy Minimization: Methods Non-gradient based methods: systematic numeration; simplex; direction set (Powell’s). Gradient based methods: the steepest descents; conjugate gradients Hessian based methods: Newton-Raphson; quasi-Newton

Gradient Based Minimization: Steepest Descents To start, walk straight downhill along the gradient direction at the initial point. Perform line search along the gradient direction. The next direction to take is orthogonal to the previous direction, and so on. Use adaptive step sizes (small) in the line search. Step size will be reduced if the energy goes up, otherwise it will be increased.

Hessian Based Minimization Newton-Raphson: Idea For any 1-d quadratic function U(x), Taylor expansion at x k gives U(x) = U(x k ) + (x-x k )U’(x k ) + (x-x k ) 2 U”(x k )/2, U’(x k ) = U’(x k ) + (x-x k ) U”(x k ). At the minimum x*: U’(x*) = 0, so that x* = x k - U ’(x k )/U”(x k ) For an n-d quadratic function: x* = x k - U’(x k ) U”(x k ) -1

Hessian Based Minimization Newton-Raphson Note that U”(x k ) -1 is the inverse of a matrix (Hessian), very slow to compute if n is large. Also note that n is 3*no_of_atom for the Hessian matrix. However, the minimization can be performed in one-step for quadratic functions! For force field energy functions, it still takes less steps to minimize than other methods, but each step is much slower. The burden is shifted to the inversion of the Hessian Matrix.

Hessian Based Minimization Newton-Raphson: Pros and Cons Newton-Raphson works well for portion of the PES where the quadratic approximation is good, i.e. near a local minimum. It does not work well far away from a minimum. It does not work if the Hessian matrix has negative eigenvalues.

Which minimization should I use? If gradient is not possible, Powell’s method would be a reasonable choice. If gradient is possible to get, as in force field functions, steepest descents can be used to relax initial bad geometry. This is usually followed by a conjugate gradient method, then a quasi-Newton method. If a small system, less than a few hundred atoms, the Newton-Raphson method may be used if it is sufficiently close to a local minimum.

Energy Minimization: Limitations Extrema (stationary points) are located by most methods; this includes maxima, minima, and saddle points. Among the minima, local minima are found, not necessarily the global minimum. With a flat PES, a lot of cpu time can be spent seeking the lowest energy structure.

Energy Minimization: Limitations What does the global minimum energy structure mean? Does reaction/interaction of interest necessarily occur via lowest energy conformations? What other low energy conformations are available? Minimization only corresponds to motions of a molecule at 0 K temperature.

Normal Mode Analysis Useful for studies of vibrational frequencies. In macromolecules, the lowest frequency modes correspond to delocalized motions. These modes can be used to understand function- related slow molecular motions, such as allosteric motions.

Normal Mode Analysis The lowest normal mode of the complex of H2BF with formaldehyde

Beyond Static Structures Molecular motion is inherent to all biochemical processes. Simple vibrations, like bond stretching and angle bending, give rise to IR spectra. Biochemical reactions, hormone-receptor binding, and other complex processes are associated with many kinds of intra- and intermolecular motions.

Understanding the Mechanisms

The driving force for biochemical processes is described by thermodynamics. Thermodynamics dictates the energetic relationships between different chemical states. The mechanism by which chemical processes occur is described by kinetics. The sequence or rate of events that occur as molecules transform between their various possible states is described by kinetics.

Characteristic Motions in Proteins Type of MotionFunctionality ExamplesTime and Amplitude Scales Local Motions: Atomic fluctuation Side chain motion Ligand docking flexibility Temporal diffusion pathways fs - ps ( s) less than 1 A Medium Scale Motions: Loop motion Terminal arm motion Rigid-body motion Active site conformation adaptation Binding specificity ns - µ s ( s) A Large Scale Motions: Domain motion Subunit motion Hinge bending motion Allosteric transitions µ - ms ( s) A Global Motions: Folding/unfolding Subunit association Hormone activation Protein functionality ms - h ( s) more than 5 A

A 2.6 ps Simulation of DNA

The Basics of Molecular Dynamics Recall Newton’s equation of motion given E(r): A dynamics trajectory can tell us how a process involves over time, i.e. kinetics.

Time-Integration Algorithm: Leap Frog

Time Integration Algorithms: Requirements It should be fast, and require little memory. It should permit the use of a long time step. It should duplicate the classical trajectory as closely as possible (analytical). It should satisfy the known conservation laws for energy and momentum It should be time-reversible. It should be simple in form and easy to code.

Time Integration Algorithms Dynamics simulation is intrinsically chaotic. The finite accuracy of any computer program on any computer hardware will make any trajectory deviate from analytical result on long time scales. This is actually very helpful for thermodynamics analysis. The important requirement is the conservation of energy and momentum.

Minimization v.s. Dyanmics A dynamics calculation alters the intramolecular degrees of freedom in a step-wise fashion, analogous to energy minimization. However, the steps in molecular dynamics meaningfully represent the changes in atomic positions over time. The individual steps in energy minimization are merely directed at establishing a down-hill direction to a minimum.