Why Proteins Fold? (Parts of this presentation are based on work of Ashok Kolaskar) CS490B: Introduction to Bioinformatics Mar. 25, 2002
Molecular Dynamics: Introduction At physiological conditions, the biomolecules undergo several movements and changes The time-scales of the motions are diverse, ranging from few femtoseconds to few seconds These motions are crucial for the function of the biomolecules
Molecular Dynamics: Introduction Newton s second law of motion
Molecular Dynamics: Introduction We need to know The motion of the atoms in a molecule, x(t) and therefore, the potential energy, V(x)
Molecular Dynamics: Introduction How do we describe the potential energy V(x) for a molecule? Potential Energy includes terms for Bond stretching Angle Bending Torsional rotation Improper dihedrals
Molecular Dynamics: Introduction Potential energy includes terms for (contd.) Electrostatic Interactions van der Waals Interactions
Molecular Dynamics: Introduction Equation for covalent terms in P.E. ( ) )] cos( [1 ) ( ) ( ) ( 0 2 0 2 0 2 0 φ φ ω ω θ θ ω θ + + + + = n A k k l l k R V torsions n impropers angles bonds l bonded
Molecular Dynamics: Introduction Equation for non-bonded terms in P.E. V nonbonded ( R) min min rij r 12 ij 6 = ( ε [( ) 2( ) ] + ij r r i< j ij ij q q i 4πε r j ε r 0 ij
Molecular Dynamics: Introduction Each of these interactions exerts a force onto a given atom of the molecule The total resulting force on each atom is calculated using the PE function Knowing the force on an atom, its movement due to the force is then calculated:
Molecular Dynamics: Introduction To do this, we should know at given time t, initial position of the atom x 1 its velocity v 1 = dx 1 /dt and the acceleration a 1 = d 2 x 1 /dt 2 = m -1 F(x 1 )
Molecular Dynamics: Introduction The position x 2, of the atom after time interval t would be, x 2 = x 1 + v 1 t and the velocity v 2 would be, v = v + a t = v + 1 m F x t = v ( m 2 1 1 1 1) 1 1 dv dx x 1 t
How a molecule changes during MD
Molecular Dynamics: Introduction In general, given the values x 1, v 1 and the potential energy V(x), the molecular trajectory x(t) can be calculated, using, t dx x dv m v v t v x x x i i i i i i = + = 1 ) ( 1 1 1 1
Generalizing these ideas, the trajectories for all the atoms of a molecule can be calculated.
The Necessary Ingredients Description of the structure: atoms and connectivity Initial structure: geometry of the system Potential Energy Function: force field AMBER CVFF CFF95 Universal
Protein-specific Applications of MD Calculation of thermodynamic properties such as internal energy, free energy Studying the protein folding / unfolding process Studying conformational properties and transitions due to environmental conditions Studying conformational distributions in molecular system.
An overview of various motions in proteins (1) Motion Relative vibration of bonded atoms Elastic vibration of globular region Rotation of side chains at surface Spatial extent (nm) 0.2 to 0.5 1 to 2 0.5 to 1 Log 10 of characteristic time (s) -14 to 13-12 to 11-11 to 10
An overview of various motions in proteins (2) Motion Relative motion of different globular regions (hinge bending) Rotation of medium-sized side chains in interior Protein Folding Spatial Extent (nm) 1 to 2 0.5??? Log 10 of characteristic time (s) -11 to 7-4 to 0-5 to 2
A typical MD simulation protocol Initial random structure generation Initial energy minimization Equilibration Dynamics run with capture of conformations at regular intervals Energy minimization of each captured conformation
Essential Parameters for MD (to be set by user) Temperature Pressure Time step Dielectric constant Force field Durations of equilibration and MD run ph effect (addition of ions)
WHAT IS AMBER? AMBER (Assisted Model Building with Energy Refinement). Allows users to carry out molecular dynamics simulations Updated forcefield for proteins and nucleic acids Parallelized dynamics codes Ewald sum periodicity New graphical and text-based tools for building molecules Powerful tools for NMR spectral simulations New dynamics and free energy program
CASE STUDY Type II restriction endonucleases recognize DNA sequences of 4 to 8 base pairs in length and require Mg 2+ to hydrolyse DNA. The recognition of DNA sequences by endonucleases is still an open question. PvuII endonuclease, recognizes the sequence 5 - CAGCTG-3 and cleaves between the central G and C bases in both strands. Though crystal structure of the PvuII-DNA complex have been reported, very little is known about the steps involved in the recognition of the cleavage site by the PvuII enzyme. Molecular dynamics (MD) simulation is a powerful computational approach to study the macromolecular structure and motions.
CASE STUDY: METHODS (MD Simulations) Simulations were carried out on the sequence 5 -TGACCAGCTGGTC-3 Rectangular box (60 X 48 X 54 Å 3 ) containing 24 Na +, using PBC SHAKE algorithm Integration time step of 1 fs 283 K with Berendsen coupling Particle Mesh Ewald (PME) method 9.0 Å cutoff was applied to the Lennard-Jones interaction term. Equilibration was performed by slowly raising the temperature from 100 to 283 K. Production run was initiated for 1.288 ns and the structures were saved at intervals of one picosecond. The trajectory files were imaged using the RDPARM program and viewed and analysed using the MOIL-VIEW and CURVES packages respectively.
STARTING DNA MODEL
DNA MODEL WITH IONS
DNA in a box of water
SNAPSHOTS
SNAPSHOTS
DOCKING
DOCKING
Computing Potentials Boundary Conditions Truncating Potentials Approximating Potentials
Computing Potentials Fast decaying potentials can be truncated (for example, Lennart-Jones Potentials) Other potentials such as Coloumbic potentials must be approximated.
Fast Multipole Method What is the gravitational potential where we are sitting because of the sun? Can we treat the sun as a point mass or should we integrate over the volume of the sun. Fast Multipole Method works on the principle of hierarchical aggregation.
Fast Multipole Method Collect spatially proximate objects and represent it with a series -- a multipole series. For an evaluation point far from this group, compute the series as opposed to point-wise evaluation. Do this hierarchically. Reduces complexity from O(n 2 ) to O(n).
Ewald Summations Decompose the potential into two terms: an infinite term that whose FFT can be computed. a fast converging local term that can be truncated. Reduces complexity to O(n log n)
Software CHARMM NAMD many other in-grown codes.