Molecular Mechanics, Dynamics & Docking Lawrence Hunter, Ph.D. Director, Computational Bioscience Program University of Colorado School of Medicine Larry.Hunter@uchsc.edu http://compbio.uchsc.edu/hunter
Molecular Mechanics Back to Physics: proteins are completely described by the physicochemical properties of their constituent atoms and bonds. Computational statistical mechanics: Calculate the dynamics by repeated integration of the forces acting on each atom. Minimum energy conformation in solution is (assumed to be) the native state, so relevant to protein folding.
Energy Minimization Many forces act on a protein Hydrophobic: inside of protein wants to avoid water Packing: atoms can't be too close, nor too far away Bond angle/length constraints Long distance, e.g. Electrostatics & Hydrogen bonds Disulphide bonds Salt bridges Can calculate all of these forces, and minimize Intractable in general case, but can be useful
More and less than folding... MM provides physical (predictive) model atomic fluctuations, heat capacity, configurational changes (and rates), etc. Can (theoretically) predict chemical characteristics (e.g. energy of formation, bond making/breaking, etc.) Energy minimization is an important part of both empirical and predicted structures MM could be used to calculate large scale conformational changes over long periods of time, but currently computationally infeasible.
How does MM work? Three aspects: Functions that describe the forces acting on the atoms Numerical integration methods, to calculate the motion of the atoms due to the forces acting on them Long time propagation of the equations of motion Computational demands are intense Accuracy (small errors propagate!) Stability Lots of techniques for approximation (e.g. rigid bodies) and handling artifacts (resonance).
Hydrophobic packing models Dill's HP model Two classes of amino acids, hydrophobic (H) and polar (P) Lattice model for position of (point) amino acids. Thread chain of H's and P's through lattice to maximize number of H-H contacts 3D 2D
But... Even the 2D HP packing problem (which is easier than the 3D one) turns out to be NP complete! Good approximation results exist. 3/8 of optimal approximation (3D) In triangular lattice, algorithm for >60% of optimal packing Other interesting results in the model, e.g. Which sequences have a single optimal fold?
Real energetics Steric (conformational) energy. Additive combination of Bonded: stretching, bending, stretching and bending Non-bonded: Van der Waals, electrostatic and torsional Minimum energy conformation minimizes these energies Rosetta energy function is an empirical attempt to capture most of this energy function without having to calculate it fully.
The Force Fields How do atoms stretch, vibrate, rotate, etc.? Must represent the constraints on atomic motion (e.g. van der Waals, electrostatic, bonds, etc.) Must also represent solvation effects, anharmonic interactions, etc. Quantum solutions exist, but are too complex to calculate for such large systems Empirical (approximate) energy functions must be used. No single best function exists.
Bond length Spring-like term for energy based on distance E str = ½k s,ij (r ij -r o ) 2 where k s,ij is the stretching force constant for the bond between i and j, r ij is the length, and r o is the equilibrium bond length
Bond bend Same basic idea for bending E bend = ½k b,ij (θ ij θ o ) 2 where where k b,ij is the bending force constant, θ ij is the instantaneous bond angle, and θ o is the equilibrium bond angle
Stretch-bend When a bond is bent, the two associated bond lengths increase, with interaction term: E str-bend =½k sb,ijk (r ij -r o )(θ ik - θ o ) where k sb,ijk is the stretch-bend force constant for the bond between atoms i and j with the bend between atoms i, j, and k.
Van der Waals A non-bonded interaction capturing the preferred distance between atoms where A and B are constants depending on the atoms. For two hydrogen atoms, A=70.4kCÅ 6 and B=6286kCÅ 12
Electrostatics If bonds in the molecule are polar, some atoms will have partial electrostatic charges, which attract if opposite and repel otherwise. where Q i and Q j are the partial atomic charges for i and j separated by distance r ij, ε is the dielectric constant of the solute, and k is a units constant (k=2086 kcal/mol)
Torsional energy Torsion is the energy needed to rotate about bonds. Only relevant to single bonds, since others are too stiff to rotate at all E tor = ½k tor,1 (1 - cos θ) + ½k tor,2 (1-2cos θ) + ½k tor,3 (1-3cos θ) where θ is the dihedral angle around the bond, and k tor,1, k tor,2 and k tor,3 are constants for one-, two- and three-fold barriers. energy of 3-fold torsional barrier in ethane
Notes on energy Torsional energy is not well understood. Originally added as fudge factor to make butane model work. Although theoretically possible, impractical to calculate constants from quantum mechanics. We use empirical estimates (derived from molecules with known energies). Lots of competing estimates... Often, additional terms are added. Anharmonicity: quadratic bond length term means bonds can never break! Add cubic term to fix
Energy minimization Given some energy function and initial conditions, we want to find the minimum energy conformation. Optimization problem, various methods: Steepest descent Conjugate gradient descent Newton-Raphson Various programs: Charmm, Amber are two most widely used (and packaged)
Time steps Need time steps of roughly 1/10 the period of the smallest time scale of interest, or about a femtosecond (10-15 s). A million computational steps per nanosecond of simulation...
Issues in Molecular Mechanics Solvation models: water & salt are very important to molecular behavior. Must model as many water atoms as protein atoms. Initial conditions: velocity & position Equilibration: simulated heating and cooling Chaos: sensitivity to initial conditions, and statistical characterization of states Boundary effects Computational issues (e.g. parallelization)
Molecular Dynamics Molecules, especially proteins, are not static. Dynamics can be important to function Trajectories, not just minimum energy state. MM ignores kinetic energy, does only potential energy MD takes same force model, but calculates F=ma and calculates velocities of all atoms (as well as positions)
Classical MD versus Quantum Dynamics Classical MD calculates only nuclear motion, not electronic Quantum dynamics necessary for electronic phenomena, e.g. bond formation/breaking, polarization, bonding of metal ions, etc. Classical MD works at higher temperatures and longer time scales Quantum Dynamics necessary at low temperature and short time scales Hybrid methods exist.
MM/MD Resources Takes powerful computers and complex software; no easy web sites Software is available, but generally has licensing fees (even for academics) AMBER (Scripps/UCSF) http://amber.scripps.edu/ CHARMM (Harvard) http://yuri.harvard.edu/ Acceryls (commercial): http://www.accelrys.com/ Other useful programs SDSC http://www.sdsc.edu/ccms/ (free!) More links on the course web site
MD example Alcohol dehydrogenase, from Gerstein lab at Yale
Docking Computation to assess binding affinity Looks for conformational and electrostatic "fit" between proteins and other molecules Optimization again: what position and orientation of the two molecules minimizes energy? Large computations, since there are many possible positions to check, and the energy for each position may involve many atoms
Virtual Screening Docking small ligands to proteins is a way to find potential drugs. Industrially important A small region of interest (pharmacophore) can be identified, reducing computation Empirical scoring functions are not universal Various search methods: Rigid provides score for whole ligand (accurate) Flexible breaks ligands into pieces and docks them individually
Docking example Biotin docking with Streptavidin, from Olsen lab at Scripps
Macromolecular docking Docking of proteins to proteins or to DNA Important to understanding macromolecular recognition, genetic regulation, etc. Conceptually similar to small molecule docking, but practically much more difficult Score function can't realistically compute energies Use either shape complementarity alone or some kind of mean field approximation
Docking Resources AutoDock http://autodoc.scripps.edu/ FlexX http://www.biosolveit.de/flexx/ and commercially at http://www.tripos.com Dock http://www.cmpharm.ucsf.edu/kuntz/ dock.html 3D-Dock http://www.bmm.icnet.uk/docking/ which uses an unusual Fourier correlation method and is aimed at protein-protein interactions