1. Molecular Modelling Dr Michelle Kuttel Department of Computer Science University of Cape Town

2. Aim Brief introduction to/overview of molecular modelling  why  theory  how Hand-on experience of molecular modelling package (NAMD) and visualization software (VMD)

3. Resources Slides Molecular Modelling. Principles and Applications. A. R. Leach, Addison Wesley Longman Limited,1996 (in library) Essentials of Computational Chemistry - Theories and Models, 2nd Edition, Christopher J. Cramer NAMD website - http://www.ks.uiuc.edu/

4. Computational Chemistry Chemical/physical problem ? Molecular Simulations Analytical Tools Comparison with experiment Validation Insight simulation data Predictions

5. Why is computational Chemistry Increasingly Popular? Chemical waste disposal and computational technology  which keeps getting cheaper and cheaper and which more and more expensive?

6. Simulations: Modelling Strategies Chemical/physical problem ? Molecular Simulations Force Field Methods Ab initio QM Methods

7. Quantum Mechanics postulates and theorems of quantum mechanics form the rigorous foundation for the prediction of observable chemical properties from first principles.  microscopic systems are described by wave functions that completely characterise all the physical properties of the system  operators applied to the wave function allow one to predict the probability of the system having a value or range of values.

8. Quantum mechanics vs Force Field methods QM deals with electrons in system  Accurate  Can deal with reactions (bond breaking etc.)  Often used to parameterize force fields  Large number of particles means infeasibly time-consuming for molecules as large as proteins  Static models only (no time) FF methods  Molecular mechanics  Cannot answer questions that depend on electron distribution in a molecule  But fast and surprisingly useful

9. Computable properties I Structure  determination of “best” structure very common application of Comp Chem. lowest possible energy care when comparing theory with experiment - thermal averaging for measured structure

10. Computable properties II Potential Energy Surfaces  fully characterize the potential energy surface (PES) for a given chemical formula 3N-6 coordinate dimensions, where N is the number of atoms >= 3  points of interest are local minima (optimal structures), saddle points (lowest energy barriers on the paths connecting minima - transition state)  Typically, take slices through PES, involving 1 or 2 coordinates, as hard to visualize otherwise

11. Computable properties III Chemical Properties  Single molecule properties e.g. spectral quantities - NMR shifts and coupling constants etc.  Thermodynamic quantities. Enthalpy, free energy. theory extensively used to estimate equilibrium constants, which are derived from free energy differences between minima on a PES and connected transition state structures. reaction thermochemistries, heats of formation and combustion, hydrogen bonding strengths etc. etc.

12. Molecular Mechanics Approach to understanding structure-function relationships Applications:  Structure determination and refinement  Homology modelling  Structure-based ligand design  Pharmacore modelling  Mutant structure prediction  Enzyme mechanism  Protein folding pathways  Protein design  Molecular dynamics  Normal mode analysis (“characteristic motions”)

13. Molecular Mechanics Force Fields Classical Mechanical approximation

14. Molecular mechanics Potential describe deviation from a reference value

15. Force Fields - Parameterization Can had more/fewer terms  CHARMM and similar force fields seek to have a minimal set of easily comprehensible terms To define a force field, one must specify not only the functional form, but also the parameters.  Two force fields may have identical functional form but very different parameters e.g. “CHARMM” force field has many possible parameter sets Force field parameter terms expressed in terms of “atom type”  Distinguish e.g. between sp, sp 2 and sp 3 hybridized carbons

16. Force Fields - Parameterization Force field parameters are not necessarily transferable  “energy” is relative Typically parameterized for a specific class of molecules - protein, DNA/RNA, carbohydrates, etc.  “general” force fields - CVFF etc.  Usually designed to predict structural properties Force field parameterization is a full-time job

17. Force Field examples AMBER GROMOS CHARMM - topology and parameter files MM2/MM3

18. Force Fields How can I pick the best force field for my problem? How can I trust the results?

19. Coordinate files: PDB Simulations start with atomic structures from the Protein Data Bank, in standard PDB file format PDB files contain a lot of data - species, tissue, authorship,citations secondary structure etc.  Only interested in atomic data

20. Molecular Visualization Packages Huge number of molecular visualization packages  RasMol  The PyMOL Molecular Graphics System  Gopenmol  VMD Similar approaches to visualization  may combine with structure refinement/molecular modelling etc.  My recommendation is to use one product and get to know it well.

21. Molecular Modelling Software Commercial:  Cerius2, Insight II (from Accelrys) Academic:  CHARMM  AMBER  GROMOS  NAMD

22. Why NAMD? “VMD, NAMD, and BioCoRE represent a broad effort by the Theoretical and Computational Biophysics Group, an NIH Resource for Macromolecular Modeling and Bioinformatics, to develop and freely distribute effective tools (with source code) for molecular dynamics studies in structural biology.”  Scales best …  Uses CHARMM force field  Written in C++  Good tutorial and user guide

23. What type of simulation? What are the most stable/probable conformations?  Energy minimization  Molecular dynamics  Monte Carlo methods  Hybrid MD-MC methods  Simulated annealing What are the functional motions?  Molecular (deterministic) dynamics  Stochastic dynamics  Normal modes There are of course other questions… optimization sampling dynamics

24. Energy Minimization/Geometry Optimization Aim : find the lowest-energy conformation  problem in applied mathematics  Given a function f with independent variables x 1,x 2,…,x n find the values of those variables where f has a minimum value Can’t use analytical methods because of the complicated way energy varies with coordinates Minima located using numerical methods which gradually change the coordinates to produce configurations with lower and lower energies until minimum is found  Various optimization procedures

25. Energy Minimization - Algorithms Most algorithms only go downhill on surface - multiple minima problem

26. Energy Minimization - Algorithms Some algorithms use derivatives of energy, others do not Usually one uses a combination of methods  More robust (but less efficient) first  Less robust (more efficient) second Convergence criteria  Usually monitor energy from one iteration to next and stop when difference between successive measurements falls beneath a certain threshold

27. Energy Minimization - Simplex Method Non-derivative method A simplex is a geometrical figure with M+1 interconnected vertices, where M is the dimensionality of the energy function Each vertex corresponds to a set of coordinates for which the energy can be calculated

28. Energy Minimization - Simplex Method 3 moves possible:  Reflect (1)  Contract in one dimension (2)  Contract around lowest point (3)

29. Energy Minimization - Simplex Method Need to generate vertices of initial simplex  Add constant increment to each coordinate in turn Expensive Algorithm - many energy evaluations  most useful where initial configuration very high in energy (i.e. very far from minimum)  Often used for initial few minimization steps

30. Energy Minimization - Steepest Descent Algorithm 1st derivative method, Moves in direction parallel to net force  “straight downhill” Need to decide how far to move Most implementations have a step size with predetermined default value If 1st step has decrease in E, increase the step size by a factor for next iteration If energy increases, decrease step size by a factor Both gradients and direction of successive steps are orthogonal

31. Energy Minimization - Steepest Descent Algorithm Advantages:  Direction of gradient is determined by largest inter-atomic forces, good for relieving highest energy features in initial configuration  robust Disadvantage:  Numerous small steps when proceeding down long narrow valley - right angled turn at each step “tacking into the wind”

32. Energy Minimization - Conjugate gradients method 1st-derivative method Does not show oscillatory behaviour in narrow valleys. Similar to steepest descents, but while gradients are orthogonal, directions are conjugate

33. Energy Minimization - Newton- Rhapson Algorithm Simplest second derivative method  Uses information about the curvature of the function  Should be familiar from 1st year… Computationally demanding and more suited to smaller molecules  Has problems with structures far from minimum minimization can become unstable.

34. Work for tomorrow Install NAMD and VMD perform minimization of a suitable protein have “before” and “after”.pdb files - send to me via email

35. Energy Minimization - problems Only local minima found  Multiple minima problem Only minimum of potential energy, not free energy

36. Energy Minimization - applications Widely used in molecular modelling  Prior to Monte Carlo simulations or MD to remove any unfavourable interactions in initial configuration of system Often used for optimizing experimental structures

37. NAMD minimization procedure Conjugate gradient parameters The default minimizer uses a sophisticated conjugate gradient and line search algorithm with much better performance than the older velocity quenching method. The method of conjugate gradients is used to select successive search directions (starting with the initial gradient) which eliminate repeated minimization along the same directions. Along each direction, a minimum is first bracketed (rigorously bounded) and then converged upon by either a golden section search, or, when possible, a quadratically convergent method using gradient information. For most systems, it just works. * minimization $ $ Acceptable Values: on or off Default Value: off Description: Turns efficient energy minimization on or off. * minTinyStep $ $ Acceptable Values: positive decimal Default Value: 1.0e-6 Description: If your minimization is immediately unstable, make this smaller. * minBabyStep $ $ Acceptable Values: positive decimal Default Value: 1.0e-2 Description: If your minimization becomes unstable later, make this smaller. * minLineGoal $ $ Acceptable Values: positive decimal Default Value: 1.0e-4 Description: Varying this might improve conjugate gradient performance.

38. Grid search with energy minimization Approach to “mapping” the energy surface  Produce an adiabatic map as a function of the chief conformational coordinates  E.g Ramachandran map of conformational energy as function of   Search through conformational coordinates in increments  Very time-consuming if done thoroughly, only practicable in a few dimensions  

39. Energy Minimization - Normal Mode Analysis Normal modes of vibration are simple harmonic oscillations about a local energy minimum, characteristic of a system's structure and its energy function. For a purely harmonic function any motion can be exactly expressed as a superposition of normal modes. For an anharmonic function, the potential near the minimum will still be well approximated by a harmonic potential, and any small-amplitude motion can still be well described by a sum of normal modes.

40. Energy Minimization - Normal Mode Analysis The normal mode spectrum of a 3- dimensional system of N atoms contains 3N - 6 normal modes ( for linear molecules in 3D).  In general, the number of modes is the system's total number of degrees of freedom minus the number of degrees of freedom that correspond to pure rigid body motion (rotation or translation).

42. Energy Minimization - Normal Mode Analysis Each mode is defined by an eigenvector and its corresponding eigenfrequency.  The eigenvector contains the amplitude and direction of motion for each atom (same frequency of vibration for all atoms)

43. Energy Minimization - Normal Mode Analysis In macromolecules, the lowest frequency modes correspond to delocalized motions, in which a large number of atoms oscillate with considerable amplitude. The highest frequency motions are more localized, with appreciable amplitudes for fewer atoms, e.g., the stretching of bonds between carbon and hydrogen atoms.

44. Energy Minimization - Normal Mode Analysis Normal modes useful because they correspond to collective motions of atoms in a coupled system that can be individually excited Frequencies of normal modes and displacements may be calculated from a molecular mechanics force field using the Hessian matrix of second derivatives Molecule must be at a minimum

45. Energy Minimization - Normal Mode Analysis Results can be:  Used to calculate thermodynamic quantities  Compared to spectroscopic experiments Used in parameterization of force fields For large molecules, low-energy vibrations are of most interest  Correspond to large-scale conformational motions  Can be compared to molecular dynamics simulations

46. Energy Minimization - Docking the prediction of the strength and specificity with which a small to medium sized molecule can bind to a biological macromolecule docking - evaluating the energy of binding between two molecules for various relative positions of the two  simiplification: use rigid molecules