Molecular simulations in chemistry Adam Liwo Room B325 adam@sun1.chem.univ.gda.pl

30 lecture hours 2 hrs/week; Tuesdays, 8:15 – 10:00 am Completion requirements Project Exam

Scope Purpose, time-, and size-scales of molecular simulations. Energy surfaces of molecules. All-atom force fields: purpose, derivation, and parameterization Treatment of solvent in force fields. Models of water. Metropolis Monte Carlo. Molecular dynamics. Calculating ensemble-averages and error estimation in simulations. Umbrella-sampling simulations and the weighted-histogram analysis method. Generalized-ensemble simulations. Enlarging the time- and size-scale of simulations: coarse-grained models. The CABS and UNRES force fields. Thermodynamics and kinetics of protein folding from simulations. QM/MM simulations.

Literature Daan Frenkel, Berend Smit, „Understanding Molecular Simulation: From Algorithms to Applications” Academic Press, San Diego, 1996 D.C.A. Rapaport, „The Art of Molecular Dynamics Simulations”, Cambridge University Press, 1998. A.R. Leach: „Molecular Modeling: Principles and Applications”, Pearson Education EMA, 2001.

Learning Nature – how does Science work? No model (pysicochemical tables) Model (equations) Experiment Exact solution Simulations

Equations (approximate) – exact solutions „I am really longing for those good old times when a theorist didn’t need anything but a piece of paper, a pencil, and own brains”. Quotation from a late Professor of Physical Chemistry. Not possible anymore…unless we want to consider spherical horses in vacuo to model horse race. Feynman’s dream that we will be able to ‘see’ the solutions of equations someday does not seem to ever come true.

Successful examples of the „exact solution” approach Chemical Thermodynamics (phenomenological). Chemical Kinetics. Modeling electrochemical processess. Quantum Chemistry. Kinetic theory of gases. Application of Statistical Mechanics in Chemistry.

What are ‘Simulations’? ‘Das ganze Tschechische Volk ist eine Simulantenbande’ – Dr. Gruenstein of K.u.K military draft office Modeling (computing) the behavior of complex systems by applying a given description (e.g., Newton’s equations of motion).

Where do the ‘molecular simulations’ enter into play? Condensed systems composed of many particles (e.g., a protein + solvent). Strong interactions between system’s components. The partition function cannot be separated. The time evolution has Lyapunov instability depending on the initial conditions. Therefore, we actually need to compute system’s behavior for given initial/boundary condition rather than analyze the solutions in terms of those.

Are simulations another version of experiment? No, we do not deal with a real system but with a ‘virtual’ one. However, the results depend on starting point and are subject to statistical error as the experiemental results.

(Pre) History Lord Kelvin (early 1900’s): hand computations of hard- sphere collisions. Manhattan Project (Ulam; 1940’s – 1950’s): hand and computer simulations of nuclear fission (ENIAC computer). J.D. Bernal (1950’s): mechanical models of liquid particles from rubber/styrofoam balls connected with metal rods. G. Vineyard (1950’s): computer simulation of radiation damage in crystalline Cu. Rosenbluth, Rosenbluth, Metropolis, Teller (1950’s): Formulation of the Metropolis Monte Carlo algorithm. Alder and Wainwright (1957): MD simulations of hard- sphere liquids.

Types of simulations Monte Carlo (MC): need only energy). Molecular dynamics (MD): time evolution; need forces). Combination thereof.

What systems do we treat and what are the limits?

Description level System level (Networks) Individual components Fully-detailed QM Averaging over „less important” degrees of freedom QM/MM Averaging over individual components Individual components Atomistically-detailed All-atom United-atom Description level Residue level Coarse-grained PDEs to describe reaction/diffusion Molecule/domain level System level (Networks) Network graphs

10-15 10-12 10-9 10-6 10-3 100 femto pico nano micro milli seconds sidechain rotation helix formation protein folding 10-15 femto 10-12 pico 10-9 nano 10-6 micro 10-3 milli 100 seconds bond vibration loop closure folding of -hairpins all atom MD step

Time step Dt for some standard MD packages Explicit Solvent Implicit Solvent AMBERa 1 fs 2 fs CHARMMb 3 fs 4-5 fs TINKERc a http://amber.scripps.edu/ b http://www.charmm.org/ c http:// dasher.wustl.edu/tinker/

Energy surfaces of molecular systems and their properties

From Schrödinger equation to analytical all-atom potentials

The Born-Oppenheimer approximation Figure 3b). The Born-Oppenheimer approximation

Conversion of iso-hydrogen cyanide into hydrogen cyainde HNC®HCN

Transition structure AM1 energy hypersurface of the conversion of iso-hydrogen cyanide into hydrogen cyanide Energy [kcal/mol] HNC HCN

Contour plot of the PES HCN struktura przejściowa HNC

H N -C energy [kcal/mol] DE╪ HNC HCN DE reaction coordinate

Jean-Louis David Napoleon

Propane PES as a function of the two dihedral angles

Conformational-energy map of terminally-blocked alanine (degrees) (degrees)

Energy expansion about the stationary point The derivatives are zero at a stationary point but it need not be a stable point (Coulomb’s egg problem).

Matrix H is termed energy Hessian V – eigenvector matrix

Case study: the układu HCN – HNC system Energy [kcal/mol] x h h x Neighborhood of the minimum corresponding to the HCN molecule Neighborhood of the transition point

Generalization on n coordinates

Summary pf critical points Minimum: all Hessian eigenvalues > 0 Corresponds to a stable state of a system. First-order saddle point: l1<0, l2, …, ln >0 Corresponds to the transition state in a reaction. Higher-order transition points are not interesting. A maximum: all Hessian eigenvalues < 0.