August MOBIL Summer School Lea Thøgersen

 Model based on observations and theory. Used to predict and explain new observations  Molecular Modeling  Use the computer as a laboratory  Do you know any methods?  What are they used for?  Today: Molecular Dynamics  Experimental observations and simple physical rules combined to simulate how different atoms move wrt each other.

 Topics: Conformational energy, force field and molecular dynamics  Literature: “Part 3” (Chap. 8 Diffraction and Simulation) p (first 4 lines), p , p  Goal: Obtain basic feeling for the possibilities and limitations of molecular dynamics  Means: active participation from you  First session “Conformational Energy and Force Fields”  ends with an exercise  Second session “Molecular Dynamics”  includes discussion of a current research study ?

 E tot =  E kin for a molecule  e.g. vibration, diffusion  coupled with temperature and atom velocities, but independent of atom positions  E pot for molecule  atoms affect each other dependent on atom type and distance => E pot coupled with atom positions  conformational energy E kin + E pot {½mv 2 } {mgh (gravity) ; ½kx 2 (spring)} ? ? ? ?

C 4 H 10

 Atoms  nuclei (protons+neutrons)  electrons  Quantum Mechanics:  when chemical bonds are formed electrons redistribute on all atoms in the molecule  a carbon (e.g.) would be different from molecule to molecule  the distribution of both the electrons and the nuclei in a molecule determines the conformational energy  Experimentally:  atoms of particular type and in particular functional groups behave similar independent of the molecule  IR wave lengths and NMR chemical shifts have characteristic values for certain atom types and groups independent of which molecule they are a part of  Molecular Mechanics:  Conformational energy from distribution of only the nuclei  Not without problems ? ?

 Energy as function of the relative positions of the atoms => conformational energy  Additive energy contributions  Spectroscopy of small molecules suggest that energy contributions from individual internal coordinates are independent, to a good approximation  Energy function as sum of independent contributions  Relative energies instead of absolutes  Easier to define energy penalty than absolute energy  Constant contributions can be ignored  Divided in “bonding” and “non-bonding” contributions

Describing the physics and chemistry of the atom interactions r or θ E eq E φ bond stretch angle bend bond rotation => dihedral

Describing the physics and chemistry of the atom interactions ++ ÷÷ ÷ ÷ + + Van der Waals interactions Electrostatic interactions

 Constants in the energy expression should be determined ex.  Based on experimental observations and QM computations.  Hard and tedious work to construct a good and general force field. ?

E pot ||F|| Gravitymghmg Spring½k(Δx)2½k(Δx)2 k Δxk Δx GenerallyEpEp ∂Ep/∂x∂Ep/∂x x equilibrium ∆x > 0 ∆x < 0 F = 0 F < 0 F > 0 F = - ∂E p /∂x ? ? ? ? ? ? ? ?

 The form of the potential energy function defines a force field  Function describing the potential energy of the molecule as a function of atom positions - conformational energy  +Parameterization of this energy function  Examples: MMFF, CHARMM, OPLS, GROMOS… Force Field

 Complex energy surface  Molecule specific  Only two out of 3N-6 variables shown here. Potential energy surface Energy Coord 1 Coord 2  Minima correspond to equilibrium structures

 Q1 Bond Stretch: Which of the three lines represent the stretching of the double C=C bond in propene and why?  Q2 Bond Rotation: Which line represents the single bond, which represents the double bond and why? How many interactions contribute in fact to the rotation around the single and the double bonds?  Q3 vdW Interactions: Which line represents the H-H interaction, which represents the C-H interaction and why?  Q4: What constitutes a force field, and why does it make sense to call it a ”force field”? Number 2. The equilibrium is found for a shorter distance (than for the solid line), and the graph is steeper, meaning the force constant is higher, meaning the bond is stronger. Number 1 = single bond, number 2 = double bond. Number 1 has three minima (characteristic of an sp3 bond) and a low rotation barrier. Number 2 has two minima (characteristic of an sp2 bond) and a high rotation barrier. The double bond rotation has four contributions ( , , , ) The single bond rotation has six contributions (6-2-3-{7,8,9} and {7,8,9}) Number 1 = H-H interaction, number 2 = C-H interaction. Hydrogen is a smaller atom than carbon, and therefore the minimum vdW distance is smaller for H-H than for H-C. A force field consists of a potential energy function and the parameters for the function. It is called a force field since the first derivative of the potential energy wrt the position of an atom gives the force acting on this atom from the rest of the atoms in the system.

A Virtual Experiment

 Both potential and kinetic energy  Given a start structure and a force field an MD simulation output the development of the system over time (nanosecond time scale)

,000 atoms 100 ps ,000 atoms 10 ns ,000,000 atoms 14 ns Satellite tobacco mosaic virus, complete with protein, RNA, ions ER DNA- binding domain LacI-DNA complex

ri(t)vi(t)ai(t)ri(t)vi(t)ai(t) Time line time step ∆t, δt typical ∆t ≈ 1· s = 1 fs r i (t+ δt) v i (t+ δt) a i (t+δt) atom positions atom velocities atom accelerations r i (0) v i (0) a i (0) ? ?

 Find initial coordinates r(t=0) for all atoms in the system  For proteins an X-ray or NMR structure is used or modified  Water and lipid can be found pre-equilibrated from the modeling software or on the web  Smaller molecules can be sketched naively and pre-optimized within the modeling software

 Avoid boundary effects  Every atom ’sees’ at most one picture of the other atoms.  Cutoff less than half the shortest box side  At least 10Å cutoff.

E pot ||F|| Spring½k(Δx)2½k(Δx)2 k Δxk Δx GenerallyEpEp ∂Ep/∂x∂Ep/∂x x equilibrium ∆x > 0 ∆x < 0 F = 0 F < 0 F > 0 F = - ∂E p /∂x = -G F = m a r(t=0) => F(r(t=0)) => a(t=0) ?

 Maxwell-Boltzmann distribution for kinetic energy ε k = ½mv 2 => v(t=0)  Initial distribution of speed reproducing the requested temperature  random directions of the velocities

Time line time step ∆t, δt typical ∆t ≈ 1· s = 1 fs ri(t)ai(t)vi(t)ri(t)ai(t)vi(t)

Good ? Bad Total simulation time e.g. 10 ns = conformations  Collisions should occur smoothly!  Time step ~ 1/10 T fast motion period  T C-H vib ~ 10 fs => Time step = 1 fs Time line time step ∆t, δt typical ∆t ≈ 1· s = 1 fs ?

 Build the system  Clean pdb-structure for unwanted atoms  Add missing atoms  Add the environment  Make a structure file describing connections  Minimization of the system  Some 2000 steps, gradient < 5 or so  To remove clashes  Equilibration of the system  Maybe constraining some atoms to their initial position too keep overall structure  Maybe starting from low temperature, and slowly increasing it to the wanted  Maybe letting the volume adjust properly to the size of the system  Energy and RMSD should level out  Production run  Constant temp, vol, pressure?

 Experimenting with different setups to see what happens – is the system stable?  Mutations, temperature, pressure, environment....  Test out hypotheses based on experiment  Detailed information at the atomic level  Free energy differences – site-directed mutagenesis  Other thermodynamics stuff  Poke it / steer it

 X-ray, NMR and various biophysical studies and mutation studies and more?  Model the hypothesis, does the modelled response fit the experiment? If so, both the experiment and simulation conclusion is strengthen and a higher level of understanding is gained  Shortcomings of MD:  Timescale - ns is very short – no conformational changes  System size – the dimensions of the model are less than nm  No electrons – polarization cannot be described

 6 simulation setups. 10 ns simulations of SERCA in a membrane consisting of either short, POPC, long, DMPC, or DOPC lipids, and SERCA in a membrane of 2:1 C12E8:POPC atoms.  X-ray low resolution scattering from bilayer leaflets. The bilayers in the crystals consist of 16:7 detergent:lipid (detergent C12E8, lipids from native membrane).  Try to come up with relevant and interesting things to study from the MD simulations.

Long POPC POPC+detergent

Membrane type Avg. Hydrophobic thickness (8-10 ns) (Å) Avg. Overall tilt (8-10 ns) (°) < 7 Å from protein> 7 Å from protein POPC:C12E8 (1:2)* Short DMPC DOPC POPC Long purePOPC31.3- purePOPC:C12E8 (1:2)*24.0- α

From Theoretical and Computational Biophysics Group, University of Illinois at Urbana-Champaign

 K + permeation  Voltage bias  Conduction via knock-on mechanism  Selective filter

 transmembrane pore of alpha- hemolysin  Electrophoretically-driven  58-nucleotide DNA strand

 Full structure of satellite tobacco mosaic virus, complete with protein, RNA, ions, and a small water box