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August 13 2009 MOBIL Summer School Lea Thøgersen. Modeling of Membrane Proteins. Modeling in Science. 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?

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august 13 2009 mobil summer school lea th gersen
August 13 2009

MOBIL Summer SchoolLea Thøgersen

Modeling of Membrane Proteins

modeling in science
Modeling in Science
  • 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.
the plan
The Plan
  • Topics:Conformational energy, force field and molecular dynamics
  • Literature: “Part 3” (Chap. 8 Diffraction and Simulation) p.196-200 (first 4 lines), p. 203-207, p. 210-212.
  • 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

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energy
Energy

?

  • Etot =
  • Ekin for a molecule
    • e.g. vibration, diffusion
    • coupled with temperature and atom velocities, but independent of atom positions
  • Epot for molecule
    • atoms affect each other dependent on atom type and distance=> Epot coupled with atom positions
    • conformational energy

Ekin + Epot

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{½mv2}

?

?

{mgh(gravity) ; ½kx2 (spring)}

molecular subparts
Molecular Subparts

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  • 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 boththe 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

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potential energy function
Potential Energy Function
  • 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
bonding interactions
Bonding Interactions

Describing the physics and chemistry of the atom interactions

bond stretch

angle bend

bond rotation

=> dihedral

E

E

eq

r or θ

φ

non bonding interactions
Non-bonding Interactions

Describing the physics and chemistry of the atom interactions

Electrostatic interactions

Van der Waals interactions

+

+

÷

÷

+

÷

÷

+

parameters
Parameters
  • Constants in the energy expression should be determinedex.
  • Based on experimental observations and QM computations.
  • Hard and tedious work to construct a good and general force field.

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energy force
Energy => Force

?

?

x

?

F = 0

equilibrium

?

F = - ∂Ep/∂x

?

?

F < 0

∆x > 0

?

?

F > 0

∆x < 0

force field
Force Field
    • The form of the potentialenergy 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

potential energy surface as function of atom coordinates
Potential Energy Surfaceas function of atom coordinates
  • 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
force field exercise
Force Field Exercise
  • 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 (5-1-2-6, 5-1-2-3, 4-1-2-6, 4-1-2-3)

The single bond rotation has six contributions (6-2-3-{7,8,9} and 1-2-3-{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.

molecular dynamics md
Molecular Dynamics (MD)
  • 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)
the advancements of md
The Advancements of MD

2005

314,000 atoms

10 ns

199736,000 atoms100 ps

LacI-DNA complex

ER DNA-binding domain

2007-8

1,000,000 atoms

14 ns

Satellite tobacco mosaic virus, complete with protein, RNA, ions

moving in time
Moving in Time

ri(0)vi(0)ai(0)

ri(t)vi(t)ai(t)

ri(t+ δt)vi(t+ δt)ai(t+δt)

atom positionsatom velocitiesatom accelerations

?

?

Time line

time step

∆t, δt

typical ∆t ≈ 1·10-15s = 1 fs

build system setup
Build System Setup
  • 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
periodic boundary conditions
Periodic Boundary Conditions
  • 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.
energy force acceleration
Energy => Force => Acceleration

x

F = 0

equilibrium

F = - ∂Ep/∂x = -G

F = m a

F < 0

∆x > 0

?

F > 0

∆x < 0

r(t=0) => F(r(t=0)) => a(t=0)

initial velocity
Initial Velocity
  • Maxwell-Boltzmann distribution for kinetic energy εk = ½mv2 => v(t=0)
  • Initial distribution of speed reproducing the requested temperature
  • random directionsof the velocities
moving in time1
Moving in Time

ri(t) ai(t)vi(t)

Time line

time step

∆t, δt

typical ∆t ≈ 1·10-15s = 1 fs

time step size

Bad

?

Good

Time Step Size

Time line

time step

∆t, δt

typical ∆t ≈ 1·10-15s = 1 fs

  • Collisions should occur smoothly!
    • Time step ~ 1/10 Tfast motion period
    • TC-H vib ~ 10 fs => Time step = 1 fs

?

Total simulation time e.g. 10 ns = 10.000.000 conformations

running an md simulation
Running an MD Simulation
  • 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?
what to use it for
What to use it for?
  • 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
comparing to experiments
Comparing to Experiments
  • 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
ca 2 atpase serca interaction with lipid membrane
Ca2+-ATPase (SERCA) interaction with lipid membrane
  • 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. 200-240.000 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.
slide34
From Theoretical and Computational Biophysics Group, University of Illinois at Urbana-Champaign

http://www.ks.uiuc.edu/Gallery/

Flashy Examples

kv1 2 channel
Kv1.2 Channel
  • K+ permeation
  • Voltage bias
  • Conduction via knock-on mechanism
  • Selective filter
dna translocation
DNA Translocation
  • transmembrane pore of alpha-hemolysin
  • Electrophoretically-driven
  • 58-nucleotide DNA strand
full virus
Full Virus
  • Full structure of satellite tobacco mosaic virus, complete with protein, RNA, ions, and a small water box