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Molecular Dynamics simulations of biological ion channels. Titus A. Beu University ”Babeş-Bolyai” Department of Theoretical and Computational Physics Cluj-Napoca, Romania. Interest for ion channels. Ion channels - proteins that control the passage of ions across cell membranes.

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molecular dynamics simulations of biological ion channels

Molecular Dynamics simulations of biological ion channels

Titus A. Beu

University ”Babeş-Bolyai”

Department of Theoretical and Computational Physics

Cluj-Napoca, Romania

interest for ion channels
Interest for ion channels
  • Ion channels - proteins that control the passage of ions across cell membranes.
  • Responsible for:
    • generation of action potentials in nerves and muscles
    • regulation of hormone release from endocrine cells etc.
  • High selectivity for a particular ion type (Na+, K+, Ca2+, Cl-).
  • High transport rates (~108 ions per second).
  • The simplest proteins to which statistical mechanics may be applied.
  • Highly inhomogeneous electrolyte – challenging aspect.
  • ”Toolbox” of theoretical models and methods may be validated.
slide3
The Nobel Prize in Chemistry for 2003:“for discoveries concerning channels in cell membranes.”
  • Peter Agre:“for the discovery of water channels.”
  • Roderick MacKinnon:“for structural and mechanistic studies of ion channels.”
high resolution structure of an ion channel d doyle and r mackinnon science 280 69 77 1998
High-resolution structure of an ion channelD. Doyle, ..., and R. MacKinnon, Science 280, 69-77 (1998).
  • The KcsA K+ channel
  • (Streptomyces lividans)
  • Selectivity filter
  • High K+ selectivity - adapted to desolvating K ions.
  • Below and above – fully or partially hydrated K ions.
  • Inside – binding sites (O atoms) mimic the hydration shell
  • Gate
  • Opened by sensor domains
literature
Literature

Most detailed ion channel model and simulations:

  • P.S. Crozier et al., Phys. Rev. Lett. 86, 2467 (2001).
  • P.S. Crozier et al., Biophys. J. 81, 3077 (2001).

FFT-accelerated mesh-based Ewald sums (P3M method):

  • R.W. Hockney and J.W. Eastwood,Computer Simulation Using Particles (IOP, Bristol, 1988).
  • M. Deserno and C. Holm, J. Chem. Phys. 109, 7678 (1998).
  • M. Deserno and C. Holm, J. Chem. Phys. 109, 7694 (1998).
the model membrane channel p s crozier et al phys rev lett 86 2467 2001
The model membrane channelP.S. Crozier et al., Phys. Rev. Lett. 86, 2467 (2001).

Length = 25 Å, diameter = 10.625 Å, atom-atom distance = 2.5 Å

Embedded in a rigid nonpolar membrane (similar to nicotine acetycloline receptor)

388 sites: charges (-0.5e, -0.35e, +0.35e, +0.5e, neutral) + LJ interactions

11 20-atom rings

relative rotation 9°

the simulation cell

300 K

25 Å

0.02 V/Å

55 Å

The simulation cell
  • The electrolyte – 1M NaCl solution: 600 H2O molecules, 8 Na+ and 8 Cl-
  • Periodic boundary conditions in all three directions.
  • Simulation cell: 25 Å x 25 Å x 55 Å
modeling options for water
Modeling options for water
  • Molecules made up of atoms subject to constraints
    • Intermolecular atom-atom potential
    • Material-point dynamics for atoms
    • ”Shake”-algorithm - to preserve molecular structure
  • Rigid molecules:
    • Intermolecular site-site potential
    • Rigid-body dynamics for molecules:
      • Translation of CM – material-point dynamics governed by total force
      • Rotation about the CM – governed by total torqueQuaternions – alternative to Euler-angles
the tip4p model potential for h 2 o w l jorgensen et al chem phys 79 926 1983

O

0.9572 Å

0.15 Å

S

104.52°

H

H

The TIP4P model potential for H2OW.L. Jorgensen et al., Chem. Phys. 79, 926 (1983).
  • 4 interaction sites:O, H atoms + site S
  • Electrostatic charges:H atoms + site S
  • Lennard-Jones interactions:only between O atoms
equations of motion for rigid molecules motion of the cm
Equations of motion for rigid moleculesMotion of the CM
  • Total force acting on molecule I:
  • Newton’s law – equation of motion :
  • The Verlet propagator – from t to t+∆t:
equations of motion for rigid molecules definition of quaternions
Equations of motion for rigid moleculesDefinition of quaternions
  • Euler angles – sequence of rotations:
  • Quaternions – equivalent description – numerically more convenient!
  • Rotation matrix:
equations of motion for rigid molecules the quaternion representation
Equations of motion for rigid moleculesThe quaternion representation
  • Angular velocities:
  • Angular accelerations: Verlet propagator:
  • Quaternion accelerations:
gaussian thermostat
Gaussian thermostat
  • System in contact with a heat bath (T = const)
  • Gauss’s principle of least constraint: for the constrained motion
  • The motion is no longer Newtonian
  • The constrained equations of motion of the molecules:
  • The Lagrange multipliers:
periodic boundary conditions
Periodic boundary conditions

Minimizing surface effects for bulk phase

  • Simulation region replicated to infinity to fill the space.
  • "Image" particles move solidary with the "real" ones.

Correcting particle coordinates

  • When a particle exits the simulation region – an image enters through the opposite boundary.
  • Number of particles is conserved.

Interactions for short-range potentials

  • For distances rij > Rcut – interactions can be ignored.
  • For box size > 2Rcut – particle i lies within Rcut with at most one of all periodic images of another particle j.

Minimum image criterion

  • Among all images of a particle, interactions are considered only with the closest one.
the ewald sum
The Ewald Sum
  • Long-range potentials - interactions with distant images cannot be neglected.
  • Electrostatic energy for all charges and their periodic images:
  • Slowly decaying - straightforward summation is impracticable.
  • The trick: split the problem:
  • f(r)/r - negligible for r > Rcut (rapidly converging in the real space).
  • [1 - f(r)]/r - slowly varying (rapidly converging in the k-space).Fourier transform can be represented by only a few k vectors.
the ewald sum the smeared charge density
The Ewald SumThe smeared charge density
  • Equivalent interpretation:Charge density - very sharpperiodic function:
  • The Fourier representation would never converge
  • We "smear" the charges - replace them with Gaussian functions:
  • Smearing parameter a – (inverse length) tunes the relative weight of the real and reciprocal space contributions.
  • Large a - sharply defined charges.
the ewald sum the electrostatic energy
The Ewald SumThe electrostatic energy
  • The Ewald formula for the electrostatic energy:
  • Real space contribution:
  • Reciprocal space contribution:
  • Self energy correction:
the ewald sum the electrostatic forces
The Ewald SumThe electrostatic forces
  • The Ewald formula for the electrostatic forces:
  • Real space contribution – interaction of smeared charges:
  • Reciprocal space contribution – interaction of point and smeared charges:
the ewald sum the p 3 m fft accelerated method of hockney and eastwood
The Ewald SumThe P3M FFT-accelerated method of Hockney and Eastwood
  • Mesh-based charge density and charge assignment function of order P :
the ewald sum the p 3 m fft accelerated method of hockney and eastwood20
The Ewald SumThe P3M FFT-accelerated method of Hockney and Eastwood
  • Mesh-based charge density and charge assignment function of order P :
  • Optimal influence function – computed only once:
  • Mesh-based electrostatic field:
  • Reciprocal-space contributions to energy and forces:
simulation details 1m nacl solution 600 h 2 o molecules 8 na and 8 cl
Simulation details1M NaCl solution: 600 H2O molecules, 8 Na+ and 8 Cl-

Initial state:

Equilibration time: 0.25 ns

Time step: 2.5 fs

Total simulation time: 200 ns

Data storage interval: 2.5 ps

electrostatic potential
Electrostatic potential
  • Uniform electric field - applied in the z-direction to produce the membrane potential: 0.02 V/Å
  • Potential due to particles and channel sites - Poisson‘s equation for ensemble-averaged mesh-based charge distribution in reciprocal space - using FFT
ion passages
Ion passages
  • Ion passages are accompanied by an increase of the water polarization, followed by relaxation
  • Polarizationangle q –between the water dipole and the electric field (channel axis)
  • Polarization in the channel:
    • large fluctuations
    • quick relaxation
    • reverses sign after ion passage
    • lower on the average
average density distributions
Average density distributions

Structured channel – H2O molecules form boundary layers

Spatial density profiles depend little on the applied magnetic field

net current
Net current
  • The magnetic fields
    • cause aslight increase of the ion current (up to 10%), not a decrease
    • enhance ion transport indirectly, by enhancing water polarization.
conclusions
Conclusions
  • The ion channel
    • structured – water forms boundary layers in the channel and at the membrane walls
    • high transport rates ~3x108 ions/s (agree with experiments), ion currents ~50 pA
    • high selectivity – Na+ passages are ~60 times more probable than Cl- passages.
  • The magnetic fields (1-20 T) and gradients (~100 T/m) technologically available
    • cause aslight increase of the ion current (up to 10%), not a decrease
    • enhance ion transport indirectly, by enhancing water polarization.
  • The channel selectivity is not affected by magnetic fields.
  • Ion passages cause a pronounced water polarization - importance of water model.
  • Water polarization in magnetic fields- enhanced in reservoirs, unchanged in the channel.
  • Cost of “experiment“:
    • ~1.25 ns/day on up-to-date Compaq workstations or PC (3 GHz)
    • ~1600 ns simulated  ~1300 CPU days (~3.5 CPU years)