Particle-Based Simulation of
Bio-Electronic Systems
Alex Smolyanitsky, and Marco Saraniti
Center for Computational Nanoscience
Arizona State University
ARIZONA INSTITUTE
FOR
NANO-ELECTRONICS
Center for Computational Nanoscience
Outline
Particle-based Brownian dynamics simulations for
bioelectronic systems
•
•
•
•
•
•
•
Complex-field DC-electrophoresis of charged proteins
Simulations of molecule: constraints and general
computational framework
SHAKE and LINCS algorithms
RATTLE and general velocity correction
Results and discussion for OmpF ion channel
Preliminary results for Kv1.2 ion channel
Visualization of simple protein folding
Conclusions and future work
ARIZONA INSTITUTE
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NANO-ELECTRONICS
Center for Computational Nanoscience
Complex field electrophoresis: system description
300 nm
hole
Top view
•α-Hemolysin protein molecules are
driven by DC electrophoresis.
•Protein modeled as charged rigid sphere
(r = 5 nm) suspended in water (ε = 78.0).
•External field, stokesian drag, stochastic
contribution explicitly included in the
simulation.
ab
Teflon sl
buried electrode (1.25 V)
30
0n
m
20 nm
300 nm
40 nm
•Driving fields obtained via application of
constant potentials, not constant fields.
•Electric charge calculated from
protonation states of individual residues in
α-Hemolysin at a given pH value.
•T = 300K, q = +65|e| at pH = 5.0;
diffusion coefficient, mobility, and settling
time used in simulation, respectively:
The simulation setup is a 300 nm x 300
nm x 300 nm water-filled box split by a 30
nm thick teflon membrane (ε = 2.0).
ARIZONA INSTITUTE
FOR
NANO-ELECTRONICS
Center for Computational Nanoscience
Complex field electrophoresis: visualization
•The distance from the protein’s
initial position is calculated at
approx. 115 nm.
•Total focusing time is about 4
microseconds.
•The protein with effective diameter
of 10 nm is successfully focused into
a 20 nm x 20 nm hole.
ARIZONA INSTITUTE
FOR
NANO-ELECTRONICS
Center for Computational Nanoscience
Constrained dynamics: basic constraints
dij
i
j
φ
i
k
j
a) simple linear bond
b) 2-D bond angle
l
j
k
i
c) 3-D dihedral angle
ARIZONA INSTITUTE
FOR
NANO-ELECTRONICS
Center for Computational Nanoscience
Constrained dynamics: flowchart
Flowchart of the Brownian dynamics simulation tool without (left) and with (right) the
constrained dynamics corrections.
Find potential distribution
Find potential distribution
Calculate electrical
fields and forces
Calculate electrical
fields and forces
Update particle velocities
and positions
Update particle velocities
and positions
Correct positions
and velocities
of constrained particles
ARIZONA INSTITUTE
FOR
NANO-ELECTRONICS
Center for Computational Nanoscience
Constraint algorithm review
General Framework
•
•
Based on Lagrange multiplier method
For a system containing N particles requires inversion of N x N matrix at every
timestep
SHAKE algorithm
•
•
Approximate iterative method to avoid direct matrix inversion
Guaranteed to converge within 50 iterations with timesteps up to 10 fs
LINCS algorithm
•
•
•
Non-iterative, uses matrix form of Taylor expansion to avoid direct matrix inversion
Timesteps up to 20 fs, twice as large compared to SHAKE
Applicable only to systems with low connectivity, limiting use for constraining the
angles using artificial bonds and demanding use of angle-constraining potentials
rather than artificial bonds
RATTLE and general velocity correction
•
•
•
Removes bond strain by minimizing relative velocity along the constraint
Applied sequentially
Improves SHAKE convergence
ARIZONA INSTITUTE
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NANO-ELECTRONICS
Center for Computational Nanoscience
avg. number of SHAKE iterations
Constrained dynamics: SHAKE algorithm
bonds, angles, dihedrals constrained
bonds and angles constrained
bonds constrained
60
40
20
0
500
1000
1500
2000
2500
3000
3500
number of bound particles
Average number of SHAKE iterations vs. number of bound particles required for
convergence to relative SHAKE tolerance of 0.001 for various types of constraints.
Verlet unconstrained integrator with free flight timestep of 8 fs used.
ARIZONA INSTITUTE
FOR
NANO-ELECTRONICS
Center for Computational Nanoscience
Constrained dynamics: velocity correction
BPTI protein constrained dynamics velo
Euler
velocity correction off (1 fs st
Verletint,
unconstrained
Euler
int,
velocity correction on (1 fs st
integrator,
8 fs
timestep
Pred/corr int,
velocity
correction off (4
Pred/corr int, velocity correction on (4
Verlet int,Velocity
velocity
correction
on & off (
correction on
0.045
0.08
Velocity correction off
0.04
0.06
0.035
0.04
0.03
0.02
0.025
0
0.020
0
0.02
0.02
0.04
0.06
time, ns 0.06
0.04
0.08
0.08
0.1
0.1
0.05
average bound atom energy [eV]
average
bound atom energy [eV]
average bound atom energy, eV
0.1
0.05
Predictor-corrector unconstrained
integrator, 8 fs timestep
0.045
Velocity correction off
Velocity correction on
0.04
0.035
0.03
0.025
0.02
simulated time [ns]
0
0.02
0.04
0.06
0.08
0.1
simulated time [ns]
average bound atom energy [eV]
0.07
Euler unconstrained
integrator, 2 fs timestep
Time evolution of the average energy of the
bound atoms for various unconstrained
integrator algorithms.
After velocity correction, avg. kinetic energy
around 30meV for all algorithms.
No spurious heating/cooling of molecule.
0.06
Velocity correction off
Velocity correction on
0.05
0.04
0.03
0.02
0
0.02
0.04
0.06
0.08
0.1
simulated time [ns]
ARIZONA INSTITUTE
FOR
NANO-ELECTRONICS
Center for Computational Nanoscience
OmpF ion channel simulation: general structure
A
•
•
•
•
Three 16-strand barrel subunits (340 residues each)
Permeation region constricted to 7 x 11 Å
Transverse fields in permeation region due to charged residues
Cation-selective, depending on salt concentration selectivity ratio 1.5 to 2.5
ARIZONA INSTITUTE
FOR
NANO-ELECTRONICS
Center for Computational Nanoscience
OmpF ion channel simulation: system description
14
12
lipid membrane
ε=4.0
8
6
8
6
4
4
2
2
water
ε=78.0
2
4
6
8
x [nm]
10
12
14
2
4
6
8
10
12
14
8
10
12
14
x [nm]
14
5
12
5
y [nm
]
10
15 15
10
5
12
10
0
m]
x [n
xy-plane slice
z = 7.5 nm
bottom
contact
dielectric
constant
5 15 25 35 45 55 65 75
xy-plane slice
z = 8.5 nm
10
y [nm]
00
y [nm]
z [nm]
15
xy-plane slice
z = 7.0 nm
10
y [nm]
top
contact
10
12
10
y [nm]
protein region
ε=6.0
14
xy-plane slice
z = 6.5 nm
8
6
8
6
4
4
2
2
2
4
6
8
x [nm]
10
12
14
2
3-D dielectric map of the system (left) and
dielectric contour planes at various z-coordinates (right).
ARIZONA INSTITUTE
FOR
NANO-ELECTRONICS
Center for Computational Nanoscience
4
6
x [nm]
OmpF ion channel simulation: conductance
and selectivity
4
BD, εprotein=6.0
experimental data ***
4
IK/ICl current ratio
NK/NCl ion number ratio
3.5
selectivity ratio
OmpF trimer conductance [nS]
5
3
3
2.5
2
1
0
2
1.5
0.2
0.4
0.6
0.8
KCl concentration [M]
1
1
0.2
0.4
0.6
0.8
KCl concentration [M]
1
Simulated OmpF conductance vs. KCl concentration compared to
experimental data, and simulated ionic selectivity based on currents and ion numbers (right).
*** S. J. Wilk, S. Aboud, L. Petrossian, M. Goryll, J. M. Tang, R. S. Eisenberg, M.Saraniti,
S. M. Goodnick, and T. J. Thornton. Ion channel conductance measurements on a siliconbased platform. Journal of Physics Conference Series, 37(1):21-24, 2006.
ARIZONA INSTITUTE
FOR
NANO-ELECTRONICS
Center for Computational Nanoscience
OmpF ion channel simulation: axial potential
and ion distribution profiles
3
-0.2
Arg-168, Lys-80
constriction zone
Asp-113, Glu-117
-0.4
intracellular
region
extracellular
region
-0.8
-1
0.25M KCl
0.5M KCl
1.0M KCl
no ions, no bias
-1.2
2
4
6
8
10
12
14
axial position [nm]
avg. concentration [M]
avg. potential [V]
0
-0.6
0.25M KCl, cations
0.25M KCl, anions
0.5M KCl, cations
0.5M KCl, anions
1.0M KCl, cations
1.0M KCl, anions
channel region
0.2
2.5
2
channel region
intracellular
region
extracellular
region
constriction zone
Asp-113, Glu-117
1.5
Arg-168, Lys-80
1
0.5
0
2
4
6
8
10
12
axial position [nm]
Simulated distributions of potential (left) and ionic concentration (right)
along the axis of an OmpF monomer for various KCl concentrations.
ARIZONA INSTITUTE
FOR
NANO-ELECTRONICS
Center for Computational Nanoscience
14
OmpF ion channel simulation: visualization of
conduction through isolated monomer
The potassium and chlorine ions are shown in grey and green, respectively.
The OmpF monomer is shown as semi-transparent, inserted in lipid
membrane (impermeable dielectric slab, not shown). The transmembrane
potential is 100mV.
ARIZONA INSTITUTE
FOR
NANO-ELECTRONICS
Center for Computational Nanoscience
Kv1.2 voltage-dependent potassium channel
•
•
•
•
•
Belongs to large family of voltage-dependent potassium channels
Regulates potassium flow across cell membrane in neuron synapse of
mammals
Transmembrane portion is a tetramer, each subunit consisting of six helices S1-S6
forming voltage sensor (S1-S4) and pore domain (S5 and S6)
Channel can be in open and closed conformations, depending on transmembrane
voltage, the exact electromechanical process still unknown
Conformation transition on millisecond timescale
Top view (RCSB code 2R9R)
ARIZONA INSTITUTE
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NANO-ELECTRONICS
Center for Computational Nanoscience
Side view
Kv1.2 potential profile (side)
Dielectric constant of protein
region and implicit lipid
membrane set to 2.0.
Dielectric smoothing of the
protein-water contact using the
results in [1].
1. Cyril Azuara, Henri Orland, Michael Bon,
Patrice Koehl, and Marc Delarus,
Incorporating Dipolar Solvents in PoissonBoltzmann Electrostatics, Biophysical Journal,
Vol. 95, Dec. 2008.
XZ-plane slices at y = 5.0 nm of simulated distributions of potential (left)
and dielectric constant (right). No added KCl, single Poisson step.
ARIZONA INSTITUTE
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NANO-ELECTRONICS
Center for Computational Nanoscience
Kv1.2 potential profile (top)
4eV-deep potential well in the
selectivity filter.
Considerable positive charge in the
voltage sensor domains.
XY-plane slice at z = 6.5 nm of simulated distribution of potential.
No added KCl, single Poisson step.
ARIZONA INSTITUTE
FOR
NANO-ELECTRONICS
Center for Computational Nanoscience
Kv1.2 axial energy and potassium distribution
1
2
selectivity filter
0
-1
K+ ion energy [eV]
-3
-4
1
-5
+
K ion energy [eV]
+
axial K [M]
-6
accumulation at
the mouth
0.5
-7
-8
-9
K+ distribution [M]
1.5
-2
Peaks in ion distribution spatially
coincide with near-zero axial field
regions.
0
-10
-11
-12
Ion distribution consistent
with molecular dynamics
simulation results revealing two
potassium ions inside the
selectivity filter and one at the
mouth of KcsA channel with
similar selectivity filter [2].
2
4
6
8
axial position [nm]
10
12
-0.5
2. Simon Berneche and Benoit Roux, Molecular
Dynamics of the KcsA K+ Channel in a Bilayer
Membrane, Biophysical Journal, Vol 78, June
2000.
Potential energy of a potassium ion and potassium ion
distribution along the channel axis. Bulk KCl concentration 1mM,
40 ns simulation, results averaged over the last 20 ns.
ARIZONA INSTITUTE
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NANO-ELECTRONICS
Center for Computational Nanoscience
Visualization: Chicken Villin Headpiece folding
One of the few protein subdomains
obtaining stable conformation within
microseconds (see, for example,
RCSB code 1VII).
200 ns simulated, starting from thermally unstable linear conformation. LINCS
bond constraint algorithm with angle-constraining potentials used.
ARIZONA INSTITUTE
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NANO-ELECTRONICS
Center for Computational Nanoscience
Conclusions and future work
•
•
Constrained dynamics with velocity correction implemented
Conduction in OmpF ion channel studied, good agreement with
experiment
•
OmpF selectivity reveals combination of electrostatic and steric
effects
•
Preliminary data on Kv1.2 voltage-dependent potassium
channel obtained, consistent with experimental data and MD
simulations
Future work
•
Developing a Monte-Carlo based mechanism mimicking ion
adsorption by chemically active solid surfaces in aqueous
environment
•
Moving closer to MD and electrically polarizable forcefield
•
Modeling ionic conduction in nanostructures, including manmade and biological structures
ARIZONA INSTITUTE
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NANO-ELECTRONICS
Center for Computational Nanoscience