Chicago, July 22- 23, 2002 DARPA Simbiosys Review 1 Monte Carlo Particle Simulation of Ionic Channels Trudy van der Straaten Umberto Ravaioli Beckman Institute University of Illinois at Urbana-Champaign
Dec 31, 2015
Chicago, July 22-23, 2002 DARPA Simbiosys Review 1
Monte Carlo Particle Simulation of Ionic Channels
Trudy van der Straaten
Umberto Ravaioli
Beckman InstituteUniversity of Illinois at Urbana-Champaign
Chicago, July 22-23, 2002 DARPA Simbiosys Review 2
Outline
• Introduction to Transport Monte Carlo simulation
• Extensions for the treatment of ionic transport
– inclusion of ions as charged spheres
• Validation of the transport model for microfluidics:
– pair correlation function results
• Contact injection issues and model development
• Future work
Chicago, July 22-23, 2002 DARPA Simbiosys Review 3
Introduction to Transport Monte Carlo simulation
A Transport Monte Carlo model simulates transport of charged particles as a sequence of trajectories interrupted by random scattering events.
The free flights have random duration, obtained by selecting
a uniform random number “r”. For uniform scattering rate , the flight has duration given by
l- = = G Þ =- Gò ò0 0
ln( )ln( ) ( )T T
rr t d t d t T
Chicago, July 22-23, 2002 DARPA Simbiosys Review 4
Introduction to Transport Monte Carlo simulation
The charged particle trajectories are evaluated from a detailed electric field distribution in space, obtained by solving the Poisson equation.
A charge assignment procedure associates the carrier charge to the mesh, to generate the right-hand side of the discretized equation.
Dirichlet Boundary Conditions
Neuman Boundary Conditions
e f rÑ×Ñ =-
Chicago, July 22-23, 2002 DARPA Simbiosys Review 5
Introduction to Transport Monte Carlo simulation
Short-range charge-charge interaction may be introduced by evaluating the Coulomb force in a range surrounding the particles.
The total force is obtained by adding the short-range Coulomb force (particle-particle) with the Poisson equation force (particle-mesh), plus a correction to eliminate double counting in the overlap region between the two domains.
Chicago, July 22-23, 2002 DARPA Simbiosys Review 6
Extension for the treatment of ionic channels
Water is assumed to be a continuum background with a dielectric permittivity distribution.
Interaction between charged ions and water is modeled by the scattering rate.
Gramicidin - Successful Na+ trajectory - Point particle model
Chicago, July 22-23, 2002 DARPA Simbiosys Review 7
Extension for the treatment of ionic channels
Ion size is introduced by dressing the particles with a Lennard-Jones model potential.
This creates an additional force that maintains the ions separated from each other and from the boundary.
Lennard-Jones 6-12 potential
6 12
4LJ LJ r r
Na Cl K
LJ [ kCal/mol ]
[ eV ]
0.1
0.004336
0.15
0.006504
0.36
0.01561
[ Å ] 2.7297 4.2763 3.3605
short range repulsion
-0.50
0.00
0.50
1.00
Vol
ts
0.0 2.0 4.0 6.0 8.0 10.0
ion separation (Angstroms)
long rangeweak attraction
point-particle Coulomb potential
Lennard-Jones potential
Chicago, July 22-23, 2002 DARPA Simbiosys Review 8
Validation of the transport model for microfluidics
The Transport Monte Carlo model needs to be validated by comparing with other models that treat water interaction in greater detail.
We compare against benchmarks obtained from Molecular Dynamics simulations (Rush group) and Metropolis Monte Carlo simulations (Utah group), which calculate the ion-ion pair correlation function in space.
Reproducing the ion-ion pair correlation function is a crucial prerequisite to obtaining the correct thermodynamic properties of the system.
Chicago, July 22-23, 2002 DARPA Simbiosys Review 9
Validation of the transport model for microfluidics
The pair correlation function is the radial distribution function that measures how atoms organize themselves around one another.
It is a measure of the probability of finding two atoms separated by a distance .
It can be measured from x-ray and neutron diffraction experiments and is readily computed from simulations of trajectories
We compared results for simulation of a bulk monovalent electrolyte solution.
r r
2( ) ( )
distance between ions and
N N
iji j i
ij
Volg r r rN
r i j
Chicago, July 22-23, 2002 DARPA Simbiosys Review 10
Validation of the transport model for microfluidics
The benchmarks were provided for a simplifed (shifted-truncated) Lennard-Jones potential
12 61/6
1/6
( ) 4 , 2
0 , 2
Lennard-Jones energy p
ij ij ij ij ij ij ijLJ LJ LJ
ij ij
LJ
U r d r d r r d
r d
arameter ( / 1)
where and are the Lennard-Jones 2
distance parameters for ions and
LJ
i jij i j
kT
d dd d d
i j
Chicago, July 22-23, 2002 DARPA Simbiosys Review 11
Validation of the transport model for microfluidics
distance ( Å )
(
eV )
shift-truncated potential
0 2 4 6 8 10-1.0
1.0
3.0
5.0
7.0
9.0
Lennard-Jones potential
Chicago, July 22-23, 2002 DARPA Simbiosys Review 12
Validation of the transport model for microfluidics
shift-truncated potential
( eV
)
2 4 6 8 10
distance ( Å )
-0.100
0.000
0.100
0.200
0.300
Lennard-Jones potential
Magnified view
Chicago, July 22-23, 2002 DARPA Simbiosys Review 13
Validation of the transport model for microfluidics
Simulation conditions
1 Molar monovalent electrolyte solution q+ = +1; q-= -1
Lennard-Jones distance parameter + = -= 3Å
Simulation cell LLL
Metropolis MC Transport MCL ( Å ) 69.251 70.0 or 72.0T ( K ) 298.15 300.0
r (water) 78.46 80.0N cations 200 206 or 225N anions 200 206 or 225
Chicago, July 22-23, 2002 DARPA Simbiosys Review 14
Validation of the transport model for microfluidics
CATION-CATION BULK1
t=10fs
Poisson update 10t
Tsim=5nsPCF updated every 10ps
0.00
0.50
1.00
1.50
2.00
Pair
Cor
rela
tion
Func
tion
0.0 5.0 10.0 15.0 20.0 25.0 30.0
r (Angstrom)
= 2Å
Transport MC
Metropolis MC
Chicago, July 22-23, 2002 DARPA Simbiosys Review 15
Validation of the transport model for microfluidics
ANION-ANION BULK1
t=10fs
Poisson update 10t
Tsim=5nsPCF updated every 10ps
0.00
0.50
1.00
1.50
2.00
Pai
r C
orre
lati
on F
unct
ion
0.0 5.0 10.0 15.0 20.0 25.0 30.0
r (Angstrom)
= 2Å
Transport MC
Metropolis MC
Chicago, July 22-23, 2002 DARPA Simbiosys Review 16
Validation of the transport model for microfluidics
CATION-ANION BULK1
t=10fs
Poisson update 10t
Tsim=5nsPCF updated every 10ps
0.00
1.00
2.00
3.00
Pair
Cor
rela
tion
Func
tion
0.0 5.0 10.0 15.0 20.0 25.0 30.0
r (Angstrom)
= 2Å
Transport MC
Metropolis MC
Chicago, July 22-23, 2002 DARPA Simbiosys Review 17
Validation of the transport model for microfluidics
t=10fs
Poisson update 10t
Tsim=5nsPCF updated every 10ps
= 5A (SRC) = 5A = 2A
ANION-ANION BULK1
0.0 2.0 4.0 6.0 8.0 10.0
r/
0.00
0.50
1.00
1.50
2.00Pa
ir C
orre
latio
n F
unct
ion
Chicago, July 22-23, 2002 DARPA Simbiosys Review 18
Validation of the transport model for microfluidics
CATION-CATION BULK1
t=10fs
Poisson update 10t
Tsim=5nsPCF updated every 10ps
= 5A (SRC) = 5A = 2A
0.0 2.0 4.0 6.0 8.0 10.0
r/
0.00
0.50
1.00
1.50
2.00
Pai
r C
orre
lati
on F
unct
ion
Chicago, July 22-23, 2002 DARPA Simbiosys Review 19
Validation of the transport model for microfluidics
CATION-ANION BULK1
t=10fs
Poisson update 10t
Tsim=5nsPCF updated every 10ps
= 5A (SRC) = 5A = 2A
0.00
1.00
2.00
3.00
Pair
Cor
rela
tion
Func
tion
0.0 2.0 4.0 6.0 8.0 10.0
r/
Chicago, July 22-23, 2002 DARPA Simbiosys Review 20
Validation of the transport model for microfluidics
t=10fs
Tsim=5nsPCF updated every 10ps
=2A, no SRC
tpoisson (ps)0.10.51210100
ANION-ANION BULK1
0.0 5.0 10.0 15.0 20.0 25.0 30.0
r (Angstroms)
0.00
0.50
1.00
2.00Pa
ir C
orre
latio
n Fu
nctio
n
1.50
Chicago, July 22-23, 2002 DARPA Simbiosys Review 21
Validation of the transport model for microfluidics
ANION-ANION BULK1
t=10fs
Tsim=5nsPCF updated every 10ps
=2A, no SRC
tpoisson (ps)0.10.51210100
0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0
r (Angstroms)
0
1
2
3
4
5
6
7
8Pa
ir C
orre
latio
n Fu
nctio
n
Chicago, July 22-23, 2002 DARPA Simbiosys Review 22
Validation of the transport model for microfluidics
t=10fs
Tsim=5nsPCF updated every 10ps
=2A, no SRC
tpoisson (ps)0.10.51210100
CATION-ANION BULK1
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
Pair
Cor
rela
tion
Func
tion
0.0 5.0 10.0 15.0 20.0 25.0 30.0
r (Angstroms)
Chicago, July 22-23, 2002 DARPA Simbiosys Review 23
Validation of the transport model for microfluidics
ANION-ANION BULK1
t=10fs
Poisson update 10tCoulomb update every dt
Tsim=5nsPCF updated every 10ps
poisson, 5A mesh + SRCcoulomb
0.00
0.50
1.00
1.50Pa
ir C
orre
latio
n Fu
nctio
n
0.0 2.0 4.0 6.0 8.0 10.0
r/
Chicago, July 22-23, 2002 DARPA Simbiosys Review 24
Validation of the transport model for microfluidics
CATION-CATION BULK1
t=10fs
Poisson update 10tCoulomb update every dt
Tsim=5nsPCF updated every 10ps
poisson, 5A mesh + SRCcoulomb
0.00
0.50
1.00
1.50Pa
ir C
orre
latio
n Fu
nctio
n
0.0 2.0 4.0 6.0 8.0 10.0
r/
Chicago, July 22-23, 2002 DARPA Simbiosys Review 25
Validation of the transport model for microfluidics
t=10fs
Poisson update 10tCoulomb update every dt
Tsim=5nsPCF updated every 10ps
poisson, 5A mesh + SRCcoulomb
CATION-ANION BULK1
0.00
1.00
2.00
3.00
Pair
Cor
rela
tion
Func
tion
0.0 2.0 4.0 6.0 8.0 10.0
r/
Chicago, July 22-23, 2002 DARPA Simbiosys Review 26
Validation of the transport model for microfluidics
CATION-CATION
CATION-ANION
ANION-ANION
0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0
Z(Angstroms)
0.0
5.0
10.0
15.0
20.0
Pai
r C
orre
lati
on F
unct
ion
Results obtained using untruncated Lennard-Jones
Chicago, July 22-23, 2002 DARPA Simbiosys Review 27
Contact injection issues and model development
Work in progress is addressing the formulation of contact boundary conditions in the full ion channel simulation by Transport Monte Carlo.
For realistic simulation, one has to reach a trade-off between computational cost and model complexity. The actual ion population in the simulation domain is small and the real computational bottleneck in 3-D simulation becomes the solution of Poisson equation.
The most natural way to implement contacts, avoiding excessive fluctuations or spurious effects, is to provide buffer layers where the bath concentration is maintained constant.
Chicago, July 22-23, 2002 DARPA Simbiosys Review 28
Contact injection issues and model development
z ( Å )
x ( Å )
Gramicidin
0.0
24.0
0 50 100
Chicago, July 22-23, 2002 DARPA Simbiosys Review 29
Future Work
Completion of comparisons for pair-correlation function over a wide range of benchmarks.
Inclusion of parallelized Poisson solver (several developed and tested on SGI Origin and PC clusters).
Completion and testing of buffer layer contacts.
Design of a reduced Monte Carlo algorithm based on Local Iterative procedure.
Optimization of the overall Transport Monte Carlo algorithm, based on different grids for Poisson solution and ion dynamics.