chicago, july 22-23, 2002darpa simbiosys review 1 monte carlo particle simulation of ionic channels...

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


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


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



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 =-



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.


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


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


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.



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



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



distance ( Å )

(

eV )

shift-truncated potential

0 2 4 6 8 10-1.0

1.0

3.0

5.0

7.0

9.0




shift-truncated potential

( eV

)

2 4 6 8 10

distance ( Å )

-0.100

0.000

0.100

0.200

0.300


Magnified view



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



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



ANION-ANION BULK1

t=10fs

Poisson update 10t


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



CATION-ANION BULK1

t=10fs

Poisson update 10t


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



t=10fs

Poisson update 10t


= 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



CATION-CATION BULK1

t=10fs

Poisson update 10t


= 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



CATION-ANION BULK1

t=10fs

Poisson update 10t


= 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/



t=10fs


=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



ANION-ANION BULK1

t=10fs


=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



t=10fs


=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)



ANION-ANION BULK1

t=10fs

Poisson update 10tCoulomb update every dt


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/



CATION-CATION BULK1

t=10fs




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/



t=10fs




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/



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


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.


Contact injection issues and model development

z ( Å )

x ( Å )

Gramicidin

0.0

24.0

0 50 100


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.

chicago, july 22-23, 2002darpa simbiosys review 1 monte carlo particle simulation of ionic channels...

Documents

transport model

transport of charged

lennardjones model potential

particle trajectories

charged ions

radial distribution

charged spheres validation

additional force