three-dimensional numerical simulation of ion nanochannels in … · 2015-07-13 ·...

Three-Dimensional Numerical Simulationof Ion Nanochannels in the presence ofMechanical, Thermal, and Fluid Forces

A.G.Mauri1,2, P.Airoldi2, R.Sacco2 and J.W.Jerome3

1 Micron Technology2 Dipartimento di Matematica, Politecnico di Milano

3 Northwestern University, Evanston, Il

July 22, 2015

IMA Workshop: Mathematics of Biological Charge Transport

Mauri,Airoldi,Sacco,Jerome () July 22, 2015 1 / 28

Outline

1 The mathematical models:

thermal PNP (thermally uni-directional: thermal gradients in PNPequations; no thermal feedback to the heat equation for the ionenvironment);velocity-extended thermal PNP (Stokes coupling);mechanical effects via linear elasticity (implemented by meshdisplacement; mechanical model not currently coupled to the system).

2 A preliminary methodology validation based on the Gramicidin A channel.

3 Simulations of the mathematical models based upon a channel with twocation carriers. This channel is discussed at length in [Keener-Sneyd,Mathematical Physiology, Sec. 2.8] as an example of a channel whichmaintains volume regulation and ion transport. An example cited in[Keener-Sneyd] is that of Na+-transporting epithelial cells.

4 3D geometry: hexahedral and cylindrical channel structures are compared.


Introduction

Ion channels are the gateways that allow electrical signal transmissionamong excitable cells in the human body. For this reason they have beensubject to extensive biophysical investigation.

The most commonly used approach has been the identification of the cellwith an equivalent electric circuit containing a non-constant conductanceelement; for example, the Hodgkin-Huxley model involves the solution of anonlinear system of ordinary differential equations.

Current mathematical methods and enhanced computer power open thepossibility of exploring at much smaller physical scales the complexities ofthe phenomena occurring through the cell membrane; also, they permit thecoupling of ion transport with thermal, fluid and mechanical effects.

Recently, with the help of the FEMOS-MP platform (Finite ElementMethods Object Oriented Simulation for Multi-Physics Simulation), we wereable to extend such a complex physical description into a 3D framework.


Thermal Poisson-Nernst-Planck System

Ionic concentrations ni , electrostatic potential ϕ and (ambient) temperature T

Mass balance:

∂ni

∂t+ div fi = 0

fi = nivi (includes a perturbation from thermal equilibrium)vi : average particle velocity.

∀i = 1, ...,M

∂ni

∂t+ div fi = 0

fi =qziDi

kBTniE− Di∇ni − Dini

∇TT

div (εf E) = q∑M

j=1 zjnj

E = −∇ϕ∂(ρcT )

∂t+ div (−k∇T ) = 0

q: elementary charge [C ]zi : effective charge number [-]kB : Boltzmann constant [JK−1]Di : diffusion coefficient [m2s−1]εf : dielectric permittivity [Fm−1]ρ: density [Kg m−3]c : specific heat [m2s−2K−1]k : thermal conductivity [Wm−1K−1]


Stokes Problem: Initial Formulation

Variables: velocity u and pressure p

Navier-Stokes equations:ρf

(∂u

∂t+ (u · ∇)u

)− div σ(u, p) = F

div u = 0

σ(u, p) = 2µf ε(u)− pδ

ε(u) = 12 (∇u + (∇u)T )

ρf : fluid density [Kg m−3]µf : dynamic viscosity [Pa s]F: external volume force[N m−3]δ: identity tensor [-]σ: stress tensor [Pa]

ε: strain rate tensor [s−1]


Stokes Problem

Navier-Stokes equations reduced to the Stokes system:ρf

(∂u

∂t+(u · ∇)u

)− div σ(u, p) = F

div u = 0

F =∑

i qziniE

Re is O(10−3) << 1 → Stokes time-dependent problem


Thermal Poisson-Nernst-Planck Velocity-Extended Model

∀i = 1, ...,M

∂ni

∂t+ div fi = 0

fi =qzi Di

kB TniE− Di∇ni − Di ni

∇T

T+ niu

div (εf E) = q∑M

j=1 zj nj

E = −∇ϕ∂(ρcT )

∂t+ div (−k∇T ) = 0

ρf∂u

∂t− div σ(u, p) = q

M∑j=1

zj njE

div u = 0

+ B.C . + I .C .

where, in the PNP system,

niu

is the additional translationcontribution to ion flow due to thefluid motion

where, in the Stokes equation,

qM∑

j=1

zjnjE

is the electric pressure exerted bythe ionic charge on the electrolytefluid


Flowchart for Thermal PNP Velocity-Extended Model

In view of the numerical solution ofthe VE-Thermal-PNP Stokessystem, we introduce a uniformpartition to obtain N time intervalsτk = (tk−1, tk ) of width∆t = Tfin/N where Tfin is the totalsimulation time. Then we denote by:

UkPNP =

[nk

1 , nk2 , ..., n

kM , ϕ

k ,T k]

UkStokes =

[uk , pk

]Uk =

[Uk

PNP ,UkStokes

]the solution vector of the ThermalPNP block, of the Stokes block andof the whole coupled problem attime level tk , k ≥ 0


Navier-Lame Mechanical Model for Mesh Displacement

Mechanical equilibrium equation:

divσ(u) + f = 0 in Ω

σ(u;T ) = C(ε− εth)

ε := ∇s(u)

u = uD on ΓD

σ(u)n = g on ΓN

- u displacement [m]- σ stress tensor [Pa]

- f volumetric force [N/m3]- g surface force [N/m2]- ε total strain tensor

- εth thermal strain tensor- C elastic tensor [Pa]

Thermal strain tensor εth

εth =

α1 0 00 α2 00 0 α3

· (T (x , y , z , t)− Tref )

with α1,2,3 the linear thermal expansion coefficients, and with Tref and T thereference and ambient temperatures [K ], resp.


Navier-Lame Mechanical Model

Lame coefficients λ and µ:

λ =νE

(1 + ν)(1− 2ν)µ =

E

2(1 + ν)

0 < ν < 0.5 is the Poisson ratio, and E > 0 Young’s modulus [Pa].

Elastic tensor C (Equivalent 6× 6 matrix representation):

C =

λ+ 2µ λ λ 0 0 0

λ+ 2µ λ 0 0 0λ+ 2µ 0 0 0

µ 0 0µ 0

µ


Full Solution Maps and Flowchart

As before, we have introduced auniform partition of thesimulation time Tfin. Then wedenote by:

uMEC = [ux , uy , uz ]

Updated Mesh = θ (ux , uy , uz )

UkPNP =

[nk

1 , nk2 , ..., n

kM , ϕ

k ,T k]

UkStokes =

[uk , pk

]Uk =

[Uk

PNP ,UkStokes

]the solution vector of theMechanics, ThermalPNP blockand Stokes block and of theentire coupled problem at timelevel tk , k ≥ 0


Methodology Validation: PNP

This simulation represents the gramicidin-A channel (assumed cylindrical) inwhich sodium and chloride ions are being transported. It is highly selective, dueto the negative profile of permanent charge ρfixed . We compare one-dimensionaland three-dimensional simulations for the validation.

Parameter ValueL[nm] 2.5r [nm] 0.2zNa+ +1zCl− -1

DNa+ [cm2s−1] 1 · 10−5

DCl− [cm2s−1] 2.03 · 10−5

Na+

[cm−3] 6.022 · 1022

Cl−

[cm−3] 6.022 · 1016

T [K] 300ϕleft [V] -0.079ϕright [V] -0.179

BC :

Na+ = Na+

on left ∪ right

Cl− = Cl−

on left ∪ right

ϕ = ϕleft on left face

ϕ = ϕright on right face

homogeneous Neumann elsewhere


Methodology Validation: PNP

The 3D simulation values are restricted to the cylindrical z-axis for comparison inthe graphs.


3D Extensions for PNP/Stokes: Two-Cation Channel

We have reproduced a well known example of a volume regulation channel whichbalances ionic transport with two species K+ and Na+ (basis1 ≡ left face; basis2≡ right face in the figures): zK + = zNa+ = +1,n

(1)

K + = 2.41 · 1020, n(1)

Na+ = 3.01 · 1019

n(2)

K + = 1.2 · 1019, n(2)

Na+ = 2.65 · 1020

2D Model

3D HEXA Model

σn = 0 on basis1

σn = [0, 0, 100 N cm−2] on basis2

σn = 0 on ∂Ω \ basis1 ∪ basis2ϕ = 0.2 V on basis1

ϕ = 0 V on basis2

∇ϕ · n = 0 on ∂Ω \ basis1 ∪ basis2ni = n

(1)i on basis1

ni = n(2)i on basis2

fi · n = 0 on ∂Ω \ basis1 ∪ basis2

3D CYL Model


Cylindrical or Hexahedral Extension for PNP/Stokes?

Hexahedral and cylindricalextensions are not equivalentfor the physical quantitieswe are interested in, such asthe electrostatic potential,electric field, and velocity.

Remark: The simulation length here is 10 nm. This includes parts of theintracellular and extracellular regions. This allows for more effective boundaryconditions.


Ionic Concentrations for PNP/Stokes

Ionic concentration profiles, at the endof the simulation time. The 3Dcylindrical geometry gives results closerto 2D simulations.


Effects of the Thermal Gradient

This is the only set of experimentsutilizing the thermal gradient; its effectsare negligible, even when temperaturesare elevated outside the biologicalregime. The boundary conditions aredeliberately exaggerated to show this.The contribution of the thermal force tothe diffusion of ionic species i is given by

fthi = −Di ni∇T

T

where the Temperature is the solutionof the proper heat equation

∂(ρcT )

∂t+ div (−k∇T ) = 0

T = T = 293.75 on basis1

T = T = 343.75 on basis2 : case1

T = T = 373.75 on basis2 : case2


Effects of Thermal Gradient on Concentrations


Effects of Mechanical Stress: Full Model

The following mechanical problems (deformation 1 and 2) have been solved.

uD = [0, 0, 0]T (µm) on Top ∪ Bot

g = [0, 0, 0]T (N/cm2) on ΓN

f = [0, 0, 0]T (N/cm3) in Ω1 ∪ Ω2 ∪ Ω3

σ0 = [γ, γ, 0, 0, 0, 0]T (N/cm2) in Ω2

γ = −1 · 107deformation 1

γ = −2 · 107deformation 2

Material E [N/cm2] ν

Ω1 and Ω3 70 · 105 0.40Ω2 150 · 105 0.20

Table: (Typical) Material parameter valuesRemark: In σ0, the vanishing of the third componentimplies planar displacement.



Electrostatic potential and z-velocity component for the two different mechanicalproblems

Remark: The far left figures are the simulations of PNP/Stokes.


Effects of Mechanical Stress and Electric Pressure

The first picture shows the effectson the velocity profile induced bythe geometric deformation of theion channel (without solving PNP)

In the second picture (larger scale)the geometric effect has beensubstantially modified by theelectric pressure exerted on theStokes equation by the ionicspecies and the electric field


Effects of Velocity in VE-PNP (PNP/Stokes)

The inclusion of the additional translational contribution to the ion flow is clearlyvisible in the two picture below where the PNP system has been solved in thedeformed grid (after inclusion of the mechanical stress effect) with and withoutthe coupling term.


Existence Results for PNP/Navier-Stokes

Mathematical model:

1 A local smooth solution result for the Cauchy problem was obtained via(equivalent of) evolution operators. [J, Trans. Th. Stat. Phys. 31,

333--366 (2002)]. The interval of existence/uniqueness was shown stableunder vanishing viscosity η → 0.

2 There are several results for the initial/boundary-value problem.

Rubinstein model:[J. Jerome and R. Sacco, Nonlinear Anal. 71, e2487--e2497

(2009)].[M. Schmuck, MMM Appl. Sc. 19, 993--1015 (2009)].[R. Ryham, ArXiv preprint (2009) ].Electro-osmosis: [D. Bothe et al. , SIAM J. Math. Anal. 46,

1263--1316 (2014)].

3 For the steady problem, weak solutions have been demonstrated [J,Nonlin. Anal. 74, 7486--7498 (2011)] by a switching fixed point map.

4 No inclusion of thermal effects yet.


PNP Subsystem Discretization

Linear Poisson Equation

P1 conforming elements on simplicial mesh are used. RHS of the equation isnumerically integrated by means of a 3D trapezoidal rule.

Continuity Equations

1 Current density is approximated by edge-averaged finite elements withexponential fitting (Xu-Zikatanov [1999; Math. Comp]), and ensuresM-matrices in the calculations. This was implemented in the thesis ofC. DeFalco (2005), under the supervision of R. Sacco.

2 Time discretization is (implicit) backward Euler, although current workuses TR-BDF2.


Stokes Subsystem/Mechanical Subsystem Discretization

1 Stokes subsystem

Two-field format: velocity and pressure in the stress tensor.P2/P1 Taylor-Hood finite element pair is used (LBB stability).Uzawa iteration with pre-conditioning for the algebraic saddle-pointproblems.

2 Mechanical subsystem

As implemented in the graphics, a P1 displacement method was used. Afunctional calculation repositions each mesh node without changing bulk orinterface properties, or the vertex index. As a result, all quantities(potential, field, concentrations, etc.) are still connected to the originalvertex. Drawbacks include the re-calculation of element matrices and theeffect on the Xu-Zikatanov M-matrix property.

FEMOS-MP has a two-field capability which could be used in futuremechanical simulations.

Remark: The code warns the user of the possible breakdown of the M-matrixproperty; however, it was not experienced in these simulations.


Conclusions and Future Perspectives

The Poisson-Nernst-Planck velocity-extended model has been successfullyimplemented and validated in a 3D framework. Effects of a Thermal gradient andmechanical stress induced by an external agent are also considered and applied toa realistic geometric situation.

This talk is based upon an article of the same title and authors, and willsoon be submitted for publication.

Future work for consideration includes:

Coupling of the thermal and mechanical subsystems–analysis;

Use of a Mixed Finite Element Method;

PNP model extension considering the electroneutrality constraint.


References: FEMOS-MP

1 A.G. Mauri, A. Bortolossi, G. Novielli, and R. Sacco. Three-dimensionalfinite element modeling of industrial semiconductor devices, including impactionization. J. Math. (in) Industry 5 (1) (2015). doi:10.1186/s13362-015-00015-z.

2 A.G. Mauri, R. Sacco, and M. Verri. Electro-Thermo-Chemicalcomputational models for 3D heterogeneous semiconductor devicesimulation. Appl. Math. Modelling. doi: 10.1016/j.apm.2014.12.008.

3 P. Airoldi, A.G. Mauri, R. Sacco, and J. Jerome. Three dimensionalnumerical simulation of ion nano-channels. J. Coupled Sys. Multiscale Dyn.3 (1) (2015) doi: 10.1166/jcsmd.2015.1065.


three-dimensional numerical simulation of ion nanochannels in … · 2015-07-13 ·...

Documents