multiscale modeling of ion transport through nanopores ...ex + z αeΦ(r) poisson ... dezsőboda...

Multiscale modeling of ion transport through nanopores:the case study of a rectifying bipolar nanofluidic diode

Dezső Boda

Department of Physical Chemistry, University of Pannonia, Veszprém, HungaryInstitute of Advanced Studies Kőszeg (iASK), Hungary

[email protected]

Hybrid Methods in Molecular SimulationCagliari, Italy

Supported by the National Research, Development and Innovation Office – NKFIH NN113527 in theframework of ERA Chemistry.

Cagliari, 2017.04.03-04.

Dezső Boda (U. Pannonia, iASK, Hungary) Multiscale modeling of nanopores Cagliari, 2017.04.03-04. 1 / 28

People on board: international multiscale modeling network

The University of Pannonia / iASK team: Dezső Boda, Tamás Kristóf, Mónika Valiskó, Zoltán Ható

The Chicago wizard: Dirk Gillespie The Italian connection: Simone Furini, Claudio Berti

The Mathematicians: The Carloni Group (Jülich):Bart Matejczyk, M.-T. Wolfram, Jan Pietschmann Paolo Carloni, Justin Finnerty


Devices

Study nanodevices and peek into the black box

Many devices are black boxes, in the sense that input-output relations(response functions) are known, but we know much less about their internalstructures.Their inner mechanisms are usually studied with continuum theories (forexample, PNP for semiconductor devices)Dimensions of the devices, however, continuously shrink (nanodevices)The molecular behavior of the core unit of the device (that can be verydifferent from bulk behavior) determines device behaviorWe need to study the core unit on the molecular level with modeling andcomputation


Devices

Core units and the devices „around” them havedifferent length/time scales

Different length and timescalesDifferent techniques to dealwith themWhat is the relation of thedifferent levels?


Devices

In our studes, the core units are porous materials

Bath Left Bath Right

Channel in a membrane

-20 -10 0 10 20


Membrane with pores

-20 -10 0 10 20


Porous membrane

Examples: ion channelsin biological membranes,synthetic nanopores,silicate membranes(zeolite, silicalite,kaolinite)

Phenomena of interest:selectivity, permeation,rectification

Models: from all-atom tocoarse-grained (reduced)

Motivation: technology,biology


Nanopores

Nanopore as a device – a rectifying diode

Pore etched into plastic foilDimensions: µm length, nmradiusInput: voltageOutput: currentEngineering point of view:voltage-current relation

Black box response functionBut what is in the black box?


Nanopores

Nanopore fabrication

Motivation: semiconductorp-n junctions

PDMS or PET nanopores


Multiscaling

Resolution of the microscopic model – multiscaling

Detailed model vs. reduced model to handle subsystems with different scalesAdvantages/disadvantages (too many details vs. too much approximation)Reduced models for large-scale device properties, all-atom models for moleculardetailsThe challenge is to link the various modeling levels

1 Z. Ható, D. Boda, D. Gillepie, J. Vrabec, G. Rutkai, and T. Kristóf. Simulation study of a rectifying bipolar ion channel: detailed model versusreduced model. Cond. Matt. Phys., 19(1):13802, 2016.


Multiscaling

Hierarchy of models and methods

Explicit water (MD) → implicit water (BD, LEMC, PNP)Direct simulation of tranport (BD) → transport equation (NP)Particle simulation (LEMC) → continuum method (PNP)

{rN}

{pN}

Decreasing resolution (and computation time)

All-atom

DMCMD

Reduced (e.g., implicit solvent)

Newton equations

BD

Langevin equations

Molecular Dynamics Brownian Dynamics

Monte Carlo

ci(r) ↔µ

i(r)

LEMC PB

ci(r) ↔µ

i(r)

Local Equilibrium

Nernst-Planck

of transport

Direct simulation

NP+LEMC

Direct simulation

of transport

-kT ji(r) = D

i(r) c

i(r) ∇µ

i(r)

Mimictrajectories

Poisson-Boltzmann

transport equation

Dynamic Monte Carlo

PNP

WEEKS DAYS HOURS SECONDSDAYS


NP+LEMC

Electrodiffusion: the Nernst-Planck equation

jα(r) = − 1kT Dα(r)cα(r)∇µα(r)

Flux: output of the calculation – jα(r)Concentration: availability of the transported particles – cα(r)Diffusion coefficient: mobility of the transported particles – Dα(r)Driving force: difference/gradient of the electrochemical potential – µα(r)What is the relation of cα(r) and µα(r) in the transport (non-equilibrium) region?Non-equilibrium statistical mechanics is needed. Not well-established.General solution: divide the system into volume elements and assume localequilibrium (LE) in them.Use the procedures of equilibrium statistical mechanics in the volumeelements.

2 D. Boda, D. Gillespie. Steady state electrodiffusion from the Nernst-Planck equation coupled to Local Equilibrium Monte Carlo simulations. J.Chem. Theory Comp., 8(3):824-829, 2012.


NP+LEMC

Statistical mechanical methods that can be used

The electrochemical potential:

µα(r) = µα0 + kT ln cα(r) + µαEX + zαeΦ(r)

Poisson-Boltzmann: used for the ideal solution (µαEX(r) = 0) – NP+PB

(traditionally called Poisson-Nernst-Planck (PNP) theory)Density Functional Theory (Tarazona, Gillespie) – NP+DFT (1D)Simulation (our suggestion): perform GCMC simulation using differentelectrochemical potentials in the volume elements (each element is an open systemin the GC ensemble) – Local Equilibrium Monte Carlo (LEMC)

2 D. Boda, D. Gillespie. Steady state electrodiffusion from the Nernst-Planck equation coupled to Local Equilibrium Monte Carlo simulations. J.Chem. Theory Comp., 8(3):824-829, 2012.

3 D. Boda, R. Kovács, D. Gillespie, T. Kristóf. Selective transport through a model calcium channel studied by Local Equilibrium Monte Carlosimulations coupled to the Nernst-Planck equation J. Mol. Liq., 189:100, 2014.

4 D. Boda. Monte Carlo simulation of electrolyte solutions in biology: In and out of equilibrium. Annual Reports in Computational Chemistry, 10,Chapter 5, Pages 127–163, Elsevier, 2014.


NP+LEMC

Local Equilibrium Monte Carlo (LEMC)

-40 -30 -20 -10 0 10 20 30 40

z / Å

-40

-30

-20

-10

0

10

20

30

40

r /

Å

Lef

t b

ulk

reg

ion

Rig

ht

acc

ess

regio

nProtein

Lef

t acc

ess

regio

n

Rig

ht

bu

lk r

egio

n

ProteinM

emb

ran

eM

emb

ran

e

HB

Hacc

Racc

RB

R

Solution domain (pore + acess regions)is divided into volume elements (Vi)Input: µα(ri ), Output: cα(ri )Acceptance probability for inserting(χ = 1) or deleting (χ = −1) an ion in avolume element (below).Nα

i is the number of ions in Vi beforeinsertion/deletionThe energy contains the interactionswith everything in the whole simulationcell including the applied field (r ∈ Vi).

pαi,χ(r) = Nαi !V χ

i(Nα

i + χ)! exp(−∆U(r)− χµαi

kT

)


Rectifying OmpF ion channel

Failure for a mutant rectifying ion channel with MDsimulations

0 10 20 30 40 50 60

t / ns

50

100

150

200

250

I / pA

Cl- 200mV (1 trimer)

Cl- 200mV (4 trimers)

Cl- -200mV

Cl- -200mV (4 trimers)

RREE

0 10 20 30 40 50 600

20

40

60

80

100

Num

ber

of io

n

cro

ssin

gs

Cl- 200mV (1 trimer)

Cl- 200mV (4 trimers)

Cl- -200mV (1 trimer)

Cl- -200mV (4 trimers)

NaCl

Experiments (Miedema et al.Nano Lett., 7, 2886, 2007.)showed rectification for theRREE mutantOur large-scale all-atom MDsimulations did not

1 Z. Ható, D. Boda, D. Gillepie, J. Vrabec, G. Rutkai, and T. Kristóf. Simulation study of a rectifying bipolar ion channel: detailed model versusreduced model. Cond. Matt. Phys., 19(1):13802, 2016.


MD vs. NP+LEMC

Bipolar nanopore models

Nanopore is CNT, membrane isCNSMD: GROMACS, CHARMM27Explicit water: SPC

ON OFF

Na+

Cl-

Na+

Cl-

Nanopore and membrane walls arehard wallsIons: charged hard spheresImplicit water: ε = 78.5

-6 -3 0 3 6

z / nm

6

3

0

3

6

r /

nm

Le

ft e

qu

ilib

riu

m b

ulk

Rig

h n

on

-eq

uil

ibri

um

tra

ns

po

rt r

eg

ion

+ + + + + +

Le

ft n

on

-eq

uil

ibri

um

tra

sn

po

rt r

eg

ion

Rig

ht

eq

uil

ibri

um

bu

lk

- - - - - - -

+ + + + + +

membrane

- - - - - - -

membrane

N region P region

σ -σ

5 Z. Ható, M. Valiskó, T. Kristóf, D. Gillespie, D. Boda. Multiscale modeling of a rectifying bipolar nanopore: explicit-water versus implicit-watersimulations. PCCP submitted, 2017.


MD vs. NP+LEMC

Explicit vs. implicit water comparison (MD vs. NP+LEMC)

Results for device function - qualitative agreement

Current vs. voltage

-300 -200 -100 0 100 200 300

U / mV

-200

0

200

400

600

I /

pA

c = 0.5 Mc = 1 M

ON

OFF

Symbols: all atom

Lines: Reduced

Current vs. pore charge

50

100

150

200

250

| I/pA

|

0 0.5 1 1.5 2

σ / e nm-2

50

100

150

200

| I/pA

|

ON

OFF

ON

OFF

Cl-

Na+



MD vs. NP+LEMC


Concentration profiles - different agreement in z and r dimensions

Axial profiles – agreeRelevant for device function

-3 0 3

z / nm

0

0.5

1

1.5

2

ci(z

) / M

0

0.5

1

1.5

2

ci(z

) / M

all atom

OFF

ON

Na+

Cl-

ON

OFF

reduced

Radial profiles – disagreeIrrelevant for device function

(A) (B)

0

1

2

3

4

5

ci(r

) /

M

solid lines w symbols: MD, dashed: NP+LEMC

0 0.2 0.4 0.6 0.8 1

r / nm

0

1

2

3

4

ci(r

) /

M

water

water

Na+

Cl-

Cl-

Na+

N region

P region



MD vs. NP+LEMC


Potential profiles - not even close



MD vs. NP+LEMC

Screening in the explicit solvent model

Potential profiles of ions and water are opposite



BD vs. NP+LEMC

Direct dynamics vs. transport equation (BD vs. NP+LEMC)

Both are for implicit water


50

100

150

200

250

| I/

pA

|

Solid: NP+LEMC Dashed: BD

0 0.5 1 1.5 2

σ / e nm-2

50

100

150

| I/

pA

|

ON

OFF

ON

OFF

Cl-

Na+

Axial concentration profiles

-5 0 5

z / nm

0

0.5

1

1.5

2

2.5

Concentr

ation p

rofile

s / M

-5 0 5

z / nm

Cl-

Na+

ON

OFF

ON

OFF

Solid: NP+LEMC Dashed. BD

Using the same diffusion coefficient in the pore in the two casesCurrents are different: Di has different meanings in BD and NP


NP+LEMC vs. PNP

Molecular simulation vs. continuum theory (LEMC vs. PNP)

Device properties

Current vs. voltage

-200 -100 0 100 200

U / mV

0

50

100

150

200

I /

pA

NP+LEMC

lin. PNP

0

10

20

30

40

I /

pA

R = 1 nm, σ = 1 e/nm2

0 100 2000

5

10

0 100 2000

50

100

c = 0.1 M

c = 1 M

Rectification

Rectification


0 0.5 1 1.5

σ / e/nm2

0

50

100

150

200

| I

/ p

A |

NP+LEMCPNP

c = 1 M, R = 1 nm, U = ±200 mV

0 0.5 1 1.50

5

10

15

20

Rectification

ON

OFF

6 B. Matejczyk, M. Valiskó, M.-T. Wolfram, J.-F. Pietschmannn, D. Boda. Multiscale modeling of a rectifying bipolar nanopore: ComparingPoisson-Nernst-Planck to Monte Carlo. J. Chem. Phys. 146, 124125, 2017.


NP+LEMC vs. PNP

Molecular simulation vs. continuum theory (LEMC vs. PNP)

Device properties

Current vs. concentration

0.01 0.1 1

c / M

0

50

100

150

200

250

| I

/ p

A |

NP+LEMCPNP

R = 1 nm, U = ±200 mV, σ = 1 e/nm2

0.01 0.1 1

0

100

200

300

400

500

Rectification

ON

OFF

Current vs. pore radius

1 2 3

R / nm

0

500

1000

1500

| I

/ p

A |

NP+LEMCPNP

c = 1 M, U = ±200 mV, σ = 1 e/nm2

1 2 3

10

20

30

40

Rectificaio

n

OFF

ON

6 B. Matejczyk, M. Valiskó, M.-T. Wolfram, J.-F. Pietschmannn, D. Boda. Multiscale modeling of a rectifying bipolar nanopore: ComparingPoisson-Nernst-Planck to Monte Carlo. J. Chem. Phys. 146, 124125, 2017.


Results for an ion channel

RyR calcium-release ion channel

Ca2+-release Ryanodine Receptor(RyR) calcium channel channelThe released Ca2+ initiates musclecontraction



Models of the RyR Ca2+-release channel

1-D model of Dirk Gillespie forNP+DFTJPCB 109, 15598, BJ, 94, 1169, 2008; BJ, 97, 2212, 2009.

Our 3-D model for NP+LEMCbased on Dirk’s model

-20 -10 0 10 20

z / Å

sele

ctiv

ity fi

lter

SR

lum

en

cyto

sol

D4945

D4938

D4899

E4900

E4902

RyR(b)

εw

εw



I-V curves for NaCl and NaCl-CaCl2Symmetric 250 mM NaCl (basis of fitof Dα in the channel).

-150 -100 -50 0 50 100 150

U / mV

-50

0

50

I / m

A

Exp.

NP+DFTNP+LEMC

250mM NaCl | 250mM NaCl

NaCl-CaCl2 mixture - non-symmetricI-V curve.

-150 -100 -50 0 50 100 150

U / mV

-20

0

20

40

60

80

I /

mA

Exp.

NP+LEMCNP+DFT

250mM NaCl | 250mM NaCl

4µM CaCl2 | 50mM CaCl

2

DFT: Gillespie et al. JPCB 109, 15598, BJ, 94, 1169, 2008; BJ, 97, 2212, 2009.

4 D. Boda. Monte Carlo simulation of electrolyte solutions in biology: In and out of equilibrium. Annual Reports in Computational Chemistry,Volume 10, Chapter 5, Pages 127–163, Elsevier, 2014.



Anomalous mole fraction effects for the RyR channel

CaCl2 is added to 100 mM CsCl

-6 -5 -4 -3 -2

lg[CaCl2] added to 100 mM CsCl

0

2

4

6

8

Curr

ent / pA

Experiment

NP+DFTNP+LEMC total

NP+LEMC Ca2+

NP+LEMC Cs+

NaCl-CsCl mole fraction experiment

0 0.2 0.4 0.6 0.8 1

Mole fraction of Na+

470

480

490

500

510

520

Conducta

nce / p

S

Exp.

NP+DFTNP+LEMC

DFT: Gillespie et al. JPCB 109, 15598, BJ, 94, 1169, 2008; BJ, 97, 2212, 2009.

4 D. Boda. Monte Carlo simulation of electrolyte solutions in biology: In and out of equilibrium. Annual Reports in Computational Chemistry,Volume 10, Chapter 5, Pages 127–163, Elsevier, 2014.


Why do reduced models work?

Reduced models reproduce experimental data for device behavior properly.

Why is that?How can reduced models work while theyignore seemingly important molecular details(water, for example)?Our answer is that they work, because theyget those properties right that are importantfor device behavior.In the case of nanopores/channels, it is theaxial behavior of concentration profiles.When we build a reduced model by ignoring certain degrees of freedom, wecan ignore the „unimportant” ones and we must include the „important”ones in the model.How do you distinguish „important” and „uniportant” details is the art ofmultiscaling. Sometimes, it is obvious. Sometimes, it is not.


Nanopore sensor

Nanopore sensor of biomolecules – preliminary NP+LEMC results

Pore contains selective square-wellbinding sites for the analyte molecule(X+) that is present in the sample insmall (possibly trace) concentrationX+ ions continuously replace Na+ ionsas [X+] increases

-9 -6 -3 0 3 6 9

z / nm

-6

-4

-2

0

2

4

6

r / nm

SW binding site for X+

X+ (analyte)

- - - -

Na+ (current carrier)

Cl-

- - - - - -

- - - -

PET

- - - - - -

PET

σ

σ

Calibration curves: Currentreducement vs. [X+]Small [X+] can be detectedNoise/sign relation is important

1e-05 1e-04 1e-03 1e-02 1e-01

[X+] / [K

+]

0.4

0.5

0.6

0.7

0.8

0.9

1

I(X

+)

/ I(

X+=

0)

[K+] = 0.1 M

[K+] = 0.01 M

Possibility for multiscaling: connect reduced model of binding site to MD or QM simulationsDezső Boda (U. Pannonia, iASK, Hungary) Multiscale modeling of nanopores Cagliari, 2017.04.03-04. 27 / 28

Thanks

Thanks for your attention!


multiscale modeling of ion transport through nanopores ...ex + z αeΦ(r) poisson ... dezsőboda...

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