Coupling the Nernst-Planck Equation to Monte Carlo Simulations to Explain Selectivity in Ca

Channels and in Charged Pores in Plastic

Dezső Boda

1 Department of Physical Chemistry, University of Pannonia, Veszprem, Hungary

2 Department of Molecular Biophysics and Physiology, Rush University Medical Center, Chicago, IL

A typical selectivity experiment

-log10

[CaCl2] added to 32 mM NaCl

1 M Ca2+ blocks Na+ current in the L-type Ca channel (Almers et al.)

Minimal structural information we know about Ca channels:

EEEE amino acids in the selectivity filter

-1e -1e

EEEE (-4e) We do not know where are they: no structure.

-1e-1e

Assumed cartoon-theory of selectivity and permeation in Ca channels

Sather and McCleskey, 2003, Annu. Rev. Physiol., 65: 133.

The charge/space competition (CSC) mechanism

Nonner et al. 2000, Biophys. J. 79: 1976.

The selectivity filter is small and crowded with the side chains of the glutamates. Cations are attracted to the filter by the negative charges of the carboxyl

oxygens. This decreases electrostatic energy (U). In the meantime, the filter becomes more crowded. It is difficult for the cations to find space. Steric repulsion (volume exclusion) effects become more important. This decreases entropy. Which ion enters the filter with higher probability? The one for which the free energy is smaller:

F = U -TS

The ion which can decrease U without increasing -TS too much will win the competition.

The charge/space competition (CSC) mechanism

F = U -TSExamples Ca2+ provides twice the charge of Na+ (this decreases U) while occupying about the same space (-TS is unchanged). Ions with higher charge win. Smaller ions find space easier than larger ions (of the same charge). U is unchanged while -TS is smaller for smaller ions. Smaller ions win.

In this picture, the balance of electrostatic attraction and entropic repulsion determines selectivity.

For a more detailed analysis: see the next talk by Dirk Gillespie

The problem is studied in equilibrium

The system in the selectivity filter is in equilibrium with the bath where the concentrations of the competing ionic species (eg. Na+ and Ca2+) are changed.

The question: which ion enters the selectivity filter with higher probability as the bath [CaCl

2] is

gradually increased?

The answer requires fairly correct treatment of both the filter and the bath.

Challenges for the calculation

The electrolyte should be simulated at micromolar concentrations.

The equilibration of the small (crowded) selectivity filter with the large (dilute) bath.

Energy calculation: the accurate solution of Poisson's equation is required in every simulation step.

What model and what method can be used to cope with these challenges?

MODEL

Model of solvent

We have to simulate trace concentrations as low as 10-10 M.

The solvent has to be simulated as a dielectric continuum

Model of the protein: the doughnut

Model of the selectivity filter: flexible confinement © Eduardo Rios

Structural ions are confined to the filter but free to move inside it.

D:

E:

K:

The mobile structural ions provide a flexible environment for the passing ions

Effective channel diameter is smaller than the diameter of the confining cylinder

r

z

Oxygen ions form a `binding pocket' in the selectivity filter

lg[c(r,z)] of oxygenspore wall

pore wall

entr

ance

entr

ance

Primitive model of the electrolyte

Ions: hard spheres with a point charge in the center (polarization charge around ions is ignored)

Solvent: continuum dielectrics

Engineering variables of the reduced model

Composition of structural ions in the selectivity filter (EEEE for Ca channel)

Radius of the selectivity filter Dielectric constant of the protein

These parameters can be changed by protein structure, namely, by genetic code.

Small ε and R favor Ca2+-selectivity: Boda, D; Valisko, M; Eisenberg, B; Nonner, W; Henderson, D; Gillespie, G, The effect

of protein dielectric coefficient on the ionic selectivity of a calcium channel, J CHEM PHYS, 125 (3): 034901 JUL 21 2006

Boda, D; Valisko, M; Eisenberg, B; Nonner, W; Henderson, D; Gillespie, G, The combined effect of pore radius and protein dielectric coefficient on the selectivity of a calcium channel, PHYS REV LETT, 98 (16): 168102 APR 17 2007

Values ε=10 and R=3.5A provide and optimal Ca2+-selectivity and they have been fixed.

METHOD

Method: equilibrium Grand Canonical Monte Carlo simulation

We have to simulate trace concentrations as low as 10-10 M.

We use the grand canonical ensemble.

Method: equilibrium Grand Canonical Monte Carlo simulation

Efficient sampling of channel vs. bath is needed.

1. Preference sampling and2. Targeted GCMC insertions/deletions are applied

Biased ion exchange between channel and bath We prefer ion moves

between channel and bath: if the selected ion is in the channel, we move it into the bath and vice versa.

It is a nonuniform, biased sampling, so it has to be unbiased in the acceptance probability:

Efficiency of the preference sampling

Improved GCMC sampling

Original method:

ions go from bulk to channel in two steps

GC

biased jump

GC

Ions are inserted directly into the channel.

External bulkSimulation box

Improved GCMC sampling

In the preferential/targeted GCMC steps we insert/delete the cation into the channel/filter.

This GCMC step considerably accelerates the convergence of the number of ions in the filter

Beware the logarithmic scale of abscissa!

Energy calculation The differential equation

with the boundary condition

at the dielectric boundary is transformed into an integral equation.

Its variable is the induced charge instead of the electric field.

It self-contains the boundary condition.

Energy calculation The integral equation:

The solution is performed numerically The surface is divided into surface elements

(boundary element methods). A matrix equation is obtained, where the matrix

can be precalculated at the beginning of the simulation.

Vector c (the electric field at the surface) changes as ions move, and the induced charge (h) can be computed from the matrix equation.

Outputs of simulations:Equilibrium concentration profiles: 1. The baths on the two sides of the channel are equivalent (no chemical potential gradient)2. No voltage (no potential gradient)

Outputs of experiments: current

How can we relate our simulation results to experimental data?

(Gillespie, D; Boda, D, The anomalous mole fraction effect in calcium channels: A measure of preferential selectivity, BIOPHYS J, Published ahead of print on May 30, 2008 as doi:10.1529/biophysj.107.127977.)

The resistor-in-series model

Current carried by different ions flows though resistors connected in parallel. Different regions of the channel along the ionic pathway corresponds to resistors in series. Bath: low resistance region = “the wire” Channel: high resistance region, where flux is limited Inspiration: Nonner, Chen, and Eisenberg. 1998. Biophys. J. 74: 2327.

Nernst-Planck equation:

0th assumption: the Nernst-Planck treatment is valid for the Ca channel.

Ji – particle flux

Di(x) – diffusion coefficient profile

A(x) – cross section

i(x) – chemical potential profile

V(x) – potential profile

zi – ionic valence

T – temperature, k – Boltzmann constant, e – elementary charge

1st assumption: the two baths are symmetrical: no concentration difference, chemical potentials in the two baths are the same

The potential gradient is the only driving force.Let us integrate from bath1 to bath2:

2nd assumption: the current-voltage relation is linear – conductance of the channel is constant:

3rd assumption: flux is limited in the bottleneck, in the selectivity filter:we integrate only over the filter, where D

i and A are constants

This approach decouples - the number of available charge carriers (the integral of reciprocal of concentration profiles) and - their mobility (diffusion coefficient).

Concentration profiles come from GCMC simulations, diffusion coefficients are adjustable parameters of this NP model.

Because we compute normalized currents, there is only one adjustable parameter: the ratio of diffusion coefficients of competing ions

RESULTS

Concentration profiles for Ca2+ vs. Na+ competition in

a Ca channel

CaCl2 is added to 30 mM

NaCl

Ca2+ quickly replaces Na+ in the filter

Occupancy results for the Ca channel

1μM Ca2+ squeezes half of the Na+ out from the filter

Applying the integrated Nernst-Planck equation on the concentration profiles given by the simulation, experimental data are reproduced.

Half Na+ conducts half the current.

Ca2+ does not conduct, because of the depletion

zones (low concentration – high resistance zone).

Ca2+ starts to conduct when there is enough of it (>1mM) and the depletion

zones vanish.

Depletion zone

Size selectivity: results for Ba2+ vs. Ca2+ AMFE(Friel and Tsien, 1989, PNAS, 86: 5207; Yue and Marban, 1990, J. Gen. Physiol. 95: 911.)

Competition of trivalent, divalent, and monovalent ions(Experiment: Babich, Reeves, and Shirokov, 2007, J. Gen. Physiol. 129: 461.)

AMFE in a wide pore

double-conical pore etched in plastic

diameter in the bottleneck is 50 A

its wall is negatively charged (-e/10x10A)

Gillespie, D; Boda, D; He, Y; Apel, P; Siwy, ZS, Synthetic nanopores as a test case for ion channel theories: The anomalous mole fraction effect without single filing, BIOPHYS J, Published ahead of print on April 4, 2008 as doi:10.1529/biophysj.107.127985.

Monovalents against Ca2+ AMFEExperiment: Siwy et al.

Lines: our GCMC+NP calculationsSymbols: experiments of Siwy et al.

Radial concentration profiles in the center of the poreCa2+ is favored

The pore center is Ca2+-selective

The pore selectively adsorbs Ca2+ at its highly charged wall.

There is no single-filing in this wide pore. AMFE is still found.

This proves that the textbook-assumption that the presence of AMFE indicates single-filing is wrong.

We used the same equation in the case of Ca channels and the wide pore: the mechanism of AMFE is probably the same in both cases.

MUTATIONS

Mutation of the DEKA Na channel into a Ca channel

The Heinemann-experiment in computer: mutating K into E the Na channel turns into a Ca channel

Mutation of the DEKA Na channel into a Ca channel

The Heinemann-experiment in computer: mutating K into E the Na channel turns into a Ca channel

Heinemann et al., 1992, Nature, 356, 441.

Acknowledgment

Bob Eisenberg, Rush

Dirk Gillespie, Rush

Wolfgang Nonner, Miami

Doug Henderson, Provo

Mónika Valiskó, VeszprémTHANKS!!!