# Development of an Ion-Channel Model-Framework for in-vitro assisted interpretation of current voltage relations

Project heads : Manuel Landstorfer, Jürgen Fuhrmann, Barbara Wagner

Cooperation partner: Robert S. Eisenberg (Rush Univ. Chicago)

Project staff : Christine Keller

## Summary

We develop a PDE based model framework to predict current-voltage relations of calcium channels for varying concentrations, voltages and channel elasticity to help interpret measured results. Theory and numerical approaches treat progressively more complex models covering mechano-sensitivity and selectivity.

## Methods

Ion channels control a large fraction of biological processes and are important targets for curing medication.

Like valves and transistors, they are devices’’ which allow to control macroscopic system behavior with small energetical input.

The behaviour of ion channels, even single ion channels, can be studied in laboratory conditions by measuring the current response to some time dependent voltage difference.

This signal is a unique signature of a channel and its environment and depends on various local and global quantities.

The interpretation of measured current voltage (IV) relations, and especially their variation with the structure of the channel protein and bulk ion concentrations is of great importance for biology, physiology and medicine.

Sketch of a deformable ion channel modeled as a multidomain heterostructure.

Theoretical investigations of ion channels use methods from quantum chemical approaches via molecular dynamics (MD) simulations to PDE based models. While MD simulations resolve an ion channel on the atomic scale, enabling detailed investigations of the channel protein, they lack predictability on IV relations and bulk concentration variations, of Ca$$^{2+}$$ concentrations from $$10^{-8}$$ to $$10^{-6}$$ M.

% Continuum based, molecular mean field theories essentially yield averages of trajectories of MD simulations . They treat the ion channel as a geometric domain without resolving all its atoms explicitly in space and allow to yield IV curves via numerical simulations.

The project aims at the development of a continuum based mathematical and numerical modeling framework from first principles of non-equilibrium thermodynamics.

## Literature

[1] J. Fuhrmann, “A numerical strategy for Nernst-Planck systems with solvation effect,” Fuel cells, vol. 16, no. 6, pp. 704–714, 2016. DOI: 10.1002/fuce.201500215.

[2] C. Cancès, C. Chainais-Hillairet, J. Fuhrmann, and B. Gaudeul, “A numerical-analysis-focused comparison of several finite volume schemes for a unipolar degenerate drift-diffusion model,” IMA Journal of Numerical Analysis, vol. 41, no. 1, pp. 271–314, 2021. DOI: 10.1093/imanum/draa002.

[3] J. Fuhrmann, C. Guhlke, A. Linke, C. Merdon, and R. Müller, “Models and numerical methods for electrolyte flows,” in Topics in applied analysis and optimisation, 2019, pp. 183–209. DOI: 10.1007/978-3-030-33116-0_8.

[4] M. Landstorfer, C. Guhlke, and W. Dreyer, “Theory and structure of the metal-electrolyte interface incorporating adsorption and solvation effects,” Electrochimica Acta, vol. 201, pp. 187–219, 2016. DOI: 10.1016/j.electacta.2016.03.013.

[5] M. Landstorfer, “Boundary conditions for electrochemical interfaces,” Journal of The Electrochemical Society, vol. 164, no. 11, p. E3671, 2017. DOI: 10.1149/2.0641711jes.

[6] W. DREYER, C. GUHLKE, M. LANDSTORFER, and R. MÜLLER, “New insights on the interfacial tension of electrochemical interfaces and the lippmann equation,” European Journal of Applied Mathematics, vol. 29, no. 4, pp. 708–753, 2018. DOI: 10.1017/S0956792517000341.

[7] G. L. Celora, M. G. Hennessy, A. Münch, B. Wagner, and S. L. Waters, “A kinetic model of a polyelectrolyte gel undergoing phase separation.” WIAS Preprint 2802, 2020. DOI: 10.20347/WIAS.PREPRINT.2802.

[8] G. L. Celora, M. G. Hennessy, Münch, A., B. Wagner, and S. L. Waters, “A kinetic model of a polyelectrolyte gel undergoing phase separation.” WIAS Preprint 2731, 2020. DOI: 10.20347/WIAS.PREPRINT.2731.

[9] J.-L. Liu and B. Eisenberg, “Molecular mean-field theory of ionic solutions: A poisson-nernst-planck-bikerman model,” Entropy, vol. 22, no. 5, 2020. DOI: 10.3390/e22050550.

[10] D. Boda, M. Valiskó, D. Henderson, B. Eisenberg, D. Gillespie, and W. Nonner, “Ionic selectivity in L-type calcium channels by electrostatics and hard-core repulsion,” Journal of General Physiology, vol. 133, no. 5, pp. 497–509, Apr. 2009. DOI: 10.1085/jgp.200910211.

[11] Y. Wang, C. Liu, P. Liu, and B. Eisenberg, “Field theory of reaction-diffusion: Law of mass action with an energetic variational approach,” Phys. Rev. E, vol. 102, no. 6, p. 062147, 2020. DOI: 10.1103/PhysRevE.102.062147.