Mathematical Model Investigating the Effects of Neurostimulation Therapies on Neural Functioning: Comparing the Effects of Neuromodulation Techniques on Ion Channel Gating and Ionic Flux Using Finite Element Analysis

Neurostimulation therapies demonstrate success as a medical intervention for individuals with neurodegenerative diseases, such as Parkinson’s and Alzheimer’s disease. Despite promising results from these treatments, the influence of an electric current on ion concentrations and subsequent transmembrane voltage is unclear. This project focuses on developing a unique cellular-level mathematical model of neurostimulation to better understand its e↵ects on neuronal electrodynamics. The mathematical model presented here integrates the Poisson-Nernst-Planck system of PDEs and Hodgkin-Huxley based ODEs to model the e↵ects of this neurotherapy on transmembrane voltage, ion channel gating, and ionic mobility. This system is decoupled using the Gauss-Seidel method and then the equations are solved using the finite element method on a biologically-inspired discretized domain. Results demonstrate the influence of transcranial electrical stimulation on membrane voltage, ion channel gating, and transmembrane flux. Simulations also compare the e↵ects of two di↵erent types of neurostimulation (transcranial electrical stimulation and deep brain stimulation) showcasing cellular-level di↵erences resulting from these distinct forms of electrical therapy. Hopefully this work will ultimately help elucidate the principles by which neurostimulation alleviates disease symptoms

Electrokinetic Lattice Boltzmann solver coupled to Molecular Dynamics: application to polymer translocation

We develop a theoretical and computational approach to deal with systems that involve a disparate range of spatio-temporal scales, such as those comprised of colloidal particles or polymers moving in a fluidic molecular environment. Our approach is based on a multiscale modeling that combines the slow dynamics of the large particles with the fast dynamics of the solvent into a unique framework. The former is numerically solved via Molecular Dynamics and the latter via a multi-component Lattice Boltzmann. The two techniques are coupled together to allow for a seamless exchange of information between the descriptions. Being based on a kinetic multi-component description of the fluid species, the scheme is flexible in modeling charge flow within complex geometries and ranging from large to vanishing salt concentration. The details of the scheme are presented and the method is applied to the problem of translocation of a charged polymer through a nanopores. In the end, we discuss the advantages and complexities of the approach

Testing the Applicability of Nernst-Planck Theory in Ion Channels: Comparisons with Brownian Dynamics Simulations

The macroscopic Nernst-Planck (NP) theory has often been used for predicting ion channel currents in recent years, but the validity of this theory at the microscopic scale has not been tested. In this study we systematically tested the ability of the NP theory to accurately predict channel currents by combining and comparing the results with those of Brownian dynamics (BD) simulations. To thoroughly test the theory in a range of situations, calculations were made in a series of simplified cylindrical channels with radii ranging from 3 to 15 Å, in a more complex ‘catenary’ channel, and in a realistic model of the mechanosensitive channel MscS. The extensive tests indicate that the NP equation is applicable in narrow ion channels provided that accurate concentrations and potentials can be input as the currents obtained from the combination of BD and NP match well with those obtained directly from BD simulations, although some discrepancies are seen when the ion concentrations are not radially uniform. This finding opens a door to utilising the results of microscopic simulations in continuum theory, something that is likely to be useful in the investigation of a range of biophysical and nano-scale applications and should stimulate further studies in this direction

Interacting Ions in Biophysics: Real is not Ideal

Ions in water are important in biology, from molecules to organs. Classically, ions in water are treated as ideal noninteracting particles in a perfect gas. Excess free energy of ion was zero. Mathematics was not available to deal consistently with flows, or interactions with ions or boundaries. Non-classical approaches are needed because ions in biological conditions flow and interact. The concentration gradient of one ion can drive the flow of another, even in a bulk solution. A variational multiscale approach is needed to deal with interactions and flow. The recently developed energetic variational approach to dissipative systems allows mathematically consistent treatment of bio-ions Na, K, Ca and Cl as they interact and flow. Interactions produce large excess free energy that dominate the properties of the high concentration of ions in and near protein active sites, channels, and nucleic acids: the number density of ions is often more than 10 M. Ions in such crowded quarters interact strongly with each other as well as with the surrounding protein. Non-ideal behavior has classically been ascribed to allosteric interactions mediated by protein conformation changes. Ion-ion interactions present in crowded solutions--independent of conformation changes of proteins--are likely to change interpretations of allosteric phenomena. Computation of all atoms is a popular alternative to the multiscale approach. Such computations involve formidable challenges. Biological systems exist on very different scales from atomic motion. Biological systems exist in ionic mixtures (extracellular/intracellular solutions), and usually involve flow and trace concentrations of messenger ions (e.g., 10-7 M Ca2+). Energetic variational methods can deal with these characteristic properties of biological systems while we await the maturation and calibration of all atom simulations of ionic mixtures and divalents

Rectification properties of conically shaped nanopores: consequences of miniaturization

Nanopores attracted a great deal of scientific interest as templates for biological sensors as well as model systems to understand transport phenomena at the nanoscale. The experimental and theoretical analysis of nanopores has been so far focused on understanding the effect of the pore opening diameter on ionic transport. In this article we present systematic studies on the dependence of ion transport properties on the pore length. Particular attention was given to the effect of ion current rectification exhibited for conically shaped nanopores with homogeneous surface charges. We found that reducing the length of conically shaped nanopores significantly lowered their ability to rectify ion current. However, rectification properties of short pores can be enhanced by tailoring the surface charge and the shape of the narrow opening. Furthermore we analyze the relationship of the rectification behavior and ion selectivity for different pore lengths. All simulations were performed using MsSimPore, a software package for solving the Poisson-Nernst-Planck (PNP) equations. It is based on a novel finite element solver and allows for simulations up to surface charge densities of -2 e/nm^2. MsSimPore is based on 1D reduction of the PNP model, but allows for a direct treatment of the pore with bulk electrolyte reservoirs, a feature which was previously used in higher dimensional models only. MsSimPore includes these reservoirs in the calculations; a property especially important for short pores, where the ionic concentrations and the electric potential vary strongly inside the pore as well as in the regions next to pore entrance