Steady-State Solution Of One-Dimensional Poisson-Nernst-Planck Model With Permanent Charge And Its Applications In Ion Channels

Electrodiffusion,the diffusion and migration of electric charge,plays a central role in a wide range of technology and science,such as physics,chemistry and biology.One of the most popular models for the transport of ions under the influence of electrochemical gradient is the Poisson-Nernst-Planck(PNP)model.The PNP model is a combination of Poisson equation,describing the electrostatic field produced by mobile ions and fixed charges,and NernstPlanck equations,describing the electrodiffusion of ions under the influence of ionic concentration gradient in the electrostatic field.The PNP model has been widely used in many fields,such as carrier transport in semiconductors,ion migration in electrolyte solutions,in particular in ion transport through ion channels in cells.Ion channels are proteins with holes embedded in membranes,which provide a major pathway for electrodiffusion of selected ions through biological membranes and for communications between cells and the external environment.In this way,ion channels control a large number of biological functions.Permanent charge are the major structural quantities and play the main role for functions of ion channels.The properties of ionic flows through ion channels depend further on other important physical parameters such as boundary concentrations and boundary potential.In this thesis,based on the simplified settings used in many biological experiments,we apply the geometric singular perturbation theory and dynamical system theory to one-dimensional steadystate PNP model and its solution.Furthermore,for ionic flows through ion channels involving three ion species(two cations with different valences and one anion),we examine the effects of permanent charges,channel geometry and boundary conditions on individual fluxes.This thesis consists of four chapters,the third and forth chapters are main work.In Chapter 1,we first briefly introduce the background and development of PNP model,then focus on the applications of PNP model in ion channel problems,and finally outline the main work of this thesis.In Chapter 2,we give some preliminary results,such as Fenichel Theorem and Exchange Lemma,which will be used in this thesis.In Chapter 3,we apply the geometric singular perturbation theory to the steady-state solution of the boundary value problem of one-dimensional PNP model with permanent charges.We first convert the boundary value problem to a connecting orbit problem and construct a singular orbit on the whole interval by matching the singular orbits(each consists of two boundary layers and one regular layer)over each sub-interval.Then by Exchange Lemma,we prove that there is a unique solution nearby the singular orbit we construct.At last,we obtain the first order approximate solution in small permanent charge of the governing system involving three ion species.In Chapter 4,we focus on the effects of small permanent charges on individual fluxes of three ion species with different valences via one-dimensional PNP model with permanent charges.We first recall the concept of flux ratio for permanent charge effects on ionic fluxes,and obtain the first order approximation of the flux ratio in small permanent charge.Then,we study how flux ratio depends on boundary conditions and channel geometry.Finally,for ionic flows through ion channels involving three ion species(two cations with different valences and one anion),we examine as systematically as possible behaviors of flux ratios,influenced by small permanent charge,boundary conditions and channel geometry,in two cases with special boundary conditions.