Abstract. In this work, we examine effects of permanent charges on ionic flows through ion channels via a quasi-one-dimensional classical Poisson-Nernst-Planck (PNP) model. The geometry of the three-dimensional channel is presented in this model to a certain extent, which is crucial for the study in this paper. Two ion species, one positively charged and one negatively charged, are considered with a simple profile of permanent charges: zeros at the two end regions and a constant Q 0 over the middle region. The classical PNP model can be viewed as a boundary value problem (BVP) of a singularly perturbed system. The singular orbit of the BVP depends on Q 0 in a regular way. Assuming |Q 0 | is small, a regular perturbation analysis is carried out for the singular orbit. Our analysis indicates that effects of permanent charges depend on a rich interplay between boundary conditions and the channel geometry. Furthermore, interesting common features are revealed: for Q 0 = 0, only an average quantity of the channel geometry plays a role; however, for Q 0 = 0, details of the channel geometry matter; in particular, to optimize effects of a permanent charge, the channel should have a short and narrow neck within which the permanent charge is confined. The latter is consistent with structures of typical ion channels.Key words. ionic flow, permanent charge, channel geometry AMS subject classifications. 34A26, 34B16, 34D15, 37D10, 92C35 DOI. 10.1137/140992527 Introduction.In this work, we analyze effects of permanent charges on ionic flows through ion channels, based on a quasi-one-dimensional classical PoissonNernst-Planck (PNP) model. The geometry of the three-dimensional channel is presented in this model to a certain extent, which is crucial for the study in this paper. We start with a brief discussion of the biological background of ion channel problems, a quasi-one-dimensional PNP model, and the main concern of our work in this paper.
We investigate Ussing's unidirectional fluxes and flux ratios of charged tracers motivated particularly by the insightful proposal of Hodgkin and Keynes on a relation between flux ratios and channel structure. Our study is based on analysis of quasi-one-dimensional Poisson-Nernst-Planck type models for ionic flows through membrane channels. This class of models includes the Poisson equation that determines the electrical potential from the charges present and is in that sense consistent. Ussing's flux ratios generally depend on all physical parameters involved in ionic flows, particularly, on bulk conditions and channel structures. Certain setups of ion channel experiments result in flux ratios that are universal in the sense that their values depend on bulk conditions but not on channel structures; other setups lead to flux ratios that are specific in the sense that their values depend on channel structures too. Universal flux ratios could serve some purposes better than specific flux ratios in some circumstances and worse in other circumstances. We focus on two treatments of tracer flux measurements that serve as estimators of important properties of ion channels. The first estimator determines the flux of the main ion species from measurements of the flux of its tracer. Our analysis suggests a better experimental design so that the flux ratio of the tracer flux and the main ion flux is universal. The second treatment of tracer fluxes concerns ratios of fluxes and experimental setups that try to determine some properties of channel structure. We analyze the two widely used experimental designs of estimating flux ratios and show that the most widely used method depends on the spatial distribution of permanent charge so this flux ratio is specific and thus allows estimation of (some of) the properties of that permanent charge, even with ideal ionic solutions. The work presented in this paper is a first step showing how measurements of fluxes and flux ratios can give important insights into channel structure and function. ). 1 Tracer c [t,o] 1 z 1 > 0 J [t,o] 1 Counter ion c [o] 2 z 2 < 0 J [o] 2 Concerning the flux ratio between J [t,i] 1 and J [t,o] 1 , we have the following formula. [t,o] 1 associated to the boundary condition (5.16) is, with s = L 1 /R 1 , J [t,i] 1 J [t,o] 1 , F 1 F 0 = J [i] 11 J [i] 10 − J [o] 11 J [o] 10
In this paper, we study the nonlinear one-dimensional periodic wave equation with x-dependent co-, 2) and the periodic conditions y(x, t + T ) = y(x, t), y t (x, t + T ) = y t (x, t). Such a model arises from the forced vibrations of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media. A main concept is the notion "weak solution" to be given in Section 2. For T is the rational multiple of π , we prove some important properties of the weak solution operator. Based on these properties, the existence and regularity of weak solutions are obtained.
This paper is devoted to the study of time-periodic solutions to the nonlinear one-dimensional wave equation with x-dependent coefficients (π, t). Such a model arises from the forced vibrations of a non-homogeneous string and the propagation of seismic waves in non-isotropic media. Our main concept is that of the 'weak solution'. For T , the rational multiple of π, we prove some important properties of the weak solution operator. Based on these properties, the existence and regularity of weak solutions are obtained.
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