In this paper, we study effects of permanent charges on ion flows through membrane channels via a quasi-one-dimensional classical Poisson-Nernst-Planck sys- tem. This system includes three ion species, two cations with different valences and one anion, and permanent charges with a simple structure, zeros at the two end re- gions and a constant over the middle region. For small permanent charges, our main goal is to analyze the effects of permanent charges on ionic flows, interacting with the boundary conditions and channel structure. Continuing from a previous work, we investigate the problem for a new case toward a more comprehensive understanding about effects of permanent charges on ionic fluxes.

The movement of ionic solutions is an essential part of biology and technology. Fluidics, from nano-to microfluidics, is a burgeoning area of technology which is all about the movement of ionic solutions, on various scales. Many cells, tissues, and organs of animals and plants depend on osmosis, as the movement of fluids is called in biology. Indeed, the movement of fluids through channel proteins (that have a hole down their middle) is fluidics on an atomic scale. Ionic fluids are complex fluids, with energy stored in many ways. Ionic fluid flow is driven by gradients of concentration, chemical and electrical potential, and hydrostatic pressure. In this paper, a series of sharp interface models are derived for ionic solution with permeable membranes. By using the energy variation method, the unknown flux and interface conditions are derived consistently. We start from the derivation the generic model for the general case that the density of solution varies with ionic solvent concentrations and membrane is deformable. Then the constant density and fix membrane cases are derived as special cases of the generic model.

Conformation changes control the function of many proteins and thus much of biology. But it is not always clear what conformation means: is it the distribution of mass? Is it the distribution of permanent charge, like that on acid and base side chains? Is it the distribution of dielectric polarization? Here we point out that one of the most important conformation changes in biology can be directly measured and the meaning of conformation is explored in simulations and theory. The conformation change that underlies the main signal of the nervous system produces a displacement current— NOT an ionic current—that has been measured. Macroscopic measurements of atomic scale currents are possible because total current (including displacement current) is everywhere exactly the same in a one dimensional series system like a voltage clamped nerve membrane, as implied by the mathematical properties of the Maxwell Ampere law and the Kirchhoff law it implies. We use multiscale models to show how the change of a single side chain is enough to modulate dielectric polarization and change the speed of opening of voltage dependent channels. The idea of conformation change is thus made concrete by experimental measurements, theory, and simulations.

In this note, the boundedness below of linear relation matrix $M_{C}=\left(\begin{array}{cc}A & C \\ 0 & B\end{array}\right) \in L R(H \oplus K)$ is considered, where $A \in C L R(H), B \in C L R(K), C \in B L R(K, H), H, K$ are separable Hilbert spaces. By suitable space decompositions, a necessary and sufficient condition for diagonal relations A,B is given so that $M_{C}$ is bounded below for some $C \in B L R(K, H)$. Besides, the characterization of $\sigma_{a p}\left(M_{C}\right)$ and $\sigma_{s u}\left(M_{C}\right)$ are obtained, and the relationship between $\sigma_{a p}\left(M_{0}\right)$ and $\sigma_{a p}\left(M_{C}\right)$ is further presented.

In this paper we obtain the boundedness of non-regular pseudo-differential operators with symbols in Besov spaces on matrix-weighted Besov-Triebel-Lizorkin spaces. These symbols include the classical Hormander classes.

We study the global well-posedness of the initial-value problem for the 2D Boussinesq-Navier-Stokes equations with dissipation given by an operator L that can be defined through both an integral kernel and a Fourier multiplier. When the operator L is represented by $ \frac{|\xi|}{a(|\xi|)}$ with a satisfying $ \lim _{|\xi| \rightarrow \infty} \frac{a(|\xi|)}{|\xi|^{\sigma}}=0$ for any σ>0, we obtain the global well-posedness. A special consequence is the global well-posedness of 2D Boussinesq-Navier-Stokes equations when the dissipation is logarithmically supercritical.