Formation of clumps and patches in self-aggregation of finite size particles

New model equations are derived for dynamics of self-aggregation of finite-size particles. Differences from standard Debye-Huckel and Keller-Segel models are: a) the mobility $\mu$ of particles depends on the locally-averaged particle density and b) linear diffusion acts on that locally-averaged particle density. The cases both with and without diffusion are considered here. Surprisingly, these simple modifications of standard models allow progress in the analytical description of evolution as well as the complete analysis of stationary states. When $\mu$ remains positive, the evolution of collapsed states in our model reduces exactly to finite-dimensional dynamics of interacting particle clumps. Simulations show these collapsed (clumped) states emerging from smooth initial conditions, even in one spatial dimension. If $\mu$ vanishes for some averaged density, the evolution leads to spontaneous formation of \emph{jammed patches} (weak solution with density having compact support). Simulations confirm that a combination of these patches forms the final state for the system.Comment: 38 pages, 8 figures; submitted to Physica

Water alignment, dipolar interactions, and multiple proton occupancy during water-wire proton transport

A discrete multistate kinetic model for water-wire proton transport is constructed and analyzed using Monte-Carlo simulations. The model allows for each water molecule to be in one of three states: oxygen lone pairs pointing leftward, pointing rightward, or protonated (H$_{3}$O$^{+}$). Specific rules for transitions among these states are defined as protons hop across successive water oxygens. We then extend the model to include water-channel interactions that preferentially align the water dipoles, nearest-neighbor dipolar coupling interactions, and coulombic repulsion. Extensive Monte-Carlo simulations were performed and the observed qualitative physical behaviors discussed. We find the parameters that allow the model to exhibit superlinear and sublinear current-voltage relationships and show why alignment fields, whether generated by interactions with the pore interior or by membrane potentials {\it always} decrease the proton current. The simulations also reveal a ``lubrication'' mechanism that suppresses water dipole interactions when the channel is multiply occupied by protons. This effect can account for an observed sublinear-to-superlinear transition in the current-voltage relationship

(In)validity of the constant field and constant currents assumptions in theories of ion transport.

Constant electric fields and constant ion currents are often considered in theories of ion transport. Therefore, it is important to understand the validity of these helpful concepts. The constant field assumption requires that the charge density of permeant ions and flexible polar groups is virtually voltage independent. We present analytic relations that indicate the conditions under which the constant field approximation applies. Barrier models are frequently fitted to experimental current-voltage curves to describe ion transport. These models are based on three fundamental characteristics: a constant electric field, negligible concerted motions of ions inside the channel (an ion can enter only an empty site), and concentration-independent energy profiles. An analysis of those fundamental assumptions of barrier models shows that those approximations require large barriers because the electrostatic interaction is strong and has a long range. In the constant currents assumption, the current of each permeating ion species is considered to be constant throughout the channel; thus ion pairing is explicitly ignored. In inhomogeneous steady-state systems, the association rate constant determines the strength of ion pairing. Among permeable ions, however, the ion association rate constants are not small, according to modern diffusion-limited reaction rate theories. A mathematical formulation of a constant currents condition indicates that ion pairing very likely has an effect but does not dominate ion transport

Integral weak diffusion and diffusion approximations applied to ion transport through biological ion channels

In this article a theory is presented to calculate integral properties of biological ion channels (like currentvoltage and conductance-concentration relations). The qualitative form of these relations predicted by the theory agrees well with data measured in experiments. For instance, the saturation of the channel conductance with increasing external ion concentration is predicted for a class of ion channels (as, for instance, found for the gramicidin A, acetylcholine receptors, NMDA, and sarcoplasmic reticulum channels). In contrast to commonly used approaches such as the Eyring rate theory, this method is directly related to physical parameters of the ion channel such as the channel length and diameter, dielectric constant, ionic mobility, and minimal ionic concentration inside the channel. The theory starts from Nernst-Planck and Poisson equations. Using the method of phase trajectory (as proposed by Schottky) and the regional approximation, rather general expressions can be derived for integral channel quantities in the drift limit (|Vl > kBT/eo) in the presence of multiple ionic species. The theory predicts two typical types of conductance-concentration relations found experimentally: a monotone saturating conductance and a maximum in the conductance. The realized type of relation depends on the minimal ionic concentration inside the channel. In the present form the theory is restricted to narrow ion channels where the length exceeds its diameter. The ions are assumed to behave like structureless point charges at not too high ionic concentration

Integral weak diffusion and diffusion approximations applied to ion transport through biological ion channels

In this article a theory is presented to calculate integral properties of biological ion channels (like currentvoltage and conductance-concentration relations). The qualitative form of these relations predicted by the theory agrees well with data measured in experiments. For instance, the saturation of the channel conductance with increasing external ion concentration is predicted for a class of ion channels (as, for instance, found for the gramicidin A, acetylcholine receptors, NMDA, and sarcoplasmic reticulum channels). In contrast to commonly used approaches such as the Eyring rate theory, this method is directly related to physical parameters of the ion channel such as the channel length and diameter, dielectric constant, ionic mobility, and minimal ionic concentration inside the channel. The theory starts from Nernst-Planck and Poisson equations. Using the method of phase trajectory (as proposed by Schottky) and the regional approximation, rather general expressions can be derived for integral channel quantities in the drift limit (|Vl > kBT/eo) in the presence of multiple ionic species. The theory predicts two typical types of conductance-concentration relations found experimentally: a monotone saturating conductance and a maximum in the conductance. The realized type of relation depends on the minimal ionic concentration inside the channel. In the present form the theory is restricted to narrow ion channels where the length exceeds its diameter. The ions are assumed to behave like structureless point charges at not too high ionic concentration