Multiple Scales in the Simulation of Ion Channels and Proteins

Dynamics accelerate the kinetics of ion diffusion through channels: Continuous-time random walk models beyond the mean field approximation

Ion channels are proteins that play a significant role in physiological processes, including neuronal excitability and signal transduction. However, the precise mechanisms by which these proteins facilitate ion diffusion through cell membranes are not well understood. This is because experimental techniques to characterize ion channel activity operate on a time scale too large to understand the role of the various protein conformations on diffusion. Meanwhile, computational approaches operate on a time scale too short to rationalize the observed behavior at the microscopic scale. In this paper, we present a continuous-time random walk model that aims to bridge the scales between the atomistic models of ion channels and the experimental measurement of their conductance. We show how diffusion slows down in complex systems by using 3D lattices that map out the pore geometry of two channels: Nav1.7 and gramicidin. We also introduce spatial and dynamic site disorder to account for system heterogeneity beyond the mean field approximation. Computed diffusion coefficients show that an increase in spatial disorder slows down diffusion kinetics, while dynamic disorder has the opposite effect. Our results imply that microscopic or phenomenological models based on the potential of mean force data overlook the functional importance of protein dynamics on ion diffusion through channels.

Setting Boundaries for Statistical Mechanics

Statistical mechanics has grown without bounds in space. Statistical mechanics of noninteracting point particles in an unbounded perfect gas is widely used to describe liquids like concentrated salt solutions of life and electrochemical technology, including batteries. Liquids are filled with interacting molecules. A perfect gas is a poor model of a liquid. Statistical mechanics without spatial bounds is impossible as well as imperfect, if molecules interact as charged particles, as nearly all atoms do. The behavior of charged particles is not defined until boundary structures and values are defined because charges are governed by Maxwell's partial differential equations. Partial differential equations require boundary structures and conditions. Boundary conditions cannot be defined uniquely 'at infinity' because the limiting process that defines 'infinity' includes such a wide variety of structures and behaviors, from elongated ellipses to circles, from light waves that never decay, to dipolar fields that decay steeply, to Coulomb fields that hardly decay at all. Boundaries and boundary conditions needed to describe matter are not prominent in classical statistical mechanics. Statistical mechanics of bounded systems is described in the EnVarA system of variational mechanics developed by Chun Liu, more than anyone else. EnVarA treatment does not yet include Maxwell equations.

Marf-mediated mitochondrial fusion is imperative for the development and functioning of indirect flight muscles (IFMs) in drosophila

Dynamic changes in mitochondrial shape and size are vital for mitochondrial health and for tissue development and function. Adult Drosophila indirect flight muscles contain densely packed mitochondria. We show here that mitochondrial fusion is critical during early muscle development (in pupa) and that silencing of the outer mitochondrial membrane fusion gene, Marf, in muscles results in smaller mitochondria that are functionally defective. This leads to abnormal muscle development resulting in muscle dysfunction in adult flies. However, post-developmental silencing of Marf has no obvious effects on mitochondrial and muscle phenotype in adult flies, indicating the importance of mitochondrial fusion during early muscle development.

Investigating ion transport inside the pentameric ion channel encoded in COVID-19 E protein

Ion flow inside an ion channel can be described through continuum based Born-Poisson-Nernst-Planck (BPNP) equations in conjunction with the Lennard-Jones potential. Keeping in mind the ongoing pandemic, in this study, an attempt has been made to understand the selectivity and the current voltage relation of the COVID-19 E protein pentameric ion channel. Two ionic species, namely Na^{+} and Cl^{-}, have been considered here. E protein is one of the smallest structural protein which is embedded in the outer membrane of the virus. Once the virus is inside the host cell, this protein is expressed abundantly and is responsible for activities such as replication and budding of the virus. In the literature, we can find a few experimental studies focusing on understanding the activity of the channel formed by E proteins of different viruses. Here, we attempt the same study for the COVID-19 E protein ion channel through mathematical modeling. The channel geometry is calculated from the protein data bank file which was provided by NARLabs, Taiwan, using the hole program. Further, it was used to obtain the charge distribution using the pdbtopqr online program. The immersed boundary-lattice Boltzmann method (IB-LBM) has been implemented to numerically solve the system of equations in the channel generated by the protein data bank file. Further, an in-house code which operates on multiple GPUs and uses the cuda platform has been developed to achieve the goal of performing the current investigation.

Molecular Mean-Field Theory of Ionic Solutions: A Poisson-Nernst-Planck-Bikerman Model

We have developed a molecular mean-field theory—fourth-order Poisson–Nernst–Planck–Bikerman theory—for modeling ionic and water flows in biological ion channels by treating ions and water molecules of any volume and shape with interstitial voids, polarization of water, and ion-ion and ion-water correlations. The theory can also be used to study thermodynamic and electrokinetic properties of electrolyte solutions in batteries, fuel cells, nanopores, porous media including cement, geothermal brines, the oceanic system, etc. The theory can compute electric and steric energies from all atoms in a protein and all ions and water molecules in a channel pore while keeping electrolyte solutions in the extra- and intracellular baths as a continuum dielectric medium with complex properties that mimic experimental data. The theory has been verified with experiments and molecular dynamics data from the gramicidin A channel, L-type calcium channel, potassium channel, and sodium/calcium exchanger with real structures from the Protein Data Bank. It was also verified with the experimental or Monte Carlo data of electric double-layer differential capacitance and ion activities in aqueous electrolyte solutions. We give an in-depth review of the literature about the most novel properties of the theory, namely Fermi distributions of water and ions as classical particles with excluded volumes and dynamic correlations that depend on salt concentration, composition, temperature, pressure, far-field boundary conditions etc. in a complex and complicated way as reported in a wide range of experiments. The dynamic correlations are self-consistent output functions from a fourth-order differential operator that describes ion-ion and ion-water correlations, the dielectric response (permittivity) of ionic solutions, and the polarization of water molecules with a single correlation length parameter.

Solution of Ion Channel Flow Using Immersed Boundary-Lattice Boltzmann Methods

Poisson-Nernst-Planck (PNP) model has been extensively used for the study of channel flow under the influence of electrochemical gradients. PNP theory is a continuum description of ion flow where ionic distributions are described in terms of concentrations. Nonionic interparticle interactions are not considered in this theory as in continuum framework, their impact on the solution is minimal. This theory holds true for dilute flows or flows where channel radius is significantly larger than ion radius. However, for ion channel flows, where channel dimensions and ionic radius are of similar magnitude, nonionic interactions, particularly related to the size of the ions (steric effect), play an important role in defining the selectivity of the channel, concentration distribution of ionic species, and current across the channel, etc. To account for the effect of size of ions, several modifications to PNP equations have been proposed. One such approach is the introduction of Lennard-Jones potential to the energy variational formulation of PNP system. This study focuses on understanding the role of steric effect on flow properties. To discretize the system, Lattice Boltzmann method has been used. The system is defined by modified PNP equations where the steric effect is described by Lennard-Jones potential. In addition, boundary conditions for the complex channel geometry have been treated using immersed boundary method.

Application of a bipolar nanopore as a sensor: rectification as an additional device function

We model and simulate a nanopore sensor that selectively binds analyte ions. This binding leads to the modulation of the local concentrations of the ions of the background electrolyte (KCl), and, thus, to the modulation of the ionic current flowing through the pore. The nanopore's wall has a bipolar charge pattern with a larger positive buffer region determining the anions as the main charge carriers and a smaller negative binding region containing binding sites. This charge pattern proved to be an appropriate one as shown by a previous comparative study of varying charge patterns (Mádai et al. J. Mol. Liq., 2019, 283, 391-398.). Binding of the positive analyte ions attracts more anions in the pore thus increasing the current. The asymmetric nature of the pore results in an additional device function, rectification. Our model, therefore, is a dual response device. Using a reduced model of the nanopore studied by a hybrid computer simulation method (Local Equilibrium Monte Carlo coupled with the Nernst-Planck equation) we show that we can create a sensor whose underlying mechanisms are based on the changes in the local electric field as a response to changing thermodynamic conditions. The change in the electric field results in changes in the local ionic concentrations (depletion zones), and, thus, changes in ionic currents.

Multiscale analysis of the effect of surface charge pattern on a nanopore's rectification and selectivity properties: From all-atom model to Poisson-Nernst-Planck

We report a multiscale modeling study for charged cylindrical nanopores using three modeling levels that include (1) an all-atom explicit-water model studied with molecular dynamics, and reduced models with implicit water containing (2) hard-sphere ions studied with the Local Equilibrium Monte Carlo simulation method (computing ionic correlations accurately), and (3) point ions studied with Poisson-Nernst-Planck theory (mean-field approximation). We show that reduced models are able to reproduce device functions (rectification and selectivity) for a wide variety of charge patterns, that is, reduced models are useful in understanding the mesoscale physics of the device (i.e., how the current is produced). We also analyze the relationship of the reduced implicit-water models with the explicit-water model and show that diffusion coefficients in the reduced models can be used as adjustable parameters with which the results of the explicit- and implicit-water models can be related. We find that the values of the diffusion coefficients are sensitive to the net charge of the pore but are relatively transferable to different voltages and charge patterns with the same total charge.

Golden aspect ratio for ion transport simulation in nanopores

Access resistance indicates how well current carriers from a bulk medium can converge to a pore or opening and is an important concept in nanofluidic devices and in cell physiology. In simplified scenarios, when the bulk dimensions are infinite in all directions, it depends only on the resistivity and pore radius. These conditions are not valid in all-atom molecular dynamics simulations of transport, due to the computational cost of large simulation cells, and can even break down in micro- and nanoscale systems due to strong confinement. Here, we examine a scaling theory for the access resistance that predicts a special simulation cell aspect ratio-the golden aspect ratio-where finite-size effects are eliminated. Using both continuum and all-atom simulations, we demonstrate that this golden aspect ratio exists and that it takes on a universal value in linear response and moderate concentrations. Outside of linear response, it gains an apparent dependence on characteristics of the transport scenario (concentration, voltages, etc.) for small simulation cells, but this dependence vanishes at larger length scales. These results will enable the use of all-atom molecular dynamics simulations to study contextual properties of access resistance-its dependence on protein and molecular-scale fluctuations, the presence of charges, and other functional groups-and yield the opportunity to quantitatively compare computed and measured resistances.

Numerical methods for a Poisson-Nernst-Planck-Fermi model of biological ion channels

Numerical methods are proposed for an advanced Poisson-Nernst-Planck-Fermi (PNPF) model for studying ion transport through biological ion channels. PNPF contains many more correlations than most models and simulations of channels, because it includes water and calculates dielectric properties consistently as outputs. This model accounts for the steric effect of ions and water molecules with different sizes and interstitial voids, the correlation effect of crowded ions with different valences, and the screening effect of polarized water molecules in an inhomogeneous aqueous electrolyte. The steric energy is shown to be comparable to the electrical energy under physiological conditions, demonstrating the crucial role of the excluded volume of particles and the voids in the natural function of channel proteins. Water is shown to play a critical role in both correlation and steric effects in the model. We extend the classical Scharfetter-Gummel (SG) method for semiconductor devices to include the steric potential for ion channels, which is a fundamental physical property not present in semiconductors. Together with a simplified matched interface and boundary (SMIB) method for treating molecular surfaces and singular charges of channel proteins, the extended SG method is shown to exhibit important features in flow simulations such as optimal convergence, efficient nonlinear iterations, and physical conservation. The generalized SG stability condition shows why the standard discretization (without SG exponential fitting) of NP equations may fail and that divalent Ca(2+) may cause more unstable discrete Ca(2+) fluxes than that of monovalent Na(+). Two different methods-called the SMIB and multiscale methods-are proposed for two different types of channels, namely, the gramicidin A channel and an L-type calcium channel, depending on whether water is allowed to pass through the channel. Numerical methods are first validated with constructed models whose exact solutions are known. The experimental data of both channels are then used to verify and explain novel features of PNPF as compared with previous PNP models. The PNPF currents are in accord with the experimental I-V (V for applied voltages) data of the gramicidin A channel and I-C (C for bath concentrations) data of the calcium channel with 10(-8)-fold bath concentrations that pose severe challenges in theoretical simulations.

Interacting ions in biophysics: real is not ideal

Ions in water are important throughout biology, from molecules to organs. Classically, ions in water were treated as ideal noninteracting particles in a perfect gas. Excess free energy of each ion was zero. Mathematics was not available to deal consistently with flows, or interactions with other ions or boundaries. Nonclassical 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 the bio-ions Na(+), K(+), Ca(2+), 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, ion channels, and nucleic acids: the number density of ions is often >10 M. Ions in such crowded quarters interact strongly with each other as well as with the surrounding protein. Nonideal behavior found in many experiments has classically been ascribed to allosteric interactions mediated by the protein and its conformation changes. The ion-ion interactions present in crowded solutions-independent of conformation changes of the protein-are likely to change the interpretation of many 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 (like extracellular and intracellular solutions), and usually involve flow and trace concentrations of messenger ions (e.g., 10(-7) M Ca(2+)). Energetic variational methods can deal with these characteristic properties of biological systems as we await the maturation and calibration of all-atom simulations of ionic mixtures and divalents.

Ionic Interactions Are Everywhere

Ionic solutions are dominated by interactions because they must be electrically neutral, but classical theory assumes no interactions. Biological solutions are rather like seawater, concentrated enough so that the diameter of ions also produces important interactions. In my view, the theory of complex fluids is needed to deal with the interacting reality of biological solutions.

Ionizable side chains at catalytic active sites of enzymes

Catalytic active sites of enzymes of known structure can be well defined by a modern program of computational geometry. The CASTp program was used to define and measure the volume of the catalytic active sites of 573 enzymes in the Catalytic Site Atlas database. The active sites are identified as catalytic because the amino acids they contain are known to participate in the chemical reaction catalyzed by the enzyme. Acid and base side chains are reliable markers of catalytic active sites. The catalytic active sites have 4 acid and 5 base side chains, in an average volume of 1,072 Å(3). The number density of acid side chains is 8.3 M (in chemical units); the number density of basic side chains is 10.6 M. The catalytic active site of these enzymes is an unusual electrostatic and steric environment in which side chains and reactants are crowded together in a mixture more like an ionic liquid than an ideal infinitely dilute solution. The electrostatics and crowding of reactants and side chains seems likely to be important for catalytic function. In three types of analogous ion channels, simulation of crowded charges accounts for the main properties of selectivity measured in a wide range of solutions and concentrations. It seems wise to use mathematics designed to study interacting complex fluids when making models of the catalytic active sites of enzymes.