Volume 5, Issue 1

In this review paper we discuss fatgraphs as a conceptual framework for RNA structures. We discuss various notions of coarse-grained RNA structures and relate them to fatgraphs.We motivate and discuss the main intuition behind the fatgraph model and showcase its applicability to canonical as well as noncanonical base pairs. Recent discoveries regarding novel recursions of pseudoknotted (pk) configurations as well as their translation into context-free grammars for pk-structures are discussed. This is shown to allow for extending the concept of partition functions of sequences w.r.t. a fixed structure having non-crossing arcs to pk-structures. We discuss minimum free energy folding of pk-structures and combine these above results outlining how to obtain an inverse folding algorithm for PK structures.

Knot polynomials have been used to detect and classify knots in biomolecules. Computation of knot polynomials in DNA and protein molecules have revealed the existence of knotted structures, and provided important insight into their topological structures. However, conventional knot polynomials are not well suited to study RNA molecules, as RNA structures are determined by stem regions which are not taken into account in conventional knot polynomials. In this study, we develop a new class of knot polynomials specifically designed to study RNA molecules, which considers stem regions. We demonstrate that our knot polynomials have direct structural relation with RNA molecules, and can be used to classify the topology of RNA secondary structures. Furthermore, we point out that these knot polynomials can be used to model the topological effects of disulfide bonds in protein molecules.

Cryo-electron microscopy (Cryo-EM) is a powerful technique to produce 3-dimensional density maps for large molecular complexes. Although many atomic structures have been solved from cryo-EM density maps, it is challenging to derive atomic structures when the resolution of density maps is not sufficiently high. Geometrical shape representation of secondary structural components in a medium-resolution density map enhances modeling of atomic structures.We compare two methods in producing surface representation of the β-sheet component of a density map. Given a 3-dimensional volume of β-sheet that is segmented from a density map, the performance of a polynomial fitting was compared with that of an iterative Bézier fitting. The results suggest that the iterative Bézier fitting is more suitable for β-sheets, since it provides more accurate representation of the corners that are naturally twisted in a β-sheet.

Nonlocal continua have been proposed to offer a more realistic model for the electrostatic response of solutions such as the electrolyte solvents prominent in biology and electrochemistry. In this work, we review three nonlocal models based on the Landau-Ginzburg framework which have been proposed but not directly compared previously, due to different expressions of the nonlocal constitutive relationship. To understand the relationships between these models and the underlying physical insights from which they are derive, we situate these models into a single, unified Landau-Ginzburg framework. One of the models offers the capacity to interpret how temperature changes affect dielectric response, and we note that the variations with temperature are qualitatively reasonable even though predictions at ambient temperatures are not quantitatively in agreement with experiment. Two of these models correctly reproduce overscreening (oscillations between positive and negative polarization charge densities), and we observe small differences between them when we simulate the potential between parallel plates held at constant potential. These computations require reformulating the two models as coupled systems of local partial differential equations (PDEs), and we use spectral methods to discretize both problems. We propose further assessments to discriminate between the models, particularly in regards to establishing boundary conditions and comparing to explicit-solvent molecular dynamics simulations.

We study a quasi-one-dimensional steady-state Poisson-Nernst-Planck model for ionic flows through membrane channels. Excess chemical potentials are included in this work to account for finite ion size effects. This is the main difference from the classical Poisson-Nernst-Planck models, which treat ion species as point charges and neglect ion-to-ion interactions. Due to the fact that most experiments (with some exceptions) can only measure the total current while individual fluxes contain much more information on channel functions, our main focus is to study the qualitative properties of ionic flows in terms of individual fluxes under electroneutrality conditions. Our result shows that, in addition to ion sizes, the property depends on multiple physical parameters such as boundary concentrations and potentials, diffusion coe-cients, and ion valences. For the relatively simple setting and assumptions of the model in this paper, we are able to characterize, almost completely, the distinct effects of the nonlinear interplay between these physical parameters. The boundaries of different parameter regions are identified through a number of critical potential values that are explicitly expressed in terms of the physical parameters.Numerical simulations are performed to detect the critical potentials and investigate the quantitative properties of ionic flows over different potential regions. In particular, a special case is studied in Section 5 without the assumption of electroneutrality conditions.

Electricity plays a special role in our lives and life. The dynamics of electrons allow light to flow through a vacuum. The equations of electron dynamics are nearly exact and apply from nuclear particles to stars. These Maxwell equations include a special term, the displacement current (of a vacuum). The displacement current allows electrical signals to propagate through space. Displacement current guarantees that current is exactly conserved from inside atoms to between stars, as long as current is defined as the entire source of the curl of the magnetic field, as Maxwell did.We show that the Bohm formulation of quantum mechanics allows the easy definition of the total current, and its conservation, without the dificulties implicit in the orthodox quantum theory. The orthodox theory neglects the reality of magnitudes, like the currents, during times that they are not being explicitly measured.We show how conservation of current can be derived without mention of the polarization or dielectric properties of matter. We point out that displacement current is handled correctly in electrical engineering by ‘stray capacitances’, although it is rarely discussed explicitly. Matter does not behave as physicists of the 1800’s thought it did. They could only measure on a time scale of seconds and tried to explain dielectric properties and polarization with a single dielectric constant, a real positive number independent of everything. Matter and thus charge moves in enormously complicated ways that cannot be described by a single dielectric constant,when studied on time scales important today for electronic technology and molecular biology. When classical theories could not explain complex charge movements, constants in equations were allowed to vary in solutions of those equations, in a way not justified by mathematics, with predictable consequences. Life occurs in ionic solutions where charge is moved by forces not mentioned or described in the Maxwell equations, like convection and diffusion. These movements and forces produce crucial currents that cannot be described as classical conduction or classical polarization. Derivations of conservation of current involve oversimplified treatments of dielectrics and polarization in nearly every textbook. Because real dielectrics do not behave in that simple way-not even approximately-classical derivations of conservation of current are often distrusted or even ignored. We show that current is conserved inside atoms. We show that current is conserved exactly in any material no matter how complex are the properties of dielectric, polarization, or conduction currents. Electricity has a special role because conservation of current is a universal law.Most models of chemical reactions do not conserve current and need to be changed to do so. On the macroscopic scale of life, conservation of current necessarily links far spread boundaries to each other, correlating inputs and outputs, and thereby creating devices.We suspect that correlations created by displacement current link all scales and allow atoms to control the machines and organisms of life. Conservation of current has a special role in our lives and life, as well as in physics. We believe models, simulations, and computations should conserve current on all scales, as accurately as possible, because physics conserves current that way. We believe models will be much more successful if they conserve current at every level of resolution, the way physics does.We surely need successful models as we try to control macroscopic functions by atomic interventions, in technology, life, and medicine. Maxwell’s displacement current lets us see stars. We hope it will help us see how atoms control life.

We present a nonlocal electrostatic formulation of nonuniform ions and water molecules with interstitial voids that uses a Fermi-like distribution to account for steric and correlation efects in electrolyte solutions. The formulation is based on the volume exclusion of hard spheres leading to a steric potential and Maxwell’s displacement field with Yukawa-type interactions resulting in a nonlocal electric potential. The classical Poisson-Boltzmann model fails to describe steric and correlation effects important in a variety of chemical and biological systems, especially in high field or large concentration conditions found in and near binding sites, ion channels, and electrodes. Steric effects and correlations are apparent when we compare nonlocal Poisson-Fermi results to Poisson-Boltzmann calculations in electric double layer and to experimental measurements on the selectivity of potassium channels for K+ over Na+.

We investigate the effects of the relative dielectric coefficient on ionic flows in open ion channels, using mathematical analysis of reasonably general Poisson-Nernst-Planck type models that can include the finite sizes of ions. The value of the relative dielectric coefficient is of course a crucial parameter for ionic behavior in general. Using the powerful theory of singularly perturbed problems in applied mathematics, we show that some properties of open channels are quite insensitive to variation in the relative dielectric coefficient, thereby explaining such effects seen unexpectedly in simulations. The ratio between the total number of one ion species and that of another ion species, and the ratio between the flux of one ion species and that of another ion species do not depend significantly on the relative dielectric coefficient.

Classical Poisson-Boltzman and Poisson-Nernst-Planck models can only work when ion concentrations are very dilute, which often mismatches experiments. Researchers have been working on the modification to include finite-size effect of ions, which is non-negelible when ion concentrations are not dilute. One of modified models with steric effect is Bikerman model, which is rather popular nowadays. It is based on the consideration of ion size by putting additional entropy term for solvent in free energy. However, ion size is non-specific in original Bikerman model, which did not consider specific ion sizes. Many researchers have worked on the extension of Bikerman model to have specific ion sizes. A direct extension of original Bikerman model by simply replacing the non-specific ion size to specific ones seems natural and has been acceptable to many researchers in this field.Herewe prove this straight forward extension, in some limiting situations, fails to uphold the basic requirement that ion occupation sites must be identical. This requirement is necessary when computing entropy via particle distribution on occupation sites.We derived a new modified Bikerman model for using specific ion sizes by fixing this problem, and obtained its modified Poisson-Boltzmann and Poisson-Nernst-Planck equations.