Out-of-plane deformability and its coupling with electrostatics in biomembranes.

Cell membranes are quasi-bidimensional soft systems formed by multipoles in an ordered array that can be polarized in an electric field. Consequently, electrostatic potentials emerge inside membranes, and membranes respond to external electric fields. From a mechanical perspective, membranes can be easily compressed-expanded, laterally deformed, and curved. Bending is particularly easy, and this kind of deformation translates to changes in the relative positions of the negative and positive charges, leading to strain gradient-induced polarization. Conversely, an external electric field gradient will exert a bending stress that translates to mechanical membrane deformation. These phenomena are described through membrane flexoelectricity. Here, we describe this property in lipid bilayers and cell membranes and summarize the studies in the field with emphasis on the effects promoted by membrane asymmetry.

Field emitter electrostatics: a review with special emphasis on modern high-precision finite-element modelling.

This review of the quantitative electrostatics of field emitters, covering analytical, numerical and 'fitted formula' approaches, is thought the first of its kind in the 100 years of the subject. The review relates chiefly to situations where emitters operate in an electronically ideal manner, and zero-current electrostatics is applicable. Terminology is carefully described and is 'polarity independent', so that the review applies to both field electron and field ion emitters. It also applies more generally to charged, pointed electron-conductors-which exhibit the 'electrostatic lightning-rod effect', but are poorly discussed in general electricity and magnetism literature. Modern electron-conductor electrostatics is an application of the chemical thermodynamics and statistical mechanics of electrons. In related theory, the primary role of classical electrostatic potentials (rather than fields) becomes apparent. Space and time limitations have meant that the review cannot be comprehensive in both detail and scope. Rather, it focuses chiefly on the electrostatics of two common basic emitter forms: the needle-shaped emitters used in traditional projection technologies; and the post-shaped emitters often used in modelling large-area multi-emitter electron sources. In the post-on-plane context, we consider in detail both the electrostatics of the single post and the interaction between two identical posts that occurs as a result of electrostatic depolarization (often called 'screening' or 'shielding'). Core to the review are discussions of the 'minimum domain dimensions' method for implementing effective finite-element-method electrostatic simulations, and of the variant of this that leads to very precise estimates of dimensionless field enhancement factors (error typically less than 0.001% in simple situations where analytical comparisons exist). Brief outline discussions, and some core references, are given for each of many 'related considerations' that are relevant to the electrostatic situations, methods and results described. Many areas of field emitter electrostatics are suggested where further research and/or separate mini-reviews would probably be useful.

Towards optimal boundary integral formulations of the Poisson-Boltzmann equation for molecular electrostatics.

The Poisson-Boltzmann equation offers an efficient way to study electrostatics in molecular settings. Its numerical solution with the boundary element method is widely used, as the complicated molecular surface is accurately represented by the mesh, and the point charges are accounted for explicitly. In fact, there are several well-known boundary integral formulations available in the literature. This work presents a generalized expression of the boundary integral representation of the implicit solvent model, giving rise to new forms to compute the electrostatic potential. Moreover, it proposes a strategy to build efficient preconditioners for any of the resulting systems, improving the convergence of the linear solver. We perform systematic benchmarking of a set of formulations and preconditioners, focusing on the time to solution, matrix conditioning, and eigenvalue spectrum. We see that the eigenvalue clustering is a good indicator of the matrix conditioning, and show that they can be easily manipulated by scaling the preconditioner. Our results suggest that the optimal choice is problem-size dependent, where a simpler direct formulation is the fastest for small molecules, but more involved second-kind equations are better for larger problems. We also present a fast Calderón preconditioner for first-kind formulations, which shows promising behavior for future analysis. This work sets the basis towards choosing the most convenient boundary integral formulation of the Poisson-Boltzmann equation for a given problem.

Debye-Hückel Free Energy of an Electric Double Layer with Discrete Charges Located at a Dielectric Interface.

Poisson-Boltzmann theory provides an established framework to calculate properties and free energies of an electric double layer, especially for simple geometries and interfaces that carry continuous charge densities. At sufficiently small length scales, however, the discreteness of the surface charges cannot be neglected. We consider a planar dielectric interface that separates a salt-containing aqueous phase from a medium of low dielectric constant and carries discrete surface charges of fixed density. Within the linear Debye-Hückel limit of Poisson-Boltzmann theory, we calculate the surface potential inside a Wigner-Seitz cell that is produced by all surface charges outside the cell using a Fourier-Bessel series and a Hankel transformation. From the surface potential, we obtain the Debye-Hückel free energy of the electric double layer, which we compare with the corresponding expression in the continuum limit. Differences arise for sufficiently small charge densities, where we show that the dominating interaction is dipolar, arising from the dipoles formed by the surface charges and associated counterions. This interaction propagates through the medium of a low dielectric constant and alters the continuum power of two dependence of the free energy on the surface charge density to a power of 2.5 law.

Physical interactions driving the activation/inhibition of calcium/calmodulin dependent protein kinase II.

CaMKII is a protein kinase whose function is regulated by the binding of the Calcium/Calmodulin complex (Ca2+/CaM). It is a major player in the Long Term Potentiation process where it acts as a molecular switch, oscillating between inhibited and active conformations. The mechanism for the switching is thought to be initiated by Ca2+/CaM binding, which allows the trans-phosphorylation of a subunit of CaMKII by a neighboring kinase, leading to the active state of the system. A combination of all-atom and coarse-grained MD simulations with free energy calculations, led us to reveal an interplay of electrostatic forces exerted by Ca2+/CaM on CaMKII, which initiate the activation process. The highly electrically charged Ca2+/CaM neutralizes basic regions in the linker domain of CaMKII, facilitating its opening and consequent activation. The emerging picture of CaMKII's behavior highlights the preponderance of electrostatic interactions, which are modulated by the presence of Ca2+/CaM and the phosphorylation of key sites.

A Boundary-Integral Approach for the Poisson-Boltzmann Equation with Polarizable Force Fields.

Implicit-solvent models are widely used to study the electrostatics in dissolved biomolecules, which are parameterized using force fields. Standard force fields treat the charge distribution with point charges; however, other force fields have emerged which offer a more realistic description by considering polarizability. In this work, we present the implementation of the polarizable and multipolar force field atomic multipole optimized energetics for biomolecular applications (AMOEBA), in the boundary integral Poisson-Boltzmann solver PyGBe. Previous work from other researchers coupled AMOEBA with the finite-difference solver APBS, and found difficulties to effectively transfer the multipolar charge description to the mesh. A boundary integral formulation treats the charge distribution analytically, overlooking such limitations. This becomes particularly important in simulations that need high accuracy, for example, when the quantity of interest is the difference between solvation energies obtained from separate calculations, like happens for binding energy. We present verification and validation results of our software, compare it with the implementation on APBS, and assess the efficiency of AMOEBA and classical point-charge force fields in a Poisson-Boltzmann solver. We found that a boundary integral approach performs similarly to a volumetric method on CPU. Also, we present a GPU implementation of our solver. Moreover, with a boundary element method, the mesh density to correctly resolve the electrostatic potential is the same for standard point-charge and multipolar force fields. Finally, we saw that for binding energy calculations, a boundary integral approach presents more consistent results than a finite difference approximation for multipolar force fields. © 2019 Wiley Periodicals, Inc.

Development of constant-pH simulation methods in implicit solvent and applications in biomolecular systems.

pH is a critical parameter for biological and technological systems directly related with electrical charges. It can give rise to peculiar electrostatic phenomena, which also makes them more challenging. Due to the quantum nature of the process, involving the forming and breaking of chemical bonds, quantum methods should ideally by employed. Nevertheless, due to the very large number of ionizable sites, different macromolecular conformations, salt conditions, and all other charged species, the CPU time cost simply becomes prohibitive for computer simulations, making this a quite complex problem. Simplified methods based on Monte Carlo sampling have been devised and will be reviewed here, highlighting the updated state-of-the-art of this field, advantages, and limitations of different theoretical protocols for biomolecular systems (proteins and nucleic acids). Following a historical perspective, the discussion will be associated with the applications to protein interactions with other proteins, polyelectrolytes, and nanoparticles.

Electric fingerprint of voltage sensor domains.

A dynamic transmembrane voltage field has been suggested as an intrinsic element in voltage sensor (VS) domains. Here, the dynamic field contribution to the VS energetics was analyzed via electrostatic calculations applied to a number of atomistic structures made available recently. We find that the field is largely static along with the molecular motions of the domain, and more importantly, it is minimally modified across VS variants. This finding implies that sensor domains transfer approximately the same amount of gating charges when moving the electrically charged S4 helix between fixed microscopic configurations. Remarkably, the result means that the observed operational diversity of the domain, including the extension, rate, and voltage dependence of the S4 motion, as dictated by the free energy landscape theory, must be rationalized in terms of dominant variations of its chemical free energy.