This book is a tutorial written by researchers and developers behind the FEniCS Project and explores an advanced, expressive approach to the development of mathematical software. The presentation spans mathematical background, software design and the use of FEniCS in applications. Theoretical aspects are complemented with computer code which is available as free/open source software. The book begins with a special introductory tutorial for beginners. Followingare chapters in Part I addressing fundamental aspects of the approach to automating the creation of finite element solvers. Chapters in Part II address the design and implementation of the FEnicS software. Chapters in Part III present the application of FEniCS to a wide range of applications, including fluid flow, solid mechanics, electromagnetics and geophysics.

Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book

The structure and function of macromolecules depend critically on the ionization (protonation) states of their acidic and basic groups. A number of existing practical methods predict protonation equilibrium pK constants of macromolecules based upon their atomic resolution Protein Data Bank (PDB) structures; the calculations are often performed within the framework of the continuum electrostatics model. Unfortunately, these methodologies are complex, involve multiple steps and require considerable investment of effort. Our web server http://biophysics.cs.vt.edu/H++ provides access to a tool that automates this process, allowing both experts and novices to quickly obtain estimates of pKs as well as other related characteristics of biomolecules such as isoelectric points, titration curves and energies of protonation microstates. Protons are added to the input structure according to the calculated ionization states of its titratable groups at the user-specified pH; the output is in the PQR (PDB + charges + radii) format. In addition, corresponding coordinate and topology files are generated in the format supported by the molecular modeling package AMBER. The server is intended for a broad community of biochemists, molecular modelers, structural biologists and drug designers; it can also be used as an educational tool in biochemistry courses.

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H++: a server for estimating pKas and adding missing hydrogens to macromolecules

Theory of self-assembly of lipid bilayers and vesicles

Publisher Summary This chapter discusses electrostatic potentials at membrane solution interfaces. Lipids in the membranes of all cells and subcellular organelles are arranged in the form of a bilayer with the hydrocarbon tails sequestered away from the water and the polar head groups exposed to the aqueous environment. About 10%–20% of the lipids in the membranes of many cells and organelles bear a net negative charge, whereas positively charged lipids are extremely rare. The concentration of monovalent cations at the surface of the bilayer will be an order of magnitude higher than the concentration of these ions in the bulk aqueous phase. The surface potential produced by charged lipids is dependent on the salt concentration in the bulk aqueous phase and a seminal paper. The absence of proteins, polysaccharides, and other macromolecules present in biological membranes can be considered an advantage to test how well the theory of the diffuse double layer describes the electrostatic potential produced by charges on lipids. The chapter describes hydrophobic adsorption of charged molecules to bilayer membranes, the electrostatic potential produced by molecular dipoles at membrane-solution interfaces, and the electrostatic boundary potential produced by charges located in the interior of the membrane, a few angstroms from the interface. Few examples of the possible biological significance of these electrostatic surface potentials are also described.

Electrostatic Potentials at Membrane-Solution Interfaces

Differential scanning calorimetry (DSC) and fluorescence polarization of embedded probe molecules were used to detect phase behavior of various phospholipids. The techniques were directly compared for detecting the transition of dipalmitoylphosphatidylcholine (DPPC) and dipalmitoylphosphatidic acid (DPPA) dispersed in aqueous salt solutions. Excellent agreement occurred in the case of phosphatidylcholine; however, in the case of phosphatidic acid, at pH 6.5, transitions detected by fluorescence polarization using the disc-like perylene molecule occurred about 10 degrees lower than those detected by DSC. Discrepancy between fluorescence and DSC methods is eliminated by using a rod-like molecule, diphenylhexatriene (DPH). Both techniques show that doubly ionizing the phosphate group reduces the Tc by about 9 degrees. Direct pH titration of fluidity can be accomplished and this effect is most dramatic when membranes are in their transition temperature range (ca. 50 degrees). Phosphatidic acid transitions occur at higher temperatures, and have appreciably lower transition enthalpies and entropies than phosphatidylcholine. These effect could not be explained simply on the basis of double layer electrostatics and several other factors were discussed in an attempt to rationalize the results. Addition of monovalent cations (0.01-0.5 M) is shown to increase the Tc of dipalmitoylphosphatidylglycerol by less than 3 degrees. However, addition of (1 x 10-3 M) Ca2+ abolishes the phase transition of both phosphatidyglycerol and phosphatidylserine in the range 0-70 degrees. Preliminary X-ray evidence indicates the phosphatidylserine-Ca2+ bilayers are in a crystalline state at 24 degrees. In contrast, 5 x 10-3 M Mg2+ only broadens the transition and increases the Tc indicating a considerable difference between the effects of Ca2+ and Mg2+. Neutralization of PS increases the Tc from 6 degrees (at pH 7.4) to 20-26 degrees (at pH 2.5-3.0) but does not abolish the transition, suggesting the Ca2+ effect involves more than charge neutralization. Addition of Ca2+ to mixed phosphatidylserine-phosphatidylcholine dispersions, induces a phase separation of the dipalmitoyl- (and also distearoyl-) phosphatidylcholine as seen by the appearance of a new endothermic peak at 41 degrees (58 degrees). Similarly, in mixed (dipalmitoyl) phosphatidic acid-phosphatidylcholine (2:1) dispersions, Ca2+ again can separate the phosphatidylcholine component.

Phase transitions and phase separations in phospholipid membranes induced by changes in temperature, pH, and concentration of bivalent cations.

Phospholipid bilayer membranes were bathed in a decimolar solution of monovalent ions, and the conductance produced by neutral carriers of these monovalent cations and anions was used to assess the electric potential at the surface of the membrane. When the bilayers were formed from a neutral lipid, phosphatidylethanolamine, the addition of alkaline earth cations produced no detectable surface potential, indicating that little or no binding occurs to the polar head group with these ions. When the bilayers were formed from a negatively charged lipid, phosphatidylserine, the addition of Sr and Ba decreased the magnitude of the surface potential as predicted by the theory of the diffuse double layer. In particular, the potential decreased 27 mv for a 10-fold increase in concentration in the millimolar-decimolar range. A 10-fold increase in the Ca or Mg concentration also produced a 27 mv decrease in potential in this region, which was again due to screening, but it was necessary to invoke some specific binding to account for the observation that these cations were effective at a lower concentration than Ba or Sr. It is suggested that the ability of the alkaline earth cations to shift the conductance-voltage curves of a nerve along the voltage axis by 20–26 mv for a 10-fold increase in concentration may be due to essentially a screening rather than a binding phenomenon.

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Divalent ions and the surface potential of charged phospholipid membranes.

Bilayer membranes, formed from various phospholipids, were studied to assess the influence of the charge of the polar head groups on the membrane conductance mediated by neutral „carriers” of cations and anions. The surface charge of an amphoteric lipid, phosphatidyl ethanolamine, was altered by varying the pH, and the surface charge of several lipids was screened by increasing the ionic strength of the solution with impermeant monovalent and divalent electrolytes. The surface charge should be a key parameter in defining the membrane conductance for a variety of permeation mechanisms; conductance measurements in the presence of carriers may be used to estimate the potential difference, due to surface charge, between the interior of the bilayer and the bulk aqueous phase. The large changes in conductance observed upon varying the surface charge density and the ionic strength agree with those predicted by the Gouy-Chapman theory for an aqueous diffuse double layer. Explicit expressions for the dependence of the membrane conductance on the concentrations of the carrier, the permeant ion, the surface charge density, and the ionic strength are presented.

https://www.pnas.org/content/pnas/67/3/1268.full.pdf

Surface Charge and the Conductance of Phospholipid Membranes

SUMMARY

In this paper we have examined the possibility of identifying those membrane structural variables (polar head groups and the nature of hydrocarbon tails) that modulate membrane ionic permeability. Altering the bilayer lipid composition produces variations in physical parameters (surface potential, partition coefficient, and mobility) governing the conductance mediated by neutral carriers of anions and cations. Specifically, the effects of the charged polar head groups are shown to be understandable in terms of the surface potential they produce through the formation of a diffuse double layer, whereas the effects of the viscosity may be demonstrated by “freezing” the membrane. The effects of membrane composition on membrane conductance are illustrated by a third, less well understood, example of how cholesterol alters bilayer conductances. The results indicate the possibility of using positive and negative permeant species as probes of membrane structures.

Ionic probes of membrane structures.

Ion Binding and Ion Transport Produced by Neutral Lipid-Soluble Molecules

The cyclic polyether XXXII, a neutral, lipid soluble molecule, produces large increases in the conductance of bilayer membranes formed from a variety of lipids. The conductance increases linearly with the concentration of alkali metal cation but with the square, and at higher concentrations the cube, of the polyether concentration. This implies that two or three polyether molecules combine with a single cation to carry it across the membrane. In the presence of XXXII the bilayer is permeable solely to cations and the membrane potential is described by an equation of the Goldman-Hodgkin-Katz type. The permeability ratios determined from potential measurements are independent of salt concentration, decrease in the sequence Cs>Rb>K>NH4>Na>Li(1.0,0.25, 0.15, 0.075, 0.007, 0.0013) and are equal to the conductance ratios at low (e.g. 10−3
m) salt concentration. At higher salt concentrations, the permeability and conductance ratios are not equal and maxima in the conductancevs. salt concentration curves are observed. Both these phenomena are postulated to be caused by the formation of relatively impermeant 1ν1 polyether cation complexes in the aqueous phase. The 1ν1 aqueous association constants deduced from bilayer measurements decrease in the sequence K>Rb>Na>NH4>Cs>Li (120, 34, 26, 19, 12, 4 liters per mole) and agree quantitatively with the literature values for the more water soluble polyether XXXI, which lacks only thet-butyl groups of XXXII.

S. G. A. McLaughlin

Papers

Divalent ions and the surface potential of charged phospholipid membranes.

Surface Charge and the Conductance of Phospholipid Membranes

Ionic probes of membrane structures.

Ion Binding and Ion Transport Produced by Neutral Lipid-Soluble Molecules

The effects of a cyclic polyether on the electrical properties of phospholipid bilayer membranes.