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The rich thermotropic behavior of lipid bilayers is addressed using phenomenological theory informed by many experiments. The most recent experiment not yet addressed by theory has shown that the tilt modulus in DMPC lipid bilayers decreases dramatically as the temperature is lowered toward the main transition temperature TM . It is shown that this behavior can be understood by introducing a simple free energy functional for tilt that couples to the area per molecule. This is combined with a chain melting free energy functional in which the area is the primary order parameter that is the driver of the main transition. Satisfactory agreement with experiment is achieved with values of the model parameters determined by experiments, but the transition is directly into the gel phase. The theory is then extended to include the enigmatic ripple phase by making contact with the most recent experimentally determined ripple structure., Comment: 11 pages, 9 figures

Regarding the effect on the bending modulus of adding cholesterol to lipid bilayers, recent results using neutron spin echo and nuclear magnetic resonance relaxation methods that involve linear transport properties have conflicted with earlier results from purely equilibrium experiments that do not involve linear transport properties. A general discussion indicates how one can be misled by data obtained by methods that involve linear transport properties. It is then shown specifically how the recent neutron spin echo results can be interpreted to agree with the earlier purely equilibrium experimental results, thereby resolving that conflict. Regarding the nuclear magnetic resonance relaxation method, it is noted that current interpretation of the data is unclear regarding the identity of the modulus that is involved, and an alternative interpretation is explored that does not disagree with the results of the equilibrium experiments., Comment: 7 pages, 1 figure

Assuming that partisan fairness and responsiveness are important aspects of redistricting, it is important to measure them. Many measures of partisan bias are satisfactory for states that are balanced with roughly equal proportions of voters for the two major parties. It has been less clear which metrics measure fairness robustly when the proportion of the vote is unbalanced by as little as 60% to 40%. We have addressed this by analyzing past election results for four states with Democratic preferences (CA, IL, MA, and MD), three states with Republican preferences (SC, TN, and TX) and comparing those to results for four nearly balanced states (CO, NC, OH, and PA). We used many past statewide elections in each state to build statistically precise seats for votes and rank for votes graphs to which many measures of partisan bias were applied. In addition to providing values of responsiveness, we find that five of the measures of bias provide mutually consistent values in all states, thereby providing a core of usable measures for unbalanced states. Although all five measures focus on different aspects of partisan bias, normalization of the values across the eleven states provides a suitable way to compare them, and we propose that their average provides a superior measure which we call composite bias. Regarding other measures, we find that the most seemingly plausible symmetry measure fails for unbalanced states. We also consider deviations from the proportionality ideal, but using it is difficult because the political geography of a state can entangle responsiveness with total partisan bias. We do not attempt to separate intentional partisan bias from the implicit bias that results from the interaction of the map drawing rules of a state and its political geography, on the grounds that redistricting should attempt to minimize total partisan bias whatever its provenance., Comment: 55 pages and 27 figures

High resolution low angle x-ray data are reported for the gel phase of DPPC lipid bilayers, extending the previous q range of 1.0 {\AA}-1 to 1.3 {\AA}-1, and employing a new technique to obtain more accurate intensities and form factors |F(q)| for the highest orders of diffraction. Combined with previous wide angle x-ray and volumetric data, a space filling model is employed to obtain gel phase structure at a mesoscopic level. A new conclusion from this analysis is that the hydrocarbon chains from opposing monolayers are mini-interdigitated; this would help explain the previously well-established result that the opposing monolayers are strongly coupled with respect to their chain tilt directions. Even more detailed structural features are described that have not been obtained from experiment but that could, in principle, be obtained from simulations that would first be validated by agreement with the wide angle and the new low angle |F(q)| x-ray data., Comment: 25 pages, 6 figures, supplementary material available from authors

Several measures of partisan bias are reviewed for single member districts with two dominant parties. These include variants of the simple bias that considers only deviation of seats from 50% at statewide 50% vote. Also included are equalization of losing votes and equalization of wasted votes, both of which apply directly when the statewide vote is not 50% and which require, not just partisan symmetry, but specific forms of the seats-votes curve. A new measure of bias is introduced, based on the geometric area between the seats-vote curve and the symmetrically inverted seats-votes curve. These measures are applied to recent Pennsylvania congressional elections and to abstract models of the seats-votes curves. The numerical values obtained from the various measures of bias are compared and contrasted. Each bias measure has merits for different seats-votes curves and for different elections, but all essentially agree for most cases when applied to measure only partisan bias, not conflated with competitiveness. This supports the inclusion of partisan fairness as a fundamental element for election law reform, and some options are discussed.

In order to precisely quantify the fundamental interactions between heterogeneous lipid membranes with coexisting liquid-ordered (Lo) and liquid-disordered (Ld) domains, we performed detailed osmotic stress SAXS experiments by exploiting the domain alignment in raft-mimicking lipid multibilayers. Performing a Monte Carlo (MC) based analysis allowed us to determine with high reliability the magnitude and functional dependence of interdomain forces concurrently with the bending elasticity moduli. In contrast to previous methodologies, this approach enabled us to consider the entropic undulation repulsions on a fundamental level, without having to take recourse to crudely justified mean-field like additivity assumptions. Our detailed Hamaker coefficient calculations indicated only small differences in the van der Waals attractions of coexisting Lo and Ld phases. In contrast, the repulsive hydration and undulation interactions differed significantly, with the latter dominating the overall repulsions in the Ld phase. Therefore, alignment of like domains in multibilayers appears to originate from both, hydration and undulation repulsions., Comment: 51 pages, 11 figures, 2 tables because supplementary information (SI) got attached to the end of the manuscript. Changes: Introduction enlarged to enrease readability, a few points got clarified in the Materials & methods, rationale for the presence of Lo/Ld phases added, main text and SI merged into one file; results and conclusions are unchanged. Submitted to Biophysical Journal

Quantitative theory of interbilayer interactions is essential to interpret x-ray scattering data and to elucidate these interactions for biologically relevant systems. For this purpose Monte Carlo simulations have been performed to obtain pressure P and positional fluctuations sigma. A new method, called Fourier Monte-Carlo (FMC), that is based on a Fourier representation of the displacement field, is developed and its superiority over the standard method is demonstrated. The FMC method is applied to simulating a single membrane between two hard walls, which models a stack of lipid bilayer membranes with non-harmonic interactions. Finite size scaling is demonstrated and used to obtain accurate values for P and sigma in the limit of a large continuous membrane. The results are compared with perturbation theory approximations, and numerical differences are found in the non-harmonic case. Therefore, the FMC method, rather than the approximations, should be used for establishing the connection between model potentials and observable quantities, as well as for pure modeling purposes., Comment: 10 pages, 10 figures

The liquid crystalline model biomembrane system consisting of a stack of interacting membranes is studied by the newly developed Fourier Monte Carlo simulation technique. In comparison to perturbation theory, substantial quantitative discrepancies are found that affect determination of interbilayer interactions. A harmonic theory is also routinely used to interpret x-ray scattering line shapes; this is shown to be valid because the distance dependence of the simulated correlation functions can be fairly well fit by the harmonic theory., Comment: 4 pages, 6 figures