Biological membranes are primarily composed of phospholipids-a diverse class of compounds compose... more Biological membranes are primarily composed of phospholipids-a diverse class of compounds composed of a hydrophilic head group covalently attached to a pair of hydrophobic fatty acids. This amphipathic structure leads phospholipid molecules to spontaneously form bilayers when placed in water, as the phospholipids are driven to orient their head groups towards water and shield their fatty acid tails from it via the hydrophobic effect. While these bilayers tend to exist in a fluid phase under physiological conditions, their component phospholipids can undergo phase transitions under the correct environmental conditions. Similarly to the familiar transitions between the liquid, solid, and gas phases of simpler systems, these lipid phase transitions represent changes in the entropy of the system through reorganization of the system's components in response to changes in the free energy of the system. Lipids can exist in a number of phases, which are summarized below. The liquid disordered phase, as its name implies, is a highly fluid state in which individual lipids can move laterally across the surface of the membrane relatively unhindered. Liquid-disordered bilayers are often characterized by irregular packing of individual lipid molecules, as well as the presence of kinks in unsaturated fatty acids. These kinks effectively reduce the surface area accessible to other fatty acid chains, weakening Van der Waals interactions.
bioRxiv (Cold Spring Harbor Laboratory), Jul 5, 2019
Extraction from simulations of the area compressibility moduli of the monolayers in a bilayer is ... more Extraction from simulations of the area compressibility moduli of the monolayers in a bilayer is considered theoretically. A statistical mechanical derivation shows that the bilayer modulus is the sum of the two monolayer moduli, as is often supposed, but contrary to a recent study. Seemingly plausible assumptions regarding fluctuations are tested rigorously. Prospects for future research are discussed. It is important to describe the properties of both leaflets of generally asymmetric Biomembranes. One such property is the area compressibility modulus. This MS rigorously establishes the fundamental theory that corrects a recent BJ paper. The theory is straightforward but substantial enough that it was not readily apparent why the previous theory was incorrect. This is why this MS should be considered a new paper and not just a comment. Another reason is that this MS points to an alternative method, used only once previously, for extracting the leaflet area compressibility modulus from simulations.
ADAMS has recently considered the dielectric constant of ice 1. His Monte Carlo calculations fill... more ADAMS has recently considered the dielectric constant of ice 1. His Monte Carlo calculations fill a gap in the literature and complement previous calculations. However, his theoretical discussion, while displaying an important constraint on the Kirkwood-Frohlich (K-F) theory, falls short of the 'simple resolution' claimed. Adams' primary equation 1 is (µ.1"i.µ. i), N/9e okTV Lebowitz, J. L. & Montroll, E. W.) (North-Holland, Amsterdam, in the press), ADAMS REPLIES-The 'simple resolution' which I claim is the observation that it is not necessary to choose between either the Onsager-Slater equation or the Kirkwood equation. The Onsager-Slater
Using exact methods, pair-correlation functions are studied in the dimer model defined on a brick... more Using exact methods, pair-correlation functions are studied in the dimer model defined on a brick lattice. At long distances these functions exhibit strongly anisotropic algebraic decay and, near criticality, the length scales diverge differently in the two principal directions. The critical exponents are vx = 89 and Vy = 1. These results are in agreement with deductions drawn from recent exact finite-size scaling calculations. We also interpret our results in the light of domain wall theories of commensurate-incommensurate transitions, and in particular we study the relation of the present model to the discrete version of the Pokrovsky Talapov model introduced by Villain.
Experiments on the chain melting thermal transition in simple biological membranes are briefly re... more Experiments on the chain melting thermal transition in simple biological membranes are briefly reviewed and shown to indicate that a microscopic order-disorder model may be appropriate to describe the thermodynamics of the transition. To test this conclusion further a pair of two dimensional lattice models (A and B) are introduced and the statistical mechanics is solved exactly using dimer techniques. The phenomenological parameters required by the models are evaluated from experiments on other systems. The more realistic of the two models (model A) has a second order transition at a temperature of 353°K compared to 315°K for dipalmitoyl-L-α-lecithin membranes. However, the specific heat peaks do not have the same shape and the transition is broader for model A than for the experiment. In comparison, the less realistic model B has a first order transition at 925°K, considerably higher than the experimental transition temperature. From these results it seems likely that the points of disagreement between model A and experiment may be due to the simplified features of model A. It is concluded that the basic order-disorder model is likely to be correct for the main transition. There is also a smaller transition at a lower temperature which is also discussed in terms of an order-disorder model involving the phospholipid head groups. In this case the calculation shows that the simple model is either wrong or that some additional features must be added.
It is shown that the hydrogen-bonded configurational entropy problem associated with the model of... more It is shown that the hydrogen-bonded configurational entropy problem associated with the model of Cu(HCOO)24H20 discussed by Okada is isomorphic to the exactly solvable two-dimensional dimer problem on the square lattice.
Dimer models with excluded volume interactions and classical anisotropic dispersion interactions ... more Dimer models with excluded volume interactions and classical anisotropic dispersion interactions are solved exactly in the limit of close packing on the two dimensional square, triangular and honeycomb lattices. These models exhibit nematic phase transitions for the aforementioned three lattices at temperatures of 2.55b/k, 1.51b/k and 3.10b/k respectively, where - b is the ground state energy per molecule.
It is recommended that plots of absorbance change vs. log time be considered as a way to represen... more It is recommended that plots of absorbance change vs. log time be considered as a way to represent many decades of flash spectroscopy data when several separate decays are present. The values of the rate constants k, for the decays are easily revealed by nearly linear portions of the plots centered near rj = k j ' , which are inflection points in the graphs. The amplitudes of the decays are easily seen because they are proportional to the slopes of the linear portions of the plots. Such plots introduce no spurious singularities and they are easy to construct using semi-log paper. Examples, including one from bacteriorhodopsin, are presented. A simple mathematical proof of these features is provided.
We investigate the Poisson ratio n of fluid lipid bilayers, i.e., the question how area strains c... more We investigate the Poisson ratio n of fluid lipid bilayers, i.e., the question how area strains compare to the changes in membrane thickness (or, equivalently, volume) that accompany them. We first examine existing experimental results on the area-and volume compressibility of lipid membranes. Analyzing them within the framework of linear elasticity theory for homogeneous thin fluid sheets leads us to conclude that lipid membrane deformations are to a very good approximation volume-preserving, with a Poisson ratio that is likely about 3% smaller than the common soft matter limit n ¼ 1 2. These results are fully consistent with atomistic simulations of a DOPC membrane at varying amount of applied lateral stress, for which we instead deduce n by directly comparing area-and volume strains. To assess the problematic assumption of transverse homogeneity, we also define a depth-resolved Poisson ratio n(z) and determine it through a refined analysis of the same set of simulations. We find that throughout the membrane's thickness, n(z) is close to the value derived assuming homogeneity, with only minor variations of borderline statistical significance.
The extensive surface area A and the conjugate surface pressure Π are introduced into a cooperati... more The extensive surface area A and the conjugate surface pressure Π are introduced into a cooperatively interacting hydrocarbon chain model. The exact statistical mechanical method of solution is extended to accommodate the new variables. The previously discussed 3/2 order transition becomes a critical point in phase diagrams involving Π and A. The values of the usual critical exponents are β=1, δ=2, γ=γ′=1, and α=0, and the 3/2 order exponents appear on the critical isobar, Π=Πc. The model shows Π-A phase behavior which is strikingly different from the P–V behavior, in agreement with the difference between the experimental Π-A monolayer behavior and the P–V bilayer behavior. This supports the theory that the bilayer is two weakly coupled monolayers.
Assuming that partisan fairness and responsiveness are important aspects of redistricting, it is ... more 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. 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. 6 Although this conclusion will not come as a surprise to many political scientists, e.g.
Abstract A brief survey is given of a number of exactly solved two-dimensional dimer models with ... more Abstract A brief survey is given of a number of exactly solved two-dimensional dimer models with K-type transitions and with isomorphisms to chain melting and/or domain wall, adsorbed atom models.
Critical Phenomena and Phase Transitions
Springer eBooks, 1975
The area of critical phenomena is a rather large one, both quantitatively, as measured by the num... more The area of critical phenomena is a rather large one, both quantitatively, as measured by the number of researchers and the number of publications, and qualitatively, as judged by the percentage of “hard” results such as exact calculations and highly precise experiments. This makes it impossible to write an article or even a book which is completely self-contained and which includes all that has been accomplished.
Values of the bending modulus K C and the tilt modulus K u are reported for single component lipi... more Values of the bending modulus K C and the tilt modulus K u are reported for single component lipid bilayers. The lipids studied have the common names DOPC, DMPC, diC22:1PC, SOPC, POPC, diPhyPC, DLPC, DPPC, DHPC and DEPC, listed in the order of number of samples examined. The experimental method, thus far the only one that measures the tilt modulus of lipid bilayers, first obtains diffuse X-ray scattering data from oriented stacks of bilayers. The values of the moduli emerge from fitting the data to the accepted tilt-dependent continuum model for the free energy of a single bilayer, further enhanced by interactions between bilayers in the stack. The results indicate the broad trend that the tilt modulus for these PC lipids is smaller the closer the temperature is to the main transition temperature. Another trend is that inclusion of tilt raises the value of the bending modulus more for lipids with smaller values of the tilt modulus. Values of both moduli are compared to recent literature values obtained from simulations and values of the bending modulus are compared to the literature values obtained by other experimental methods.
The application of excited-state intramolecular proton transfer can be considered as a photoinduc... more The application of excited-state intramolecular proton transfer can be considered as a photoinduced chemical laser case, because the tautomer produced is a new chemical species generated in an excited state, emitting to a ground state of zero population. The generic nature of the specific case (-OH proton transfer to O==C-, followed by pyrane ring aromatization to pyrillium) should be recognized, suggesting a whole new class of intramolecular pro-Acknowledgment. This work was supported by Contract No.
A direct nonlinear least squares method is described that obtains the true kinetic rate constants... more A direct nonlinear least squares method is described that obtains the true kinetic rate constants and the temperature-independent spectra of n intermediates from spectroscopic data taken in the visible at three or more temperatures. A theoretical analysis, which is independent of implementation of the direct method, proves that well determined local solutions are not possible for fewer than three temperatures. This analysis also proves that measurements at more than n wavelengths are redundant, although the direct method indicates that convergence is faster if n + m wavelengths are measured, where m is of order one. This suggests that measurements should concentrate on high precision for a few measuring wavelengths, rather than lower precision for many wavelengths. Globally, false solutions occur, and the ability to reject these depends upon the precision of the data, as shown by explicit example. An optimized way to analyze vibrational spectroscopic data is also presented. Such data yield unique results, which are comparably accurate to those obtained from data taken in the visible with comparable noise. It is discussed how use of both kinds of data is advantageous if the data taken in the visible are significantly less noisy.
An efficient method for extracting volumetric data from simulations is developed. The method is i... more An efficient method for extracting volumetric data from simulations is developed. The method is illustrated using a recent atomic-level molecular dynamics simulation of L,, phase 1,2-dipalmitoyl-sn-glycero-3-phosphocholine bilayer. Results from this simulation are obtained for the volumes of water (Vw), lipid (V,), chain methylenes (V2), chain terminal methyls (V3), and lipid headgroups (VH), including separate volumes for carboxyl (Vcoo), glyceryl (Vg,), phosphoryl (Vp04), and choline (VChO,) groups. The method assumes only that each group has the same average volume regardless of its location in the bilayer, and this assumption is then tested with the current simulation. The volumes obtained agree well with the values Vw and VL that have been obtained directly from experiment, as well as with the volumes VH, V2, and V3 that require certain assumptions in addition to the experimental data. This method should help to support and refine some assumptions that are neccessary when interpreting experimental data.
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