Tail Order Research Articles

Parting gravity’s tail: quadrupole tails at fifth order and beyond via integer partitions

This work studies the systematic organization of higher-order gravitational quadrupole tails using generalized unitarity methods imported from the study of scattering amplitudes. The first major result is a constructive algorithm for generic arbitrary-order tail effective actions which links the structure of their loop integral basis expansion with integer partitions, and predicts that only a single new unitarity cut needs to be evaluated at each tail order with all other contributions given in terms of lower-loop data. The algorithm is employed to compute the tail-of-tail-of-tail-of-tail-of-tail (T5) contributions to the effective action and associated energy loss to gravitational waves. Validation of the new effective action and radiated energy is done through counterterm extraction and renormalization analysis, leading to complete agreement with known counterterms and renormalization flow equations.

How well do empirical molecular mechanics force fields model the cholesterol condensing effect?

Membrane properties are determined in part by lipid composition, and cholesterol plays a large role in determining these properties. Cellular membranes show a diverse range of cholesterol compositions, the effects of which include alterations to cellular biomechanics, lipid raft formation, membrane fusion, signaling pathways, metabolism, pharmaceutical therapeutic efficacy, and disease onset. In addition, cholesterol plays an important role in non-cellular membranes, with its concentration in the skin lipid matrix being implicated in several skin diseases. In phospholipid membranes, cholesterol increases the tail ordering of neighboring lipids, decreasing the membrane lateral area and increasing the thickness. This reduction in the lateral area, known as the cholesterol condensing effect, results from cholesterol-lipid mixtures deviating from ideal mixing. Capturing the cholesterol condensing effect is crucial for molecular dynamics simulations as it directly affects the accuracy of predicted membrane properties, which are essential for understanding membrane function. We present a comparative analysis of cholesterol models across several popular force fields: CHARMM36, Slipids, Lipid17, GROMOS 53A6L, GROMOS-CKP, MARTINI 2, MARTINI 3, and ELBA. The simulations of 1,2-dimyristoyl-sn-glycero-3-phosphocholine (DMPC) and 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC) membranes with varying cholesterol concentrations were conducted to calculate the partial-molecular areas of cholesterol and other condensing parameters, which are compared to the experimental data for validation. While all tested force fields predict small negative deviations from ideal mixing in cholesterol-DOPC membranes, only all-atom force fields capture the larger deviations expected in DMPC membranes. United-atom and coarse-grained models under-predict this effect, condensing fewer neighboring lipids by smaller magnitudes, resulting in too small deviations from ideal mixing. These results suggest that all-atom force fields, particularly CHARMM36 or Slipids, should be used for accurate simulations of cholesterol-containing membranes.

A Tale of Two Tails: Tail Ordering of Stoichiometric 1:1 DTAB:SDS Pairs Adsorbed at the Oil-Water Interface.

Cationic:anionic surfactant mixtures adsorbed at an oil-water interface stabilize foams in the presence of oil, making them essential to the oil, gas, and firefighting industries. The oil tolerance of foams stabilized by surfactant mixtures, relative to pure (unmixed) cationic and anionic surfactants, results from the mixtures' enhanced flexibility in tailoring the physicochemical properties of the interface. To judiciously employ these mixtures, it is necessary to characterize the structure-function property relationship of their surfactant monolayers that lend to oil-tolerant/intolerant foams. In this work, we employ interfacial tensiometry and vibrational sum frequency spectroscopy to determine the composition (surfactant population and cationic:anionic ratio) and the structure (surfactant alkyl tail conformation) of monolayers prepared at the oil-water interface by 1:1 DTAB:SDS (dodecyltrimethylammonium bromide:sodium dodecyl sulfate) mixtures. We show that the interfacial surfactant density of 1:1 DTAB:SDS mixtures greatly exceeds that of pure DTAB and SDS at similar concentrations up to and beyond their respective critical micelle concentration. The enhanced interfacial adsorption of these mixtures is due to the adsorption of stoichiometric 1:1 DTAB:SDS surfactant pairs that form through the attractive electrostatic interactions between surfactant headgroups. We find that these paired surfactants preferentially adsorb at the interface, causing the interfacial DTAB:SDS ratio to be nearly 1:1. Additionally, we find that the SDS tail is more conformationally ordered than the DTAB tail, even though they are expected to be conformationally identical along the entire tail, since they are likely conjoined through van der Waals interactions. This leads to the conclusion that the surfactant pairs are in a staggered arrangement at the interface. These findings help to uncover molecular factors that contribute to the enhanced oil tolerance, and in some cases oil intolerance, of foams stabilized by cationic:anionic surfactant mixtures.

Investigating the Bromoform Membrane Interactions Using Atomistic Simulations and Machine Learning: Implications for Climate Change Mitigation.

Methane emissions from livestock contribute to global warming. Seaweeds used as food additive offer a promising emission mitigation strategy because seaweeds are enriched in bromoform─a methanogenesis inhibitor. Therefore, understanding bromoform storage and production in seaweeds and particularly in a cell-like environment is crucial. As a first step toward this aim, we present an atomistic description of bromoform dynamics, diffusion, and aggregation in the presence of lipid membranes. Using all-atom molecular dynamics simulations with customized CHARMM-formatted bromoform force field files, we investigate the interactions of bromoform and lipid bilayer across various concentrations. Bromoform penetrates membranes and at high concentrations forms aggregates outside the membrane without affecting membrane thickness or lipid tail order. Aggregates outside the membrane influence the membrane curvature. Within the membrane, bromoform preferentially localizes in the membrane hydrophobic core and diffuses the slowest along the membrane normal. Employing general local-atomic descriptors and unsupervised machine learning, we demonstrate the similarity of bromoform local structures between the liquid and aggregated forms.

Molecular dynamics simulations of lipid composition and its impact on structural and dynamic properties of skin membrane

The stratum corneum (SC) plays the most important role in the absorption of topical and transdermal drugs. In this study, we developed a multi-layered SC model using coarse-grained molecular dynamics (CGMD) simulations of ceramides, cholesterol, and fatty acids in equimolar proportions, starting from two different initial configurations. In the first approach, all ceramide molecules were initially in the hairpin conformation, and the membrane bilayers were pre-formed. In the second approach, ceramide molecules were introduced in either the hairpin or splayed conformation, with the lipid molecules randomly oriented at the start of the simulation. The aim was to evaluate the effects of lipid chain length on the structural and dynamic properties of SC. By incorporating ceramides and fatty acids of different chain lengths, we simulated the SC membrane in healthy and diseased states. We calculated key structural properties including the thickness, normalized lipid area, lipid tail order parameters, and spatial ordering of the lipids from each system. The results showed that systems with higher ordering and structural integrity contained an equimolar ratio of ceramides (chain length of 24 carbon atoms), fatty acids with chain lengths ≥ of 20 carbon atoms, and cholesterol. In these systems, strong apolar interactions between the ceramide and fatty acid long acyl chains restricted the mobility of the lipid molecules, thereby maintaining a compact lipid headgroup region and high order in the lipid tail region. The simulations also revealed distinct flip-flop mechanisms for cholesterol and fatty acid within the multi-layered membrane. Cholesterol is mostly diffused through the tail-tail interface region of the membrane and could flip-flop in the same bilayer. In contrast, fatty acids flip-flopped between adjacent leaflets of two bilayers in which the tails crossed the thinner headgroup region of the membrane. To conclude, our SC model provides mechanistic insights into lipid mobility and is flexible in its design and composition of different lipids, enabling studies of varying skin conditions.

Measuring Asymmetric Tails Under Copula Distributions

Extensive evidence has been gathered showcasing the prevalence of heavy-tailed distributions and asymmetric tail interdependence within equity and foreign exchange markets, particularly during times of crisis. Tail interdependence in financial markets often manifests as financial contagion, characterized by periods where declining prices and heightened volatility propagate across economic and financial sectors. This phenomenon causes markets that typically exhibit minimal or no correlation to behave similarly, often in opposition to fundamental principles. Instances of unwarranted contagion present a perplexing challenge, suggesting irrationality among market participants and defying conventional risk management strategies and optimal portfolio selections. Our objective is to construct a comprehensive framework for dissecting such occurrences, utilizing a metric of tail-nonexchangeable dependencies employing various copulas with diverse marginal distributions. We offer analytical insights into our measurement of tail order dependence to aid in understanding these events.

New Bivariate Copulas via Lomax Distribution Generated Distortions

We develop a framework for creating distortion functions that are used to construct new bivariate copulas. It is achieved by transforming non-negative random variables with Lomax-related distributions. In this paper, we apply the distortions to the base copulas of independence, Clayton, Frank, and Gumbel copulas. The properties of the tail dependence coefficient, tail order, and concordance ordering are explored for the new families of distorted copulas. We conducted an empirical study using the daily net returns of Amazon and Google stocks from January 2014 to December 2023. We compared the popular Clayton, Gumbel, Frank, and Gaussian copula models to their corresponding distorted copula models induced by the unit-Lomax and unit-inverse Pareto distortions. The new families of distortion copulas are equipped with additional parameters inherent in the distortion function, providing more flexibility, and are demonstrated to perform better than the base copulas. After analyzing the data, we have found that the joint extremes of Amazon and Google stocks are more likely for high daily net returns than for low daily net returns.

Analyzing Lipid Membrane Defects via a Coarse-Grained to Triangulated Surface Map: The Role of Lipid Order and Local Curvature in Molecular Binding.

Lipid packing defects are known to serve as quantitative indicators for protein binding to lipid membranes. This paper presents a protocol for mapping molecular lipid detail onto a triangulated continuum leaflet representation. Besides establishing the desired forward counterpart to the existing inverse TS2CG map, this coarse-grained to triangulated surface (CG2TS) map enables straightforward extraction of the defect characteristics for any membrane geometry found in nature. We have applied our protocol to investigate the role of local curvature and varying lipid packing on the defect constant π. We find that the defect size is greatly influenced by both factors, arguing strongly against the usual assignment of a single defect constant in the case of more realistic membrane conditions. An important discovery is that lipids in the gel phase produce larger defects, or a higher π, in domains of high (local) curvature than the same lipid in a liquid phase of any curvature. This finding suggests that membranes featuring very ordered lipid packing can bind proteins via large defects in curved regions. Finally, we propose a route for estimating defect constants directly from the standard membrane properties. Identifying the precise role of composition, lipid (tail) order, and (local) curvature in defects for the irregular lipid structures that are (temporally) present in many biological processes is instrumental for obtaining fundamental insight as well as for a rational design of membrane binding targets.

Origin of the nonlinear structural and mechanical properties in oppositely curved lipid mixtures.

Structural and mechanical properties of membranes such as thickness, tail order, bending modulus and curvature energetics play crucial role in controlling various cellular functions that depend on the local lipid organization and membrane reshaping. While behavior of these biophysical properties are well understood in single component membranes, very little is known about how do they change in the mixed lipid membranes. Often various properties of the mixed lipid bilayers are assumed to change linearly with the mole fractions of the constituent lipids which, however, is true for "ideal" mixing only. In this study, using molecular dynamics simulations, we show that structural and mechanical properties of binary lipid mixture change nonlinearly with the lipid mole fractions, and the strength of the nonlinearity depends on two factors - spontaneous curvature difference and locally inhomogeneous interactions between the lipid components.

Extreme Behaviors of the Tail Gini-Type Variability Measures

AbstractFor a bivariate random vector $(X, Y)$, suppose $X$ is some interesting loss variable and $Y$ is a benchmark variable. This paper proposes a new variability measure called the joint tail-Gini functional, which considers not only the tail event of benchmark variable $Y$, but also the tail information of $X$ itself. It can be viewed as a class of tail Gini-type variability measures, which also include the recently proposed tail-Gini functional. It is a challenging and interesting task to measure the tail variability of $X$ under some extreme scenarios of the variables by extending the Gini's methodology, and the two tail variability measures can serve such a purpose. We study the asymptotic behaviors of these tail Gini-type variability measures, including tail-Gini and joint tail-Gini functionals. The paper conducts this study under both tail dependent and tail independent cases, which are modeled by copulas with so-called tail order property. Some examples are also shown to illuminate our results. In particular, a generalization of the joint tail-Gini functional is considered to provide a more flexible version.

Phospholipid Monolayer/Graphene Interfaces: Curvature Effect on Lipid Morphology and Dynamics.

Phospholipids are an important class of lipids that are widely used as model platforms for the study of biological processes and interactions. These lipids can form stable interfaces with solid substrates, such as graphene, and these interfaces have potential applications in biosensing and targeted drug delivery. In this paper, we perform molecular dynamics simulations of graphene-supported lipid monolayers to characterize the lipid properties of such interfaces. We observed substantial differences between the supported monolayer and free-standing bilayer in terms of the lipid properties, such as the tail order parameters, density profiles, diffusion rates, and so on. Furthermore, we studied these interfaces on sinusoidally deformed graphene substrates to understand the effect of curvature on the supported lipids. Here, we observed that the nature of the substrate curvature, that is, concave or convex, can locally affect the lipid/substrate adhesion strength and induce structural and dynamic changes in the adsorbed lipid monolayer. Together, these results help characterize the properties of lipid/graphene interfaces and provide insights into the substrate curvature effect on these interfaces, which can enable the tuning of lipid properties for various sensor devices and drug delivery applications.

Molecular dynamics simulations of the human ocular lens with age and cataract

The human ocular lens consists primarily of elongated, static fibers characterized by high stability and low turnover, which differ dramatically in their composition and properties from other biological membranes. Cholesterol (Chol) and sphingolipids (SL) are present at high concentrations, including saturated SLs, such as dihyrosphingomyelin (DHSM). Past molecular dynamics simulations demonstrated that the presence of DHSM and high Chol concentration contributes to higher order in lipid membranes. This current study simulated more complex models of human lens membranes. Models were developed representing physiological compositions in cataractous lenses aged 74 ± 6 years and in healthy lenses aged 22 ± 4, 41 ± 6, and 69 ± 3 years. With older age, Chol and ceramide concentrations increase and glycerophospholipid concentration decreases. With cataract, ceramide concentration increases and Chol and glycerophospholipid concentrations decrease. Surface area per lipid, deuterium order parameters (SCD), sterol tilt angle, electron density profiles, bilayer thickness, chain interdigitation, two-dimensional radial distribution functions (2D-RDF), lipid clustering, and hydrogen bonding were calculated for all simulations. All systems exhibited low surface area per lipid and high bilayer thickness, indicative of strong vertical packing. SCD parameters suggest similarly, with saturated tails in the hydrophobic core of the membrane having elevated order. Vertical packing and acyl tail order increased with both age and cataract condition. Lateral diffusion decreased with age and cataracts, with the older and cataractous models demonstrating increased long-range structure by the 2D-RDF analysis. In future work examining the membrane proteins of the lens, these models can serve as a physiologically accurate representation of the lens lipidome.

Stochastic comparisons on extremes of burr type XII samples associated with Archimedean copula and heterogeneous shape parameters

This study deals with random variables from Burr Type XII samples having heterogeneous shape parameters and homogeneous frailty parameters. For samples whose survival copula (whose copula) is Archimedean copula, sufficient conditions based on the heterogeneity of shape parameter vector for the stochastic tail ordering and usual stochastic ordering between sample minimums (sample maximums) are provided. As for independent samples, the convex transform order and tail hazard rate order (tail reversed hazard rate order) between sample minimums (maximums) are discussed. Several numerical examples are also presented to illustrate the theoretical findings.

Bacterial Membranes Are More Perturbed by the Asymmetric Versus Symmetric Loading of Amphiphilic Molecules.

Characterizing the biophysical properties of bacterial membranes is critical for understanding the protective nature of the microbial envelope, interaction of biological membranes with exogenous materials, and designing new antibacterial agents. Presented here are molecular dynamics simulations for two cationic quaternary ammonium compounds, and the anionic and nonionic form of a fatty acid molecule interacting with a Staphylococcus aureus bacterial inner membrane. The effect of the tested materials on the properties of the model membranes are evaluated with respect to various structural properties such as the lateral pressure profile, lipid tail order parameter, and the bilayer’s electrostatic potential. Conducting asymmetric loading of molecules in only one leaflet, it was observed that anionic and cationic amphiphiles have a large impact on the Staphylococcus aureus membrane’s electrostatic potential and lateral pressure profile as compared to a symmetric distribution. Nonintuitively, we find that the cationic and anionic molecules induce a similar change in the electrostatic potential, which points to the complexity of membrane interfaces, and how asymmetry can induce biophysical consequences. Finally, we link changes in membrane structure to the rate of electroporation for the membranes, and again find a crucial impact of introducing asymmetry to the system. Understanding these physical mechanisms provides critical insights and viable pathways for the rational design of membrane-active molecules, where controlling the localization is key.

Remarks on the tail order on moment sequences

We consider positively supported Borel measures for which all moments exist. On the set of compactly supported measures in this class a partial order is defined via eventual dominance of the moment sequences. Special classes are identified on which the order is total, but it is shown that already for the set of distributions with compactly supported smooth densities the order is not total. In particular we construct a pair of measures with smooth density for which infinitely many moments agree and another one for which the moments alternate infinitely often. This disproves some recently published claims to the contrary. Some consequences for games with distributional payoffs are discussed.

Computational Study on the Microscopic Adsorption Characteristics of Linear Alkylbenzene Sulfonates with Different Chain Lengths on Anthracite Surface

In order to explore the influence of different lengths of hydrophobic carbon chains on the diffusion characteristics of surfactants on the surface of anthracite, six linear alkyl benzene sulfonates with different hydrophobic carbon chain lengths were selected (mC, m = 8, 10, 12, 14, 16, 18; m represents the numbers of carbon atoms in the hydrophobic carbon chain), and molecular dynamics (MD) simulations were adopted. Models of surfactant-anthracite, surfactant-graphite layer, and water-surfactant-anthracite were constructed. After analyzing a series of properties such as adsorption energy, diffusion coefficient, radial distribution function (RDF), and hydrophobic tail order parameters, it was found that 12C had the highest adsorption strength on the surface of anthracite; the reason was that 12C had the highest degree of aggregation near the oxygen-containing functional groups on the surface of anthracite. Further studies had found that the hydrophobic tail chain of 12C had the strongest isotropy. The study fills the gap in the systematic study of the diffusion characteristics of linear alkylbenzene sulfonates (LAS) with different chain lengths on the surface of anthracite, enriches and develops the basic theory of coal wettability, and also provides technical ideas for the design of new surfactants and new dust suppression agents.

Tail dependence functions of the bivariate Hüsler–Reiss model

The tail dependence coefficient, a special case of the tail dependence function, measures extremal dependence between two random variables. It is known that the tail dependence coefficient of bivariate Gaussian random vectors with constant correlation coefficient |ρ|<1 is zero with rate satisfying regular variation at zero. In this note, we consider the tail dependence function of bivariate Gaussian random vectors with dynamic correlation coefficient ρn satisfying the Hüsler–Reiss condition. We also establish the convergence rates to the tail dependence functions under some refined Hüsler–Reiss conditions. By-products are the tail orders and tail order functions with related expansions.

Stochastic comparison of parallel systems with Pareto components

AbstractPareto distribution is an important distribution in extreme value theory. In this paper, we consider parallel systems with Pareto components and study the effect of heterogeneity on skewness of such systems. It is shown that, when the lifetimes of components have different shape parameters, the parallel system with heterogeneous Pareto component lifetimes is more skewed than the system with independent and identically distributed Pareto components. However, for the case when the lifetimes of components have different scale parameters, the result gets reversed in the sense of star ordering. We also establish the relation between star ordering and dispersive ordering by extending the result of Deshpande and Kochar [(1983). Dispersive ordering is the same as tail ordering. Advances in Applied Probability 15(3): 686–687] from support $(0, \infty )$ to general supports $(a, \infty )$, $a &gt; 0$. As a consequence, we obtain some new results on dispersion of order statistics from heterogeneous Pareto samples with respect to dispersive ordering.

Tail dependence and heavy tailedness in extreme risks

In the modeling of multivariate extreme risks, the tail dependence and the heavy tailedness are the two key factors. Heavy tailedness are usually defined through the regular variation. Tail dependence can be modeled by copulas with the so-called tail order property. In this paper, we propose a new risk measure called the Joint Expected Shortfall (JES) as an alternative of quantifying extreme risks. The JES, which can be viewed as a consolidation of both Expected Shortfall (ES) and Marginal Expected Shortfall (MES) risk measures, has the desirable property of measuring risk by jointly capturing both tail dependence and heavy tailedness. The asymptotic analysis of JES is conducted to provide a simple and transparent way of studying the interplay between tail dependence and heavy tailedness. Various examples are presented to illuminate our results. In particular, risk measures such as ES and MES that ignore the joint effect of dependence and heavy tailedness may severely underestimate the underlying risk.

Molecular Simulations of Ocular Lens Membranes at Various Stages of Age and Cataract

The human ocular lens consists primarily of elongated, static fibers characterized by high stability and low turnover, which differ dramatically in their composition and properties from other biological membranes. Cholesterol (Chol) and sphingolipids (SL) are present at high concentrations, including saturated SLs, such as dihyrosphingomyelin. Past molecular dynamics (MD) simulations demonstrated that the presence of DHSM and high Chol concentration contributes to higher order in lipid membranes. Using the CHARMM36 all-atom force field, this study simulated more complex models of human lens membranes with the inclusion of lysolipids and plasmanyl lipids. Models were developed representing physiological compositions in cataractous lenses aged 74±6 years and in healthy lenses aged 22±4, 41±6, and 69±3 years. With older age, Chol and ceramide concentrations increase and glycerophospholipid concentration decreases. With cataract, ceramide concentration increases and Chol and glycerophospholipid concentrations decrease. Surface area per lipid, deuterium order parameters (SCD), sterol tilt angle, electron density profiles (EDP), bilayer thickness, chain interdigitation, two-dimensional radial distribution functions (2D-RDF), mean-square displacement, lipid clustering, and hydrogen bonding were calculated for all simulations. All systems exhibited low surface area per lipid and high bilayer thickness, indicative of strong vertical packing. SCD parameters suggest similarly, with saturated tails in the hydrophobic core of the membrane having order parameter exceeding 0.4. Vertical packing and acyl tail order increased with both age and cataract. Lateral diffusion decreased with age and cataracts, with the older and cataractous models demonstrating increased long-range structure by the 2D-RDF analysis. Experimental results have found crystallization of Chol in older lenses, which similarly indicates increased order. In future work examining the membrane proteins of the lens, these models can serve as physiologically accurate representations of the lens lipidome.