Phase behaviour and structure of a model biomolecular condensate.

Phase separation of immiscible fluids is a common phenomenon in polymer chemistry, and is recognized as an important mechanism by which cells compartmentalize their biochemical reactions. Biomolecular condensates are condensed fluid droplets in cells that form by liquid-liquid phase separation of intrinsically-disordered proteins. They have a wide range of functions and are associated with chronic neurodegenerative diseases in which they become pathologically rigid. However, it remains unclear how their material properties depend on the molecular structure of the proteins. Here we explore the phase behaviour and structure of a model biomolecular condensate composed of semi-flexible polymers with attractive end-caps using coarse-grained simulations. The model contains the minimal molecular features that are sufficient to observe liquid-liquid phase separation of soluble polymers into a porous, three-dimensional network in which their end-caps reversibly bind at junctions. The distance between connected junctions scales with the polymer length as a self-avoiding random walk over a wide range of concentration with a weak affinity-dependent prefactor. By contrast, the average number of polymers that meet at the junctions depends on the end-cap affinity but only weakly on the polymer length. The structured porosity of the condensed phase suggests a mechanism for cells to regulate biomolecular condensates. Protein interaction sites may be turned on or off to modulate the condensate's porosity and therefore the diffusion and interaction of additional proteins.

Exploring the raft-hypothesis by probing planar bilayer patches of free-standing giant vesicles at nanoscale resolution, with and without Na,K-ATPase.

The structure of functional lipid domains (rafts) in biological membranes has for long time been unresolved due to their small length scales and transient nature. These cooperative properties of the lipid bilayer matrix are modelled by free-standing giant unilammellar vesicles (GUVs) with well-characterized lipid composition. We review a series of recent advances in preparation and analysis of GUVs, which allows for characterization of small domains by high-resolution imaging techniques. These includes a new GUV preparation method with a desired overall lipid composition achieved by mixing small unilammellar vesicles (SUVs), test of the lipids compositional uniformity in GUVs and swift adsorption of GUVs to solid support by kinetically arresting the lateral structure of membrane prior to collapse for subsequent imaging. The techniques are applied to the analysis of membrane domains in GUVs formed from mixtures of DOPC/DPPC/cholesterol with and without Na,K-ATPase (NKA), a transmembrane protein known to be associated with rafts. Two mechanisms of domain formation are revealed: 1) close to lo/ld phase coexistence, domains in size up to 100nm appear as thermally induced droplet fluctuations, 2) NKA shows interfacial activity and cluster in lo/ld micro-emulsion droplets. Some perspectives for the application of the techniques and the understanding of the nature of raft domains are outlined.

Numerical insights into the phase diagram of p-atic membranes with spherical topology.

The properties of self-avoiding p-atic membranes restricted to spherical topology have been studied by Monte Carlo simulations of a triangulated random surface model. Spherically shaped p-atic membranes undergo a Kosterlitz-Thouless transition as expected with topology induced mutually repelling disclinations of the p-atic ordered phase. For flexible membranes the phase behaviour bears some resemblance to the spherically shaped case with a p-atic disordered crumpled phase and p-atic ordered, conformationally ordered (crinkled) phase separated by a KT-like transition with proliferation of disclinations. We confirm the proposed buckling of disclinations in the p-atic ordered phase, while the expected associated disordering (crumpling) transition at low bending rigidities is absent in the phase diagram.

Membrane invagination induced by Shiga toxin B-subunit: from molecular structure to tube formation.

The bacterial Shiga toxin is composed of an enzymatically active A-subunit, and a receptor-binding homopentameric B-subunit (STxB) that mediates intracellular toxin trafficking. Upon STxB-mediated binding to the glycolipid globotriaosylceramide (Gb3) at the plasma membrane of target cells, Shiga toxin is internalized by clathrin-dependent and independent endocytosis. The formation of tubular membrane invaginations is an essential step in the clathrin-independent STxB uptake process. However, the mechanism by which STxB induces these invaginations has remained unclear. Using a combination of all-atom molecular dynamics and Monte Carlo simulations we show that the molecular architecture of STxB enables the following sequence of events: the Gb3 binding sites on STxB are arranged such that tight avidity-based binding results in a small increment of local curvature. Membrane-mediated clustering of several toxin molecules then creates a tubular membrane invagination that drives toxin entry into the cell. This mechanism requires: (1) a precise molecular architecture of the STxB binding sites; (2) a fluid bilayer in order for the tubular invagination to form. Although, STxB binding to the membrane requires specific interactions with Gb3 lipids, our study points to a generic molecular design principle for clathrin-independent endocytosis of nanoparticles.

Scrotal impaling injury causing ascending colon perforation and retroperitoneal fistula.

Impalement injuries though relatively uncommon are some of the most dramatic and are known to cause significant damage due to the forces involved, the combination of blunt and penetrating mechanisms, and wound contamination. They generally occur following falls from a height, motor vehicle accidents or other high velocity mechanisms. Their management requires specific consideration to the prehospital management of the impaling object, management of any organs injured and appropriate debridement and washout of the tract. We report a case of a scrotal impalement traversing the abdominal cavity and causing a colonic injury and a rib fracture which resulted in a chronic discharging sinus from the patient's retroperitoneum.

Advantages of statistical analysis of giant vesicle flickering for bending elasticity measurements.

We show how to greatly improve precision when determining bending elasticity of giant unilamellar vesicles. Taking advantage of the well-known quasi-spherical model of liposome flickering, we analyze the full probability distributions of the configurational fluctuations instead of limiting the analysis to the second moment measurements only as usually done in previously published works. This leads to objective criteria to reject vesicles that do not behave according to the model. As a result, the confidence in the bending elasticity determination of individual vesicles that fit the model is improved and, consequently, the reproducibility of this measurement for a given membrane system. This approach uncovers new possibilities for bending elasticity studies like detection of minute influences by solutes in the buffer or into the membrane. In the same way, we are now able to detect the inhomogeneous behavior of giant vesicle systems such as the hazardous production of peroxide in bilayers containing fluorescent dyes.

Nonlinearity-generated resilience in large complex systems.

We consider a generic nonlinear extension of May's 1972 model by including all higher-order terms in the expansion around the chosen fixed point (placed at the origin) with random Gaussian coefficients. The ensuing analysis reveals that as long as the origin remains stable, it is surrounded by a "resilience gap": there are no other fixed points within a radius r_{*}>0 and the system is therefore expected to be resilient to a typical initial displacement small in comparison to r_{*}. The radius r_{*} is shown to vanish at the same threshold where the origin loses local stability, revealing a mechanism by which systems close to the tipping point become less resilient. We also find that beyond the resilience radius the number of fixed points in a ball surrounding the original point of equilibrium grows exponentially with N, making systems dynamics highly sensitive to far enough displacements from the origin.

Lipid domains in giant unilamellar vesicles and their correspondence with equilibrium thermodynamic phases: a quantitative fluorescence microscopy imaging approach.

We report a novel analytical procedure to measure the surface areas of coexisting lipid domains in giant unilamellar vesicles (GUVs) based on image processing of 3D fluorescence microscopy data. The procedure involves the segmentation of lipid domains from fluorescent image stacks and reconstruction of 3D domain morphology using active surface models. This method permits the reconstruction of the spherical surface of GUVs and determination of the area fractions of coexisting lipid domains at the level of single vesicles. Obtaining area fractions enables the scrutiny of the lever rule along lipid phase diagram's tie lines and to test whether or not the coexistence of lipid domains in GUVs correspond to equilibrium thermodynamic phases. The analysis was applied to DLPC/DPPC GUVs displaying coexistence of lipid domains. Our results confirm the lever rule, demonstrating that the observed membrane domains correspond to equilibrium thermodynamic phases (i.e., solid ordered and liquid disordered phases). In addition, the fact that the lever rule is validated from 11 to 14 randomly selected GUVs per molar fraction indicates homogeneity in the lipid composition among the explored GUV populations. In conclusion, our study shows that GUVs are reliable model systems to perform equilibrium thermodynamic studies of membranes.

Consequences of Dale's law on the stability-complexity relationship of random neural networks.

In the study of randomly connected neural network dynamics there is a phase transition from a simple state with few equilibria to a complex state characterized by the number of equilibria growing exponentially with the neuron population. Such phase transitions are often used to describe pathological brain state transitions observed in neurological diseases such as epilepsy. In this paper we investigate how more realistic heterogeneous network structures affect these phase transitions using techniques from random matrix theory. Specifically, we parametrize the network structure according to Dale's law and use the Kac-Rice formalism to compute the change in the number of equilibria when a phase transition occurs. We also examine the condition where the network is not balanced between excitation and inhibition causing outliers to appear in the eigenspectrum. This enables us to compute the effects of different heterogeneous network connectivities on brain state transitions, which can provide insights into pathological brain dynamics.

C24 Sphingolipids Govern the Transbilayer Asymmetry of Cholesterol and Lateral Organization of Model and Live-Cell Plasma Membranes.

Mammalian sphingolipids, primarily with C24 or C16 acyl chains, reside in the outer leaflet of the plasma membrane. Curiously, little is known how C24 sphingolipids impact cholesterol and membrane microdomains. Here, we present evidence that C24 sphingomyelin, when placed in the outer leaflet, suppresses microdomains in giant unilamellar vesicles and also suppresses submicron domains in the plasma membrane of HeLa cells. Free energy calculations suggested that cholesterol has a preference for the inner leaflet if C24 sphingomyelin is in the outer leaflet. We indeed observe that cholesterol enriches in the inner leaflet (80%) if C24 sphingomyelin is in the outer leaflet. Similarly, cholesterol primarily resides in the cytoplasmic leaflet (80%) in the plasma membrane of human erythrocytes where C24 sphingolipids are naturally abundant in the outer leaflet. We conclude that C24 sphingomyelin uniquely interacts with cholesterol and regulates the lateral organization in asymmetric membranes, potentially by generating cholesterol asymmetry.

The association of body height, height variability and inequality.

Body height is associated with environmental conditions. It has been suggested that under poor conditions when inequality within a population increases, also the variability in height tends to increase. We studied the association of body height, within-country variability in height and geographic and historic origin in 767 growth studies carried out in 80 countries, published between 1794 and 2013, with data on N = 78,184 infants age 2 years, and N = 2,130,729 juveniles age 7 years. The studies represent almost the whole spectrum of economic diversity in human societies since the end-18(th) century. 207 studies contained data for both infants and juveniles with 50,819 subjects (age 2), and 123,078 subjects (age 7). Multiple linear regressions showed significant interactions between height, sex, historic year of the study, geographic origin, and within-study standard deviation for height with multiple R-squared = 0.527, p < 0.001, at age 2, and multiple R-squared = 0.436, p < 0.001, at age 7. Yet, the two age groups differed in respect to within-study standard deviation for height. We found a significant association between body height and within-study standard deviation for height only at age 2: tall infant populations are less variable in height (r = -0.27, p < 0.01). There was no such association in children aged 7 years. Tall children from affluent and short children from less affluent countries do not differ in the variability of body height. The data suggest that the 'environmental adversity' hypothesis for variation in growth: small mean values for height go along with large standard deviations for height, does not apply for children at age 7.

Modelling the phase equilibria in two-component membranes of phospholipids with different acyl-chain lengths.

A phenomenological model is proposed to describe the membrane phase equilibria in binary mixtures of saturated phospholipids with different acyl-chain lengths. The model is formulated in terms of thermodynamic and thermomechanic properties of the pure lipid bilayers, specifically the chain-melting transition temperature and enthalpy, the hydrophobic bilayer thickness, and the lateral area compressibility modulus. The model is studied using a regular solution theory made up of a set of interaction parameters which directly identify that part of the lipid-lipid interaction which is due to hydrophobic mismatch of saturated chains of different lengths. It is then found that there is effectively a single universal interaction parameter which, in the full composition range, describes the phase equilibria in mixtures of DMPC/DPPC, DPPC/DSPC, DMPC/DSPC, and DLPC/DSPC, in excellent agreement with experimental measurements. The model is used to predict the variation with temperature and composition of the specific heat, as well as of the average membrane thickness and area in each of the phases. Given the value of the universal interaction parameter, the model is then used to predict the phase diagrams of binary mixtures of phospholipids with different polar head groups, e.g., DPPC/DPPE, DMPC/DPPE and DMPE/DSPC. By comparison with experimental results for these mixtures, it is shown that difference in acyl-chain lengths gives the major contribution to deviation from ideal mixing. Application of the model to mixtures with non-saturated lipids is also discussed.

Intrinsic molecules in lipid membranes change the lipid-domain interfacial area: cholesterol at domain interfaces.

A theoretical analysis of the effects of intrinsic molecules on the lateral density fluctuations in lipid bilayer membranes is carried out by means of computer simulations on a microscopic interaction model of the gel-to-fluid chain-melting phase transition. The inhomogeneous equilibrium structures of gel and fluid domains, which in previous work (Cruzeiro-Hansson, L. and Mouritsen, O.G. (1988) Biochim. Biophys. Acta 944, 63-72) were shown to characterize the transition region of pure lipid membranes, are here shown to be enhanced by intrinsic molecules such as cholesterol. Cholesterol is found to increase the interfacial area and to accumulate in the interfaces. The interfacial area, the average cluster size, the lateral compressibility, and the membrane area are calculated as functions of temperature and cholesterol concentration. It is shown that the enhancement by cholesterol of the lateral density fluctuations and the lipid-domain interfacial area is most pronounced away from the transition temperature. The implications of the results are discussed in relation to passive ion permeability and function of interfacially active enzymes such as phospholipase.

Phase equilibria and local structure in binary lipid bilayers.

A molecular interaction model is used to describe the phase diagram of two-component phospholipid bilayer membranes of saturated phospholipids, DCnPC, with different acyl-chain lengths, n = 12,14,18,20. The interaction between acyl chains of different length is formulated in terms of a hydrophobic mismatch which permits the series of binary phase diagrams to be calculated in terms of a single 'universal' interaction parameter. The properties of the model are calculated by computer-simulation techniques which not only permit determination of the specific-heat function and the phase diagram but also reveal the local structure of the mixture in the different parts of the phase diagram. The local structure is described pictorially and characterized quantitatively in terms of a correlation function. It is shown that the non-ideal mixing of lipid species due to mismatch in the hydrophobic lengths leads to a progressively increasing local ordering as the chain-length difference is increased. A pronounced local structure is found to persist deep inside the fluid phase of the mixture. The local structure is discussed in relation to the features observed in the specific-heat function, for which theoretical data, as well as experimental data obtained from differential-scanning calorimetry are presented.

A general model for the interaction of foreign molecules with lipid membranes: drugs and anaesthetics.

A general microscopic interaction model is proposed to describe the changes in the physical properties of phospholipid bilayer membranes due to foreign molecules which, to different degrees, partition between the membrane phases and the aqueous environment. The model is a multi-state lattice model for the main phase transition of lipid bilayers and the foreign molecules are assumed to intercalate as interstitials in the lattice. By varying the model parameters, the diversity in the thermodynamic properties of the model is explored using computer-simulation techniques which faithfully take account of the thermal fluctuations. The calculations are performed in both the canonical and the grand canonical ensembles corresponding to the cases where the concentration of foreign molecules in the membrane is either fixed or varies as the external conditions are changed. A classification of the diverse thermal behaviour, specifically with regard to the phase diagram, the specific heat, the density fluctuations, and the partition coefficient, is suggested with a view to rationalizing a large body of experimental measurements of the effects of different foreign molecules on membrane properties. The range of foreign molecules considered includes compounds as diverse as volatile general anaesthetics like halothane, cocaine-derived local anaesthetics like procaine, calcium-channel blocking drugs like verapamil, antidepressants like chlorpromazine, and anti-cancer agents like adriamycin.

The effects of density fluctuations on the partitioning of foreign molecules into lipid bilayers: application to anaesthetics and insecticides.

An extensive computer-simulation study is performed on a simple but general molecular model recently proposed (Jørgensen et al. (1991) Biochem. Biophys. Acta 1062, 277-238) to describe foreign molecules interacting with lipid bilayers. The model is a multi-state lattice model of the main bilayer transition in which the foreign molecules are assumed to intercalate at interstitial lattice positions. Specific as well as non-specific interactions between the foreign molecules and the lipid acyl chains are considered. Particular attention is paid to the fluctuating properties of the membrane and how the presence of the foreign molecules modulates these fluctuations in the transition region. By means of computer-stimulation techniques, a detailed account is given of the macroscopic as well as microscopic consequences of the fluctuations. The macroscopic consequences of the fluctuations are seen in the thermal anomalies of the specific heat and the passive trans-membrane permeability. Microscopically, the fluctuations manifest themselves in lipid-domain formation in the transition region which implies an effective dynamic membrane heterogeneity. Within the model it is found that certain anaesthetics and insecticides which are characterised by specific interactions with the lipids have a strong effect on the heterogeneity of the membrane inducing regions of locally very high concentration of the foreign molecules. This leads to a broadening of the specific heat peak and a maximum in the membrane/water partition coefficient. These results are in accordance with available experimental data for volatile general anaesthetics like halothane, local anaesthetics like cocain derivatives, and insecticides like lindane.

Phase equilibria in the phosphatidylcholine-cholesterol system.

A thermodynamic and a microscopic interaction model are proposed to describe the phase equilibria in the phosphatidylcholine-cholesterol system. The model calculations allow for a solid phase with conformationally ordered acyl chains and liquid phases with conformationally ordered as well as disordered chains. The resulting phase diagram is in excellent agreement with the experimental phase diagram for dipalmitoylphosphatidylcholine bilayers with cholesterol as determined by a recent NMR and calorimetry study. It is thus demonstrated that the phase behaviour of phosphatidylcholine-cholesterol mixtures can be rationalized using only a few basic assumptions: (i) Cholesterol interacts favourably with phosphatidylcholine chains in an extended conformation, (ii) the main transition of pure phosphatidylcholine bilayers takes place in terms of translational variables as well acyl-chain conformational variables, and (iii) cholesterol disturbs the translational order in the crystalline (gel) state of phosphatidylcholine. These results suggest that the occurrence of specific phosphatidylcholine-cholesterol complexes is not implied by the experimental thermodynamic data.

Theoretical analysis of protein organization in lipid membranes.

The fundamental physical principles of the lateral organization of trans-membrane proteins and peptides as well as peripheral membrane proteins and enzymes are considered from the point of view of the lipid-bilayer membrane, its structure, dynamics, and cooperative phenomena. Based on a variety of theoretical considerations and model calculations, the nature of lipid-protein interactions is considered both for a single protein and an assembly of proteins that can lead to aggregation and protein crystallization in the plane of the membrane. Phenomena discussed include lipid sorting and selectivity at protein surfaces, protein-lipid phase equilibria, lipid-mediated protein-protein interactions, wetting and capillary condensation as means of protein organization, mechanisms of two-dimensional protein crystallization, as well as non-equilibrium organization of active proteins in membranes. The theoretical findings are compared with a variety of experimental data.

Model of a sub-main transition in phospholipid bilayers.

A recently discovered submain phase transition in multi-lamellar bilayers of long-chain saturated diacyl phosphatidylcholines (Jørgensen, K. (1995) Biochim. Biophys. Acta 1240, 111-114) is discussed in terms of a theoretical molecular interaction model using computer simulation techniques. The model interprets the transition to be due to a decoupling of the acyl-chain melting from the melting of the pseudo-two-dimensional crystalline lattice of the P beta' phase. A two-stage melting process is predicted by the calculations suggesting that the sub-main transition involves a lattice melting whereas the acyl-chain melting takes place at a higher temperature at the main transition. The calculated heat contents of the two transitions as well as the chain-length dependence compare favorably with experimental data for multi-lamellar phosphatidylcholine lipid bilayers.

Characterization of the elastic properties of the nuclear envelope.

Underlying the nuclear envelope (NE) of most eukaryotic cells is the nuclear lamina, a meshwork consisting largely of coiled-coil nuclear intermediate filament proteins that play a critical role in nuclear organization and gene expression, and are vital for the structural stability of the NE/nucleus. By confocal microscopy and micromanipulation of the NE in living cells and isolated nuclei, we show that the NE undergoes deformations without large-scale rupture and maintains structural stability when exposed to mechanical stress. In conjunction with image analysis, we have developed theory for a two-dimensional elastic material to quantify NE elastic behaviour. We show that the NE is elastic and exhibits characteristics of a continuous two-dimensional solid, including connections between lamins and the embedded nuclear pore complexes. Correlating models of NE lateral organization to the experimental findings indicates a heterogeneous lateral distribution of NE components on a mesoscopic scale.