Observation of broken detailed balance in polymorphic transformation of bacterial flagellar filament

Living systems operate far from thermodynamic equilibrium, which usually manifests as broken detailed balance at the molecular scale. At larger scales with collective function of many molecules, the presence of non-equilibrium thermodynamics may not be evident. In bacterial motility, the switching dynamics of the flagellar rotary motor was recently discovered to be operating in non-equilibrium. However, the resulting motility pattern at the mesoscale, the run-and-tumble behavior, was normally considered to be a Poisson process that can be described by a two-state equilibrium model. Here, we studied the details of the run-and-tumble behavior by following the polymorphic transformation of the flagellar filaments, observing broken detailed balance that reveals its non-equilibrium nature. Evaluation of entropy production provided a direct measure of the lack of detailed balance and a quantification of the rate of energy dissipation for bacterial run-and-tumble regulation.     

Computational Screening of Phase-separating Proteins

Phase separation is an important mechanism that mediates the compartmentalization of proteins in cells. Proteins that can undergo phase separation in cells share certain typical sequence features, like intrinsically disordered regions(IDRs) and multiple modular domains. Sequencebased analysis tools are commonly used in the screening of these proteins. However, current phase separation predictors are mostly designed for IDR-containing proteins, thus inevitably overlook the phase-separating proteins with relatively low IDR content. Features other than amino acid sequence could provide crucial information for identifying possible phase-separating proteins: protein–protein interaction(PPI) networks show multivalent interactions that underlie phase separation process; post-translational modifications(PTMs) are crucial in the regulation of phase separation behavior; spherical structures revealed in immunofluorescence(IF) images indicate condensed droplets formed by phase-separating proteins, distinguishing these proteins from non-phaseseparating proteins. Here, we summarize the sequence-based tools for predicting phaseseparating proteins and highlight the importance of incorporating PPIs, PTMs, and IF images into phase separation prediction in future studies.     

Application of a thermodynamic model to the prediction of phase separations in freeze-concentrated formulations for protein lyophilization

Many of the compounds considered for use in pharmaceutical formulations demonstrate incompatibilities with other components at high enough concentrations, including pairs of polymers, polymers and salts, or even proteins in combination with polymers, salts, or other proteins. Freeze concentration can force solutions into a region where incompatibilities between solutes will manifest as the formation of multiple phases. Such phase separation complicates questions of the stability of the formulation as well as labile components, such as proteins. Yet, phase separation events are difficult to identify by common formulation screening methods. In this report, we use the osmotic virial expansion model of Edmond and Ogston (1) to describe phase-separating behavior of ternary aqueous polymer solutions. Second osmotic virial coefficients of polyethylene glycol 3350 (PEG) and dextran T500 were measured by light scattering. Assuming an equilibrium between ice and water in the freeze-concentrated solution, a degree of freeze concentration can be estimated, which, when combined with the phase separation spinodal, describes a "phase separation envelope" in which phase separation tendencies can be expected in the frozen solution. The phase separation envelope is bounded at low temperatures by the glass transition temperature of the freeze-concentrated solution. Scanning electron microscopic images and infrared spectroscopy of protein structure are provided as experimental evidence of the phase separation envelope in a freeze-dried system of PEG, dextran, and hemoglobin.     

Rich Phase Separation Behavior of Biomolecules

Phase separation is a thermodynamic process leading to the formation of compositionally distinct phases. For the past few years, numerous works have shown that biomolecular phase separation serves as biogenesis mechanisms of diverse intracellular condensates, and aberrant phase transitions are associated with disease states such as neurodegenerative diseases and cancers. Condensates exhibit rich phase behaviors including multiphase internal structuring, noise buffering, and compositional tunability. Recent studies have begun to uncover how a network of intermolecular interactions can give rise to various biophysical features of condensates. Here, we review phase behaviors of biomolecules, particularly with regard to regular solution models of binary and ternary mixtures. We discuss how these theoretical frameworks explain many aspects of the assembly, composition, and miscibility of diverse biomolecular phases, and highlight how a model-based approach can help elucidate the detailed thermodynamic principle for multicomponent intracellular phase separation.     

Statistical analysis of channel current from a membrane patch. I. Some stochastic properties of ion channels or molecular systems in equilibrium

The stochastic behavior of single-channel current in a steady-state has been interpreted as the channel's state transitions between several open and shut states, and these transitions have been regarded as a homogeneous Markov process. When a channel is in equilibrium, the principle of detailed balance holds for every step in the state transition scheme. Here we show two stochastic properties of a channel, or any molecule obeying a reversible state transition scheme, under the constraint of detailed balance. First, the distribution functions and the probability density functions of shut or open dwell-time are expressed by the sum of exponential terms with positive coefficients. The same holds for the time-dependent open (or shut) frequency after the shut (or open) transition. Second, the time course of state transition from the state SI to SJ (PI,J(t] is proportional to its reverse transition time course (PJ,I(t], even if SI and SJ are widely separated. The same relation holds also for a transition scheme having transition pathways to the absorbing states. If analysis of a channel current record shows it to be incompatible with either of these two properties, the channel is not in equilibrium but in a steady-state with an energy-consuming cyclic flow. These two properties are also useful for the analysis of any molecular process obeying a homogeneous Markov process or a network of first-order chemical reactions.     

Quality Control of Membraneless Organelles

The formation of membraneless organelles (MLOs) by phase separation has emerged as a new way of organizing the cytoplasm and nucleoplasm of cells. Examples of MLOs forming via phase separation are nucleoli in the nucleus and stress granules in the cytoplasm. The main components of these MLOs are macromolecules such as RNAs and proteins. In order to assemble by phase separation, these proteins and RNAs have to undergo many cooperative interactions. These cooperative interactions are supported by specific molecular features within phase-separating proteins, such as multivalency and the presence of disordered domains that promote weak and transient interactions. However, these features also predispose phase-separating proteins to aberrant behavior. Indeed, evidence is emerging for a strong link between phase-separating proteins, MLOs, and age-related diseases. In this review, we discuss recent progress in understanding the formation, properties, and functions of MLOs. We pay special attention to the emerging link between MLOs and age-related diseases, and we explain how changes in the composition and physical properties of MLOs promote their conversion into an aberrant state. Furthermore, we discuss the key role of the protein quality control machinery in regulating the properties and functions of MLOs and thus in preventing age-related diseases.     

Thermotropic phase behavior of model membranes composed of phosphatidylcholines containing iso-branched fatty acids. 1. Differential scanning calorimetric studies

The thermotropic phase behavior of aqueous dispersions of phosphatidylcholines containing one of a series of methyl iso-branched fatty acyl chains was studied by differential scanning calorimetry. These compounds exhibit a complex phase behavior on heating which includes two endothermic events, a gel/gel transition, involving a molecular packing rearrangement between two gel-state forms, and a gel/liquid-crystalline phase transition, involving the melting of the hydrocarbon chains. The gel to liquid-crystalline transition is a relatively fast, highly cooperative process which exhibits a lower transition temperature and enthalpy than do the chain-melting transitions of saturated straight-chain phosphatidylcholines of similar acyl chain length. In addition, the gel to liquid-crystalline phase transition temperature is relatively insensitive to the composition of the aqueous phase. In contrast, the gel/gel transition is a slow process of lower cooperativity than the gel/liquid-crystalline phase transition and is sensitive to the composition of the bulk aqueous phase. The gel/gel transitions of the methyl iso-branched phosphatidylcholines have very different thermodynamic properties and depend in a different way on hydrocarbon chain length than do either the "subtransitions" or the "pretransitions" observed with linear saturated phosphatidylcholines. The gel/gel and gel/liquid-crystalline transitions are apparently concomitant for the shorter chain iso-branched phosphatidylcholines but diverge on the temperature scale with increasing chain length, with a pronounced odd/even alternation of the characteristic temperatures of the gel/gel transition.(ABSTRACT TRUNCATED AT 250 WORDS)     

Phase behavior of lipid bilayers under tension

Given the proposed importance of membrane tension in regulating cellular functions, we explore the effects of a finite surface tension on phase equilibrium using a molecular theory that captures the quantitative structure of the phase diagram of the tensionless DPPC/DOPC/Cholesterol lipid bilayer. We find that an increase in the surface tension decreases the temperature of the transition from liquid to gel in a pure DPPC system by ～1.0 K/(mN/m), and decreases the liquid-disordered to liquid-ordered transition at constant chemical potentials by approximately the same amount. Our results quantitatively isolate the role of tension in comparison to other thermodynamic factors, such as pressure, in determining the phase behavior of lipid bilayers.     

Effect of binding on melting transitions in polymer–diluent systems

An expression is derived for the melting point of a polymer when in equilibrium with a solution in which binding of low molecular weight compounds to the polymer takes place. Allowance is made for the possibility that the crystalline polymer itself is a complex. The argument is a purely thermodynamic one and is based on a consideration of the change in free energy as a result of a change in binding. Allowance is made also for non-specific polymer–solvent interactions, in which the mixture of low molecular weight solvents is treated as a single solvent. Special attention is paid to “inverted” melting transitions, i.e., cases in which the melting point increases with increasing dilution of the polymer. It is shown that as a rule this is accompanied by a corresponding, inverted effect of the solvent composition on the melting point. It is further shown that-in the absence of binding, “normal” behavior at the critical point (i.e., phase separation is induced by lowering the temperature) is always accompanied by “normal” melting behavior (i.e., a decrease in melting point when the polymer is diluted). Also, “inverted” melting always implies that phase separation at the critical point is induced by heating, but the reverse is not necessarily true.     

Isothermal lipid phase transitions

In liotropic lipid systems phase transitions can be induced isothermally by changing the solvent concentration or composition; alternatively, lipid composition can be modified by (bio)chemical means. The probability for isothermal phase transitions increases with the decreasing transition entropy; it is proportional to the magnitude of the transition temperature shift caused by transformation-inducing system variation. Manipulations causing large thermodynamic effects, such as lipid (de)hydration, binding of protons or divalent ions and macromolecular adsorption, but also close bilayer approach are, therefore, likely to cause structural lipid change(s) at a constant temperature. Net lipid charges enhance the membrane susceptibility to salt-induced isothermal phase transitions; a large proportion of this effect is due to the bilayer dehydration, however, rather than being a consequence of the decreased Coulombic electrostatic interactions. Membrane propensity for isothermal phase transitions, consequently, always increases with the hydrophilicity of the lipid heads, as well as with the desaturation and shortening of the lipid chains. Upon a phase change at a constant temperature, some of the interfacially bound solutes (e.g. protons or calcium) are released in the solution. Membrane permeability and fusogenicity simultaneously increase. In mixed systems, isothermal phase transitions, moreover, may result in lateral phase separation. All this opens up ways for the involvement of isothermal phase transitions in the regulation of biological processes.     

Self-assembly of rationally designed peptides under two-dimensional confinement

The rational design of interfacially confined biomolecules offers a unique opportunity to explore the cooperative relationship among self-assembly, nucleation, and growth processes. This article highlights the role of electrostatics in the self-assembly of β-sheet-forming peptides at the air-water interface. We characterize the phase behavior of a periodically sequenced sheet-forming peptide by using Langmuir techniques, Brewster angle microscopy, attenuated total reflection Fourier transform infrared spectroscopy, and circular dichroism spectroscopy. We find that peptides with an alternating binary sequence transition at high pressures from discrete circular domains to fibrous domains. The qualitative behavior is independent of surface pressure but dependent on molecular areas. In addition, thermodynamic models are employed to specifically quantify differences in electrostatics by obtaining parameters for the critical aggregation area, the limiting molecular area, and the dimensionless ratio of line tension/dipole density. Using these parameters, we are able to relate localized charge distribution to phase transitions, which will allow us to apply these molecules to examine how the dynamics of self-assembly can be directly coupled to the formation of composite nanostructures in biology.     

Mechanistic Inferences From Analysis of Measurements of Protein Phase Transitions in Live Cells

The combination of phase separation and disorder-to-order transitions can give rise to ordered, semi-crystalline fibrillar assemblies that underlie prion phenomena namely, the non-Mendelian transfer of information across cells. Recently, a method known as Distributed Amphifluoric Förster Resonance Energy Transfer (DAmFRET) was developed to study the convolution of phase separation and disorder-to-order transitions in live cells. In this assay, a protein of interest is expressed to a broad range of concentrations and the acquisition of local density and order, measured by changes in FRET, is used to map phase transitions for different proteins. The high-throughput nature of this assay affords the promise of uncovering sequence-to-phase behavior relationships in live cells. Here, we report the development of a supervised method to obtain automated and accurate classifications of phase transitions quantified using the DAmFRET assay. Systems that we classify as undergoing two-state discontinuous transitions are consistent with prion-like behaviors, although the converse is not always true. We uncover well-established and surprising new sequence features that contribute to two-state phase behavior of prion-like domains. Additionally, our method enables quantitative, comparative assessments of sequence-specific driving forces for phase transitions in live cells. Finally, we demonstrate that a modest augmentation of DAmFRET measurements, specifically time-dependent protein expression profiles, can allow one to apply classical nucleation theory to extract sequence-specific lower bounds on the probability of nucleating ordered assemblies. Taken together, our approaches lead to a useful analysis pipeline that enables the extraction of mechanistic inferences regarding phase transitions in live cells.     

Molecular interactions and thermotropic behavior of glycosphingolipids in model membrane systems

The oligosaccharide chain of glycosphingolipids (GSLs) has a marked influence on their thermotropic behavior, intermolecular packing and surface electrical potential. The transition temperature and enthalpy of GSLs decrease proportionally to the complexity of the polar head group and show a linear dependence with the intermolecular spacings. Interactions occurring among GSLs and phospholipids induce changes of the molecular area and surface potential that depend on the type of GSLs. Increasing proportions of phospholipids perturb the thermodynamic properties of the GSLs up to a point where phase separated phospholipid domains separate out but no phase separation of pure GSLs occurs. Heterogeneous equilibria among different structures occur for some systems. Large changes of the molecular free energy, eccentricity, asymmetry ratio and phase state of the GSLs-containing structure can be triggered by small changes of the molecular parameters, lipid composition and lateral surface pressure. The thermotropic behavior of GSLs is considerably perturbed by myelin basic protein. Phase separation occurs depending on the amount of protein and type of GSLs. The protein induces a decrease of the lipid molecular area, the more so the more complex the oligosaccharide chain in the GSLs. These membrane systems can not be described only on the basis of the individual properties of the molecules involved in a simple causal manner. Still scarcely explored long range thermodynamic, geometric and field effects that belong simultaneously to the intervening molecules, to the morphological properties of the structure involved and to the aqueous environment, are important determinants of their behavior.     

Head-on collisions of waves in an excitable FitzHugh-Nagumo system: a transition from wave annihilation to classical wave behavior

For the particular case of an excitable FitzHugh-Nagumo system with diffusion, we investigate the transition from annihilation to crossing of the waves in the head-on collision. The analysis exploits the similarity between the local and the global phase portraits of the system. We find that the transition has features typical of the nucleation theory of first-order phase transitions, and may be understood through purely geometrical arguments. In the case of periodic boundary conditions, the transition is an infinite-dimensional analog of the creation and the vanishing of limit cycles via a homoclinic Andronov bifurcation. Both before and after the transition, the behavior of a single cell continues to be typical for excitable systems: a stable equilibrium state, and a threshold above which an excitation pulse can be induced. The generality and qualitative character of our argument shows that the phenomenon described can be observed in excitable systems well beyond the particular case presented here.     

Targeted modulation of protein liquid–liquid phase separation by evolution of amino-acid sequence

Rationally and efficiently modifying the amino-acid sequence of proteins to control their ability to undergo liquid–liquid phase separation (LLPS) on demand is not only highly desirable, but can also help to elucidate which protein features are important for LLPS. Here, we propose a computational method that couples a genetic algorithm to a sequence-dependent coarse-grained protein model to evolve the amino-acid sequences of phase-separating intrinsically disordered protein regions (IDRs), and purposely enhance or inhibit their capacity to phase-separate. We validate the predicted critical solution temperatures of the mutated sequences with ABSINTH, a more accurate all-atom model. We apply the algorithm to the phase-separating IDRs of three naturally occurring proteins, namely FUS, hnRNPA1 and LAF1, as prototypes of regions that exist in cells and undergo homotypic LLPS driven by different types of intermolecular interaction, and we find that the evolution of amino-acid sequences towards enhanced LLPS is driven in these three cases, among other factors, by an increase in the average size of the amino acids. However, the direction of change in the molecular driving forces that enhance LLPS (such as hydrophobicity, aromaticity and charge) depends on the initial amino-acid sequence. Finally, we show that the evolution of amino-acid sequences to modulate LLPS is strongly coupled to the make-up of the medium (e.g. the presence or absence of RNA), which may have significant implications for our understanding of phase separation within the many-component mixtures of biological systems.     

Beta turn propensity and a model polymer scaling exponent identify intrinsically disordered phase-separating proteins

The complex cellular milieu can spontaneously demix, or phase separate, in a process controlled in part by intrinsically disordered (ID) proteins. A protein''s propensity to phase separate is thought to be driven by a preference for protein–protein over protein–solvent interactions. The hydrodynamic size of monomeric proteins, as quantified by the polymer scaling exponent (v), is driven by a similar balance. We hypothesized that mean v, as predicted by protein sequence, would be smaller for proteins with a strong propensity to phase separate. To test this hypothesis, we analyzed protein databases containing subsets of proteins that are folded, disordered, or disordered and known to spontaneously phase separate. We find that the phase-separating disordered proteins, on average, had lower calculated values of v compared with their non-phase-separating counterparts. Moreover, these proteins had a higher sequence-predicted propensity for β-turns. Using a simple, surface area-based model, we propose a physical mechanism for this difference: transient β-turn structures reduce the desolvation penalty of forming a protein-rich phase and increase exposure of atoms involved in π/sp² valence electron interactions. By this mechanism, β-turns could act as energetically favored nucleation points, which may explain the increased propensity for turns in ID regions (IDRs) utilized biologically for phase separation. Phase-separating IDRs, non-phase-separating IDRs, and folded regions could be distinguished by combining v and β-turn propensity. Finally, we propose a new algorithm, ParSe (partition sequence), for predicting phase-separating protein regions, and which is able to accurately identify folded, disordered, and phase-separating protein regions based on the primary sequence.     

A thermodynamic model for extended complexes of cholesterol and phospholipid

Studies of monolayer mixtures of certain phospholipids with cholesterol by epifluorescence microscopy and measurement of cholesterol desorption show evidence for the formation of "condensed complexes." A thermodynamic model of these complexes has been developed and has been shown to be generally consistent with observed phase diagrams, cholesterol desorption rates, and electric field susceptibility. Previous work has shown that complexes comprising 10-50 molecules provide good agreement with experimental results. The present study examines the calculated properties of complexes containing very large numbers of molecules and extends the condensed complex model to incorporate the formation of complexes of variable size. Trends in equilibrium composition are similar to those calculated for small complexes. Thermal transitions are continuous, with a strong composition dependence of the breadth of the transition. The average number of molecules in a large complex shows a pronounced dependence on the composition of the reaction mixture. Large complexes have properties of a separate thermodynamic phase.     

Sterol structure determines miscibility versus melting transitions in lipid vesicles

Lipid bilayer membranes composed of DOPC, DPPC, and a series of sterols demix into coexisting liquid phases below a miscibility transition temperature. We use fluorescence microscopy to directly observe phase transitions in vesicles of 1:1:1 DOPC/DPPC/sterol within giant unilamellar vesicles. We show that vesicles containing the "promoter" sterols cholesterol, ergosterol, 25-hydroxycholesterol, epicholesterol, or dihydrocholesterol demix into coexisting liquid phases as temperature is lowered through the miscibility transition. In contrast, vesicles containing the "inhibitor" sterols androstenolone, coprostanol, cholestenone, or cholestane form coexisting gel (solid) and liquid phases. Vesicles containing lanosterol, a sterol found in the cholesterol and ergosterol synthesis pathways, do not exhibit coexisting phases over a wide range of temperatures and compositions. Although more detailed phase diagrams and precise distinctions between gel and liquid phases are required to fully define the phase behavior of these sterols in vesicles, we find that our classifications of promoter and inhibitor sterols are consistent with previous designations based on fluorescence quenching and detergent resistance. We find no trend in the liquid-liquid or gel-liquid transition temperatures of membranes with promoter or inhibitor sterols and measure the surface fraction of coexisting phases. We find that the vesicle phase behavior is related to the structure of the sterols. Promoter sterols have flat, fused rings, a hydroxyl headgroup, an alkyl tail, and a small molecular area, which are all attributes of "membrane active" sterols.     

Phase transition in superdense hydrogen and deuterium

Results are presented from numerical simulations of the thermodynamic properties of superdense hydrogen and deuterium plasmas by the Monte Carlo method and from calculations by a multicomponent chemical model. The results obtained reveal the anomalous behavior of the thermodynamic functions and composition of molecular gas plasmas in the submegabar and megabar pressure ranges. Such behavior is interpreted as a dissociative phase transition. The results of calculations by the chemical model are compared with the experimental data on the equation of state and conductivity of hydrogen and deuterium plasmas.     

Phase Behavior of Whey Protein Aggregates/κ-Carrageenan Mixtures: Experiment and Theory

The phase separation behavior of whey protein isolate (WPI) aggregates and κ-carrageenan (κ-car) mixtures was studied using the Vrij's theory and image analysis method. The intrinsic parameter (molecular mass and radius of gyration) for κ-car and the WPI aggregates was determined using intrinsic viscosity and reduced viscosity of each biopolymer. Confocal microscopy observations revealed the appearance of protein aggregate domains when phase separation occurred, with microgel droplets of WPI included in a continuous κ-car phase. The occurrence of aggregate droplet has not been reported before for the phase-separating WPI/κ-car mixtures. So far, network emulsion-like microstructures have been observed with WPI in a network structure. By using different WPI concentrations (4% or 6%), the microstructure of the systems changes while increasing the κ-car concentration. The size of the microgels (1–2.5 μm) depends on both κ-car and WPI concentration. Confocal microscopy combined with image analysis (method of the variance) was used effectively as objective means to determine the phase boundary of the phase-separating systems. Additional information on the depletion layer thickness, Δ, was obtained using self-consistent field theory. The results show that Δ has a constant value of 80.5 nm for c_{k - car} \prec 2 g/l {{\hbox{c}}_{\kappa {\rm{ - car}}}} \prec {\hbox{2 g}}/{l} , in agreement with ∆ ≈ R _g (radius of gyration). Above this concentration, Δ decreases as a function of κ-car concentration. The experimental phase boundary was well predicted using Vrij's theory. This work showed a new approach to generate phase diagrams (e.g., under shear) of phase-separating systems.