Surface Densities Prewet a Near-Critical Membrane
SSurface Densities Prewet a Near-CriticalMembrane
Mason Rouches a,b , Sarah Veatch c , and Benjamin Machta b,d a Department of Molecular Biophysics and Biochemistry, Yale University, New Haven, Connecticut 06511, USA; b Systems Biology Institute, Yale University, West Haven,Connecticut 06516, USA; c Department of Biophysics, University of Michigan, Ann Arbor, MI 48109, USA; d Department of Physics, Yale University, New Haven, Connecticut06511, USAFebruary 19, 2021
Recent work has highlighted roles for thermodynamic phase behav-ior in diverse cellular processes. Proteins and nucleic acids canphase separate into three-dimensional liquid droplets in the cyto-plasm and nucleus and the plasma membrane of animal cells ap-pears tuned close to a two-dimensional liquid-liquid critical point.In some examples, cytoplasmic proteins aggregate at plasma mem-brane domains, forming structures such as the post-synaptic densityand diverse signaling clusters. Here we examine the physics of thesesurface densities, employing minimal simulations of co-acervatingpolymers coupled to an Ising membrane surface in conjunction witha complementary Landau theory. We argue that these surface densi-ties are a novel phase reminiscent of pre-wetting, in which a molec-ularly thin three-dimensional liquid forms on a usually solid surface.However, in surface densities the solid surface is replaced by a mem-brane with an independent propensity to phase separate. We showthat proximity to criticality in the membrane dramatically increasesthe parameter regime in which a pre-wetting-like transition occurs,leading to a broad region where coexisting surface phases can formeven when a bulk phase is unstable. Our simulations naturally ex-hibit three surface phase coexistence even though both the mem-brane and the polymer bulk can only display two phase coexistenceon their own. We argue that the physics of these surface densitiesenables diverse functions seen in Eukaryotic cells.
Eukaryotic cells are heterogeneous at scales far larger thanindividual macromolecules, yet smaller than classically definedorganelles. Proteins, RNA, and DNA can self-organize intothree-dimensional, liquid-like droplets in the cytoplasm andnucleus (1) and lipids and proteins in the plasma membranesimilarly organize into two-dimensional domains, often termed‘rafts’ (2). These domains and droplets are thought to formin part due to a thermodynamic tendency of their compo-nents to phase separate into coexisting liquids. Proteins andother molecules within three-dimensional droplets are heldtogether through weak but specific multi-valent interactions (3–5). Lipids and other membrane components interact throughless specific effective forces that arise from hydrophobic mis-match, from the interaction of lipid headgroups and fromsteric interactions between lipid tails (6). Cell derived vesiclesseparate into coexisting phases termed liquid-ordered ( l o ) andliquid-disordered ( l d ), passing through a critical point whencooled somewhat below growth temperature (7). At growthtemperatures, domains arising from proximity to this criticalpoint likely resemble corresponding low temperature phasesat small scales but with finite size and lifetime.Some surface densities appear to form due to a combinationof these forces. In these systems proteins aggregate in a thinfilm at a membrane surface with some components stronglyattached to membrane lipids (8–12) while others are free to exchange with the bulk. The protein components of these filmscan phase separate in the bulk, but only at substantially higherconcentrations than are seen in vivo (8, 11, 13). Examplesof these surface densities include the Nephrin/Nck/NWaspsystem that plays a role in cell adhesion (8, 14), T-cell signalingclusters (9), and the post-synaptic density (10, 13).Systems that phase separate in three-dimensions can un-dergo wetting transitions (15, 16), where there is a changein the bulk phase that adheres to a surface. In addition towetting transitions which take place inside of coexistence onthe bulk phase diagram, surfaces of bulk fluids can undergo prewetting transitions (17, 18) which occur near to bulk coex-istence. In prewetting transitions, a normally unstable bulkphase is stabilized through favorable interactions with a sur-face, leading to a surface film which resembles the nearby (inthe thermodynamic phase diagram) bulk phase, but which ismolecularly thin.The behavior of membrane domains and protein dropletshave both been successfully described using theories of phasetransitions in fluid systems (6, 19), but there has been lesswork interpreting these surface aggregates. We use latticeMonte-Carlo simulations in conjunction with a minimal Lan-dau theory to explore the physical principles governing thesedroplets. We argue that surface densities are similar to prewetphases, but with subtlety arising from their adhesion to a two-dimensional liquid which is itself prone to phase separating.We predict a novel surface phase sensitive to both membraneand bulk parameters which we argue describes a wide varietyof structures which are already biochemically characterized. Simulation results
Model overview-
In our simulations we describe the mem-brane using a conserved order parameter two-dimensionalsquare-lattice Ising model (20, 21). In this model spins roughlyrepresent membrane components - proteins or lipids - whichprefer the liquid-ordered (spin up, white in figures) or liquid-disordered (spin down, dark) membrane phases. The Isingmodel introduces two parameters, the coupling between neigh-boring spins J mem , and M , the difference in the number of upand down spins. Experiments suggest that plasma membranecomposition is tuned close to the critical point of de-mixingwhich occurs in the Ising model when M = 0 (equal numberof up and down spins) at a critical coupling J = J c .We model phase-separating cytoplasmic proteins as a latticecoacervate. Here two types of polymers, each 20 monomersin length, live on a 3D cubic lattice. Unlike polymers inter- E-mail: [email protected], [email protected], [email protected] a r X i v : . [ phy s i c s . b i o - ph ] F e b ct attractively with coupling J bulk (in the range of k B T , seemethods for exact values of all parameters) when occupyingthe same lattice position, and like polymers cannot occupy thesame position. We also include a weak non-specific nearest-neighbor interaction between all polymer sites which allowsphases to localize in space (22, 23). The two polymer typesroughly represent interacting components of phase-separateddroplets in the cytoplasm or oppositely charged, syntheticpolymers such as poly-lysine and poly-glutamine (24, 25). Insynthetic systems, the coupling between polymers can be mod-ulated by salt and polymer length. Cells alter their couplingthrough post-translational modifications, changes in salt, pH,and changes in valency (26, 27).To couple our membrane and bulk models we introducetethers which may be thought of as membrane-localized pro-teins. Tethers connect to up-spins on the membrane andextend several (five) lattice spacings into the third dimensionwhere they interact with bulk polymers through an attractiveinteraction J tether . Unlike bulk polymers, tethers translatein just two-dimensions across the membrane surface. In cellstethers correspond to lipidated or transmembrane proteinsthat interact with proteins in the cytoplasm (28). In syn-thetic systems tethers have been engineered through strongnon-covalent binding attachment of peptides or proteins tolipid headgroups (8, 9, 13) Bulk phase behavior is independent of surface details-
We expect the 3D bulk polymers to have a phase diagramwhich, in the thermodynamic limit, does not depend on prop-erties of the Ising surface. In the absence of a membrane, atfixed polymer number, the bulk can can be in either a uniformstate or can display coexistence between a polymer dense stateand polymer dilute state. A phase diagram for this is sketchedin figure Figure 1B in black; at low coupling, J bulk , or hightemperature, the state is uniform for any bulk concentration.At higher coupling there is a coexistence region where tie-lineendpoints, the black circles in Figure 1B, represent physicallyaccessible polymer densities and where both endpoints havethe same chemical potentials and Gibbs Free energies. Toobserve coexistence we perform simulations at fixed polymernumber with equal numbers of red and blue polymers. Whilethe coexistence region of the composition-coupling plane doesnot depend on the properties of the membrane surface, it’sappearance in simulation does; in a ‘dry’ regime, the polymerdense droplet avoids the surface, while in a ‘wet’ regime itadheres to at least a portion of the surface. Wetting transi-tions occur when the bulk phase which adheres to the surfacechanges - here this can be achieved by altering either the bulkproperties or the membrane properties, and in particular bychanging the concentration of tethers. Our focus, however, ison the surface phases which can coexist even in a single phaseregion of the bulk. Henceforth we conduct simulations in whichbulk polymers are instead held at fixed chemical potential inthe dilute regime (see methods for parameter values). Multiple surface phases can coexist on the boundaryof a single bulk phase-
In the absence of tethers, the mem-brane can phase separate if the interaction strength J mem islower than a critical value. In this sense, it is possible for thesystem to display surface phase coexistence even when thebulk is uniform. In the absence of tethers our membrane’sphase diagram is well characterized, with a large coexistenceregion. When tethers are added which prefer one of these two Fig. 1. Bulk and Surface Phases : A ) Cartoon of the minimal model used to de-scribe surface densities. In our simulations red and blue lattice polymers have anattractive interaction in the three-dimensional bulk. An Ising model on the bulk’sboundary contains bright/dark pixels representing liquid ordered/disordered preferringcomponents of a membrane. Tethers (yellow polymers) are connected to up spins,and have an attractive interaction with components of the bulk. B ) Schematic BulkPhase diagram. On a plot of inverse interaction strength (like temperature) vs polymerdensity, the bulk phase diagram contains a single bulk phase region (blue and grey)and a region where a dense and dilute phase coexist (yellow). The shape of the bulkcoexistence curve does not depend on location within the surface phase diagram. C )The region of surface coexistence depends on both bulk and surface parameters. InB and C, we show two two-dimensional slices where surface coexistence occurs inthe blue region. The blue X in B corresponds to location in the bulk phase diagram forwhich the surface phase diagram is plotted in C; moving this location would changethe shape of the blue coexistence curve in C. The blue X in C is the location in thesurface parameters for which the surface phase diagram is plotted in B. D ) Examplephases observed in Monte-Carlo simulations. On the left are two examples withoutbulk coexistence, one with surface coexistence, one without. On the right are twoexamples of coexisting bulk phases, a wet phase adheres to the membrane and a dryphase avoids the membrane. phases, we qualitatively see that the bulk polymer distributionis different near these two phases (see Figure 1D). This impliesthat bulk properties should be able to qualitatively changethe surface phase diagram even in the absence of bulk phaseseparation. In particular, increasing bulk coupling should beable to induce phase separation at the surface even when mem-brane interactions are too weak to induce phase separation ontheir own ( J mem < J c,mem , equivalent to T > T c,mem ).We thus expect the surface phase diagram to depend onparameters of the bulk polymer solution and on the membraneand tethers which make up the surface. We sketch two 2-dimensional slices through this five-dimensional phase diagramin Figure 1B,C. At a given point in the bulk phase diagram(blue x in Figure 1B) we see a surface coexistence regionresembling that for a two-dimensional coexisting liquid proneto phase separating via an Ising transition (blue shaded region | Rouches et al. n Figure 1C). Alternatively, by fixing the surface parametersat the blue x in Figure 1C, the surface coexistence region isplotted in Figure 1B.These surface coexistence regions are analogous to prewet-ting where, for example, a liquid film adheres to a solid surfaceof a gas phase bulk. In these classical examples there can beeither an abrupt or a continuous transition to a prewet statetriggered by either increasing bulk density, or by loweringtemperature. In the limit where J mem = 0 our system isanalagous to this, albeit with the additional complexity of afluid surface quantity in membrane tethers. More substantiallydifferent, the Ising model on the surface also participates inthe prewetting transition by further enhancing the interactionsthat drive surface aggregation. Surface and bulk properties together determine thesurface phase diagram-
To more quantitatively explore thesurface phase diagram in simulation we found a region of pa-rameter space that displays two coexisting phases far fromtheir critical point so that phases could be easily identifiedin small simulations (see Figure 2A). These two phases differfrom each other in their membrane order, their density of teth-ers and the density of polymers near them. We expect to beable to move from a single phase to two-phase coexistence byincreasing either membrane interactions or bulk interactions(schematically shown in Figure 2B). This is demonstrated inFigure 2C; a single phase surface is brought into the surfacecoexistence region by increasing the coupling between bulkpolymers (lower) or by increasing the interactions betweenmembrane components (left). Each of these coexisting surfacephases has a characteristic polymer density profile with dis-tance from the surface (Figure 2D). Although we primarilyfocus on membrane and bulk couplings, we confirmed thatprewetting can additionally be triggered by increasing thenumber of tethers on the membrane (see Supplement).Our simulation results suggests that the range of J bulk inwhich we see prewetting expands significantly as the membranecritical temperature is approached (see Figure 2B) or as webring the membrane towards it’s critical composition ( M = 0)at fixed coupling strength (see Supplement). These resultsimply that the membrane critical point expands the surfacecoexistence region, which we explore more quantitatively usinga Landau theory below. Simulations demonstrate three-phase surfacecoexistence-
Three phase coexistence in our model isallowed by Gibbs phase rule; two conserved quantities on thesurface - tether and membrane composition - allow for upto 2 + 1 phase coexistence. Indeed, we see three coexistingsurface phases in simulations (Figure 3A) each with distinctmembrane compositions, as well as tether and polymer densityprofiles. Three phase coexistence generally occurs at polymercouplings that would prewet a single-phase membrane and atmembrane couplings that would phase separate even in theabsence of any bulk coupling. We extracted the tether andmembrane composition of each phase, plotted in Figure 3B.When tether and membrane composition lay inside theshaded triangle the system phase separates into phases withtether and membrane compositions given as the verticesof the triangle, each with an accompanying density profileshown in Figure 3C. We ran simulations at each of thesesurface compositions to observe individual phases, shown inFigure 3D.
Fig. 2. Prewetting of surface densities: A ) Snapshot of a simulation where apolymer-dense droplet prewets the membrane surface even though droplets areunstable in the bulk. Time averaged membrane, tether, and polymer compositionsare shown at right. B ) Schematic phase diagram in terms of membrane and bulkcouplings. A single phase system (black dot) can move into the surface coexistenceregion by increasing J bulk (purple arrow) or increasing J mem (green arrow). C )Simulations at weak bulk and membrane coupling are brought into the coexistenceregion through increasing J bulk (purple) or J mem (green). D ) Density of polymersas a function of distance from the membrane. A system at weak bulk and membranecouplings sees a single phase (purple, simulation in lower left of C) while systemsat stronger membrane couplings see two coexisting polymer density profiles (blue,simulation in right of C). We describe surface phases by their membrane and polymercompoitions. What we denote the l o -Prewet phase is composedof a l o -like membrane rich in tethers, with an adhered polymerdroplet. The l d -Dry phase is an l d membrane excluded oftethers and with lacking an enhancement of bulk polymers.The final phase, l o -Dry, consists of an l o -like membrane some-what sparse in tethers and without a significant enrichment ofbulk components. Here we assume tethers prefer l o lipids; l d preferring tethers would instead form an analogous l d -Prewetphase. Landau analysis of Surface Phase behavior
Our lattice simulations serve to give a primarily qualitativeand intuitive picture for the phases we see. To more quantita-tively understand these surface phases we introduce a Landaufree-energy functional, modifying the analysis commonly usedto theoretically describe prewetting transitions to incorporatemembrane and tethers. As in standard analysis we introduceorder parameter fields, and a Landau functional of their con-figuration, and consider the order parameter of the system totake the configuration which globally minimizes the Landaufunctional (29). Phase coexistence occurs when two configura-tions of fields both have the same minimum value of the freeenergy.Our Landau functional, L , describes a bulk system ( z > z = 0, with ~x parameterizing the planeparallel to the surface. A single bulk order parameter φ ( ~x, z )describes the local density of polymers while two surface orderparameters, ρ ( ~x ) and ψ ( ~x ) describe the density of tethers andthe membrane composition along an l o - l d tie-line. We define φ ( ~x ) = φ ( ~x, z = 0) and, suppressing coordinates, we write a Rouches et al.et al.
Our lattice simulations serve to give a primarily qualitativeand intuitive picture for the phases we see. To more quantita-tively understand these surface phases we introduce a Landaufree-energy functional, modifying the analysis commonly usedto theoretically describe prewetting transitions to incorporatemembrane and tethers. As in standard analysis we introduceorder parameter fields, and a Landau functional of their con-figuration, and consider the order parameter of the system totake the configuration which globally minimizes the Landaufunctional (29). Phase coexistence occurs when two configura-tions of fields both have the same minimum value of the freeenergy.Our Landau functional, L , describes a bulk system ( z > z = 0, with ~x parameterizing the planeparallel to the surface. A single bulk order parameter φ ( ~x, z )describes the local density of polymers while two surface orderparameters, ρ ( ~x ) and ψ ( ~x ) describe the density of tethers andthe membrane composition along an l o - l d tie-line. We define φ ( ~x ) = φ ( ~x, z = 0) and, suppressing coordinates, we write a Rouches et al.et al. ) Simulations display three-phase coexistencewhere a polymer-dense droplet prewets a phase-separated membrane. Views oftime-averaged tether density, membrane composition, and polymer density show atether and polymer-dense phase rich in ordered components, an ordered membranephase with a small amount of tethers, and a disordered membrane phase devoid oftethers. B ) Phase diagram over membrane and tether composition extracted from thesimulation in A. Membrane and tether compositions falling inside the blue triangle splitinto three-phases, each with a composition given by the vertices of the triangle. BlackX corresponds to the surface composition of the simulation in A. C ) Polymer densityprofiles, as a function of distance from the membrane in each of the three phases. D ) Snapshots of simulations ran at compositions corresponding to the endpoints ofthree-phase coexistence. Landau free energy for this system as L = L D + L D with: L D = Z V d ~xdz
12 ( ∇ φ ) + f D ( φ ) [1] L D = Z ∂V d ~x f D ( ψ, ρ, φ )Where f D and f D describe the energy of the surface andbulk systems: f D ( φ ) = t bulk φ + u bulk φ − µ bulk φf D ( ψ, ρ ) = t mem ψ + u mem ψ − λ ψ ψ | {z } f membrane [2]+ ( ρ − ρ ? ) ρ ? + ( ρ − ρ ? ) ρ ? − λ ρ ρ | {z } f tether − h ψ ρψ − h φ ρφ | {z } f int Where t bulk is the distance from the bulk critical point, µ bulk the chemical potential of the bulk system, and t mem the dis-tance from the membrane critical point. u bulk and u mem arehigher order membrane and bulk couplings. The first twoterms in f tether are taken from an expansion of ρ log ρ at ρ ? .Lagrange multipliers λ ψ and λ ρ enforce membrane and tethercomposition, respectively. Membrane-tether and tether-bulkinteractions are set by h ψ and h φ . We take h φ > µ bulk <
0, corresponding to a dilute-phase polymer mixturewhose components interact favorably with tethers. Minimizingthis Landau functional determines the value of two deriva-tives and a functional derivative, ∂ L /∂ψ = ∂ L /∂ρ = 0 and δ L /δφ ( z ) = 0.In the thermodynamic limit the 2D Ising model and tethersact as a boundary condition for the bulk, and thus cannotinfluence which bulk phases are stable. The bulk phase isthe value of φ ∞ which globally minimizes f D ( φ ), defining f bulk = f D ( φ ∞ ). The resulting bulk phase diagram recapit-ulates Figure 1A, but with mean field exponents. At hightemperatures, or low concentration of polymers there is a sin-gle dilute phase, which can coexist with a second dense phaseat lower temperature. Analysis of Surface behavior-
Outside of bulk coexistence, L D is globally minimized by a unique φ ( ~x, z ) = φ ∞ , where L D = V f bulk , with V the system volume and where A is itsarea. The free energy of the surface, f surf , contains mem-brane contributions and contributions from surface induceddistortions of the bulk field ∆ f bulk : L surf = L −
V f bulk = Af surf ( ρ, ψ, { φ ( z ) } ) f surf = f D ( ρ, ψ, φ ) + Z dz
12 ( ∇ φ ) + f D ( φ ) − f bulk | {z } ∆ f bulk [3]For a given location in the bulk phase diagram the surface canexhibit its own set of phases and transitions which are localminima of f surf . While ∆ f bulk and f D cannot be indepen-dently minimized, they can be independently minimized fora given value of φ . Local minima of f D | φ satisfy the con-ditions that ∂f D /∂ρ = ∂f D /∂ψ = 0. Minima of ∆ f bulk | φ satisfy the differential equation ∂ φ ( z ) /∂z = df D /dφ withboundary conditions φ (0) = φ and φ ( ∞ ) = φ bulk . The valueof f D ( φ ) = min ρ,ψ f D and ∆ f bulk ( φ ) = min { φ ( x ) } ∆ f bulk are plot-ted in Figure 4A, along with their sum, f surf ( φ ). The valuesof ψ and ρ that minimize f surf ( φ ) are visualized simultane-ously in Figure 4C, each corresponding to the local minima inFigure 4A.Minima can be identified more systematically using thegraphical construction in Figure 4B, plotting − df D ( φ ) /dφ and ∂ ∆ f bulk /∂φ , derivatives of the curves in 4A. Local min-ima of the surface free energy occur when these curves cross.In general, two local minima are separated by a local maxi-mum. For two minima to have the same free energy, the areabetween the two curves (blue shaded regions in Figure 4B)must be equal. Surface enhancement of bulk interactions divergesnear the membrane critical point-
We plotted the re-gions of surface phase coexistence as a function of bulk andmembrane coupling for fixed values of φ ∞ , ψ , and ρ , in Fig-ure 5A. As with simulations we notice that the two-phaseregion expands significantly as J mem → J c,mem .In the absence of interactions with tethers ( h ψ = 0) themembrane of our model ( f mem ) has a line of abrupt phasetransitions when t mem < λ ψ = 0, terminating in a criticalphase transition at t mem = 0, λ ψ = 0. For weak interactions,the location of this first order line and critical point can shift,but it’s topology is unchanged - in particular, the locationof the critical point shifts towards higher (positive) valuesof t mem , signifying that the critical point in our simulations | Rouches et al. ig. 4. Landau Theory of surface phases: A ) Bulk ∆ f bulk (blue), membrane f D (red, already minimized over ψ, ρ ), and surface f surf = ∆ f bulk + f D (brown)free energies as a function of surface polymer density φ . There are two energyminimia, φ low and φ high in the combined f surf even in the absence of multipleminima in ∆ f bulk or f D . B ) Gradient construction used to visualize solutions.Intersections of derivatives of f D (red) and ∆ f bulk (blue) give possible surfacesolutions φ low and φ high . The free energy difference between these solutions isgiven by the area between these curves, visualized as the shaded regions. Changingthe position or slope of surface or bulk lines changes the surface solutions. C ) Surfacefree energy f surf calculated over values of ψ and ρ , minimized first over φ . Twominima (purple and green) correspond to surface compositions that minimize the freeenergy of the membrane and tethers along with their resulting contributions to bulkenergy. D ) Density profiles and energy density (inset) as a function of distance fromthe membrane z for the two physical phases. Both φ high and φ low decay to the bulkdensity φ ∞ . This adds unfavorable contributions to the free energy ∆ f bulk ( φ ( z )) that are balanced by contributions from f surf should occur at weaker membrane coupling. Thus the surfacecoexistence line should meet the membrane only transitionline where J bulk = 0 as in Figure 2/Figure 5A, with bulkinteractions supplementing membrane ones away from it.We can also understand the enlargement of the prewettingregime using the language of classical prewetting theories. Inprewetting to a solid surface, f D is typically assumed to takethe simple form f s = f − µ φ − m φ . Here µ is the surfacechemical potential and m is the surface enhancement (15)quantifying increased attractive interactions between bulkcomponents in proximity to the surface. In most examplesthe surface enhancement is negative due to loss of effectiveinteractions mediated through negative values of z. However,small positive surface enhancements are possible, for examplewhen magnetic spins interact through contact with a surfacewith a larger magnetic susceptibility (30, 31).While our theory only explicitly includes first-order terms in φ , higher order terms are generated by minimizing over ψ and ρ contributions, generating an effective surface enhancement .In the graphical construction in Figure 4B, we can interpretthe surface enhancement as the slope of the red − df D /dφ line. Near the critical point, components embedded in themembrane feel long range effective forces mediated by themembrane, sometimes called critical Casimir forces (32, 33).In surface densities these long range critical Casimir forcesprovide an effective surface enhancement, mediating an in-creased interaction between bulk components. The magnitudeof this membrane mediated effective surface enhancement can be understood quantitatively as arising from the integral ofthe pairwise potential between tethers on the surface. Thisyields a quantity proportional to the susceptibility (29) whichdiverges near the critical point. This manifests as a steepeningof the surface line as the membrane critical point is approachedalong increasing J mem (blue to green curves in Figure 5B).Below the membrane critical point we see the surface line foldback on itself, with two local minima and a local maxima atsome values of φ , implying the membrane can phase separatewithout bulk interactions.While we expect our phase diagram to be topologicallycorrect, our Landau theory fails to accurately predict the formof these phase boundaries. Mean-field theories like ours gener-ally underestimate fluctuation effects, especially close to thecritical point (29). We expect that a more sophisticated renor-malization group treatment would predict a larger criticalitymediated enhancement and resulting surface coexistence re-gion as well as a surface coexistence curve with Ising exponentsrather than mean field ones. Fig. 5. Critical Point Enhancement: A ) Phase diagram over J bulk and J mem near the membrane critical point. The surface coexistence region (blue) extendsto very weak J bulk near J c,mem , marked by the dashed line. Outside of thecoexistence region the surface is single-phase (yellow). B ) Gradient constructionshowing how the 2D curve changes on varying membrane coupling along the thegreen line in A (colors from points in A). The slope of the surface curve increases as J mem → J c,mem , diverging like the susceptibility near the membrane critical point. C ) Gradient construction varying J bulk along the purple line in A. Increasing J bulk decreases the slope of the bulk curve, promoting surface phase coexistence. Landau theory predicts coexistence of three surfacephases-
In general, each local minimum has a different valueof ψ and ρ . We expect to have two-phase coexistence whenthe chemical potentials λ ρ and λ ψ are such that the globalminimum is doubly degenerate and three phase coexistencewhen the global minimum is triply degenerate. Coexistenceadditionally implies that the chemical potentials of each phaseare identical. We minimized L over a range of chemical poten-tials searching for regions of two and three phase coexistence,shown in Figure 6B. We find a single point where three phasescoexist and three lines of two-phase coexistence when we tunethe two chemical potentials while fixing other parameters.This is permitted by Gibbs phase rule, as three phases are al-lowed to coexist at a single point when tuning two parameters,recapitulating the qualitative findings from our simulations. Rouches et al.et al.
In general, each local minimum has a different valueof ψ and ρ . We expect to have two-phase coexistence whenthe chemical potentials λ ρ and λ ψ are such that the globalminimum is doubly degenerate and three phase coexistencewhen the global minimum is triply degenerate. Coexistenceadditionally implies that the chemical potentials of each phaseare identical. We minimized L over a range of chemical poten-tials searching for regions of two and three phase coexistence,shown in Figure 6B. We find a single point where three phasescoexist and three lines of two-phase coexistence when we tunethe two chemical potentials while fixing other parameters.This is permitted by Gibbs phase rule, as three phases are al-lowed to coexist at a single point when tuning two parameters,recapitulating the qualitative findings from our simulations. Rouches et al.et al. ) Phase diagram overmembrane compositions ( ψ ) and tether compositions ( ρ ) calculated from Landautheory. Three phases coexist in the blue triangle, with the surface composition of eachphase give by the vertices of the triangle. The positions within the triangle, (black x)sets the area fraction of phases. Three two-phase coexistence regions (red, purple,green) border the three-phase region and are plotted as tie lines. Surfaces constructedon a tie-line split into two phases with compositions given by the ends of the tie line.Single-phase regions border each two phase region. B ) Phase diagram over thesurface chemical potentials λ ρ , λ ψ for the same system shown in A. The three phasetriangle is represented as the blue point, and each colored line corresponds to the twophase regions in A. Outside of these lines and the point of three-phase coexistencethe system is single-phase. C ) Gradient construction within the three-phase region ofA. The surface line is phase separated, folding in on itself and intersecting at the blueand yellow points. It additionally intersects with the bulk curve at high densities, greenpoint. Discussion
We have presented a model for surface densities in which bulkcomponents, a membrane order parameter and membranebound tethers phase separate together in a manner reminiscentof prewetting. In our simulations the membrane is composedof a lattice Ising model, while the bulk is composed of co-acervating lattice polymers. The stability of these surfacedensities can be modulated by membrane interaction strength,by the density and interactions between bulk componentsand through the density of tethers which couple membraneand bulk. We see that when the membrane is held closeto it’s critical point, the regime where we see surface phasecoexistence widens dramatically which we trace to membranemediated enhancement of bulk polymer interactions. Thesesurface densities are stable thermodynamic phases and theirputative roles should be distinguished from roles the membranemay play in nucleating droplets that are already stable in thebulk but which face substantial nucleation barriers to theirassembly.While our model is not microscopically detailed, we believeit captures the coarse-grained behavior of a wide range ofsurface densities seen in cells. Building on these ideas, wepropose that the unique physics of surface densities supportbiological function both by acting as dynamic scaffolds andby triggering cellular responses.
Stable surface densities enable dynamic functionaldomains-
Prewet phases likely facilitate organization of pro-teins and lipids into stable, long lived complexes which performspecific functions at distinct sites. The post-synaptic densityis composed of phase-separating bulk proteins (10) adheredto a membrane domain enriched in particular ion channels,receptors and other components of the excitatory synapse.Some of these proteins, like PSD-95 are heavily palmitoylated,a modification which is dynamically regulated and confers apreference for ordered membrane lipids (34). PalmitoylatedPSD-95 likely plays a role analogous to the tethers in ourmodel, connecting liquid ordered membrane components to cy-toplasmic components of the post-synaptic density. The liquidnature of surface densities allows the post-synaptic density todynamically exchange components with the bulk and with sur-rounding membrane, facilitating the mechanisms of learningwhich take place in neuronal synapses. Other structure withcharacteristics of surface densities play roles in neuronal me-chanics. On the presynaptic side, RIM/RIM-BP condensatescluster calcium channels and machinery mediating synapticvesicle release (12). The inhibitory post-synaptic density dis-plays broadly similar organization to its excitatory counterpart,but with different protein-protein interactions which insteadlocalize inhibitory ion channels and the overlapping machineryrequired there (35). Common across these examples are liquidstructures at the membrane whose components undergo con-stant turnover yet whose organization and function persistsover longer time scales. As in our model, the combination ofmembrane mediated forces and bulk interactions allows for astable domain highly enriched in particular components evenwhile individual components remain mobile.
Surface density formation can initiate signaltransduction-
Immune receptor signaling is often de-pendent upon membrane lipids and long-lived associationsbetween receptors and scaffolding elements, some of which aremembrane bound. Measurements in reconstituted systemsmimicking T cell receptor (TCR) signaling support the ideathat these scaffolds are surface densities whose formationis triggered by the phosphorylation of membrane-boundLAT (9). Some of these phosphorylations enhance interactionswith soluble binding partners, equivalent to strengtheninginteractions with tethers within our model. Moreover, LATis itself palmitoylated (36), likely conferring a l o characterto LAT tethered surface densities in T cell membranes (37).In B cell receptor signaling, it is argued that clusteringreceptors enhances receptor phosphorylation by stabilizinga more l o -like local environment (38). Similar to TCR,we anticipate that phosphorylation dependent interactionsbetween membrane-bound and soluble proteins could triggerthe formation of a surface density that commits to acellular-level response. In this case, the primary functionof the membrane phase transition is to enhance tether-bulkinteractions via phosphorylation and not to enhance effectiveinteractions between tethers, although this may play a role.In the Endoplasmic Reticulum (ER) the integral membraneprotein IRE1 forms phase-separated clusters in response tounfolded protein stress (39, 40) and when lipid metabolismis disrupted (41). IRE1 is thought to have an affinity fordisordered lipids, but it also interacts with unfolded proteinsin the ER lumen, possibly playing a role analogous to a tetherin our model but in the disordered phase. | Rouches et al. n the above examples, a signal is transduced in part by ac-tivated and sometimes cross-linked receptors seeding domainswhich bring downstream components into close proximity.While the specific proteins involved in these initial signalingsteps are diverse, their commonality may be that in each casesignal leads to increased interactions, either between membranebound components (like increasing J mem in our model) be-tween membrane and bulk components (like increasing J tether or ρ ) or between particular bulk components (like changing J bulk ). In general, we propose that prewet phases serve natu-ral roles in signaling networks owing to their unique physics.Surface transitions depend on bulk, membrane, and tetherproperties allowing the cell several mechanisms to regulatea single response. Moreover, prewetting can be a first-order(abrupt) transition, providing a natural mechanism to trans-duce a continuous signal into a discrete, switch-like cellularresponse. The interactions that drive surface densities play rolesin more complex cellular structures-
Many examples inbiology display some of the phenomena we investigate herebut with additional subtleties and complications. Recentstudies have shown that condensates can induce membranedeformations in clathrin-mediated endocytosis (42) and insynthetic systems (43). While we expect that surface densitiescan substantially deform the membrane, and that deformationsmay influence the interactions between membrane and bulk,we don’t allow this in our model and so cannot explore it’sconsequences here. In other cases prewet phases may mediateadhesive interactions between multiple surfaces. Componentsof tight junctions (11) have been shown to phase separate in thebulk, and their condensation in vivo likely includes interactionswith membrane components as well as contributions from othereffective forces. The assembly of the Golgi apparatus (44,45) and synaptic vesicles clustering in the presynapse (46)are both thought to include proteins that phase-separate onthe surface of these organelles, possibly sorting vesicles fortransport. While our focus is on membranes, prewetting canoccur on other interesting biological surfaces. Phase separationhas been proposed to play prominent roles in transcriptionalregulation (47, 48), where DNA has been proposed to act as aone-dimensional ‘surface’ for prewetting (49).
Surface densities can be driven by membrane or bulkinteractions alone or through a combination-
The lipidcomposition of plasma membranes appears to be tuned closeto a thermodynamic critical point (7, 50, 51), which we haveargued has important consequences for surface phases. Nearthe membrane critical point, the bulk coupling needed to seesurface coexistence rapidly decreases (Figure 5). We expectthat surface densities could be stabilized entirely throughmembrane criticality mediated interactions, solely throughprewetting interactions between bulk components, or througha mixture of these two forces. The two extremes have beenexplored in synthetic systems. Synthetic membranes can phaseseparate into coexisting two-dimensional liquid phases in theabsence of any proteins. More recently, two-dimensional coex-isting phases have been observed on single-component mem-branes (52, 53), driven by interactions between bulk proteinssome of which adhere. Similar experiments in multi-componentmembranes (54, 55) highlight the bulk’s ability to mediateinteractions between membrane lipids. Because these interac-tions are stable outside of the regime of bulk coexistence, they are most likely prewet. In cells, proximity of the membrane toits critical point likely allows for weak and diverse interactionsbetween sparse proteins leading to surface phases far outsideof their coexistence regime and even far outside of the regimein which they would prewet a single component membrane.
Prewetting appears to be more common than wet-ting in cellular phase separation-
The phase-separated β -catenin destruction complex (56), integral to Wnt signal-ing, is recruited to the plasma membrane on induction of theWnt pathway, remaining a dynamic assembly on the mem-brane (57). This is likely analogous to a transition between adry and wet phases in our model. However, nearly all recentlydescribed cytoplasmic condensates are observed away frommembranes (1, 27), even though our model suggests that onlyweak, tether-driven interactions are required for membranewetting. By contrast, a large number of cellular structuresappear to be prewet - forming thin films on specific membranedomains outside of bulk coexistence. This may suggest thatattractive interactions between droplets and cytoskeletal ele-ments outcompete interactions with membrane components,or that these interactions are limited by material propertiesof the cortex (58).The prediction of prewetting (17) significantly preceded itsfirst experimental realizations (59, 60). Prewet phases outsideof biology typically require fine-tuning and subtle experimentalconsiderations to observe. By contrast, in biological contexts,surface densities appear to be common, owing to the presenceof a complex membrane with a propensity to phase separateinteracting with a dense polymer solution. Our conception ofsurface densities includes membrane dominated phases, closeto the usual concept of a lipid raft, bulk driven phases thatclosely match the classical concept of prewetting, as well asphases which make use of a combination of these interactions.We hope that future work will clarify the roles these surfacedensities play in diverse cellular functions. Materials and Methods
Simulation code, Landau Theory calculations, and supplementaltext and videos can be found on GitHub at critical membraneprewetting
Monte-Carlo Simulations
Monte-Carlo simulations were imple-mented on a 3-Dimensional lattice ( D x L x L ) populated with poly-mers, tethers, and a membrane simulated by an Ising model. Thelattice is periodic in the two L dimensions and has free boundaryconditions at D = 0 , L , with the Ising model located at the D = 0boundary. Our model is described by a simple Hamiltonian: H bulk = J bulk X i σ bluei σ redi + J nn X i,j ∈ nn σ i σ j − µ bulk N bulk H ising = J ising X i,j ∈ nn s i s j H tether = J tether X i ∈ tethers σ bulki σ tetheri [4]Where J bulk is the interaction strength between polymers ofdifferent types (‘red’ and ‘blue’), J nn is a nearest neighbor energy,and µ bulk is the chemical potential of the 3D system. The spinswithin the Ising model interact with coupling J ising and componentsof the bulk interact with tethers through J tether Bulk Polymers:
Cytoplasmic proteins are simulated as a mixtureof lattice polymers. Bulk polymers occupy the vertices of a 3D cubiclattice. Snake-like moves where the tail of the polymer is moved toa free space adjacent to the head (and vice-versa) allow polymers to
Rouches et al.et al.
Rouches et al.et al. xplore the lattice. Here we simulate just two bulk polymer speciesand a single tether species. Polymers of the same type cannotinhabit the same lattice position while polymers of opposite typeinteract through J bulk when occupying the same lattice site. All bulkpolymers interact equally with tethers. Additionally, all polymersand tethers have a small, favorable nearest-neighbor interactions J nn = 0 . k b T . This nearest-neighbor energy is required to give thedroplets tension, without which they do not condense (22, 23). Tethers:
Tethers move in two dimensions across the surface of theIsing model. Proposed moves translate a tether one lattice spacein a random direction. Proposals that move the tether off of anup spin or result in two tethers occupying the same lattice site areimmediately rejected.
Membrane:
The membrane is simulated as a conserved orderparameter Ising model, implemented on a 2D cubic lattice withperiodic boundary conditions. To conserve the total magnetization,or lipid composition, we use a non-local Kawasaki moves whereIsing spins are exchanged rather than flipped. We fix up-spins atevery tether-occupied site during each sweep.
Simulation Scheme:
Each simulation consists of sweeps throughpolymers, tethers, and membrane spins. We proposed movesthrough a randomized sequence of polymers and tethers in thesystem, followed by a sweep through all Ising spins, and proposalof particle exchanges. We equilibrate simulations by raising bulkcoupling and tether coupling in increments of 0 . − . k b T with1 × - 5 × Monte-Carlo sweeps per temperature step. Simu-lations were sometimes extended from the previous endpoints, forup to 5 × Monte-Carlo sweeps at a single set of parameters, toensure equilibration.Single polymer, tether, and Ising moves are accepted with theMetropolis probability e − ( H f − H i ) /k b T where H f , H i are the ener-gies of the final and initial system configurations. To accelerateequilibration we propose cluster moves where a connected set ofpolymers translate one lattice spacing. Cluster moves are proposedwith probability (1 /N poly ) and are only accepted if the move doesnot form or break any bonds, satisfying detailed balance.In simulations at fixed µ bulk , polymers are exchanged betweenthe system and a non-interacting reservoir. The amount of polymersexchanged per Monte Carlo step is sampled from a Poisson distri-bution where λ = N sys + N res N init + N res , ensuring that chemical potentialremains constant as particles are added to the system. Exchangesare accepted with probability e − ∆ H nn − µ bulk , where ∆ H nn is thechange in nearest neighbor energy. Exchanges that remove or addbonds to the system are immediately rejected. Swapping a particlefrom the reservoir to the system simply copies the reservoir particleinto the system while moving a particle from the system to thereservoir removes the particle from the system but does not placean additional particle in the reservoir. This scheme of ‘virtual’exchanges is done so the reservoir is effectively infinite while weonly simulate a finite amount of particles. Extracting Surface Composition from Simulations : To ob-tain the membrane and tether compositions of simulations thatappeared to have 3 coexisting phases, we analyzed histograms ofmembrane and tether composition. First we averaged the membranespins and tether positions over 50000 MCS. From this time-average,we scanned the surface with a 5x5 grid, computing the averagemembrane and tether compositions within. These values are col-lected over the last half of simulation run, 2,500,000 MCS, and usedto construct a two-dimensional histogram of tether and membranecompositions. We defined the surface composition of coexistingphases as the peaks of this histogram. Because there were multiplepeaks likely corresponding to a single phase, we required that thedifference in tether density between peaks was greater than 0.05.
Simulation Parameters used in figures
Figure Tether Density Membrane Order J bulk ,k B T J mem ,T c J tether µ bulk / φ bulk Mean-Field Theory : To minimize the free-energy of our systemwe sought to express the contributions from bulk terms in terms of surface and bulk densities φ and φ ∞ , as these alone determinethe density profile. Following previous work (15, 17), we identifyspatial gradients ∇ φ with distance from φ bulk ∇ φ = ± p f D ( φ ) − f bulk )Where this follows from the functional derivative δL D δφ z . We usethis identity to express the contributions from spatial variations of∆ φ ( z ) in terms of φ and φ bulk ∆ f bulk = Z ∞ dz
12 ( ∇ φ ) + f D ( φ ) − f bulk = Z ∞ dz (cid:16) dφdz (cid:17) (cid:16) dzdφ (cid:17)
12 ( ∇ φ ) + f D ( φ ) − f bulk = Z φ ( ∞ ) φ (0) dφ ( ∇ φ ) −
12 ( ∇ φ ) + f D ( φ ) − f bulk = Z φ bulk φ dφ ( ∇ φ ) −
12 ( ∇ φ ) + f D ( φ ) − f bulk | {z } ∇ φ = Z φ bulk φ dφ ( ∇ φ ) − ( ∇ φ ) ∆ f bulk ( φ , φ ∞ ) = Z φ bulk φ dφ p f D ( φ ) − f bulk )) [5]The total free energy of the bulk and surface terms, f surf , can nowbe written as: f surf = f D ( ρ, ψ, φ ) + Z φ bulk φ dφ p f D ( φ ) − f bulk )) [6]Which we minimize numerically over values of φ , ψ , ρ to obtainresults throughout the text. f surf can be minimized independentlyover φ , ψ , or ρ values to obtain the surface free energy as a functionof the remaining terms, as plotted in Figure 4A,C. Numerical Phase Diagrams : We minimized f surf numericallywith Mathematica. We calculated solutions at over a range λ ρ , λ ψ values to find coexistence regions. When there were multiple so-lutions with near-identical energies, within < . k b T , we declaredthem coexisting phases. Values of ψ , ρ that minimize the free energyat these points terminate tie lines in a fixed composition system.This procedure is visualized in Figure 6A,B where the ψ , ρ valuesin A correspond to λ ρ , λ ψ values in B. Multiple phase diagrams inthe space of J bulk , J mem were constructed through combining thetie lines and three phase regions of of phase diagrams calculated atvalues of t mem and t bulk . At a specific ψ , ρ values we determinedwhether the system was in a one, two, or three phase region of thephase diagram. Landau theory parameters used in figures
Figure t mem t bulk φ ∞ ρ ? λ ρ λ ψ ρ ψ h φ h ψ Figure 4 1.1 -0.1 -2.0 1 -0.1 0.5 N/A N/A 1 1Figure 5A N/A N/A -2.0 1 N/A N/A -0.5 0 1 1Figure 5B 1, 0.75, 0.45, 0.25, 0.1 2 -2.0 1 -0.05 0.25 N/A N/A 1 1Figure 5C 0.45 10, 5, 2, 1, 0.5 -2.0 1 -0.05 0.25 N/A N/A 1 1Figure 6A,B -0.2 -0.4 -2.0 1 N/A N/A N/A N/A 1 1Figure 6C -0.2 -0.4 -2.0 1 -2 1 N/A N/A 1 1
ACKNOWLEDGMENTS.
We thank Isabella Graf and Jon Machtafor helpful comments on a draft, and Ilya Levental and Hong-YinWang for useful discussions. This work was supported by NSFBMAT 1808551 (MR, SLV and BBM), NIH R35 GM138341 (BBM)and NSF 1522467 (MR).
1. S Alberti, Phase separation in biology.
Curr. Biol . , R1097–R1102 (2017).2. P Sengupta, B Baird, D Holowka, Lipid rafts, fluid/fluid phase separation, and their relevanceto plasma membrane structure and function. Semin. Cell & Dev. Biol . , 583–590 (2007).3. CP Brangwynne, et al., Germline P Granules Are Liquid Droplets That Localize by ControlledDissolution/Condensation. Science , 1729–1732 (2009).4. P Li, et al., Phase transitions in the assembly of multivalent signalling proteins.
Nature ,336–340 (2012). | Rouches et al. . D Priftis, M Tirrell, Phase behaviour and complex coacervation of aqueous polypeptide solu-tions. Soft Matter , 9396–9405 (2012).6. AR Honerkamp-Smith, SL Veatch, SL Keller, An introduction to critical points for biophysicists;observations of compositional heterogeneity in lipid membranes. Biochimica et Biophys. Acta- Biomembr . , 53–63 (2009).7. SL Veatch, et al., Critical fluctuations in plasma membrane vesicles. ACS Chem Bio , 287–293 (2008).8. S Banjade, MK Rosen, Phase transitions of multivalent proteins can promote clustering ofmembrane receptors. eLife , e04123 (2014).9. X Su, et al., Phase separation of signaling molecules promotes T cell receptor signal trans-duction. Science , 595–599 (2016).10. M Zeng, et al., Phase Transition in Postsynaptic Densities Underlies Formation of SynapticComplexes and Synaptic Plasticity.
Cell , 1163–1175 (2016).11. O Beutel, R Maraspini, K Pombo-García, Cl Martin-Lemaitre, A Honigmann, Phase Sepa-ration of Zonula Occludens Proteins Drives Formation of Tight Jun ctions.
Cell , 923–936.e11 (2019).12. X Wu, et al., RIM and RIM-BP Form Presynaptic Active-Zone-like Condensates via PhaseSeparation.
Mol. Cell , 971–984.e5 (2019).13. M Zeng, et al., Reconstituted Postsynaptic Density as a Molecular Platform for UnderstandingSynapse Formation and Plasticity. Cell , 1172–1187 (2018).14. LB Case, X Zhang, JA Ditlev, MK Rosen, Stoichiometry controls activity of phase-separatedclusters of actin signaling proteins.
Science , 1093–1097 (2019).15. PG de Gennes, Wetting: statics and dynamics.
Rev. Mod. Phys . , 827–863 (1985).16. D Bonn, J Eggers, J Indekeu, J Meunier, E Rolley, Wetting and spreading. Rev. Mod. Phys . , 739–805 (2009).17. JW Cahn, Critical point wetting. The J. Chem. Phys . , 3667–3672 (1977).18. H Nakanishi, ME Fisher, Multicriticality of Wetting, Prewetting, and Surface Transitions. Phys.Rev. Lett . , 1565–1568 (1982).19. CP Brangwynne, P Tompa, RV Pappu, Polymer physics of intracellular phase transitions. Nat.Phys . , 899–904 (2015).20. AR Honerkamp-Smith, et al., Line Tensions, Correlation Lengths, and Critical Exponents inLip id Membranes Near Critical Points. Biophys. J . , 236–246 (2008).21. BB Machta, S Papanikolaou, JP Sethna, SL Veatch, Minimal model of plasma membraneheterogeneity requires coupling cortical actin to criticality. Biophys J , 1668–1677 (2011).22. ES Freeman Rosenzweig, et al., The Eukaryotic CO2-Concentrating Organelle Is Liquid-likeand Exhibits Dynamic Reorganization.
Cell , 148–162.e19 (2017).23. B Xu, et al., Rigidity enhances a magic-number effect in polymer phase separation.
Nat.Commun . , 1561 (2020).24. D Priftis, M Tirrell, Phase behaviour and complex coacervation of aqueous polypeptide solu-tions. Soft Matter , 9396–9405 (2012).25. D Priftis, N Laugel, M Tirrell, Thermodynamic Characterization of Polypeptide Complex Coac-ervation. Langmuir , 15947–15957 (2012).26. WT Snead, AS Gladfelter, The Control Centers of Biomolecular Phase Separation: HowMembrane Surfaces, PTMs, and Active Processes Regulate Condensation. Mol. Cell ,295–305 (2019).27. Y Shin, CP Brangwynne, Liquid phase condensation in cell physiology and disease. Science , eaaf4382 (2017).28. H Jiang, et al., Protein Lipidation: Occurrence, Mechanisms, Biological Functions, and En-abling Technologies.
Chem. Rev . , 919–988 (2018).29. N Goldenfeld, Lectures on phase transitions and the renormalization group , Frontiers inphysics. (Addison-Wesley, Advanced Book Program, Reading, Mass) No. v. 85, (1992).30. K Binder, DP Landau, Wetting and layering in the nearest-neighbor simple-cubic Ising lattice:A Monte Carlo in vestigation.
Phys. Rev. B , 1745–1765 (1988).31. K Binder, DP Landau, S Wansleben, Wetting transitions near the bulk critical point: MonteCarlo simulations for the Ising model. Phys. Rev. B , 6971–6979 (1989).32. BJ Reynwar, M Deserno, Membrane composition-mediated protein-protein interactions. Biointerphases , FA117–FA124 (2008).33. BB Machta, SL Veatch, JP Sethna, Critical casimir forces in cellular membranes. Phys RevLett , 138101 (2012).34. K Tulodziecka, et al., Remodeling of the postsynaptic plasma membrane during neural devel-opment.
Mol. Biol. Cell , 3480–3489 (2016).35. G Bai, Y Wang, M Zhang, Gephyrin-mediated formation of inhibitory postsynaptic densitysheet via phase separation. Cell Res . (2020).36. W Zhang, RP Trible, LE Samelson, LAT Palmitoylation.
Immunity , 239–246 (1998).37. I Levental, D Lingwood, M Grzybek, U Coskun, K Simons, Palmitoylation regulates raft affinityfor the majority of integral raft proteins. Proc. Natl. Acad. Sci . , 22050–22054 (2010).38. MB Stone, SA Shelby, MF Núñez, K Wisser, SL Veatch, Protein sorting by lipid phase-likedomains supports emergent signaling function in b lymphocyte plasma membranes. eLife ,e19891 (2017).39. T Aragón, et al., Messenger RNA targeting to endoplasmic reticulum stress signalling sites. Nature , 736–740 (2009).40. V Belyy, NH Tran, P Walter, Quantitative microscopy reveals dynamics and fate of clusteredIRE1 α . Proc. Natl. Acad. Sci . , 1533–1542 (2020).41. K Halbleib, et al., Activation of the Unfolded Protein Response by Lipid Bilayer Stress. Mol.Cell , 673–684.e8 (2017).42. LP Bergeron-Sandoval, et al., Endocytosis caused by liquid-liquid phase separation of pro-teins. BioRxiv (2017).43. F Yuan, et al., Membrane bending by protein phase separation.
BioRxiv , 22 (2020).44. JE Rothman, Jim’s View: Is the Golgi stack a phase-separated liquid crystal?
FEBS Lett . , 2701–2705 (2019).45. AA Rebane, et al., Liquid–liquid phase separation of the Golgi matrix protein GM130. FEBSLett . , 1132–1144 (2020).46. D Milovanovic, Y Wu, X Bian, P De Camilli, A liquid phase of synapsin and lipid vesicles. Science , 604–607 (2018). 47. D Hnisz, K Shrinivas, RA Young, AK Chakraborty, PiA Sharp, A Phase Separation Model forTranscriptional Control.
Cell , 13–23 (2017).48. W Bialek, T Gregor, G Tkaˇcik, Action at a distance in transcriptional regulation. arXiv:1912.08579 [cond-mat, physics:physics, q-bio] (2019) arXiv: 1912.08579.49. JA Morin, et al., Surface condensation of a pioneer transcription factor on DNA.
BioRxiv (2020).50. E Gray, J Karslake, BB Machta, SL Veatch, Liquid General Anesthetics Lower Critical Tem-peratures in Plasma Membrane Vesicles.
Biophys. J . , 2751–2759 (2013).51. EM Gray, G Díaz-Vázquez, SL Veatch, Growth Conditions and Cell Cycle Phase ModulatePhase Transition Temperatures in RBL-2H3 Derived Plasma Membrane Ves icles. PLOSONE , e0137741 (2015).52. MGF Last, S Deshpande, C Dekker, pH-Controlled Coacervate–Membrane Interactionswithin Liposomes. ACS Nano , 4487–4498 (2020).53. C Love, et al., Reversible pH-Responsive Coacervate Formation in Lipid Vesicles ActivatesDormant Enzymatic Reactions. Angewandte Chemie Int. Ed . , 5950–5957 (2020).54. IH Lee, MY Imanaka, EH Modahl, AP Torres-Ocampo, Lipid Raft Phase Modulation byMembrane-Anchored Proteins with Inherent Phase Sepa ration Properties. ACS Omega , 6551–6559 (2019).55. JK Chung, et al., Coupled membrane lipid miscibility and phosphotyrosine-driven proteinconden sation phase transitions. Biophys. J . , S0006349520307281 (2020).56. KN Schaefer, M Peifer, Wnt/Beta-Catenin Signaling Regulation and a Role for BiomolecularCondensates. Dev. Cell , 429–444 (2019).57. T Schwarz-Romond, C Merrifield, BJ Nichols, M Bienz, The Wnt signalling effector Dishev-elled forms dynamic protein assemblies rather than stable associations with cytoplasmic vesi-cles. J. Cell Sci . , 5269–5277 (2005).58. P Ronceray, S Mao, A Košmrlj, MP Haataja, Liquid demixing in elastic networks: cavitation,permeation, or size selecti on? arXiv:2102.02787 [cond-mat, physics:physics] (2021) arXiv:2102.02787.59. JE Rutledge, P Taborek, Prewetting phase diagram of He 4 on cesium. Phys. Rev. Lett . ,937–940 (1992).60. H Kellay, D Bonn, J Meunier, Prewetting in a binary liquid mixture. Phys. Rev. Lett . , 2607–2610 (1993). Rouches et al.et al.