Binding and segregation of proteins in membrane adhesion: Theory, modelling, and simulations
BBinding and segregation of proteins in membraneadhesion: Theory, modelling, and simulations
Thomas R. Weikl , Jinglei Hu , Batuhan Kav , and Bartosz R´o˙zycki Max Planck Institute of Colloids and Interfaces, Theory and Bio-Systems, Potsdam, Germany Kuang Yaming Honors School & Institute for Brain Sciences, Nanjing University, Nanjing, China Institute of Physics, Polish Academy of Sciences, Warsaw, Poland
Contents
Abstract
The adhesion of biomembranes is mediated by the binding of membrane-anchoredreceptor and ligand proteins. The proteins can only bind if the separation between1 a r X i v : . [ q - b i o . S C ] N ov pposing membranes is sufficiently close to the length of the protein complexes,which leads to an interplay between protein binding and membrane shape. In thisarticle, we review current models of biomembrane adhesion and novel insightsobtained from the models. Theory and simulations with elastic-membrane andcoarse-grained molecular models of biomembrane adhesion indicate that the bind-ing of proteins in membrane adhesion strongly depends on nanoscale shape fluctu-ations of the apposing membranes, which results in binding cooperativity. A lengthmismatch between protein complexes leads to repulsive interactions that are causedby membrane bending and act as a driving force for the length-based segregationof proteins during membrane adhesion. Keywords cell adhesion; membrane bending energy; membrane shape fluctuations; proteinbinding; binding cooperativity
Cell adhesion and the adhesion of vesicles to the biomembranes of cells or or-ganelles are central processes in immune responses, tissue formation, and cellsignaling [1]. These processes are mediated by a variety of receptor and ligandmolecules that are anchored in the membranes. Biomembrane adhesion is inher-ently multivalent – the two apposing membranes contain numerous receptors andligands and adhere via numerous receptor-ligand complexes, which “clamp” themembranes together. A striking feature are the largely different length scales fromnanometers to micrometers involved in cell adhesion. Cells have typical diam-eters of tens of micrometers, but the separation of the apposing membranes inequilibrated cell adhesion zones is only about 15 to 40 nanometer [2]. For typicalconcentrations of receptor-ligand complexes of the order of µ m − , the averageseparation between neighboring complexes is of the order of (cid:112) / µ m − = 100 nm. The membrane shape and shape fluctuations between neighboring receptor-ligand complexes are then dominated by the bending energy of the membranes withbending rigidity κ . On length scales of several hundred nanometers or microns, incontrast, the shape of cell membranes is dominated by the membrane tension σ andby the membrane-connected actin cortex. The crossover from the dominance of thebending energy on small length scales to the dominance of membrane tension andactin cortex on large scales depends on the characteristic mesh size of the actincortex of about 50 to 100 nm [3–5] and on the characteristic length (cid:112) κ/σ . Fortypical values of the bending rigidity κ of lipid membranes in the range from 102o 40 k B T [6, 7] and typical tensions σ of a few µ N / m [8–10], the characteristiclength (cid:112) κ/σ adopts values between 100 and 400 nm.Receptor and ligand molecules anchored in apposing membranes can only bindif the membrane separation at the site of these molecules is sufficiently close tothe length of the receptor-ligand complex. Understanding biomembrane adhesionrequires therefore an understanding of the interplay between (i) the specific bind-ing of membrane-anchored receptor and ligand molecules and (ii) the membraneshape as well as shape fluctuations between and around the receptor-ligand com-plexes. This review is focused on theory, modelling, and simulations of the inter-play of protein binding and membrane shape in biomembrane adhesion. In chapter2, we review various models of biomembrane adhesion that differ in their repre-sentation of the membranes and of the membrane-anchored receptors and ligands.In elastic-membrane models of biomembrane adhesion, the two adhering mem-branes are described as discretized elastic surfaces, and the receptors and ligandsas molecules that diffuse along these surfaces. In molecular models of biomem-brane adhesion, the membrane lipids are captured as individual molecules, eithercoarse-grained or atomistic, on the same level as the receptors and ligands. Fi-nally, multiscale modelling combines detailed molecular modelling of individualreceptor-ligand complexes on short length scales with elastic-membrane modellingon large length scales. In chapter 3, we review insights on the binding equilib-rium and binding constant of receptors and ligands in membrane adhesion zonesobtained from elastic-membrane, coarse-grained molecular, and multiscale mod-elling. A central result confirmed by recent experiments is the binding cooper-ativity of membrane-anchored receptors and ligands from membrane shape fluc-tuations on nanoscales [11–13]. In chapter 4, we consider the length-based seg-regation and domain formation of receptor-ligand complexes in membrane adhe-sion zones. Short receptor-ligand complexes tend to segregate from long receptor-ligand complexes or molecules because the membranes need to curve to compen-sate the length mismatch, which costs bending energy [14, 15]. The article endswith a summary and outlook. The adhesion of cells or vesicles leads to an adhesion zone in which receptorsanchored in the cell or vesicle membrane interact with ligands anchored in the ap-posing membrane (see Fig. 1). The challenge of modelling biomembrane adhesionis to model the interplay between the binding of these receptors and ligands andthe membrane shape and shape fluctuations in the membrane adhesion zone. Thetheoretical models reviewed here differ in their spatial resolution and describe this3 igure 1:
Cartoon of a cell that adheres to a substrate-supported membrane via receptor-ligand complexes. interplay for smaller or larger segments of the adhesion zone and for mobile re-ceptors that can diffuse within the membranes and are not strongly coupled to thecytoskeleton of an adhering cell. We first introduce the bending energy of adher-ing membranes, which is the basis of elastic-membrane and multiscale models ofbiomembrane adhesion.
Biomembranes can be seen as thin elastic shells because their lateral extensions upto micrometers greatly exceed their thickness of a few nanometers. In general, suchthin shells can be deformed by shearing, stretching, or bending. Biomembranesare hardly stretchable and pose no shear resistance because of their fluidity. Thedominant deformation of biomembranes therefore is bending. The bending energydepends on the local curvature of the membranes, which can be characterized bythe two principal curvatures c and c , or alternatively by the mean curvature M = ( c + c ) and Gaussian curvature K = c c . The overall bending energy of amembrane is the integral [16] E be = (cid:90) (cid:20) κ (cid:16) M − m (cid:17) + ¯ κK (cid:21) dA (1)over the membrane area A . The characteristic elastic parameters of the membraneare the bending rigidity κ and the modulus of Gaussian curvature ¯ κ . The spon-taneous curvature m reflects an intrinsic preference for bending, which can resultfrom an asymmetry in either the composition of the two monolayers or in the con-centration of solutes on both sides of the membranes [17].Our focus here is on symmetric membranes with zero spontaneous curvature.For typical κ values in the range from 10 to 40 k B T [6, 7] and typical ¯ κ val-ues between − . κ and − κ of lipid membranes [18], the bending energy (1)4f symmetric membranes with m = 0 is minimal in the planar state [18, 19].Membrane shape changes and fluctuations relative to the planar minimum-energystate with zero bending energy can be described by the perpendicular deviation h ( x, y ) out of a reference x - y -plane. For typically small angles between the mem-brane and the reference x - y -plane, the mean curvature M can be approximatedas M (cid:39) ∆ h ( x, y ) = ∂ h ( x, y ) /∂x + ∂ h ( x, y ) /∂y , and the bending energyis [20] E be = (cid:90) κ (∆ h ) d x d y (2)because the integral over the Gaussian curvature K in Eq. (1) is zero for all devia-tions and fluctuations that do not change the membrane topology of the planar state,according to the Gauss-Bonnet theorem. Contributions from boundary terms of theGauss-Bonnet theorem vanish for appropriate choices of the boundary conditions,e.g. for periodic boundaries.The overall bending energy of two adhering membranes with negligible spon-taneous curvature is the sum E be = (cid:82) (cid:104) κ (∆ h ) + κ (∆ h ) (cid:105) d x d y of thebending energies (2) of the membranes. Here, κ and κ are the bending rigidi-ties of the two membranes, and h ( x, y ) and h ( x, y ) are the deviation fieldsout of a reference x - y -plane. In analogy to the two-body problem, a transfor-mation of variables to the separation field l = h − h and ‘center-of-mass’ field l cm = κ h + κ h allows to rewrite the overall bending energy as E be = E ef + E cm with the effective bending energy [21] E ef = (cid:90) κ ef (∆ l ) d x d y (3)for the separation field and the ‘center-of-mass’ energy E cm = (cid:82) ( κ + κ ) − (∆ l cm ) d x d y . The effective binding rigidity in Eq. (3) is κ ef = κ κ / ( κ + κ ) .Because the binding of receptors and ligands typically depends only on the localmembrane separation l [22], the energy E cm is irrelevant for the specific bindingof receptors and ligands. If one of the membranes, e.g. membrane 2, is a planarsupported membrane, the effective bending rigidity κ ef equals the rigidity κ ofthe apposing membrane because the rigidity κ of the supported membrane can betaken to be much larger than κ .In simulations, elastic membranes are usually discretized. A discretization ofthe effective bending energy (3) of two adhering membranes can be achieved bydiscretizing the reference x - y -plane into a square lattice of lattice sites i with latticeconstant a and local separations l i . A discretized version of the effective bendingenergy (3) is [23, 24] E ef = κ ef a (cid:88) i (∆ d l i ) (4)5ith the discretized Laplacian ∆ d l i = l i + l i + l i + l i − l i . Here, l i to l i are the membrane separations at the four nearest-neighbor sites of site i on thesquare lattice. The linear size a of the membrane patches is typically chosen to bearound 5 nm to capture the whole spectrum of bending deformations of the lipidmembranes [25]. The discretization of the reference plane in the effective bending energy (4) im-plies a discretization of two adhering membranes into apposing pairs of nearlyquadratic membrane patches. Discrete elastic models of biomembrane adhesioncan be constructed by including receptor and ligand molecules in these membranepatches [12,23,24,26–37]. If the adhesion is mediated by a single type of receptor-ligand complexes, the distribution of receptors in one of the membranes can bedescribed by the occupation numbers n i = 1 or , which indicate whether a re-ceptor is present or absent in patch i of this membrane. In the same way, thedistribution of ligands in the apposing membrane can be described by occupationnumbers m i = 1 or (see Fig. 2(a)). The specific interactions of the receptors andligands can then be taken into account by the interaction energy [12, 26] E int = (cid:88) i n i m i V ( l i ) (5)Here, V ( l i ) is the interaction potential of a receptor and a ligand that are present atthe apposing membrane patches i , which implies n i = 1 and m i = 1 . The interac-tion potential V ( l i ) depends on the local separation l i of the apposing patches. Asimplified interaction potential is the square-well potential V ( l i ) = − U for l o − l we / < l i < l o + l we /
2= 0 otherwise (6)with the binding energy
U > , effective length l o , and binding range l we ofa receptor-ligand complex. For this interaction potential, a receptor and ligandare bound with energy − U if the local separation l i of their apposing membranepatches is within the binding range l o ± l we / .The total energy of the model is the sum E tot = E be + E int of the effectivebending energy (4) of the membranes and the interaction energy (5) of the recep-tors and ligands. The binding equilibrium of the biomembranes can be determinedfrom the free energy F = − k B T ln Z , where Z is the partition function of the sys-tem, k B is Boltzmann’s constant, and T is the temperature. The partition function Z is the sum over all possible membrane configurations, with each configuration6 i (a)(b) Figure 2:
Elastic-membrane models of biomembrane adhesion. In both models, the mem-branes are described as discretized elastic surfaces (see section 2.1). In (a), the distri-bution of receptors and ligands in the quadratic patches of the discretized membranes isdescribed by occupation numbers n i and m i that can adopt the values 0 and 1 [24, 26].Here, n i = 1 indicates that a receptor is present in membrane patch i of the upper mem-brane (red patches), and m i = 1 indicates a ligand in patch i of the lower membrane (bluepatches). The values n i = 0 and m i = 0 indicate patches without receptors and ligands(grey patches). Receptors and ligands at apposing membrane patches i interact via a poten-tial V ( l i ) that depends on the local separation l i of the patches (see Eq. (5)). The model canbe generalized to membranes with different types of receptors and ligands (see Eq. (9)).In (b), the receptors and ligands are modelled as anchored rods that diffuse continuouslyalong the membranes and rotate around their anchoring points [22]. The binding and un-binding of the receptors and ligands is taken into account by distance- and angle-dependentinteractions of the binding sites, which are located at the tips of the receptors and ligands. exp [ − E tot /k B T ] . A membrane configurationis characterized by the local separations { l i } of the apposing membrane patchesand by the distributions { m i } and { n i } of the receptors and ligands. In this model,the partial summation in the partition function Z over all possible distributions { m i } and { n i } of receptors and ligands can be performed exactly, which leads toan effective adhesion potential. The effective adhesion potential V ef ( l i ) is a square-well potential with the same binding range l we as the receptor-ligand interaction (6)and an effective potential depth U ef that depends on the concentrations and bindingenergy U of receptors and ligands [12, 24, 26]: V ef ( l i ) = − U ef for l o − l we / < l i < l o + l we /
2= 0 otherwise (7)For typical concentrations of receptors and ligands up to hundred or several hun-dred molecules per square micrometer in cell adhesion zones, the average distancebetween neighboring pairs of receptor and ligand molecules is much larger thanthe lattice spacing a (cid:39) nm of the discretized membranes. For these small con-centrations, the effective binding energy of the membranes is U ef (cid:39) k B T K [ R ][ L ] (8)where [ R ] is the area concentration of unbound receptors, [ L ] is the area concen-tration of unbound ligands, and K = a e U/k B T is the binding constant for localseparations inside the binding well [12]. The binding equilibrium in the contactzone can therefore be determined from considering two membranes with the dis-crete elastic energy (5) that interact via the effective adhesion potential (7) withwell depth U ef and width l we .An important quantity is the fraction P b of membrane patches that are bound inthe square well of the effective adhesion potential (7). The bound membrane frac-tion P b can be determined in Monte Carlo simulations [24, 26] and continuouslydecays to zero when the effective binding energy U ef approaches the critical bind-ing energy U c ef for the unbinding of the membranes [38]. The unbinding transitionof the membranes results from an interplay of the effective adhesion (7) and thesteric repulsion of the membranes that is caused by thermally excited membraneshape fluctuations [39]. For effective binding energies smaller than U c ef , the effec-tive adhesion potential (7) is not strong enough to hold the membranes togetheragainst their steric, fluctuation-induced repulsion.Biomembrane adhesion via a single type of receptor-ligand bond occurs inexperiments with a single type of ligand that binds to a single type of recep-tor in cells [40–42], reconstituted vesicles [31, 43–54], or cell-membrane-derived8esicles [13]. The ligands in these experiments are either anchored in substrate-supported membranes, or directly deposited on substrates. But cell adhesion isoften mediated by several types of receptor-ligand complexes. The adhesion of Tcells, natural killer (NK) cells, and B cells is mediated by receptor-ligand com-plexes of different lengths [55–60]. Cell adhesion via long and short receptor-ligand complexes has been investigated in experiments in which the cells adhere tosupported membranes that contain two types of ligands [57, 61–65].Adhesion via two types of receptors and ligands can be modeled with occupa-tion numbers n i and m i that adopt the values 0, 1, and 2, where n i = 1 and m i = 1 indicate the presence of a receptor and ligand of type 1 at the apposing pair i ofmembrane patches, and n i = 2 and m i = 2 indicate the presence of a receptor andligand of type 2. The interaction energy E int = (cid:88) i δ n i , δ m i , V ( l i ) + δ n i , δ m i , V ( l i ) (9)of the membranes then includes two interaction potentials V and V for the twotypes of receptor-ligand complexes [14, 29]. The Kronecker symbol δ j,k in Eq. (9)adopts the value for j = k and the value for j (cid:54) = k . For square-well inter-action potentials V and V with binding energies U and U , effective lengths l and l of the complexes, and well widths l (1) we and l (2) we , a summation over all possi-ble distributions { m i } and { n i } of the receptors and ligands leads to an effectivedouble-well adhesion potential if the length difference | l − l | of the receptor-ligand complexes is larger than the potential widths l (1) we and l (2) we . The two wells ofthis effective adhesion potential are located at l and l , have the widths l (1) we and l (2) we , and the effective depths U ef1 (cid:39) k B T K [ R ][ L ] (10)and U ef2 (cid:39) k B T K [ R ][ L ] (11)which depend on the concentrations [ R ] , [ R ] , [ L ] , and [ L ] of unbound receptorsand ligands of type 1 and 2, and on the binding constants K = a e U /k B T and K = a e U /k B T of the receptors and ligands for local membrane separationsinside their binding wells [14].In elastic-membrane models with the interaction energies (5) and (9), the bind-ing of receptors and ligands is described by interaction potentials of membranepatches that contain receptors and ligands. In a more recent elastic-membranemodel, the receptors and ligands are modelled as anchored rods that diffuse contin-uously along the discretized membranes and rotate around their anchoring points9 igure 3: Snapshot from a molecular dynamics simulation of a coarse-grained molecu-lar model of biomembrane adhesion with transmembrane receptors and ligands [11, 81].Each of the membranes contains 22136 lipids, 25 receptors or ligands, and has an area of × nm . The lipid molecules consist of three hydrophilic head beads (dark gray)and two hydrophobic chains with four beads each (light grey) [82–85]. The transmembranereceptors and ligands consist of 84 beads (12 layers of 7 beads) arranged in a cylindricalshape and have hydrophobic anchors that are embedded in the lipid bilayer. The interac-tion domain of the receptors (green) and ligands (red) consists of six layers of hydrophilicbeads, with an interaction bead or ‘binding site’ located in the center of the top layer ofseven beads. The transmembrane anchors of the receptors or ligands mimic the transmem-brane segments of membrane proteins and are composed of four layers of hydrophobiclipid-chain-like beads (not visible) in between two layers of lipid-head-like beads (blue).For clarity, the water beads of the model are not displayed in the snapshot. The boundariesof the simulation box (not shown) are periodic. (see Fig. 2(b)) [22]. The total energy in this model is the sum E tot = E (1)be + E (2)be + E int + E anc of the discretized bending energies (2) of both membranes, theoverall interaction energy E int of the rod-like receptors as ligands, and the overallanchoring energy E anc . The overall anchoring energy is the sum of the anchor-ing potentials V anc of the receptors and ligands. A simple anchoring potential isthe harmonic potential V anc = k a θ a with anchoring strength k a and anchoringangle θ a of the receptors or ligands relative to the local membrane normal [22].The overall interaction energy E int in this model is the sum over the distance- andangle-dependent interaction potentials of the binding sites located at the tips of thereceptors and ligands.In both types of elastic-membrane models illustrated in Fig. 2, the receptors andligands are modelled as individual molecules that can bind and unbind. In otherelastic-membrane models, receptors and ligands have been described via concen-tration fields and not as individual molecules [66–75], or receptor-ligand bondshave been treated as constraints on the local membrane separation [76–80].10 .3 Molecular models of biomembrane adhesion In molecular models of biomembrane adhesion, both the membrane lipids and theanchored receptors and ligands are modelled as individual molecules. Coarse-grained modelling and simulations have been widely used to investigate the self-assembly [82, 86–88], fusion [84, 85, 89–92], and lipid domains [93–97] of mem-branes as well as the diffusion [98, 99], aggregation [100], and curvature gener-ation [101–104] of membrane inclusions and membrane proteins. In the coarse-grained molecular model of biomembrane adhesion [11, 81] shown in Fig. 3, thelipid molecules consist of three hydrophilic head beads (dark gray) and two hy-drophobic chains with four beads each (light grey) [82–85]. The receptors andligands are composed of an interaction domain that protrudes out of the membrane(green, red), and a membrane anchor (not visible). The interaction domain consistsof hydrophilic beads arranged in a cylindrical shape, with an ‘interaction bead’ asbinding site at the center of the tip. This interaction domain is either rigidly con-nected to a cylindrical transmembrane anchor that contains hydrophobic beads, oris flexibly connected to a lipid molecule [81]. The specific binding of the recep-tors and ligands is modeled via a distance- and angle-dependent binding potentialbetween two interaction beads at the tip of the molecules. The binding potentialhas no barrier to ensure an efficient sampling of binding and unbinding events ofreceptors and ligands in our simulations. The average binding times of receptorsand ligands in this potential are of the order of 10 µs . Total simulation times of tensof milliseconds then lead to typically thousands of binding and unbinding events,which allows to determine the binding equilibrium and kinetics with high preci-sion [11, 81]. The length and time scales of this coarse-grained model have beenestimated by comparing to experimental data for the bilayer thickness and lipiddiffusion constant of fluid membranes.Atomistic simulation trajectories of protein binding and biomembrane adhe-sion are currently limited to typical timescales of microseconds [105–107], whichis orders of magnitude smaller than the binding times of receptor and ligand pro-teins in membrane adhesion [63–65,108,109]. The binding equilibrium of proteinsin membrane adhesion systems is therefore beyond the scope of current atomisticsimulations. But in some systems, membrane adhesion can also be mediated byglycolipids [110] with binding times of nanoseconds in the atomistic simulationsof Fig. 4 [105]. Simulation times of one microsecond are therefore sufficient toreach binding equilibrium. In the atomistic simulations of Fig. 4, the two mem-branes are composed of 1620 lipids and 180 glycolipids each and have an area ofabout × nm . The glycolipids contain five sugar rings that are connected tolipid tails [110].Besides reaching the binding equilibrium of receptors and ligands, a challenge11 orem ipsum Figure 4:
Snapshot from an atomistic simulation of two apposing membranes that interact via lipid-anchored Lewis-X (LeX) saccharides [105]. The fucose and galactose rings atthe tip of LeX are represented in red and orange. The remaining three sugar rings that areconnected to the lipid tails are given in yellow. Lipid heads and tails are shown in blackand gray, respectively. for coarse-grained and atomistic simulations of biomembrane adhesion is to equi-librate the thermal shape fluctuations of the membranes. The equilibration timedepends on the lateral correlation length ξ (cid:107) of the membranes [111, 112], whichin turn depends on the concentration [RL] of the receptor-ligand bonds [22], andon the lateral size of the membrane in simulations of small membrane systemswith few bonds [11]. In the membrane systems of Figs. 3 and 4, the correlationlength is smaller than the lateral extension of the membranes and, thus, dominatedby the concentration [RL] of bonds, which constrain the membrane shape fluctu-ations. Membrane shape fluctuations lead to variations in the local separation ofthe membranes, which can be quantified by the relative roughness ξ ⊥ . The relativeroughness ξ ⊥ is the standard deviation of the variations in the local separation. Forfluid membranes, the thermal roughness is proportional to the correlation length ξ (cid:107) [113], which in turn is proportional to the average distance / (cid:112) [RL] betweenneighboring bonds for sufficiently large membrane adhesion systems. Theory and12imulations indicate the approximate scaling relation [22] ξ ⊥ (cid:39) . (cid:112) ( k B T /κ ef ) (cid:46)(cid:112) [RL] (12)with the effective bending rigidity κ ef = κ κ / ( κ + κ ) introduced in Eq. (3). Thescaling relation (12) holds for roughnesses smaller than the length of the receptor-ligand bonds. For such roughnesses, the fluctuation-induced, steric repulsion of themembranes is negligible. For a typical concentration [RL] (cid:39) /µ m of receptor-ligand bonds in cell adhesion and for typical values of the bending rigidities κ and κ of lipid membranes between 10 k B T and 40 k B T [6, 7], the relative membraneroughness ξ ⊥ attains values between 3 nm and 6 nm according to Eq. (12). Thesecalculated roughness values are smaller than the lengths of receptor-ligand bondsin cell adhesion, which are typically between 15 and 40 nanometers [2]. The detailed modelling of protein binding in membrane adhesion is complicatedby the relatively long binding times of proteins from milliseconds to seconds [63–65, 108, 109] and by the relatively large lateral membrane sizes of tens or hundredsof nanometers that are required to capture the full spectrum of membrane shapefluctuations up to the correlation length of the adhering membranes. A promis-ing approach is multiscale modelling, i.e. a combination of molecular modellingof proteins on short length scales and elastic modelling of membranes on longerlength scales. Fig. 5 illustrates a multiscale-modelling approach for the adhesionof vesicle membranes with the marker-of-self protein CD47 to SIRP α proteinsimmobilized on a surface. This approach combines (i) coarse-grained molecularmodelling and simulations of a single CD47-SIRP α complex (see Fig. 5(a)) to de-termine the effective spring constant of the complex, and (ii) simulations of largeelastic membrane segments adhering via CD47-SIRP α complexes, modelled aselastic springs (see Fig. 5(b)).The molecular simulations of the CD47-SIRP α complex are based on the Kim-Hummer model, in which large multi-domain proteins are divided into rigid do-mains and unstructured linkers that connect these domains [114]. CD47 is dividedinto a rigid trans-membrane domain (green) and a rigid binding domain (yellow),which are connected by a rather short linker segment and an additional harmonicbond that mimics a disulfide bridge. The rigid SIRP α interaction domain (blue) isconnected to a surface-bound GST domain by a relatively long linker. In the molec-ular simulations, the CD47 transmembrane domain can rotate in a plane parallel tothe substrate surfaces but keeps its orientation relative to the substrate in order tomimic the membrane embedding of this domain. The separation between the CD4713 μ m μ m μ m nm nm μ m μ m μ m (a) (b) Figure 5:
Multiscale modelling of membrane adhesion mediated by CD47-SIRP α com-plexes [13]: (a) Snapshots from coarse-grained molecular simulations of a CD47-SIRP α complex. The separation between the membrane patch (light blue) and the substrate (grey)varies in the simulations mainly due to conformational changes of the unstructured linkerthat covalently connects SIRP α (blue) to substrate-bound GST proteins (red). The trans-membrane and interaction domain of CD47 are shown in green and yellow. (b) Snapshotfrom a simulation of an adhering membrane segment with area × µ m for the concen-tration [RL] = 30 µ m − of CD47-SIRP α complexes (black dots) at the parameter value ∆ l = 4 nm of the repulsive membrane-substrate interactions. The complexes are mod-elled as elastic springs with a spring constant determined from the molecular simulations.The repulsive interactions between the protein layer on the substrate and the membraneare taken into account by allowing only local separations l > l ∆ l between the mem-brane patches and the substrate, where l is the preferred separation of the CD47-SIRP α complexes for binding. α complex. The flexi-bility of the complex is dominated by the rather long unstructured linker segmentthat connects SIRP α and GST, which leads to the standard deviation σ (cid:39) . nmin the separation of the CD47 transmembrane domain and the substrate. This stan-dard deviation of the membrane-substrate separation corresponds to an effectivespring constant k S = k B T /σ of the complex. The mean value l (cid:39) . nm ofthe separation is the preferred membrane-substrate separation of the complex.The elastic-membrane simulations of Fig. 5(b) are based on the discretizedbending energy described in section 2.1. The large adhering membrane segmentsin these simulations are discretized into quadratic patches with linear size a (cid:39) nm. This larger patch size allows to equilibrate the largescale membrane shapefluctuations, which dominate the membrane roughness, for the rather small com-plex concentrations of [RL] = 30 µ m − of CD47-SIRP α . Membrane patchesthat contain CD47-SIRP α complexes (black dots) are bound to the substrate via aharmonic potential with the preferred separation l and spring constant k S of thecomplexes obtained from the molecular simulations. Repulsive interactions be-tween the protein layer on the substrate and the membrane are taken into accountby allowing only local separations l > l ∆ l between the membrane patches andthe substrate, with ∆ l > . The CD47-SIRP α complexes in the simulations aremobile and diffuse along the membrane by hopping from patch to patch. In theexperiments, the mobility of complexes results from unbinding and rebinding ofthe relatively few CD47 proteins to the many SIRP α proteins on the substrate [13]. Adhesion processes of cells depend sensitively on the binding affinity of the membrane-anchored receptor and ligand molecules that mediate adhesion. Binding affinitiesare typically quantified by binding equilibrium constants. For soluble receptorand ligand molecules that are free to diffuse in three dimensions, the binding con-stant K = [RL] / [R] [L] is fully determined by the binding free energyof the receptor-ligand complex RL. Here, [RL] , [R] , and [L] are the vol-ume concentrations of the complex RL and of the unbound receptors R and lig-ands L. A corresponding quantity for membrane-anchored receptors and ligands is K = [RL] / [R] [L] where [RL] , [R] , and [L] are the area concen-15 oluble proteins membrane-anchored proteins [ RL ] = K [ R ] [ L ] [ RL ] = K [ R ] [ L ] Figure 6:
The binding constant K of soluble receptor and ligand molecules is deter-mined by the binding free energy of the molecules. The corresponding quantity K ofmembrane-anchored receptor and ligand molecules, in contrast, does not only depend onproperties of the molecules, but depends also on the membrane separation because the re-ceptors and ligands can only bind if this separation is sufficiently close to the length of thereceptor-ligand complexes. trations of the complex RL and of the unbound receptors R and ligands L [115,116](see Fig. 6). In contrast to K , the quantity K is not fully determined by thereceptors and ligands, but depends also on properties of the membranes. For exam-ple, K is zero if the membrane separation is significantly larger than the lengthof the receptor-ligand complex, because RL complexes cannot form at such largemembrane separations.A membrane-anchored receptor can only bind to an apposing membrane-an-chored ligand if the local membrane separation l at the site of the receptor andligand is within an appropriate range. This local separation l of the membranesvaries – along the membranes, and in time – because of thermally excited mem-brane shape fluctuations. Measurements of K imply an averaging in space andtime over membrane adhesion regions and measurement durations. Theory andsimulations with the elastic-membrane model of Fig. 2(b) and the coarse-grainedmolecular model of Fig. 3 indicate that this averaging can be expressed as [22, 81] K = (cid:90) k ( l ) P ( l )d l (13)where k ( l ) is the binding equilibrium constant of the receptors and ligands as afunction of the local membrane separation l , and P ( l ) is the distribution of lo-cal membrane separations that reflects the spatial and temporal variations of l .16he binding constant function k ( l ) is determined by molecular properties ofthe membrane-anchored receptors and ligands and is, thus, the proper equivalentof the binding konstant K of soluble receptors and ligands. However, the setof molecular features that affect k ( l ) is rather large and includes the lengths,flexibility, and membrane anchoring of the receptors and ligands, besides the in-teractions at the binding site [22, 81]. A softer membrane anchoring, for example,leads to smaller values of k ( l ) because softly anchored receptors and ligandshave more rotational entropy to loose during binding, compared to more rigidlyanchored receptors and ligands. The anchoring elasticity depends on whether thereceptors and ligands are anchored via lipids or trans-membrane protein domains,and on the length and flexibility of unstructured linker segments that connect theextracellular domains of the receptors and ligands to these membrane anchors. Ifthe adhesion is mediated by a single type of receptor-ligand complex with a lengththat is larger than the relative roughness ξ ⊥ of the fluctuating membranes fromthermally excited shape fluctuations, the distribution P ( l ) of the local separationcan be approximated by the Gaussian function [22, 81] P ( l ) (cid:39) exp (cid:2) − ( l − ¯ l ) / ξ ⊥ (cid:3) / ( √ πξ ⊥ ) (14)The mean of the distribution P ( l ) is the average separation ¯ l of the membranes,and the standard deviation of this distribution is the relative roughness ξ ⊥ .Fig. 7 illustrates the modelling of simulation results for K based on Eq. (13).The data points in Fig. 7(a) and (b) result from MD simulations with the coarse-grained molecular model of Fig. 3. In these simulations, K has been measuredfor a variety of membrane systems that differ in membrane area, in the numberof receptors and ligands, or in the membrane potential [81], at ‘optimal’ averagemembrane separations ¯ l at which K is maximal. These optimal average sep-arations correspond to average separations in equilibrated adhesion zones of cell,because maxima in K correspond to minima in the adhesion free energy [11,22].In the MD simulations, the average separation is determined by the number of wa-ter beads between the membranes and is, thus, constant. The relative membraneroughness ξ ⊥ depends on the area L x × L y of the membranes because the peri-odic boundaries of the simulation box suppress membrane shape fluctuations withwavelength larger than L x / π . In membrane systems with several anchored recep-tors and ligands, the roughness is affected by the number of receptor-ligand bondsbecause the bonds constrain the membrane shape fluctuations. For the small num-bers of receptors and ligands in the MD simulations, the binding constants can bedetermined from the times spent in bound and unbound states [11, 81].With increasing size of the membrane systems, the relative roughness ξ ⊥ stronglyincreases, while K strongly decreases. The decrease of K with increasing17
15 16 17 1800.20.40.60.80.2 0.4 0.6 0.8 1.0 1.2 1.4 1.60.50.60.70.80.91.0 = = = = relative membrane roughness ξ ⟂ [ nm ] K D [ K D n m - ] = = = = relative membrane roughness ξ ⟂ [ nm ] k D ( l ) / K D [ n m - ] k (l) ξ ⟂ = ξ ⟂ = ξ ⟂ = ξ ⟂ = ξ ⟂ = ξ ⟂ = ξ ⟂ = ξ ⟂ = ξ ⟂ = ξ ⟂ = k (l) d i s t r i bu t i on P ( l ) P(l)
P(l) k D ( l ) / K D [ n m - ] d i s t r i bu t i on P ( l ) lipid-anchored R and L transmembrane R and L local separation l [ nm ] local separation l [ nm ] (c)(a) (b)(d) membrane area A=80×80 nm²A=30×30 nm²A=14×14 nm² 15R+15L1R+1L1R+1L (e) lipid-anchored transmembrane K D [ K D n m - ] Figure 7: (a) and (b) K for lipid-anchored and transmembrane receptors and ligandsversus relative membrane roughness ξ ⊥ at the ‘optimal’ average separation for bind-ing [81]. The MD data points result from a variety of membrane systems with trans-membrane and lipid-anchored receptors and ligands of a coarse-grained molecular modelof biomembrane adhesion. In these systems, the area A of the two apposing membranesranges from × nm to × nm , and the number of receptors R and ligandsL varies between 1 and 25 (see figure legends). The units of K are K / nm where K (cid:39) nm is the binding constant of soluble variants of the receptors and ligandswithout membrane anchors. The full lines in (a) and (b) represents fits based on Eqs.(13) and (14) with functions k ( l ) from modelling the receptor-ligand bonds as harmonicsprings that can tilt [22,81]. These functions k ( l ) are shown in (c) and (d), together withthe distributions P ( l ) of local separations for various relative membrane roughnesses ξ ⊥ .(e) Snapshots from MD simulations of three different membrane systems with transmem-brane receptors and ligands. ξ ⊥ can be understood from Eq. (13). An increase in ξ ⊥ implies a broad-ening of P ( l ) , which leads to smaller values of the integral (cid:82) k ( l ) P ( l )d l at theoptimal average separation of binding. In Fig. 7(c) and (d), the distributions P ( l ) at the optimal average separation are illustrated for several values of the membraneroughness ξ ⊥ . The function k ( l ) for the lipid-anchored receptors and ligands inFig. 7(c) is significantly broader than k ( l ) for the transmembrane receptors andligands in Fig. 7(d). The functions k ( l ) are asymmetric because the receptor-ligand complexes can tilt at local separations l smaller than the ‘preferred’ separa-tion l at which the functions are maximal, but need to stretch at local separationslarger than l .Because K is difficult to measure in adhesion experiments, an importantquestion is how K is related to the binding equilibrium constant K of solublevariants of the receptors and ligands that lack the membrane anchors [11,12,22,81,115–120]. The binding constant K can be quantified with standard experimentalmethods [121–123]. In Fig. 7, K is given in units of K / nm where K (cid:39) nm is determined by the binding potential of the coarse-grained model [11, 81].For the rod-like receptors and ligands of this coarse-grained molecular model andof the elastic-membrane model of Fig. 2(b), the ratio k ( l ) /K can be calculatedfrom the loss of rotational and translational entropy during binding [22, 81]: k ( l ) /K (cid:39) √ π A b V b Ω RL ( l )Ω R Ω L (15)Here, Ω R , Ω L , and Ω RL ( l ) are the rotational phase space volumes of the unboundreceptors R, unbound ligands L, and bound receptor-ligand complex RL relative tothe membranes, and A b and V b are the translational phase space area and transla-tional phase space volume of the bound ligand relative to the receptor in 2D and3D. The rotational phase space volume Ω RL ( l ) depends on the local separation l of the membranes and can be calculated by modelling the membrane-anchoredcomplex as a harmonic spring that can tilt [22, 81]. The Eqs. (13), (14), and (15)provide a general relation between K and K . The ratio K /K has unitsof an inverse length, but depends on several length scales: the average separation ¯ l and relative roughness ξ ⊥ of the membranes, the width of the function k ( l ) , andthe ratio V b /A b in Eq. (15). The ratio V b /A b is a characteristic length for the bind-ing interface of the receptor-ligand complex and can be estimated as the standarddeviation of the binding-site distance in the direction of the complex [22].19 igure 8: The relative roughness of adhering membranes decreases with increasing con-centration of receptorligand bonds because the bonds constrain membrane shape fluctu-ations (see also Eq.12). The binding constant K of the receptors and ligands in turnincreases with decreasing roughness (see Eq. (17)), which leads to cooperative binding. .2 Cooperative protein binding from membrane shape fluctuationson nanoscales As discussed in the previous section, two important length scales in the generaltheory for K are the relative membrane roughness ξ ⊥ and the average mem-brane separation ¯ l , i.e. the standard deviation and mean of the distribution P ( l ) of local membrane separations. A third important length scale is the width of thefunction k ( l ) in Eq. (13), which reflects the membrane confinement exerted bya single receptor-ligand complex. Coarse-grained molecular modelling indicatesthat the standard deviation of the function k ( l ) is about 1 nm or less, dependingon the anchoring of the receptors and ligands [13, 81] (see Figs. 7 and 9). The rela-tive membrane roughness ξ ⊥ depends on the concentration [RL] of receptor-ligandcomplexes, which ‘clamp’ the membranes together and thus suppress membraneshape fluctuations, and on the bending rigidities κ and κ of the adhering mem-branes. The scaling relation (12) leads to ξ ⊥ values between 3 nm and 6 nm for atypical concentration [RL] (cid:39) /µ m of receptor-ligand bonds in cell adhesionand for typical values of the bending rigidities κ and κ of lipid membranes be-tween 10 k B T and 40 k B T [6, 7]. For such relative membrane roughnesses, thedistribution P ( l ) of local membrane separations is significantly broader than thefunction k ( l ) . Eq. (13) with the Gaussian distribution P ( l ) of Eq. (14) can thenbe approximated as [22] K (cid:39) P ( l o ) (cid:90) k ( l )d l ∼ /ξ ⊥ (16)for average separations ¯ l close to the preferred separation l of the bonds. Eq.(16) implies an inverse proportionality between K and ξ ⊥ at the optimal averageseparation ¯ l for binding, which has been first observed in coarse-grained MD sim-ulations with transmembrane receptors and ligands [11]. For these transmembranereceptors and ligands, the standard deviation of the function k ( l ) in Fig. 7(d) isabout 0.37 nm and, thus, smaller than all roughness values of Fig. 7(b), and theblack line in Fig. 7(b) is K /K (cid:39) . /ξ ⊥ [124]. For the lipid-anchored re-ceptors and ligands of Fig. 7, the standard deviation of the function k ( l ) in Fig.7(c) is about 1.4 nm, and the inverse proportionality of Eq. (16) holds for rough-ness values larger than about 2 nm. Together with the scaling relation (12) betweenthe relative roughness ξ ⊥ and the concentration [RL] of receptor-ligand bonds, theinverse proportionality of K and ξ ⊥ leads to the law of mass action [11, 12]: [RL] = K [R][L] ∼ [R] [L] (17)The quadratic dependence of the bond concentration [RL] on the concentrations [R] and [L] of unbound receptors and ligands in Eq. (17) indicates cooperative21 concentration [RL] of complexes [ μ m -2 ] [RL] = 80 μ m -2 [RL] = 70 μ m -2 [RL] = 60 μ m -2 [RL] = 50 μ m -2 [RL] = 40 μ m -2 [RL] = 30 μ m -2 k (l) K D [ R ] d i s t r i bu t i on P ( l ) (b)(a)
40 50 60 70 8050060070080090010001100 local separation l [ nm ] P(l)
Figure 9: (a) K of CD47-SIRP α complexes as a function of complex concentration [RL] from experiments (data points) and multiscale modelling (line) based on Eq. (13). (b)Shape of the function k ( l ) determined from coarse-grained molecular simulations of asingle CD47-SIRP α complex (see Fig. 5(a)), and distributions P ( l ) of local membrane sep-arations from elastic-membrane simulations with different complex concentrations [RL] (see Fig. 5(b)). In the elastic-membrane simulations, the complexes are modelled as har-monic springs with a spring constant determined from the coarse-grained molecular simu-lations. Repulsive interactions between the protein layer on the substrate and the membranein these simulations are taken into account by allowing only local separations l > l ∆ l where l (cid:39) . nm is the preferred separation of the complexes for binding. The pa-rameter value for the repulsive membrane-substrate interactions here is ∆ l = 4 nm. Themaximum value K max = (378 ± / [R] of the function k ( l ) is fitted to the data in (a),where [R] is the concentration of unbound SIRP α on the substrate. binding. The binding cooperativity results from a smoothening of the membraneswith increasing bond concentration [RL] , which facilitates the binding of additionalreceptors and ligands (see Fig. 8).The cooperative binding of receptors and ligands in membrane adhesion hasbeen recently confirmed in experiments in which CD47 proteins in giant vesi-cles generated from cell membranes bind to SIRP α proteins immobilized on aplanar substrate [13]. The experimental data points in Fig. 9(a) show that K increases with increasing concentration [RL] of receptor-ligand bonds. The mul-tiscale modelling illustrated in Fig. 5 confirms that this experimentally observedincrease in K results from a smoothening of the membranes with increasing [RL] . In the multiscale modelling approach, the shape of the function k ( l ) is de-termined from coarse-grained simulations of a single CD47-SIRP α complex (seeFig. 5(a)). The distribution of membrane-substrate separations at the site of com-plex obtained from these simulations is equivalent to the shape of k ( l ) , becauseof k ( l ) ∼ exp ( − ∆ G ( l ) /k B T ) where ∆ G ( l ) is the binding free energy ofthe complex [22], and because the Boltzmann factor exp ( − ∆ G ( l ) /k B T ) deter-22ines the distributions of local separations at the site of the complex. The coarse-grained molecular simulations indicate that the variations of the local separation atthe site of the complex are dominated by the rather long unstructured linker seg-ment that connects SIRP α to the substrate-bound GST domain in Fig. 5(a). Theresulting distribution of local separations is approximately Gaussian, with a stan-dard deviation σ (cid:39) . nm that corresponds to the width of k ( l ) . In the elastic-membrane simulations of Fig. 5(b), bound CD47-SIRP α complexes are thereforemodelled as harmonic springs with effective spring constant k S = k B T /σ . Thedistributions P ( l ) for the large adhering membrane segments of these simulationsare shown in Fig. 9(b) for various concentrations [RL] of CD47-SIRP α complexes.The distributions P ( l ) are asymmetric because the width of these distributions, therelative membrane roughness ξ ⊥ , is larger than the separation ∆ l = 4 nm betweenthe membrane and the protein layer in the simulations. Steric interactions betweenthe protein layer on the substrate and the membrane are taken into account byallowing only local separations l > l ∆ l where l is the preferred membrane-substrate separation of CD47-SIRP α complex. The steric interactions lead toa fluctuation-induced repulsion of the membrane from the protein layer on thesubstrate. Because of this fluctuation-induced repulsion, the average membrane-substrate separation ¯ l , i.e. the mean of P ( l ) , is larger than the preferred membrane-substrate separation of CD47-SIRP α complex, which is l (cid:39) . nm accordingto the coarse-grained molecular simulations of Fig. 5(a). With increasing concen-tration [RL] , the distributions P ( l ) of Fig. 9(b) become narrower and shift towards l , which both leads to larger values of K according to Eq. (13) due to a largeroverlap of P ( l ) with k ( l ) . The line in Fig. 9(a) results from the distributions in(b) with the fit parameter K max = (378 ± / [R] for the maximal value of k ( l ) ,where [R] is the concentration of unbound SIRP α on the substrate. Similar linesare obtained for values of the modelling parameter ∆ l between 1 nm and 5 nm [13]. The adhesion of T cells and other immune cells is mediated by receptor-ligandcomplexes of different lengths, which tend to segregate into domains of long andshort complexes [55–60]. Long and short receptor-ligand complexes in membraneadhesion zones repel each other because the membranes need to curve to compen-sate the length mismatch (see Fig. 10). The strength of this curvature-mediatedrepulsion and segregation depends both on the length difference and the concen-23 igure 10:
Long and short receptor-ligand complexes repel each other because the mem-branes need to curve to compensate the length mismatch. This curvature-mediated repul-sion leads to domains of long and short receptor-ligand complexes for sufficiently largecomplex concentrations. trations of the receptor-ligand complexes.Calculations and simulations with the elastic-membrane model of Fig. 2 withthe interaction energy (9) for short complexes R L of length l and long com-plexes R L of length l indicate that the curvature-mediated segregation of thecomplexes is stable for bond concentrations [R L ] = [R L ] > c k B Tκ ef ( l − l ) (18)with the numerical prefactor c = 0 . ± . [14, 15, 29]. Here, κ ef = κ κ / ( κ + κ ) is the effective bending rigidity of the two membranes with rigidities κ and κ (see section 2.1). In equilibrium, the concentration [R L ] of the short com-plexes in domain 1 is equal to the concentration [R L ] of the long complexes indomain 2. For complex concentrations [R L ] and [R L ] smaller than the criticalconcentration c k B T /κ ef ( l − l ) of Eq. (18), the domains are unstable becausethe entropy of mixing of the bonds then dominates over the curvature-mediatedrepulsion.The derivation of Eq. (18) is based on the effective double-well potential ofthe elastic-membrane model with well depths U ef1 and U ef1 that depend on the con-centrations of the receptors and ligands and on their binding constants K and K for membrane separations within the potential wells (see Eq. (10) and (11)).Elastic membranes interacting via this double-membrane potential exhibit stabledomains for U (1) ef l (1) we = U (2) ef l (2) we > c a ( k B T ) /κ ( l − l ) with c a = 0 . ± . where l (1) we and l (2) we are the widths the two wells [29]. For typical relativemembrane roughnesses in the domains larger than the well widths l (1) we and l (2) we ,the bond concentrations [R L ] = c b ( κ/k B T ) (cid:16) l (1) we K [R ][L ] (cid:17) and [R L ] = a) (b) (c) (d) Figure 11:
Domains patterns in the adhesion zone of T cells: (a) Final pattern of helper Tcells with a central TCR domain (green) surrounded by an integrin domain (red) [56, 57].The pattern results from cytoskeletal transport of TCRs towards the adhesion zone center[28, 61]. (b) Simulated final pattern in the absence of TCR transport [28]. The length ofthe boundary line between the TCR and the integrin domain is minimal in this pattern. (c)and (d) The two types of intermediate patterns observed in the first minutes of adhesion[55]. In simulations, both patterns result from the nucleation of TCR clusters in the firstseconds of adhesion and the subsequent influx of unbound TCR and MHC-peptide ligandsinto the adhesion zone [28]. The closed TCR ring in pattern (c) forms from fast-growingTCR clusters in the periphery of the adhesion zone at sufficiently large TCR-MHC-peptideconcentrations. The pattern (d) forms at smaller TCR-MHC-peptide concentrations. c b ( κ/k B T ) (cid:16) l (2) we K [R ][L ] (cid:17) with c b = (13 ± are proportional to the squareof concentrations of the unbound receptors and ligands [12, 14] (see also Eq. (17).Together, these equations lead to Eq. (18) [15].In T cell adhesion zones, nanoclusters and domains of short TCR-MHCp com-plexes and long integrin complexes form at sufficiently large concentrations ofMHCp [57, 61, 125–127]. For the preferred membrane separations l (cid:39) nmand l (cid:39) nm of the TCR-MHCp and integrin complexes [2] and for typicalmembrane bending rigidities between 10 and 40 k B T , the critical concentrationsof Eq. (18) for curvature-mediated segregation into domains vary between 25 and200 complexes per square micron. These bond concentrations are smaller thanthe experimentally measured concentrations of TCR and integrin complexes [57],which indicates that the curvature-mediated repulsion of these complexes is suffi-ciently strong to drive the domain formation.The domains of long and short receptor-ligand complexes formed during T celladhesion evolve in characteristic patterns. The final domain pattern in the T-cellcontact zone is formed within 15 to 30 minutes and consists of a central TCR do-main surrounded by a ring-shaped integrin domain [56,57] (see Fig. 11(a)). In con-trast, the intermediate patterns formed within the first minutes of T-cell adhesionare either the inverse of the final pattern, with a central integrin domain surroundedby a ring-shaped TCR domain (see fig. 11(c)), or exhibit several nearly circularTCR domains in the adhesion zone (see fig. 11(d)) [55, 57]. To understand this25attern evolution, several groups have modelled and simulated the time-dependentpattern formation during adhesion [27, 28, 30, 36, 54, 68, 70, 71, 128–131]. MonteCarlo simulations with a discretized elastic-membrane model indicate that the cen-tral TCR cluster is only formed if TCR molecules are actively transported by thecytoskeleton towards the center of the adhesion zone [28]. The active transport hasbeen simulated by a biased diffusion of TCRs towards the adhesion zone center,which implies a weak coupling of TCRs to the cytoskeleton. In the absence of ac-tive TCR transport, the Monte Carlo simulations lead to the final, equilibrium pat-tern shown in Fig. 11(b), which minimizes the energy of the boundary line betweenthe TCR and the integrin domain [28]. In agreement with these simulations, T-celladhesion experiments on patterned substrates indicate that cytoskeletal forces drivethe TCRs towards the center of the adhesion zone [61, 132], by a weak frictionalcoupling of the TCRs to the cytoskeletal flow initiated by T cell activation. Theintermediate patterns formed in the Monte Carlo simulations closely resemble theintermediate T cell patterns shown in Figs. 11(c) and (d). In the simulations, thesepatterns emerge from small TCR clusters that are formed within the first secondsof adhesion [28]. The diffusion of free TCR and MHC-peptide molecules intothe adhesion zone leads to faster growth of TCR clusters close to the peripheryof the adhesion zone. For sufficiently large TCR-MHC-peptide concentrations,the peripheral TCR clusters grow into the ring-shaped domain of Fig. 11(c). Atsmaller TCR-MHC-peptide concentration, the initial clusters evolve into the mul-tifocal pattern of Fig. 11(d).Experiments on reconstituted membrane systems and simulations show thatcurvature-mediated segregation also occurs between small receptor-ligand com-plexes and larger membrane-anchored proteins that sterically prevent the preferredlocal separation of the receptors and ligands for binding [133]. Calculations andsimulations with discretized elastic-membrane models indicate segregation of ster-ically repulsive proteins of length l r and receptor-ligand complexes with preferredlocal separation l < l r if the concentration of the repulsive proteins exceeds thecritical concentration c r k B T /κ ef ( l r − l ) with a numerical prefactor c r [27]. Theinterplay of receptor-ligand binding and steric repulsion from glycocalyx proteinshas also been investigated in models for the clustering of integrin receptors dur-ing cell adhesion [75, 134–136]. In addition, segregation of long and short bondshas been observed in experiments with reconstituted membranes that adhere via anchored DNA [137]. Segregation can also result from the interplay of specificreceptor-ligand binding and generic adhesion if the minimum of the generic adhe-sion potential is located at membrane separations that are significantly larger thanthe lengths of the receptor-ligand complexes [54, 66, 138].26 .2 Fluctuation-mediated attraction of adhesion complexes Complexes of receptor and ligand proteins that are anchored in apposing mem-branes ‘clamp’ the membranes together and constrain the shape fluctuations ofthe membranes. These constraints lead to attractive fluctuation-induced interac-tions between identical receptor-ligand bonds, or receptor-ligand bonds of equallengths. The interaction of two neighboring bonds is attractive because the mem-brane shape fluctuations are less constrained if the bonds are close to each other.The membranes then are effectively clamped together at a single site, and not attwo sites as for larger bond distances. A single receptor-ligand bond that pins twofluctuating membranes locally together leads to a cone-shaped average membraneprofile around the pinning site, i.e. the average local separation of the membranesincreases linearly with the distance from the receptor-ligand bond [139, 140] be-cause of the steric, fluctuation-induced repulsion of the membranes [38, 39]. Pairinteractions of two isolated receptor-ligand bonds can be understood as interactionsof two cone-shaped profiles around the bonds [141], but this interaction appears ar-tificial because membrane adhesion is typically mediated by many bonds, and theapposing membranes are on average planar and parallel to each other in adhesionzones with many bonds of the same type or length. As an alternative, pair interac-tions of receptor-ligand bonds can be calculated for membranes held together byan additional generic harmonic potential, but these pair interactions then dependon the potential strength and the location of the potential minimum [78].A more suitable approach is to quantify the overall strength of the fluctuation-mediated interactions between receptor-ligand bonds from the phase behavior andaggregation tendency of many bonds [26,142]. The overall strength of the fluctuation-mediated attraction between identical receptor-ligand complexes depends on thelength of the complexes, on the strength of the membrane confinement exerted bya complex, and on the concentration of the complexes. Because of the fluctuation-induced attraction, the receptor-ligand bonds in Fig. 5(b) tend to be in the vicin-ity of other receptor-ligand complexes. However, the fluctuation-mediated inter-actions are not sufficiently strong to induce aggregation and domain formationon their own, irrespective of the confinement strength and concentration of thebonds [24, 26]. Aggregation only due to fluctuation-mediated interactions does notoccur in the elastic-membrane model of Fig. 2(a) because the unbinding transi-tion of membranes in the effective adhesion potential (7) of this model is continu-ous [38], which precludes a discontinuity in concentrations during adhesion that isnecessary for aggregation and domain formation [24, 26]. Bond aggregation eitherrequires additional direct attractive interactions [26, 79], an additional local stiff-ening of the membranes by the membrane anchors of the receptors or ligands [26],or a preclustering into multimeric receptors and ligands [142].27
Summary and outlook
Biomembrane adhesion involves an interplay between protein binding and mem-brane shape fluctuations, because the binding of the proteins that mediate the ad-hesion depends on the separation of the two adhering membranes. The elastic-membrane and coarse-grained molecular models of biomembrane adhesion re-viewed here indicate that the binding equilibrium resulting from this interplay canbe understood based on Eq. (13) for the apparent binding constant K of theproteins (see section 3.1). The apparent binding constant K depends on prop-erties of the protein molecules as well as on properties of the membranes and istherefore not a proper binding constant. The proper quantification of the bindingaffinity of proteins in membrane adhesion is the function k ( l ) in Eq. (13), i.e.the binding constant of the proteins as a function of the local membrane separa-tion l . The function k ( l ) depends only on molecular properties of the proteins,i.e. on their membrane anchoring, lengths, flexibility, and binding interaction, andis thus an equivalent of the binding constant K of soluble proteins. The depen-dence of K on overall properties of the membranes is captured by the distribution P ( l ) of local membrane separations in Eq. (13). The distribution P ( l ) depends onthe concentration of the protein complexes because the protein complexes con-strain the membrane shape fluctuations, which leads to a binding cooperativity(see section 3.2). Membrane shape fluctuations also lead to attractive interactionsbetween receptor-ligand bonds of similar lengths (see section 4.2), besides cooper-ative binding. However, these fluctuation-mediated interactions are not sufficientlystrong to induce domains of receptor-ligand bonds on their own. In contrast, thecurvature-mediated repulsion between receptor-ligand bonds of different lengthsis a strong driving force for segregation and domain formation (see section 4.1).This curvature-mediated repulsion results from the membrane curvature requiredto compensate the length mismatch between short protein bonds and long bonds orproteins.A future challenge of simulations and experiments is to determine the bindingconstant function k ( l ) of membrane-anchored proteins. For the CD47-SIRP α complex of Fig. 5(a), the shape of the function k ( l ) has been obtained from sim-ulations with a coarse-grained molecular model of the bound complex (see Fig.9(b)). In principle, values of k ( l ) at selected local separations l can be obtainedfrom modelling and simulations by determining the probability that the membrane-anchored proteins are bound at these local separations. A promising approach isto focus on the ratio k ( l ) /K because this ratio is determined by the moleculararchitecture and anchoring flexibility of the membrane-anchored proteins, whichcan be well described with coarse-grained modelling (see also Eq. (15) for rod-like receptors and ligands). The free-energy contribution of the binding site drops28ut from this ratio because the binding site can be assumed to be identical for thesoluble and membrane-anchored proteins. Absolute values for k ( l ) can then beobtained if the binding constant K of soluble variants of the proteins withoutmembrane anchors is known from experiments. A further experimental challengeis to measure the relative roughness of the adhering membranes. Experimentalmeasurements of the relative membrane roughness require a spatial resolution inthe nanometer range both in the directions parallel and perpendicular to the mem-branes, which is beyond the scope of current methods [143–145]. Acknowledgements
We would like to thank Mesfin Asfaw, Heinrich Krobath, Emanuel Schneck, Guang-Kui Xu, and in particular Reinhard Lipowsky for numerous discussions and jointwork on topics of this review.BR has been supported by the National Science Centre, Poland, grant number2016/21/B/NZ1/00006. JH has been supported by the National Science Foundationof China, grant numbers 21504038 and 21973040. Part of the simulations havebeen performed on the computing facilities in the High Performance ComputingCenter (HPCC) of Nanjing University.
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