Monte Carlo study of single molecule diffusion can elucidate the mechanism of B cell synapse formation

Lattice-based mesoscale simulations and mean-field theory of cell membrane adhesion

Adhesion of cell membranes involves multi-scale phenomena, ranging from specific molecular binding at Angstrom scale all the way up to membrane deformations and phase separation at micrometer scale. Consequently, theory and simulations of cell membrane adhesion require multi-scale modeling and suitable approximations that capture the essential physics of these phenomena. Here, we present a mesoscale model for membrane adhesion which we have employed in a series of our recent studies. This model quantifies, in particular, how nanoscale lipid clusters physically affect and respond to the intercellular receptor-ligand binding that mediates membrane adhesion. The goal of this Chapter is to present all details and subtleties of the mean-field theory and Monte Carlo simulations of this mesoscale model, which can be used to further explore physical phenomena related to cell membrane adhesion.

Membrane-Mediated Cooperative Interactions of CD47 and SIRPα

The specific binding of the ubiquitous 'marker of self' protein CD47 to the SIRPα protein anchored in the macrophage plasma membrane results in the inhibition of the engulfment of 'self' cells by macrophages and thus constitutes a key checkpoint of our innate immune system. Consequently, the CD47-SIRPα protein complex has been recognized as a potential therapeutic target in cancer and inflammation. Here, we introduce a lattice-based mesoscale model for the biomimetic system studied recently in fluorescence microscopy experiments where GFP-tagged CD47 proteins on giant plasma membrane vesicles bind to SIRPα proteins immobilized on a surface. Computer simulations of the lattice-based mesoscale model allow us to study the biomimetic system on multiple length scales, ranging from single nanometers to several micrometers and simultaneously keep track of single CD47-SIRPα binding and unbinding events. Our simulations not only reproduce data from the fluorescence microscopy experiments but also are consistent with results of several other experiments, which validates our numerical approach. In addition, our simulations yield quantitative predictions on the magnitude and range of effective, membrane-mediated attraction between CD47-SIRPα complexes. Such detailed information on CD47-SIRPα interactions cannot be obtained currently from experiments alone. Our simulation results thus extend the present understanding of cooperative effects in CD47-SIRPα interactions and may have an influence on the advancement of new cancer treatments.

A theory of coalescence of signaling receptor clusters in immune cells

A theory of coalescence of signal receptor clusters in mast cells is developed in close connection with experiments. It is based on general considerations involving a feedback procedure and a time-dependent capture as part of a reaction-diffusion process. Characteristic features of observations that need to be explained are indicated and it is shown why calculations available in the literature are not satisfactory. While the latter involves static centers at which the reaction part of the phenomenon occurs, by its very nature, coalescence involves dynamically evolving centers. This is so because the process continuously modifies the size of the cluster aggregate which then proceeds to capture more material. We develop a procedure that consists of first solving a static reaction-diffusion problem and then imbuing the center with changing size. The consequence is a dependence of the size of the signal receptor cluster aggregate on time. A preliminary comparison with experiment is shown to reveal a sharp difference between theory and data. The observation indicates that the reaction occurs slowly at first and then picks up rapidly as time proceeds. Parameter modification to fit the observations cannot solve the problem. We use this observation to build into the theory an accumulation rate that is itself dependent on time. A memory representation and its physical basis are explained. The consequence is a theory that can be fit to observations successfully.

From affinity selection to kinetic selection in Germinal Centre modelling

Affinity maturation is an evolutionary process by which the affinity of antibodies (Abs) against specific antigens (Ags) increases through rounds of B-cell proliferation, somatic hypermutation, and positive selection in germinal centres (GC). The positive selection of B cells depends on affinity, but the underlying mechanisms of affinity discrimination and affinity-based selection are not well understood. It has been suggested that selection in GC depends on both rapid binding of B-cell receptors (BcRs) to Ags which is kinetically favourable and tight binding of BcRs to Ags, which is thermodynamically favourable; however, it has not been shown whether a selection bias for kinetic properties is present in the GC. To investigate the GC selection bias towards rapid and tight binding, we developed an agent-based model of GC and compared the evolution of founder B cells with initially identical low affinities but with different association/dissociation rates for Ag presented by follicular dendritic cells in three Ag collection mechanisms. We compared an Ag collection mechanism based on association/dissociation rates of B-cell interaction with presented Ag, which includes a probabilistic rupture of bonds between the B-cell and Ag (Scenario-1) with a reference scenario based on an affinity-based Ag collection mechanism (Scenario-0). Simulations showed that the mechanism of Ag collection affects the GC dynamics and the GC outputs concerning fast/slow (un)binding of B cells to FDC-presented Ags. In particular, clones with lower dissociation rates outcompete clones with higher association rates in Scenario-1, while remaining B cells from clones with higher association rates reach higher affinities. Accordingly, plasma cell and memory B cell populations were biased towards B-cell clones with lower dissociation rates. Without such probabilistic ruptures during the Ag extraction process (Scenario-2), the selective advantage for clones with very low dissociation rates diminished, and the affinity maturation level of all clones decreased to the reference level.

Adaptive immunity is one of the vital defence mechanisms of the human body to fight virtually unlimited types of pathogens by producing antigen-specific high-affinity antibodies that bind to pathogens and neutralise them or mark them for further elimination. Affinity is a quantity used to measure and report the strength of interaction between antibodies and antigens that depends both on how fast antibodies bind to antigens (association rate) and how long the bond lasts (dissociation rate). The affinity of produced antibodies for a specific antigen increases in germinal centres through a process called affinity maturation, during which B cells with higher affinities have a competitive advantage and get positively selected to differentiate to antibody-producing plasma cells. Our research shows that the mechanism by which B cells capture Ag affects GC dynamics and GC output with respect to B-cell receptor kinetics. Notably, in a mechanism where rupture of CC-FDC bonds is possible during Ag extraction, B-cell clones with low dissociation rates outcompete clones with high association rates over time. Understanding how B cells get selected in germinal centres could help to develop an optimised and effective immune response against a disease through vaccination for a fast-operating and long-lasting immune response.

Biomechanics as driver of aggregation of tethers in adherent membranes

Cell adhesion is an important cellular process and is mediated by adhesion proteins residing on the cell membrane. Sometimes, two types of linker proteins are involved in adhesion, and they can segregate to form domains through a poorly understood size-exclusion process. We present an experimental and theoretical study of adhesion via linkers of two different sizes, realised in a mimetic model-system, based on giant unilamellar vesicles interacting with supported lipid bilayers. Here, adhesion is mediated by DNA linkers with two different lengths, but with the same binding enthalpy. We study the organisation of these linkers into domains as a function of relative fraction of long and short DNA constructs. Experimentally, we find that, irrespective of the composition, the adhesion domains are uniform with coexisting DNA bridge types, despite their relative difference in length of 9 nm. However, simulations suggest formation of nanodomains of the minority fraction at short length scales, which is below the optical resolution of the microscope. The nano-aggregation is more significant for long bridges, which are also more stable.

Interplay Between Receptor-Ligand Binding and Lipid Domain Formation Depends on the Mobility of Ligands in Cell-Substrate Adhesion

Cell-cell adhesion and the adhesion of cells to extracellular matrix are mediated by the specific binding of receptors on the cell membrane to their cognate ligands on the opposing surface. The adhesion receptors can exhibit affinity for nanoscale lipid clusters that form in the cell membrane. Experimental studies of such adhesion systems often involve a cell adhering either to a solid surface with immobile ligands or a supported lipid bilayer with mobile ligands. A central question in these cell-substrate adhesions is how the mobility of the ligands physically affects their binding to the adhesion receptors and thereby the behavior of the nanoscale lipid clusters associated with the receptors. Using a statistical mechanical model and Monte Carlo simulations for the adhesion of cells to substrates with ligands, we find that, for mobile ligands, binding to adhesion receptors can promote the formation of mesoscale lipid domains, which in turn enhances the receptor-ligand binding. However, in the case of immobile ligands, the receptor-ligand binding and the tendency for the nanoscale lipid clusters to further coalesce depend on the distribution of the ligands on the substrate. Our findings help to explain why different adhesion experiments for identifying the interplay between receptor-ligand binding and heterogeneities in cell membranes led to contradictory results.

Molecular Biomechanics Controls Protein Mixing and Segregation in Adherent Membranes

Cells interact with their environment by forming complex structures involving a multitude of proteins within assemblies in the plasma membrane. Despite the omnipresence of these assemblies, a number of questions about the correlations between the organisation of domains and the biomechanical properties of the involved proteins, namely their length, flexibility and affinity, as well as about the coupling to the elastic, fluctuating membrane, remain open. Here we address these issues by developing an effective Kinetic Monte Carlo simulation to model membrane adhesion. We apply this model to a typical experiment in which a cell binds to a functionalized solid supported bilayer and use two ligand-receptor pairs to study these couplings. We find that differences in affinity and length of proteins forming adhesive contacts result in several characteristic features in the calculated phase diagrams. One such feature is mixed states occurring even with proteins with length differences of 10 nm. Another feature are stable nanodomains with segregated proteins appearing on time scales of cell experiments, and for biologically relevant parameters. Furthermore, we show that macroscopic ring-like patterns can spontaneously form as a consequence of emergent protein fluxes. The capacity to form domains is captured by an order parameter that is founded on the virial coefficients for the membrane mediated interactions between bonds, which allow us to collapse all the data. These findings show that taking into account the role of the membrane allows us to recover a number of experimentally observed patterns. This is an important perspective in the context of explicit biological systems, which can now be studied in significant detail.

Interplay between cooperativity of intercellular receptor-ligand binding and coalescence of nanoscale lipid clusters in adhering membranes

Adhesion of biological cells is mediated by the specific binding of receptors and ligands which are typically large proteins spanning through the plasma membranes of the contacting cells. The receptors and ligands can exhibit affinity for nanoscale lipid clusters that form within the plasma membrane. A central question is how these nanoscale lipid clusters physically affect and respond to the receptor-ligand binding during cell adhesion. Within the framework of classical statistical mechanics we find that the receptor-ligand binding reduces the threshold energy for lipid clusters to coalesce into mesoscale domains by up to ∼50%, and that the formation of these domains induces significant cooperativity of the receptor-ligand binding. The interplay between the receptor-ligand binding cooperativity and the lipid domain formation manifests acute sensitivity of the membrane system to changes in control parameters. This sensitivity can be crucial in cell signaling and immune responses.

Intercellular Receptor-Ligand Binding and Thermal Fluctuations Facilitate Receptor Aggregation in Adhering Membranes

Nanoscale molecular clusters in cell membranes can serve as platforms to recruit membrane proteins for various biological functions. A central question is how these nanoclusters respond to physical contacts between cells. Using a statistical mechanics model and Monte Carlo simulations, we explore how the adhesion of cell membranes affects the stability and coalescence of clusters enriched in receptor proteins. Our results show that intercellular receptor-ligand binding and membrane shape fluctuations can lead to receptor aggregation within the adhering membranes even if large-scale clusters are thermodynamically unstable in nonadhering membranes.

Binding constant of membrane-anchored receptors and ligands that induce membrane curvatures

Cell adhesion is crucial for immune response, tissue formation, and cell locomotion. The adhesion process is mediated by the specific binding of membrane-anchored receptor and ligand proteins. These adhesion proteins are in contact with the membranes and may generate curvature, which has been shown for a number of membrane proteins to play an important role in membrane remodeling. An important question remains of whether the local membrane curvatures induced by the adhesion proteins affect their binding. We've performed Monte Carlo simulations of a mesoscopic model for membrane adhesion via the specific binding of curvature-inducing receptors and ligands. We find that the curvatures induced by the adhesion proteins do affect their binding equilibrium constant. We presented a theory that takes into account the membrane deformations and protein-protein interactions due to the induced curvatures, and agrees quantitatively with our simulation results. Our study suggests that the ability to induce membrane curvatures represents a molecular property of the adhesion proteins and should be carefully considered in experimental characterization of the binding affinity.

Dimensionality of Motion and Binding Valency Govern Receptor-Ligand Kinetics As Revealed by Agent-Based Modeling

Mathematical modeling and computer simulations have become an integral part of modern biological research. The strength of theoretical approaches is in the simplification of complex biological systems. We here consider the general problem of receptor–ligand binding in the context of antibody–antigen binding. On the one hand, we establish a quantitative mapping between macroscopic binding rates of a deterministic differential equation model and their microscopic equivalents as obtained from simulating the spatiotemporal binding kinetics by stochastic agent-based models. On the other hand, we investigate the impact of various properties of B cell-derived receptors—such as their dimensionality of motion, morphology, and binding valency—on the receptor–ligand binding kinetics. To this end, we implemented an algorithm that simulates antigen binding by B cell-derived receptors with a Y-shaped morphology that can move in different dimensionalities, i.e., either as membrane-anchored receptors or as soluble receptors. The mapping of the macroscopic and microscopic binding rates allowed us to quantitatively compare different agent-based model variants for the different types of B cell-derived receptors. Our results indicate that the dimensionality of motion governs the binding kinetics and that this predominant impact is quantitatively compensated by the bivalency of these receptors.

Lipid rafts enhance the binding constant of membrane-anchored receptors and ligands

Gaining insights into the binding of membrane-anchored receptors and ligands that mediate cell adhesion and signal transduction is of great significance for understanding numerous physiological processes driven by intercellular communication. Lipid rafts, microdomains in cell membranes enriched in cholesterol and saturated lipids such as sphingomyelin, are believed to serve as the essential platforms to recruit protein molecules for biological functions. An important question remains how the lipid rafts affect the binding constant of membrane-anchored receptors and ligands. We have investigated the adhesion of multicomponent membranes by using Monte Carlo simulations of a mesoscopic model with biologically relevant parameters. We find that the preferential partitioning of membrane-anchored receptor and ligand proteins in the lipid rafts significantly increases the binding constant of those proteins, in cooperation with the shape fluctuations of the membranes caused by thermal excitations. The binding constant can even be greater than that of the same receptors and ligands anchored to two apposing supported, planar membranes without shape fluctuations. The membrane shape fluctuations facilitate the binding of the anchored receptors and ligands, in contrast to the case of homogeneous membranes. Our results suggest that cells might regulate the binding of membrane-anchored receptor and ligand proteins by modulating the properties of lipid rafts such as area fraction, size and the affinity of rafts to the proteins.

Crowding of receptors induces ring-like adhesions in model membranes

The dynamics of formation of macromolecular structures in adherent membranes is a key to a number of cellular processes. However, the interplay between protein reaction kinetics, diffusion and the morphology of the growing domains, governed by membrane mediated interactions, is still poorly understood. Here we show, experimentally and in simulations, that a rich phase diagram emerges from the competition between binding, cooperativity, molecular crowding and membrane spreading. In the cellular context, the spontaneously-occurring organization of adhesion domains in ring-like morphologies is particularly interesting. These are stabilized by the crowding of bulky proteins, and the membrane-transmitted correlations between bonds. Depending on the density of the receptors, this phase may be circumvented, and instead, the adhesions may grow homogeneously in the contact zone between two membranes. If the development of adhesion occurs simultaneously with membrane spreading, much higher accumulation of binders can be achieved depending on the velocity of spreading. The mechanisms identified here, in the context of our mimetic model, may shed light on the structuring of adhesions in the contact zones between two living cells. This article is part of a Special Issue entitled: Mechanobiology.

How B cells capture, process and present antigens: a crucial role for cell polarity

B cells are key components of the adaptive immune response. Their differentiation into either specific memory B cells or antibody-secreting plasma cells is a consequence of activation steps that involve the processing and presentation of antigens. The engagement of B cell receptors by surface-tethered antigens leads to the formation of an immunological synapse that coordinates cell signalling events and that promotes antigen uptake for presentation on MHC class II molecules. In this Review, we discuss membrane trafficking and the associated molecular mechanisms that are involved in antigen extraction and processing at the B cell synapse, and we highlight how B cells use cell polarity to coordinate the complex events that ultimately lead to efficient humoral responses.

Discrimination of membrane antigen affinity by B cells requires dominance of kinetic proofreading over serial engagement

B-cell receptor signaling in response to membrane-bound antigen increases with antigen affinity, a process known as affinity discrimination. We use computational modeling to show that B-cell affinity discrimination requires that kinetic proofreading predominate over serial engagement. We find that if B-cell receptors become signaling-capable immediately upon antigen binding, which results in decreasing serial engagement as affinity increases, then increasing affinity can lead to weaker signaling. Rather, antigen must stay bound to B-cell receptors for a threshold time of several seconds before becoming signaling-capable, a process similar to kinetic proofreading. This process overcomes the loss in serial engagement due to increasing antigen affinity, and replicates the monotonic increase in B-cell signaling with increasing affinity that has been observed in B-cell activation experiments. This finding matches well with the experimentally observed time (∼20 s) required for the B-cell receptor signaling domains to undergo antigen and lipid raft-mediated conformational changes that lead to Src-family kinase recruitment. We hypothesize that the physical basis for a threshold time of antigen binding might lie in the formation timescale of B-cell receptor dimers. The time required for dimer formation decreases with increasing antigen affinity, thereby resulting in shorter threshold antigen binding times as affinity increases. Such an affinity-dependent kinetic proofreading requirement results in affinity discrimination very similar to that observed in biological experiments. B-cell affinity discrimination is critical to the process of affinity maturation and the production of high-affinity antibodies, and thus our results have important implications in applications such as vaccine design.

Multiphoton fluorescence recovery after photobleaching in bounded systems

Multiphoton fluorescence recovery after photobleaching (MP-FRAP) is a laser microscopy technique used to measure diffusion coefficients of macromolecules in biological systems. The three-dimensional resolution and superior depth penetration within scattering samples offered by MP-FRAP make it an important tool for investigating both in vitro and in vivo systems. However, biological systems frequently confine diffusion within solid barriers, and to date the effect of such barriers on the measurement of absolute diffusion coefficients via MP-FRAP has not been studied. We have used Monte Carlo simulations of diffusion and MP-FRAP to understand the effect of barriers of varying geometries and positions relative to the two-photon focal volume. Furthermore, we supply ranges of barrier positions within which MP-FRAP can confidently be employed to measure accurate diffusion coefficients. Finally, we produce two new MP-FRAP models that can produce accurate diffusion coefficients in the presence of a single plane boundary or parallel infinite plane boundaries positioned parallel to the optical axis, up to the resolution limit of the multiphoton laser scanning microscope.

Monte Carlo study of B-cell receptor clustering mediated by antigen crosslinking and directed transport

It is known from experiments that in the presence of soluble antigen, B-cell receptors (BCRs) assemble into microclusters and then collect into a macrocluster known as a 'cap'. However, the mechanisms of BCR cluster formation during recognition of soluble antigens remain unclear. In previous work, we demonstrated that effective intrinsic attractions among BCRs can lead to the formation of small microclusters of BCR molecules. The effective intrinsic attractions could be caused by multivalent antigen binding, association with lipid rafts, or other biochemical factors. In the present study, we have developed and studied a Monte Carlo model of BCR clustering mediated by explicit binding and crosslinking of soluble bivalent antigens. Antigen crosslinking is shown to microcluster BCRs in an affinity-dependent manner and also in a biologically relevant timescale; however, antigen crosslinking alone does not appear to be sufficient for the formation of a single macrocluster of receptor molecules. We show that directed transport of BCRs is needed to drive the formation of large macroclusters. We constructed a simple model of directed transport, where BCR molecules diffuse towards the largest cluster or towards a random BCR microcluster, which results in a single macrocluster of receptor molecules. The mechanisms for both types of directed transport are compared using network-based metrics. We also develop and use appropriate network measures to analyze the effect of BCR and antigen concentration on BCR clustering, the stability of the formed clusters over time and the size of BCR-antigen crosslinked chains.

Monte Carlo investigation of diffusion of receptors and ligands that bind across opposing surfaces

Studies of receptor diffusion on a cell surface show a variety of behaviors, such as diffusive, sub-diffusive, or super-diffusive motion. However, most studies to date focus on receptor molecules diffusing on a single cell surface. We have previously studied receptor diffusion to probe the molecular mechanism of receptor clustering at the cell–cell junction between two opposing cell surfaces. Here, we characterize the diffusion of receptors and ligands that bind to each other across two opposing cell surfaces, as in cell–cell and cell–bilayer interactions. We use a Monte Carlo method, where receptors and ligands are simulated as independent agents that bind and diffuse probabilistically. We vary receptor–ligand binding affinity and plot the molecule-averaged mean square displacement (MSD) of ligand molecules as a function of time. Our results show that MSD plots are qualitatively different for flat and curved interfaces, as well as between the cases of presence and absence of directed transport of receptor–ligand complexes toward a specific location on the interface. Receptor–ligand binding across two opposing surfaces leads to transient sub-diffusive motion at early times provided the interface is flat. This effect is entirely absent if the interface is curved, however, in this instance we observe sub-diffusive motion. In addition, a decrease in the equilibrium value of the MSD occurs as affinity increases, something which is absent for a flat interface. In the presence of directed transport of receptor–ligand complexes, we observe super-diffusive motion at early times for a flat interface. Super-diffusive motion is absent for a curved interface, however, in this case we observe a transient decrease in MSD with time prior to equilibration for high-affinity values.

Modeling of B cell Synapse Formation by Monte Carlo Simulation Shows That Directed Transport of Receptor Molecules Is a Potential Formation Mechanism

The formation of the protein segregation structure known as the "immunological synapse" in the contact region between B cells and antigen presenting cells appears to precede antigen (Ag) uptake by B cells. The mature B cell synapse consists of a central cluster of B cell receptor/Antigen (BCR/Ag) complexes surrounded by a ring of LFA-1/ICAM-1 complexes. In this study, we used an in silico model to investigate whether cytoskeletally driven transport of molecules toward the center of the contact zone is a potential mechanism of immunological synapse formation in B cells. We modeled directed transport by the cytoskeleton in an effective manner, by biasing the diffusion of molecules toward the center of the contact zone. Our results clearly show that biased diffusion of BCR/Ag complexes on the B cell surface is sufficient to produce patterns similar to experimentally observed immunological synapses. This is true even in the presence of significant membrane deformation as a result of receptor-ligand binding, which in previous work we showed had a detrimental effect on synapse formation at high antigen affinity values. Comparison of our model's results to those of experiments shows that our model produces synapses for realistic length, time, and affinity scales. Our results also show that strong biased diffusion of free molecules has a negative effect on synapse formation by excluding BCR/Ag complexes from the center of the contact zone. However, synapses may still form provided the bias in diffusion of free molecules is an order-of-magnitude weaker than that of BCR/Ag complexes. We also show how diffusion trajectories obtained from single-molecule tracking experiments can generate insight into the mechanism of synapse formation.

The network of receptors characterize B cell receptor micro- and macroclustering in a Monte Carlo model

During the recognition of soluble antigens, B cell receptors (BCR) are known to form signaling clusters that can crucially modulate intracellular activation pathways and B cell response. Little is known about the precise nature of receptor cluster and its formation mechanism for the case of soluble antigens. Initial experiments have shown that B cell receptors first microcluster upon ligation with soluble antigens, and then coarsen into a macroscopic cap structure at one pole of a B cell. Such a mutual receptor-receptor attraction can arise locally due to cross-linking by soluble antigens among other possibilities. We develop an energy based Monte Carlo model to investigate the mechanism of B-cell receptor clustering upon ligation with soluble antigens. Our results show that mutual attraction between nearest neighbor receptor pairs can lead to microclustering of B cell receptors, but it is not sufficient for receptor macroclustering. A simple model of biased diffusion where BCR molecules experience a biased directed motion toward the largest cluster is then applied, which results in a single macrocluster of receptor molecules. The various types of receptor clusters are analyzed using the developed network-based metrics such as the average distance between any pairs of receptors.