The effect of receptor clustering on diffusion-limited forward rate constants.

The effect of receptor clustering on the diffusion-limited forward rate constant (k+) is studied theoretically by modeling cell surface receptors by hemispheres distributed on a plane. We give both exact results and bounds. The exact results are obtained using an electrostatic analogue and applying the method of the images. Accurate upper bounds on k+ are found from a variational principle.     

Transmembrane interactions and the mechanisms of transport of proteins across membranes

We have made observations, by double fluorescence staining of the same cell, of the distributions of surface receptors, and of intracellular actin and myosin, on cultured normal fibroblasts and other flat cells, and on lymphocytes and other rounded cells. The binding of multivalent ligands (a lectin or specific antibodies) to a cell surface receptor on flat cells clusters the cell receptors into small patches, which line up directly over the actin- and myosin-containing stress fibers inside the cell. Similar ligands binding to rounded cells can cause their surface receptors to be collected into caps on the surface, and these caps are invariably found to be associated with concentrations of actin and myosin under the capped membrane. Although these ligand-induced surface phenomena appear to be different on flat and rounded cells, we propose that in both cases clusters of receptors become linked across the membrane to actin- and myosin-containing structures. In flat cells these structures are very long stress fibers; therefore, when clusters of receptors become linked to these fibers, the clusters are immobilized. In round cells, membrane-associated actin- and myosin-containing structures are apparently much less extensive than in flat cells; therefore, clusters of receptors linked to these structures are still mobile in the plane of the membrane. We suggest that in this case the clusters are then actively collected into a cap by an analogue of the muscle sliding filament mechanism. To explain the transmembrane linkage, we propose that actin is associated with the plasma membrane as a peripheral protein which is directly or indirectly bound to an integral protein (or proteins) X of the membrane. Individual molecules of any receptor are not bound to X, but after they are specifically clustered into patches, a patch of receptors then becomes bound to S and hence to actin/myosin. Patching and capping of specific receptors on rounded cells is often accompanied by a specific endocytosis of the ligand-receptor complexes. This represents one common transport mechanism of a protein (the ligand) across the plasma membrane. The possibility is discussed that this type of endocytosis is mediated by a transmembrane linkage of the clustered receptor to actin/myosin. Another mechanism of endocytosis involves the “coated pit” structures that are observed by electron microscopy of plasma membranes. The possible relationships between an actin/myosin and a coated pit mechanism of endocytosis are explored.     

Clustering, mobility, and triggering activity of small oligomers of immunoglobulin E on rat basophilic leukemia cells

We have recently shown that small oligomers of IgE bound to univalent receptors for IgE on the surface of rat basophilic leukemia cells induce extensive aggregation of the receptors at 4 degrees C into patches resolvable by fluorescence microscopy and that this does not occur with monomeric IgE (Menon, A. K., D. Holowka, and B. Baird, 1984, J. Cell Biol. 98:577-583). Here we use fluorescence photobleaching recovery measurements to show that receptor oligomerization by this means is accompanied by a dramatic reduction of receptor lateral mobility, and that this immobilization occurs even when the clustering is not microscopically detectable. Furthermore, the degree of immobility induced by a particular oligomer fraction from a gel filtration column correlates positively with its ability to trigger cellular degranulation, whereas receptors labeled with monomeric IgE have no triggering activity and exhibit typical membrane protein mobility. The slow, large-scale oligomer-induced clustering appears to be a long term consequence of earlier selective interactions that result in receptor immobilization, and this highly clustered state provides a competent, noninhibitory triggering signal resulting in cellular degranulation upon warming to 37 degrees C. We conclude that even limited clustering of IgE receptors on rat basophilic leukemia cells induces interactions with other cellular components that constrain receptor mobility and eventually cause massive coalescence of the clusters. These primary selective interactions occurring at the level of receptor oligomers or small clusters of oligomers that result in immobilization may play a role in triggering cellular degranulation.     

The effect of agrin and laminin on acetylcholine receptor dynamics in vitro

Using optical imaging assays, we investigated the dynamics of acetylcholine receptors (AChRs) at laminin-associated clusters on cultured myotubes in the absence or presence of the nerve-derived clustering factor, agrin. Using fluorescence recovery after photobleaching (FRAP) on fluorescent bungarotoxin-labeled receptors, we found that approximately 9% of original fluorescence was recovered after 8 h as surface AChRs were recruited into clusters. By quantifying the loss of labeled receptors and the recovery of fluorescence after photobleaching, we estimated that the half-life of clustered receptors was approximately 4.5 h. Despite the rapid removal of receptors, the accumulation of new receptors at clusters was robust enough to maintain receptor density over time. We also found that the AChR half-life was not affected by agrin despite its role in inducing the aggregation of AChRs. Interestingly, when agrin was added to myotubes grown on laminin-coated substrates, most new receptors were not directed into preexisting laminin-induced clusters but instead formed numerous small aggregates on the entire muscle surface. Time-lapse imaging revealed that the agrin-induced clusters could be seen as early as 1 h, and agrin treatment resulted in the complete dissipation of laminin-associated clusters by 24 h. These results reveal that while laminin and agrin are involved in the clustering of receptors they are not critical to the regulation of receptor metabolic stability at these clusters, and further argue that agrin is able to rapidly and fully negate the laminin substrate clustering effect while inducing the rapid formation of new clusters.     

Power calculation for cross-sectional stepped wedge cluster randomized trials with variable cluster sizes

Standard sample size calculation formulas for stepped wedge cluster randomized trials (SW-CRTs) assume that cluster sizes are equal. When cluster sizes vary substantially, ignoring this variation may lead to an under-powered study. We investigate the relative efficiency of a SW-CRT with varying cluster sizes to equal cluster sizes, and derive variance estimators for the intervention effect that account for this variation under a mixed effects model—a commonly used approach for analyzing data from cluster randomized trials. When cluster sizes vary, the power of a SW-CRT depends on the order in which clusters receive the intervention, which is determined through randomization. We first derive a variance formula that corresponds to any particular realization of the randomized sequence and propose efficient algorithms to identify upper and lower bounds of the power. We then obtain an “expected” power based on a first-order approximation to the variance formula, where the expectation is taken with respect to all possible randomization sequences. Finally, we provide a variance formula for more general settings where only the cluster size arithmetic mean and coefficient of variation, instead of exact cluster sizes, are known in the design stage. We evaluate our methods through simulations and illustrate that the average power of a SW-CRT decreases as the variation in cluster sizes increases, and the impact is largest when the number of clusters is small.     

Coupling of agonist-induced AMPA receptor internalization with receptor recycling

Excitatory post-synaptic currents in the CNS are primarily mediated by alpha-amino-3-hydroxy-5-methyl-isoxazole-4-propionic acid (AMPA) receptors in response to glutamate. Internalization of cell-surface receptors has been shown to be one mechanism by which to control receptor function. To test for agonist control of AMPA receptor plasma membrane expression we used biochemical assays to study AMPA receptor internalization and insertion processes. In heterologous cells, we observed a slow constitutive internalization and a rapid agonist-induced internalization of AMPA receptors. To our surprise, however, agonist treatment had no effect on the steady-state levels of AMPA receptors on the cell surface. To examine whether this could be explained by an agonist-induced increase in the insertion rate of AMPA receptors into the plasma membrane we developed an assay to independently measure receptor insertion. Remarkably, agonist treatment of cells also dramatically increased AMPA receptor plasma membrane insertion rates. In addition, using an assay to measure recycling of internalized pools we found that internalized receptors are rapidly recycled to the cell surface. These results suggest that agonist-induced receptor internalization is coupled to increases in receptor recycling. This increase in receptor flux through intracellular pools may allow for rapid changes in receptor surface expression by independent regulatory control of internalization and insertion.     

Reconstitution of the receptor for immunoglobulin E into liposomes. Reincorporation of purified receptors

Mast cells and related cells have on their surface receptors that bind immunoglobulin E (IgE) with high affinity and which, when aggregated, trigger exocytosis. We recently demonstrated that when these receptors are solubilized with mild detergents, their subunits dissociate unless an appropriate lipid:detergent ratio is maintained. The conditions required to maintain the receptors' integrity appeared to parallel those previously determined as necessary to obtain adequate incorporation of unpurified IgE-receptor complexes from detergent extracts into liposomes. We now show that purified IgE-receptor complexes having the full complement of subunits become preferentially inserted into liposomes. If the receptor subunits are chemically cross-linked to each other, at least some of such receptors can be incorporated, even though lipid is omitted during their purification. The findings suggest that the IgE-binding alpha subunit of the receptor is anchored to the bilayer by means of one or both of the other subunits.     

Modeling synaptic dynamics driven by receptor lateral diffusion

The synaptic weight between a pre- and a postsynaptic neuron depends in part on the number of postsynaptic receptors. On the surface of neurons, receptors traffic by random motion in and out from a microstructure called the postsynaptic density (PSD). In the PSD, receptors can be stabilized at the membrane when they bind to scaffolding proteins. We propose a mathematical model to compute the postsynaptic counterpart of the synaptic weight based on receptor trafficking. We take into account the receptor fluxes at the PSD, which can be regulated by neuronal activity, and the interactions of receptors with the scaffolding molecules. Using a Markovian approach, we estimate the mean and the fluctuations of the number of bound receptors. When the number of receptors is large, a deterministic system is also derived. Moreover, these equations can be used, for example, to fit fluorescence-recovery-after-photobleaching experiments to determine, in living neurons, the chemical binding constants for the receptors/scaffolding molecules interaction at synapses.     

Diffusion-limited forward rate constants in two dimensions. Application to the trapping of cell surface receptors by coated pits.

A variety of receptors are known to aggregate in specialized cell surface structures called coated pits, prior to being internalized when the coated pits close off. At 37 degrees C on human fibroblasts, as well as on other cell types, a recycling process maintains a constant number of coated pits on the cell surface. In this paper, we explore implications for receptor aggregation and internalization of the two types of recycling models that have been proposed for the maintenance of the coated pit concentration. In one model, coated pits alternate between accessible and inaccessible states at fixed locations on the cell surface, while in the other model, coated pits recycle to random locations on the cell surface. We consider receptors that are randomly inserted in the membrane, move by pure diffusion with diffusion coefficient D, and are instantly and irreversibly trapped when they reach a coated pit boundary (the diffusion limit). For such receptors, we calculate for each of the two models: the mean time tau to reach a coated pit, the forward rate constant k+ for the interaction of a receptor with a coated pit, and the fraction phi of receptors aggregated in coated pits. We show that for the parameters that characterize coated pits on human fibroblasts, the way in which coated pits return to the surface has a negligible effect on the values of tau, k+, and phi for mobile receptors, D greater than or equal to 1.0 X 10(-11) cm2/s, but has a substantial effect for "immobile" receptors, D much less than 1 X 10(-11) cm2/s. We present numerical examples to show that it may be possible to distinguish between these models if one can monitor slowly diffusing receptors (D less than 1 X 10(-11) cm2/s) on cells whose coated pits have relatively short lifetimes (less than or equal to 1 min). Finally, we show that for the low-density lipoprotein (LDL) receptor on human fibroblasts (D = 4.5 X 10(-11) cm2/s), the predicted and observed values of K+ and phi are in close agreement. Therefore, even for slowly diffusing LDL receptor, unaided diffusion as the transport mechanism of receptors to coated pits is consistent with measured rates of LDL internalization.     

Isolation and analysis of Pseudomonas aeruginosa PAO mutants resistant to nonlysogenic bacteriophages

Nonlysogenizing Pseudomonas aeruginosa PAO bacteriophages were studied. According to morphology of the plaques, they were distributed into three groups: phi k, phi m and phi mn. The mutants of P. aeruginosa PAO resistant to these bacteriophages were selected. On the basis of cris-cross resistance analysis of the mutants, a formal scheme of the receptor sites on the P. aeruginosa PAO bacterial cell surface is drawn. It is shown that bacteriophages phi k and phi m use different receptors for their adsorption. The receptors of phi m and phi mn phages are specifically interconnected. Thus, the receptor for phi k phages is connected with the receptor for phage phi 11. It appears that the receptor for bacteriophage E79 is identical to those of phi m phages. The phi m receptor is of a composite structure: it includes two different receptors used by phi mn phages.     

Effects of receptor clustering on ligand dissociation kinetics: theory and simulations

Receptor-ligand binding is a critical first step in signal transduction and the duration of the interaction can impact signal generation. In mammalian cells, clustering of receptors may be facilitated by heterogeneous zones of lipids, known as lipid rafts. In vitro experiments show that disruption of rafts significantly alters the dissociation of fibroblast growth factor-2 (FGF-2) from heparan sulfate proteoglycans (HSPGs), co-receptors for FGF-2. In this article, we develop a continuum stochastic formalism to address how receptor clustering might influence ligand rebinding. We find that clusters reduce the effective dissociation rate dramatically when the clusters are dense and the overall surface density of receptors is low. The effect is much less pronounced in the case of high receptor density and shows nonmonotonic behavior with time. These predictions are verified via lattice Monte Carlo simulations. Comparison with FGF-2-HSPG experimental results is made and suggests that the theory could be used to analyze similar biological systems. We further present an analysis of an additional cooperative internal-diffusion model that might be used by other systems to increase ligand retention when simple rebinding is insufficient.     

Quantum dot fluorescence characterizes the nanoscale organization of T cell receptors for antigen

Changes in the clustering of surface receptors modulate cell responses to ligands. Hence, global measures of receptor clustering can be useful for characterizing cell states. Using T cell receptor for antigen as an example, we show that k-space image correlation spectroscopy of quantum dots blinking detects T cell receptor clusters on a scale of tens of nanometers and reports changes in clustering after T cell activation. Our results offer a general approach to the global analysis of lateral organization and receptor clustering in single cells, and can thus be applied when the cell type of interest is rare.     

Clustering and immobilization of acetylcholine receptors by the 43-kD protein: a possible role for dystrophin-related protein

Recombinant acetylcholine receptors (AChRs) expressed on the surface of cultured fibroblasts become organized into discrete membrane domains when the 43-kD postsynaptic protein (43k) is co-expressed in the same cells (Froehner, S.C., C. W. Luetje, P. B. Scotland, and J. Patrick, 1990. Neuron. 5:403-410; Phillips, W. D., M. C. Kopta, P. Blount, P. D. Gardner, J. H. Steinbach, and J. P. Merlie. 1991. Science (Wash. DC). 251:568-570). Here we show that AChRs present on the fibroblast cell surface prior to transfection of 43k are recruited into 43k-rich membrane domains. Aggregated AChRs show increased resistance to extraction with Triton X-100, suggesting a 43k-dependent linkage to the cytoskeleton. Myotubes of the mouse cell line C2 spontaneously display occasional AChR/43k-rich membrane domains that ranged in diameter up to 15 microns, but expressed many more when 43k was overexpressed following transfection of 43k cDNA. However, the membrane domains induced by recombinant 43k were predominantly small (< or = 2 microns). We were then interested in whether the cytoskeletal component, dystrophin related protein (DRP; Tinsley, J. M., D. J. Blake, A. Roche, U. Fairbrother, J. Riss, B. C. Byth, A. E. Knight, J. Kendrick-Jones, G. K. Suthers, D. R. Love, Y. H. Edwards, and K. E. Davis, 1992. Nature (Lond.). 360:591-593) contributed to the development of AChR clusters. Immunofluorescent anti-DRP staining was present at the earliest stages of AChR clustering at the neuromuscular synapse in mouse embryos and was also concentrated at the large AChR-rich domains on nontransfected C2 myotubes. Surprisingly, anti-DRP staining was concentrated mainly at the large, but not the small AChR clusters on C2 myotubes suggesting that DRP may be principally involved in permitting the growth of AChR clusters.     

Image correlation spectroscopy. II. Optimization for ultrasensitive detection of preexisting platelet-derived growth factor-beta receptor oligomers on intact cells.

Previously we introduced image correlation spectroscopy (ICS) as an imaging analog of fluorescence correlation spectroscopy (FCS). Implementation of ICS with image collection via a standard fluorescence confocal microscope and computer-based autocorrelation analysis was shown to facilitate measurements of absolute number densities and determination of changes in aggregation state for fluorescently labeled macromolecules. In the present work we illustrate how to use ICS to quantify the aggregation state of immunolabeled plasma membrane receptors in an intact cellular milieu, taking into account background fluorescence. We introduce methods that enable us to completely remove white noise contributions from autocorrelation measurements for individual images and illustrate how to perform background corrections for autofluorescence and nonspecific fluorescence on cell population means obtained via ICS. The utilization of photon counting confocal imaging with ICS analysis in combination with the background correction techniques outlined enabled us to achieve very low detection limits with standard immunolabeling methods on normal, nontransformed human fibroblasts (AG1523) expressing relatively low numbers of platelet-derived growth factor-beta (PDGF-beta) receptors. Specifically, we determined that the PDGF-beta receptors were preaggregated as tetramers on average with a mean surface density of 2.3 clusters micrometer(-2) after immunolabeling at 4 degreesC. These measurements, which show preclustering of PDGF-beta receptors on the surface of normal human fibroblasts, contradict a fundamental assumption of the ligand-induced dimerization model for signal transduction and provide support for an alternative model that posits signal transduction from within preexisting receptor aggregates.     

Revising Berg-Purcell for finite receptor kinetics

From nutrient uptake to chemoreception to synaptic transmission, many systems in cell biology depend on molecules diffusing and binding to membrane receptors. Mathematical analysis of such systems often neglects the fact that receptors process molecules at finite kinetic rates. A key example is the celebrated formula of Berg and Purcell for the rate that cell surface receptors capture extracellular molecules. Indeed, this influential result is only valid if receptors transport molecules through the cell wall at a rate much faster than molecules arrive at receptors. From a mathematical perspective, ignoring receptor kinetics is convenient because it makes the diffusing molecules independent. In contrast, including receptor kinetics introduces correlations between the diffusing molecules because, for example, bound receptors may be temporarily blocked from binding additional molecules. In this work, we present a modeling framework for coupling bulk diffusion to surface receptors with finite kinetic rates. The framework uses boundary homogenization to couple the diffusion equation to nonlinear ordinary differential equations on the boundary. We use this framework to derive an explicit formula for the cellular uptake rate and show that the analysis of Berg and Purcell significantly overestimates uptake in some typical biophysical scenarios. We confirm our analysis by numerical simulations of a many-particle stochastic system.     

Cross-talk and co-trafficking between rho1/GABA receptors and ATP-gated channels

Gamma-aminobutyric-acid (GABA) and ATP ionotropic receptors represent two structurally and functionally different classes of neurotransmitter-gated channels involved in fast synaptic transmission. We demonstrate here that, when the inhibitory rho1/GABA and the excitatory P2X2 receptor channels are co-expressed in Xenopus oocytes, activation of one channel reduces the currents mediated by the other one. This reciprocal inhibitory cross-talk is a receptor-mediated phenomenon independent of agonist cross-modulation, membrane potential, direction of ionic flux, or channel densities. Functional interaction is disrupted when the cytoplasmic C-terminal domain of P2X2 is deleted or in competition experiments with minigenes coding for the C-terminal domain of P2X2 or the main intracellular loop of rho1 subunits. We also show a physical interaction between P2X2 and rho1 receptors expressed in oocytes and the co-clustering of these receptors in transfected hippocampal neurons. Co-expression with P2X2 induces retargeting and recruitment of mainly intracellular rho1/GABA receptors to surface clusters. Therefore, molecular and functional cross-talk between inhibitory and excitatory ligand-gated channels may regulate synaptic strength both by activity-dependent current occlusion and synaptic receptors co-trafficking.     

The effect of diffusion on the binding of membrane-bound receptors to coated pits.

We have formulated a kinetic model for the primary steps that occur at the cell membrane during receptor-mediated endocytosis. This model includes the diffusion of receptor molecules, the binding of receptors to coated pits, the loss of coated pits by invagination, and random reinsertion of receptors and coated pits. Using the mechanistic statistical theory of nonequilibrium thermodynamics, we employ this mechanism to calculate the two-dimensional radial distribution of receptors around coated pits at steady state. From this we obtain an equation that describes the effect of receptor diffusion on the rate of binding to coated pits. Our equation does not assume that ligand binding is instantaneous and can be used to assess the effect of diffusion on the binding rate. Using experimental data for low density lipoprotein receptors on fibroblast cells, we conclude that the effect of diffusion on the binding of these receptors to coated pits is no more than 84% diffusion controlled. This corresponds to a dissociation rate constant for receptors on coated pits (k-) that is much less than the rate constant for invagination of the pits (lambda = 3.3 X 10(-3)/s) and a correlation length for the radial distribution function of six times the radius of a coated pit. Although the existing experimental data are compatible with any value of k-, we obtain a lower bound for the value of the binding constant (k+) of 2.3 X 10(-2)(micron)2/s. Comparison of the predicted radial distributions with experiment should provide a clear indication of the effect of diffusion on k+.     

Kinetics of cell detachment: peeling of discrete receptor clusters.

Clustering of cell surface adhesion receptors is an essential step in the development of focal contacts, specialized cell-substrate attachment sites where receptors are simultaneously linked to extracellular ligand and cytoskeletal proteins. Previously, we examined the effect of receptor clustering on attachment strength. Here, we employ the numerical methodology developed by Dembo and colleagues (Dembo, M., D.C. Torney, K. Saxman, and D. Hammer. 1988. Proc. R. Soc. Lond. B. 234:55-83) to investigate the kinetics of cell detachment when receptors are clustered into discrete patches. We show that the membrane peeling velocity decreases if receptors are clustered within a patch located inside the contact region. Peeling of clusters is influenced by the chemistry and mechanics of receptor-ligand bonds within the patch. Detachment is also prohibited if the applied tension equals the critical tension of the patch, unless the patch length is small compared with the boundary length over which membrane bending occurs, in which case the patch will peel. Peeling of these short patches only occurs when the mechanical stiffness of clustered bonds is within an optimal range. We compare our model predictions with experimental measurements of T lymphocyte detachment from ICAM-1 substrates. We demonstrate that if discrete patches of receptors are present, detachment occurs through intervals of slow and fast peeling, similar to the dynamics of T lymphocyte peeling, indicating that clustering of LFA-1 receptors is one possible explanation for the observed detachment kinetics in this system.     

Relationship of IgE receptor topography to secretion in RBL-2H3 mast cells.

In RBL-2H3 rat leukemic mast cells, cross-linking IgE-receptor complexes with anti-IgE antibody leads to degranulation. Receptor cross-linking also stimulates the redistribution of receptors on the cell surface, a process observed here by labeling the anti-IgE with 15 nm protein A-gold particles that are visible by back-scattered electron imaging in the scanning electron microscope. We report that anti-IgE binding stimulates the redistribution of IgE-receptor complexes at 37 degrees C from a dispersed topography to distributions dominated sequentially by short chains, small clusters, and large aggregates of cross-linked receptors. Cells incubated with 1 microgram/ml anti-IgE, a concentration that stimulates maximum net secretion, redistribute receptors into chains and small clusters during a 15 min incubation period. At 3 and 10 micrograms/ml anti-IgE, net secretion is reduced and the majority of receptors redistribute rapidly into clusters and large aggregates. The addition of Fab fragments with the high anti-IgE concentrations, to reduce cross-linking, delays receptor aggregation and enhances secretion. The progression of receptors from small clusters to large aggregates is prevented in cells treated with dihydrocytochalasin B to prevent F-actin assembly. These results establish that characteristic patterns of receptor topography are correlated with receptor activity. In particular, they link the formation of large receptor aggregates to reduced signalling activity. Cytoskeleton-membrane interaction is implicated in the formation or stabilization of the large receptor clusters.