Mathematical modelling and computational study of two-dimensional and three-dimensional dynamics of receptor–ligand interactions in signalling response mechanisms

Cell signalling processes involve receptor trafficking through highly connected networks of interacting components. The binding of surface receptors to their specific ligands is a key factor for the control and triggering of signalling pathways. But the binding process still presents many enigmas and, by analogy with surface catalytic reactions, two different mechanisms can be conceived: the first mechanism is related to the Eley–Rideal (ER) mechanism, i.e. the bulk-dissolved ligand interacts directly by pure three-dimensional (3D) diffusion with the specific surface receptor; the second mechanism is similar to the Langmuir–Hinshelwood (LH) process, i.e. 3D diffusion of the ligand to the cell surface followed by reversible ligand adsorption and subsequent two-dimensional (2D) surface diffusion to the receptor. A situation where both mechanisms simultaneously contribute to the signalling process could also occur. The aim of this paper is to perform a computational study of the behavior of the signalling response when these different mechanisms for ligand-receptor interactions are integrated into a model for signal transduction and ligand transport. To this end, partial differential equations have been used to develop spatio-temporal models that show trafficking dynamics of ligands, cell surface components, and intracellular signalling molecules through the different domains of the system. The mathematical modeling developed for these mechanisms has been applied to the study of two situations frequently found in cell systems: (a) dependence of the signal response on cell density; and (b) enhancement of the signalling response in a synaptic environment.     

Trafficking of the ErbB receptors and its influence on signaling

Although members of the ErbB receptor family are found predominantly at the cell surface, these receptors undergo constant cycling between the plasma membrane and the endosomal compartment. In the absence of an activating ligand, these receptors are slowly internalized (t(1/2) approximately 30 min) but are quickly recycled. The constitutive degradation rate of the epidermal growth factor (EGF) receptor (EGFR) is slower than other ErbB family members and only the EGFR appears to alter its trafficking pattern in response to ligand binding. This altered pattern is characterized by accelerated internalization and enhanced lysosomal targeting. Ligand-regulated trafficking of the EGFR is mediated by a series of motifs distributed through the cytoplasmic domain of the receptor that are exposed by a combination of activation-mediated conformation changes and the binding of proteins such as Grb2. As a consequence of induced internalization, most EGFR signaling occurs within endosomes whereas signaling by the other members of the ErbB family appear to be generated predominantly from the cell surface. Overexpression of ErbB family members can disrupt normal receptor trafficking by driving heterodimerization of receptors with disparate trafficking patterns. Because different ErbB receptor substrates are localized in different cellular compartments, disrupted trafficking could be an important factor in the altered signaling patterns observed as a consequence of receptor overexpression.     

Cell surface receptors for signal transduction and ligand transport: a design principles study

Receptors constitute the interface of cells to their external environment. These molecules bind specific ligands involved in multiple processes, such as signal transduction and nutrient transport. Although a variety of cell surface receptors undergo endocytosis, the systems-level design principles that govern the evolution of receptor trafficking dynamics are far from fully understood. We have constructed a generalized mathematical model of receptor–ligand binding and internalization to understand how receptor internalization dynamics encodes receptor function and regulation. A given signaling or transport receptor system represents a particular implementation of this module with a specific set of kinetic parameters. Parametric analysis of the response of receptor systems to ligand inputs reveals that receptor systems can be characterized as being: i) avidity-controlled where the response control depends primarily on the extracellular ligand capture efficiency, ii) consumption-controlled where the ability to internalize surface-bound ligand is the primary control parameter, and iii) dual-sensitivity where both the avidity and consumption parameters are important. We show that the transferrin and low-density lipoprotein receptors are avidity-controlled, the vitellogenin receptor is consumption-controlled, and the epidermal growth factor receptor is a dual-sensitivity receptor. Significantly, we show that ligand-induced endocytosis is a mechanism to enhance the accuracy of signaling receptors rather than merely serving to attenuate signaling. Our analysis reveals that the location of a receptor system in the avidity-consumption parameter space can be used to understand both its function and its regulation.     

Cellular signaling is potentially regulated by cell density in receptor trafficking networks

Previous work has shown that receptor trafficking is a potential site for the control of signaling pathways. In most biological experiments, the ligand concentration and cell density vary within a wide range among different systems. However, there is less attention to systematically analyze how much cellular signal response is affected by cell densities. Here, we use a quantitative mathematical model to investigate signal responses in different receptor trafficking networks by simultaneous variations of ligand concentration and cell density. Computational analysis of the model revealed that receptor trafficking networks have potential sigmoid responses to ratio between ligand and surface receptor number per cell, which is a key factor to control the signaling responses in receptor trafficking networks. Furthermore, cell density also affects the robustness of dose-response curve upon the variation of binding affinity.     

Membrane sorting of toll-like receptor (TLR)-2/6 and TLR2/1 heterodimers at the cell surface determines heterotypic associations with CD36 and intracellular targeting

Toll-like receptors (TLRs) are receptors of the innate immune system responsible for recognizing pathogen-associated molecular patterns. TLR2 seems to be the most promiscuous TLR receptor able to recognize the most diverse set of pathogen-associated patterns. Its promiscuity has been attributed to its unique ability to heterodimerize with TLRs 1 and 6 and, most recently, to its association with CD36 in response to diacylated lipoproteins. Thus, it seems that TLR2 forms receptor clusters in response to different microbial ligands. In this study we investigated TLR2 cell surface heterotypic interactions in response to different ligands as well as internalization and intracellular trafficking. Our data show that TLR2 forms heterodimers with TLR1 and TLR6 and that these heterodimer pre-exist and are not induced by the ligand. Upon stimulation by the specific ligand, these heterodimers are recruited within lipid rafts. In contrast, heterotypic associations of TLR2/6 with CD36 are not preformed and are ligand-induced. All TLR2 receptor clusters accumulate in lipid rafts and are targeted to the Golgi apparatus. This localization and targeting is ligand-specific. Activation occurs at the cell surface, and the observed trafficking is independent of signaling.     

Interactions of the low density lipoprotein receptor gene family with cytosolic adaptor and scaffold proteins suggest diverse biological functions in cellular communication and signal transduction

The members of the low density lipoprotein (LDL) receptor gene family bind a broad spectrum of extracellular ligands. Traditionally, they had been regarded as mere cargo receptors that promote the endocytosis and lysosomal delivery of these ligands. However, recent genetic experiments in mice have revealed critical functions for two LDL receptor family members, the very low density lipoprotein receptor and the apoE receptor-2, in the transmission of extracellular signals and the activation of intracellular tyrosine kinases. This process regulates neuronal migration and is crucial for brain development. Signaling through these receptors requires the interaction of their cytoplasmic tails with the intracellular adaptor protein Disabled-1 (DAB1). Here, we identify an extended set of cytoplasmic proteins that might also participate in signal transmission by the LDL receptor gene family. Most of these novel proteins are adaptor or scaffold proteins that contain PID or PDZ domains and function in the regulation of mitogen-activated protein kinases, cell adhesion, vesicle trafficking, or neurotransmission. We show that binding of DAB1 interferes with receptor internalization suggesting a mechanism by which signaling through this class of receptors might be regulated. Taken together, these findings imply much broader physiological functions for the LDL receptor family than had previously been appreciated. They form the basis for the elucidation of the molecular pathways by which cells respond to the diversity of ligands that bind to these multifunctional receptors on the cell surface.     

Slit-Dependent Endocytic Trafficking of the Robo Receptor Is Required for Son of Sevenless Recruitment and Midline Axon Repulsion

Understanding how axon guidance receptors are activated by their extracellular ligands to regulate growth cone motility is critical to learning how proper wiring is established during development. Roundabout (Robo) is one such guidance receptor that mediates repulsion from its ligand Slit in both invertebrates and vertebrates. Here we show that endocytic trafficking of the Robo receptor in response to Slit-binding is necessary for its repulsive signaling output. Dose-dependent genetic interactions and in vitro Robo activation assays support a role for Clathrin-dependent endocytosis, and entry into both the early and late endosomes as positive regulators of Slit-Robo signaling. We identify two conserved motifs in Robo’s cytoplasmic domain that are required for its Clathrin-dependent endocytosis and activation in vitro; gain of function and genetic rescue experiments provide strong evidence that these trafficking events are required for Robo repulsive guidance activity in vivo. Our data support a model in which Robo’s ligand-dependent internalization from the cell surface to the late endosome is essential for receptor activation and proper repulsive guidance at the midline by allowing recruitment of the downstream effector Son of Sevenless in a spatially constrained endocytic trafficking compartment.     

Transduction of basolateral-to-apical signals across epithelial cells: ligand-stimulated transcytosis of the polymeric immunoglobulin receptor requires two signals

Transcytosis of the polymeric immunoglobulin receptor (pIgR) is stimulated by binding of its ligand, dimeric IgA (dIgA). During this process, dIgA binding at the basolateral surface of the epithelial cell transmits a signal to the apical region of the cell, which in turn stimulates the transport of dIgA-pIgR complex from a postmicrotubule compartment to the apical surface. We have previously reported that the signal of stimulation was controlled by a protein-tyrosine kinase (PTK) activated upon dIgA binding. We now show that this signal of stimulation moves across the cell independently of pIgR movement or microtubules and acts through the tyrosine kinase activity by releasing Ca++ from inositol trisphosphate-sensitive intracellular stores. Surprisingly we have found that a second independent signal is required to achieve dIgA-stimulated transcytosis of pIgR. This second signal depends on dIgA binding to the pIgR solely at the basolateral surface and the ability of pIgR to dimerize. This enables pIgR molecules that have bound dIgA at the basolateral surface to respond to the signal of stimulation once they reach the postmicrotubule compartment. We propose that the use of two signals may be a general mechanism by which signaling receptors maintain specificity along their signaling and trafficking pathways.     

Autocrine ligand binding to cell receptors. Mathematical analysis of competition by solution "decoys".

Autocrine ligands have been demonstrated to regulate cell proliferation, cell adhesion, and cell migration in a number of different systems and are believed to be one of the underlying causes of malignant cell transformation. Binding of these ligands to their cellular receptors can be compromised by diffusive transport of ligand away from the secreting cell. Exogenous addition of antibodies or solution receptors capable of competing with cellular receptors for these autocrine ligands has been proposed as a means of inhibiting autocrine-stimulated cell behavioral responses. Such "decoys" complicate cellular binding by offering alternative binding targets, which may also be capable of aiding or abating transport of the ligand away from the cell surface. We present a mathematical model incorporating autocrine ligand production and the presence of competing cellular and solution receptors. We elucidate effects of key system parameters including ligand diffusion rate, binding rate constants, cell density, and secretion rate on the ability of solution receptors to inhibit cellular receptor binding. Both plated and suspension cell systems are considered. An approximate analytical expression relating the key parameters to the critical concentration of solution "decoys" required for inhibition is derived and compared to the numerical calculations. We find that in order to achieve essentially complete inhibition of surface receptor binding, the concentration of decoys may need to be as much as four to eight orders of magnitude greater than the equilibrium disociation constant for ligand binding to surface receptors.     

Antidepressant-induced switch of beta 1-adrenoceptor trafficking as a mechanism for drug action

Reduction in surface beta(1)-adrenoceptor (beta1AR) density is thought to play a critical role in mediating the therapeutic long term effects of antidepressants. Since antidepressants are neither agonists nor antagonists for G protein-coupled receptors, receptor density must be regulated through processes independent of direct receptor activation. Endocytosis and recycling of the beta1AR fused to green fluorescent protein at its carboxyl-terminus (beta1AR-GFP) were analyzed by confocal fluorescence microscopy of live cells and complementary ligand binding studies. In stably transfected C6 glioblastoma cells, beta1AR-GFP displayed identical ligand-binding isotherms and adenylyl cyclase activation as native beta1AR. Upon exposure to isoproterenol, a fraction of beta1AR-GFP (10-15%) internalized rapidly and colocalized with endocytosed transferrin receptors in an early endosomal compartment in the perinuclear region. Chronic treatment with the tricyclic antidepressant desipramine (DMI) did not affect internalization characteristics of beta1AR-GFP when challenged with isoproterenol. However, internalized receptors were not able to recycle back to the cell surface in DMI-treated cells, whereas recycling of transferrin receptors was not affected. Endocytosed receptors were absent from structures that stained with fluorescently labeled dextran, and inhibition of lysosomal protease activity did not restore receptor recycling, indicating that beta1AR-GFP did not immediately enter the lysosomal compartment. The data suggest a new mode of drug action causing a "switch" of receptor fate from a fast recycling pathway to a slowly exchanging perinuclear compartment. Antidepressant-induced reduction of receptor surface expression may thus be caused by modulation of receptor trafficking routes.     

A response calculus for immobilized T cell receptor ligands

To address the molecular mechanism of T cell receptor (TCR) signaling, we have formulated a model for T cell activation, termed the 2D-affinity model, in which the density of TCR on the T cell surface, the density of ligand on the presenting surface, and their corresponding two-dimensional affinity determine the level of T cell activation. When fitted to T cell responses against purified ligands immobilized on plastic surfaces, the 2D-affinity model adequately simulated changes in cellular activation as a result of varying ligand affinity and ligand density. These observations further demonstrated the importance of receptor cross-linking density in determining TCR signaling. Moreover, it was found that the functional two-dimensional affinity of TCR ligands was affected by the chemical composition of the ligand-presenting surface. This makes it possible that cell-bound TCR ligands, despite their low affinity in solution, are of optimal two-dimensional affinity thereby allowing effective TCR binding under physiological conditions, i.e. at low ligand densities in cellular interfaces.     

Regulation of serotonin receptor function in the nervous system by lipid rafts and adaptor proteins

5-HT is a phylogenetically conserved monoaminergic neurotransmitter which is crucial for a number of physiological processes and is dysregulated in several disease states including depression, anxiety and schizophrenia. 5-HT neurons in the central nervous system are localized in the raphe nuclei and project to a wide range of target areas. 5-HT exerts its functions through 14 subtypes of 5-HT receptors. The tertiary structures of seven transmembrane 5-HT receptors contain several important features, including cholesterol consensus motifs, prominent intracellular loops and free C-termini. Alterations of cholesterol levels affect binding of ligands to 5-HT receptors and cholesterol-enriched microdomains in the cell membrane, termed lipid rafts, regulate 5-HT receptor internalization and signaling. The intracellular loops and the C-termini of 5-HT receptors provide binding sites for interacting adaptor proteins. Adaptor proteins affect internalization, desensitization as well as G-protein dependent and independent signaling via 5-HT receptors. We will here briefly review recent progress on the role of lipid rafts and adaptor proteins in the regulation of localization, trafficking, signaling and ligand bias of 5-HT receptors.     

Cohort movement of different ligands and receptors in the intracellular endocytic pathway of alveolar macrophages

The rate of movement of different receptors and ligands through the intracellular endocytic apparatus was studied in alveolar macrophages. Cells were exposed to iodinated alpha-macroglobulin-protease complexes, mannose terminal glycoproteins, diferric transferrin, and maleylated proteins. By use of the diaminobenzidine density shift procedure, we demonstrated that these ligands were internalized into the same endocytic vesicle. We then compared the rates of transfer to the lysosome or recycling to the cell surface of different ligands/receptors contained in the same endosome. We found that although the rate constant for degradation was ligand specific, the lag time prior to the initiation of degradation was the same for all three ligands. We also found that molecules taken up nonspecifically by fluid-phase pinocytosis had the same lag time prior to degradation as ligands internalized via receptor-mediated endocytosis. These data suggest that different molecules within the same endocytic compartment are transferred to the lysosome (or degradative compartment) at the same rate. We measured the rate of return of receptors to the cell surface by either inactivating surface receptors by protease treatment at 0 degrees C, or by incubating cells with saturating amounts of nonradioactive ligand at 37 degrees C. We then measured the rate of appearance of "new" receptors on the cell surface. Using these approaches, we found that three different receptors were transferred from internal pools to the cell surface at the same rate. The rate of transfer was independent of whether receptors were initially occupied or unoccupied. Our observations indicate that receptor/ligands, once inside alveolar macrophages, are transported by vesicles which transfer their contents as a cohort from one compartment to another. The rate of movement of these receptors is determined by the movement of vesicles and is independent of their content.     

Mixed extracellular matrix ligands synergistically modulate integrin adhesion and signaling

Cell adhesion to extracellular matrix (ECM) components through cell-surface integrin receptors is essential to the formation, maintenance and repair of numerous tissues, and therefore represents a central theme in the design of bioactive materials that successfully interface with the body. While the adhesive responses associated with a single ligand have been extensively analyzed, the effects of multiple integrin subtypes binding to multivalent ECM signals remain poorly understood. In the present study, we generated a high throughput platform of non-adhesive surfaces presenting well-defined, independent densities of two integrin-specific engineered ligands for the type I collagen (COL-I) receptor alpha(2)beta(1) and the fibronectin (FN) receptor alpha(5)beta(1) to evaluate the effects of integrin cross-talk on adhesive responses. Engineered surfaces displayed ligand density-dependent adhesive effects, and mixed ligand surfaces significantly enhanced cell adhesion strength and focal adhesion assembly compared to single FN and COL-I ligand surfaces. Moreover, surfaces presenting mixed COL-I/FN ligands synergistically enhanced FAK activation compared to the single ligand substrates. The enhanced adhesive activities of the mixed ligand surfaces also promoted elevated proliferation rates. Our results demonstrate interplay between multivalent ECM ligands in adhesive responses and downstream cellular signaling.     

Role of Cre-loxP-mediated conditional gene targeting in understanding the function of Notch receptors

The Notch family of cell surface receptors and their ligands constitute an evolutionarily conserved signaling pathway that is used by invertebrates and vertebrates to regulate a broad spectrum of cell specification events through local cell interactions. After ligand binding Notch receptor undergoes proteolytic processing ultimately liberating the cytoplasmic domain of the Notch receptor which translocates to the nucleus and activates target genes. In all animal models tested, mutations in Notch genes invariably resulted in developmental abnormalities. In mammals, Notch signaling controls key stages of lymphocyte differentiation as well as activation and several abnormalities in Notch pathway have been suggested to cause human leukemias. Cre-loxP mediated conditional gene targeting significantly contributed to our current understanding of the physiological roles of different Notch family members in hematopoietic compartment. This technique helped to overcome embryonic lethality of Notch mutants providing the opportunity to inactivate specific Notch gene in adulthood.     

Regulation of G protein‐coupled receptor signaling by plasma membrane organization and endocytosis

The trafficking of G protein coupled‐receptors (GPCRs) is one of the most exciting areas in cell biology because of recent advances demonstrating that GPCR signaling is spatially encoded. GPCRs, acting in a diverse array of physiological systems, can have differential signaling consequences depending on their subcellular localization. At the plasma membrane, GPCR organization could fine‐tune the initial stages of receptor signaling by determining the magnitude of signaling and the type of effectors to which receptors can couple. This organization is mediated by the lipid composition of the plasma membrane, receptor‐receptor interactions, and receptor interactions with intracellular scaffolding proteins. GPCR organization is subsequently changed by ligand binding and the regulated endocytosis of these receptors. Activated GPCRs can modulate the dynamics of their own endocytosis through changing clathrin‐coated pit dynamics, and through the scaffolding adaptor protein β‐arrestin. This endocytic regulation has signaling consequences, predominantly through modulation of the MAPK cascade. This review explores what is known about receptor sorting at the plasma membrane, protein partners that control receptor endocytosis, and the ways in which receptor sorting at the plasma membrane regulates downstream trafficking and signaling.      

Tie2 and Eph Receptor Tyrosine Kinase Activation and Signaling

The Eph and Tie cell surface receptors mediate a variety of signaling events during development and in the adult organism. As other receptor tyrosine kinases, they are activated on binding of extracellular ligands and their catalytic activity is tightly regulated on multiple levels. The Eph and Tie receptors display some unique characteristics, including the requirement of ligand-induced receptor clustering for efficient signaling. Interestingly, both Ephs and Ties can mediate different, even opposite, biological effects depending on the specific ligand eliciting the response and on the cellular context. Here we discuss the structural features of these receptors, their interactions with various ligands, as well as functional implications for downstream signaling initiation. The Eph/ephrin structures are already well reviewed and we only provide a brief overview on the initial binding events. We go into more detail discussing the Tie-angiopoietin structures and recognition.     

Localization of receptors in lipid rafts can inhibit signal transduction

Processes of cell survival, division, differentiation, and death are guided by the binding of signal molecules to receptors, which activates intracellular signaling networks and ultimately elicits genetic, biochemical, or biomechanical responses within the cell. While intracellular mechanisms for these processes have been well studied, little attention has been given to the role extracellular ligand transport and binding may play in signal initiation. Recent studies have found that the localization of receptors in lipid rafts is critical for the functions of many signaling pathways. By concentrating membrane components, rafts may promote essential interactions for signaling. Lipid rafts can also have negative effects on signaling, but mechanisms remain elusive. We propose that raft-mediated receptor clustering can reduce signaling by prolonging the diffusion of ligands to their receptors. We quantify this effect using a simple diffusion-limited binding model that accounts for the spatial distribution of lipid rafts and receptors on the cell surface. We find that receptor clustering can reduce the apparent rate of receptor binding by up to 80%, consistent with observed increases in epidermal growth factor (EGF) binding by up to 100% following disruption of lipid rafts (Pike and Casey 2002 Biochemistry 41:10315-10322; Roepstorff et al. 2002 J Biol Chem 277:18954-18960). Failure to account for the effects of receptor clustering on rates of ligand binding can skew the interpretation of current methods of cancer diagnosis and treatment. Finally, we discuss how the activation of particular signaling pathways can change over time, depending, in part, on the overall level and spatial distribution of the receptors.