Aggregation of membrane proteins by cytosolic cross-linkers: theory and simulation of the LAT-Grb2-SOS1 system

Transmembrane signaling through single-spanning receptors modulated by phase separation at the cell surface

A wide variety of transmembrane signals are transduced by cell-surface receptors that activate intracellular signaling molecules. In particular, receptor clustering in the plasma membrane plays a critical role in these processes. Single-spanning or single-pass transmembrane proteins are among the most significant types of membrane receptors, which include adhesion receptors, such as integrins, CD44, cadherins, and receptor tyrosine kinases. Elucidating the molecular mechanisms underlying the regulation of the activity of these receptors is of great significance. Liquid-liquid phase separation (LLPS) is a recently emerging paradigm in cellular physiology for the ubiquitous regulation of the spatiotemporal dynamics of various signaling pathways. This study describes the emerging features of transmembrane signaling through single-spanning receptors from the perspective of phase separation. Possible physicochemical modulations of LLPS-based transmembrane signaling are also discussed.

The membrane surface as a platform that organizes cellular and biochemical processes

Membranes are essential for life. They act as semi-permeable boundaries that define cells and organelles. In addition, their surfaces actively participate in biochemical reaction networks, where they confine proteins, align reaction partners, and directly control enzymatic activities. Membrane-localized reactions shape cellular membranes, define the identity of organelles, compartmentalize biochemical processes, and can even be the source of signaling gradients that originate at the plasma membrane and reach into the cytoplasm and nucleus. The membrane surface is, therefore, an essential platform upon which myriad cellular processes are scaffolded. In this review, we summarize our current understanding of the biophysics and biochemistry of membrane-localized reactions with particular focus on insights derived from reconstituted and cellular systems. We discuss how the interplay of cellular factors results in their self-organization, condensation, assembly, and activity, and the emergent properties derived from them.

Discrete LAT condensates encode antigen information from single pMHC:TCR binding events

LAT assembly into a two-dimensional protein condensate is a prominent feature of antigen discrimination by T cells. Here, we use single-molecule imaging techniques to resolve the spatial position and temporal duration of each pMHC:TCR molecular binding event while simultaneously monitoring LAT condensation at the membrane. An individual binding event is sufficient to trigger a LAT condensate, which is self-limiting, and neither its size nor lifetime is correlated with the duration of the originating pMHC:TCR binding event. Only the probability of the LAT condensate forming is related to the pMHC:TCR binding dwell time. LAT condenses abruptly, but after an extended delay from the originating binding event. A LAT mutation that facilitates phosphorylation at the PLC-γ1 recruitment site shortens the delay time to LAT condensation and alters T cell antigen specificity. These results identify a function for the LAT protein condensation phase transition in setting antigen discrimination thresholds in T cells.

Kinetic frustration by limited bond availability controls the LAT protein condensation phase transition on membranes

LAT is a membrane-linked scaffold protein that undergoes a phase transition to form a two-dimensional protein condensate on the membrane during T cell activation. Governed by tyrosine phosphorylation, LAT recruits various proteins that ultimately enable condensation through a percolation network of discrete and selective protein-protein interactions. Here, we describe detailed kinetic measurements of the phase transition, along with coarse-grained model simulations, that reveal that LAT condensation is kinetically frustrated by the availability of bonds to form the network. Unlike typical miscibility transitions in which compact domains may coexist at equilibrium, the LAT condensates are dynamically arrested in extended states, kinetically trapped out of equilibrium. Modeling identifies the structural basis for this kinetic arrest as the formation of spindle arrangements, favored by limited multivalent binding interactions along the flexible, intrinsically disordered LAT protein. These results reveal how local factors controlling the kinetics of LAT condensation enable formation of different, stable condensates, which may ultimately coexist within the cell.

Systems-level conservation of the proximal TCR signaling network of mice and humans

We exploited traceable gene tagging in primary human T cells to establish the composition and dynamics of seven canonical TCR-induced protein signaling complexes (signalosomes) using affinity purification coupled with mass spectrometry (AP-MS). It unveiled how the LAT adaptor assembles higher-order molecular condensates and revealed that the proximal TCR-signaling network has a high degree of qualitative and quantitative conservation between human CD4+ and CD8+ T cells. Such systems-level conservation also extended across human and mouse T cells and unexpectedly encompassed protein–protein interaction stoichiometry. Independently of evolutionary considerations, our study suggests that a drug targeting the proximal TCR signaling network should behave similarly when applied to human and mouse T cells. However, considering that signaling differences likely exist between the distal TCR-signaling pathway of human and mouse, our fast-track AP-MS approach should be favored to determine the mechanism of action of drugs targeting human T cell activation. An opportunity is illustrated here using an inhibitor of the LCK protein tyrosine kinase as a proof-of-concept.

Height, but not binding epitope, affects the potency of synthetic TCR agonists

Under physiological conditions, peptide-major histocompatibility complex (pMHC) molecules can trigger T cell receptors (TCRs) as monovalent ligands that are sparsely distributed on the plasma membrane of an antigen-presenting cell. TCRs can also be triggered by artificial clustering, such as with pMHC tetramers or antibodies; however, these strategies circumvent many of the natural ligand discrimination mechanisms of the T cell and can elicit nonphysiological signaling activity. We have recently introduced a synthetic TCR agonist composed of an anti-TCRβ Fab′ antibody fragment covalently bound to a DNA oligonucleotide, which serves as a membrane anchor. This Fab′-DNA ligand efficiently triggers TCR as a monomer when membrane associated and exhibits a potency and activation profile resembling agonist pMHC. In this report, we explore the geometric requirements for efficient TCR triggering and cellular activation by Fab′-DNA ligands. We find that T cells are insensitive to the ligand binding epitope on the TCR complex but that length of the DNA tether is important. Increasing, the intermembrane distance spanned by Fab′-DNA:TCR complexes decreases TCR triggering efficiency and T cell activation potency, consistent with the kinetic-segregation model of TCR triggering. These results establish design parameters for constructing synthetic TCR agonists that are able to activate polyclonal T cell populations, such as T cells from a human patient, in a similar manner as the native pMHC ligand.

The Clinical Aspect of Adaptor Molecules in T Cell Signaling: Lessons Learnt From Inborn Errors of Immunity

Adaptor molecules lack enzymatic and transcriptional activities. Instead, they exert their function by linking multiple proteins into intricate complexes, allowing for transmitting and fine-tuning of signals. Many adaptor molecules play a crucial role in T-cell signaling, following engagement of the T-cell receptor (TCR). In this review, we focus on Linker of Activation of T cells (LAT) and SH2 domain-containing leukocyte protein of 76 KDa (SLP-76). Monogenic defects in these adaptor proteins, with known roles in T-cell signaling, have been described as the cause of human inborn errors of immunity (IEI). We describe the current knowledge based on defects in cell lines, murine models and human patients. Germline mutations in Adhesion and degranulation adaptor protein (ADAP), have not resulted in a T-cell defect.

PLCγ1 promotes phase separation of T cell signaling components

The T cell receptor (TCR) pathway receives, processes, and amplifies the signal from pathogenic antigens to the activation of T cells. Although major components in this pathway have been identified, the knowledge on how individual components cooperate to effectively transduce signals remains limited. Phase separation emerges as a biophysical principle in organizing signaling molecules into liquid-like condensates. Here, we report that phospholipase Cγ1 (PLCγ1) promotes phase separation of LAT, a key adaptor protein in the TCR pathway. PLCγ1 directly cross-links LAT through its two SH2 domains. PLCγ1 also protects LAT from dephosphorylation by the phosphatase CD45 and promotes LAT-dependent ERK activation and SLP76 phosphorylation. Intriguingly, a nonmonotonic effect of PLCγ1 on LAT clustering was discovered. Computer simulations, based on patchy particles, revealed how the cluster size is regulated by protein compositions. Together, these results define a critical function of PLCγ1 in promoting phase separation of the LAT complex and TCR signal transduction.

Suppression of human T cell activation by derivatives of glycerol monolaurate

Glycerol monolaurate (GML), a naturally occurring monoglyceride, is widely used commercially for its antimicrobial properties. Interestingly, several studies have shown that GML not only has antimicrobial properties but is also an anti-inflammatory agent. GML inhibits peripheral blood mononuclear cell proliferation and inhibits T cell receptor (TCR)-induced signaling events. In this study, we perform an extensive structure activity relationship analysis to investigate the structural components of GML necessary for its suppression of human T cell activation. Human T cells were treated with analogs of GML, differing in acyl chain length, head group, linkage of acyl chain, and number of laurate groups. Treated cells were then tested for changes in membrane dynamics, LAT clustering, calcium signaling, and cytokine production. We found that an acyl chain with 12-14 carbons, a polar head group, an ester linkage, and a single laurate group at any position are all necessary for GML to inhibit protein clustering, calcium signaling, and cytokine production. Removing the glycerol head group or replacing the ester linkage with a nitrogen prevented derivative-mediated inhibition of protein cluster formation and calcium signaling, while still inhibiting TCR-induced cytokine production. These findings expand our current understanding of the mechanisms of action of GML and the of GML needed to function as a novel immunosuppressant.

How the T cell signaling network processes information to discriminate between self and agonist ligands

T cells exhibit remarkable sensitivity and selectivity in detecting and responding to agonist peptides (p) bound to MHC molecules in a sea of self pMHC molecules. Despite much work, understanding of the underlying mechanisms of distinguishing such ligands remains incomplete. Here, we quantify T cell discriminatory capacity using channel capacity, a direct measure of the signaling network's ability to discriminate between antigen-presenting cells (APCs) displaying either self ligands or a mixture of self and agonist ligands. This metric shows how differences in information content between these two types of peptidomes are decoded by the topology and rates of kinetic proofreading signaling steps inside T cells. Using channel capacity, we constructed numerically substantiated hypotheses to explain the discriminatory role of a recently identified slow LAT Y132 phosphorylation step. Our results revealed that in addition to the number and kinetics of sequential signaling steps, a key determinant of discriminatory capability is spatial localization of a minimum number of these steps to the engaged TCR. Biochemical and imaging experiments support these findings. Our results also reveal the discriminatory role of early negative feedback and necessary amplification conferred by late positive feedback.

Understanding the Dynamics of T-Cell Activation in Health and Disease Through the Lens of Computational Modeling

T cells in the immune system are activated by binding to foreign peptides (from an external pathogen) or mutant peptide (derived from endogenous proteins) displayed on the surface of a diseased cell. This triggers a series of intracellular signaling pathways, which ultimately dictate the response of the T cell. The insights from computational models have greatly improved our understanding of the mechanisms that control T-cell activation. In this review, we focus on the use of ordinary differential equation–based mechanistic models to study T-cell activation. We highlight several examples that demonstrate the models’ utility in answering specific questions related to T-cell activation signaling, from antigen discrimination to the feedback mechanisms that initiate transcription factor activation. In addition, we describe other modeling approaches that can be combined with mechanistic models to bridge time scales and better understand how intracellular signaling events, which occur on the order of seconds to minutes, influence phenotypic responses of T-cell activation, which occur on the order of hours to days. Overall, through concrete examples, we emphasize how computational modeling can be used to enable the rational design and optimization of immunotherapies.

The Control Centers of Biomolecular Phase Separation: How Membrane Surfaces, PTMs, and Active Processes Regulate Condensation

Biomolecular condensation is emerging as an essential process for cellular compartmentalization. The formation of biomolecular condensates can be driven by liquid-liquid phase separation, which arises from weak, multivalent interactions among proteins and nucleic acids. A substantial body of recent work has revealed that diverse cellular processes rely on biomolecular condensation and that aberrant phase separation may cause disease. Many proteins display an intrinsic propensity to undergo phase separation. However, the mechanisms by which cells regulate phase separation to build functional condensates at the appropriate time and location are only beginning to be understood. Here, we review three key cellular mechanisms that enable the control of biomolecular phase separation: membrane surfaces, post-translational modifications, and active processes. We discuss how these mechanisms may function in concert to provide robust control over biomolecular condensates and suggest new research avenues that will elucidate how cells build and maintain these key centers of cellular compartmentalization.

A molecular assembly phase transition and kinetic proofreading modulate Ras activation by SOS

The guanine nucleotide exchange factor (GEF) Son of Sevenless (SOS) is a key Ras activator that is autoinhibited in the cytosol and activates upon membrane recruitment. Autoinhibition release involves structural rearrangements of the protein at the membrane and thus introduces a delay between initial recruitment and activation. In this study, we designed a single-molecule assay to resolve the time between initial receptor-mediated membrane recruitment and the initiation of GEF activity of individual SOS molecules on microarrays of Ras-functionalized supported membranes. The rise-and-fall shape of the measured SOS activation time distribution and the long mean time scale to activation (~50 seconds) establish a basis for kinetic proofreading in the receptor-mediated activation of Ras. We further demonstrate that this kinetic proofreading is modulated by the LAT (linker for activation of T cells)-Grb2-SOS phosphotyrosine-driven phase transition at the membrane.

Mapping the stochastic sequence of individual ligand-receptor binding events to cellular activation: T cells act on the rare events

T cell receptor (TCR) binding to agonist peptide major histocompatibility complex (pMHC) triggers signaling events that initiate T cell responses. This system is remarkably sensitive, requiring only a few binding events to successfully activate a cellular response. On average, activating pMHC ligands exhibit mean dwell times of at least a few seconds when bound to the TCR. However, a T cell accumulates pMHC-TCR interactions as a stochastic series of discrete, single-molecule binding events whose individual dwell times are broadly distributed. With activation occurring in response to only a handful of such binding events, individual cells are unlikely to experience the average binding time. Here, we mapped the ensemble of pMHC-TCR binding events in space and time while simultaneously monitoring cellular activation. Our findings revealed that T cell activation hinges on rare, long-dwell time binding events that are an order of magnitude longer than the average agonist pMHC-TCR dwell time. Furthermore, we observed that short pMHC-TCR binding events that were spatially correlated and temporally sequential led to cellular activation. These observations indicate that T cell antigen discrimination likely occurs by sensing the tail end of the pMHC-TCR binding dwell time distribution rather than its average properties.

MCell-R: A Particle-Resolution Network-Free Spatial Modeling Framework

Spatial heterogeneity can have dramatic effects on the biochemical networks that drive cell regulation and decision-making. For this reason, a number of methods have been developed to model spatial heterogeneity and incorporated into widely used modeling platforms. Unfortunately, the standard approaches for specifying and simulating chemical reaction networks become untenable when dealing with multistate, multicomponent systems that are characterized by combinatorial complexity. To address this issue, we developed MCell-R, a framework that extends the particle-based spatial Monte Carlo simulator, MCell, with the rule-based model specification and simulation capabilities provided by BioNetGen and NFsim. The BioNetGen syntax enables the specification of biomolecules as structured objects whose components can have different internal states that represent such features as covalent modification and conformation and which can bind components of other molecules to form molecular complexes. The network-free simulation algorithm used by NFsim enables efficient simulation of rule-based models even when the size of the network implied by the biochemical rules is too large to enumerate explicitly, which frequently occurs in detailed models of biochemical signaling. The result is a framework that can efficiently simulate systems characterized by combinatorial complexity at the level of spatially resolved individual molecules over biologically relevant time and length scales.

Cooperative assembly of a four-molecule signaling complex formed upon T cell antigen receptor activation

The T cell antigen receptor encounters foreign antigen during the immune response. Receptor engagement leads to activation of specific protein tyrosine kinases, which then phosphorylate multiple enzymes and adapter proteins. One such enzyme, phospholipase-Cγ1, is responsible for cleavage of a plasma membrane lipid substrate, a phosphoinositide, into two second messengers, diacylglycerol, which activates several enzymes including protein kinase C, and an inositol phosphate, which induces intracellular calcium elevation. In T cells, phospholipase-Cγ1 is recruited to the plasma membrane as part of a four-protein complex containing three adapter molecules. We have used recombinant proteins and synthetic phosphopeptides to reconstitute this quaternary complex in vitro. Extending biophysical tools to study concurrent interactions of the four protein components, we demonstrated the formation and determined the composition of the quaternary complex using multisignal analytical ultracentrifugation, and we characterized the thermodynamic driving forces of assembly by isothermal calorimetry. We demonstrate that the four proteins reversibly associate in a circular arrangement of binding interfaces, each protein interacting with two others. Three interactions are of high affinity, and the fourth is of low affinity, with the assembly of the quaternary complex exhibiting significant enthalpy-entropy compensation as in an entropic switch. Formation of this protein complex enables subsequent recruitment of additional molecules needed to activate phospholipase-Cγ1. Understanding the formation of this complex is fundamental to full characterization of a central pathway in T cell activation. Such knowledge is critical to developing ways in which this pathway can be selectively inhibited.

Generalizing Gillespie's Direct Method to Enable Network-Free Simulations

Gillespie's direct method for stochastic simulation of chemical kinetics is a staple of computational systems biology research. However, the algorithm requires explicit enumeration of all reactions and all chemical species that may arise in the system. In many cases, this is not feasible due to the combinatorial explosion of reactions and species in biological networks. Rule-based modeling frameworks provide a way to exactly represent networks containing such combinatorial complexity, and generalizations of Gillespie's direct method have been developed as simulation engines for rule-based modeling languages. Here, we provide both a high-level description of the algorithms underlying the simulation engines, termed network-free simulation algorithms, and how they have been applied in systems biology research. We also define a generic rule-based modeling framework and describe a number of technical details required for adapting Gillespie's direct method for network-free simulation. Finally, we briefly discuss potential avenues for advancing network-free simulation and the role they continue to play in modeling dynamical systems in biology.

Dynamic Scaling Analysis of Molecular Motion within the LAT:Grb2:SOS Protein Network on Membranes

Biochemical signaling pathways often involve proteins with multiple, modular interaction domains. Signaling activates binding sites, such as by tyrosine phosphorylation, which enables protein recruitment and growth of networked protein assemblies. Although widely observed, the physical properties of the assemblies, as well as the mechanisms by which they function, remain largely unknown. Here we examine molecular mobility within LAT:Grb2:SOS assemblies on supported membranes by single-molecule tracking. Trajectory analysis reveals a discrete temporal transition to subdiffusive motion below a characteristic timescale, indicating that the LAT:Grb2:SOS assembly has the dynamical structure of a loosely entangled polymer. Such dynamical analysis is also applicable in living cells, where it offers another dimension on the characteristics of cellular signaling assemblies.

Allosteric Modulation of Grb2 Recruitment to the Intrinsically Disordered Scaffold Protein, LAT, by Remote Site Phosphorylation

Tyrosine phosphorylation of membrane receptors and scaffold proteins followed by recruitment of SH2 domain-containing adaptor proteins constitutes a central mechanism of intracellular signal transduction. During early T-cell receptor (TCR) activation, phosphorylation of linker for activation of T cells (LAT) leading to recruitment of adaptor proteins, including Grb2, is one prototypical example. LAT contains multiple modifiable sites, and this multivalency may provide additional layers of regulation, although this is not well understood. Here, we quantitatively analyze the effects of multivalent phosphorylation of LAT by reconstituting the initial reactions of the TCR signaling pathway on supported membranes. Results from a series of LAT constructs with combinatorial mutations of tyrosine residues reveal a previously unidentified allosteric mechanism in which the binding affinity of LAT:Grb2 depends on the phosphorylation at remote tyrosine sites. Additionally, we find that LAT:Grb2 binding affinity is altered by membrane localization. This allostery mainly regulates the kinetic on-rate, not off-rate, of LAT:Grb2 interactions. LAT is an intrinsically disordered protein, and these data suggest that phosphorylation changes the overall ensemble of configurations to modulate the accessibility of other phosphorylated sites to Grb2. Using Grb2 as a phosphorylation reporter, we further monitored LAT phosphorylation by TCR ζ chain-recruited ZAP-70, which suggests a weakly processive catalysis on membranes. Taken together, these results suggest that signal transmission through LAT is strongly gated and requires multiple phosphorylation events before efficient signal transmission is achieved.

Automated visualization of rule-based models

Frameworks such as BioNetGen, Kappa and Simmune use "reaction rules" to specify biochemical interactions compactly, where each rule specifies a mechanism such as binding or phosphorylation and its structural requirements. Current rule-based models of signaling pathways have tens to hundreds of rules, and these numbers are expected to increase as more molecule types and pathways are added. Visual representations are critical for conveying rule-based models, but current approaches to show rules and interactions between rules scale poorly with model size. Also, inferring design motifs that emerge from biochemical interactions is an open problem, so current approaches to visualize model architecture rely on manual interpretation of the model. Here, we present three new visualization tools that constitute an automated visualization framework for rule-based models: (i) a compact rule visualization that efficiently displays each rule, (ii) the atom-rule graph that conveys regulatory interactions in the model as a bipartite network, and (iii) a tunable compression pipeline that incorporates expert knowledge and produces compact diagrams of model architecture when applied to the atom-rule graph. The compressed graphs convey network motifs and architectural features useful for understanding both small and large rule-based models, as we show by application to specific examples. Our tools also produce more readable diagrams than current approaches, as we show by comparing visualizations of 27 published models using standard graph metrics. We provide an implementation in the open source and freely available BioNetGen framework, but the underlying methods are general and can be applied to rule-based models from the Kappa and Simmune frameworks also. We expect that these tools will promote communication and analysis of rule-based models and their eventual integration into comprehensive whole-cell models.

Absolute Ligand Discrimination by Dimeric Signaling Receptors

Many signaling pathways act through shared components, where different ligand molecules bind the same receptors or activate overlapping sets of response regulators downstream. Nevertheless, different ligands acting through cross-wired pathways often lead to different outcomes in terms of the target cell behavior and function. Although a number of mechanisms have been proposed, it still largely remains unclear how cells can reliably discriminate different molecular ligands under such circumstances. Here we show that signaling via ligand-induced receptor dimerization-a very common motif in cellular signaling-naturally incorporates a mechanism for the discrimination of ligands acting through the same receptor.

A Perspective on the Role of Computational Models in Immunology

This is an exciting time for immunology because the future promises to be replete with exciting new discoveries that can be translated to improve health and treat disease in novel ways. Immunologists are attempting to answer increasingly complex questions concerning phenomena that range from the genetic, molecular, and cellular scales to that of organs, whole animals or humans, and populations of humans and pathogens. An important goal is to understand how the many different components involved interact with each other within and across these scales for immune responses to emerge, and how aberrant regulation of these processes causes disease. To aid this quest, large amounts of data can be collected using high-throughput instrumentation. The nonlinear, cooperative, and stochastic character of the interactions between components of the immune system as well as the overwhelming amounts of data can make it difficult to intuit patterns in the data or a mechanistic understanding of the phenomena being studied. Computational models are increasingly important in confronting and overcoming these challenges. I first describe an iterative paradigm of research that integrates laboratory experiments, clinical data, computational inference, and mechanistic computational models. I then illustrate this paradigm with a few examples from the recent literature that make vivid the power of bringing together diverse types of computational models with experimental and clinical studies to fruitfully interrogate the immune system.

2016: Signaling Breakthroughs of the Year

Signaling breakthroughs of 2016 clustered mainly in the areas of neuroscience, immunology, and metabolism, with excursions into plant hormone signaling and bacterial manipulation of host signaling pathways. Perhaps reflecting the growing maturity of the discipline of cell signaling, many of this year's breakthroughs have implications for the pathogenesis or treatment of human disease.

Phosphotyrosine-mediated LAT assembly on membranes drives kinetic bifurcation in recruitment dynamics of the Ras activator SOS

The assembly of cell surface receptors with downstream signaling molecules is a commonly occurring theme in multiple signaling systems. However, little is known about how these assemblies modulate reaction kinetics and the ultimate propagation of signals. Here, we reconstitute phosphotyrosine-mediated assembly of extended linker for the activation of T cells (LAT):growth factor receptor-bound protein 2 (Grb2):Son of Sevenless (SOS) networks, derived from the T-cell receptor signaling system, on supported membranes. Single-molecule dwell time distributions reveal two, well-differentiated kinetic species for both Grb2 and SOS on the LAT assemblies. The majority fraction of membrane-recruited Grb2 and SOS both exhibit fast kinetics and single exponential dwell time distributions, with average dwell times of hundreds of milliseconds. The minor fraction exhibits much slower kinetics, extending the dwell times to tens of seconds. Considering this result in the context of the multistep process by which the Ras GEF (guanine nucleotide exchange factor) activity of SOS is activated indicates that kinetic stabilization from the LAT assembly may be important. This kinetic proofreading effect would additionally serve as a stochastic noise filter by reducing the relative probability of spontaneous SOS activation in the absence of receptor triggering. The generality of receptor-mediated assembly suggests that such effects may play a role in multiple receptor proximal signaling processes.

Phase separation of signaling molecules promotes T cell receptor signal transduction

Activation of various cell surface receptors triggers the reorganization of downstream signaling molecules into micrometer- or submicrometer-sized clusters. However, the functional consequences of such clustering have been unclear. We biochemically reconstituted a 12-component signaling pathway on model membranes, beginning with T cell receptor (TCR) activation and ending with actin assembly. When TCR phosphorylation was triggered, downstream signaling proteins spontaneously separated into liquid-like clusters that promoted signaling outputs both in vitro and in human Jurkat T cells. Reconstituted clusters were enriched in kinases but excluded phosphatases and enhanced actin filament assembly by recruiting and organizing actin regulators. These results demonstrate that protein phase separation can create a distinct physical and biochemical compartment that facilitates signaling.

The linker for activation of T cells (LAT) signaling hub: from signaling complexes to microclusters

Since the cloning of the critical adapter, LAT (linker for activation of T cells), more than 15 years ago, a combination of multiple scientific approaches and techniques continues to provide valuable insights into the formation, composition, regulation, dynamics, and function of LAT-based signaling complexes. In this review, we will summarize current views on the assembly of signaling complexes nucleated by LAT. LAT forms numerous interactions with other signaling molecules, leading to cooperativity in the system. Furthermore, oligomerization of LAT by adapter complexes enhances intracellular signaling and is physiologically relevant. These results will be related to data from super-resolution microscopy studies that have revealed the smallest LAT-based signaling units and nanostructure.

Modeling biomolecular site dynamics in immunoreceptor signaling systems

The immune system plays a central role in human health. The activities of immune cells, whether defending an organism from disease or triggering a pathological condition such as autoimmunity, are driven by the molecular machinery of cellular signaling systems. Decades of experimentation have elucidated many of the biomolecules and interactions involved in immune signaling and regulation, and recently developed technologies have led to new types of quantitative, systems-level data. To integrate such information and develop nontrivial insights into the immune system, computational modeling is needed, and it is essential for modeling methods to keep pace with experimental advances. In this chapter, we focus on the dynamic, site-specific, and context-dependent nature of interactions in immunoreceptor signaling (i.e., the biomolecular site dynamics of immunoreceptor signaling), the challenges associated with capturing these details in computational models, and how these challenges have been met through use of rule-based modeling approaches.

Building a patchwork - The yeast plasma membrane as model to study lateral domain formation

The plasma membrane (PM) has to fulfill a wide range of biological functions including selective uptake of substances, signal transduction and modulation of cell polarity and cell shape. To allow efficient regulation of these processes many resident proteins and lipids of the PM are laterally segregated into different functional domains. A particularly striking example of lateral segregation has been described for the budding yeast PM, where integral membrane proteins as well as lipids exhibit very slow translational mobility and form a patchwork of many overlapping micron-sized domains. Here we discuss the molecular and physical mechanisms contributing to the formation of a multi-domain membrane and review our current understanding of yeast PM organization. Many of the fundamental principles underlying membrane self-assembly and organization identified in yeast are expected to equally hold true in other organisms, even for the more transient and elusive organization of the PM in mammalian cells. This article is part of a Special Issue entitled: Nanoscale membrane organisation and signalling.

Binding and function of phosphotyrosines of the Ephrin A2 (EphA2) receptor using synthetic sterile α motif (SAM) domains

The sterile α motif (SAM) domain of the ephrin receptor tyrosine kinase, EphA2, undergoes tyrosine phosphorylation, but the effect of phosphorylation on the structure and interactions of the receptor is unknown. Studies to address these questions have been hindered by the difficulty of obtaining site-specifically phosphorylated proteins in adequate amounts. Here, we describe the use of chemically synthesized and specifically modified domain-length peptides to study the behavior of phosphorylated EphA2 SAM domains. We show that tyrosine phosphorylation of any of the three tyrosines, Tyr(921), Tyr(930), and Tyr(960), has a surprisingly small effect on the EphA2 SAM structure and stability. However, phosphorylation at Tyr(921) and Tyr(930) enables differential binding to the Src homology 2 domain of the adaptor protein Grb7, which we propose will lead to distinct functional outcomes. Setting up different signaling platforms defined by selective interactions with adaptor proteins thus adds another level of regulation to EphA2 signaling.

An Interaction Library for the FcεRI Signaling Network

Antigen receptors play a central role in adaptive immune responses. Although the molecular networks associated with these receptors have been extensively studied, we currently lack a systems-level understanding of how combinations of non-covalent interactions and post-translational modifications are regulated during signaling to impact cellular decision-making. To fill this knowledge gap, it will be necessary to formalize and piece together information about individual molecular mechanisms to form large-scale computational models of signaling networks. To this end, we have developed an interaction library for signaling by the high-affinity IgE receptor, FcεRI. The library consists of executable rules for protein–protein and protein–lipid interactions. This library extends earlier models for FcεRI signaling and introduces new interactions that have not previously been considered in a model. Thus, this interaction library is a toolkit with which existing models can be expanded and from which new models can be built. As an example, we present models of branching pathways from the adaptor protein Lat, which influence production of the phospholipid PIP₃ at the plasma membrane and the soluble second messenger IP₃. We find that inclusion of a positive feedback loop gives rise to a bistable switch, which may ensure robust responses to stimulation above a threshold level. In addition, the library is visualized to facilitate understanding of network circuitry and identification of network motifs.

Live-cell fluorescence imaging reveals high stoichiometry of Grb2 binding to the EGF receptor sustained during endocytosis

Activation of epidermal growth factor (EGF) receptor (EGFR) leads to its interaction with Grb2, a dual-function adapter mediating both signaling through Ras and receptor endocytosis. We used time-lapse three-dimensional imaging by spinning disk confocal microscopy to analyze trafficking of EGFR and Grb2 in living HeLa cells stimulated with low, physiological concentrations of EGFR ligands. Endogenous Grb2 was replaced in these cells by Grb2 fused to yellow fluorescent protein (YFP). After transient residence in the plasma membrane, Rhodamine-conjugated EGF (EGF-Rh) and Grb2-YFP were rapidly internalized and accumulated in endosomes. Quantitative image analysis revealed that on average two Grb2-YFP molecules were colocalized with one EGF-Rh in cells stimulated with 2 ng/ml EGF-Rh, and the excess of Grb2-YFP over EGF-Rh was even higher when a receptor-saturating concentration of EGF-Rh was used. Therefore, we hypothesize that a single EGFR molecule can be simultaneously associated with functionally distinct Grb2 interaction partners during and after endocytosis. Continuous presence of Grb2-YFP in endosomes was also observed when EGFR was activated by transforming growth factor-α and amphiregulin, suggesting that endosomal EGFRs remain ligand occupied and signaling competent, despite the fact that these growth factors are thought to dissociate from the receptor at acidic pH. The prolonged localization and activity of EGFR-Grb2 complexes in endosomes correlated with the sustained activation of extracellular stimulus-regulated kinase 1/2, suggesting that endosomal EGFRs contribute significantly to this signaling pathway. We propose that endosomal EGFRs function to extend signaling in time and space to compensate for rapid downregulation of surface EGFRs in cells with low receptor expression levels.

Efficient stochastic simulation of chemical kinetics networks using a weighted ensemble of trajectories

We apply the "weighted ensemble" (WE) simulation strategy, previously employed in the context of molecular dynamics simulations, to a series of systems-biology models that range in complexity from a one-dimensional system to a system with 354 species and 3680 reactions. WE is relatively easy to implement, does not require extensive hand-tuning of parameters, does not depend on the details of the simulation algorithm, and can facilitate the simulation of extremely rare events. For the coupled stochastic reaction systems we study, WE is able to produce accurate and efficient approximations of the joint probability distribution for all chemical species for all time t. WE is also able to efficiently extract mean first passage times for the systems, via the construction of a steady-state condition with feedback. In all cases studied here, WE results agree with independent "brute-force" calculations, but significantly enhance the precision with which rare or slow processes can be characterized. Speedups over "brute-force" in sampling rare events via the Gillespie direct Stochastic Simulation Algorithm range from ~10(12) to ~10(18) for characterizing rare states in a distribution, and ~10(2) to ~10(4) for finding mean first passage times.

Regulation of ras exchange factors and cellular localization of ras activation by lipid messengers in T cells

The Ras-MAPK signaling pathway is highly conserved throughout evolution and is activated downstream of a wide range of receptor stimuli. Ras guanine nucleotide exchange factors (RasGEFs) catalyze GTP loading of Ras and play a pivotal role in regulating receptor-ligand induced Ras activity. In T cells, three families of functionally important RasGEFs are expressed: RasGRF, RasGRP, and Son of Sevenless (SOS)-family GEFs. Early on it was recognized that Ras activation is critical for T cell development and that the RasGEFs play an important role herein. More recent work has revealed that nuances in Ras activation appear to significantly impact T cell development and selection. These nuances include distinct biochemical patterns of analog versus digital Ras activation, differences in cellular localization of Ras activation, and intricate interplays between the RasGEFs during distinct T cell developmental stages as revealed by various new mouse models. In many instances, the exact nature of these nuances in Ras activation or how these may result from fine-tuning of the RasGEFs is not understood. One large group of biomolecules critically involved in the control of RasGEFs functions are lipid second messengers. Multiple, yet distinct lipid products are generated following T cell receptor (TCR) stimulation and bind to different domains in the RasGRP and SOS RasGEFs to facilitate the activation of the membrane-anchored Ras GTPases. In this review we highlight how different lipid-based elements are generated by various enzymes downstream of the TCR and other receptors and how these dynamic and interrelated lipid products may fine-tune Ras activation by RasGEFs in developing T cells.

Multipoint binding of the SLP-76 SH2 domain to ADAP is critical for oligomerization of SLP-76 signaling complexes in stimulated T cells

The adapter molecules SLP-76 and LAT play central roles in T cell activation by recruiting enzymes and other adapters into multiprotein complexes that coordinate highly regulated signal transduction pathways. While many of the associated proteins have been characterized, less is known concerning the mechanisms of assembly for these dynamic and potentially heterogeneous signaling complexes. Following T cell receptor (TCR) stimulation, SLP-76 is found in structures called microclusters, which contain many signaling complexes. Previous studies showed that a mutation to the SLP-76 C-terminal SH2 domain nearly abolished SLP-76 microclusters, suggesting that the SH2 domain facilitates incorporation of signaling complexes into microclusters. S. C. Bunnell, A. L. Singer, D. I. Hong, B. H. Jacque, M. S. Jordan, M. C. Seminario, V. A. Barr, G. A. Koretzky, and L. E. Samelson, Mol. Cell. Biol., 26:7155-7166, 2006). Using biophysical methods, we demonstrate that the adapter, ADAP, contains three binding sites for SLP-76, and that multipoint binding to ADAP fragments oligomerizes the SLP-76 SH2 domain in vitro. These results were complemented with confocal imaging and functional studies of cells expressing ADAP with various mutations. Our results demonstrate that all three binding sites are critical for SLP-76 microcluster assembly, but any combination of two sites will partially induce microclusters. These data support a model whereby multipoint binding of SLP-76 to ADAP facilitates the assembly of SLP-76 microclusters. This model has implications for the regulation of SLP-76 and LAT microclusters and, as a result, T cell signaling.

Systems model of T cell receptor proximal signaling reveals emergent ultrasensitivity

Receptor phosphorylation is thought to be tightly regulated because phosphorylated receptors initiate signaling cascades leading to cellular activation. The T cell antigen receptor (TCR) on the surface of T cells is phosphorylated by the kinase Lck and dephosphorylated by the phosphatase CD45 on multiple immunoreceptor tyrosine-based activation motifs (ITAMs). Intriguingly, Lck sequentially phosphorylates ITAMs and ZAP-70, a cytosolic kinase, binds to phosphorylated ITAMs with differential affinities. The purpose of multiple ITAMs, their sequential phosphorylation, and the differential ZAP-70 affinities are unknown. Here, we use a systems model to show that this signaling architecture produces emergent ultrasensitivity resulting in switch-like responses at the scale of individual TCRs. Importantly, this switch-like response is an emergent property, so that removal of multiple ITAMs, sequential phosphorylation, or differential affinities abolishes the switch. We propose that highly regulated TCR phosphorylation is achieved by an emergent switch-like response and use the systems model to design novel chimeric antigen receptors for therapy.

Recognition of antigen by the T cell antigen receptor (TCR) is a central event in the initiation of adaptive immune responses and for this reason the TCR has been extensively studied. Multiple studies performed over the past 20 years have revealed intriguing findings that include the observation that the TCR has multiple phosphorylation sites that are sequentially phosphorylated by the kinase Lck and that ZAP-70, a cytosolic kinase, binds to these sites with different affinities. The purpose of multiple sites, their sequential phosphorylation by Lck, and the differential binding affinities of ZAP-70 are unknown. Using a novel mechanistic model that incorporates a high level of molecular detail, we find, unexpectedly, that all factors are critical for producing ultrasensitivity (switch-like response) and therefore this signaling architecture exhibits systems-level emergent ultrasensitivity. We use the model to study existing therapeutic chimeric antigen receptors and in the design of novel ones. The work also has direct implications to the study of many other immune receptors.

Analyzing the homeostasis of signaling proteins by a combination of Western blot and fluorescence correlation spectroscopy

The determination of intracellular protein concentrations is a prerequisite for understanding protein interaction networks in systems biology. Today, protein quantification is based either on mass spectrometry, which requires large cell numbers and sophisticated measurement protocols, or on quantitative Western blotting, which requires the expression and purification of a recombinant protein as a reference. Here, we present a method that uses a transiently expressed fluorescent fusion protein of the protein-of-interest as an easily accessible reference in small volumes of crude cell lysates. The concentration of the fusion protein is determined by fluorescence correlation spectroscopy, and this concentration is used to calibrate the intensity of bands on a Western blot. We applied this method to address cellular protein homeostasis by determining the concentrations of the plasma membrane-located transmembrane scaffolding protein LAT and soluble signaling proteins in naïve T cells and transformed T-cell lymphoma (Jurkat) cells (with the latter having nine times the volume of the former). Strikingly, the protein numbers of soluble proteins scaled with the cell volume, whereas that of the transmembrane protein LAT scaled with the membrane surface. This leads to significantly different stoichiometries of signaling proteins in transformed and naïve cells in concentration ranges that may translate directly into differences in complex formation.

Modeling and simulation of aggregation of membrane protein LAT with molecular variability in the number of binding sites for cytosolic Grb2-SOS1-Grb2

The linker for activation of T cells (LAT), the linker for activation of B cells (LAB), and the linker for activation of X cells (LAX) form a family of transmembrane adaptor proteins widely expressed in lymphocytes. These scaffolding proteins have multiple binding motifs that, when phosphorylated, bind the SH2 domain of the cytosolic adaptor Grb2. Thus, the valence of LAT, LAB and LAX for Grb2 is variable, depending on the strength of receptor activation that initiates phosphorylation. During signaling, the LAT population will exhibit a time-varying distribution of Grb2 valences from zero to three. In the cytosol, Grb2 forms 1∶1 and 2∶1 complexes with the guanine nucleotide exchange factor SOS1. The 2∶1 complex can bridge two LAT molecules when each Grb2, through their SH2 domains, binds to a phosphorylated site on a separate LAT. In T cells and mast cells, after receptor engagement, receptor phosphoyrlation is rapidly followed by LAT phosphorylation and aggregation. In mast cells, aggregates containing more than one hundred LAT molecules have been detected. Previously we considered a homogeneous population of trivalent LAT molecules and showed that for a range of Grb2, SOS1 and LAT concentrations, an equilibrium theory for LAT aggregation predicts the formation of a gel-like phase comprising a very large aggregate (superaggregate). We now extend this theory to investigate the effects of a distribution of Grb2 valence in the LAT population on the formation of LAT aggregates and superaggregate and use stochastic simulations to calculate the fraction of the total LAT population in the superaggregate.

Quantifying intramolecular binding in multivalent interactions: a structure-based synergistic study on Grb2-Sos1 complex

Numerous signaling proteins use multivalent binding to increase the specificity and affinity of their interactions within the cell. Enhancement arises because the effective binding constant for multivalent binding is larger than the binding constants for each individual interaction. We seek to gain both qualitative and quantitative understanding of the multivalent interactions of an adaptor protein, growth factor receptor bound protein-2 (Grb2), containing two SH3 domains interacting with the nucleotide exchange factor son-of-sevenless 1 (Sos1) containing multiple polyproline motifs separated by flexible unstructured regions. Grb2 mediates the recruitment of Sos1 from the cytosol to the plasma membrane where it activates Ras by inducing the exchange of GDP for GTP. First, using a combination of evolutionary information and binding energy calculations, we predict an additional polyproline motif in Sos1 that binds to the SH3 domains of Grb2. This gives rise to a total of five polyproline motifs in Sos1 that are capable of binding to the two SH3 domains of Grb2. Then, using a hybrid method combining molecular dynamics simulations and polymer models, we estimate the enhancement in local concentration of a polyproline motif on Sos1 near an unbound SH3 domain of Grb2 when its other SH3 domain is bound to a different polyproline motif on Sos1. We show that the local concentration of the Sos1 motifs that a Grb2 SH3 domain experiences is approximately 1000 times greater than the cellular concentration of Sos1. Finally, we calculate the intramolecular equilibrium constants for the crosslinking of Grb2 on Sos1 and use thermodynamic modeling to calculate the stoichiometry. With these equilibrium constants, we are able to predict the distribution of complexes that form at physiological concentrations. We believe this is the first systematic analysis that combines sequence, structure, and thermodynamic analyses to determine the stoichiometry of the complexes that are dominant in the cellular environment.

Guidelines for visualizing and annotating rule-based models

Rule-based modeling provides a means to represent cell signaling systems in a way that captures site-specific details of molecular interactions. For rule-based models to be more widely understood and (re)used, conventions for model visualization and annotation are needed. We have developed the concepts of an extended contact map and a model guide for illustrating and annotating rule-based models. An extended contact map represents the scope of a model by providing an illustration of each molecule, molecular component, direct physical interaction, post-translational modification, and enzyme-substrate relationship considered in a model. A map can also illustrate allosteric effects, structural relationships among molecular components, and compartmental locations of molecules. A model guide associates elements of a contact map with annotation and elements of an underlying model, which may be fully or partially specified. A guide can also serve to document the biological knowledge upon which a model is based. We provide examples of a map and guide for a published rule-based model that characterizes early events in IgE receptor (FcεRI) signaling. We also provide examples of how to visualize a variety of processes that are common in cell signaling systems but not considered in the example model, such as ubiquitination. An extended contact map and an associated guide can document knowledge of a cell signaling system in a form that is visual as well as executable. As a tool for model annotation, a map and guide can communicate the content of a model clearly and with precision, even for large models.

Dynein-driven transport of T cell receptor microclusters regulates immune synapse formation and T cell activation

When T cells recognize a peptide-major histocompatibility complex on antigen-presenting cells (APCs), T cell receptor microclusters (TCR-MCs) are generated and move to the center of the T cell-APC interface to form the central supramolecular activation cluster (cSMAC). cSMAC formation depends on stimulation strength and regulates T cell activation. We demonstrate that the dynein motor complex colocalized and coimmunoprecipitated with the TCR complex and that TCR-MCs moved along microtubules (MTs) toward the center of the immune synapse in a dynein-dependent manner to form cSMAC. MTs are located in close proximity to the plasma membrane at the activation site. TCR-MC velocity and cSMAC formation were impaired by dynein or MT inhibitors or by ablation of dynein expression. T cells with impaired cSMAC formation exhibited enhanced cellular activation including protein phosphorylation and interleukin-2 production. These results indicate that cSMAC formation by TCR-MC movement depends on dynein and MTs, and the movement regulates T cell activation.

The efficiency of reactant site sampling in network-free simulation of rule-based models for biochemical systems

Rule-based models, which are typically formulated to represent cell signaling systems, can now be simulated via various network-free simulation methods. In a network-free method, reaction rates are calculated for rules that characterize molecular interactions, and these rule rates, which each correspond to the cumulative rate of all reactions implied by a rule, are used to perform a stochastic simulation of reaction kinetics. Network-free methods, which can be viewed as generalizations of Gillespie's method, are so named because these methods do not require that a list of individual reactions implied by a set of rules be explicitly generated, which is a requirement of other methods for simulating rule-based models. This requirement is impractical for rule sets that imply large reaction networks (i.e. long lists of individual reactions), as reaction network generation is expensive. Here, we compare the network-free simulation methods implemented in RuleMonkey and NFsim, general-purpose software tools for simulating rule-based models encoded in the BioNetGen language. The method implemented in NFsim uses rejection sampling to correct overestimates of rule rates, which introduces null events (i.e. time steps that do not change the state of the system being simulated). The method implemented in RuleMonkey uses iterative updates to track rule rates exactly, which avoids null events. To ensure a fair comparison of the two methods, we developed implementations of the rejection and rejection-free methods specific to a particular class of kinetic models for multivalent ligand-receptor interactions. These implementations were written with the intention of making them as much alike as possible, minimizing the contribution of irrelevant coding differences to efficiency differences. Simulation results show that performance of the rejection method is equal to or better than that of the rejection-free method over wide parameter ranges. However, when parameter values are such that ligand-induced aggregation of receptors yields a large connected receptor cluster, the rejection-free method is more efficient.

Scaffold-mediated nucleation of protein signaling complexes: elementary principles

Proteins with multiple binding sites play important roles in cell signaling systems by nucleating protein complexes in which, for example, enzymes and substrates are co-localized. Proteins that specialize in this function are called by a variety names, including adapter, linker and scaffold. Scaffold-mediated nucleation of protein complexes can be either constitutive or induced. Induced nucleation is commonly mediated by a docking site on a scaffold that is activated by phosphorylation. Here, by considering minimalist mathematical models, which recapitulate scaffold effects seen in more mechanistically detailed models, we obtain analytical and numerical results that provide insights into scaffold function. These results elucidate how recruitment of a pair of ligands to a scaffold depends on the concentrations of the ligands, on the binding constants for ligand-scaffold interactions, on binding cooperativity, and on the milieu of the scaffold, as ligand recruitment is affected by competitive ligands and decoy receptors. For the case of a bivalent scaffold, we obtain an expression for the unique scaffold concentration that maximally recruits a pair of monovalent ligands. Through simulations, we demonstrate that a bivalent scaffold can nucleate distinct sets of ligands to equivalent extents when the scaffold is present at different concentrations. Thus, the function of a scaffold can potentially change qualitatively with a change in copy number. We also demonstrate how a scaffold can change the catalytic efficiency of an enzyme and the sensitivity of the rate of reaction to substrate concentration. The results presented here should be useful for understanding scaffold function and for engineering scaffolds to have desired properties.

Ultrasensitivity in multisite phosphorylation of membrane-anchored proteins

Cellular signaling is initially confined to the plasma membrane, where the cytoplasmic tails of surface receptors and other membrane-anchored proteins are phosphorylated in response to ligand binding. These proteins often contain multiple phosphorylation sites that are regulated by membrane-confined enzymes. Phosphorylation of these proteins is thought to be tightly regulated, because they initiate and regulate signaling cascades leading to cellular activation, yet how their phosphorylation is regulated is poorly understood. Ultrasensitive or switchlike responses in their phosphorylation state are not expected because the modifying enzymes are in excess. Here, we describe a novel mechanism of ultrasensitivity exhibited by multisite membrane-anchored proteins, but not cytosolic proteins, even when enzymes are in excess. The mechanism underlying this concentration-independent ultrasensitivity is the local saturation of a single enzyme by multiple sites on the substrate. Local saturation is a passive process arising from slow membrane diffusion, steric hindrances, and multiple sites, and therefore may be widely applicable. Critical to this ultrasensitivity is the brief enzymatic inactivation that follows substrate modification. Computations are presented using ordinary differential equations and stochastic spatial simulations. We propose a new role, to our knowledge, for multisite membrane-anchored proteins, discuss experiments that can be used to probe the model, and relate our findings to previous theoretical work.

Hierarchical graphs for rule-based modeling of biochemical systems

Background

In rule-based modeling, graphs are used to represent molecules: a colored vertex represents a component of a molecule, a vertex attribute represents the internal state of a component, and an edge represents a bond between components. Components of a molecule share the same color. Furthermore, graph-rewriting rules are used to represent molecular interactions. A rule that specifies addition (removal) of an edge represents a class of association (dissociation) reactions, and a rule that specifies a change of a vertex attribute represents a class of reactions that affect the internal state of a molecular component. A set of rules comprises an executable model that can be used to determine, through various means, the system-level dynamics of molecular interactions in a biochemical system.

Results

For purposes of model annotation, we propose the use of hierarchical graphs to represent structural relationships among components and subcomponents of molecules. We illustrate how hierarchical graphs can be used to naturally document the structural organization of the functional components and subcomponents of two proteins: the protein tyrosine kinase Lck and the T cell receptor (TCR) complex. We also show that computational methods developed for regular graphs can be applied to hierarchical graphs. In particular, we describe a generalization of Nauty, a graph isomorphism and canonical labeling algorithm. The generalized version of the Nauty procedure, which we call HNauty, can be used to assign canonical labels to hierarchical graphs or more generally to graphs with multiple edge types. The difference between the Nauty and HNauty procedures is minor, but for completeness, we provide an explanation of the entire HNauty algorithm.

Conclusions

Hierarchical graphs provide more intuitive formal representations of proteins and other structured molecules with multiple functional components than do the regular graphs of current languages for specifying rule-based models, such as the BioNetGen language (BNGL). Thus, the proposed use of hierarchical graphs should promote clarity and better understanding of rule-based models.

Shaping the response: the role of FcεRI and Syk expression levels in mast cell signaling

Many receptor systems initiate cell signaling through ligand-induced receptor aggregation. For bivalent ligands binding to mono- or bivalent receptors, a plot of the equilibrium concentration of receptors in aggregates against the log of the free ligand concentration, the cross-linking curve, is symmetric and bell shaped. However, steady state cellular responses initiated through receptor cross-linking may have a different dependence on ligand concentration than the aggregated receptors that initiate and maintain these responses. The authors illustrate by considering the activation of the protein kinase Syk that rapidly occurs after high affinity receptors for IgE, FcεRI, are aggregated on the surface of mast cells and basophils. Using a mathematical model of Syk activation the authors investigate two effects, one straightforward and one less so, that result in Syk activation not qualitatively following the cross-linking curve. Model predictions show that if the mechanism by which Syk is fully activated involves the transphosphorylation of Syk by Syk, then Syk activation curves can be either bell shaped or double humped, depending on the cellular concentrations of Syk and FcεRI. The model also predicts that the Syk activation curve can be non-symmetric with respect to the ligand concentration. The cell can exhibit differential Syk activation at two different ligand concentrations that produce identical distributions of receptor aggregates that form and dissociate at the same rates. The authors discuss how, even though it is only receptor aggregates that trigger responses, differences in total ligand concentration can lead to subtle kinetic effects that yield qualitative differences in the levels of Syk activation.