Calculation of diffusion-limited kinetics for the reactions in collision coupling and receptor cross-linking

Measuring the Adsorption Cross Section of YOYO-1 to Immobilized DNA Molecules

Many interactions between small molecules and particles occur in solutions. They are surrounded by other molecules that do not react, for example, biological processes in water, chemical reactions in gas or liquid solutions, and environmental reactions in air and water. However, predicting the rate of such diffusive interactions remains challenging, due to the random motion of molecules in solutions, as exampled by the famous Brownian motion of pollen particles. In this report, we experimentally confirmed that a disruptive rate equation we have published before can predict the association rate of typical adsorption at interfaces, which has a surprising fraction order of 4/3 that has not been considered before. This could be an important step toward a generalized method to predict the adsorption rate of many reactions.

Hypericin, a potential new BH3 mimetic

Many types of cancer such as prostate cancer, myeloid leukemia, breast cancer, glioblastoma display strong chemo resistance, which is supported by enhanced expression of multiple anti-apoptotic Bcl-2, Bcl-XL and Mcl-1 proteins. The viable anti-cancer strategies are based on developing anti-apoptotic Bcl-2 proteins inhibitors, BH3 mimetics. Our focus in past years has been on the investigating a new potential BH3 mimetic, Hypericin (Hyp). Hyp is a naturally occurring photosensitive compound used in photodynamic therapy and diagnosis. We have demonstrated that Hyp can cause substantial effects in cellular ultrastructure, mitochondria function and metabolism, and distribution of Bcl2 proteins in malignant and non-malignant cells. One of the possible mechanisms of Hyp action could be the direct interactions between Bcl-2 proteins and Hyp. We investigated this assumption by in silico computer modelling and in vitro fluorescent spectroscopy experiments with the small Bcl2 peptide segments designed to correspond to Bcl2 BH3 and BH1 domains. We show here that Hyp interacts with BH3 and BH1 peptides in concentration dependent manner, and shows the stronger interactions than known BH3 mimetics, Gossypol (Goss) and ABT-263. In addition, interactions of Hyp, Goss and ABT263, with whole purified proteins Bcl-2 and Mcl-1 by fluorescence spectroscopy show that Hyp interacts stronger with the Bcl-2 and less with Mcl-1 protein than Goss or ABT-263. This suggest that Hyp is comparable to other BH3 mimetics and could be explore as such. Hyp cytotoxicity was low in human U87 MG glioma, similar to that of ABT263, where Goss exerted sufficient cytotoxicity, suggesting that Hyp acts primarily on Bcl-2, but not on Mcl-1 protein. In combination therapy, low doses of Hyp with Goss effectively decreased U87 MG viability, suggesting a possible synergy effect. Overall, we can conclude that Hyp as BH3 mimetic acts primarily on Bcl-2 protein and can be explored to target cells with Bcl-2 over-expression, or in combination with other BH3 mimetics, that target Mcl-1 or Bcl-XL proteins, in dual therapy.

A Collision Coupling Model Governs the Activation of Neuronal GIRK1/2 Channels by Muscarinic-2 Receptors

The G protein-activated Inwardly Rectifying K⁺-channel (GIRK) modulates heart rate and neuronal excitability. Following G-Protein Coupled Receptor (GPCR)-mediated activation of heterotrimeric G proteins (Gαβγ), opening of the channel is obtained by direct binding of Gβγ subunits. Interestingly, GIRKs are solely activated by Gβγ subunits released from Gα_i/o-coupled GPCRs, despite the fact that all receptor types, for instance Gα_q-coupled, are also able to provide Gβγ subunits. It is proposed that this specificity and fast kinetics of activation stem from pre-coupling (or pre-assembly) of proteins within this signaling cascade. However, many studies, including our own, point towards a diffusion-limited mechanism, namely collision coupling. Here, we set out to address this long-standing question by combining electrophysiology, imaging, and mathematical modeling. Muscarinic-2 receptors (M2R) and neuronal GIRK1/2 channels were coexpressed in Xenopus laevis oocytes, where we monitored protein surface expression, current amplitude, and activation kinetics. Densities of expressed M2R were assessed using a fluorescently labeled GIRK channel as a molecular ruler. We then incorporated our results, along with available kinetic data reported for the G-protein cycle and for GIRK1/2 activation, to generate a comprehensive mathematical model for the M2R-G-protein-GIRK1/2 signaling cascade. We find that, without assuming any irreversible interactions, our collision coupling kinetic model faithfully reproduces the rate of channel activation, the changes in agonist-evoked currents and the acceleration of channel activation by increased receptor densities.

Molecular basis of signaling specificity between GIRK channels and GPCRs

Stimulated muscarinic acetylcholine receptors (M2Rs) release Gβγ subunits, which slow heart rate by activating a G protein-gated K⁺ channel (GIRK). Stimulated β2 adrenergic receptors (β2ARs) also release Gβγ subunits, but GIRK is not activated. This study addresses the mechanism underlying this specificity of GIRK activation by M2Rs. K⁺ currents and bioluminescence resonance energy transfer between labelled G proteins and GIRK show that M2Rs catalyze Gβγ subunit release at higher rates than β2ARs, generating higher Gβγ concentrations that activate GIRK and regulate other targets of Gβγ. The higher rate of Gβγ release is attributable to a faster G protein coupled receptor – G protein trimer association rate in M2R compared to β2AR. Thus, a rate difference in a single kinetic step accounts for specificity.

Dynamic disorder in simple enzymatic reactions induces stochastic amplification of substrate

A growing amount of evidence over the last two decades points to the fact that many enzymes exhibit fluctuations in their catalytic activity, which are associated with conformational changes on a broad range of timescales. The experimental study of this phenomenon, termed dynamic disorder, has become possible thanks to advances in single-molecule enzymology measurement techniques, through which the catalytic activity of individual enzyme molecules can be tracked in time. The biological role and importance of these fluctuations in a system with a small number of enzymes, such as a living cell, have only recently started being explored. In this work, we examine a simple stochastic reaction system consisting of an inflowing substrate and an enzyme with a randomly fluctuating catalytic reaction rate that converts the substrate into an outflowing product. To describe analytically the effect of rate fluctuations on the average substrate abundance at steady state, we derive an explicit formula that connects the relative speed of enzymatic fluctuations with the mean substrate level. Under fairly general modelling assumptions, we demonstrate that the relative speed of rate fluctuations can have a dramatic effect on the mean substrate, and lead to large positive deviations from predictions based on the assumption of deterministic enzyme activity. Our results also establish an interesting connection between the amplification effect and the mixing properties of the Markov process describing the enzymatic activity fluctuations, which can be used to easily predict the fluctuation speed above which such deviations become negligible. As the techniques of single-molecule enzymology continuously evolve, it may soon be possible to study the stochastic phenomena due to enzymatic activity fluctuations within living cells. Our work can be used to formulate experimentally testable hypotheses regarding the nature and magnitude of these fluctuations, as well as their phenotypic consequences.

Effect of Hydrodynamic Interactions on Reaction Rates in Membranes

The Brownian motion of two particles in three dimensions serves as a model for predicting the diffusion-limited reaction rate, as first discussed by von Smoluchowski. Deutch and Felderhof extended the calculation to account for hydrodynamic interactions between the particles and the target, which results in a reduction of the rate coefficient by about half. Many chemical reactions take place in quasi-two-dimensional systems, such as on the membrane or surface of a cell. We perform a Smoluchowski-like calculation in a quasi-two-dimensional geometry, i.e., a membrane surrounded by fluid, and account for hydrodynamic interactions between the particles. We show that rate coefficients are reduced relative to the case of no interactions. The reduction is more pronounced than the three-dimensional case due to the long-range nature of two-dimensional flows.

Rebinding in biochemical reactions on membranes

The behavior of many biochemical processes depends crucially on molecules rapidly rebinding after dissociating. In the case of multisite protein modification, the importance of rebinding has been demonstrated both experimentally and through several recent computational studies involving stochastic spatial simulations. As rebinding stems from spatio-temporal correlations, theorists have resorted to models that explicitly include space to properly account for the effects of rebinding. However, for reactions in three space dimensions it was recently shown that well-mixed ordinary differential equation (ODE) models can incorporate rebinding by adding connections to the reaction network. The rate constants for these new connections involve the probability that a pair of molecules rapidly rebinds after dissociation. In order to study biochemical reactions on membranes, in this paper we derive an explicit formula for this rebinding probability for reactions in two space dimensions. We show that ODE models can use the formula to replicate detailed stochastic spatial simulations, and that the formula can predict ultrasensitivity for reactions involving multisite modification of membrane-bound proteins. Further, we compute a new concentration-dependent rebinding probability for reactions in three space dimensions. Our analysis predicts that rebinding plays a much larger role in reactions on membranes compared to reactions in cytoplasm.

Nanoscale diffusion in the synaptic cleft and beyond measured with time-resolved fluorescence anisotropy imaging

Neural activity relies on molecular diffusion within nanoscopic spaces outside and inside nerve cells, such as synaptic clefts or dendritic spines. Measuring diffusion on this small scale in situ has not hitherto been possible, yet this knowledge is critical for understanding the dynamics of molecular events and electric currents that shape physiological signals throughout the brain. Here we advance time-resolved fluorescence anisotropy imaging combined with two-photon excitation microscopy to map nanoscale diffusivity in ex vivo brain slices. We find that in the brain interstitial gaps small molecules move on average ~30% slower than in a free medium whereas inside neuronal dendrites this retardation is ~70%. In the synaptic cleft free nanodiffusion is decelerated by ~46%. These quantities provide previously unattainable basic constrains for the receptor actions of released neurotransmitters, the electrical conductance of the brain interstitial space and the limiting rate of molecular interactions or conformational changes in the synaptic microenvironment.

Modeling bispecific monoclonal antibody interaction with two cell membrane targets indicates the importance of surface diffusion

We have developed a mathematical framework for describing a bispecific monoclonal antibody interaction with two independent membrane-bound targets that are expressed on the same cell surface. The bispecific antibody in solution binds either of the two targets first, and then cross-links with the second one while on the cell surface, subject to rate-limiting lateral diffusion step within the lifetime of the monovalently engaged antibody-antigen complex. At experimental densities, only a small fraction of the free targets is expected to lie within the reach of the antibody binding sites at any time. Using ordinary differential equation and Monte Carlo simulation-based models, we validated this approach against an independently published anti-CD4/CD70 DuetMab experimental data set. As a result of dimensional reduction, the cell surface reaction is expected to be so rapid that, in agreement with the experimental data, no monovalently bound bispecific antibody binary complexes accumulate until cross-linking is complete. The dissociation of the bispecific antibody from the ternary cross-linked complex is expected to be significantly slower than that from either of the monovalently bound variants. We estimate that the effective affinity of the bivalently bound bispecific antibody is enhanced for about 4 orders of magnitude over that of the monovalently bound species. This avidity enhancement allows for the highly specific binding of anti-CD4/CD70 DuetMab to the cells that are positive for both target antigens over those that express only one or the other We suggest that the lateral diffusion of target antigens in the cell membrane also plays a key role in the avidity effect of natural antibodies and other bivalent ligands in their interactions with their respective cell surface receptors.

The membrane environment can promote or suppress bistability in cell signaling networks

Many key biochemical reactions that mediate signal transduction in cells occur at the cell membrane, yet how the two-dimensional membrane environment influences the collective behavior of signaling networks is poorly understood. We study models of two topologically different signaling pathways that exhibit bistability, examining the effects of reduced protein mobility and increased concentration at the membrane, as well as effects due to differences in spatiotemporal correlations between the membrane environment and three-dimensional cytoplasm. The two model networks represent the distributive enzymatic modification of a protein at multiple sites and the positive feedback-mediated activation of a protein. In both cases, we find that confining proteins to a membrane-like environment can markedly alter the emergent dynamics. For the distributive protein modification network, increased concentration promotes bistability through enhanced protein-protein binding, while lower mobility and membrane-enhanced spatiotemporal correlations suppress bistability. For the positive feedback-mediated activation network, confinement to a membrane environment enhances protein activation, which can induce bistability or stabilize a monostable, active state. Importantly, the influence of the membrane environment on signaling dynamics can be qualitatively different for signaling modules with different network topologies.

Impact of receptor clustering on ligand binding

Background

Cellular response to changes in the concentration of different chemical species in the extracellular medium is induced by ligand binding to dedicated transmembrane receptors. Receptor density, distribution, and clustering may be key spatial features that influence effective and proper physical and biochemical cellular responses to many regulatory signals. Classical equations describing this kind of binding kinetics assume the distributions of interacting species to be homogeneous, neglecting by doing so the impact of clustering. As there is experimental evidence that receptors tend to group in clusters inside membrane domains, we investigated the effects of receptor clustering on cellular receptor ligand binding.

Results

We implemented a model of receptor binding using a Monte-Carlo algorithm to simulate ligand diffusion and binding. In some simple cases, analytic solutions for binding equilibrium of ligand on clusters of receptors are provided, and supported by simulation results. Our simulations show that the so-called "apparent" affinity of the ligand for the receptor decreases with clustering although the microscopic affinity remains constant.

Conclusions

Changing membrane receptors clustering could be a simple mechanism that allows cells to change and adapt its affinity/sensitivity toward a given stimulus.

Quantitative understanding of cell signaling: the importance of membrane organization

Systems biology modeling of signal transduction pathways traditionally employs ordinary differential equations, deterministic models based on the assumptions of spatial homogeneity. However, this can be a poor approximation for certain aspects of signal transduction, especially its initial steps: the cell membrane exhibits significant spatial organization, with diffusion rates approximately two orders of magnitude slower than those in the cytosol. Thus, to unravel the complexities of signaling pathways, quantitative models must consider spatial organization as an important feature of cell signaling. Furthermore, spatial separation limits the number of molecules that can physically interact, requiring stochastic simulation methods that account for individual molecules. Herein, we discuss the need for mathematical models and experiments that appreciate the importance of spatial organization in the membrane.

Analysis of reaction-diffusion systems with anomalous subdiffusion

Reaction-diffusion equations are the cornerstone of modeling biochemical systems with spatial gradients, which are relevant to biological processes such as signal transduction. Implicit in the formulation of these equations is the assumption of Fick's law, which states that the local diffusive flux of species i is proportional to its concentration gradient; however, in the context of complex fluids such as cytoplasm and cell membranes, the use of Fick's law is based on empiricism, whereas evidence has been mounting that such media foster anomalous subdiffusion (with mean-squared displacement increasing less than linearly with time) over certain length scales. Particularly when modeling diffusion-controlled reactions and other systems where the spatial domain is considered semi-infinite, assuming Fickian diffusion might not be appropriate. In this article, two simple, conceptually extreme models of anomalous subdiffusion are used in the framework of Green's functions to demonstrate the solution of four reaction-diffusion problems that are well known in the biophysical context of signal transduction: fluorescence recovery after photobleaching, the Smolochowski limit for diffusion-controlled reactions in solution, the spatial range of a diffusing molecule with finite lifetime, and the collision coupling mechanism of diffusion-controlled reactions in two dimensions. In each case, there are only subtle differences between the two subdiffusion models, suggesting how measurements of mean-squared displacement versus time might generally inform models of reactive systems with partial diffusion control.

Lipid raft-mediated regulation of G-protein coupled receptor signaling by ligands which influence receptor dimerization: a computational study

G-protein coupled receptors (GPCRs) are the largest family of cell surface receptors; they activate heterotrimeric G-proteins in response to ligand stimulation. Although many GPCRs have been shown to form homo- and/or heterodimers on the cell membrane, the purpose of this dimerization is not known. Recent research has shown that receptor dimerization may have a role in organization of receptors on the cell surface. In addition, microdomains on the cell membrane termed lipid rafts have been shown to play a role in GPCR localization. Using a combination of stochastic (Monte Carlo) and deterministic modeling, we propose a novel mechanism for lipid raft partitioning of GPCRs based on reversible dimerization of receptors and then demonstrate that such localization can affect GPCR signaling. Modeling results are consistent with a variety of experimental data indicating that lipid rafts have a role in amplification or attenuation of G-protein signaling. Thus our work suggests a new mechanism by which dimerization-inducing or inhibiting characteristics of ligands can influence GPCR signaling by controlling receptor organization on the cell membrane.

Adhesive dynamics simulation of G-protein-mediated chemokine-activated neutrophil adhesion

To reach sites of inflammation, a blood-borne neutrophil first rolls over the vessel wall, becoming firmly adherent on activation, and then transmigrates through the endothelium. In this study, we simulate the transition to firm adhesion via chemokine-induced integrin activation. To recreate the transition from rolling to firm adhesion, we use an integrated signaling adhesive dynamics simulation that includes selectin, integrin, and chemokine interactions between the cell and an adhesive substrate. Integrin bonds are of low affinity until activated by chemokine binding to G-protein coupled receptors on the model cell. The signal propagates within the cell through probabilistic diffusion and reaction of the signaling elements to induce the high-affinity integrins required for firm adhesion. This model showed that integrins become progressively active as cells roll and interact with chemokines, leading to a slight slowing before firm adhesion on a timescale similar to that observed in experiments. Increasing the density of chemokine resulted in decreases in the rolling time before stopping, consistent with experimental observations. However, a limit is reached where further increases in chemokine density do not increase adhesion. We found that the timescale for integrin activation correlated with the time to stop. Further, altering parameters within the intracellular signaling cascade that changed the speed of integrin activation, such as effector activation and dissociation rates, correspondingly affected the time to firm adhesion. For all conditions tested, the number of active integrin bonds at the point of firm adhesion was relatively constant. The model predicts that the time to stop would be relatively independent of selectin or integrin density, but strongly dependent on the shear rate because higher shear rates limit the intrinsic activation rate of integrins and require more integrins for adhesion.

Collision coupling, crosstalk, and compartmentalization in G-protein coupled receptor systems: can a single model explain disparate results?

The collision coupling model describes interactions between receptors and G-proteins as first requiring the molecules to find each other by diffusion. A variety of experimental data on G-protein activation have been interpreted as suggesting (or not) the compartmentalization of receptors and/or G-proteins in addition to a collision coupling mechanism. In this work, we use a mathematical model of G-protein activation via collision coupling but without compartmentalization to demonstrate that these disparate observations do not imply the existence of such compartments. In experiments with GTP analogs (commonly GTPgammaS), the extent of G-protein activation is predicted to be a function of both receptor number and the rate of GTP analog hydrolysis. The sensitivity of G-protein activation to receptor number is shown to be dependent upon the assay used, with the sensitivity of phosphate production assays (GTPase) >GTPgammaS-binding assays >cAMP inhibition assays. Finally, the amount of competition or crosstalk between receptor species activating the same type of G-proteins is predicted to depend on receptor and G-protein number, but in some (common) experimental regimes this dependence is expected to be minimal. Taken together, these observations suggest that the collision coupling model, without compartments of receptors and/or G-proteins, is sufficient to explain a variety of observations in literature data.

Signal transduction at point-blank range: analysis of a spatial coupling mechanism for pathway crosstalk

The plasma membrane provides a physical platform for the orchestration of molecular interactions and biochemical conversions involved in the early stages of receptor-mediated signal transduction in living cells. In that context, we introduce here the concept of spatial coupling, wherein simultaneous recruitment of different enzymes to the same receptor scaffold facilitates crosstalk between different signaling pathways through the local release and capture of activated signaling molecules. To study the spatiotemporal dynamics of this mechanism, we have developed a Brownian dynamics modeling approach and applied it to the receptor-mediated activation of Ras and the cooperative recruitment of phosphoinositide 3-kinase (PI3K) by activated receptors and Ras. Various analyses of the model simulations show that cooperative assembly of multimolecular complexes nucleated by activated receptors is facilitated by the local release and capture of membrane-anchored signaling molecules (such as active Ras) from/by receptor-bound signaling proteins. In the case of Ras/PI3K crosstalk, the model predicts that PI3K is more likely to be recruited by activated receptors bound or recently visited by the enzyme that activates Ras. By this mechanism, receptor-bound PI3K is stabilized through short-range, diffusion-controlled capture of active Ras and Ras/PI3K complexes released from the receptor complex. We contend that this mechanism is a means by which signaling pathways are propagated and spatially coordinated for efficient crosstalk between them.

Diffusion-limited reactions in G-protein activation: unexpected consequences of antagonist and agonist competition

In this work, we ask whether the simultaneous movement of agonist and antagonist among surface receptors (i.e. continually associating and dissociating from individual receptors according to specified kinetics) has any unexpected consequences for G-protein activation and receptor desensitization. A Monte Carlo model framework is used to track the diffusion and reaction of individual receptors, allowing the requirement for receptors and G-proteins or receptors and kinases to find each other by diffusion (collision coupling) to be implemented explicitly. We find that at constant agonist occupancy the effect of an antagonist on both G-protein activation and the ratio of G-protein activation to receptor desensitization can be modulated by varying the antagonist dissociation kinetics. The explanation for this effect is that antagonist dissociation kinetics influence the ability of agonists to access particular receptors and thus reach G-proteins and kinases near those receptors. Relevant parameter ranges for observation of these effects are identified. These results are useful for understanding experimental and therapeutic situations when both agonist and antagonist are present, and in addition may offer new insights into insurmountable antagonism.

Endogenous regulators of G protein signaling differentially modulate full and partial mu-opioid agonists at adenylyl cyclase as predicted by a collision coupling model

Regulator of G protein signaling (RGS) proteins accelerate the endogenous GTPase activity of Galpha(i/o) proteins to increase the rate of deactivation of active Galpha-GTP and Gbetagamma signaling molecules. Previous studies have suggested that RGS proteins are more effective on less efficiently coupled systems such as with partial agonist responses. To determine the role of endogenous RGS proteins in functional responses to mu-opioid agonists of different intrinsic efficacy, Galpha(i/o) subunits with a mutation at the pertussis toxin (PTX)-sensitive cysteine (C351I) and with or without a mutation at the RGS binding site (G184S) were stably expressed in C6 glioma cells expressing a mu-opioid receptor. Cells were treated overnight with PTX to inactivate endogenous G proteins. Maximal inhibition of forskolin-stimulated adenylyl cyclase by the low-efficacy partial agonists buprenorphine and nalbuphine was increased in cells expressing RGS-insensitive Galpha(o)(CIGS), Galpha(i2)(CIGS), or Galpha(i3)(CIGS) compared with their Galpha(CI) counterparts, but the RGS-insensitive mutation had little or no effect on the maximal inhibition by the higher efficacy agonists DAMGO and morphine. The potency of all the agonists to inhibit forskolin-stimulated adenylyl cyclase was increased in cells expressing RGS-insensitive Galpha(o)(CIGS), Galpha(i2)(CIGS), or Galpha(i3)(CIGS), regardless of efficacy. These data are comparable with predictions based on a collision coupling model. In this model, the rate of G protein inactivation, which is modulated by RGS proteins, and the rate of G protein activation, which is affected by agonist intrinsic efficacy, determine the maximal agonist response and potency at adenylyl cyclase under steady state conditions.

Signaling by committee: receptor clusters determine pathways of cellular activation

Receptor clustering is a common signaling mechanism for cell surface receptors. Exogenous ligands such as antibodies or synthetic analogues can be used to artificially induce clustering. New studies using defined synthetic ligands suggest that the spatial organization of these clusters attenuates signaling in one pathway but has no effect in another.

Rapid-mix flow cytometry measurements of subsecond regulation of G protein-coupled receptor ternary complex dynamics by guanine nucleotides

We have used rapid-mix flow cytometry to analyze the early subsecond dynamics of the disassembly of ternary complexes of G protein-coupled receptors (GPCRs) immobilized on beads to examine individual steps associated with guanine nucleotide activation. Our earlier studies suggested that the slow dissociation of Galpha and Gbetagamma subunits was unlikely to be an essential component of cell activation. However, these studies did not have adequate time resolution to define precisely the disassembly kinetics. Ternary complexes were assembled using three formyl peptide receptor constructs (wild type, formyl peptide receptor-Galpha(i2) fusion, and formyl peptide receptor-green fluorescent protein fusion) and two isotypes of the alpha subunit (alpha(i2) and alpha(i3)) and betagamma dimer (beta(1)gamma(2) and beta(4)gamma(2)). At saturating nucleotide levels, the disassembly of a significant fraction of ternary complexes occurred on a subsecond time frame for alpha(i2) complexes and tau(1/2)< or =4s for alpha(i3) complexes, time scales that are compatible with cell activation. beta(1)gamma(2) isotype complexes were generally more stable than beta(4)gamma(2)-associated complexes. The comparison of the three constructs, however, proved that the fast step was associated with the separation of receptor and G protein and that the dissociation of the ligand or of the alpha and betagamma subunits was slower. These results are compatible with a cell activation model involving G protein conformational changes rather than disassembly of Galphabetagamma heterotrimer.

Both ligand- and cell-specific parameters control ligand agonism in a kinetic model of g protein-coupled receptor signaling

G protein–coupled receptors (GPCRs) exist in multiple dynamic states (e.g., ligand-bound, inactive, G protein–coupled) that influence G protein activation and ultimately response generation. In quantitative models of GPCR signaling that incorporate these varied states, parameter values are often uncharacterized or varied over large ranges, making identification of important parameters and signaling outcomes difficult to intuit. Here we identify the ligand- and cell-specific parameters that are important determinants of cell-response behavior in a dynamic model of GPCR signaling using parameter variation and sensitivity analysis. The character of response (i.e., positive/neutral/inverse agonism) is, not surprisingly, significantly influenced by a ligand's ability to bias the receptor into an active conformation. We also find that several cell-specific parameters, including the ratio of active to inactive receptor species, the rate constant for G protein activation, and expression levels of receptors and G proteins also dramatically influence agonism. Expressing either receptor or G protein in numbers several fold above or below endogenous levels may result in system behavior inconsistent with that measured in endogenous systems. Finally, small variations in cell-specific parameters identified by sensitivity analysis as significant determinants of response behavior are found to change ligand-induced responses from positive to negative, a phenomenon termed protean agonism. Our findings offer an explanation for protean agonism reported in β₂--adrenergic and α_2A-adrenergic receptor systems.

G protein–coupled receptors (GPCRs) are transmembrane proteins involved in physiological functions ranging from vasodilation and immune response to memory. The binding of both endogenous ligands (e.g., hormones, neurotransmitters) and exogenous ligands (e.g., pharmaceuticals) to these receptors initiates intracellular events that ultimately lead to cell responses. We describe a dynamic model for G protein activation, an immediate outcome of GPCR signaling, and use it together with efficient parameter variation and sensitivity analysis techniques to identify the key cell- and ligand-specific parameters that influence G protein activation. Our results show that although ligand-specific parameters do strongly influence cell response (either causing increases or decreases in G protein activation), cellular parameters may also dictate the magnitude and direction of G protein activation. We apply our findings to describe how protean agonism, a phenomenon in which the same ligand may induce both positive and negative responses, may result from changes in cell-specific parameters. These findings may be used to understand the molecular basis of different responses of cell types and tissues to pharmacological treatment. In addition, these methods may be applied generally to models of cellular signaling and will help guide experimental resources toward further characterization of the key parameters in these networks.

Dimerization of VEGF receptors and implications for signal transduction: a computational study

Vascular endothelial growth factor (VEGF) is a potent cytokine involved in the induction of neovascularization. Secreted as a cysteine-linked dimer, it has two binding sites at opposite poles through which it may bind VEGF receptors (VEGFRs), receptor tyrosine kinases found on the surface of endothelial and other cells. The binding of a VEGF molecule to two VEGFR molecules induces transphosphorylation of the intracellular domains of the receptors, leading to signal transduction. The dominant mechanism of receptor dimerization is not clear: the receptors may be present in an inactive pre-dimerized form, VEGF binding first to one of the receptors, the second receptor then ideally located for dimerization; or VEGF may bind receptor monomers on the cell surface, which then diffuse and bind to available unligated receptor monomers to complete the activation. Both processes take place and one or other may dominate on different cell types. We demonstrate the impact of dimerization mechanism on the binding of VEGF to the cell surface and on the formation of active signaling receptor complexes. We describe two methods to determine which process dominates, based on binding and phosphorylation assays. The presence of two VEGF receptor populations, VEGFR1 and VEGFR2, can result in receptor heterodimer formation. Our simulations predict that heterodimers will comprise 10-50% of the active, signaling VEGF receptor complexes, and that heterodimers will form at the expense of homodimers of VEGFR1 when VEGFR2 populations are larger. These results have significant implications for VEGF signal transduction and interpretation of experimental studies. These results may be applicable to other ligand-receptor pairs, in particular PDGF.

The development of quantum dot calibration beads and quantitative multicolor bioassays in flow cytometry and microscopy

The use of fluorescence calibration beads has been the hallmark of quantitative flow cytometry. It has enabled the direct comparison of interlaboratory data as well as quality control in clinical flow cytometry. In this article, we describe a simple method for producing color-generalizable calibration beads based on streptavidin functionalized quantum dots. Based on their broad absorption spectra and relatively narrow emission, which is tunable on the basis of dot size, quantum dot calibration beads can be made for any fluorophore that matches their emission color. In an earlier publication, we characterized the spectroscopic properties of commercial streptavidin functionalized dots (Invitrogen). Here we describe the molecular assembly of these dots on biotinylated beads. The law of mass action is used to readily define the site densities of the dots on the beads. The applicability of these beads is tested against the industry standard, namely commercial fluorescein calibration beads. The utility of the calibration beads is also extended to the characterization surface densities of dot-labeled epidermal growth factor ligands as well as quantitative indicators of the binding of dot-labeled virus particles to cells.

Some mechanistic insights into GPCR activation from detergent-solubilized ternary complexes on beads

The binding of full and partial agonist ligands (L) to G protein-coupled receptors (GPCRs) initiates the formation of ternary complexes with G proteins [ligand-receptor-G protein (LRG) complexes]. Cyclic ternary complex models are required to account for the thermodynamically plausible complexes. It has recently become possible to assemble solubilized formyl peptide receptor (FPR) and beta(2)-adrenergic receptor (beta(2)AR) ternary complexes for flow cytometric bead-based assays. In these systems, soluble ternary complex formation of the receptors with G proteins allows direct quantitative measurements which can be analyzed in terms of three-dimensional concentrations (molarity). In contrast to the difficulty of analyzing comparable measurements in two-dimensional membrane systems, the output of these flow cytometric experiments can be analyzed via ternary complex simulations in which all of the parameters can be estimated. An outcome from such analysis yielded lower affinity for soluble ternary complex assembly by partial agonists compared with full agonists for the beta(2)AR. In the four-sided ternary complex model, this behavior is consistent with distinct ligand-induced conformational states for full and partial agonists. Rapid mix flow cytometry is used to analyze the subsecond dynamics of guanine nucleotide-mediated ternary complex disassembly. The modular breakup of ternary complex components is highlighted by the finding that the fastest step involves the departure of the ligand-activated GPCR from the intact G protein heterotrimer. The data also show that, under these experimental conditions, G protein subunit dissociation does not occur within the time frame relevant to signaling. The data and concepts are discussed in the context of a review of current literature on signaling mechanism based on structural and spectroscopic (FRET) studies of ternary complex components.

Computational modelling of ErbB family phosphorylation dynamics in response to transforming growth factor alpha and heregulin indicates spatial compartmentation of phosphatase activity

Members of the ErbB receptor family are associated with several cancers and appear to be providing useful targets for pharmacological therapeutics for tumours of the lung and breast. Further improvements of these therapies may be guided by a quantitative, dynamic integrative systems understanding of the complexities of ErbB dimerisation, trafficking and activation, for it is these complexities that render difficult intuiting how perturbations such as drug intervention will affect ErbB signalling activities. Towards this goal, we have developed a computational model implementing commonly accepted principles governing ErbB receptor interaction, trafficking, phosphorylation and dephosphorylation. Using this model, we are able to investigate several hypotheses regarding the compartmental localisation of dephosphorylation. Model results applied to experimental data on ErbB 1, ErbB2 and ErbB3 phosphorylation in H292 human lung carcinoma cells support a hypothesis that key dephosphorylation activity for these receptors occurs largely in an intracellular, endosomal compartment rather than at the cell surface plasma membrane. Thus, the endocytic trafficking-related compartmentalisation of dephosphorylation may define a critical aspect of the ErbB signalling response to ligand.

Kinetics of bimolecular reactions in model bilayers and biological membranes. A critical review

The quantitative study of the probability of molecular encounters giving rise to a reaction in membranes is a challenging discipline. Model systems, model in the sense that they use model bilayers and model reactants, have been widely used for this purpose, but the methodologies employed for the analysis of the results obtained in experiments, and for experimental design, are so disparate that a concerned experimentalist has difficulty in deciding about the value of each approach. This review intends to examine the several approaches that can be found in the literature showing, when feasible, the weakness, strengths and limits of application of each of them. There is not, so far, a full experimental validation of the most promising theories for the analysis of reactions in two dimensions, what leaves open a large field for new research. The major challenge resides in the time range in which the processes take place, but the possibilities of the existing techniques for these studies are far from exhausted. We review also the attempts of several authors to quantitatively analyze the kinetics of reactions in biological membranes. Especially in this field, the recently developed microspectroscopies enclose a still unexplored potential.

Quantitative model of Ras-phosphoinositide 3-kinase signalling cross-talk based on co-operative molecular assembly

In growth-factor-stimulated signal transduction, cell-surface receptors recruit PI3Ks (phosphoinositide 3-kinases) and Ras-specific GEFs (guanine nucleotide-exchange factors) to the plasma membrane, where they produce 3'-phosphorylated phosphoinositide lipids and Ras-GTP respectively. As a direct example of pathway networking, Ras-GTP also recruits and activates PI3Ks. To refine the mechanism of Ras-PI3K cross-talk and analyse its quantitative implications, we offer a theoretical model describing the assembly of complexes involving receptors, PI3K and Ras-GTP. While the model poses the possibility that a ternary receptor-PI3K-Ras complex forms in two steps, it also encompasses the possibility that receptor-PI3K and Ras-PI3K interactions are competitive. In support of this analysis, experiments with platelet-derived growth factor-stimulated fibroblasts revealed that Ras apparently enhances the affinity of PI3K for receptors; in the context of the model, this suggests that a ternary complex does indeed form, with the second step greatly enhanced through membrane localization and possibly allosteric effects. The apparent contribution of Ras to PI3K activation depends strongly on the quantities and binding affinities of the interacting molecules, which vary across different cell types and stimuli, and thus the model could be used to predict conditions under which PI3K signalling is sensitive to interventions targeting Ras.

Spatial modeling of dimerization reaction dynamics in the plasma membrane: Monte Carlo vs. continuum differential equations

Bimolecular reactions in the plasma membrane, such as receptor dimerization, are a key signaling step for many signaling systems. For receptors to dimerize, they must first diffuse until a collision happens, upon which a dimerization reaction may occur. Therefore, study of the dynamics of cell signaling on the membrane may require the use of a spatial modeling framework. Despite the availability of spatial simulation methods, e.g., stochastic spatial Monte Carlo (MC) simulation and partial differential equation (PDE) based approaches, many biological models invoke well-mixed assumptions without completely evaluating the importance of spatial organization. Whether one is to utilize a spatial or non-spatial simulation framework is therefore an important decision. In order to evaluate the importance of spatial effects a priori, i.e., without performing simulations, we have assessed the applicability of a dimensionless number, known as second Damköhler number (Da), defined here as the ratio of time scales of collision and reaction, for 2-dimensional bimolecular reactions. Our study shows that dimerization reactions in the plasma membrane with Da approximately >0.1 (tested in the receptor density range of 10(2)-10(5)/microm(2)) require spatial modeling. We also evaluated the effective reaction rate constants of MC and simple deterministic PDEs. Our simulations show that the effective reaction rate constant decreases with time due to time dependent changes in the spatial distribution of receptors. As a result, the effective reaction rate constant of simple PDEs can differ from that of MC by up to two orders of magnitude. Furthermore, we show that the fluctuations in the number of copies of signaling proteins (noise) may also depend on the diffusion properties of the system. Finally, we used the spatial MC model to explore the effect of plasma membrane heterogeneities, such as receptor localization and reduced diffusivity, on the dimerization rate. Interestingly, our simulations show that localization of epidermal growth factor receptor (EGFR) can cause the diffusion limited dimerization rate to be up to two orders of magnitude higher at higher average receptor densities reported for cancer cells, as compared to a normal cell.

Multiscale spatial Monte Carlo simulations: Multigriding, computational singular perturbation, and hierarchical stochastic closures

Monte Carlo (MC) simulation of most spatially distributed systems is plagued by several problems, namely, execution of one process at a time, large separation of time scales of various processes, and large length scales. Recently, a coarse-grained Monte Carlo (CGMC) method was introduced that can capture large length scales at reasonable computational times. An inherent assumption in this CGMC method revolves around a mean-field closure invoked in each coarse cell that is inaccurate for short-ranged interactions. Two new approaches are explored to improve upon this closure. The first employs the local quasichemical approximation, which is applicable to first nearest-neighbor interactions. The second, termed multiscale CGMC method, employs singular perturbation ideas on multiple grids to capture the entire cluster probability distribution function via short microscopic MC simulations on small, fine-grid lattices by taking advantage of the time scale separation of multiple processes. Computational strategies for coupling the fast process at small length scales (fine grid) with the slow processes at large length scales (coarse grid) are discussed. Finally, the binomial tau-leap method is combined with the multiscale CGMC method to execute multiple processes over the entire lattice and provide additional computational acceleration. Numerical simulations demonstrate that in the presence of fast diffusion and slow adsorption and desorption processes the two new approaches provide more accurate solutions in comparison to the previously introduced CGMC method.

Decreased internalisation of ErbB1 mutants in lung cancer is linked with a mechanism conferring sensitivity to gefitinib

A majority of gefitinib (IRESSA)-responsive tumours in non-small cell lung cancer have been found to carry mutations in ErbB1. Previously, it has been observed that internalisation-deficient ErbB1 receptors are strong drivers of oncogenesis. Using a computational model of ErbB1 trafficking and signalling, it is found that a deficiency in ErbB1 internalisation is sufficient to explain the observed signalling phenotype of these gefitinib-responsive ErbB1 mutants in lung cancer cell lines. Experimental tests confirm that gefitinib-sensitive cell lines with and without ErbB1 mutations exhibit markedly slower internalisation rates than gefitinib-insensitive cell lines. Moreover, the computational model demonstrates that reduced ErbB1 internalisation rates are mechanistically linked to upregulated AKT signalling. Experimentally it is confirmed that impaired internalisation of ErbB1 is associated with increased AKT activity, which can be blocked by gefitinib. On the basis of these experimental and computational results, it is surmised that gefitinib sensitivity is a marker of a reliance on AKT signalling for cell survival that may be brought about by impaired ErbB1 internalisation.

Computational modeling reveals molecular details of epidermal growth factor binding

BACKGROUND

The ErbB family of receptors are dysregulated in a number of cancers, and the signaling pathway of this receptor family is a critical target for several anti-cancer drugs. Therefore a detailed understanding of the mechanisms of receptor activation is critical. However, despite a plethora of biochemical studies and recent single particle tracking experiments, the early molecular mechanisms involving epidermal growth factor (EGF) binding and EGF receptor (EGFR) dimerization are not as well understood. Herein, we describe a spatially distributed Monte Carlo based simulation framework to enable the simulation of in vivo receptor diffusion and dimerization.

RESULTS

Our simulation results are in agreement with the data from single particle tracking and biochemical experiments on EGFR. Furthermore, the simulations reveal that the sequence of receptor-receptor and ligand-receptor reaction events depends on the ligand concentration, receptor density and receptor mobility.

CONCLUSION

Our computer simulations reveal the mechanism of EGF binding on EGFR. Overall, we show that spatial simulation of receptor dynamics can be used to gain a mechanistic understanding of receptor activation which may in turn enable improved cancer treatments in the future.

Kinetic modeling of Na(+)-induced, Gbetagamma-dependent activation of G protein-gated K(+) channels

G protein-activated K(+)(GIRK) channels are activated by numerous neurotransmitters that act on Gi/o proteins, via a direct interaction with the Gbetagamma subunit of G proteins. In addition, GIRK channels are positively regulated by intracellular Na(+) via a direct interaction (fast pathway) and via a GGbetagamma-dependent mechanism (slow pathway). The slow modulation has been proposed to arise from the recently described phenomenon of Na(+)-induced reduction of affinity of interaction between GalphaGDP and Gbetagamma subunits of G proteins. In this scenario, elevated Na(+) enhances basal dissociation of G protein heterotrimers, elevating free cellular Gbetagamma and activating GIRK. However, it is not clear whether this hypothesis can account for the quantitative and kinetic aspects of the observed regulation. Here, we report the development of a quantitative model of slow, Na(+)-dependent, G protein-mediated activation of GIRK. Activity of GIRK1F137S channels, which are devoid of direct interaction with Na(+), was measured in excised membrane patches and used as an indicator of free GGbetagamma levels. The change in channel activity was used to calculate the Na(+)-dependent change in the affinity of G protein subunit interaction. Under a wide range of initial conditions, the model predicted that a relatively small decrease in the affinity of interaction of GalphaGDP and GGbetagamma (about twofold under most conditions) accounts for the twofold activation of GIRK induced by Na(+), in agreement with biochemical data published previously. The model also correctly described the slow time course of Na(+) effect and explained the previously observed enhancement of Na(+)-induced activation of GIRK by coexpressed Galphai3. This is the first quantitative model that describes the basal equilibrium between free and bound G protein subunits and its consequences on regulation of a GGbetagamma effector.

Reactions on cell membranes: comparison of continuum theory and Brownian dynamics simulations

Biochemical transduction of signals received by living cells typically involves molecular interactions and enzyme-mediated reactions at the cell membrane, a problem that is analogous to reacting species on a catalyst surface or interface. We have developed an efficient Brownian dynamics algorithm that is especially suited for such systems and have compared the simulation results with various continuum theories through prediction of effective enzymatic rate constant values. We specifically consider reaction versus diffusion limitation, the effect of increasing enzyme density, and the spontaneous membrane association/dissociation of enzyme molecules. In all cases, we find the theory and simulations to be in quantitative agreement. This algorithm may be readily adapted for the stochastic simulation of more complex cell signaling systems.

Differential binding of VEGF isoforms to VEGF receptor 2 in the presence of neuropilin-1: a computational model

A comprehensive, biophysically accurate, computational model of vascular endothelial growth factor (VEGF) family member interactions with endothelial cell surface receptors was developed to study angiogenesis. Neuropilin-1 (NRP1) and the signaling VEGF receptor, VEGFR2, do not interact directly but are bridged by one VEGF isoform, VEGF165. Using the model and published experimental data, we estimated the kinetic rate of this VEGFR2-NRP1 coupling in vitro. With the use of this rate, our model gives predictions in good quantitative agreement with several independent in vitro experiments involving VEGF121and VEGF165isoforms, confirming that VEGFR2-NRP1 coupling through VEGF165can fully explain the observed differences in receptor binding and phosphorylation in response to these isoforms. Model predictions also determine the mechanism of action of a commonly used NRP1 antibody and predict the results of potential future experiments. This is the first model to include VEGF isoforms or NRPs, and it is a necessary step toward a quantitative molecular level description of VEGF that can be extended to in vivo situations. The model has applications for both proangiogenic and antiangiogenic therapies, such as for heart disease and cancer, as well as in tissue engineering.

Monte Carlo simulations of receptor dynamics: insights into cell signaling

Many receptor-level processes involve the diffusion and reaction of receptors with other membrane-localized molecules. Monte Carlo simulation is a powerful technique that allows us to track the motions and discrete reactions of individual receptors, thus simulating receptor dynamics and the early events of signal transduction. In this paper, we discuss simulations of two receptor processes, receptor dimerization and G-protein activation. Our first set of simulations demonstrates how receptor dimerization can create clusters of receptors via partner switching and the relevance of this clustering for receptor cross-talk and integrin signaling. Our second set of simulations investigates the activation and desensitization of G-protein coupled receptors when either a single agonist or both an agonist and an antagonist are present. For G-protein coupled receptor systems in the presence of an agonist alone, the dissociation rate constant of agonist is predicted to affect the ratio of G-protein activation to receptor phosphorylation. Similarly, this ratio is affected by the antagonist dissociation rate constant when both agonist and antagonist are present. The relationship of simulation predictions to experimental findings and potential applications of our findings are also discussed.

Spatial analysis of 3' phosphoinositide signaling in living fibroblasts: I. Uniform stimulation model and bounds on dimensionless groups

Fluorescent protein probes now permit spatial distributions of specific intracellular signaling molecules to be observed in real time. Mathematical models have been used to simulate molecular gradients and other spatial patterns within cells, and the output of such models may be compared directly with experiments if the binding of the fluorescent probe and the physics of the imaging technique are each incorporated. Here we present a comprehensive model describing the dynamics of 3' phosphoinositides (PIs), lipid second messengers produced in the plasma membrane in response to stimulation of the PI 3-kinase signaling pathway, as monitored in the cell-substratum contact area using total internal reflection fluorescence microscopy. With this technique it was previously shown that uniform stimulation of fibroblasts with platelet-derived growth factor elicits the formation of axisymmetric 3' PI gradients, which we now characterize in the context of our model. We find that upper and lower bounds on the relevant dimensionless model parameter values for an individual cell can be calculated from four well-defined fluorescence measurements. Based on our analysis, we expect that the key dimensionless group, the ratio of 3' PI turnover and diffusion rates, can be estimated within approximately 20% or less.

HER2-mediated effects on EGFR endosomal sorting: analysis of biophysical mechanisms

Overexpression of HER2, a receptor-like tyrosine kinase and signaling partner for the epidermal growth factor receptor (EGFR), has been implicated in numerous experimental and clinical studies as promoting the progression of many types of cancer. One avenue by which HER2 overexpression may dysregulate EGFR-mediated cell responses, such as proliferation and migration, downstream of EGF family ligand binding, is by its modulation on EGFR endocytic trafficking dynamics. EGFR signaling is regulated by downregulation and compartmental relocalization arising from endocytic internalization and endosomal sorting to degradation versus recycling fates. HER2 overexpression influences both of these processes. At the endosomal sorting stage, increased HER2 levels elicit enhanced EGFR recycling outcomes, but the mechanism by which this transpires is poorly understood. Here, we determine whether alternative mechanisms for HER2-mediated enhancement of EGFR recycling can be distinguished by comparison of corresponding mathematical models to experimental literature data. Indeed, we find that the experimental data are clearly most consistent with a mechanism in which HER2 directly competes with EGFR for a stoichiometrically-limited quantity of endosomal retention components (ERCs), thereby reducing degradation of ERC-coupled EGFR. Model predictions based on this mechanism exhibited qualitative trends highly similar to data on the fraction of EGF/EGFR complexes sorted to recycling fate as a function of the amount of internalized EGF/EGFR complexes. In contrast, model predictions for alternative mechanisms-blocking of EGFR/ERC coupling, or altering EGF/EGFR dissociation-were inconsistent with the qualitative trends of the experimental data.

Ligand-induced coupling versus receptor pre-association: cellular automaton simulations of FGF-2 binding

The binding of basic fibroblast growth factor (FGF-2) to its cell surface receptor (CSR) and subsequent signal transduction is known to be enhanced by heparan sulfate proteoglycans (HSPGs). HSPGs bind FGF-2 with low affinity and likely impact CSR-mediated signaling via stabilization of FGF-2-CSR complexes via association with both the ligand and the receptor. What is unknown is whether HSPG associates with CSR in the absence of FGF-2. In this paper, we determine conditions by which pre-association would impact CSR-FGF-2-HSPG triad formation assuming diffusion-limited surface reactions. Using mean-field rate equations, we show that (i) when [HSPG] is much higher than [CSR], the presence of pre-formed complexes does not affect the steady state of FGF-2 binding, and (ii) when the concentrations are comparable, the presence of pre-formed complexes substantially increases the steady-state concentration of FGF-2 bound to CSR. These findings are supported by explicit cellular automaton simulations, which justify the mean-field treatment. We discuss the advantages of such a two-receptor system compared to a single-receptor model, when the parameters are comparable. Further, we speculate that the observed high concentration of HSPG in intact cells ([HSPG]-100[CSR]) provides a way to ensure that the binding levels of FGF-2 to its signaling receptor remains high, irrespective of the presence of pre-formed CSR-HSPG complexes on the cell surface, while allowing the cell to finely tune the response to FGF-2 via down-regulation of the signaling receptor.

Quantitative analysis of HER2-mediated effects on HER2 and epidermal growth factor receptor endocytosis: distribution of homo- and heterodimers depends on relative HER2 levels

Endocytic trafficking plays an important role in the regulation of the epidermal growth factor receptor (EGFR) family. Many cell types express multiple EGFR family members (including EGFR, HER2, HER3, and/or HER4) that interact to form an array of homo- and heterodimers. Differential trafficking of these receptors should strongly affect signaling through this system by changing substrate access and heterodimerization efficiency. Because of the complexity of these dynamic processes, we used a quantitative and computational model to understand their integrated operation. Parameters characterizing EGFR and HER2 interactions were determined using experimental data obtained from mammary epithelial cells constructed to express different levels of HER2, enabling us to estimate receptor-specific internalization rate constants and dimer uncoupling rate constants. Significant novel results obtained from this work are as follows: first, that EGFR homodimerization and EGFR/HER2 heterodimerization occur with comparable affinities; second, that EGFR/HER2 heterodimers traffic as single entities. Furthermore, model predictions of the relationship of HER2 expression levels to consequent distribution of EGFR homodimers and EGFR/HER2 heterodimers suggest that the levels of HER2 found on normal cells are barely at the threshold necessary to drive efficient heterodimerization. Thus, altering HER2 concentrations, either overall or local, could provide an effective mechanism for regulating EGFR/HER2 heterodimerization and may explain why HER2 overexpression found in some cancers has such a profound effect on cell physiology.

Monte carlo simulations of enzyme reactions in two dimensions: fractal kinetics and spatial segregation

Conventional equations for enzyme kinetics are based on mass-action laws, that may fail in low-dimensional and disordered media such as biological membranes. We present Monte Carlo simulations of an isolated Michaelis-Menten enzyme reaction on two-dimensional lattices with varying obstacle densities, as models of biological membranes. The model predicts that, as a result of anomalous diffusion on these low-dimensional media, the kinetics are of the fractal type. Consequently, the conventional equations for enzyme kinetics fail to describe the reaction. In particular, we show that the quasi-stationary-state assumption can hardly be retained in these conditions. Moreover, the fractal characteristics of the kinetics are increasingly pronounced as obstacle density and initial substrate concentration increase. The simulations indicate that these two influences are mainly additive. Finally, the simulations show pronounced S-P segregation over the lattice at obstacle densities compatible with in vivo conditions. This phenomenon could be a source of spatial self organization in biological membranes.

Drug efficacy at G protein-coupled receptors

Efficacy has been defined in receptor pharmacology as a proportionality factor denoting the amount of physiological response a given ligand imparts to a biological system for a given amount of receptor occupancy. While first defined in terms of response, the concept can be expanded to a wide variety of G protein-coupled receptor (GPCR) behaviors, which includes pleiotropic interaction with multiple G proteins, internalization, oligomerization, desensitization, and interaction with membrane auxilliary proteins. Thus, there can be numerous types of efficacy, and different ligands can have a range of efficacies for different receptor behaviors. This review discusses the use of the efficacy concept in GPCR models based on the thermodynamic linkage theory and also in terms of the protein ensemble theory, in which macroaffinity of ligands for an ensemble of receptor microstates produces a new ligand-bound ensemble. The pharmacological characteristics of the ligand emerge from the intersection of the ligand-bound ensemble with the various ensembles defining pharmacological receptor behaviors. Receptor behaviors discussed are activation of G proteins; ability to be phosphorylated, desensitized, and internalized; formation of dimers and oligomers; and the interaction with auxiliary membrane and cytosolic proteins. The concepts of ligand-specific receptor conformation and conditional efficacy are also discussed in the context of ligand control of physiological response.

A unified model for signal transduction reactions in cellular membranes

An analytical solution is obtained for the steady-state reaction rate of an intracellular enzyme, recruited to the plasma membrane by active receptors, acting upon a membrane-associated substrate. Influenced by physical and chemical effects, such interactions are encountered in numerous signal-transduction pathways. The generalized modeling framework is the first to combine reaction and diffusion limitations in enzyme action, the finite mean lifetime of receptor-enzyme complexes, reactions in the bulk membrane, and constitutive and receptor-mediated substrate insertion. The theory is compared with other analytical and numerical approaches, and it is used to model two different signaling pathway types. For two-state mechanisms, such as activation of the Ras GTPase, the diffusion-limited activation rate constant increases with enhanced substrate inactivation, dissociation of receptor-enzyme complexes, or crowding of neighboring complexes. The latter effect is only significant when nearly all of the substrate is in the activated state. For regulated supply and turnover pathways, such as phospholipase C-mediated lipid hydrolysis, an additional influence is receptor-mediated substrate delivery. When substrate consumption is rapid, this process significantly enhances the effective enzymatic rate constant, regardless of whether enzyme action is diffusion limited. Under these conditions, however, enhanced substrate delivery can result in a decrease in the average substrate concentration.

Bioengineering models of cell signaling

Strategies for rationally manipulating cell behavior in cell-based technologies and molecular therapeutics and understanding effects of environmental agents on physiological systems may be derived from a mechanistic understanding of underlying signaling mechanisms that regulate cell functions. Three crucial attributes of signal transduction necessitate modeling approaches for analyzing these systems: an ever-expanding plethora of signaling molecules and interactions, a highly interconnected biochemical scheme, and concurrent biophysical regulation. Because signal flow is tightly regulated with positive and negative feedbacks and is bidirectional with commands traveling both from outside-in and inside-out, dynamic models that couple biophysical and biochemical elements are required to consider information processing both during transient and steady-state conditions. Unique mathematical frameworks will be needed to obtain an integrated perspective on these complex systems, which operate over wide length and time scales. These may involve a two-level hierarchical approach wherein the overall signaling network is modeled in terms of effective "circuit" or "algorithm" modules, and then each module is correspondingly modeled with more detailed incorporation of its actual underlying biochemical/biophysical molecular interactions.