Dynamic Transition States of ErbB1 Phosphorylation Predicted by Spatial Stochastic Modeling

Abstract

ErbB1 overexpression is strongly linked to carcinogenesis, motivating better understanding of erbB1 dimerization and activation. Recent single-particle-tracking data have provided improved measures of dimer lifetimes and strong evidence that transient receptor coconfinement promotes repeated interactions between erbB1 monomers. Here, spatial stochastic simulations explore the potential impact of these parameters on erbB1 phosphorylation kinetics. This rule-based mathematical model incorporates structural evidence for conformational flux of the erbB1 extracellular domains, as well as asymmetrical orientation of erbB1 cytoplasmic kinase domains during dimerization. The asymmetric dimer model considers the theoretical consequences of restricted transactivation of erbB1 receptors within a dimer, where the N-lobe of one monomer docks with the C-lobe of the second monomer and triggers its catalytic activity. The dynamic nature of the erbB1 phosphorylation state is shown by monitoring activation states of individual monomers as they diffuse, bind, and rebind after ligand addition. The model reveals the complex interplay between interacting liganded and nonliganded species and the influence of their distribution and abundance within features of the membrane landscape.

Introduction

ErbB1 (epidermal growth factor receptor) is the canonical member of the erbB receptor family (1) and a critical player in normal growth and development, as well as carcinogenesis (1). ErbB1 signaling is initiated by ligand-induced homo- and heterodimerization mediated primarily by engagement of extracellular dimerization arms (2). Structural evidence also suggests that the erbB1 extracellular domain fluctuates between the closed and open conformation in the absence of ligand (2), transiently exposing the erbB1 dimerization arm and permitting transient preformed dimers to occur (3). In a previous work, we used spatial stochastic modeling to predict the impact of receptor density, through local receptor trapping in membrane domains or receptor overexpression, on the rate of preformed dimers (4). The ability of nonligand-bound erbB1 monomers to partner with each other and with ligand-bound monomers leads to a complex mix of dimer configurations. Once dimers form, the signal is propagated by activation of integral tyrosine kinase activity in the receptor cytoplasmic tail, transphosphorylation of tyrosine residues in receptor tails, and recruitment of cytosolic signaling partners (1). Both deterministic and stochastic mathematical models have been developed to consider the complexity of erbB1 signaling, with successive generations of erbB1 models building on ever richer data sets for binding kinetics, phosphorylation/dephosphorylation dynamics, and adaptor recruitment (4–12).

Not yet considered in mathematical models is the asymmetrical docking and activation of erbB1 cytoplasmic kinase domains, which accompanies extracellular domain dimer formation (13–16). In an asymmetric dimer, the N-terminal lobe of one kinase domain in the dimer pair interacts with the C-lobe of the other (13). Mutagenesis and biochemical studies support an unusual transactivation model, where activation of catalytic activity is restricted to the monomer whose C-lobe has been engaged. Thus, one monomer in the dimer pair is considered to be the receiver and the monomer contributing the N-lobe is considered to be the activator. A novel aspect of this study is the consideration of restrictions that asymmetrical docking theoretically imposes upon ErbB transphosphorylation into the spatial stochastic model, taking advantage of the flexibility of the model’s rule-based framework.

Our model also builds on improved measures of erbB1 diffusive behavior and dimerization kinetics, made possible through remarkable advances in single-particle-tracking (SPT) methodology (17). This recent study by Low-Nam et al. (17) provides important new parameters for the spatial stochastic model. Among these values are the differential lifetimes of dimer pairs, based upon the occupancy of the ligand-binding site in each monomer. For example, the authors showed that dimer pairs comprised of two ligand-bound monomers have the longest lifetimes, compared to lifetimes of pairs comprised of one ligand-bound and one unliganded monomer or two unliganded monomers (17). In addition, data from SPT experiments provided strong evidence for repeated interactions between two receptors while coconfined in specialized features of the plasma membrane, referred to as membrane domains or corrals. Since SPT relies on sparse labeling and captures only a minute fraction of receptor dimer events, an important aspect of the spatial model presented here is the explicit consideration of the impact of these new measurements on population dynamics. The spatial model also yields new (to our knowledge) insight into the activation states of individual monomers after ligand addition, as they cycle through rounds of dimerization, asymmetrical kinase activation, and phosphorylation/dephosphorylation.

Materials and Methods

Mathematical modeling

Detailed descriptions of the three-module platform are presented in Fig. S1 in the Supporting Material. In brief, the first module is a Matlab-based, image-processing step, since the simulation space is initialized from a micrograph of cell membrane sheets that have been immunogold-labeled for the receptor of interest and imaged by transmission electron microscopy (TEM) (18). These data provide an estimate of the number of receptors and their initial membrane distribution. Confinement zones are estimated from the area occupied by receptor clusters. The second module is the Brownian motion and reaction simulator, which uses a modified Smoluchowski model (19) and is executed in FORTRAN. Postprocessing of simulated trajectories, protein-protein interactions, and receptor activation state is performed in a third module in Matlab.

Decisions for each receptor over the simulation time course are rule-based. At each time step, individual monomers can diffuse or react. If unrestricted by domains, receptors freely diffuse. Receptors also freely enter domains, with defined values for restriction from exit from domain boundaries (Table 1). Escape probabilities are determined through fitting, to arrive at similar receptor cluster distributions at any point during the simulation process; this parameter was validated by the Hopkins spatial statistic (4,20) and by comparison of jump-size distributions with experimental values from SPT (17). Four reactions are possible: dimerization, dissociation, phosphorylation, and dephosphorylation. After motion, each monomer’s position is scanned for other receptors within the binding radius; dimerization likelihood within this radius is based upon a modification of the Smoluchowski approach (19) and calculated from dimer estimates in Martin-Fernandez et al. (21). Dimer dissociation is implemented through a probability calculated using the dimer off rate and the time step since binding. When dimers dissociate, the monomers are assigned an unbinding radius to minimize the occurrence of an unrealistic amount of repeated interactions. These binding and unbinding radii take into account the kinetics and diffusion of the receptors. Using the binding and unbinding radius as the inner and outer bound, respectively, of this region minimizes instantaneous rebinding (19). Computational time for these values is minimized by use of look-up tables (19). Phosphorylation occurs only during dimerization intervals and is based upon a rule where one monomer in the dimer is assigned at random as the activator and the other monomer in the dimer is the receiver. Dephosphorylation occurs at the same rate irrespective of monomer or dimer state (Table 1). Equations and mathematical methodologies are available in the Supporting Material.

Table 1.

Model parameters

Species	Diffusion coefficient (μm2/s)a	Dimer on rate (μm3/s) (∗1E-04)c,d	Dimer off rate (1/s)a	Domain exit rate (1/frame)a	Phos. rate (1/s)b	Dephos. rate (1/s)c
LR	0.0512	—	—	0.0121	—	—
LRP	0.0512	—	—	0.0121	—	0.13
R	0.0512	—	—	0.0183	—	—
RP	0.0512	—	—	0.0183	—	0.13
LRLR	0.0191	0.9138	0.273	0.00874	0.0733	—
LRR	0.0191	0.9138	0.738	0.00874	0.0733	—
RR	0.0191	0.9138	1.24	0.00874	0.0733	—
LRPLR	0.0191	0.9138	0.273	0.00874	0.0733	0.13
LRPR	0.0191	0.9138	0.738	0.00874	0.0733	0.13
RPR	0.0191	0.9138	1.24	0.00874	0.0733	0.13
LRPLRP	0.00563	0.9138	0.273	0.00874	0.0733	0.13
LRPRP	0.00563	0.9138	0.738	0.00874	0.0733	0.13
RPRP	0.00563	0.9138	1.24	0.00874	0.0733	0.13

Set of experimental parameters used for each of the species during simulations.

^aValues taken from Low-Nam et al. (17).

^bValues back-calculated using SMOLDYN (19).

^cValues taken from Kleiman et al. (10).

^dValues in this column should be multiplied by 10^-4.

Single-particle tracking

Detailed methods for tracking and analyzing erbB1 motion are described in Low-Nam et al. (17). In brief, erbB1 receptors were tracked with two-color quantum dots (585 and 685 QDs, Molecular Probes, Eugene, OR) conjugated with either VHH monomeric antibody fragments (noncompeting with ligand) or with epidermal-growth-factor -conjugated QDs. A431 breast cancer cells were serum starved for a minimum of 2 h and observed on an Olympus IX71 inverted microscope equipped with a 60× 1.3 NA water objective and an electron multiplying CCD camera (iXon 887, Andor Technology, Belfast, United Kingdom). Samples were maintained at 34–36°C by an objective heater (Bioscience Tools, San Diego, CA). QD probes were applied at picomolar concentrations to achieve sparse labeling required for SPT. A three-state Hidden Markov model was used to identify transition rates between two distant monomers (free), coconfined pairs, and dimerized receptors. From these rates, the states of receptor pairs in the raw data could be extracted.

Results and Discussion

In this work, we began with modifications to our existing spatial stochastic model (22). Receptors are represented as discrete particles and move through a 2D simulation space with Brownian motion and under periodic boundary conditions. An improvement in the model is the use of modified Smoluchowski dynamics to govern reactions, as described in Materials and Methods. Similar to that applied by Hsieh et al. (4,22), this simulation approach follows the molecular transformations and Brownian motion of individual particles; however, each dimerization and dissociation reaction type is implemented using a single, precalculated geometric parameter, yielding both faster execution and increased physical accuracy. Additional details, including a schematic of the overall framework, are found in Fig. S1 A in the Supporting Material.

Dimerization reactions in this simulation environment are diffusion-limited. Individual particles move independently and randomly at each time step, with normally distributed jump sizes. At the end of each move, a scan of the surrounding area within a defined radius of the particle determines where a binding event will occur. This binding radius was based upon simulations that reproduce results of Martin-Fernandez et al. (21) and takes into account measured reaction rates and diffusion coefficients (17), with 1 μs simulation time steps. The model tracks all particles in the simulation at every step. Instead of monitoring the binding of ligands, these simulations are initiated with a predetermined percentage of ligated receptors as a simplification strategy. Table 1 summarizes the experimental values for dimer off rates, phosphorylation, and dephosphorylation used to calculate the probability of events occurring at each time step in the simulation.

The conformational states of erbB1 are specifically represented by dimerization rules in the model, as illustrated schematically in Fig. 1 A. In the absence of bound ligand, receptors (R) are presumed to predominantly assume the bent state, with a 1% probability at each time step of fluxing to the open state that exposes the dimerization arm (3,4). Ligand-bound receptors (LR) are assumed to be in the extended conformation as long as ligand is bound. Thus, there are three possible types of erbB1 homodimers: two ligand-bound receptors (LRLR), one ligand-bound receptor and one nonliganded receptor (LRR), and two nonliganded receptors (RR), the latter representing the preformed dimer state.

Figure 1.

ErbB1 species, simulation space, diffusion, and off-rate validation. (A) Monomer and dimer species accounted for in the spatial stochastic model. R is a resting monomer with no ligand bound. Resting receptors spend 99% of simulation time in a tethered conformation, with 1% probability of flux to the extended conformation. LR is a ligand-bound monomer and is stabilized in the open conformation. RR is a preformed dimer, formed by encounters between two monomers in the open conformation. LRR is comprised of one unliganded monomer and one liganded monomer. The LRLR dimer is comprised of two ligand-bound monomers. (B) TEM image used to initialize the starting positions of erbB1 receptors and estimate size and density of confinement zones. (C) Simulation interpretation of the TEM image, including static confinement zones in black boxes. (D) Sample trajectory of three different receptors over a 4 min simulation. (E) Monomer diffusion coefficient calculated from simulation data. Simulation diffusion coefficients match the diffusion coefficients from SPT experiments. (F) Histograms of dimer lifetimes for 2:2 dimers. Each histogram is fit to determine the specific dimer off rate. The red line is the simulation data fit and the blue line is the experimental data fit (17).

An important feature of the model is the introduction of membrane domains that transiently confine receptors. Fig. 1, B and C, illustrates how the area and distribution of domains are initialized based upon immunogold labeling of erbB1 decorating the membrane of A431 breast cancer cells. As described in Materials and Methods, both domain location and receptor density are imported directly from EM images through a graphical user interface (Fig. S1 B). By limiting the probability of exit from domains, receptors are confined within the domains for discrete periods but explore much of the membrane landscape over a period of seconds to minutes (Fig. 1 D). Diffusion coefficients used in the model are based upon SPT measurements (17). ErbB1 monomers are assigned the fast diffusion rate of 0.0512 μm²/s for unconfined receptors, slowing to 0.0191 μm²/s upon dimerization. Fully phosphorylated dimers further slow to 0.00563 μm²/s, approximating the slowdown attributed to assembly of docking partners and remodeling of the local environment (17). The diffusion rates for unphosphorylated dimers were based upon tracking of erbB1 dimers in the presence of the kinase inhibitor PDI53035 (17). It is noteworthy that we did not assign the slowest diffusion rate to partially phosphorylated dimers, which could also slow further when recruiting docking partners, based on comparisons indicating that implementation of further slowdown had no significant impact on the results.

Fig. 1 E summarizes the spread of jump sizes for receptors diffusing and dimerizing within the domain-studded simulation landscape, reported as a cumulative probability analysis plot. This analysis compares favorably with cumulative probability analysis plots generated from SPT data for erbB1 bound to QD probes (17). Fig. 1 F shows that the distribution of lifetimes for simulated dimers also closely matches experimental data. The model thus captures the essential features of anomalous diffusion, as well as the stochastic nature of dimer dissociation, observed for erbB1 receptors in living membranes.

Membrane domains promote repeated interactions between monomer pairs

Fig. 2 A illustrates the reproducible observation that pairs of erbB1 monomers, tracked with two colors of QD epidermal growth factor, can bind and rebind multiple times during live-cell imaging. This characteristic behavior has been attributed to coconfinement, based upon the unlikely probability of repeat encounters if dissociated monomers diffuse rapidly away from their original contact site (17). We tested this notion by examining the trajectories and binding events between receptors in the spatial stochastic model, using a simulation space with membrane domains and 50% ligand-bound receptors. Representative results are shown in Fig. 2 B, where two receptors interact multiple times during a 50-s simulation.

Figure 2.

Membrane domains influence repeat interactions between receptors. (A) Separation distance over time between two QD-labeled receptors during an SPT experiment. The receptors, initially in a dimer state, dissociate and redimerize several times, as indicated by the state line overlay. (B) Separation distance over time between two ligand-bound receptors during a simulation demonstrating the same pattern of repeat interactions. (C) Summary of repeated interactions over an entire simulation for all possible receptor pairs. A few individual receptors interact with one another >100 times during a single 4 min simulation. (D) Times between rebinding interactions of two receptors are shown for LRLR dimers. Many rebinding interactions occur below the frame rate, 20 frames/s, used in SPT experiments (17) (arrow). Although most rebinding incidents occur within 50 s, time to rebinding can occur >150 s later.

Fig. 2 C reports results of this analysis applied to the entire population of receptors in the simulation space over a 4 min time course. The number of repeat interactions between each pair of receptors varies broadly, with a high value of 141 binding interactions between a given pair.

Another prediction arising from these simulations is the average time to rebinding. A large number of rebinding reactions (28%) occur within 0.05 s (Fig. 2 D, arrow), which is equivalent to the frame rate of the data collection in Low-Nam et al. (17). Since simulation results are analyzed with millisecond resolution, this suggests that the number of repeated encounters may be underrepresented during image acquisition.

Implications of the asymmetric model for receptor transphosphorylation

We next consider the implications of asymmetric kinase orientation within erbB1 dimers. The cartoon in Fig. 3 A illustrates the basic scheme used to create rules for transphosphorylation when kinase activation is restricted to only one monomer in a given pair. Here, the N-lobe of the activator monomer is in contact with the C-lobe of the receiver monomer. We make the theoretical assumption that the now active receiver then transphosphorylates its partner; the probability of this enzymatic modification is a function of the dimer lifetime for the pair.

Figure 3.

Impact of asymmetric receptor phosphorylation. (A) A model for receptor phosphorylation shuffle. When a dimer forms, due to the configuration of the N and C lobes, only one tail of the dimer can be phosphorylated. For a dually phosphorylated dimer to occur, a phosphorylated receptor and an unphosphorylated receptor must dimerize in the correct configuration such that the unphosphorylated receptor is phosphorylated. (B) Example of a ligand-bound receptor state over a 4 min simulation (upper) and the average percentage of time spent in each state for all of the ligand-bound receptors during the simulation (lower). Ligand-bound receptors spend the majority of time in the dimer state. (C) Sample receptor state of a nonligand-bound receptor (upper) and the average percentage of time spent in each state for all of the nonligand-bound receptors during the simulation (lower). Nonligand-bound receptors also spend a large fraction of time in the dimer state, but they are also found to be in the monomer state much more often than are ligand-bound receptors. (D) Phosphorylation state of nonliganded and liganded species, independent of receptor state. Fewer than 40% of nonliganded species are phosphorylated, on average, compared to the almost 60% of phosphoylated liganded species. (E) Percentage of LRLR dimers in different phosphorylation states. LRLR is an unphosphorylated dimer (blue), LRPLR is a singly phosphorylated dimer (green), and LRPLRP is a dually phosphorylated dimer (red). A quasi-steady state is reached in the first minute of the simulation.

This fundamental premise leads to an interesting prediction: the dissociation and rebinding of dimers in a stochastic process improves the likelihood that each erbB1 monomer has the opportunity to be both receiver and activator. The predicted outcome of this receptor shuffle process is illustrated in Fig. 3 B in the context of a simulation with 50% of receptors bound to ligand at the onset. The graph traces the transition states of a single ligand-bound erbB1 receptor in the simulation space over 250 s. Collectively, the ligand-bound receptors in this simulation achieved the dimer state ∼90% of the time. The predominant dimer type is LRR, due to an equilibrium shift from equal amounts of available LR and R monomers to an equilibrium that favors R monomers (see Fig. S2). Receptors cycle rapidly through all possible dimerization and phosphorylation states, spending 58% of the time as a phosphorylated species (Fig. 3 D). Movie S1 illustrates the rapid succession of transition states for a single ligand-bound erbB1 during this type of simulation.

In contrast, Fig. 3 C tracks the transition states of an unliganded receptor in the same simulation. The unliganded receptors in this simulation participated in dimer events frequently, spending only 9% of the simulation period as free monomers. However, due to the short dimer lifetimes for RR and LRR, only 35% of unliganded receptors are phosphorylated on average (Fig. 3 D).

To reconcile transient interactions with sustained signaling, we next analyzed the potential for accumulation of phosphorylated dimers over the same stochastic simulation time course (Fig. 3 E). At early times, the asymmetric model predicts that the predominant dimer state is LRPLR, where only one erbB1 monomer is phosphorylated. Dimers achieving phosphorylation of both liganded monomers (LRPLRP) reach similar levels with a short delay (Fig. 3 E, red traces). We conclude that rapid receptor re-encounters permit the system to quickly reach equilibrium, providing a significant pool of phosphorylated receptors for recruitment of signaling partners.

In Fig. 4, we compare steady-state phosphorylation for liganded (LRP) and unliganded (RP) receptors over a range of ligand doses and for two different receptor densities. Results for the high-density situation are shown in Fig. 4 A, again for A431 cells where the erbB1 gene is amplified and there are an estimated 4 million receptors/cell. At low ligand doses (10–20% occupancy), between 30% and 40% of the phosphorylated species are unliganded receptors (RP) that interacted with liganded receptors (LRP). As the ligand dose increases, the ratio drops dramatically without raising the overall levels of phosphorylation. The failure to achieve 100% phosphorylation is due to the combined effects of phosphatase activity and the lower availability of free monomers. Plots in Fig. 4 B report ratios of phosphorylated species where erbB1 expression levels were more normal, at 30,000 receptors/cell. In this case, the simulation landscape was initialized with erbB1 receptor distributions acquired from immunogold-labeled Hec50 cells (see Fig. 5 C). At the lowest doses of ligand (10–30%) occupancy, almost 50% of the phosphorylated species are unliganded receptors. We attribute this to the lower availability of liganded monomers in the sparsely populated membrane. These results offer insight into the lateral propagation hypothesis of Bastiaens and colleagues (23) and the observations that 1:2 dimers are signaling competent (24). They suggest that initiation of a global response by low doses of ligand is unlikely when the fast dephosphorylation rates measured by Kleiman et al. (10) are coupled with fast off rates for LRR (17).

Figure 4.

Ratio of liganded (LRP) to unliganded (RP) phosphorylated receptors. (A) Percentage of phosphorylated LR and R for increasing amounts of receptor ligand occupancy for A431 cells. Initially, at low levels of ligand-bound receptor, the percentage of phosphorylated nonligand-bound receptors increases. As liganded receptor percentage increases, unliganded receptor phosphorylation decreases. (B) Percentage of phosphorylated LR and R for increasing amounts of receptor ligand occupancy for HEC50 cells. A similar trend of LRP and RP is seen for HEC50, although the sparseness of the receptors on the membrane creates a larger deviation between simulations.

Figure 5.

Membrane landscape impacts receptor state. (A–D) Receptor density and distribution for simulations. (A) Initial simulation space imported from an immunogold-labeled EM image of an A431 cell. Static confinement zones based on receptor cluster size are included. (B) Randomized distribution of the same number of receptors as in A after diffusion simulations in the absence of confinement zones. (C) Initial simulation space imported from an immunogold-labeled EM image of an HEC50 cell. Static confinement zones based on receptor cluster size are included. (D) Randomized distribution of the same number of receptors as in C after diffusion simulations in the absence of confinement zones. (E) Receptor state for simulation conditions represented in the matching simulation space and 0% ligand-bound receptors. The presence of domains impacts how often receptors will encounter one another. The simulations with domains present have a higher rate of dimer occurrence, as well as a large number of phosphorylated receptor species. (F) Receptor state for simulation conditions of 10% ligand-bound receptors and corresponding simulation space. Similar to the 0%-ligand-bound simulations, the occurrence of dimers and phosphorylated species is increased in the presence of domains. The presence of ligand also allows for the formation of species not present in 0%-ligand-bound receptor simulations. (G and H) Number of phosphorylated species present on average during a simulation, scaled to whole-cell values, for A431 cells (G) and HEC50 cells (H). The presence of domains has a clear impact on the number of phosphorylated species. (I) Repeated interactions of receptors on HEC50 cells with 10% ligand-bound receptors present. Simulations with domains and without domains were performed.

Membrane landscape impacts receptor state

Our next goal was to evaluate the impact of membrane domains and receptor density upon phosphorylation efficiency, integrating both the improved dimer-lifetime measurements and the asymmetric model. Results are shown in Fig. 5, A–D, where the images illustrate the four different conditions initialized into the simulation landscape. We first compared the impact of domains upon the rate of so-called preformed dimers that occur in the absence of ligand. These events rely on encounters between monomers that have both randomly fluxed to the extended conformation; since each monomer is assumed to flux at a rate of 1%, there is a 0.01% probability for dimerization at each encounter. Results are compared for ligandless erbB1 diffusing and becoming transiently trapped in domains versus the same number of receptors diffusing with unrestricted Brownian motion. Results show that domains are particularly influential on the predicted levels of preformed dimers. As described above, the binding radius was parameterized based upon the observations of Martin-Fernando et al. (21), who estimated steady-state levels of preformed dimers on A431 cells at 14%. Thus, as expected, simulations are consistent with this value in the corresponding domain landscape (Fig. 5 E). When domains are removed, dimerization reaches levels of only 2%. At normal receptor density, represented by Hec50 cells, up to 10% percent of receptors achieve dimerization in the domain-studded landscape. Remarkably, dimerization of unliganded receptors at this lower density is a very rare event in the absence of domains, with estimates below 0.2%.

Fig. 5 F next compares the dimerization frequencies where 10% of the receptors are ligand-bound and subject to the same four initial conditions (high density with and without domains, and normal density with and without domains). Results again demonstrate the potential influence of domains, which is particularly impactful on the rate of dimer formation at normal receptor expression levels.

Results in Fig. 5, G and H, illustrate the relative impact of domains and receptor density on signaling output, represented by the number of receptors predicted to be phosphorylated at steady state. In the case of high receptor density (Fig. 5 G), up to 3% (∼150,000) of receptors are phosphorylated in the absence of ligand on A431 membranes with domains. At 10% ligand occupancy, this value rises dramatically to an estimated 1 million phosphorylated receptors. In the absence of domains, these estimates drop to 18,000 and 587,000 phosphorylated receptors, respectively.

For the case of Hec50 cells with normal erbB1 receptor density (Fig. 5 H), predicted values of phosphorylation attributed to preformed dimers are modest even in the presence of domains, at only 1400 phosphorylated receptors. Without domains, receptor phosphorylation of ligandless receptors is exquisitely low (17 total). Values in the case of 10% ligand occupancy are also reported in Fig. 5 H, with 8500 phosphorylated receptors in the domain landscape and only 1200 in the absence of domains.

As a final demonstration of the impact of domains, Fig. 5 I compares the predicted frequencies of repeated interactions in Hec50 membranes with (left) and without (right) domains at 10% ligand occupancy. Repeated interactions occur often when receptors are coconfined, even at this low density of receptors. In the absence of domains, repeated interactions between the same pair of receptors are much more rare events.

Concluding Remarks

Dimerization is a key event for many growth factor receptors, including erbB1 and its closely related family members (25). Previous work by us and others have established that erbB1 dimerization is rapidly reversible (17,26,27), leaving open important questions regarding the sustainability of signaling. Here we specifically consider receptor dimerization as a diffusion-limited process, with an emphasis on the impact of receptor coconfinement in plasma membrane domains or rafts. Our approach is based upon mathematical modeling, using a spatial stochastic framework that incorporates the concepts of membrane domains. This approach was validated by its close approximation of receptor diffusion characteristics, including the range of jump distributions measured by SPT. In addition to experimentally determined diffusion behavior, model parameters for dimer dissociation and phosphorylation/dephosphorylation are estimated from quantitative measurements in live cells (10,17,22). As suggested in our earlier work (4,12), simulations confirm that transient receptor domain confinement can effectively raise local receptor density and enhance the likelihood for productive receptor encounters.

This work has strong implications for the field of membrane biology, where the influence of receptor clustering remains a matter of considerable debate. ErbB1 and its family members are among the best studied examples of plasma membrane nanoclustering, with evidence for erbB1 homoclustering in resting cells from a wide variety of techniques, including EM (4,18,22), scanning near-field optical microscopy (28), homo-fluorescence-resonance energy transfer (29), cross-correlation (30,31), proximity ligation assay (32), multispectral plasmon coupling microscopy (33), number and brightness (34), and single-molecule techniques (17,35,36). The phenomenon of membrane protein clustering crosses many cell types. A partial list of examples includes MHC molecules (37,38), C-type lectins and viral proteins (39,40), TCR, BCR, and Fc receptors (41–43), CD36 scavenger receptors (44), and GPI-anchored proteins (45,46).

The observation of nanometer-scale proximity of membrane proteins, typically from microscopy methods, is sometimes interpreted as a reflection of oligomerization state. Here, we do not make the assumption that clusters observed by immunoelectron microscopy are accurate reporters of the oligomeric state of erbB1. Rather, we assume that these images capture a mix of nonrandom receptor distributions that principally result from monomers diffusing in and out of membrane domains. Productive encounters between monomers can lead to formation of dimers. Since we have yet to experimentally observe or quantify larger erbB1 oligomers (47) with SPT, we do not explicitly consider that interesting possibility here.

There is evidence that membrane domains arise through complex mechanisms, including cytoskeletal barriers (38,44,48,49), the partitioning of saturated lipids and cholesterol (50), and ionic protein-lipid or protein-lipid interactions (41,51–53). Due to this complexity, we do not make assumptions here about the primary mechanism underlying the domains that cause erbB1 clustering. The assignment of membrane domain area based upon EM images can be considered a coarse-graining approach, where clustering is maintained throughout the simulation period and satisfies the essential characteristics observed experimentally for receptor motion. In our current simulation framework, domains are held to be static in size and location. This strategy lowers computational costs and follows the observation of Douglass and Vale (53) that some slow-diffusing membrane proteins can serve as reporters for relatively stable domains. However, this simplification likely does not reflect the true dynamic nature of protein-rich domains, which may diffuse as entities in the membrane, themselves encountering cytoskeletal barriers and cycling between growth and dispersion at the nanometer or submicron scale (38).

Our model explicitly considers the mounting experimental evidence for erbB structural rearrangements associated with dimerization. Like the integrins, the extracellular domains of erbB receptors are now well known to exist in both bent and extended confirmations (54–59), where ligand binding stabilizes the upright form and exposes the dimerization arm. To integrate this concept into mathematical models, we assume that unliganded receptors are predominantly in the bent confirmation and that ligand receptors are fixed in the dimerization-competent conformation (4,22). A primary goal of this study was to incorporate the critical discovery that erbB catalytic activation is dependent upon an asymmetrical orientation of erbB kinase domains (13,14,60). These landmark studies established that contact of the N-lobe of the activator with the dimer partner’s C-lobe relieves autoinhibition of the kinase domain solely in the receiver (13). Conclusions of these crystallographic structure studies are supported by EM analysis of negatively stained full-length epidermal growth factor receptor in the presence and absence of ligand and/or kinase inhibitors (15,16). These studies indicate that the conformational orientations of dimerized erbB kinase domains are dominated by the active asymmetrical orientation, as opposed to the inactive symmetrical orientation (61), although kinase inhibitors can shift the class averages for the two orientations.

We consider the implications of the asymmetric erbB1 activation scheme in its simplest form, by assuming that during the lifetime of the dimer only one member of the dimer pair becomes catalytically competent for transphosphorylation of its partner. Consistent with evidence that dimers composed of 1:2 ligand/receptor ratios are signaling-competent (24), the activation state of erbB1 in our simulations is governed not by ligand occupancy per se but by the lifetime of dimers determined experimentally (17). The probability for productive interactions is highest for 2:2 receptors, since the off rate in this case is slowest, followed by 1:2 receptor pairs and then 0:2 preformed dimers, which have very fast off rates. We do note that the number of phosphorylated receptors is increased above the total value for ligand-bound receptors, through repeated interactions and the productivity of 1:2 dimers. This amplification, combined with transient dimerization, does allow for phosphorylation of unliganded receptors in 1:2 dimers that then dissociate and later interact with other unliganded monomers. However, the shorter lifetimes and reduced interaction probability associated with unliganded receptors results in very few productive 0:2 dimer events.

Since our simulations are initiated with a fraction of monomers bound to ligand, the model presented here does not consider the potential for negative cooperativity (14,62,63). If ligand binding were to be considered in the spatial stochastic model, it would lower the probability for an additional ligand to bind to a 1:2 receptor pair only during its relatively short lifetime (k_off = 0.738 s⁻¹) (17).

One notable prediction of the simple asymmetric model considered here is that fast dissociation of dimers effectively promotes signaling, because reencounters increase the likelihood that each monomer has repeated, equal opportunities to become phosphorylated by the receiver. We note the recent work of Pike and colleagues, who used a novel luciferase fragment complementation assay to provide compelling evidence for asymmetric and sequential activation of kinases in erbB homo- and heterodimers (64). These authors also raise the possibility that reciprocity could occur during the lifetime of the same dimer event, if the kinase domains can reorient while the monomers remain bound. This intriguing possibility is not explored here, due to lack of information about energetic requirements and feasibility of such a reorientation on the timescale relevant to even the most stable 2:2 dimer (<10 s).

This work adds to a growing appreciation that cell signaling is markedly influenced by the spatial organization of the plasma membrane, where lateral segregation in the 2D environment influences interactions between signaling proteins and the propagation of positive signaling or associated negative regulatory networks (65–67). Our simulations predict that ligand-bound erbB1 cycle rapidly through all possible receptor states, generating pulses of signaling competent states. The potential for short-lived components to generate robust, system-level output has been termed digital signaling (65). We expect that the impact of membrane spatial organization will vary widely in disease and normal settings, even for a single species of receptor such as erbB1. For example, we show here that cells expressing very high levels of erbB1 (typical of gene amplification in certain cancers) are less dependent on domain coconfinement for productive encounters than cells with modest levels of surface receptors. Cell-type variable factors that could alter the stability of domains and extend receptor capture events include lipid composition, the extent and dissociation kinetics of cortical cytoskeletal connections with membrane anchors, and the lipid/protein ratio. Since lipid remodeling, protein macromolecular assembly, and cytoskeletal rearrangements often accompany signaling, the organization of the plasma membrane is subject to alterations over important time- and lengthscales. Highly diffusible products of signaling cascades, such as reactive oxygen species proposed to inhibit phosphatases acting on erbB1 and enhance lateral propagation (68), would not be subject to the same 2D restrictions. Exploring the impact of the evolving 2D and 3D landscape through creative imaging and mathematical approaches is a future challenge for the field.