Input–output behavior of ErbB signaling pathways as revealed by a mass action model trained against dynamic data

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William W Chen^1,a, Birgit Schoeberl^2,a, Paul J Jasper^1,a, Mario Niepel¹, Ulrik B Nielsen², Douglas A Lauffenburger³ & Peter K Sorger¹

^aThese authors contributed equally to this work

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Synopsis

The four transmembrane receptors of the ErbB family, of which the epidermal growth factor tyrosine kinase (EGFR or ErbB1) is the founding member, are widely expressed in human tissues and stimulate diverse cellular responses, such as proliferation, survival and motility. ErbB receptors have been studied extensively at a molecular level and several have been shown to play a central role in human cancer. ErbB2 (Her2), for example, is overexpressed in a subset of breast cancers and is the target of trastuzumab, an important anticancer drug. Signaling through ErbB receptors, and the downstream proteins they activate, is complex because ErbB1–4 combine to form both homo- and heterodimers having distinct affinities for 13 known ligands and intracellular adaptor proteins. Understanding and predicting how signaling varies with receptor mutation or overexpression is essentially impossible using informal reasoning and pictorial representations of signaling pathways.

To address this challenge, we have constructed, trained and analyzed a mass action model of immediate-early signaling involving four ErbB receptors, as well as the downstream MAP kinase and PI3K–Akt signaling pathways involved in regulating cell proliferation (Figure 1). Our model aims to capture as much mechanistic information as possible, based on an extensive literature, while accurately representing the responses of cells to two major ligand subclasses (EGF and heregulin in multiple tumor cell types). In assembling such a model, we encounter a number of challenges, among which the processes of entraining models to experimental data are the most serious. Mass action models are governed by two kinds of parameters: the concentrations of individual protein species, and the rate constants governing protein–protein association and enzyme catalysis. Models of cell signaling that are realistic at a molecular level may inevitably contain many parameters that are unknown a priori. These parameters can be estimated either by direct measurement (usually in vitro, raising the question how values obtained in dilute solution should be translated to a crowded intracellular environment) or through an inverse process in which the model is fitted to data. The processes of training a kinetic model on experimental data constrain only a subset of parameters. It is assumed by some that the presence of parametric uncertainty in trained models makes them useless; in fact the aim of fitting is to determine only the subset of parameters that determine model performance with sufficient accuracy from which meaningful hypotheses can be drawn. Methods to accomplish this with cell signaling models are in their infancy, and most biochemical models are parameterized based on generic or theoretical assumptions; systematic or rigorous analysis of parameters variation and uncertainty has been restricted to small idealized models. In the current study, we use iterative fitting to generate families of model fits with similar biochemical connectivity, but different values for unconstrained parameters. We then attempt to draw well-substantiated inferences from the families of models.

Figure 1

Simplified schematic representation of the ErbB model. Receptor interaction, internalization, recycling and activation of MAPK and PI3K/Akt pathways are shown. Phosphatase, Ras-GTP hydrolysis and feedback regulation are indicated by red arrows. Matrix shows properties of each receptor dimer relevant to the topology of the model; see Figure 1 in Supplementary information for further information. The prefix C on certain proteins denotes an abbreviated representation of multiple species in the model, e.g. C:Shc is composed of all receptor-bound Shc molecules, which may include ErbB1/1-Shc, ErbB1/2-Shc and so on. However, each of these Shc complexes is represented explicitly by one or more dynamic variables in the model.

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To ascertain which features of our ErbB model are conserved when parameterized by different sets of rate constants and concentrations, we calculated dynamic sensitivities of multiple observables; this serves to measure the 'importance' of each rate or species in affecting measurable outputs. Sensitivity analysis showed that despite degeneracy in parameter values among families of fits, many important features are conserved. However, the rank order of sensitive parameters is strongly influenced by the feature being examined. For example, the proteins and rate constants that influence ERK activation differed from those that influence Akt, and factors important at high ligand concentrations differ from those important at low concentrations. A picture of ErbB signaling emerges in which different physiological outputs can be mapped back to specific biochemical characteristics of signaling proteins under varying input conditions.

One striking aspect of this input–output behavior involves the relationship between ligand concentration (the input) and the activities of ERK or Akt (the output). A priori, we might assume little activation of the ErbB pathway when the concentration of ligand is below receptor K_d. However, experiments demonstrate significant Akt and ERK activation (20% maximal) and 100-fold lower ligand. Moreover, as ligand levels rise we would expect the response to rise 9-fold for every 81-fold increase in ligand concentrations (representing standard binding thermodynamics). In contrast, we observe a log-linear amplification relationship between ligand and output over a nearly 10⁶ concentration range (Figure 7). To understand how this might arise, we have examined the performance of our parameterized ErbB model of and subsets of model comprising coupled reactions such as the MAP kinase cascade. We conclude that the unexpected log-linear input–output behavior of the ErbB model arises from interactions between enzyme cascades that in isolation exhibit canonical Hill-like behavior. Thus, we must now think carefully about the ways in which signaling modules interact so as to generate behavior that is quite different from what we would expect of the isolated molecules or small cascades.

Figure 7

(A) Dose–response of pErbB1, pShc, pERK and pAkt (5 min after ligand addition) over a 10⁶ range of EGF concentrations in A431 cells. (B) Amplification of signal calculated with equation (5) (5 min after ligand addition) over a 10⁶ range of EGF concentrations in A431 cells for pERK, pAkt and pShc. Points represent data and lines represent simulation; error bars denote standard error of the mean. Gray boxes show high error measurements at low EGF concentrations, which arise due to phospho-ErbB1 signals being at the limit of detection.

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In conclusion, we describe an approach in constructing and analyzing models of cell signaling pathways that can incorporate substantial molecular detail, while remaining sensitive to the inability of experiments to fully constrain parametric complexity. By focusing on well-substantiated model features, we attempt to understand the origins of an unexpected and physiologically significant characteristic of the ErbB network with regard to dose–response behavior. Looking forward we anticipate building on the current model by including more accurate representations of more reactions, while developing a rigorous approach to generating model-derived hypotheses, the degree of belief of which is determined by the underlying uncertainty in the model and the data. We will then be in a position to reliably understand cellular physiology in terms of molecular mechanism.

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