Refinement-based modeling of the ErbB signaling pathway

https://doi.org/10.1016/j.compbiomed.2019.01.016Get rights and content

Highlights

  • Refinement helps to add details to biomodels while preserving their data fit.

  • Our method allows to integrate experimental evidence with mathematical constraints.

  • Our model on the ErbB signaling pathway is the largest model built with refinement.

  • Formal methods can be used for ensuring the consistency of a model.

Abstract

The construction of large scale biological models is a laborious task, which is often addressed by adopting iterative routines for model augmentation, adding certain details to an initial high level abstraction of the biological phenomenon of interest. Refitting a model at every step of its development is time consuming and computationally intensive. The concept of model refinement brings about an effective alternative by providing adequate parameter values that ensure the preservation of its quantitative fit at every refinement step. We demonstrate this approach by constructing the largest-ever refinement-based biomodel, consisting of 421 species and 928 reactions. We start from an already fit, relatively small literature model whose consistency we check formally. We then construct the final model through an algorithmic step-by-step refinement procedure that ensures the preservation of the model's fit.

Introduction

Mechanistic control of cellular activity is intricate and making predictions about its system-level behavior is highly difficult. Our ability to make such predictions can be essential not only in reversing the dynamics of cellular impairment, but also in directing cellular activity towards a more favorable behavior. Mathematical modeling is essential in making such predictions, but its use as a standard procedure in the field of practical applications is severely limited due to large numbers of parameters that are required either to be fixed or estimated, see Ref. [1].

A massive number of parameters to estimate requires the availability of a large volume of data and makes model fitting computationally intensive. For this reason, we focus on refinement-based model construction as an intermediary step in the model development cycle. Stepwise refinement emerged from the field of software engineering. It was introduced at first as a concept in parallel computing and it expanded quickly, giving rise to the framework of refinement calculus, where it is promoted as a refinement method to ensure correctness preservation, see Ref. [2].

In the field of systems biology, model refinement becomes crucial in the model development cycle. Model fit is greatly affected by changes in the number of reactants, reactions, modules, etc. The entire process of model fitting for considerably large models is not only a tedious task for the modeler as such, but it is computationally intensive since most parameter estimation routines take considerable time to complete and require massive amounts of computational resources. Hence, an iterative approach which relies on the conventional reiteration of the entire model fitting procedure is not feasible for large models. As an alternative, we consider an approach which ensures model fit preservation at every refinement step. The approach was discussed in the literature for rule-based models, see Refs. [3,4]. For reaction-based models with a quantitative dynamic described by ODEs, the method was referred to as quantitative model refinement, see Ref. [5] and then extended and called fit-preserving data refinement [6].

We discuss in this paper the implementation of the largest-ever model built through model refinement, describing the ErbB signaling pathway. Our refinement approach is based on data refinement, where a finite set of subspecies of a given species in the initial model are substituted in the refined model for their corresponding ‘parent’ species in the initial model. We started with a model of the EGFR (ErbB1) signaling pathway proposed in Refs. [7,8]. Throughout the paper, the model from Ref. [7] is referred to as the basic model. We refined this model to include four different types of receptor tyrosine kinases, ErbB14, structurally related to the epidermal growth factor receptor, EGFR, and two types of ligands, EGF and HRG, and we compared the computational effort needed to build it with that of [9]. We used logic-based formal methods support based on modeling with Event-B [10] to make sure that the basic model is consistent. An Event-B specification is written as an abstract machine that consists of variables and events. An event has a logical guard, which specifies under what condition the event will occur, and some actions. This allowed us to ensure that all 110 species (variables in Event-B) are properly defined and that the 148 reactions (events in Event-B) are consistently written. We then refined this basic model to its full version, consisting of 421 species and 928 reactions. This part of the modeling was done based on our methods on quantitative model refinement and implemented in COPASI [11], resulting in a mass-action, ODE-based model. We describe in this article the modeling process and discuss its implications.

Section snippets

Quantitative model refinement

The refinement of reaction-based models was proposed in Ref. [5] and later extended in Ref. [6] to address both the construction of the refined model and the assignment of its kinetic rate constants in such a way that it captures the same dynamics as the original model.

We consider a mass-action reaction-based model M consisting of m species, Σ={A1, …, Am} and n reactions R={r1, …, rn}, where m,n*. We write the reactions of M as:rj:cj1A1+cj2A2++cjmAmkjdj1A1+dj2A2++djmAm,where, for each k=1,n

Case-study: the ErbB signaling pathway

In this section we describe our case study, the EGFR signaling network, and its fit-preserving data refinement.

The ErbB signaling pathway is an evolvable pathway, responsible for the regulation of various physiological responses of the mammalian cell such as growth, survival, proliferation, differentiation and motility, [9,16,17]. Due to its intrinsic complexity and association with the progression of various cancer types, the pathway was extensively analyzed and found to be key to the

Ensuring the consistency of the basic model

The basic model of [7] consists of 148 reactions and 103 reactants. Ensuring the consistency of such a large model is non-trivial as small errors in the variable names or in the stoichiometric coefficients are often only indicated by standard softwares such as COPASI after the model is fully implemented; in such cases it is typically difficult to trace the source of the error. We decided to use Event-B to check the consistency of the basic model. We took advantage of the Rodin platform's

Discussion

When building an extensive system-level biological model, refinement becomes a crucial step in the model development cycle. Starting with a high level abstraction of a biological process of interest, one very often needs to include more details regarding its reactants, reactions or constituent modules. A conventional approach which would involve a reiteration of the entire model development cycle is highly ineffective, since it involves running parameter estimation routines over large sets of

Acknowledgments

This work was partially supported by Academy of Finland under grant 267915.

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