Limitations

Computational (systems) biology models validated using model checking ap- proaches usually encode biological processes/subsystems from the (intra-)cellular level (e.g. cell cycle (Brim et al., 2013a; Chabrier and Fages, 2003; Fages and Rizk, 2009; Gong and Feng, 2014; Maria et al., 2009; Rizk et al., 2008; Van Goethem et al., 2013), gene (expression/regulatory) networks (Batt et al., 2008, 2005; Cioc- chetta et al., 2009; Giacobbe et al., 2015; Yordanov and Belta, 2011), signalling pathways (Ballarini et al., 2014; Calder et al., 2006; Clarke et al., 2008; Donaldson and Gilbert, 2008a; Gilbert et al., 2007; Gong et al., 2012; Guerriero, 2009; Heath et al., 2008; Heiner et al., 2008; Kwiatkowska et al., 2007; Rizk et al., 2008))

where most experimental data is available. One common characteristic of these computational models is that the system behaviour is described as numeric values (e.g. concentrations) changing over time, without explicitly considering the system

representation in space and/or across multiple levels of organization.

Consequently one of the main limitations of the corresponding model checking approaches is that they have been similarly defined only relative to how numeric values change over time. However in order to gain a systems level understanding of how biological organisms function it is essential to consider computational models of larger scale systems (e.g. multicellular populations). Such computational models additionally capture how properties of (emergent) spatial structures (e.g. area of multicellular population) change over time and/or across multiple levels of organization, which are not considered by existing non-spatial uniscale model checking approaches.

To address this limitation existing model checking methods need to be extended with two types of functions, namely functions that enable describing how properties of spatial structures change over time, and functions that enable associating both numeric and spatial state variables with specific levels of organization.

Summary

This chapter has provided a brief description of the formal method called model checking employed to validate models of reactive systems relative to formal speci- fications. Non-probabilistic systems were represented as labelled state transition systems (LSTS), and probabilistic systems were represented as probabilistic LSTSs (PLSTS). Two classes of temporal logics were described for encoding the formal specifications, linear time logics (e.g. LTL, BLTL and P(B)LTL) which assume a linear representation of time, and branching time logics (e.g. CTL, CTL*, PCTL, CSL) which assume a branching structure of time. Depending on the model and formal specification considered different types of model checking algorithms have been presented for both non-probabilistic and probabilistic systems, which were either exhaustive (i.e. considering the entire state space) or approximate (i.e. exploring the state space only partially). Model checking approaches specifically employed for validating computational models of biological systems were described in the end, including one of their main limitations i.e. that they only capture how numeric values (e.g. concentrations) change over time, without explicitly considering the evolution of the system in space and/or across multiple levels of organization.

CHAPTER

3

Multidimensional

spatio-temporal model checking

Introduction

In this chapter a novel multidimensional spatio-temporal model checking method- ology is introduced which enables validating computational models of biological systems with respect to how both numeric and spatial properties change over time. The methodology comprises a theoretical model for abstractly representing biological systems, a spatio-temporal analysis method for automatically detecting and analysing spatial structures in the model simulation output, a standard representation format for time series data comprising numeric and spatial prop- erties, a formal language for encoding the specification against which the model is validated, and corresponding model checking algorithms. A brief description of the model checking method implementation, and a comparison with related approaches from other domains of science are provided in the end.

3.1

Spatial computational models of biological

systems

Different types of computational models are employed to represent biological systems depending on the level of organization considered.

At intracellular or more fine-grained levels it is often assumed that species (e.g. proteins/molecules) are uniformly distributed in space. Therefore computational models only capture how their average concentration changes over time without explicitly taking space into account.

Conversely at cellular and more coarse-grained levels it is assumed that the het- erogeneity of species (e.g. cells) is important because it can lead to the development of different structures in space. Therefore corresponding computational models usually explicitly record how the number/density of species evolves both over time and space and are called (multidimensional) spatial(-temporal) computational models.

In order to support the development of such spatial computational models appropriate modelling formalisms have been developed; they represent the spatial domain in either a continuous or discrete fashion.

Continuous spatial models are usually encoded as partial differential equa- tions (Schaff et al., 1997) and have been used to represent variations of reaction- diffusion (Kondo and Miura, 2010) or predator-prey (Arditi et al., 2001) systems, and the chemotactic movement of cells (Hillen and Painter, 2009). The main reason for modelling processes such as diffusion (reaction-diffusion) or population variation (predator-prey, chemotaxis) using continuous approaches is that only the average density of the species is of interest for each time point and position in space.

Conversely, if the interactions between individual species are of interest discrete spatial models could be employed instead. Representative discrete spatial mod- elling formalisms which employ a lattice-based representation of space and local rules to specify how the system changes from one state to the next are Cellular Automata (Deutsch and Dormann, 2007, Chapters 5-11) and Glazier-Graner- Hogeweg (Balter et al., 2007; Graner and Glazier, 1992) models (also known as Cellular Potts). In contrast individual-based models (An et al., 2009; Thorne et al., 2007) can employ either an on-lattice or off-lattice spatial representation, and their evolution over time is determined by rules specific to individuals (or agents) instead of lattice positions. Modelling formalisms which are not inherently spatial but have been extended with spatial attributes recording the species’ position in space (e.g. coordinates in Euclidean space) include Petri nets (Gao et al., 2013; Gilbert et al., 2013; Liu et al., 2014b; Pˆarvu et al., 2013), process algebras (Feng and Hillston, 2014; John et al., 2010), rule-based modelling languages (Blinov et al., 2004; Danos et al., 2007; John et al., 2011; Maus et al., 2011; Nikoli´c et al., 2012) and P (or membrane) systems (Barbuti et al., 2011; Besozzi et al., 2008).

Examples of biologically relevant case studies encoded using these spatial modelling formalisms include the cardiac and gastrointestinal tissue electrophysiol- ogy (Corrias et al., 2012), chemo-/photo-taxis (John et al., 2008, 2010), the growth of microbial populations (Ferrer et al., 2008; Pˆarvu et al., 2015), host-pathogen interactions (Bauer et al., 2009), organisms development or morphogenesis (Mar´ee

et al., 2007; Merks and Glazier, 2005) and tumour growth (Mallet and De Pillis, 2006; Moreira and Deutsch, 2002; Norton and Popel, 2014).

Similarly to computational models of intracellular networks, spatial computa- tional models of biological systems need to be validated before they are employed for real-life applications. However there is a lack of corresponding model checking approaches.

The main reason why existing model checking approaches cannot be em- ployed to validate spatial computational models is that they do not consider how properties of (emergent) spatial structures change over time.

These spatial structures are not hardcoded into the models but are emergent behaviours i.e. they are dynamic behaviours that occur at simulation time as a result of the interaction between their constituent entities (e.g. cells). Therefore one of the main challenges of validating spatial computational models is to automatically detect spatial structures in the model simulation output and analyse how their properties change over time. Moreover a suitable spatio-temporal formal language needs to be defined for encoding the specifications against which the models are validated. Finally the employed model checking algorithms need to be updated accordingly.