A Graphical Approach for Hybrid Simulation of 3D Diffusion Bio-Models via Coloured Hybrid Petri Nets

Abstract

Three-dimensional modelling of biological systems is imperative to study the behaviour of dynamic systems that require the analysis of how their components interact in space. However, there are only a few formal tools that offer a convenient modelling of such systems. The traditional approach to construct and simulate 3D models is to build a system of partial differential equations (PDEs). Although this approach may be computationally efficient and has been employed by many researchers over the years, it is not always intuitive since it does not provide a visual depiction of the modelled systems. Indeed, a visual modelling can help to conceive a mental image which eventually contributes to the understanding of the problem under study. Coloured Hybrid Petri Nets (HPNC) are a high-level representation of classical Petri nets that offer hybrid as well as spatial modelling of biological systems. In addition to their graphical representations, HPNC models are also scalable. This paper shows how HPNC can be used to construct and simulate systems that require three-dimensional as well as hybrid (stochastic/continuous) modelling. We use calcium diffusion in three dimensions to illustrate our main ideas. More specifically, we show that creating 3D models using HPNC can yield more flexible models as the structure can be easily scaled up and down by just modifying a few parameters. This advantage of convenient model configuration facilitates the design of different experiments without the need to alter the model structure.

Design patterns for the construction of computational biological models

“…In both cases, the diffusion pattern used here is 1D diffusion. Other examples that utilize calcium diffusion in 2D grid can be found in [ 34 ] and for 3D grids in [ 49 ]. Moreover, calcium binding and unbinding with the buffer is modelled as a reversible reaction (green colour), calcium input is modelled via a synthesis reaction (blue colour), and calcium pumps are modelled by degradation reactions (purple colour).…”

Protocol for biomodel engineering of unilevel to multilevel biological models using colored Petri nets

“… 15 Modeling 3D diffusion of calcium as a hybrid (stochastic and deterministic) model. 16 Modeling and analyzing the multicellular biological system of C. elegans vulval development. 17 Highlighted features include: one- dimensional (1D) space encoding, and patter formation.…”

From Epidemic to Pandemic Modelling

“…The use of colour enables us to encode matrices, where colours (natural numbers) represent indices; multidimensional matrices can be represented by appropriate colour tuples. Previously we have used this approach to encode one, two and three dimensional Cartesian space where coordinates were represented as tuples over the entire matrix (Gilbert et al, 2013), for example, modelling diffusion and cell movement in biological systems: planar cell polarity in the Drosophila wing (Gao et al, 2013), phase variation in bacterial colonies (Pârvu et al, 2015), bacterial quorum sensing (Gilbert et al, 2019) and intra-cellular calcium dynamics (Ismail et al, 2020).…”