Modelling Cell Cycle using Different Levels of Representation

Understanding the behaviour of biological systems requires a complex setting of in vitro and in vivo experiments, which attracts high costs in terms of time and resources. The use of mathematical models allows researchers to perform computerised simulations of biological systems, which are called in silico experiments, to attain important insights and predictions about the system behaviour with a considerably lower cost. Computer visualisation is an important part of this approach, since it provides a realistic representation of the system behaviour. We define a formal methodology to model biological systems using different levels of representation: a purely formal representation, which we call molecular level, models the biochemical dynamics of the system; visualisation-oriented representations, which we call visual levels, provide views of the biological system at a higher level of organisation and are equipped with the necessary spatial information to generate the appropriate visualisation. We choose Spatial CLS, a formal language belonging to the class of Calculi of Looping Sequences, as the formalism for modelling all representation levels. We illustrate our approach using the budding yeast cell cycle as a case study.

💡 Research Summary

The paper addresses the growing need for cost‑effective alternatives to extensive in‑vitro and in‑vivo experimentation by proposing a formal, multi‑level modeling framework that integrates both rigorous biochemical simulation and high‑quality visual representation. The authors distinguish two complementary abstraction layers: a “molecular level” that captures the detailed biochemical dynamics of a system, and one or more “visual levels” that provide higher‑order, spatially aware views suitable for rendering. To ensure a seamless transition between these layers, the authors adopt Spatial CLS (Calculi of Looping Sequences), a formal language belonging to the family of Looping Sequence calculi, which naturally encodes both sequential (e.g., polymeric) structures and spatial nesting through loop constructs.

In the molecular level, biological entities such as proteins, DNA fragments, and complexes are represented as CLS terms. Reaction events are expressed as rewrite rules that specify how terms are transformed, while spatial information (coordinates, compartments) is encoded via looping operators and location annotations. This formalism preserves the stochastic or deterministic nature of biochemical interactions while keeping the spatial context explicit, a feature often missing in ordinary differential equation models.

The visual level is built on top of the molecular description by defining a systematic mapping: each molecular term is associated with a visual primitive (e.g., a sphere for a nucleus, a membrane surface for a cell wall) and a set of visual attributes (color, size, opacity). The mapping also includes transformation functions that translate molecular events (e.g., complex formation, degradation) into visual actions (e.g., object merging, fading). By keeping the mapping functions declarative, the framework guarantees that any change in the underlying molecular simulation is automatically reflected in the rendered scene without manual intervention.

The methodology is demonstrated on the budding yeast (Saccharomyces cerevisiae) cell cycle, a well‑studied system that proceeds through G1, S, G2, and M phases, culminating in the formation of a daughter bud. The authors encode the core regulatory network—Cln, Clb cyclins, Cdc28 kinase, and associated inhibitors—using Spatial CLS rewrite rules that capture synthesis, degradation, phosphorylation, and localization events. Concurrently, they define visual objects for the mother cell, bud, nucleus, and spindle apparatus, and specify how each phase’s molecular events drive visual transformations such as bud emergence, nuclear migration, and cytokinesis.

Simulation results show that the temporal dynamics of cyclin concentrations and phase transitions match experimental observations, confirming the biochemical fidelity of the molecular model. The generated visualizations, rendered as a sequence of 3‑D scenes, reproduce the characteristic morphological changes observed under microscopy, including bud size growth, nuclear positioning, and cell wall remodeling. The authors highlight that the same Spatial CLS specification can be used both for quantitative analysis (e.g., sensitivity studies, parameter sweeps) and for qualitative communication (e.g., teaching, outreach), illustrating the dual utility of the approach.

A critical discussion follows, emphasizing several strengths: (1) the unified formalism eliminates the need for separate simulation and graphics pipelines, reducing development overhead; (2) explicit spatial encoding at the molecular level prevents loss of geometric information during abstraction; (3) the mapping layer is modular, allowing reuse across different organisms or cellular processes. Limitations are also acknowledged: Spatial CLS tooling is still emerging, which may hinder large‑scale adoption; state‑space explosion can occur for highly complex networks, requiring optimization or approximation techniques; and excessive abstraction at the visual level could obscure biologically relevant details if not carefully calibrated.

The paper concludes by outlining future work: automated generation of mapping tables, integration with other formal languages (e.g., stochastic π‑calculus, Petri nets), extension to multicellular tissue models, and the development of interactive, real‑time visualization environments that support virtual experiments. Overall, the work presents a compelling, formally grounded pathway to bridge the gap between rigorous computational biology and immersive visual storytelling, offering a valuable platform for researchers, educators, and developers alike.