[Paper Review] Modelling Spatial Interactions in the Arbuscular Mycorrhizal Symbiosis using the Calculus of Wrapped Compartments

Cristina Calcagno, Mario Coppo+5 more|Sep 7, 2011

Cellular Automata and Applications45 references5 citations

TL;DR

This paper proposes a spatial modeling approach for arbuscular mycorrhizal symbiosis using the Calculus of Wrapped Compartments (CWC), leveraging compartment labeling to implicitly represent spatial topology without explicit geometry. By modeling hyphal growth, strigolactone diffusion, and arbuscule formation across soil and root layers, the framework captures key biological dynamics, showing that low phosphate conditions promote earlier and more frequent arbuscule formation (60% in 8–9 days) compared to high phosphate (10% by day 15), aligning with in vivo observations.

ABSTRACT

Arbuscular mycorrhiza (AM) is the most wide-spread plant-fungus symbiosis on earth. Investigating this kind of symbiosis is considered one of the most promising ways to develop methods to nurture plants in more natural manners, avoiding the complex chemical productions used nowadays to produce artificial fertilizers. In previous work we used the Calculus of Wrapped Compartments (CWC) to investigate different phases of the AM symbiosis. In this paper, we continue this line of research by modelling the colonisation of the plant root cells by the fungal hyphae spreading in the soil. This study requires the description of some spatial interaction. Although CWC has no explicit feature modelling a spatial geometry, the compartment labelling feature can be effectively exploited to define a discrete surface topology outlining the relevant sectors which determine the spatial properties of the system under consideration. Different situations and interesting spatial properties can be modelled and analysed in such a lightweight framework (which has not an explicit notion of geometry with coordinates and spatial metrics), thus exploiting the existing CWC simulation tool.

Motivation & Objective

• To model spatial interactions in arbuscular mycorrhizal symbiosis, particularly fungal hyphal colonization of plant roots, using a formalism without explicit spatial coordinates.
• To demonstrate that compartment labeling in CWC can effectively encode topological spatial structure for biological systems with limited spatial complexity.
• To simulate the impact of phosphate availability on strigolactone signaling and hyphal branching, reflecting in vivo dynamics.
• To enable stochastic simulation of symbiosis processes using Continuous Time Markov Chains (CTMCs) within a lightweight, extensible framework.
• To lay the foundation for future integration of geometric spatial features into CWC for more detailed systems biology modeling.

Proposed method

• Utilizes the Calculus of Wrapped Compartments (CWC), a formalism for modeling biological systems via multisets of atoms and labeled compartments.
• Represents spatial regions as labeled compartments with fixed topological relationships, enabling modeling of hyphal movement and cell penetration without explicit coordinates.
• Employs rewrite rules to model molecular interactions, such as strigolactone release and Ca2+ oscillations, with rates derived from CTMC semantics.
• Integrates a hybrid simulation algorithm based on Gillespie’s method for exact stochastic simulation, accelerated via the FastFlow framework for multi-core execution.
• Models phosphate availability as a variable input (10, 50, 100 P atoms) to assess its effect on symbiosis progression.
• Uses compartment adjacency and label-based rules to simulate directional hyphal branching and fungal penetration into root tissues.

Experimental results

Research questions

• RQ1 Can the Calculus of Wrapped Compartments (CWC) effectively model spatial interactions in the arbuscular mycorrhizal symbiosis without explicit geometric coordinates?
• RQ2 How does phosphate availability influence the timing and frequency of arbuscule formation in a simulated environment?
• RQ3 To what extent can compartment labeling in CWC represent topological spatial structure relevant to fungal hyphal growth and root colonization?
• RQ4 How do strigolactone diffusion and fungal hyphal branching patterns vary under different phosphate conditions in the model?
• RQ5 Can the model reproduce known in vivo dynamics of AM symbiosis, such as delayed arbuscule formation under high phosphate?

Key findings

• In low phosphate conditions (10 P atoms), arbuscules formed in 60% of simulations, with the first appearing around days 8–9 of the simulation.
• Under medium phosphate (50 P atoms), arbuscule formation occurred in 30% of simulations, with onset delayed to days 9–15.
• In high phosphate conditions (100 P atoms), arbuscule formation was observed in only 10% of simulations, with the first appearance occurring between days 9 and 15.
• Strigolactone diffusion and hyphal branching were significantly reduced in high phosphate conditions, reflecting reduced fungal response to plant signals.
• The model successfully reproduced known in vivo dynamics, including the inverse relationship between phosphate availability and arbuscule formation frequency.
• The simulation framework demonstrated that topological compartment labeling in CWC is sufficient to model key spatial behaviors in symbiosis, such as directional hyphal growth and root penetration.