A mathematical model to predict network growth in physarum polycephalum as a function of extracellular matrix viscosity, measured by a novel viscometer

Physarum polycephalum is a slime mould that forms complex networks, making it an ideal model organism for studying network formation and adaptation. We introduce a novel viscometer capable of accurately measuring extracellular matrix ECM viscosity in small biological samples, overcoming the limitations of conventional instruments. Using this device, we measured the relative kinematic viscosity and developed continuous models to predict network size over time as a function of ECM viscosity and network complexity. Our results show that increased ECM viscosity, driven by higher salt (MgCl2·6H2O) concentrations, significantly slows network expansion but does not affect the final network complexity. Fractal Dimension (FD) analysis revealed that network complexity converged to a similar value across all viscosity conditions during the equilibrium stage. The models demonstrated strong predictive power, with a Mean Squared Error below 0.4 %, closely aligning with experimental data. These findings highlight the critical role of ECM viscosity in influencing network expansion while demonstrating that complexity remains stable across varying conditions. This study advances our understanding of the physical parameters shaping P. polycephalum networks and provides a foundation for exploring network dynamics in other adaptive systems. These insights offer new tools for research in biological systems where sample material is limited.

Methods
This dataset was collected using a custom-engineered viscometer designed to measure the kinematic viscosity of extracellular slime matrix (ECM) samples from Physarum polycephalum. The experimental setup involved precise mechanical control and sensor readouts (force, distance, temperature and humidity) to record how Physarum polycephalum responded to varying concentrations of magnesium chloride (MgCl2·6H2O). In parallel, growth measurements were recorded to study the organism's network expansion, while fractal dimension analysis was performed to assess network complexity.

Data Collection

Viscosity Data: Force and distance measurements were collected using load cells and distance sensors connected to a Raspberry Pi 4B. The data were processed with custom Python scripts. Measurements were taken across different concentrations of MgCl2·6H2O over time.

Growth Data: Network expansion was measured in mm² over time using time-lapse imaging and image analysis. Data were collected from multiple samples across different experimental conditions.

Fractal Dimension Data: To quantify their complexity over time, fractal dimension analysis was performed on the recorded network structures.

Data Processing
The collected raw data were processed using custom Python scripts. These scripts calculate key parameters such as kinematic viscosity, dynamic viscosity and network complexity. Data from each experimental condition are organised into CSV files for further analysis. The Python code for both the viscometer data acquisition and analysis is provided.