Modelling count data with partial differential equation models in biology.

Area of Science:

• Mathematical Biology
• Computational Biology
• Biophysics

Background:

• Partial differential equation (PDE) models are crucial for studying biological movement, birth-death processes, and population dynamics.
• Biological count data is non-negative and often bounded by carrying capacity due to competition.
• Standard additive Gaussian measurement error models are widely used but their assumptions are often unexamined.

Purpose of the Study:

• To interpret scratch assay data of cancer cell populations using a reaction-diffusion PDE model.
• To compare the standard additive Gaussian measurement error model with a more biologically realistic binomial measurement error model.
• To assess the impact of measurement error models on parameter estimation and prediction accuracy.

Main Methods:

• Interpreted scratch assay data using a reaction-diffusion PDE model for cancer cell populations.
• Compared standard additive Gaussian measurement error models with binomial measurement error models.
• Developed and utilized open-source Julia software for calculations and generalizations.

Main Results:

• Model parameter estimates showed minimal sensitivity to the choice of measurement error model.
• Model predictions were highly sensitive to the measurement error model.
• The Gaussian model produced biologically inconsistent predictions (negative counts, exceeding carrying capacity).
• The binomial model yielded biologically plausible predictions and required estimating fewer parameters.

Conclusions:

• The binomial measurement error model is superior to the standard Gaussian model for biological count data, offering greater biological realism and predictive accuracy.
• The findings highlight the importance of carefully selecting measurement error models in PDE-based biological studies.
• The developed methodology and software can be extended to more complex coupled PDE models and generalized linear models.