PDE models
Definition
Partial Differential Equation (PDE) models are mathematical frameworks that describe how biological quantities change continuously across both space and time using derivatives with respect to multiple variables. In life sciences, PDEs capture spatiotemporal dynamics of processes like morphogen gradients during development, tumor growth and invasion, neural signal propagation, and drug diffusion through tissues. Unlike ordinary differential equations (ODEs) that model temporal changes at discrete locations, PDEs account for spatial heterogeneity and transport phenomena including diffusion, advection, and reaction kinetics. These models are essential for understanding emergent biological patterns, tissue-level organization, and predicting therapeutic outcomes in spatially-structured systems.
Visualize PDE models in Nodes Bio
Researchers can visualize PDE model components as networks where nodes represent spatial compartments, molecular species, or cellular populations, and edges represent diffusion coefficients, reaction rates, or spatial coupling terms. This network representation helps identify critical spatial interactions, visualize parameter sensitivity across tissue regions, and map how local perturbations propagate through spatially-extended biological systems.
Visualization Ideas:
- Spatial compartment networks showing diffusion coupling between tissue regions
- Morphogen gradient networks linking source cells to target domains via transport edges
- Multi-scale networks connecting molecular PDE dynamics to cellular behavior patterns
Example Use Case
A developmental biology team models limb patterning using a PDE system describing BMP and Sonic Hedgehog morphogen gradients. The model includes diffusion terms for morphogen spread, degradation kinetics, and receptor-mediated signaling cascades. By representing the spatial domain as interconnected nodes with morphogen concentrations and signaling states, researchers identify critical diffusion rates and reaction parameters that determine digit formation patterns, helping explain polydactyly mutations and predict outcomes of genetic perturbations.