Introduction to Mathematical
Mathematical Biology Lecture 1
James A. Glazier
Instructor: James A. Glazier
Classes: Tu. Thu. 8:00AM–9:30AM
Office Hours: By appointment
1) Murray – Mathematical Biology volumes 1 & 2
2) Fall, Marland, Wagner and Tyson – Computational Cell
3) Keener and Sneyd – Mathematical Physiology
Weekly Homework (40% of Grade)
Written Project Report (40% of Grade)
Oral Presentation (20% of Grade)
No Final Exam – Late Assignments Will Be Marked Down.
Software: CompuCell 3D
• Population Dynamics and Mathematical
• Stochastic Gating
• Reaction Kinetics, Oscillating Reactions, and
• Molecular Motors
• Collective Phenomena – Flocks and Neural
• Higher Dimensional Models: Mathematics
• Excitable Media – Heart and Calcium Waves
• Turing Patterns
What is Mathematical Biology?
• Can be abstruse and self focused when it
concentrates on what is soluble
analytically rather than what is important.
• However, simplified models can teach
about general classes of behavior and
types of parameter dependence.
• Teach a set of generally useful
• Give a set of key examples
• Build a computational models (hopefully
leading you to publish something)
• Linear Stability Analysis
• Bifurcation Analysis
• Phase Plane Diagrams
• Stochastic Methods
• Fast/Slow Time-Scale Separation
• Scaling Theory and Fractals
What is Computational Biology?
• Modeling, Not just Curve Fitting
• Must have a mechanistic basis
• Can address multiscale structures and
feedback between elements.
• Not Bioinformatics/Genomics (primarily
• Not Cluster Analysis, Image Processing,
• To explain biological processes that result
in an observed phenomena.
• To predict previously unobserved
• To identify key generic reactions.
• To guide experiments:
– Suggest new experiments.
– Eliminate unneeded experiments.
– Help interpret experiments.
• A huge gap between level of molecular data and
• Most Modern Biology is descriptive rather than
• Epistemology – Car parts metaphor.
• Simplify impossible complexity by forcing a
hierarchy of importance – identifying key
• In a model know what all processes are.
• Failure of models can identify missing
components or concepts.
Scale Examples Methods
Atomic DNA; Protein Structure, Binding and Quantum Chemistry
Ion Channels and Photosynthesis
Molecular Receptor-ligand binding Molecular Dynamics
Signal Transduction (Classical); BIOSYM
Networks Genetic Regulatory Networks Coupled ODE Models;
Metabolic Networks Stochastic ODEs;
Diabetes Network Analysis;
Macromolecular Molecular Motors; Actin; Microtubules; Simplified Molecular
Intermediate Filaments; Chromosomes; Dynamics
DNA Coiling; DNA Transcription; Protein Langevin Equation
Synthesis Fokker-Planck Equation
Molecular Junctions; Stress Fibers; Cilia; Flagella; VirtualCell; Karyote; e-
Systems Pseudopods; Fliopodia; Mitotic Spindles; Cell; M-Cell
Growth Cones; Endoplasmic Reticulum;
Cell Membranes; Cell Polarity; Cell Motility
Scale Examples Methods
Cellular Cell Adhesion; Chemotaxis; Haptotaxis; Cell Cellular Potts Model;
Differentiation Center Models;
Epithelia; Cell Sorting; Bacterial patterning; Boundary Models;
Dictyostelium Hodgkin-Huxley Model;
Neurons; Myofibers; Cancer; Stem Cells
Tissue [Including Wound Healing; Angiogenesis; Kidney Cellular Automata; Reaction-
Diffusion Models; Fitz-Hugh-
Individual Cells] Development; Lung Development; Neural Nagumo Equation; Coupled
Circuirts; Tumor Growth PDEs; Stochastic PDEs
Organ Heart; Circulatory System; Bone; Continuum Mechanics;
Cartilage; Neural Networks; Organ Finite Element Methods;
[Neglecting Navier-Stokes Equations;
Individual Cells] Development
Individual Flocks; Theories of Learning Agent-based Models;
Population Infections/Epidemiology; Population Continuum ODEs;
Modeling; Predator-Prey Models; PDEs; Iterated maps;
Evolutionary Models; Bacterial and Kaufmann Nets; Delay
Eukaryotic Communities; Traffic ODEs; AVIDA