P Mathematical Biology

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					Introduction to Mathematical
  Mathematical Biology Lecture 1

          James A. Glazier
P548/M548 Mathematical
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
              Course Topics
• Population Dynamics and Mathematical
• Stochastic Gating
• Reaction Kinetics, Oscillating Reactions, and
  Reactor Networks
• 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)
             Main Methods
•   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,
  Pattern Recognition
• 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.
             Why Needed?
• A huge gap between level of molecular data and
  observed patterns.
• 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.
                 Biological Scales
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;
                                                                BioSpice; PhysioLab
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
     Biological Scales—Continued
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
                                                                     GENESIS; NEURON
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
                                                                     Coupled-Map Lattice;

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