Introduction to Mathematical Biology Mathematical Biology Lecture 1 James A. Glazier

P548/M548 Mathematical Biology Instructor: James A. Glazier Classes: Tu. Thu. 8:00AM–9:30AM Office Hours: By appointment Texts: 1) Murray – Mathematical Biology volumes 1 & 2 2) Fall, Marland, Wagner and Tyson – Computational Cell Biology 3) Keener and Sneyd – Mathematical Physiology Requirements: 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 Background Stochastic Gating Reaction Kinetics, Oscillating Reactions, and Reactor Networks Molecular Motors Collective Phenomena – Flocks and Neural Networks 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.

Goals Teach a set of generally useful methodologies. 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 statistics) Not Cluster Analysis, Image Processing, Pattern Recognition

Goals To explain biological processes that result in an observed phenomena. To predict previously unobserved phenomena. 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 predictive. Epistemology – Car parts metaphor. Simplify impossible complexity by forcing a hierarchy of importance – identifying key mechanisms. In a model know what all processes are. Failure of models can identify missing components or concepts.

Biological Scales ScaleExamplesMethods Atomic DNA; Protein Structure, Binding and Conformation Ion Channels and Photosynthesis Quantum Chemistry Molecular Receptor-ligand binding Signal Transduction Molecular Dynamics (Classical); BIOSYM Networks Genetic Regulatory Networks Metabolic Networks Diabetes Coupled ODE Models; Stochastic ODEs; Network Analysis; BioSpice; PhysioLab Macromolecular Molecular Motors; Actin; Microtubules; Intermediate Filaments; Chromosomes; DNA Coiling; DNA Transcription; Protein Synthesis Simplified Molecular Dynamics Langevin Equation Fokker-Planck Equation Molecular Systems Junctions; Stress Fibers; Cilia; Flagella; Pseudopods; Fliopodia; Mitotic Spindles; Growth Cones; Endoplasmic Reticulum; Cell Membranes; Cell Polarity; Cell Motility VirtualCell; Karyote; e- Cell; M-Cell

Biological Scales— Continued ScaleExamplesMethods Cellular Cell Adhesion; Chemotaxis; Haptotaxis; Cell Differentiation Epithelia; Cell Sorting; Bacterial patterning; Dictyostelium Neurons; Myofibers; Cancer; Stem Cells Cellular Potts Model; Center Models; Boundary Models; Hodgkin-Huxley Model; GENESIS; NEURON Tissue [Including Individual Cells] Wound Healing; Angiogenesis; Kidney Development; Lung Development; Neural Circuirts; Tumor Growth Cellular Automata; Reaction- Diffusion Models; Fitz-Hugh- Nagumo Equation; Coupled PDEs; Stochastic PDEs Organ [Neglecting Individual Cells] Heart; Circulatory System; Bone; Cartilage; Neural Networks; Organ Development Continuum Mechanics; Finite Element Methods; Navier-Stokes Equations; Coupled-Map Lattice; PHYSIOME Individual Flocks; Theories of Learning Agent-based Models; SWARM Population Infections/Epidemiology; Population Modeling; Predator-Prey Models; Evolutionary Models; Bacterial and Eukaryotic Communities; Traffic Continuum ODEs; PDEs; Iterated maps; Kaufmann Nets; Delay ODEs; AVIDA