Transcript P548/M548 Mathematical Biology

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
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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
Scale
Examples
Methods
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; eCell; M-Cell
Biological Scales—Continued
Scale
Examples
Methods
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 Wound Healing; Angiogenesis; Kidney
Individual Cells] Development; Lung Development; Neural
Circuirts; Tumor Growth
Cellular Automata; ReactionDiffusion Models; Fitz-HughNagumo 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