A mathematical way to think about biology

Course Curriculum

Lecture 1: Welcome to mathematics for insightful biology

Chapter 1: Using deterministic models to study aspects of stochastic systems

Lecture 1: Stochasticity a: Incommensurate periods

Lecture 2: Stochasticity b: Practically unpredictable deterministic dynamics

Lecture 3: Stochasticity c: Fundamentally indeterministic processes

Lecture 4: Stochasticity d: Memory-free (Markov) processes and their visual representations

Lecture 5: Canonical protein dynamics a: Translation and degradation events occur over time

Lecture 6: Canonical protein dynamics b: Differential equation and flowchart

Lecture 7: Canonical protein dynamics c: Qualitative graphical solution

Lecture 8: Canonical protein dynamics d: Analytic solution and rise time

Lecture 9: Mass action 1a: Law of mass action

Lecture 10: Mass action 1b: Cooperativity and Hill functions

Lecture 11: Mass action 1c: Bistability

Lecture 12: Evolutionary game theory Ia: Population dynamics

Lecture 13: Evolutionary game theory 1b: Preview comparison with tabular game theory

Lecture 14: Evolutionary game theory IIa: Cells repeatedly playing games

Lecture 15: Evolutionary game theory IIb: Relationship between time and sophisticated comput

Chapter 2: Probability and statistics

Lecture 1: Statistics a: Probability distributions and averages

Lecture 2: Statistics b: Identities involving averages

Lecture 3: Statistics c: Dispersion and variance

Lecture 4: Statistics d: Statistical independence

Lecture 5: Statistics e: Identities following from statistical independence

Lecture 6: Probability a: Bernoulli trial

Lecture 7: Probability b: Binomial distribution

Lecture 8: Probability c: Poisson distribution

Lecture 9: Preparation for central limit theorem: Stirling's approximation

Lecture 10: Central limit theorem a: Statement of theorem

Lecture 11: Central limit theorem b: Optional derivation (special case)

Lecture 12: Central limit theorem c: Properties of Gaussian distributions

Lecture 13: Prevalence of Gaussians a: Noise in physics labs is allegedly often Gaussian

Lecture 14: Prevalence of Gaussians b: Noise in biology is allegedly often log-normal

Chapter 3: Uncertainty propagation

Lecture 1: Uncertainty propagation a: Quadrature

Lecture 2: Uncertainty propagation b: Sample estimates

Lecture 3: Uncertainty propagation c: Square-root of sample size (sqrt(n)) factor

Lecture 4: Uncertainty propagation d: Comparing error bars visually

Lecture 5: Uncertainty propagation e: Illusory sample size

Lecture 6: Sample variance curve fitting a: Chi-squared

Lecture 7: Sample variance curve fitting b: Minimizing chi-squared

Lecture 8: Sample variance curve fitting c: Checklist for undergraduate curve fitting

Lecture 9: Sample variance curve fitting exercise for MatLab

Chapter 4: Computation of stochastic dynamics

Lecture 1: Master equation

Lecture 2: Stochastic simulation algorithm a: Specifying reaction types and stoichiometries

Lecture 3: Stochastic simulation algorithm b: Time until next event

Lecture 4: Stochastic simulation algorithm c: Determining type of next event

Lecture 5: Poissonian copy numbers a: Stochastic transcription and deterministic degradation

Lecture 6: Poissonian copy numbers b: Stochastic transcription and stochastic degradation

Chapter 5: Linear algebra

Lecture 1: Linear algebra Ia: Teaser

Lecture 2: Linear algebra Ib: Vectors

Lecture 3: Linear algebra Ic: Operators

Lecture 4: Linear algebra Id: Solution of teaser (part 1)

Lecture 5: Linear algebra Id: Solution of teaser (continued)

Lecture 6: Intro quasispecies a: Population dynamics from single-cell mechanisms

Lecture 7: Intro quasispecies b: Eigenvalue-eigenvector analysis

Lecture 8: Euler II: Complex exponentials

Lecture 9: Linear algebra II: Rotation a: Rotation matrix

Lecture 10: Linear algebra II: Rotation b: Complex eigenvalues

Chapter 6: Differential equations

Lecture 1: Numerical integration of differential equations

Lecture 2: Linear stability analysis a: Transcription-translation model

Lecture 3: Linear stability analysis b: Nullclines and critical point

Lecture 4: Linear stability analysis c: Eigenvalue-eigenvector analysis

Lecture 5: Linear stability analysis d: Cribsheet

Lecture 6: Almost linear stability analysis a: Incoherent feed-forward loop

Lecture 7: Almost linear stability analysis b: Adaptation

Lecture 8: Almost linear stability analysis c: Eigenvalue-eigenvector analysis

Lecture 9: Almost linear stability analysis d: Cribsheet

Lecture 10: Oscillations a: Romeo and Juliet

Lecture 11: Oscillations b: Twisting nullclines

Lecture 12: Oscillations c: Time delays

Lecture 13: Oscillations d: Stochastic excitation

Chapter 7: Physical oncology

Lecture 1: Introduction to physical oncology

Lecture 2: Dynamic heterogeneity a: Stochastic biochemistry

Lecture 3: Dynamic heterogeneity b: Phenotypic interconversion

Lecture 4: Dynamic heterogeneity c: Metronomogram

Chapter 8: Spatially-resolved systems

Lecture 1: Cellular automata a: Deterministic cellular automata

Chapter 9: Statistical physics

Lecture 1: Statistical physics 101a: Fundamental postulate of statistical mechanics

Lecture 2: Statistical physics 101b: Cartesian product

Lecture 3: Statistical physics 101c: Distribution of energy between a small system and a large bath

Lecture 4: Statistical physics 101d: Expressions for calculating average properties of systems connected to baths

Lecture 5: Ideal chain a: Introduction to model

Lecture 6: Ideal chain b: Hamiltonian and partition function

Lecture 7: Ideal chain c: Expectation of energy and elongation

Lecture 8: Macroscopic irreversibility a: Microstates of universe are explored over time

Lecture 9: Macroscopic irreversibility b: Microscopic reversibility

Lecture 10: Macroscopic irreversibility c: Ratio of volumes in phase space

Lecture 11: Macroscopic irreversibility d: Kinetically accessible volumes of phase space

Chapter 10: Appendix: Algebra

Lecture 1: Numbers a: Distinct manipulatives and geographic addresses

Lecture 2: Numbers b: Bose-Einstein statistics

Lecture 3: Numbers c: Visual representations of numbers

Lecture 4: Numbers d: Infinity is not a number

Lecture 5: Algebra a: Variables

Lecture 6: Algebra b: Functions