Use these tutorials to introduce quantitative sciences perspectives to your biological work. You can use this curriculum to prepare for introductory courses in undergraduate math, science, and systems biology. Know how to explain the canonical model for protein dynamics when it appears on the first day of systems biology 101. Understand why some systems of interacting molecules oscillate. Know how to estimate uncertainties and perform curve fitting. Know the difference between a normal and log-normal distribution and which is more intuitively expected in biology. If you find these tutorials helpful, these external resources and textbooks might interest you as well.

Pre-algebra, algebra, geometry, and precalculus

Topic	Slides	Video	Description

"What is a number?"			Street numbers, money in bank accounts, points on number lines, quantum particles as contrasted with distinguishable manipulatives For our purposes, infinity is not a number

Algebra			Variables: At once arbitrary, yet specific and particular (a.s.a.p.) Functions, composition, and inverse f(x) is not a function. f(x) is not the function f. f(x) is the particular value associated, in the big picture of the function f, with x, a number that is at once arbitrary, yet particular and specific Inverse functions do not always exist First glimpse at the complex plane and i := √ -1  Additional activity for this module

Quadratic equation			Linear combination of terms in a polynomial Zeroes or "roots" of a function Completing the square

Euclidean geometry and trigonometry			Flat space, curved space, non-embedded curved space Pythagorean theorem (ca. 300 BCE) The unit-radius circle, the unit-hypotenuse triangle, jya-ardha (sine), koti-jya (cosine) (ca. 510) Geometric construction for approximating π Angle-sum identities Additional activity for this module

Summation			Geometric series Harmonic series; sums do not always exist Gauss summation trick

Enumerative combinatorics			Permutations and factorials Combinations Binomial theorem Small parameter expansion

Calculus (ca. 1700s)

In this unit we define constructions in calculus (derivatives and integrals) so that we can eventually identify them, using explicit and concrete figures and words, with the processes of (1) describing the kinetic processes and (2) inferring long-term time-course outcomes for collections of chemically reacting biological molecules. This line of discussion presents differential equation models of biology as potentially useful approximations. Biochemistry is not always idiomatic for calculus; in fact, cellular biochemistry is an example of where applying calculus can be very dicey.

Limits			ε-δ definition of limit, notion of "arbitrarily close" Example of calculating a limit Limits do not always exist

Differentiation			Slopes and derivatives Derivatives do not always exist Common derivatives: power law, chain rule, product rule, quotient rule, sine, and cosine Infinitessimals describe processes; in this course they are not numbers

Taylor expansions			Second derivative and curvature Local approximations and Taylor series (Successful) power-series representations do not always exist Power-series expansion of sine and cosine, iterative calculation of π

Taylor expansions II			l'Hôpital's rule ("0/0" version) Newton-Raphson method for approximating zeros of a function

Integration			Anti-derivatives, Riemann sums, and integrals Example kosher calculation of a simple integral Deductive inference of integral by definition as anti-derivative "Backwards chain rule"--u substitution

Integration II			Integration by parts

Integration III			Integrals do not always exist Triangle/trigonometric substitution Partial fractions

Separation of variables			Two wrongs make a right Tear two differentials apart as though they retained meaning in isolation Slap on the smooth S integral sign as though it were a unit of meaning itself, even without a differential You get the same integral expression you would obtain long-hand using u-substitution or "change of variables" in integrals

Euler's number			Euler's number 1a: Compound interest Compounding interest with arbitrarily short compounding periods Power series representation of ex
			Euler's number 1b: e to the zero e0 = 1
			Euler's number 1c: Exponent multiplication identity (ex)p = epx
			Euler's number 1d: Exponent addition identity exey = ex+y
			Euler's number 1e: Andrew Jackson Mnemonic for memorizing e = 2.718281828459045...
			Euler's number 1f: Natural logarithm The natural logarithm is the inverse of the exponential ln(ex) = eln(x) = 1
			Euler's number 1g: Integral of 1/x ∫(1/x)dx = ln(x) + C

Euler's number projects day			Make a slide rule Equal-tempered tuning in European musical tradition

Non-dimensionalization

Stochasticity			(This section is a prerequisite for "protein dynamics 101"). Many of the homework problems from undergraduate calculus and differential equations involve notions of stochasticity. * For-real stochasticity: Fundamental indeterminism * Fake stochasticity: Periodic, deterministic hidden variables * Fake stochasticity: Aperiodic, deterministic (chaos) Markov models Additional activity for this module

Protein dynamics 101			This is a canonical worked problem from introductory systems biology (Alon, Ch. 2.4, pp. 18-21). We will explain one way to fantasize about the classic protein dynamics equation dx/dt = β - αx and analytically demonstrate that protein "rise time" depends on degradation rate only.

Mass action			Why mass action reaction rates look like terms from polynomials Cooperativity of the simple kind Hill functions

Gompertzian dynamics	PDF notes		A given trajectory can often be easily accommodated by a variety of physically inconsistent microscopic models Derivation from time-invariant, resource-limited governing equation Derivation from time-variant governing equation Negative sign floating around in Gompertzian age distribution (1825)

Linear algebra

LA I			Motivating example: Modeling dynamics of web start-up company customer base Vector space and basis Linear operator and representation Matrix multiplication Using eigenvalues and eigenvectors to avoid computational inefficiency Additional activity for this module

Euler's number II			Small-argument expansion Euler's formula: Expanding the exponential function in terms of sine and cosine

LA II			Euclidean rotation matrix Complex eigenvectors and eigenvalues

LA III			Orthogonality, dot product, and length Determinants Invertibility Euclidean rotation matrix

Quasispecies			Simple quasispecies eigendemographics and eigenrates

Differential equations

DEs I			Direction fields, quiver plots, and integral curves Numerical integration of systems of differential equations

DEs II			Integrating factor method Solution by power-series expansion

DEs III			Eigenvector-eigenrate solution method for canonical 2-by-2 system of linear differential equations with constant coefficients Phase portrait stability analysis

Transcription-translation			Canonical mRNA-protein system from systems biology 101

DEs IV			Using linear algebra to study nonlinear differential equations Local linearization: Jacobian

DEs V			Intuitive introduction to 2-d oscillations (Romeo and Juliet) Twisting nullclines Time-delays Stochastic resonance

Adaptation			Adaptation is not absence of change; instead it is the presence of eventually compensatory changes

Biological oscillators

Evolutionary game theory			Replicator dynamics Prisoner's dilemma Simultaneous survival of the relatively most fit with decrease in overall fitness

Additional homework			TBA

Ignoring small parameters (guess and check)
Fourier analysis
Example: Audio compression (irrelevance coding) on iPods and other MP3 players

Probability, statistics, and stochastic processes

Coin toss			Bernoulli (ca. 1700s) coin-toss process Binomial distribution

Limit of rare events			Poisson distribution

Luria-Delbrück fluctuation analysis			P0 method

Limit of many independent parts			Stirling's approximation Statement of central limit theorem, heuristic study of binomial distribution in limit of many coin tosses Biology's central limit theorem: Log-normal distributions

Working with brackets			Variances of independently fluctuating variables add Experimental uncertainty propagation

Parameters and sample estimators			Standard deviation vs. sample standard deviation Mean vs. sample mean Standard deviation of the mean vs. standard error of the mean

Apparent sample size			"I quantitated staining intensity for 1 million cells in contiguous fields of view, everything I measure is statistically significant!" Your data quite possibly fail to meet the fine print permitting application of the √ N  formula for calculating the standard error.

Curve fitting			Reduced chi-square χ2 Best-fit parameters are not independent (Gutenkunst, Sorger)

Describing heterogeneity			Why on earth have we multiple measurements of distribution heterogeneity? Some measures increase, some flatline, and some decrease with "effective number" size Standard deviation Coefficient of variation Fano factor Entropy, Shannon information, diversity

Stochastic models and their implementation in MatLab

Master equation and SSA			Interpretation of master equation Derivation of exponential distribution of waiting times

Numerical stochastic simulation			Pseudo-random number generators Example stochastic birth-death process script

Langevin integration method			Stochastic differential equations Gaussian white noise

Path integrals

Spatially-resolved models

Fast-Fourier transform

Efficient computation of local linear interactions			FFT convolution trick

Newtonian physics

Vectors			Review of vector addition and vector calculus

Force			Velocity v := dx/dt Momentum p := mv Momentum exchange for an isolated pair of bodies Superposition of pairwise momentum exchange FNET = ma

Newtonian gravitation			r2-law Approximate uniform gravitational acceleration for small relative changes in radii

Uniformly accelerated motion

Non-fundamental forces			Normal force Tension Kinetic and static friction

Simple-harmonic-oscillation (pendulum)

Lab: Measuring the gravitational field with a pendulum

Elastic materials (Hookean)
Elastic modulus
Propagation of vibrations on a taut string
Standing waves
Lab: Measure the speed of sound using a flute
Lab: Build a graduated cylinder water harmonica (equal-tempered tuning)

Derivation of the shape of a suspension bridge
Lab: Calculate the linear mass-density of the Golden Gate bridge

Momentum conservation
Work and energy conservation

Newton's law of gravitation
Rocket science
Electricity and magnetism

Modern physics

Light has wave-like properties

Lab: Location of phantom laser spots seen while wearing diffraction-grating glasses

Postulates of quantum mechanics
Heisenberg uncertainty relationship (not a principle, but a result): You postulate that eigenvalues of Hermitian operators represent experimental measurement outcomes; you define some operators in terms of other operators in ways such that they necessarily cannot share eigenvectors, and then you are surprised that dispersion in some observables can only be squished at the expense of precision in other observables
Ehrenfest theorem and correspondence principle, Newtonian physics from quantum mechanics
Particle in a box

Hydrogen atom			Single-electron solution

Multi-electron atom			Taylor-expansion of interaction potential See also: http://arxiv.org/abs/physics/0407126

Multi-electron molecules

Symmetries, conserved quantities, and extremized quantities
Path-integral formulation of quantum mechanics and wave optics, extremizing functions of path
Entanglement, Einstein-Podolsky-Rosen paradox, and Bell's inequalities

General relativity

Sine-Gordon equation and solitons

Statistical physics

Modern statistical mechanics			Heuristic justification of equal a priori probability postulate Second "law" of thermodynamics

Boltzmann distribution			Ways of sharing energy between a small system and a large reservoir Entropy Free energy

Ising model			Phase transition Hysteresis

Bose-Einstein statistics			Bose-Einstein systems obey Boltzmann distributions! A quantum index need not be interpreted as referring to a collection of distinct particles. Lagrange multiplier

Blackbody radiation
Diffusion, Brownian motion, and fluctuation-dissipation (Smoluchowski treatment)
Self-organized criticality
Chaos is not a lack of determinism

Biological physics and biophysics

Optical coherence microscopy
Fluorescence correlation spectroscopy and auto-correlation function
Lateral displacement mechanical flow cell sorter (Loutherback-Sturm)

Introduction to population dynamics topics			See table below

Fancy name	Descriptive name	Example insight
Fixation time of mutants	Allele proportions fluctuate stochastically in finite populations of true-breeding replicators	Rapid loss of heterogeneity is possible in small populations
Quasispecies	Interplay between dispersive forces that generate genotypic heterogeneity and populating forces that favor replication of fit genotypes determine whether populations expand, maintain heterogeneity, or disperse throughout genotypic space and lose macroscopic population numbers	Fitness, in the sense of rapid net production of progeny, is not sufficient to ensure survival. It also matters whether the progeny actually look like their parents.
Evolutionary game theory	Collections of (mostly) true-breeding cell subpopulations interact by modulating each other's net replication rates in a pairwise, linear manner	In a population where population composition influences the absolute fitness of both relatively unfit and relatively fit subpopulations, survival of the relatively fit can lead to decline in fitness for all individuals. The rich and the poor both get poorer.
Cellular automata	Probabilistic or deterministic prescription for changing the discrete state of a lattice point into the future is a function of the current configuration of its local spatial neighborhood	Cell subpopulations that would otherwise die out in a well-mixed population can survive by huddling together and minimizing their interactions with hostile neighbors. Spatiality can help sustain diversity.