Pre-algebra, algebra, geometry, and pre-calculus

We want to emphasize the following ideas about numbers and functions. (1) The number line is a sequenced collection of distinct addresses like the arrangment of house numbers along a street. (2) Infinity is not a number. (3) A variable, such as "x," is a stand in for a number from a set of interest, at once arbitrary in the sense that x could be any number from that set, but at the same time specific and particular in that x is considered at any one time only one number. (4) f(x) is the single result of applying function f to a particular x. f(x) can be plotted for a collection of values of x, as well as expressed in writing as an "association rule" using arithmetic and other operations on variable x. (5) Functions can be composed. (6) Plots of inverse functions are mirror images reflected about the y = x diagonal. If you are familiar with these ideas you can probably comfortably skip to the first set of blue videos.

"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

The following four videos are required for describing simple dynamic systems. As we will see much later, solving for the dynamics of systems with more than one moving part often reduces to solving for the roots of a polynomial expression. The simplest toy models solved this way involve polynomials of order 2. This is why we must understand how to solve a quadratic equation. We need trigonometry so we can use sinusoidal functions to describe oscillations. The videos on summation and combinatorics provide the underpinnings of the binomial theorem, which we later use to talk about the exponential function e^x, which appears when dynamics involve repetitive proportional scalings.

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

Partial differentiation			When a function depends on multiple independent variables, the ∂ denotes slopes calculated by jiggling only one independent variable at a time

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

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

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

Applications of calculus

Euler's number I			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

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. 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. Additional activity: See textbook presentation by Alon, An Introduction to Systems Biology: Design Principles of Biological Circuits, Boca Raton: Chapman & Hall/CRC, 2007 (p. 18-22).

Mass action			Mass action a: Law of mass action Collision picture
			Mass action b: Cooperativity Cooperativity of the simple kind and Hill functions
			Mass action c: Bistability Combining molecular production rates with nonlinear dose-dependence with unimolecular degradation can generate systems with multiple stable steady states
			Additional activity: See textbook presentation by Alon, An Introduction to Systems Biology: Design Principles of Biological Circuits, Boca Raton: Chapman & Hall/CRC, 2007 (sections 2.3-2.3.4, p. 7-16).

Evolutionary game theory I			Collisional population dynamics and tabular game theory An outcome of the prisoner's dilemma is simultaneous survival of the relatively most fit with decrease in overall fitness Additional activity: Access McKenzie, A.J., "Evolutionary Game Theory", The Stanford Encyclopedia of Philosophy (Fall 2009 Edition), Zalta, E.N. (ed.) (online) and compare the replicator dynamics described there with the collisional population dynamics in this tutorial. Watch Deborah Gordon talk about colony expansion, task allocation, and organization without central control in ant colonies (TED-talk video online).

Probability, statistics, and stochastic processes

Statistics 101			Distributions, averages, variances, useful identities Statistical independence Routinely-exploited expressions: Covariances vanish and variances of sums are sums of variances

Probability 101			Probability 101a: Binomial distribution Bernoulli (ca. 1700s) coin-toss process Independent events
			Probability 101b: Poisson limit Independent + "rare" events
			Probability 101c: Stirling's approximation Comparison with integral of natural logarithm
			Probability 101d: Central limit theorem Many independent events Binomial distribution in limit of many coin tosses Gaussian distribution
			Probability 101e: Stories that relate central limit theorem to physics and biology Physics: Taylor expansion in the context of tightly-controlled, narrow instrument noise Biology: Logarithm of product of fluctuating factors: Log-normal distributions

Uncertainty propagation			Uncertainty propagation a: Quadrature Quadrature formula is a result of Taylor expanding functions of multiple fluctuating variables, assuming that fluctuations are independent, and then applying the identity "variances of sums are sums of variances"
			Uncertainty propagation b: Sample estimates Standard deviation vs. sample standard deviation Mean vs. sample mean Standard deviation of the mean vs. standard error of the mean Rule of thumb for thinking about whether error bars overlap
			Uncertainty propagation c: Illusory sample size "I quantitated staining intensity for 1 million cells from 5 patients, everything I measure is statistically significant!" It is quite possible that you need to use n = 5, instead of 5 million, for the √ n  factor in the standard error.
			Uncertainty propagation d: Sample variance curve fitting Reduced chi-square χ2 fitting Do not assume that parameter fit uncertainties from black-box software packages are appropriate to interpret in a "covariance = zero" context (Gutenkunst, Sorger) Additional resource: Web page on data fitting from the Harvey Mudd College physics kiosk (online)

Stochastic dynamics

Basic stochastic simulation			Basic stochastic simulation a: Master equation
			Basic stochastic simulation b: Stochastic simulation algorithm 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

Linear algebra

LA I			LA 1a: Teaser Motivating example: Modeling dynamics of web start-up company customer base
			LA 1b: Vectors Vectors, vector spaces, and coordinate systems
			LA 1c: Operators Linear operators, matrix representation, matrix multiplication
			LA 1d: Solution of teaser Using eigenvalue-eigenvector analysis to solve for the dynamics of the demographics of the web-startup customer base
			Additional activity for this module

Quasispecies			Simple quasispecies eigendemographics and eigenrates Additional activity: Read the green box on p. 0454 from Bull, Meyers, and Lachmann, "Quasispecies made simple," PLoS Comp Biol, 1(6):e61 (2005) (online).

Euler's number II			Euler's formula: Expanding the exponential function in terms of sine and cosine Complex exponentials in the complex plane Euler's identity eiπ = -1

LA II			Rotation matrix Complex eigenvectors and eigenvalues

Differential equations

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

DEs III			DEs IIIa: Transcription-translation Canonical mRNA-protein system from systems biology 101 Additional activity: See textbook presentation by Alon, An Introduction to Systems Biology: Design Principles of Biological Circuits, Boca Raton: Chapman & Hall/CRC, 2007 (problem 2.2, p. 23).
			DEs IIIb: Eigenvector-eigenvalue analysis Determine the directions of "unbending" trajectories for a more precise hand sketch of the phase portrait
			DEs IIIc: The cribsheet of linear stability analysis Use eigenvalue-eigenvector analysis to find analytic solutions for linear systems and describe the qualitative features of trajectories approaching, side-swiping, or departing from steady state.

DEs IV			DEs IVa: Adaptation Adaptation is not absence of change; instead it is the presence of eventually compensatory changes Additional activity: Read Ma, Trusina, El-Samad, Lim, and Tang, "Defining network topologies that can achieve biochemical adaptation," Cell, 138: 760-773 (2008) (online).
			DEs IVb: Cribsheet of almost linear stability analysis Linear analysis of nonlinear systems Local linearization: Jacobian
			Additional activity: See "Sketching non-linear systems," from Differential Equations: Unit IV First-Order Systems, MIT OpenCourseWare (online) and Harris, K., "Perturbations in linear systems (2008 November 12)," Math 216: Differential Equations, University of Michigan (online).

DEs V			Heuristic picture of oscillations in 2-d Intuitive introduction to 2-d oscillations (Romeo and Juliet) Twisting nullclines Time-delays Stochastic resonance
			Additional activity: You may skim Ferrell, Jr., Tsai, and Yang, "Modeling the cell cycle: Why do certain circuits oscillate?" Cell, 144: 874-885 (2011)(online). Comment on how the positive-feedback term in Eqtn. 25 (pg. 882) contributes to the difference between the phase portraits in Fig. 4B (pg. 878) and Fig. 8B (pg. 883). The article describes the positive-feedback in terms of a time delay. Please describe the contribution of the positive-feedback term to stable oscillations instead in terms of "twisting nullclines" from the video tutorial.

Spatially-resolved models, cellular automata, and preview of agent-based models

Develop expectations			Seeing what computers can do In this activity, you will play against the computer in Blizzard's StarCraft for 2 hrs and in Sid Meier's Civilization for 2 hrs. WARNING: This activity might require rehabilitation and video game addiction treatment (PubMed).
			Cellular automata a Deterministic cellular automata In this video, we see that limiting dispersal of seeds of annual plants can increase the proportion of the copper-colored subpopulation, whereas thorough mixing instead allows the denim plant subpopulation to dominate quickly. Additional activities: Refer to a similar model in Nowak and May, "Evolutionary games and spatial chaos," Nature 359:826-829 (1992) (online). Watch Athena Aktipis talk about the walk-away model, which can contribute to the evolution of cooperation in highly-mobile populations (University of California, Los Angeles, Center for Behavior, Evolution, and Culture 2009, 1-hr video online)
			Cellular automata b Stochastic cellular automata
			Toy agent-based model You will program a simple ABM For more extensive discussion, see Athena Aktipis's page on agent-based modeling (online).

Newtonian physics

Simple-harmonic-oscillation (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

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

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

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

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