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A mathematical way to think about biology

Interdisciplinary scientists can use these videos to investigate biological systems using a physical sciences perspective: training intuition by deriving equations from graphical illustrations.

Contents
Algebra Calculus and its Biological applications Probability and statistics Uncertainty propagation Stochastic dynamics	Linear algebra Differential equations Physical oncology ^NEW Spatiality Statistical physics

Target audience
Undergraduate students Graduate students Postdocs Principal investigators	PS-OCs ICBP Centers for Systems Biology Physics of Living Systems

“	Excellent site for both basic and advanced lessons on applying mathematics to biology. - Tweeted by the National Cancer Institute Office of Physical Sciences-Oncology

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

Track	Topic	Description
1	Numbers	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 Permanent link
2	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 Permanent link
3	Quadratic formula	Linear combination of terms in a polynomial Zeroes or "roots" of a function Completing the square Permanent link
4	Geometry	Geometry a: Euclidean geometry Flat space, curved space, non-embedded curved space Pythagorean theorem Permanent link
5		Geometry b: Trigonometry The unit-radius circle, the unit-hypotenuse triangle, jya-ardha (sine), koti-jya (cosine) Geometric construction for approximating π Angle-sum identities
6	Summation	Sums a: Summation notation Greek-letter Σ notation Gauss summation trick Permanent link
7		Sums b: Introduction to infinite series Geometric series Harmonic series; sums do not always exist
8	Combinatorics	Combinatorics a: Permutations and factorials We find that there are n (n - 1) (n - 2) . . . 2 * 1 ways can we arrange n distinct objects in n slots. Because this kind of calculation appears often in the study of probabilities, we give it a symbol called the factorial: n! = n (n - 1) (n -2) . . . 2 * 1. Permanent link
9		Combinatorics b: Combinations We obtain the famous (L + N)! / (L! N!) formula for counting the number of ways to arrange L indistinguishable objects and N indistinguishable objects together in a row. This is also the number of combinations of L objects that can be drawn from a container of L + N objects.
10		Combinatorics c: Binomial theorem We use the formula for combinations from the previous video to write an expression for the binomial quantity (x + y)^p. In some applications, only a small number of terms in the resulting sum are necessary for approximate calculations.

Calculus

Track	Topic	Description
11	Limits	ε-δ definition of limit, notion of "arbitrarily close" Example of calculating a limit Limits do not always exist Permanent link
12	Differentiation	Differentiation a: Derivatives and differentials We define the derivative, caution against interpreting differentials as numbers, and remark that derivatives do not always exist. It is important to become familiar with derivatives because they provide a basic vocabulary for talking about dynamical systems in the natural sciences (including in biology). Permanent link
13		Differentiation b: Power rule We will later learn that many seemingly complicated functions can be approximated using sums of power law terms. To study the slopes of these terms, we use the power rule that we derive in this video, which is written d(xⁿ)/dx = nx^n-1.
14		Differentiation c: Chain rule (for composite functions) One way to combine functions is to nest functions within each other. The chain rule is used to study the slopes of "composite" functions. The rule is written d(g(f))/dx = dg/df df/dx.
15		Differentiation d: Products and quotients Another way to put basic functions together is to write their expressions next to each other as a product. In this video, we derive the product rule, which is used in such situations. The product rule is written d(fg)/dx = (df/dx)g + f(dg/dx).
16		Differentiation e: Sinusoidal functions The derivative of sine is cosine, and the derivative of cosine is negative sine. This back-and-forth relationship is a hallmark of dynamical systems that might support oscillations. Thus, this pattern, which you will derive in this video, is important to keep in mind when you later study biological oscillations.
17	Partial differentiation	When a function depends on multiple independent variables, the curly-d symbol, ∂, denotes slopes calculated by jiggling only one independent variable at a time Permanent link
18	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 Permanent link
19	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 Permanent link
20	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 Permanent link
21	Euler's number I	Euler's number 1a: Compound interest Compounding interest with arbitrarily short compounding periods Power series representation of e^x Permanent link
22		Euler's number 1b: e to the zero e⁰ = 1
23		Euler's number 1c: Exponent multiplication identity (e^x)^p = e^px
24		Euler's number 1d: Exponent addition identity e^xe^y = e^x+y
25		Euler's number 1e: Andrew Jackson Mnemonic for memorizing e = 2.718281828459045...
26		Euler's number 1f: Natural logarithm The natural logarithm is the inverse of the exponential ln(e^x) = e^ln(x) = x
27		Euler's number 1g: Integral of 1/x ∫(1/x)dx = ln(x) + C

Biological applications of calculus

Track	Topic	Description
28	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 Permanent link
29	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). Permanent link
30	Mass action	Mass action a: Law of mass action Collision picture Permanent link
31		Mass action b: Cooperativity Cooperativity of the simple kind and Hill functions
32		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).
33	Evolutionary game theory I	EGT 1a: Population dynamics with interactions Equations for collisional population dynamics using law of mass action 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). Permanent link
34		EGT 1b: Introduction to tabular game theory Tabular game theory An outcome of the prisoner's dilemma is simultaneous stability of D with, as a consequence, lower than maximum possible payoff for D Our first verbal suggestion (1) that payoffs from tabular game theory can be associated with rate coefficients from the population dynamics in part 1a, and (2) that part 1a should be referred to as evolutionary game theory

Probability, statistics, and stochastic processes

Track	Topic	Description
35	Statistics 101	Distributions, averages, variances, useful identities Statistical independence Routinely-exploited expressions: Covariances vanish and variances of sums are sums of variances Permanent link
36	Probability 101	Probability 101a: Binomial distribution Bernoulli coin-toss process Independent events Permanent link
37		Probability 101b: Poisson limit Independent + "rare" events
38		Probability 101c: Stirling's approximation Comparison with integral of natural logarithm
39	Probability 101: Central limit theorem	Probability 101d: Central limit theorem Many independent events Binomial distribution in limit of many coin tosses Gaussian distribution Permanent link
40		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

Track	Topic	Description
41	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" Permanent link
42	Sample estimates	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 Permanent link
43		Uncertainty propagation c: Square-root of sample size (√ n ) factor Origin of the famous √ n factor by which the standard deviation of the sample means is smaller than the standard deviation of the measurements (parent distribution)
44		Uncertainty propagation d: Comparing error bars visually Are error bars non-overlapping, barely touching, or tightly overlapping? What p-value do people associate with the situation in which error bars barely touch?
45		Uncertainty propagation e: 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.
46	Curve fitting	Uncertainty propagation f: 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 activity: Sample-variance curve fitting exercise for MatLab (PDF) Additional resource: Web page on data fitting from the Harvey Mudd College physics kiosk (online) Permanent link

Stochastic dynamics

Track	Topic	Description
47	Basic stochastic simulation	Basic stochastic simulation a: Master equation Permanent link
48		Basic stochastic simulation b: Stochastic simulation algorithm Derivation of exponential distribution of waiting times Permanent link
49	Poissonian copy numbers	Poissonian copy numbers a: Stochastic synthesis Model: RNA polymerase makes many (usually unsuccessful) independent attempts to initiate transcription and mRNA strands degrade after a precise lifetime Outcome: mRNA copy numbers are Poisson distributed Permanent link
50		Poissonian copy numbers b: Stochastic synthesis and degradation Model: RNA polymerase makes many (usually unsuccessful) independent attempts to initiate transcription. Once a mRNA strand is produced, it begins to make independent (usually many unsuccessful) attempts to be degraded. Result: As in part a, mRNA copy numbers are Poisson distributed

Linear algebra

Track	Topic	Description
51	LA I	LA 1a: Teaser Motivating example: Modeling dynamics of web start-up company customer base Permanent link
52		LA 1b: Vectors Vectors, vector spaces, and coordinate systems
53		LA 1c: Operators Linear operators, matrix representation, matrix multiplication
54		LA 1d: Solution of teaser Using eigenvalue-eigenvector analysis to solve for the dynamics of the demographics of the web-startup customer base
55	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) (open-access online). Permanent link
56	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 e^iπ = -1 Permanent link
57	LA II	Rotation matrix Complex eigenvectors and eigenvalues Permanent link

Differential equations

Track	Topic	Description
58	DEs I	Direction fields, quiver plots, and integral curves Numerical integration of systems of differential equations Permanent link
59	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). Permanent link
60		DEs IIIb: Eigenvector-eigenvalue analysis Determine the directions of "unbending" trajectories for a more precise hand sketch of the phase portrait
61		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.
62	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 (2009) (online). Permanent link
63		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 (open-access online) and Harris, K., "Perturbations in linear systems (2008 November 12)," Math 216: Differential Equations, University of Michigan (online).
64	DEs V	Heuristic picture of oscillations in 2-d Intuitive introduction to 2-d oscillations (Romeo and Juliet) Twisting nullclines Time-delays Stochastic resonance Permanent link
		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.

Applications in physical oncology

Track	Topic	Slides	Video	Description
65	Dynamic heterogeneity for the physical oncologist			Physical oncology Nicole M. Moore, Nastaran Z. Kuhn, Sean E. Hanlon, Jerry S.H. Lee, and Larry A. Nagahara, "De-convoluting cancer's complexity: using a 'physical sciences lens' to provide a different (clearer) perspective of cancer," Phys. Biol. 8(1):010302 (2011) (online) Timothy J. Newman and Alastair M. Thompson, "Beyond detection: Biological physics informing progression and treatment of cancer," Phys. Biol. 9:060301 (2012) (online) Phenotypic stochasticity David Liao, Luis Estévez-Salmerón, and Thea D. Tlsty, "Conceptualizing a tool to optimize therapy based on dynamic heterogeneity," Phys. Biol. 9:065005 (2012) (open-access online) David Liao, Luis Estévez-Salmerón, and Thea D. Tlsty, "Generalized principles of stochasticity can be used to control dynamic heterogeneity," Phys. Biol. 9:065006 (2012) (open-access online) Permanent link

Spatially-resolved models and cellular automata

Track	Topic	Description
	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).
66	Cellular automata	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) Permanent link
		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).
	Fast-Fourier transform
	Efficient computation of local linear interactions	FFT convolution trick

Statistical physics

Track	Topic	Description
67	Statistical physics 101	Statistical physics 101a: Fundamental postulate of statistical mechanics Systems have states and energy levels Energy can be exchanged between parts of a world If the Hamiltonian of the world is time-independent, the overall energy of the world is conserved Fundamental postulate of statistical mechanics: In an isolated system, all accessible microstates are accessed equally Permanent link
68		Statistical physics 101b: Notating configurations of a system with multiple parts Direct product
69		Statistical physics 101c: Distribution of energy between a small system and a large bath Bath: many parts Number of ways to find the bath configured exponentially decays with increasing system energy Boltzmann factor
70		Statistical physics 101d: Expressions for calculating average properties of systems connected to baths The system energy most typically observed is the one that corresponds to the greatest number, W, of configurations of the world Ways (W), entropy (sigma), free energy (F), probability (P), partition function (Z), taking derivative of Z Maximizing ways of the world Maximizing entropy of the world Minimizing free energy of the system
71	Ideal chain	Ideal chain a: Introduction to model A series of links pointed up or down Permanent link
72		Ideal chain b: Hamiltonian and partition function Writing the partition function for a collection of independent links
73		Ideal chain c: Expectation of energy and elongation For heavy weights, the chain tends to be found extended fully. For lesser weights, the chain can be found partially crumpled, with the weight lifted, and with energy given to the bath.
		Ideal chain homework: Entropic elasticity Bustamante, Smith, Liphardt, and Smith, "Single-molecule studies of DNA molecules" Curr. Opin. Struct. Biol. 10(3):279--285 (2000) (PDF online)

© Copyright 2011-2013 David Liao. These videos and slides are open course ware made available under a Creative Commons license (CC BY-SA 3.0). The lightbox and social sharing effects are scripts by Stéphane Caron (CC BY 2.5).