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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.

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Excellent site for both basic and advanced lessons on applying mathematics to biology.  Tweeted by the National Cancer Institute Office of Physical SciencesOncology 
Track  Topic  Slides  Video  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, nonembedded curved space Pythagorean theorem Permanent link 

5 
Geometry b: Trigonometry The unitradius circle, the unithypotenuse triangle, jyaardha (sine), kotijya (cosine) Geometric construction for approximating π Anglesum identities 

6  Summation 
Sums a: Summation notation Greekletter 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. 
Track  Topic  Slides  Video  Description 

11  Limits 
εδ definition of limit, notion of "arbitrarily close" Example of calculating a limit Limits do not always exist For an example of a strategy for writing εδ proofs useful for plots of functions that have curvature, please see Yosen Lin's examples (example # 4 on p. 34). 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(x^{n})/dx = nx^{n1}. 

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 backandforth 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 curlyd symbol, ∂, denotes slopes calculated by jiggling only one independent variable at a time Permanent link 

18  Power series representations 
Power series representations a: Second derivative and curvature Using a power series representation is like using decimal representation. Both techniques organize the description of the target object at levels of increasing refinement. In this first video, we show that the second derivative corresponds to the curvature of a plot. In this way, we strengthen intuition that higherorder derivatives can also have geometric interpretations. Permanent link 

19 
Power series representations b: Determining power series terms We imitate a function by combining the descriptions of its geometric properties as embodied in its value and the values of its higher derivatives at an expansion point. 

20 
Power series representations c: Power series for sine We obtain a power series representation for the sine function expanded about the point θ = 0. 

21 
Power series representations d: Decimal approximation for π Using the first three terms of the power series representation for sine we obtained in the previous video, we iteratively approximate π to four decimal places. 

22  Integration 
Integration a: Area under a curve In these four videos, we develop a familiar with integration that will later be useful for deducing functions of time (e.g. number of copies of a molecule as a function of time) using rates of change (e.g. the first derivative of the number of copies of a molecule with respect to time). In this first video, we develop the concept of the definite integral in terms of the area under a curve. Permanent link 

23 
Integration b: First fundamental theorem of calculus In this video, we demonstrate that differentiation undoes integration. This is called the first fundamental theorem of calculus. 

24 
Integration c: Second fundamental theorem of calculus We demonstrate that integration undoes differentiation. This is called the second fundamental theorem of calculus. This theorem allows us to construct a table of integrals using differentiation rules we previously learned. 

25 
Integration d: Change of variables rule Sometimes, superficial differences can make it seem that a listing in an integration table does not match the integral we want to study. We develop a change of variables (also called a "usubstitution") rule that can sometimes help us to identify a match between an integral we want to study and a listing in a table. 

26  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 longhand using usubstitution or "change of variables" in integrals Permanent link 

27  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 

28 
Euler's number 1b: e to the zero e^{0} = 1 

29 
Euler's number 1c: Exponent multiplication identity (e^{x})^{p} = e^{px} 

30 
Euler's number 1d: Exponent addition identity e^{x}e^{y} = e^{x+y} 

31 
Euler's number 1e: Andrew Jackson Mnemonic for memorizing e = 2.718281828459045... 

32 
Euler's number 1f: Natural logarithm The natural logarithm is the inverse of the exponential ln(e^{x}) = e^{ln(x)} = x 

33 
Euler's number 1g: Integral of 1/x ∫(1/x)dx = ln(x) + C 
Track  Topic  Slides  Video  Description 

34  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. * Forreal stochasticity: Fundamental indeterminism * Fake stochasticity: Periodic, deterministic hidden variables * Fake stochasticity: Aperiodic, deterministic (chaos) Markov models Permanent link 

35  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. 1822). Permanent link 

36  Mass action 
Mass action a: Law of mass action Collision picture Permanent link 

37 
Mass action b: Cooperativity Cooperativity of the simple kind and Hill functions 

38 
Mass action c: Bistability Combining molecular production rates with nonlinear dosedependence 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 2.32.3.4, p. 716).  
39  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 (TEDtalk video online). Permanent link 

40 
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 

41  Evolutionary game theory II 
EGT 2a: Evolution resulting from repeated game play In the previous slide deck, we noted similarities between population dynamics and business transaction payoff pictures. In this and the next video, we provide deeper understanding of these connections. In this video, we derive the population dynamics equations in such a way that it is natural to say that cells being modeled repeatedly play games and are subject to game outcomes. Permanent link 

42 
EGT 2b: Relationship between time and sophisticated computation Repeated simple interactions in a population of robotic replicators can achieve results seemingly related to results obtained from sophisticated computations. The use of population dynamics and business transaction payoff matrix analyses from the previous slide deck to obtain this understanding is an example of quantitative reasoning. 
Track  Topic  Slides  Video  Description 

43  Statistics 101 
Distributions, averages, variances, useful identities Statistical independence Routinelyexploited expressions: Covariances vanish and variances of sums are sums of variances Permanent link 

44  Probability 101 
Probability 101a: Binomial distribution Bernoulli cointoss process Independent events Permanent link 

45 
Probability 101b: Poisson limit Independent + "rare" events 

46  Probability 101: Central limit theorem 
Probability 101c: Stirling's approximation Comparison with integral of natural logarithm Permanent link 

47 
Probability 101d: Central limit theorem Many independent events Binomial distribution in limit of many coin tosses Gaussian distribution 

48 
Probability 101e: Stories that relate central limit theorem to physics and biology Physics: Taylor expansion in the context of tightlycontrolled, narrow instrument noise Biology: Logarithm of product of fluctuating factors: Lognormal distributions 
Track  Topic  Slides  Video  Description 

49  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 

50  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 

51 
Uncertainty propagation c: Squareroot 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) 

52 
Uncertainty propagation d: Comparing error bars visually Are error bars nonoverlapping, barely touching, or tightly overlapping? What pvalue do people associate with the situation in which error bars barely touch? 

53 
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. 

54  Curve fitting 
Uncertainty propagation f: Sample variance curve fitting Reduced chisquare χ^{2} fitting Do not assume that parameter fit uncertainties from blackbox software packages are appropriate to interpret in a "covariance = zero" context (Gutenkunst, Sorger) Additional activity: Samplevariance curve fitting exercise for MatLab (PDF) Additional resource: Web page on data fitting from the Harvey Mudd College physics kiosk (online) Permanent link 
Track  Topic  Slides  Video  Description 

55  Basic stochastic simulation 
Basic stochastic simulation a: Master equation Permanent link 

56 
Basic stochastic simulation b: Stochastic simulation algorithm Derivation of exponential distribution of waiting times Permanent link 

57  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 

58 
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 
Track  Topic  Slides  Video  Description 

59  LA I 
LA 1a: Teaser Motivating example: Modeling dynamics of web startup company customer base Permanent link 

60 
LA 1b: Vectors Vectors, vector spaces, and coordinate systems 

61 
LA 1c: Operators Linear operators, matrix representation, matrix multiplication 

62 
LA 1d: Solution of teaser Using eigenvalueeigenvector analysis to solve for the dynamics of the demographics of the webstartup customer base 

63  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) (openaccess online). Permanent link 

64  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 

65  LA II 
Rotation matrix Complex eigenvectors and eigenvalues Permanent link 
Track  Topic  Slides  Video  Description 

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

67  DEs III 
DEs IIIa: Transcriptiontranslation Canonical mRNAprotein 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 

68 
DEs IIIb: Eigenvectoreigenvalue analysis Determine the directions of "unbending" trajectories for a more precise hand sketch of the phase portrait 

69 
DEs IIIc: The cribsheet of linear stability analysis Use eigenvalueeigenvector analysis to find analytic solutions for linear systems and describe the qualitative features of trajectories approaching, sideswiping, or departing from steady state. 

70  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, ElSamad, Lim, and Tang, "Defining network topologies that can achieve biochemical adaptation," Cell, 138: 760773 (2009) (online). Permanent link 

71 
DEs IVb: Cribsheet of almost linear stability analysis Linear analysis of nonlinear systems Local linearization: Jacobian 

Additional activity: See "Sketching nonlinear systems," from Differential Equations: Unit IV FirstOrder Systems, MIT OpenCourseWare (openaccess online) and Harris, K., "Perturbations in linear systems (2008 November 12)," Math 216: Differential Equations, University of Michigan (online).  
72  DEs V 
Heuristic picture of oscillations in 2d Intuitive introduction to 2d oscillations (Romeo and Juliet) Twisting nullclines Timedelays 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: 874885 (2011)(online). Comment on how the positivefeedback 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 positivefeedback in terms of a time delay. Please describe the contribution of the positivefeedback term to stable oscillations instead in terms of "twisting nullclines" from the video tutorial. 
Track  Topic  Slides  Video  Description 

73  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, "Deconvoluting 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 EstévezSalmerón, and Thea D. Tlsty, "Conceptualizing a tool to optimize therapy based on dynamic heterogeneity," Phys. Biol. 9:065005 (2012) (openaccess online) David Liao, Luis EstévezSalmerón, and Thea D. Tlsty, "Generalized principles of stochasticity can be used to control dynamic heterogeneity," Phys. Biol. 9:065006 (2012) (openaccess online) Permanent link 
Track  Topic  Slides  Video  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). 

74  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 coppercolored 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 359:826829 (1992) (online). Watch Athena Aktipis talk about the walkaway model, which can contribute to the evolution of cooperation in highlymobile populations (University of California, Los Angeles, Center for Behavior, Evolution, and Culture 2009, 1hr video online) Permanent link 

Cellular automata b Stochastic cellular automata 

Toy agentbased model You will program a simple ABM For more extensive discussion, see Athena Aktipis's page on agentbased modeling (online). 

FastFourier transform  
Efficient computation of local linear interactions  FFT convolution trick 
Track  Topic  Slides  Video  Description 

75  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 timeindependent, 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 

76 
Statistical physics 101b: Notating configurations of a system with multiple parts Direct product 

77 
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 

78 
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 

79  Ideal chain 
Ideal chain a: Introduction to model A series of links pointed up or down Permanent link 

80 
Ideal chain b: Hamiltonian and partition function Writing the partition function for a collection of independent links 

81 
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, "Singlemolecule studies of DNA molecules" Curr. Opin. Struct. Biol. 10(3):279285 (2000) (PDF online) 
If this describes you,  then this is how you can use this digest 

I take tonight's red eye to give a talk on quantitative biology in the morning. I haven't had time to learn this field. How can I learn to use buzzwords convincingly?  Download the videos in this digest (follow the vimeo links in the video lightboxes). The indicated segments can be viewed in under 2 hours, and even an hour's sample should offer an informative taste of reasoning styles and topics commonly encountered in introductory quantitative biology. Buzzwords are like salt. Avoid unconvincing overuse. Observe that the videos in the digest never use the words "complexity" or "emergence" even though both terms could be used repeatedly. 
My institution already trains faculty in bioinformatics. How much of this website can I skip?  This is not a bioinformatics course (there's only a little bit of rudimentary probability). Going in the other direction, much of the content from this course might be missing from your bioinformatics training. To determine whether you are already familiar with styles of reasoning in quantitative biology, respond to the quiz questions below and view the accompanying videos that follow. 
I just got funded to collaborate in quantitative biology. How do I find experts to coach me to understand the mathematical models I need to use?  You could test possible instructors by asking them to help you work through some of the quiz questions below. See how they answer (it's probably best to watch the corresponding video sections ahead of time). 
Track  Topic  Slides  Video  Description 

D1  Protein dynamics 101 
Instructors use this classic surprise in introductory qbio courses to illustrate the need for mathematical reasoning to understand biological systems. Pull the citation for Alon's textbook presentation by jumping to the Permanent link in main curriculum  Permanent link in digest 
Track  Topic  Slides  Video  Description 

D2  Mass action 
Mass action a: Law of mass action Collision picture Permanent link in main curriculum  Permanent link in digest 

D3 
Mass action b: Cooperativity Cooperativity of the simple kind and Hill functions 

D4 
Mass action c: Bistability Combining molecular production rates with nonlinear dosedependence with unimolecular degradation can generate systems with multiple stable steady states 
Track  Topic  Slides  Video  Description 

D5  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 Watch the sequel to this video and get additional references by jumping to the Permanent link in main curriculum  Permanent link in digest 
Track  Topic  Slides  Video  Description 

D6  Poissonian copy numbers 
Poissonian copy numbers a: Stochastic synthesis To watch the sequel to this video, jump to the Permanent link in main curriculum  Permanent link in digest 
Track  Topic  Slides  Video  Description 

D7  Quasispecies 
Simple quasispecies eigendemographics and eigenrates, adapted from Bull, Meyers, and Lachmann, "Quasispecies made simple," PLoS Comp Biol, 1(6):e61 (2005) (openaccess online). Permanent link in main curriculum  Permanent link in digest 
Track  Topic  Slides  Video  Description 

D8  Differential Eqtns IV 
DEs IVa: Adaptation Please see the excerpt from 19 min 14 sec to 21 min 27 sec. This video is inspired by Ma, Trusina, ElSamad, Lim, and Tang, "Defining network topologies that can achieve biochemical adaptation," Cell, 138: 760773 (2009) (online). To view the short sequel to this video, jump to the Permanent link in main curriculum  Permanent link in digest 
Track  Topic  Slides  Video  Description 

D9  Differential Eqtns V 
Heuristic picture of oscillations in 2d For additional description and further reading, jump to the Permanent link in main curriculum  Permanent link in digest 
Track  Topic  Slides  Video  Description 

D10  Dynamic heterogeneity for the physical oncologist 
Description of the metronomogram developed in the Physical SciencesOncology Network. For references, jump to the Permanent link in main curriculum  Permanent link in digest 
Track  Topic  Slides  Video  Description 

D11  Cellular automata 
Cellular automata a Deterministic cellular automata For additional description and references, please jump to the Permanent link in main curriculum  Permanent link in digest 
© Copyright 20112013 David Liao. These videos and slides are open course ware made available under a Creative Commons license (CC BYSA 3.0). The lightbox and social sharing effects are scripts by Stéphane Caron (CC BY 2.5). 