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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.
Target audience
• 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 Slides Video Description
1 Numbers Numbers a: Distinguishable manipulatives and geographic addresses
In this and the following three videos, we will review the concept of quantity, which is represented by numbers. In this video, we review two ways in which we learned to think about numbers in elementary school. We used numbers to refer to the idea of having distinct manipulatives, and we used numbers to refer to the idea of labeling geographic locations with addresses.
2 Numbers b: Bose-Einstein statistics
The analysis of a system of particles display Bose-Einstein statistics is an example of a situation in which it is important to be aware whether we are thinking of numbers in terms of distinct manipulatives or in terms of addresses on a street. Incorrectly assuming that atomic and subatomic particles are just as distinct as the plastic counting manipulatives from kindergarten leads to overestimating the number of ways that particles can be excited out of the lowest energy state. In some situations, a system of particles that tends to occupy the lowest energy state in a way that is quantitatively consistent with thinking of numbers in terms of addresses (rather than thinking of particles as distinct manipulatives) is sometimes referred to as a Bose-Einstein condensate.
3 Numbers c: Visual representations of numbers
Numbers can be represented using a number line, a wedge, and place-value representation. The application of memorized rules for performing arithmetic on numerals formatted in place-value representation is called algorism.
4 Numbers d: Infinity is not a number
Infinity is not a number. There is no tick mark on the number line labeled "infinity."
5 Algebra Algebra a: Variables
This slide deck presents aspects of quantitative "vocabulary" (variables) and quantitative "grammar" (functions and function composition) that will allow us to express quantitative reasoning in future slide decks. In this first of five videos, we note that it is cumbersome to describe quantitative relationships purely through the enumeration of repetitive examples involving concrete numbers. This difficult can be addressed with the assistance of abstract "placeholder," "stand-in" symbols. A variable is a symbol that stands in for a number at once arbitrary, yet specific and particular. Using variables, we can communicate quantitative relationships concisely.
6 Algebra b: Functions
Functions are basic building-block sentences of mathematical reasoning. A function relates input values in a domain to output values in a codomain, and these associations can be depicted using plots. While different disciplines use slightly different definitions of a function, an essential stipulation familiar to scientists and mathematicians from a variety of fields is that a function associates each input value with precisely one output value.
7 Algebra c: Composition of functions
Functions can be combined by using the output of one function as the input for another function. The resulting object is a composite function, which is one way to combine mathematical ideas to derive mathematical conclusions.
8 Algebra d: Inverse functions
When two functions are called each other's inverses, they can be composed. The overall composite function has the property that the value entered as an input is returned as an output. The plot of the composition of inverse functions is the diagonal line y = x.
9 Algebra e: Square-root function and imaginary root i
When we try to think of an inverse of the squaring function, we encounter two difficulties. One problem is that the reflection of the parabola y = x2 is, in many places, double-valued, and, thus, not a function. Second, this plot does not explore negative input values. When we attempt to address this second difficulty, we develop the idea of the imaginary root i, which, when squared, gives -1. Knowledge of the imaginary root because will help us to study oscillatory dynamics in a later slide deck.
10 Quadratic formula Linear combination of terms in a polynomial
Zeroes or "roots" of a function
Completing the square
11 Geometry Geometry a: Euclidean geometry
The geometry routinely used by physical scientists on a day-to-day basis is only a small portion of the typical high school course. Useful concepts include the notion of a flat space (as opposed to a curved space), as well as the Pythagorean theorem.
12 Geometry b: Sine and cosine in relation to the unit circle
The unit circle is a circle of radius one centered at the origin of the xy-coordinate plane. The location of a point on a circle is specified by the angle θ it sweeps counterclockwise from the x axis. The location of a point is also specified using its corresponding x- and y-coordinates, which, in this context, are referred to as cos(θ) and sin(θ), respectively.
13 Geometry c: Approximating π
Using the Pythagorean theorem to relate the lengths of sides of triangles drawn in the context of a circle, we estimate π. We also provide a mnemonic for memorizing π to 6 digits. This allows us to understand that the tick marks on the horizontal axis of the function plots from the previous video correspond to numerical values.
14 Geometry d: Right triangles and trigonometric identities
Even though sine and cosine are fundamentally defined as functions that provide the y- and x-coordinates, respectively, of points on the unit circle, sine and cosine are also regarded as "trigonometric" functions, which describe the geometry of right triangles. We practice applying this perspective as we derive two examples of identities involving sine and cosine.
15 Summation Sums a: Summation notation
Greek-letter Σ notation
Gauss summation trick
16 Sums b: Introduction to infinite series
Geometric series
Harmonic series; sums do not always exist
17 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.
18 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.
19 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 Slides Video Description
20 Limits Limits a: Limit of a function
Informally, when we say that the limit of a function as x approaches a is L, we mean that as x becomes arbitrarily close to a, the function becomes arbitrarily close to L. This idea is made more precise using the ε-δ definition.

For an example of a strategy for writing ε-δ proofs useful for plots of functions that have curvature, please see Yosen Lin's examples (example # 4 on p. 3-4).
21 Limits b: Improper (infinite) limits
When we say that the limit of a function at a value of x = a is infinity, we mean that as x becomes arbitrarily close to a, the value of the function becomes arbitrarily large.
22 Limits c: Limits "at" infinity
When we say that a function has a limit of L "at" infinity, we mean that as x becomes arbitrarily large, the function becomes arbitrarily close to L.
23 Limits d: Infinite limits "at" infinity
When we say that a function has an infinite limit "at" infinity, we mean that as x becomes arbitrarily large, the function becomes arbitrarily large.
24 Limits e: Limits do not always exist
An example of a situation in which a function can fail to have a limit at a value of x = a is when the function jumps discontinuously in height at that value of x. One example of a situation in which a function can fail to have a limit at infinity is an oscillatory function that fails to approach a particular value of y = L because it keeps swinging with sustained amplitude up and down through y = L.
25 Limits f: Outline of ε-δ proof of a limit of a linear function
In this video, the outline for using the ε-δ definition to prove that the limit of a function has a particular value y = L at x = a has two main parts. First, we determine what range of y values the function takes when x is restricted to intervals on either side of the value x = a of interest. Then, we ask whether we can narrow these intervals sufficiently to ensure that the range of y values taken by the function is contained within a range of y values of interest centered at y = L. When we conclude that this can be done for any finite range of such y values, we conclude that the limit of interest exists.
26 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).
27 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 = nxn-1.
28 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.
29 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).
30 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.
31 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
32 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 higher-order derivatives can also have geometric interpretations.
33 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.
34 Power series representations c: Power series for sine
We obtain a power series representation for the sine function expanded about the point θ = 0.
35 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.
36 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.
37 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.
38 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.
39 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 "u-substitution") rule that can sometimes help us to identify a match between an integral we want to study and a listing in a table.
40 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
41 Euler's number I Euler's number 1a: Compound interest Compounding interest with arbitrarily short compounding periods
Power series representation of ex
42 Euler's number 1b: e to the zero
e0 = 1
43 Euler's number 1c: Exponent multiplication identity
(ex)p = epx
44 Euler's number 1d: Exponent addition identity
exey = ex+y
45 Euler's number 1e: Andrew Jackson
Mnemonic for memorizing e = 2.718281828459045...
46 Euler's number 1f: Natural logarithm
The natural logarithm is the inverse of the exponential ln(ex) = eln(x) = x
47 Euler's number 1g: Integral of 1/x
∫(1/x)dx = ln(x) + C

Biological applications of calculus

Track Topic Slides Video Description
48 Stochasticity Stochasticity a: Incommensurate periods
Concepts of stochasticity underlie many of the models of dynamic systems explored in quantitative biology. We describe some of these ideas in this and the following three videos. In this video, we state that systems exhibiting deterministic dynamics can sample a messy variety of waiting times between chemical reaction events even when the motions of component parts are periodic. Particularly, this can happen when the periods of motion of individual parts are incommensurate (pairs of periods form ratios that are irrational).
49 Stochasticity b: Practically unpredictable deterministic dynamics
In a deterministic system with complicated interactions, small differences in initial conditions can quickly avalanche into qualitative differences in dynamics. Since initial conditions can only be measured with finite certainty, the dynamics of such systems are, for practical purposes, unpredictable after short times.
50 Stochasticity c: Fundamentally indeterministic processes
In the previous two videos, deterministic systems displayed dynamics with aspects associated with stochasticity. In contrast, some systems not only mimic some aspects associated with stochasticity, but, instead, display indeterminism at a fundamental level. For example, when a collection of completely identical systems later displays heterogeneous outcomes, the systems are fundamentally indeterministic. They have no initial properties that can be used to discern which individual system will display which particular outcome.
51 Stochasticity d: Memory-free (Markov) processes and their representations
Markov models are often used when developing mathematical models of systems which partially or more fully display aspects associated with stochasticity (depending on how fully a system displays aspects associated with stochasticity, the use of a Markov model might need to be recognized as a conceptual approximation). Icons that can represent the use of such models include spinning wheels of fortune and rolling dice.
52 Canonical protein dynamics Protein dynamics a: Translation and degradation events occur over time
In this and the following three videos, we present a canonical worked problem that is presented in introductory systems biology coursework. In this video, we animate a time sequence of translation and degradation events that cause the number of copies of a protein of interest in a cell to change over time.

For an example of this mathematical lesson, see Alon, An Introduction to Systems Biology: Design Principles of Biological Circuits, Boca Raton: Chapman & Hall/CRC, 2007 (p. 18-22).
53 Protein dynamics b: Differential equation and flowchart
We derive a differential equation approximating the time-rate of change of the number of copies of protein in the cell modeled by the animation in the previous video. This differential equation reads, dx/dt = β - αx. We depict aspects of this differential equation with a flowchart. It is important to remember that this differential equation does not represent all aspects of the stochastic dynamics in the toy model presented in the previous video.
54 Protein dynamics c: Qualitative graphical solution to differential equation
We sketch a slope field corresponding to the differential equation derived in the previous video. We use this slope field to draw a qualitative curve describing how the number of copies of protein is expected to rise over time, when starting from an initial value of zero.
55 Protein dynamics d: Analytic solution and rise time
We obtain an analytic solution for the relationship between the number of copies of protein and time for the differential equation qualitatively investigated in the previous video. We find that the rise time, T1/2, is ln(2) divided by the degradation rate coefficient, α. The fact that the rise time is independent of the translation rate β is sometimes used as a pedagogical example of the importance of quantitative reasoning for gaining insights into biological dynamics that would be difficult to develop through natural-language and vaguely-structured notional reasoning alone.
56 Mass action Mass action a: Law of mass action
Collision picture
57 Mass action b: Cooperativity
Cooperativity of the simple kind and Hill functions
58 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).
59 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).
60 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
61 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.
62 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.
Evolutionary game theory III Training and validating population dynamics equations
David Liao and Thea D. Tlsty, "Evolutionary game theory for physical and biological scientists. I. Training and validating population dynamics equations," Interface Focus 4:20140037 (2014) (open-access online)

David Liao and Thea D. Tlsty, "Evolutionary game theory for physical and biological scientists. II. Population dynamics equations can be associated with interpretations," Interface Focus 4:20140038 (2014) (open-access online)

For additional material, please see the EGT resource page and the Interface Focus special issue on game theory and cancer from 2014.

Probability, statistics, and stochastic processes

Track Topic Slides Video Description
63 Statistics Statistics a: Probability distributions and averages
The first of five videos on introductory statistics, this module introduces probability distributions and averages. The average (also called "arithmetic mean") quantitatively expresses the notion of a central tendency among the results of an experiment.
64 Statistics b: Identities involving averages
The average of a sum is the sum of the averages. The average of a constant multiplied against a function is the constant multiplied by the average of the function. The average of a constant is the constant itself.
65 Statistics c: Dispersion and variance
The variance of a function is the average of the square of the function. For the purposes of theoretic calculations, it might be useful to express the variance using the "inside-out" computation formula described in this video.
66 Statistics d: Statistical independence
Two variables are said to be statistically independent if the outcome of an experiment tracked by one variable does not affect the relative likelihoods of different outcomes of the experiment tracked by the other variable. The two-variable probability distribution factorizes into two probability distribution functions.
67 Statistics e: Identities following from statistical independence
The covariance of statistically independent variables is zero. The variance of a sum of statistically independent variables equals the sum of the variances of the variables. This identity is often used to derive uncertainty propagation formulas.
68 Probability Probability a: Bernoulli trial
This slide deck provides examples of how hypotheses about probabilistic processes can be used to discuss probability distributions and obtain theoretical values for averages and variances. In this first video, we describe the Bernoulli trial, which corresponds to the experiment in which a coin is flipped to determine on which of two sides it lands.
69 Probability b: Binomial distribution
In this second video in this slide deck, we discuss the binomial distribution. This distribution describes the probability of getting x heads out of N coin tosses (Bernoulli trials), each individually having probability p of success.
70 Probability c: Poisson limit
In the Poisson limit, we take a series of [independent] Bernoulli trials (giving rise to a binomial distribution) and allow the number of coin flips N to increase without bound while allowing the chance p of success on a particular coin flip to decrease without bound in such a compensatory fashion that the average number of successes ("heads") is unchanged. Because the likelihood of "heads" on any given toss decreases without bound, this limit is called the limit of rare events.
71 Central limit theorem CLT a: Stirling's approximation
To study the combinatorics involved in an example where the central limit theorem applies, we will need to work with the factorials of large numbers. Stirling's approximation is an approximation for n! for large n. In this video, we motivate this approximation by comparing the expression for ln(n!) with an integral of the natural log function.
72 CLT b: Central limit theorem
The central limit theorem states that a Gaussian probability distribution arises when describing an overall variable that is a sum of a large number of independently randomly fluctuating variables, no small number of which dominate the fluctuations of the overall variable.
73 CLT c: 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 Slides Video Description
74 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"
75 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
76 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)
77 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?
78 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.
79 Curve fitting Curve fitting a: χ2
To identify theoretical curves that closely imitate a set of experimental data, it is necessary to be able to quantify to what extent a set of data and a curve look similar. To address this need, we present the definition of the quantity χ2 (chi-squared). For a given number of measurements, a smaller χ2 indicates a closer match between the data and the curve of interest. In other words, a smaller χ2 corresponds to a situation in which it looks more as though the data "came from" Gaussian distributions centered on the curve. The average χ2 across a number of experiments, each involving M measurements, is M.

80 Curve fitting b: Minimizing χ2
We slightly modify the definition of χ2 developed in the previous video for the situation in which a "correct" curve has not been theoretically determined beforehand. We choose a "best guess" curve with corresponding best guess values of fitting parameters by minimizing χ2, which corresponds to maximizing likelihood.
81 Curve fitting c: Checklist for undergraduate curve fitting
We present a checklist of steps for performing fitting of mathematical curves to data with error bars. These steps include checking whether the reduced χ2 is in the neighborhood of unity and inspecting a plot of normalized residuals to check for systematic patterns. This algorithm is appropriate for general education undergraduate "teaching laboratory" courses.
Notes

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)

Stochastic dynamics

Track Topic Slides Video Description
82 Basic stochastic simulation Basic stochastic simulation a: Master equation
83 Basic stochastic simulation b: Stochastic simulation algorithm
Derivation of exponential distribution of waiting times
84 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
85 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 Slides Video Description
86 LA I LA 1a: Teaser
Motivating example: Modeling dynamics of web start-up company customer base
87 LA 1b: Vectors
Vectors, vector spaces, and coordinate systems
88 LA 1c: Operators
Linear operators, matrix representation, matrix multiplication
89 LA 1d: Solution of teaser
Using eigenvalue-eigenvector analysis to solve for the dynamics of the demographics of the web-startup customer base
90 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).
91 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 = -1
92 LA II Rotation matrix
Complex eigenvectors and eigenvalues

Differential equations

Track Topic Slides Video Description
93 DEs I Direction fields, quiver plots, and integral curves
Numerical integration of systems of differential equations
94 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).
95 DEs IIIb: Eigenvector-eigenvalue analysis
Determine the directions of "unbending" trajectories for a more precise hand sketch of the phase portrait
96 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.
97 DEs IV DEs IVa: Adaptation
Adaptation is not absence of change; instead it is the presence of eventually compensatory changes
98 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).
99 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.

Applications in physical oncology

Track Topic Slides Video Description
100 Introduction to 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)
101 Dynamic heterogeneity Dynamic heterogeneity a: Stochastic biochemistry
The abstract organized into this and the following two videos highlights two recent papers from authors at the University of California, San Francisco working within the Princeton Physical Sciences Oncology Center. In this video, we review examples of ways that the timings of biochemical reactions can appear to be random.

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)

† The authors dedicate this paper to Dr Barton Kamen who inspired its initiation and enthusiastically supported its pursuit. The research described in these articles was supported by award U54CA143803 from the US National Cancer Institute. The content is solely the responsibility of the authors and does not necessarily represent the official views of the US National Cancer Institute or the US National Institutes of Health. Permanent link
102 Dynamic heterogeneity b: Phenotypic interconversion
Stochastic fluctuations in the levels of intracellular molecules can lead to transitions between phenotypic states in individual cells.
103 Dynamic heterogeneity c: Metronomogram
In the previous video, we asked whether phenotypic interconversion was a source of therapeutic failure or a therapeutic opportunity. In this video, we develop a graphical device, called a metronomogram, to understand that the dynamics of a phenotypically interconverting population (eventual reduction, expansion, or maintenance of population size) can depend on whether therapy is administered with sufficient time frequency.
Statistics of metastatic establishment Luis H. Cisneros and Timothy J. Newman, "Quantifying metastatic inefficiency: rare genotypes versus rare dynamics," Phys. Biol. 11:046003 (2014) (open-access online)

If one were to observe that metastatic colonies established themselves at new tissue sites with deterministic, rapid, exponential population expansion, would it be possible to conclude that the establishment of metastases primarily depended on implantation by highly-fit disseminated cells? No. As it turns out, a prevalance of deterministic, rapid, exponential population expansion at the sites of eventually successful metastatic establishment can also be explained using a model in which cells are generally of low fitness.

Spatially-resolved models and cellular automata

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).
104 Cellular automata Cellular automata a: Deterministic cellular automata
We use a simple lattice model of synchronous reproduction of annual plants to give an example of a kind of spatially-resolved modeling that is easy to program into personal computers for routine study. This example happens to use a "winner takes all" replacement rule. See Nowak and May, Nature (1992) for an article describing spatial patterns that can arise when using a "winner takes all" model. In this video, we see that heterogeneous coexistence (as distinguished from homogeneous dominance by a single subpopulation) can sometimes be promoted by spatial localization.

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).
Fast-Fourier transform
Efficient computation of local linear interactions FFT convolution trick

Statistical physics

Track Topic Slides Video Description
105 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
106 Statistical physics 101b: Notating configurations of a system with multiple parts
Direct product
107 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
108 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
109 Ideal chain Ideal chain a: Introduction to model
A series of links pointed up or down
110 Ideal chain b: Hamiltonian and partition function
Writing the partition function for a collection of independent links
111 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)

Digest for busy people

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).

Quiz 1: Protein level dynamics

In a toy model of a cell, protein X is produced according to a translation rate coefficient and eliminated according to a degradation rate coefficient. The protein copy number at which the rates for these processes balance is called the steady-state level. The time it takes for a cell initially containing zero copies of protein X to accumulate half the steady-state level is called the “rise time.” Which statement describes the rise time?
1. As translation rate coefficient increases, rise time becomes shorter. As degradation rate coefficient increases, rise time becomes longer.
2. As degradation rate coefficient increases, rise time becomes shorter. Rise time does not depend on translation rate coefficient.
3. As translation coefficient increases, rise time becomes longer, and as degradation rate coefficient increases, rise time becomes shorter.
4. As translation rate coefficient increases, rise time becomes shorter. Rise time does not depend on degradation rate coefficient.
5. As translation rate coefficient increases, rise time becomes longer. Rise time does not depend on degradation rate coefficient.

Quiz 2: Law of mass action

In this course, we use "law of mass action" to refer to an idea that chemical reaction kinetics can be modeled using rate formulas containing products of abundances of reactants raised to exponents. Which statement is best?
1. Mass-action exponents and stoichiometric coefficients can be related using a probabilistic model of molecular collisions, so the stoichiometric coefficients of reactants in the proximate reaction that generates products under study can be deduced using the exponents in an equilibrium constant.
2. Instantaneous collisions of more than two hard-spheres are exceedingly rare, so mass-action rate formulas cannot be used to model reactions involving more than two reactants.
3. The warning against reading mass action exponents off of stoichiometric coefficients derives from the low likelihood of instantaneous collisions between more than two hard spheres. This warning can be relaxed when molecules have non-spherical geometries, and instantaneous collisions between three, or even four, molecules is common, as long as one of the reactants is not spherical.
4. Taylor expansion of complicated functions of reactant concentrations can produce sums of mass-action terms. In other words, stringing together mass-action formulas can produce experimentally accurate mathematical descriptions while concealing insight in the same way that epicycles permit accurate description of celestial motion while obscuring Kepler’s laws.
5. Mass-action exponents must be empirically derived from time-course observations and cannot be deduced from stoichiometric coefficients.
Track Topic Slides Video Description
D112 Mass action Mass action a: Law of mass action
Collision picture
D113 Mass action b: Cooperativity
Cooperativity of the simple kind and Hill functions
D114 Mass action c: Bistability
Combining molecular production rates with nonlinear dose-dependence with unimolecular degradation can generate systems with multiple stable steady states

Quiz 3: Evolutionary game theory

Evolutionary game theory is an example of a modeling framework used in the NCI Physical Sciences-Oncology Network to understand social aspects of biological systems. What statement about evolutionary game theory is best?
1. Evolutionary game theory refers to a collection of mathematical models in which organisms (e.g. cells) are modeled as automated, robotic replicators. The propensity with which a replicator generates progeny is modified by the time-frequency with which it encounters other replicators (e.g. through pairwise interactions).
2. In the prisoner’s dilemma, defector population share and per capita fitness both increase. However, average fitness decreases because interaction between defectors and cooperators decreases the per capita fitness of cooperators more than enough to cancel out the fitness increase of defectors.
3. In normal tissue microenvironments, individual cells typically express rational behaviors (i.e. protecting long-term survival of the host). Following carcinogenesis, however, cells can also display irrational behaviors, such as circumvention of stress-related barriers. Because cells can now choose between two classes of strategies, it becomes necessary to employ evolutionary game theory to determine how individual cells might make these choices and alter overall tissue system dynamics.
4. John Maynard Smith understood the evolution of socially interacting biological individuals by describing the possible behaviors that individual agents could choose. The notion of choice in this context refers to the essential assumption that each individual organism expresses a strategy that maximizes its payoff, as computed using a payoff matrix. A startling result is the applicability of this kind of modeling, not just to organisms that perform advanced cognition, e.g. hawks and doves, but also, to individual cells, for which evolution had long been thought to proceed primarily in a Darwinian fashion.
5. In traditional models of population dynamics, cell numbers vary as though cells individually solved for anti-derivatives and then adjusted their behavior to obey governing equations. In evolutionary game theory, individual cells, instead, perform sophisticated computations using payoff matrices to decide among strategies so as to realize Nash equilibria.
Track Topic Slides Video Description
D115 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

Quiz 4: Poissonian copy numbers

Some quantitative biologists say that the copy numbers of mRNA in simple models of gene transcription are stochastic and Poisson distributed. Which of the following explains this claim?
1. The copy number of a species of mRNA can be small (~1 copy per cell or less). When a system is composed of a small number of parts, intrinsic stochastic fluctuations are generated. The copy number counts are Poisson distributed because the transcription machinery "forgets" whether it has previously generated a strand of mRNA in a time interval much shorter than needed to produce, on average, one copy of mRNA.
2. The copy number of a species of mRNA can be small (~1 copy per cell or less). This means that the molecule is "rare," so the limit of rare events, also known as the Poisson distribution, accurately models copy-number fluctuations.
3. If the transcription machinery for a species of mRNA "forgets" whether it has previously generated a copy of mRNA over a time scale much shorter than the duration that produces, on average, one mRNA strand, then the generation of mRNA is a Poisson process across time. The integral over an interval is a function evaluated on the boundary. As a rough example, the number of copies of a species of mRNA at a time of observation equals the number of copies produced up until that time since a reference time in the past (minus the number of copies generated up until a lifetime earlier than the time of observation). In this example of the second fundamental theorem of calculus, the number of copies of mRNA counted at a time is Poisson distributed because it inherits the statistical properties of the Poisson process across time described above.
4. A "Poisson distribution" refers to a distribution generated through natural random fluctuations. This is in contrast to fluctuations in engineered structures and circuits that use metabolic energy input to shape their distributions (i.e. the end products of biological signaling cascades can be log-normally distributed, rather than Poisson distributed). Hypothesizing a Poisson distribution for the copy numbers of a species of mRNA is equivalent to claiming that transcription occurs with minimal metabolic energy input.
5. Statements (A) and (C) above.
Track Topic Slides Video Description
D116 Poissonian copy numbers Poissonian copy numbers a: Stochastic synthesis

Quiz 5: Quasispecies

Quasispecies models are used to understand the compositions of populations of cells in terms of mechanisms for genetic mutation. Choose the best statement:
1. The genotype that dominates a population does not necessarily correspond to the genotype underlying those cells that most rapidly produce offspring. The dominant genotype is not necessarily the genotype with greatest fitness.
2. Classical quasispecies models are sometimes referred to as mutation-selection models. A fundamental limitation of these models is an assumption of thorough mixture. The dynamics supported by cells undergoing mutation, selection, and spatial movement are examples of emergent phenomena because they are not easily predicted from models of mutation and selection alone.
3. A mutational meltdown in cancer occurs when a sudden accumulation of dangerous mutations in cancer cells threatens the host. Dangerous mutations can include, for example, mutations that increase the likelihood of survival cancer cells (e.g. increased proliferation, insensitivity to growth control signals, etc.). Analyzing quasispecies models, mathematicians recommend decreasing the mutation rates of cancer cells to reduce the chances of a mutational meltdown in a tumor.
4. Mutation leading away from the genotype of greatest fitness must be slow enough in order for this genotype to be dominant. Otherwise, this genotype will command only minority population share (in some models, no finite number will be small enough to represent population share). Condensation onto a dominant sequence at low mutation rates is often compared to the condensation of a population of bosons into the ground orbital at low temperatures because mutation is analogous with maximizing entropy of the thermal bath and selection is analogous with maximizing the entropy of the bosons.
Track Topic Slides Video Description
D117 Quasispecies Simple quasispecies eigendemographics and eigenrates, adapted from Bull, Meyers, and Lachmann, "Quasispecies made simple," PLoS Comp Biol, 1(6):e61 (2005) (open-access online).

A gene-expression network composed precisely of nodes A, B, and C is said to exhibit adaptation. In the following, a step change in the level of node A is externally applied and maintained. Which one of the following is the best example for teaching how “adaptation” is achieved in systems biology?
1. A step increase in node A is isolated. Nodes B and C are unchanged.
2. A step increase in node A leads to a temporary increase (followed by a return to steady state) in node C. Node B is unchanged.
3. A step increase in node A eventually leads a step increase in node C and a step decrease in node B. Both nodes C and B achieve new steady states, but with a delay so that no changes in node B and C are observed in the first moments after the increase in node A.
4. A step decrease in node A initially causes a temporary decrease in node C. Node C rises and returns to its initial steady-state level owing to variation in node B.
Track Topic Slides Video Description
D118 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, El-Samad, Lim, and Tang, "Defining network topologies that can achieve biochemical adaptation," Cell, 138: 760-773 (2009) (online). To view the short sequel to this video, jump to the

Quiz 7: Oscillations

Choose the best statement regarding oscillations in mathematical biological models.
1. Cyclic movement through phase space depends on deterministic, programmed behavior. Adding stochastic fluctuations to a deterministic system does not generate oscillatory trajectories. Instead, any oscillations originally present are damped out, if not completely eliminated.
2. Oscillatory circuits can generate a variety of kinds of signals. These need not resemble sines and cosines in time plots or perfect circles and ellipses in phase planes. For example, time courses can look like momentary spikes, and phase-plane trajectories could resemble rounded squares and triangles.
3. Oscillations correspond to loops in phase space. Such a structure is essentially 2-dimensional. Oscillations occur in high-dimensional phase spaces only when most of the degrees of freedom decouple from the pair that defines the phase portrait in which loops can be drawn.
4. Even though spiral and closed-loop trajectories qualitatively resemble each other in phase portraits, they correspond to different biological network topologies. In other words, it is usually not possible to construct a protein-interaction network that can support both kinds of trajectories with merely adjustments to numerical parameters. This implies that therapies that only mildly suppress or activate a network component are unlikely to change the qualitative form of system oscillations.
5. Oscillations are often visualized as loops in two dimensional slices of phase space. This kind of depiction is, in part, the consequence of arbitrary convention. Reversing one of the two axes by multiplying it by a negative sign reverses the orientation of its back-and-forth motion, and the motions along the two axes now synchronize in a way that traces cyclic motion along on a line, rather than a loop around a plane. Thus, oscillation can be represented in a 1-d “space.”
Track Topic Slides Video Description
D119 Differential Eqtns V Heuristic picture of oscillations in 2-d

Quiz 8: Phenotypic stochasticity

A cell can execute stochastic transitions between phenotypic states without need for gene sequence alterations or large-scale genomic rearrangement. This can lead to a type of cell individuality that has been called epigenetic or “non-genetic.” Phenotypic stochasticity is actively studied in stem cell/developmental biology and cancer biology. Choose the best statement:
1. Whereas a genotype rigidly corresponding to one phenotypic state might be suited to one environment, but stressed under another, a genotype underlying a plastic collection of interconvertable phenotypes is protected from selection. Darwinian evolution occurs through selection on underlying genetic variation. For this reason, genetic evolution does not proceed in the presence of stochastic phenotypic fluctuations and non-genetic network adaptations.
2. Models based on phenotypic stochasticity and models based on cell-cycle phase specific therapeutic sensitivity both suggest adjusting dose timing according to the dynamics of non-genetic variation in the targeted cell population. However, models based on phenotypic stochasticity and cell-cycle specificity are distinct because stochastic transitions occur with fluctuating waiting times, whereas the cell-cycle proceeds in clockwork fashion through phases of well-defined duration.
3. Therapeutic failure is inevitable any time cells with a phenotype impervious to a therapeutic modality are present at the onset of treatment. Even if such a subpopulation is initially small, it will be selected for and eventually dominate. In a population regenerated by tumor [re]-initiating cells (TICs, cancer stem cells, CSCs), for example, therapy will eventually fail if the treatment modality cannot directly kill the TIC phenotype.
4. Drug-sensitive cells can enter relatively drug-resistant states through non-genetic mechanisms over time scales of days and weeks. Phenotypic stochasticity contributes to therapeutic failure.
5. Owing to the “non-genetic” character of the cell-cell individuality that phenotypic stochasticity can generate, the population dynamics resulting from phenotypic fluctuations cannot be described using principles of Darwinian evolution.

Quiz 9: Cellular automata and spatiality

Two populations of annual plants, “cooperators” and “defectors,” are sown on a field. Prisoner’s dilemmas describe the chemical and mechanical contact that occurs repeatedly between pairs of neighboring plants. These interactions determine the numbers of seeds that the plants contribute to the next generation. Which statement is true?
1. Increasing the spatial area over which offspring randomly disperse spreads the offspring of defectors too thin, making it more difficult for defectors to compete with dense pockets of cooperators. These pockets of cooperators survive and perpetuate heterogeneous co-existence.
2. Increasing the spatial area over which offspring randomly disperse promotes heterogeneous co-existence because survival of cooperators relies on their ability to move away from defectors in an ongoing cat-and-mouse chase.
3. Increasing the spatial area over which offspring randomly disperse makes it easier for defectors to take over the lattice and more difficult to realize heterogeneous co-existence.
4. Increasing the spatial area over which offspring randomly disperse promotes heterogeneous co-existence because defectors and cooperators both have increased chances of bumping into other cooperators.
5. None of the above
Track Topic Slides Video Description
D120 Cellular automata Cellular automata a
Deterministic cellular automata