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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. Permanent link
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. Permanent link
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 Permanent link
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. Permanent link
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 Permanent link
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. Permanent link
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). Permanent link
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). Permanent link
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 Permanent link
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. Permanent link
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. Permanent link
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 Permanent link
41	Euler's number I			Euler's number 1a: Compound interest Compounding interest with arbitrarily short compounding periods Power series representation of ex Permanent link
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). Permanent link
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). Permanent link
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 Permanent link
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). Permanent link
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. Permanent link
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. Permanent link
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. Permanent link
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. Permanent link
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			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. Permanent link
72				CLT a: Statement of 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 b: Optional derivation (special case) In some situations, when the number of coin tosses is large, Stirling's approximation can be applied to factorials that appear in the expression for the binomial distribution. The resulting expression is basically an exponential function of a quadratic function with a negative leading coefficient. This is the hallmark of a Gaussian distribution.
74				CLT c: Properties of Gaussian distributions For a Gaussian distribution, roughly two-thirds of the probability is found within the first standard deviation.
75	Prevalence of Gaussians			Prevalence of Gaussians a: Noise in physics labs is allegedly often Gaussian Because equipment in physics experiments is highly-engineered, individual device contributions to measurement fluctuations might be "small." The overall fluctuations in the final measured quantity might be well approximated using a first-order Taylor expansion in terms of individual device fluctuations. Fluctuations in measurements are thus sums over random variables, and thus, potentially Gaussian distributed. Permanent link
76				Prevalence of Gaussians b: Noise in biology is allegedly often log-normal The levels of molecules in biological systems can approximate "temporary" steady-state values that equal products of rate coefficients and reactant concentrations. Since logarithms convert products into sums, the logarithms of the levels of some biological molecules can be normally distributed. Hence, the levels of the biological molecules are log-normally distributed.

Uncertainty propagation

Track	Topic	Slides	Video	Description
77	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
78	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
79				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)
80				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?
81				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.
82	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. Permanent link
83				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.
84				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
85	Basic stochastic simulation			Basic stochastic simulation a: Master equation Permanent link
86				Basic stochastic simulation b: Stochastic simulation algorithm Derivation of exponential distribution of waiting times Permanent link
87	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
88				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
89	LA I			LA 1a: Teaser Motivating example: Modeling dynamics of web start-up company customer base Permanent link
90				LA 1b: Vectors Vectors, vector spaces, and coordinate systems
91				LA 1c: Operators Linear operators, matrix representation, matrix multiplication
92				LA 1d: Solution of teaser Using eigenvalue-eigenvector analysis to solve for the dynamics of the demographics of the web-startup customer base
93	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
94	Euler's number II			Euler's formula: Expanding the exponential function in terms of sine and cosine Complex exponentials in the complex plane Euler's identity eiπ = -1 Permanent link
95	LA II Rotations			Rotation a: Rotation matrix In this and the next video, we develop a familiarity with the representation of vector rotations using rotation matrices. This understanding is helpful for identify dynamical systems that support oscillations in physics, engineering, and biology. A rotation operator rotates a vector by an angle without changing the length of the vector. A rotation matrix represents the action of a rotation operator on a vector. Permanent link
96				Rotation b: Complex eigenvalues How can we determine whether a dynamical system can be represented using something that looks like a rotation matrix? Rotation matrices have complex eigenvalues. We can determine whether a dynamical system supports rotational motion by determining whether the matrix representing the system's dynamics has complex eigenvalues.

Differential equations

Track	Topic	Slides	Video	Description
97	DEs I			Direction fields, quiver plots, and integral curves Numerical integration of systems of differential equations Permanent link
98	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
99				DEs IIIb: Eigenvector-eigenvalue analysis Determine the directions of "unbending" trajectories for a more precise hand sketch of the phase portrait
100				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.
101	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
102				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).
103	DEs V Oscillations			Oscillations a: Romeo and Juliet In this and the following four videos, we present some concepts that can be used to design and recognize mathematical models that support oscillatory behavior. In this first video, we show that oscillations can be viewed as cyclic loops in a 2-dimensional plane. One way to arrange for a pair of variables R and J to perform oscillations is to let the time-derivative of each variable be proportional to the value of the other variable, with a negative sign in the coefficient of one of these differential equations. Permanent link
104				Oscillations b: Twisting nullclines The angles at which nullclines pass through the phase plane (e.g. steep vs. shallow) determine the relative arrangement of regions in which quivers point in the top-left, bottom-left, bottom-right, and top-right directions. By modifying the slopes of nullclines, and thus the relatively positions of these regions, the qualitative dynamics of a dynamical system might be modified to support a stable star, a stable spiral, a closed loop, or even an unstable spiral. One way to understand how parameters affect trajectories is to understand how parameters affect the slopes that nullclines make when drawn in the phase plane.
105				Oscillations c: Time delays A spiral sink can be modified to support a closed-loop trajectory if the system is modified so as to perform motion in the present that would, in the original dynamical system, have, instead, been performed at a previous time.
106				Oscillations d: Stochastic excitation A deterministc spiral sink that is highly skewed can support repeated oscillations when stochastic fluctuations kick the system out of the sink and onto a nearby region of rapid flow.
				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
107	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)
108	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
109				Dynamic heterogeneity b: Phenotypic interconversion Stochastic fluctuations in the levels of intracellular molecules can lead to transitions between phenotypic states in individual cells.
110				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).
111	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) 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	Slides	Video	Description
112	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
113				Statistical physics 101b: Notating configurations of a system with multiple parts Direct product
114				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
115				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
116	Ideal chain			Ideal chain a: Introduction to model A series of links pointed up or down Permanent link
117				Ideal chain b: Hamiltonian and partition function Writing the partition function for a collection of independent links
118				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

Quiz 1: Protein level dynamics

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

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
D119	Mass action			Mass action a: Law of mass action Collision picture Permanent link in main curriculum | Permanent link in digest
D120				Mass action b: Cooperativity Cooperativity of the simple kind and Hill functions
D121				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

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
D122	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

Quiz 4: Poissonian copy numbers

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
D123	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

Quiz 5: Quasispecies

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
D124	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). Permanent link in main curriculum | Permanent link in digest

Quiz 6: Adaptation

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
D125	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 Permanent link in main curriculum | Permanent link in digest

Quiz 7: Oscillations

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.”

Quiz 8: Phenotypic stochasticity

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

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
D126	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 2011-2015 David Liao. These videos and slides are open course ware made available under a Creative Commons license (CC BY-SA 4.0). The lightbox and social sharing effects are scripts by Stéphane Caron (CC BY 2.5).