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

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Excellent site for both basic and advanced lessons on applying mathematics to biology.  Tweeted by the National Cancer Institute Office of Physical SciencesOncology 
Track  Topic  Slides  Video  Description 

1  Numbers 
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 

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Numbers b: BoseEinstein statistics The analysis of a system of particles display BoseEinstein 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 BoseEinstein condensate. 

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Numbers c: Visual representations of numbers Numbers can be represented using a number line, a wedge, and placevalue representation. The application of memorized rules for performing arithmetic on numerals formatted in placevalue representation is called algorism. 

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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," "standin" 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 

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Algebra b: Functions Functions are basic buildingblock 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. 

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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. 

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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: Squareroot 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 = x^{2} is, in many places, doublevalued, 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 daytoday 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 

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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 xycoordinate 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 ycoordinates, 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 xcoordinates, 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 Greekletter Σ notation Gauss summation trick Permanent link 

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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 

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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. 

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Combinatorics c: Binomial theorem We use the formula for combinations from the previous video to write an expression for the binomial quantity (x + y)^{p}. In some applications, only a small number of terms in the resulting sum are necessary for approximate calculations. 
Track  Topic  Slides  Video  Description 

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 εδ definition. For an example of a strategy for writing εδ proofs useful for plots of functions that have curvature, please see Yosen Lin's examples (example # 4 on p. 34). Permanent link 

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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. 

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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. 

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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. 

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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 εδ proof of a limit of a linear function In this video, the outline for using the εδ 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 

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Differentiation b: Power rule We will later learn that many seemingly complicated functions can be approximated using sums of power law terms. To study the slopes of these terms, we use the power rule that we derive in this video, which is written d(x^{n})/dx = nx^{n1}. 

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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. 

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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). 

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Differentiation e: Sinusoidal functions The derivative of sine is cosine, and the derivative of cosine is negative sine. This backandforth relationship is a hallmark of dynamical systems that might support oscillations. Thus, this pattern, which you will derive in this video, is important to keep in mind when you later study biological oscillations. 

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

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 higherorder derivatives can also have geometric interpretations. Permanent link 

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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. 

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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 

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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. 

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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. 

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

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 longhand using usubstitution 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 e^{x} Permanent link 

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

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

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

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

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

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Euler's number 1g: Integral of 1/x ∫(1/x)dx = ln(x) + C 
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 

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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. 

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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. 

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Stochasticity d: Memoryfree (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. 1822). Permanent link 

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Protein dynamics b: Differential equation and flowchart We derive a differential equation approximating the timerate 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. 

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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. 

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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, T_{1/2}, 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 naturallanguage and vaguelystructured notional reasoning alone. 

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

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

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Mass action c: Bistability Combining molecular production rates with nonlinear dosedependence with unimolecular degradation can generate systems with multiple stable steady states 

Additional activity: See textbook presentation by Alon, An Introduction to Systems Biology: Design Principles of Biological Circuits, Boca Raton: Chapman & Hall/CRC, 2007 (sections 2.32.3.4, p. 716).  
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 (TEDtalk video online). Permanent link 

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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 

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

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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. 

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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 "insideout" computation formula described in this video. 

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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 twovariable probability distribution factorizes into two probability distribution functions. 

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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 

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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 

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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 

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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. 

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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. 

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CLT c: Properties of Gaussian distributions For a Gaussian distribution, roughly twothirds 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 highlyengineered, individual device contributions to measurement fluctuations might be "small." The overall fluctuations in the final measured quantity might be well approximated using a firstorder Taylor expansion in terms of individual device fluctuations. Fluctuations in measurements are thus sums over random variables, and thus, potentially Gaussian distributed. Permanent link 

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Prevalence of Gaussians b: Noise in biology is allegedly often lognormal The levels of molecules in biological systems can approximate "temporary" steadystate 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 lognormally distributed. 
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 

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Uncertainty propagation c: Squareroot of sample size (√ n ) factor Origin of the famous √ n factor by which the standard deviation of the sample means is smaller than the standard deviation of the measurements (parent distribution) 

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

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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: χ^{2} 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 χ^{2} (chisquared). For a given number of measurements, a smaller χ^{2} indicates a closer match between the data and the curve of interest. In other words, a smaller χ^{2} corresponds to a situation in which it looks more as though the data "came from" Gaussian distributions centered on the curve. The average χ^{2} across a number of experiments, each involving M measurements, is M. Permanent link 

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Curve fitting b: Minimizing χ^{2} We slightly modify the definition of χ^{2} 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 χ^{2}, which corresponds to maximizing likelihood. 

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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 χ^{2} 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 blackbox software packages are appropriate to interpret in a "covariance = zero" context (Gutenkunst, Sorger) Additional activity: Samplevariance curve fitting exercise for MatLab (PDF) Additional resource: Web page on data fitting from the Harvey Mudd College physics kiosk (online) 
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 

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Poissonian copy numbers b: Stochastic synthesis and degradation Model: RNA polymerase makes many (usually unsuccessful) independent attempts to initiate transcription. Once a mRNA strand is produced, it begins to make independent (usually many unsuccessful) attempts to be degraded. Result: As in part a, mRNA copy numbers are Poisson distributed 
Track  Topic  Slides  Video  Description 

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

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

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

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LA 1d: Solution of teaser Using eigenvalueeigenvector analysis to solve for the dynamics of the demographics of the webstartup 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) (openaccess 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 e^{iπ} = 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. 
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: Transcriptiontranslation Canonical mRNAprotein system from systems biology 101 Additional activity: See textbook presentation by Alon, An Introduction to Systems Biology: Design Principles of Biological Circuits, Boca Raton: Chapman & Hall/CRC, 2007 (problem 2.2, p. 23). Permanent link 

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DEs IIIb: Eigenvectoreigenvalue 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 eigenvalueeigenvector analysis to find analytic solutions for linear systems and describe the qualitative features of trajectories approaching, sideswiping, 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, ElSamad, Lim, and Tang, "Defining network topologies that can achieve biochemical adaptation," Cell, 138: 760773 (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 nonlinear systems," from Differential Equations: Unit IV FirstOrder Systems, MIT OpenCourseWare (openaccess online) and Harris, K., "Perturbations in linear systems (2008 November 12)," Math 216: Differential Equations, University of Michigan (online).  
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 2dimensional plane. One way to arrange for a pair of variables R and J to perform oscillations is to let the timederivative 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 topleft, bottomleft, bottomright, and topright 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 closedloop 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: 874885 (2011)(online). Comment on how the positivefeedback term in Eqtn. 25 (pg. 882) contributes to the difference between the phase portraits in Fig. 4B (pg. 878) and Fig. 8B (pg. 883). The article describes the positivefeedback in terms of a time delay. Please describe the contribution of the positivefeedback term to stable oscillations instead in terms of "twisting nullclines" from the video tutorial. 
Track  Topic  Slides  Video  Description 

107  Introduction to physical oncology 
Nicole M. Moore, Nastaran Z. Kuhn, Sean E. Hanlon, Jerry S.H. Lee, and Larry A. Nagahara, "Deconvoluting cancer's complexity: using a 'physical sciences lens' to provide a different (clearer) perspective of cancer," Phys. Biol. 8(1):010302 (2011) (online) Timothy J. Newman and Alastair M. Thompson, "Beyond detection: Biological physics informing progression and treatment of cancer," Phys. Biol. 9:060301 (2012) (online) 

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 EstévezSalmerón, and Thea D. Tlsty, "Conceptualizing a tool to optimize therapy based on dynamic heterogeneity," Phys. Biol. 9:065005 (2012) (openaccess online) David Liao, Luis EstévezSalmerón, and Thea D. Tlsty, "Generalized principles of stochasticity can be used to control dynamic heterogeneity," Phys. Biol. 9:065006 (2012) (openaccess online) † 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) (openaccess 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 highlyfit 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. 
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 spatiallyresolved 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 359:826829 (1992) (online). Watch Athena Aktipis talk about the walkaway model, which can contribute to the evolution of cooperation in highlymobile populations (University of California, Los Angeles, Center for Behavior, Evolution, and Culture 2009, 1hr video online) Permanent link 

Cellular automata b Stochastic cellular automata 

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

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

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 timeindependent, the overall energy of the world is conserved Fundamental postulate of statistical mechanics: In an isolated system, all accessible microstates are accessed equally Permanent link 

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

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

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 dosedependence with unimolecular degradation can generate systems with multiple stable steady states 
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 
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 
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) (openaccess online). Permanent link in main curriculum  Permanent link in digest 
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, ElSamad, Lim, and Tang, "Defining network topologies that can achieve biochemical adaptation," Cell, 138: 760773 (2009) (online). To view the short sequel to this video, jump to the Permanent link in main curriculum  Permanent link in digest 
Track  Topic  Slides  Video  Description 

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 20112015 David Liao. These videos and slides are open course ware made available under a Creative Commons license (CC BYSA 4.0). The lightbox and social sharing effects are scripts by Stéphane Caron (CC BY 2.5). 