|lookatphysics.com α||q-bio tutorials | oa | support | dvd | contact | about | links||On facebook | twitter | vimeo | YouTube | udemy|
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.
Excellent site for both basic and advanced lessons on applying mathematics to biology.
- Tweeted by the National Cancer Institute Office of Physical Sciences-Oncology
Street numbers, money in bank accounts, points on number lines, quantum particles
as contrasted with distinguishable manipulatives
For our purposes, infinity is not a number
Variables: At once arbitrary, yet specific and particular (a.s.a.p.)
Functions, composition, and inverse
f(x) is not a function. f(x) is not the function f. f(x) is the particular value associated, in the big picture of the function f, with x, a number that is at once arbitrary, yet particular and specific
Inverse functions do not always exist
First glimpse at the complex plane and i := √ -1
Linear combination of terms in a polynomial
Zeroes or "roots" of a function
Completing the square
Geometry a: Euclidean geometry
Flat space, curved space, non-embedded curved space
Geometry b: Trigonometry
The unit-radius circle, the unit-hypotenuse triangle, jya-ardha (sine), koti-jya (cosine)
Geometric construction for approximating π
Sums a: Summation notation
Greek-letter Σ notation
Gauss summation trick
Sums b: Introduction to infinite series
Harmonic series; sums do not always exist
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.
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.
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.
ε-δ definition of limit, notion of "arbitrarily close"
Example of calculating a limit
Limits do not always exist
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).
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.
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.
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).
Differentiation d: Sinusoidal functions
The derivative of sine is cosine, and the derivative of cosine is negative sine. This back-and-forth relationship is a hallmark of dynamical systems that might support oscillations. Thus, this pattern, which you will derive in this video, is important to keep in mind when you later study biological oscillations.
|17||Partial differentiation||When a function depends on multiple independent variables, the curly-d symbol, ∂, denotes slopes calculated by jiggling only one independent variable at a time|
Second derivative and curvature
Local approximations and Taylor series
(Successful) power-series representations do not always exist
Power-series expansion of sine and cosine, iterative calculation of π
Anti-derivatives, Riemann sums, and integrals
Example kosher calculation of a simple integral
Deductive inference of integral by definition as anti-derivative
"Backwards chain rule"--u substitution
|20||Separation of variables||
Two wrongs make a right
Tear two differentials apart as though they retained meaning in isolation
Slap on the smooth S integral sign as though it were a unit of meaning itself, even without a differential
You get the same integral expression you would obtain long-hand using u-substitution or "change of variables" in integrals
|21||Euler's number I||
Euler's number 1a: Compound interest
Compounding interest with arbitrarily short compounding periods
Power series representation of ex
Euler's number 1b: e to the zero
e0 = 1
Euler's number 1c: Exponent multiplication identity
(ex)p = epx
Euler's number 1d: Exponent addition identity
exey = ex+y
Euler's number 1e: Andrew Jackson
Mnemonic for memorizing e = 2.718281828459045...
Euler's number 1f: Natural logarithm
The natural logarithm is the inverse of the exponential ln(ex) = eln(x) = x
Euler's number 1g: Integral of 1/x
∫(1/x)dx = ln(x) + C
(This section is a prerequisite for "protein dynamics 101"). Many of the homework problems from undergraduate calculus and differential equations involve notions of stochasticity.
* For-real stochasticity: Fundamental indeterminism
* Fake stochasticity: Periodic, deterministic hidden variables
* Fake stochasticity: Aperiodic, deterministic (chaos)
|29||Protein dynamics 101||
This is a canonical worked problem from introductory systems biology. We will explain one way to fantasize about the classic protein dynamics equation dx/dt = β - αx and analytically demonstrate that protein "rise time" depends on degradation rate only.
Additional activity: See textbook presentation by Alon, An Introduction to Systems Biology: Design Principles of Biological Circuits, Boca Raton: Chapman & Hall/CRC, 2007 (p. 18-22).
Mass action a: Law of mass action
Mass action b: Cooperativity
Cooperativity of the simple kind and Hill functions
Mass action c: Bistability
Combining molecular production rates with nonlinear dose-dependence with unimolecular degradation can generate systems with multiple stable steady states
|Additional activity: See textbook presentation by Alon, An Introduction to Systems Biology: Design Principles of Biological Circuits, Boca Raton: Chapman & Hall/CRC, 2007 (sections 2.3-2.3.4, p. 7-16).|
|33||Evolutionary game theory I||
EGT 1a: Population dynamics with interactions
Equations for collisional population dynamics using law of mass action
An outcome of the prisoner's dilemma is simultaneous survival of the relatively most fit with decrease in overall fitness
Additional activity: Access McKenzie, A.J., "Evolutionary Game Theory", The Stanford Encyclopedia of Philosophy (Fall 2009 Edition), Zalta, E.N. (ed.) (online) and compare the replicator dynamics described there with the collisional population dynamics in this tutorial. Watch Deborah Gordon talk about colony expansion, task allocation, and organization without central control in ant colonies (TED-talk video online).
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
Distributions, averages, variances, useful identities
Routinely-exploited expressions: Covariances vanish and variances of sums are sums of variances
Probability 101a: Binomial distribution
Bernoulli coin-toss process
Probability 101b: Poisson limit
Independent + "rare" events
Probability 101c: Stirling's approximation
Comparison with integral of natural logarithm
Probability 101d: Central limit theorem
Many independent events
Binomial distribution in limit of many coin tosses
Probability 101e: Stories that relate central limit theorem to physics and biology
Physics: Taylor expansion in the context of tightly-controlled, narrow instrument noise
Biology: Logarithm of product of fluctuating factors: Log-normal distributions
Uncertainty propagation 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"
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
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)
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?
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.
Uncertainty propagation f: Sample variance curve fitting
Reduced chi-square χ2 fitting
Do not assume that parameter fit uncertainties from black-box software packages are appropriate to interpret in a "covariance = zero" context (Gutenkunst, Sorger)
Additional activity: Sample-variance curve fitting exercise for MatLab (PDF)
Additional resource: Web page on data fitting from the Harvey Mudd College physics kiosk (online)
|47||Basic stochastic simulation||Basic stochastic simulation a: Master equation|
Basic stochastic simulation b: Stochastic simulation algorithm
Derivation of exponential distribution of waiting times
|49||Poissonian copy numbers||
Poissonian copy numbers a: Stochastic synthesis
Model: RNA polymerase makes many (usually unsuccessful) independent attempts to initiate transcription and mRNA strands degrade after a precise lifetime
Outcome: mRNA copy numbers are Poisson distributed
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
LA 1a: Teaser
Motivating example: Modeling dynamics of web start-up company customer base
LA 1b: Vectors
Vectors, vector spaces, and coordinate systems
LA 1c: Operators
Linear operators, matrix representation, matrix multiplication
LA 1d: Solution of teaser
Using eigenvalue-eigenvector analysis to solve for the dynamics of the demographics of the web-startup customer base
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).
|56||Euler's number II||
Euler's formula: Expanding the exponential function in terms of sine and cosine
Complex exponentials in the complex plane
Euler's identity eiπ = -1
Complex eigenvectors and eigenvalues
Direction fields, quiver plots, and integral curves
Numerical integration of systems of differential equations
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).
DEs IIIb: Eigenvector-eigenvalue analysis
Determine the directions of "unbending" trajectories for a more precise hand sketch of the phase portrait
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.
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).
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).|
Heuristic picture of oscillations in 2-d
Intuitive introduction to 2-d oscillations (Romeo and Juliet)
|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.|
|65||Dynamic heterogeneity for the physical oncologist||
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)
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)
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).
Cellular automata a
Deterministic cellular automata
In this video, we see that limiting dispersal of seeds of annual plants can increase the proportion of the copper-colored subpopulation, whereas thorough mixing instead allows the denim plant subpopulation to dominate quickly.
Additional activities: Refer to a similar model in Nowak and May, "Evolutionary games and spatial chaos," Nature 359:826-829 (1992) (online). Watch Athena Aktipis talk about the walk-away model, which can contribute to the evolution of cooperation in highly-mobile populations (University of California, Los Angeles, Center for Behavior, Evolution, and Culture 2009, 1-hr video online)
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).
|Efficient computation of local linear interactions||FFT convolution trick|
|67||Statistical physics 101||
Statistical physics 101a: Fundamental postulate of statistical mechanics
Systems have states and energy levels
Energy can be exchanged between parts of a world
If the Hamiltonian of the world is time-independent, the overall energy of the world is conserved
Fundamental postulate of statistical mechanics: In an isolated system, all accessible microstates are accessed equally
Statistical physics 101b: Notating configurations of a system with multiple parts
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
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
Ideal chain a: Introduction to model
A series of links pointed up or down
Ideal chain b: Hamiltonian and partition function
Writing the partition function for a collection of independent links
Ideal chain c: Expectation of energy and elongation
For heavy weights, the chain tends to be found extended fully. For lesser weights, the chain can be found partially crumpled, with the weight lifted, and with energy given to the bath.
Ideal chain homework: Entropic elasticity
Bustamante, Smith, Liphardt, and Smith, "Single-molecule studies of DNA molecules" Curr. Opin. Struct. Biol. 10(3):279--285 (2000) (PDF online)
|© Copyright 2011-2013 David Liao. These videos and slides are open course ware made available under a Creative Commons (CC BY-SA 3.0) license.|