E4400

apma4400: introduction to biophysical modeling

Instructor: Professor Wiggins, rm 428 S. W. Mudd

Space-time: Mondays+Wednesdays 10:10-11:25. Location 503 Hamilton

Office hours: by appointment. Email me please.

URL: http://www.columbia.edu/~chw2/4400.htm

Book: "Quantitative Biology" (Chris Wiggins) students email me w/your uni (columbia ID) and your GitHub ID for access

Audience: advanced undergraduates and beginning graduate students with some exposure to probability, statistics, dynamical systems, classical mechanics, statistical mechanics, linear algebra. Previous students have come from such departments as APAM, biology, BME, CS, DBMI, EE, physics, ..., and probably several that I'm forgetting and/or don't know about.

Homework: due 2 classes after it's assigned (usually 1 week), at start of class

Grader: Bigeng Wang, [email protected]

ESTIMATED outline/schedule, i.e., the dates for checkups, tests, etc. are estimates until you hear otherwise. Don't make travel or other plans around these dates unless I tell you by email. Then it's real.

Lecture 1: Wednesday, Jan 20, 2016.

• course overview (title/audience)
  1. who are you?
  2. who am i?
• grading policy
  • checkups: 50% (hopefully there will be 2-3 of them, depending on progress of lectures)
  • homework: 10% (to be turned in, email preferred, or hard copy, at the start of class 1 week after it's assigned (or at the start of the next class after 1 week has elapsed in case of schedule irregularities)) Late HW is 0 credit.
  • final: 40%
    • check here later in the term
• course history (f01,s03,s04,s05,s06,s07,s08,s10,s11)
• prerequisites/useful background
  • math (ode,pde,lin alg,prob,stat)
  • physics
    1. no quantum mechanics
    2. stat mech will be "derived"
    3. it would be good to know some classical mechanics, but there will be no Lagrangians
    4. electrostatics will show up briefly
• books
  • Goodsell (lecture 0 only)
  • berg (lecture 1 only)
  • PCN (part I only)
  • mine
    • note: appendices to this book have as their origin students' questions. therefore, please ask questions when anything is unclear.
• things we will discuss: 3 revolutions
  1. single molecule biophysics and the physics of biological materials
  2. systems biology
  3. data-driven biology
• things we will not discuss
  1. physiology
  2. immunology
  3. neuroscience
• physics at the scale of the cell
  • the scale of the cell
  • scales of things in general
  • what's inside a cell? (cf. Goodsell's site and book)
• verbs: dynamics at the scale of the cell
  • history: brown and Brownian motion
  • movies from Eric weeks
  • demo
  This lecture will involve some simple physics, some simple mathematics, and some dimensional estimation of things.
  • Brownian motion: (image from Eric Weeks, physics department, Emory)
  • microscopic explanation a la Einstein
  • Step size model of a statistic:
    σ² = 2Dt
    
  • where does D come from?
  • T
  • D = T/b ; what are dimensions of b?
  • drag: what is it? where does it come from?
• HW #1
  1. Learn the Greek alphabet

Lecture 2: Monday, Jan 25, 2016.

• Review from last time
  • scale of the cell
  • Brownian motion, demo, diffusion
  • dimensions of temperature, viscosity, and "drag constants", suggestion of Stokes
  • dimensional suggestion of Stokes-Einstein
• Langevin approach to understanding Stokes-Einstein
• How big is viscosity? (Re)
• Universality of diffusion: stocks, polymers
• probability: 2 properties, 2 rules
• HW #2:
  1. How many cells make up you?
  2. Go to the RCSB protein data bank ( http://www.rcsb.org/ ) and look up your favorite protein. If you don't have one yet, try "kinesin", or "Gene Regulating Protein". How big is it? (in volume? in mass? in Daltons?) How many proteins could fit in a cell?
  3. Convert kT, at room temperature, into picoNewtons and nanometers
  4. what is the ``molar concentration" of 1 object in a bacterium? (a concentration is just a number of objects per volume -- irrespective of what the object is)
  5. Estimate how high you jump if you stay in the air for 1 second. (hint: use the natural acceleration given by gravity)
  6. Estimate the stall force of kinesin given that it takes 8 nm steps (hint: use the natural energy given by room temperature)
  7. Estimate deceleration time for a bacterium: Use the fact that it's decelerating with a force given by the drag force $F=-bv,$ and this balances $mass* (dv/dt).$ Then estimate the size of the exponential decay time. Assume a spherical bacterium made of water, and a drag constant given by Stokes relation (as we discussed in class).)
  8. Estimate coasting distance for a bacterium, assuming it is swimming initially at its own body length/second, and then comes to rest over one stall time

Lecture 3: Wednesday, Jan 27, 2016.

• Step size model as a probability
• Derivation of a statistic (an expectation) from a probability (a distribution)
• the flow of a probability
• derivation of diffusion equation equation and is normalized for the right choice of A and B

Lecture 4: Monday, Feb 1, 2016.

• diffusion with drift
• J=0
• Boltzmann
• partition functions $U\propto x$
• HW #3:
  1. show that $\exp(-x^2/(2Bt))/\sqrt(At)$ (the Gaussian with width given by Bt) is normalized and solves the diffusion for appropriate A and B.
  2. find ρ such that J_total = J_drift + J_diffusion = 0 and bv = −∂_xU(x).

Lecture 5: Wednesday, Feb 3, 2016.

• review:
  • 2 properties of probability
  • 2 rules of probability
  • Bayes theorem
  • random walks
  • diffusion with drift
  • conservation laws
  • statistical steady state implies Boltzmann
  • diffusion with drift
  • The Boltzmann distribution
• intuition-building / putting Boltzmann to work in biology by choices of "U"
  • $U=x$: pollen grains in gravity
• The partition function --- a moment generating function for the
• relating $Z''$ to $\sigma_U^2$. energy
• HW #4:
  1. prove ∫₀^∞dvvexp(−v)=1
  2. prove var(x)= < x² > − < x>² = <(x − (x))²>
  3. calculate <U> for U = αxⁿu, with either x=0..infinity or x=-infinity..infinity if nu=odd or even, respectively

Lecture 6: Monday, Feb 8, 2016.

• THE moment generation function; A moment generation function
• more intuition-building / putting Boltzmann to work in biology by choices of "U"
  • $U=\pm 1$: Ising model/pulled DNA/"two-state systems"

Lecture 7: Wednesday, Feb 10, 2016: fluctuations

• manning
• Debye

Lecture 8: Monday, Feb 15, 2016.

• linear response theory
• Review for checkup on Wednesday, Feb 17, 2016.

Checkup 1: Wednesday, Feb 17, 2016.

Lecture 9: Monday, Feb 22, 2016.

• J nonzero
• motor proteins (U ∝ x)
• HW #5
  1. derive J[U]
  2. derive J for U=fx
  3. derive J for Kramer's escape (double well)

Lecture 10: Wednesday, Feb 24, 2016.

• J nonzero
• Kramers and double well (U ∝ x²)
• 2-state dynamics for probability

Guest 1: Monday Feb 29, 2016.

Lecture 11: Wednesday, Mar 2, 2016.

• MMK
• standard dogma
• transcriptional regulation as a dynamical system
• Literature/context
  • Lambda
  • Lac
• HW #6:
  1. derive P₊/P₋ from rates
  2. find fixed points of x_t = λx(1 − x)
  3. find stability of each fixed point from the above
• HW for me: confirm book diagram of MMK

Lecture 12: Monday, Mar 7, 2016 (due Monday Mar 21)

• MMK and regulation
• fixed points and dynamical systems for auto-up and auto-down-regulation
• HW #7:
  • classify stability for all fixed points for the auto-up-regulating and auto-down-regulating gene. Feel free to start with y′=1/(1 + y)−ay and y′=y/(1 + y)−ay for the 2 cases.

Lecture 13: Wednesday, Mar 9, 2016 (due Wednesday Mar 23)

• Cooperativity and regulation
• Literature/Context: synthetic biology
  • Exuberance
  • Toggle
  • Oscillator
• Redo MMK-style view of up- and down-regulation but with cooperativity
• HW #8:
  • Starting with MMK-style modeling of transcription factor binding, but including cooperativity (i.e., F ↔ nX), derive y′=yⁿ/(1 + yⁿ)−αy for an auto-up-regulating gene.
  • Same question but for down-regulation, y′=1/(1 + yⁿ)−αy.

Monday, Mar 14, 2016.: spring break

Wednesday, Mar 16, 2016.: spring break

Lecture 14: Monday Mar 21

• New business: checkup #2 on April 13
• Review of MMK approach with cooperativity
• Dynamical systems with cooperativity
• Review bifurcation diagrams for n=1
• Pictoral view of cooperative autoregulation

Lecture 15: Wednesday, Mar 23, 2016.

• Critical parameter for n=2 case
• general n case
• 2D dynamics
• 2D eigenspace
  • det, trace, and stability
• comparison with 1-D case
• HW #9:
  1. show that, for a d-dimensional dynamical system y_t = g(y), near a fixed point y_* (i.e., g(y_*)=0), if the Jacobian of the velocity J_ik = ∂_yig_k admits d distinct eigenvalues λ_α with corresponding eigenvectors v_α, i.e., enjoying Jv_α = λ_αv_α, then η(t)=∑_αc_αexp(tλ_α)v_α is a solution for y(t) near a fixed point.
  2. Show that the dimensionful equations for transcription and translation (in terms of r_t and x_t) can be nondimensionalized as r_t = f_±(xⁿ)−βr, x_t = r − γx for suitable β, γ.
  3. Show that in 2 dimensions we have stability of fixed points ⇔ ∂_xf(x_*)<βγ

Lecture 16: Monday, Mar 28, 2016.

• Cellular noise
  • Master equation
• stochastics and BDP (birth death process)
  • micro-macro correspondence
  • decay
  • production
  • combination
  • steady state for 'constant' birth-death
• HW #10
  1. show that Poisson is steady state for BDP
  2. compute 1st moment from Poisson distribution
  3. compute 2nd moment from Poisson distribution

Lecture 17: Wednesday, Mar 30, 2016.

• BDP
  • general form
  • familiar conservation law form
  • dynamics of 1st moment
• Literature examples
  • Twin promotors
  • event-driven MCMC, aka Gillespie, original paper citations
  • Lambda simulation

Lecture 18: Monday, Apr 4, 2016.

• general case of BDP
  • conservation
  • dynamics of 1st moment as average creation + decay
  • general solution
  • discontinuous poisson
• matlab, n=2 auto-up-regulation
  • main code
  • qcalc code
• linear noise approximation
• HW #11

  1. Take the Fourier transform of the Poisson distribution p_n with mean x by computing G(z)≡∑_nzⁿp_n. (Note: For a general series this is called the Generating Function, but it is also seen to be the Fourier Transform if we identify z = exp(ik). The equivalence of the invertibility of the transform by integrating against exp(−ikm)/2π with the invertibility by differentiating w.r.t. z m times and evaluating at z = 0 follows from Cauchy's differentiation formula.)

  2. Next:
  • Take the same Fourier transform of the left and right hand side of the master equation ∂_tp_n = ∑_σ = ±1(E^σ − 1)ω_n^−σp_n for the vanilla birth death process (i.e., with ω_n⁺ = x, ω_n⁻ = n) to find a 1st-order in time, 1st- order in z PDE for G(z, t). Then
  • Show that the right hand side of this PDE annihilates the Poisson distribution's Fourier transform; i.e., confirm in Fourier space that the Poisson distribution is the statistical steady state of the birth death process.

Lecture 19: Wednesday, Apr 6, 2016.

• linear noise approximation
• multiple reacting species
  • solution: Van Vleck Catastrophe
  • simulation: Curse of dimensionality
• review for Checkup #2

Lecture 20: Monday, Apr 11, 2016. - review for Checkup #2

Special event: Wednesday, Apr 13, 2016. - Checkup #2

Lecture 21: Monday, Apr 18, 2016.

• Data
  • reduce
  • regression
  • regularization
  • likelihood
  • cross validation
  • HW #12
    1. Show that the minimizer of |y − β^Tx|₂ enjoys the matrix equation M₁β = b₁ (i.e., find M₁ and b₁).
    2. Show that the maximizer of P(D, β) with the prior P(β) a Gaussian (one independent Gaussian for each component) enjoys the matrix equation M₂β = b₂ (i.e., find the appropriate M₂ and b₂) (Hint: in a limit as one of your paramters goes to 0, you should recover M₁ and b₁).

• More on model selection
  • cross validation
  • "being bayesian without believing": ridge=Tikhonov=L2=Bayesian
  • BIC "The Bayesian Information Criterion is neither Bayesian, nor about Information, nor a Criterion. Discuss"
• Classification in biology
  • papers
  • 'landmark' paper using unsupervised learning for a supervised learning task: Golub 1999
  • example from viral genomics (Anil Raj, Michael Dewar, Gustavo Palacios, Raul Rabadan, CW)
  • classification loss, practical facts
  • generative approach: logistic regression
  • role of regularization
• HW #13
  1. Given P(y = +1|β)/P(y = −1|β)=exp(β^TX), show that the likelihood of all the data is P(D|β)=Π_i^N(1 + exp(−yβ^TX))⁻¹
  2. Read Schwartz's paper (very short!)

Lecture 23: Monday, Apr 25, 2016.

• Classification in biology
  • logistic regression
  • practical matters:
  • newton optimization
  • gradient descent
  • generalizing: surrogate loss functions
  • SVMs
  • boosting
• HW #14
  1. For boosting loss at iteration t, we choose a feature x_i^t ∈ { − 1, 1} with strength β. If example i has weight d_i, the loss now reads L_t(β)=∑_id_iexp(−y_iβx_i^t). Derive β_t, the value of β which minimizes this loss, by breaking the sum into two sums: one over terms such that x_i^ty_i = 1, the other such that x_i^ty_i = −1.
  2. This doesn't tell us which feature x^t would have been a good "weak rule" to introduce. Inserting β = β_t into the loss function, show that the best rule can be evaluated using two numbers D_±^t := ∑_{{i|xi^tyi = ±1}}d_i for different choices of x^t. (hint: note that D₊ + D₋ is the same for any choice of x^t.)
• Unsupervised learning in biology
  • k-means
  • GMM
  • relating the two

Lecture 24: Wednesday, Apr 27, 2016.

• Unsupervised learning in biology
  • k-means
  • GMM
  • relating the two
• HW #15
  1. derive the "maximization" steps for determining μ_z, π_z, and σ_z for a Gaussian Mixture model of K (possibly greater than 2) univariate ('one dimensional') mixtures.
  2. show that we recover k-means if we
     • replace σ_z with a common σ and
     • replace q(z) with 1 if z is the argmax of q(z) and 0 otherwise
• Reinforcement learning in biology

• Reduction from prescriptive to predictive
• Hot topics in biology: 2007-2015
  • single cell sequencing (2013 MOTY)
    • 2015 review paper
    • techniques employed: HMM, AIC, BIC, clustering, BDP
    • applications/citations
      • NB Eisen paper, discussed earlier w.r.t. reproducibility and sharing of code and data
      • Application w/BDP
• Review for final