E4400

E4400: INTRODUCTION TO BIOPHYSICAL MODELING

Instructor: Professor Wiggins, rm 205 S. W. Mudd

Space-time: Mondays+Wednesdays 11-12:15. Location 253 Engineering Terrace (Wednesdays); 214 Mudd (Mondays).

URL: http://www.columbia.edu/itc/applied/wiggins/Classes/E4400/Spring2008/

Book: "Quantitative Biology" (Chris Wiggins) (table of contents and manifesto; contact me for the full book) http://q-bio.myftp.org/

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.

Lecture 1; Wednesday, Jan 23, 2008.
- course overview (title/audience)
  1. who are you?
  2. who am i?
- grading policy
  - checkups: 50%
  - homework: 10% (to be turned in, in hard copy rather than by email, 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))
  - final: 40%
    - check here later in the term
    - projections
- course history (f01,s03,s04,s05,s06,s07)
- 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 (this is the 6th iteration of the course)
    - 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
  - java applet
  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: $\sigma^2=Dt$
  - where does D come from?
  - T
  - $D=T/b$ ; what are dimensions of b?
  - A dimensionful model, with v from ideal gas and t from drag
  - drag: what is it? where does it come from?
- Homework
  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)
Lecture 2: Monday, Jan 28, 2008.
- How big is viscosity? (Re)
- Universality of diffusion: stocks, polymers
- Step size model as a probability
- The binomial
- Derivation of a statistic (an expectation) from a probability (a distribution)
- HW
  1. Estimate how high you jump if you stay in the air for 1 second. (hint: use the natural acceleration given by gravity)
  2. Estimate the stall force of kinesin given that it takes 8 nm steps (hint: use the natural energy given by room temperature)
  3. 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).)
  4. 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
  5. Use the relationship between $p_{n+1}(x)$ and $p_n(x)$ to relate $<x^2_{n+1}>$ to $<x^2_n>$
  6. Show that $<x^2>=l^2t$ using the moment-generating function $(p+q)^t$.
Checkup 1: Wednesday, Jan 30, 2008 (This is a "getting to know you" checkup. Basic problems in probability and mechanics will be asked to give me some sense of what you know and to introduce you to what my checkups are like.)
Lecture 3: Monday, Feb 4, 2008.
- old business on modeling: viscosity?
- old business on probability: binomial distribution yields diffusive scaling?
- old business: the diffusion equation
- The Gaussian, and the Gaussian integral
- Why are the Gaussian and the binomial the same?
- Probability modeling: Transport
- conservation laws
- putting it together: diffusion with drift.
- HW
  1. Show that Gaussian solves diffusion equation ( $p_t=Dp_xx$ ) (a relation between "B" and "D")
  2. Show that the Gaussian is normalized ( $\int dx p=1$ ) (a relation between "B" and "A") (Hint: use integration by parts) (Double Hint: check the appendix to the book)
Lecture 4: Wednesday, Feb 6, 2008.
- Diffusion with drift
- The Boltzmann distribution
- HW
  1. Show that $\Gamma(z) = (z-1)\Gamma(z-1)$
  2. Derive the Gaussian from binomial ( $p=At^(-1/2)exp(-x^2/(2B^2t))$ )
  3. Derive $\<x\>$ (the expected displacement to the right) as a function of applied force $F$ for a ``two-state" model of DNA under applied tension. That is, DNA is made of N independent links, each of length $l$, and the ratios of proababilities for facing right- or left- is determined by the energy difference between these two states via the Boltzmann relation. Hint: see the book
Lecture 5: Monday, Feb 11, 2008.
- optical trapping, as in this movie)
- The partition function --- a moment generating function for the energy
- Manning condensation
- HW
  1. $<x^4>$ for $U(x)=kx^4$.
  2. express the specific heat $C=d\<U\>/dT$ in terms of the partition function Z
Lecture 6: Wednesday, Feb 13, 2008.
- Linear response
- Diffusive model of polymer coiling
- WLC
- thermal ratchet models
- HW
  1. Verify linear response for the ``two state" pollen grain with $<x>=<h>$ and $h_-=0, h_+=H, E_-=0, E_+=mgH$
  2. This is a 4-part research question about the diffusive model of polymer coiling:
    1. Look up the total length of the lambda phage DNA (i.e., the length of the entire genome of the virus)
    2. Look up the ``persistence length" of DNA (i.e., the length of one statistically independent sub-length of DNA)
    3. Look up the size of the lambda phage capsid (i.e., the ``head" of the virus, where the DNA itself is stored)
    4. Compare this size with the expected radius of gyration for the lambda phage DNA (i.e., based on the diffusion model we discussed in class) (and using parts 1 and 2).
  3. Show that the Manning parameter is dimensionless in the Manning condensation calculation
  4. Estimate this parameter for DNA at room temperature
  5. Estimate the room-temperature Bjerrum length and the Debye screening length. For the Debye screening length you will need a ``"physiological salt concentration"." Email me if you can't find an example online -- I can email back with some credible URLs. (e.g., this or this
Lecture 7: Monday, Feb 18, 2008.
- Kramers escape
- HW:
  1. thermal ratchet: derive the escape time as function of (beta*F*L)
  2. Kramers escape: Derive the 'back-reaction time'; the average time for a particle to hop from the global to the local minimum. It should have very similar dependence on temperature, and on the energy of the barrier, as we derived in class for the `forward' reaction time: the typical time for a particle to jump from the metastable well to the stable well.
  3. Show that the forward and backward rates give a dynamical system (a pair of ODEs for P_left and P_right) whose kinetic equilibrium is the thermodynamic equilibrium
Checkup #2: Wednesday, Feb 20, 2008.
Lecture 8: Monday, Feb 25, 2008.
- 2-state kinetics, steady state, entropy
- chemical kinetics, including 2nd order kinetics
- MM (enzyme) kinetics
- parts list: informatic parts
- transcriptional regulation
Lecture 9: Wednesday, Feb 27, 2008.

Lecture 10: Monday, Mar 3, 2008.

Lecture 12: Wednesday, Mar 5, 2008.

HW
1. Review the solutions to checkup 1, thoroughly, so that if, perchance, one of the questions were chosen at random to appear on future checkups, or the final, you will have mastered it.
2. Classify stability of nonzero fixed points for auto-inducing gene with cooperativity (n>1)
3. Classify stability of nonzero fixed points for auto-inducing gene without cooperativity (n=1)
4. show that the linear combination X=a+b enjoys X_t is proportional to X

Lecture 13: Monday, Mar 10, 2008.

chemical kinetics; hill form and MMK form
1. For M.M. kinetics (enzyme kinetics), with a substrate S which itself is the multimer of n units F (i.e., S<->nF ), show that the reaction rate (rate of production of phosphate in the case of motor proteins) is of the form p_dot/m_T = k2*f^n/((K_m)^n + f^n)
2. starting from the chemical kinetics equations, show that transcriptional activation and transcriptional repression, for transcription factors which multimerize with n units (e.g., dimerize with n=2, tetramerize with n=4, etc), can all be expressed by defining suitable parameters a and K, which are themselves algebraic functions of the bare reaction rates, as either
  - (for repression, a.k.a. "down-regulation") a K^n/(K^n+p^n)
  - (for activation, a.k.a. "up-regulation") a p^n/(K^n+p^n)
  note that the expressions for a and K in terms of the bare reaction rates will not be the same for the two cases above.

Checkup #3: Wednesday, Mar 12, 2008. (Spring Recess is Mar 17-21)

Lecture 14: Monday, Mar 24, 2008.

Review of all chemical kinetics; hill form and MMK form
- A <-> B
- MMK
- transcription
- repression
1D dynamical systems
HW: for f=a p^2/(K^2+p^2)-kp; c=a/k,
1. show f'(p*)/k=1-2p*/c
2. show f'(0)/k=-1

Lecture 15: Wednesday, Mar 26, 2008.

2D dynamical systems

HW:

show that (p^2/(K^2+p^2))' evaluated at c~p=K^2+p^2 equals 2/c~(1-p/c~). (c~ is some characteristic concentration)
show that bifurcation of the fixed points occurs when gamma=gamma_c=1/2.

Lecture 16: Monday, Mar 31, 2008.

Review 2D dynamical systems
Mutual repression:
- lambda
- lac
- a model
SGNs:
- oscillation
- switches
HW
1. derive the instability criterion for the symmetric solution to the N-repressor loop model, as a function of n (the hill coefficient), N (the number of genes), and alpha (the nondimensionalized transcription rate)
2. read the review paper i handed out
3. complete amy's survey letting me know what you'd like to see changed, improved, deleted, etc.

Lecture 17: Wednesday, Apr 2, 2008.

creation
HW
1. derive the poisson distribution from the binomial distribution.
2. derive the poisson distribution from d(p_j)/dt = -alpha*p_j+alpha*p_{j-1} and d(p_0)/dt = alpha*p_0
3. show that that variance of the poisson distribution is lambda.

Lecture 18: Monday, Apr 7, 2008.

decay
HW
1. show that sum(d(p_j)/dt)=0 over all j.
2. show that dX/dt=-k*X, where X=E(j)
3. when growth and decay are both involved, find the probability distribution at statistical steady state (d(p_j)/dt=0)

Lecture 19: Wednesday, Apr 9, 2008.

creation and decay
autoregulation
simulation
HW
1. Show that the binomial solves dp/dt=0 {in pg 233}.
2. Read the handout well. (email me if you didn't get it)

Lecture 20: Monday, Apr 14, 2008.

simulation
LNA: the "linear noise approximation"
network theory
(NO HW)

Lecture 21: Wednesday, Apr 16, 2008.

review for checkup
microarrays
HW
1. Show that the master equation for the reaction {null}<->R is a limit of the master equation for A<->B for k1<

Lecture 22: Monday, Apr 21, 2008.

review for checkup
REDUCE

Lecture 23: Wednesday, Apr 23, 2008.

Lecture 24: Monday, Apr 28, 2008.

Bayesian analysis
HW
1. for a vector f of unknowns, show argmax_f p(a|f)=argmin_f (1/2) f^TMf-bf (M_{k1,k2}=\sum_i x_i^{k1} x_i^{k2} is a matrix of size K * K)
2. show f-hat (the minimizer of f^TMf/2-bf) is M^{-1}b
3. show that the prior p(f)=exp(-1/2 alpha^2 \sum_k f_k^2) adds a new term to M of the form \alpha^2 \delta_{k1,k2}

Lecture 25: Wednesday, Apr 30, 2008.

Occam's Razor
HW
1. derive the razor
2. optional numerical assignment: plot xi^2 and k/2 \ln N for a randomly-generated noisy polynomial, fit to polynomials of degree k
3. optional numerical assignment: plot L0=xi^2 (for the training data) and L1=\sum_i(y_i-f(x_i))^2 (for "test data", or "hold out data") fit to polynomials of degree k

Lecture 26: Monday, May 5, 2008.

Greedy methods
Feature ranking
p values
HW
1. Show that argmax dL = argmax |a.n_k| where argmax is with respect to parameter k.
2. Derive the razor for arbitrary probabilities for noise and prior.
3. Show $f* = argmin <exp(-af)> = 0.5*ln[p(+|x)/{1-p(+|x)}]$
4. Find argmin Z_j(alpha) as a function of d+_j and d-_j (d+_j & d-_j are probabilities summing to 1).