E4400
E4400: INTRODUCTION TO BIOPHYSICAL MODELING

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

Space-time: Mondays+Wednesdays 2:40-3:55, 253 Engineering Terrace

Office hours: by appointment.

TA: Anil Raj (email)

mailing list: contact me if you a student but have not received any mails with subject line starting "[4400] "

• Lecture 1; Weds Jan 18, 2006
Nonscience:
• course overview (title/audience)
• my history
• course history (f01,s03,s04,s05,s06,s07)
prerequisites / useful background
• math (pde, ode, linear algebra, probs, stats)
• physics
• books
• 1st checkup
• 2nd checkup
• final
• homework
Science:
• 3 revolutions
• biological physics
• parts
• physics at the scale of the cell
• 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 23, 2006
This lecture will involve some simple physics, some simple mathematics, and some dimensional estimation of things.
• Brownian motion
• 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?
• 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 stall time for a bacterium (I shouldn't call this a "stall time" exactly, it's more of a "deceleration time". 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
• Lecture 3: Wednesday, Jan 25, 2006
• 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. Derive the Gaussian from binomial ( $p=At^(-1/2)exp(-x^2/(2B^2t))$ )
2. Show that Gaussian solves diffusion equation ( $p_t=Dp_xx$ ) (a relation between "B" and "D")
3. Show that the Gaussian is normalized ( $\int dx p=1$ ) (a relation between "B" and "A")
• Lecture 4: Monday, Jan 30, 2006
• Diffusion with drift
• The Boltzmann distribution
• HW
1. E(x^2) for a Gaussian
2. E(x) for pollen grain
3. E(x) + sigma_x for DNA
• Lecture 5: Wednesday, Feb 1, 2006
• Entropic forces
• Linear response
• Diffusive model of polymer coiling
• HW
1. Derive linear response $<x>_F~<x>_0+(F/T)<x^2>_0$
2. Verify linear response for the two state" pollen grain with $<x>=<h>$ and $h_-=0, h_+=H, E_-=0, E_+=FH$
3. 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).
• Lecture 6: Monday, Feb 6, 2006
• The partition function --- a moment generating function for the energy
• Manning condensation
• HW
1. Show that the exponent (i.e., the Manning parameter) is dimensionless in the Manning condensation calculation
2. Estimate this parameter for DNA at room temperature
3. Show that the expectation can be written $<r>=af(R/a)$ where $f$ is a dimensionless function, $R$ is the size of the cell (the farthest radial distance possible from the charged DNA to the counter-charged particle)
4. Show that, as $R/a$ diverges to infinity, the qualitative behavior of $<r>$ depends on whether the Manning parameter is above or below a critical value.
• Lecture 7: Wednesday, Feb 8, 2006
• No lecture: sick day
• HW
1. In honor of their 10-year anniversary, read Smith 1996 and Cluzel 1996 (Cluzel 1996 instroduces the S" phase of DNA we discussed on Monday Feb 6).
2. Check out the videos of previous lectures and let me know how CCNMTL can improve this service
• Lecture 8: Monday, Feb 13, 2006
• electrostatic surprises in vivo
• review of Manning condensation
• HW
1. 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 9: Wednesday, Feb 15, 2006
• transport; motor proteins
• HW
1. if U=Fx, derive r=r_diff*g(FL/T)
2. Show, when F=0, g(0)=1 and when F=inf, g(inf)=0
3. Show the time tau derived in the Kramers escape problem has dimensions of time.
4. 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.
• Lecture 10: Monday, Feb 20, 2006: review session for checkup
• Lecture 11: Wednesday, Feb 22, 2006 (1st checkup)
• Lecture 12: Monday, Feb 27, 2006
• 2-state kinetics, steady state, entropy
• chemical kinetics, including 2nd order kinetics
• MM (enzyme) kinetics
• parts list: informatic parts
• transcriptional regulation
• 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.
• Lecture 13: Wednesday, Mar 1, 2006
• chemical kinetics; hill form and MMK form
HW
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.
• Lecture 14: Monday, Mar 6, 2006
• 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 8, 2006
• 2D dynamical systems
• HW:
1. 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)
2. show that bifurcation of the fixed points occurs when gamma=gamma_c=1/2.
• Lecture 16: Monday, Mar 20, 2006 (Spring Recess is Mar 13-17)
• Review 2D dynamical systems
• Mutual repression:
• 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, Mar 22, 2006
• 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, Mar 27, 2006
• decay
• HW
for the decay process, d(p_j)/dt=-k*j*p_j+k*(j+1)*p_{j+1}
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, Mar 29, 2006
• 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 3, 2006
• simulation
• LNA: the "linear noise approximation"
• network theory
• (NO HW)
• Lecture 21: Wednesday, Apr 5, 2006
• 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 10, 2006
• review for checkup
• REDUCE
• Lecture 23: Wednesday, Apr 12, 2006 (2nd checkup)
• Lecture 24: Monday, Apr 17, 2006
• 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 19, 2006
• 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, Apr 24, 2006
• Greedy methods
• Feature ranking
• p values
• HW
1. Show that argmax dL = argmax |a.n_k| where argmax is with respect to parameter k.
• Lecture 27: Wednesday, Apr 26, 2006
• HW
1. Derive the razor for arbitrary probabilities for noise and prior.
2. Show $f* = argmin <exp(-af)> = 0.5*ln[p(+|x)/{1-p(+|x)}]$
3. Find argmin Z_j(alpha) as a function of d+_j and d-_j (d+_j & d-_j are probabilities summing to 1).
• Lecture 28: Monday, May 1, 2006 (Last day of Classes)