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

Space-time: Mondays+Wednesdays 11-12:15. Location 253 Engineering Terrace

Office hours: by appointment.

URL: j.mp/biophys2010

Book: "Quantitative Biology" (Chris Wiggins) (contact me for the password) http://drop.io/biophys2010

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.

Grader: Anil Raj

• Lecture 1: Wednesday, Jan 20, 2010.
  Nonscience:
  • 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, 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,s08)
  • 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.
  Science:
  • 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
    nouns:
    • 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 25, 2010.
  • How big is viscosity? (Re)
  • Universality of diffusion: stocks, polymers
  • Step size model as a probability
  • Derivation of a statistic (an expectation) from a probability (a distribution)
  • HW
    1. Show using probability that $<x^2>=jl^2$.
    2. Estimate how high you jump if you stay in the air for 1 second. (hint: use the natural acceleration given by gravity)
    3. Estimate the stall force of kinesin given that it takes 8 nm steps (hint: use the natural energy given by room temperature)
    4. 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).)
    5. 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, 2010.
  • the flow of a probability
  • derivation of diffusion-with-drift equation
  • fluxes and conservation laws
  • probabilities vs. probability densities
  • statistical steady state:
    • the bolzmann distribution
    • the stokes-einstein relation
  • HW
    1. recover the earlier relations $<x>=0$ and $<x^2>=constant*t$ from the fact that the probability density solves the diffusion equation and the initial data $<x>=0$, $<x^2>=0$.
    2. show that $exp(-x^2/(2Bt))/sqrt(A*t)$ solves the diffusion equation and is normalized for the right choice of A and B
    3. super bonus homework only if you know some fourier analysis: solve the diffusion equation for all time assuming initial data $rho_0(x)=\delta(x)$ (i.e., the initial distribution is a dirac delta function)
• Checkup 1: Monday, Feb 1, 2010. (This is a "getting to know you" checkup, with a much smaller weight than the others. 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 4: Wednesday, Feb 3, 2010.
  • review:
    • 2 properties of probability
    • 2 rules of probability
    • bayes theorem
    • random walks
    • diffusion with drift (rederived more carefully, as per mike's observation in class on monday jan 25)
    • 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
    • $U=\pm 1$: pollen grain as a "two-state system"
  • The partition function --- a moment generating function for the energy
  • HW
    1. calculate $<(x-<x>)^2>$ for a pollen grain in a test tube experiencing gravity under the two models we investigated in class:
       1. height $0<x<+infinity$
       2. height takes on two values, $x=0$ and $x=h$.
• Lecture 5: Monday, Feb 8, 2010.
  • more intuition-building / putting boltzmann to work in biology by choices of "U"
    $U=\pm 1$: ising model/pulled DNA/"two-state systems"
    • $U=x^2$: optical traps
  • old business on probability: binomial distribution yields diffusive scaling?
  • The Gaussian, and the Gaussian integral
  • The binomial
  • How are the Gaussian and the binomial related?
  • optical trapping, as in this movie)
  • HW
    1. Consider the two state system modeling DNA under external force $F$ as a series of links which can point left or right. In class we calculated $<x>$, the expected displacement for one link. Now calculate $<(x-<x>)^2$.
    2. The total extent $X$ is the sum of all the individual links. While $<X>=N<x>$, (since the individual links are independent), the sum of independent two-state variables is drawn from the binomial distribution. Use this distribution (I encourage you to invoke the moment generating function $(p+q)^N$ for a chain of $N$ links) to calculate $<(X-<X>)^2>$.
    3. Verify linear response, i.e., $<dx>=dF<x^2>/T$.
• Lecture 6: Wednesday, Feb 10, 2010.
  • HW
    1. Show that the binomial solves the equation for discrete time, discrete space updating. that is, show that $P(X=r-l|T)=T!/((r!) (T-r)!)1/2^T$ solves $P_{T+1}(X)=1/2 P_T(X-1)+1/2 P_T(X+1)$
    2. Show that the binomial asymptotically approaches the gaussian using Stirling's approximation when $r=(N/2)*(1+\delta)$ for small $\delta$. Hint: look at the back of the book.
• Lecture 7: Monday, Feb 15, 2010.
  • review of partition function tricks
  • specific heat is nonnegative
  • manning condensation
  • screening
  • HW:
    1. calculate $/a$ for the manning geometry. 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.
    2. verify that the screening function $\phi=A exp(-B*r)/r$ satisfies the linearlized poisson-boltzmann equation $(laplacian) phi-k^2 phi=-Q *delta(r)$
• Lecture 8: Wednesday, Feb 17, 2010. Thermal ratchet models of motor proteins
• Lecture 9: Monday, Feb 22, 2010. Thermal ratchet models of motor proteins
• Lecture 10: Wednesday, Feb 24, 2010. Kramers' escape
• Monday, Mar 1, 2010. Guest lecture on chemomechanical coupling
• Lecture 11: Wednesday, Mar 3, 2010. Dynamics
• Lecture 12: Monday, Mar 8, 2010. Review for checkup #2
• Wednesday, Mar 10, 2010. Checkup #2
• Lecture 13: Monday, Mar 22, 2010.
• Lecture 14: Wednesday, Mar 24, 2010.
• Lecture 15: Monday, Mar 29, 2010.
• Lecture 16: Wednesday, Mar 31, 2010.
• Lecture 17: Monday, Apr 5, 2010.
• Lecture 18: Wednesday, Apr 7, 2010.
• Lecture 20: Monday, Apr 12, 2010.
• Lecture 21: Wednesday, Apr 14, 2010.
• Lecture 23: Monday, Apr 12, 2010.
• Lecture 24: Wednesday, Apr 14, 2010.
• Lecture 25: Monday, Apr 19, 2010.
• Lecture 26: Wednesday, Apr 21, 2010.
• Lecture 27: Monday, Apr 26, 2010.
• Lecture 28: Wednesday, Apr 28, 2010.
• Lecture 29: Monday, May 3, 2010.