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

Space-time: Mondays+Wednesdays 11-12:15. Location 834 Mudd

Office hours: by appointment.

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

Book: "Quantitative Biology" (Chris Wiggins) (contact me for it)

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: TBA

• Lecture 1: Wednesday, Jan 19, 2011.
  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,s10)
  • 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.
  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: $\sigma^2=2Dt$
    • where does D come from?
    • T
    • $D=T/b$ ; what are dimensions of b?
    • 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 24, 2011.
  • How big is viscosity? (Re)
  • Universality of diffusion: stocks, polymers
  • probability: 2 properties, 2 rules
  • HW #1:
    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
• Lecture 3: Wednesday, Jan 26, 2011.
  • 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
  • HW #2:
    1. for what boundary conditions does the diffusion equation give $<x>=constant$ and $<x^2>=constant*t$
    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
• Lecture 4: Monday, Jan 31, 2011.
  • diffusion with drift
  • J=0
  • boltzmann
  • partition functions $U\propto x$
  • HW #3: using the relation $\<U\>=-\partial_\beta \ln Z(\beta)$,
    1. calculate $<x>$ for $U\propto x$, $0<x<\infinity$.
    2. calculate $<x>$ for $U\propto x$, $x\in \{0,H\}$
• Lecture 5: Wednesday, Feb 2, 2011.
  • 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
    • $U=\pm 1$: pollen grain as a "two-state system"
  • The partition function --- a moment generating function for the energy
  • HW #4:
    1. An optical trap is modeled by an energetic potential $U=(1/2)k x^2$. In an experiment, a trap can be calibrated by measuring $<x^2>$ and solving for $k$ from the expression you know relating $<U>$ to the partition function. Express $k$ as a function of $<x^2>$.
• Lecture 5: Monday, Feb 7, 2011.
  • more intuition-building / putting boltzmann to work in biology by choices of "U"
    • $U=\pm 1$: ising model/pulled DNA/"two-state systems"
    • $U=x^p$, including gravity+optical traps
  • HW #5:
    1. numbers: calculate the dimensionless ratio $mgH/T$ for
       • $m=\rho V$ where $\rho$ is density of water
       • $V$ is the volume of an orgnelle of radius $.1\mu$
       • $g$ is gravitational accelleration
       • $T$ is room temperature measured in energy
       • $H$ is the height of 1 cell
    2. manning: finish the integral we started to calculate $<r>/a$ where $U=(e\lambda/2\pi \epsilon)*log (r/a)$, and $r$ ranges from $a$ to $R$, in the limit $R/a\rightarrow\infinity$.
• Lecture 6: Wednesday, Feb 9, 2011: fluctuations
  • relating $Z''$ to $\sigma_U^2$.
  • linear response theory
  • error-free derivation of diffusion w/drift
  • HW #6:
    1. consider the pollen grain in gravity in a test tube of unbounded height. Calculate $\sigma_x^2$ from $(ln Z)''$.
    2. verify linear response for a bead in a trap defined by $U=1/2 k x^2$ to which a new spatially-constant force $F$ (e.g., a magnetic force) is introduced.
• Lecture 7: Monday, Feb 14, 2011.
  • Review for checkup on Wednesday, Feb 16, 2011.
• Checkup 1: Wednesday, Feb 16, 2011.
• Lecture 8: Monday, Feb 21, 2011. Thermal ratchet models of motor proteins
  • HW #7:
    1. show that $J=D/\int_{x_-}^{x_+}dx \int_{x}^{x_0} dx' \exp{beta(U(x')-U(x))}$
    2. calculate $J(F)$ for $U=Fx, x_-=-L, x_+=0=x_0$.
• Lecture 9: Wednesday, Feb 23, 2011.
  • J nonzero
  • \tau as a function of U
  • motor proteins
  • Kramers escape rate
  • HW #8:
    1. calculate tau for a motor protein on a dimer, where each monomer has a different energy
    2. calculate the 'inverse' time t_{c->a} for a particle moving from stable to metastable minima
• Monday, Feb 28, 2011:
  • review of calculation of flux $J$ given $D$ and $U(x)$.
  • dynamical systems, with linear 2-D systems
  • HW #9:
    1. calculate v1 and v2, the eigenvectors of the matrix describing the dynamics of $P_-$ and $P_+$.
• Lecture 10: Wednesday, Mar 2, 2011. Dynamics
  • tangent: shannon's entropy, maxent distributions, and convexity
  • reprise: 2-state model of transport
  • review: 2-d dynamical systems, trace, det, and all that
  • a geometric picture of eigenspaces
  • conserved quantities and determinants
  • enzymes
  • MM-kinetics
  • HW #10:
    1. Armed with the dynamics of enzyme, complex, and product, and under the assumption that enzyme and complex have reached equilibrium, calculate production rate, normalized by total enzyme concentration ($e+c$). Express the result as a nonlinear function of substrate concentration.
• Lecture 11: Monday, Mar 7, 2011.
• Wednesday, Mar 9, 2011.
• Monday, Mar 14, 2011.: spring break
• Wednesday, Mar 16, 2011.: spring break
• Lecture 12: Monday, Mar 21, 2011.
• Lecture 13: Wednesday, Mar 23, 2011.
• Lecture 14: Monday, Mar 28, 2011.
• Lecture 15: Wednesday, Mar 30, 2011.
• Lecture 16: Monday, Apr 4, 2011.
• Lecture 17: Wednesday, Apr 6, 2011.
• Lecture 18: Monday, Apr 11, 2011.
• Lecture 19: Wednesday, Apr 13, 2011.
• Lecture nn: Monday, Apr 11, 2011.
• Lecture nn: Wednesday, Apr 13, 2011.
• Lecture nn: Monday, Apr 18, 2011.
• Lecture nn: Wednesday, Apr 20, 2011.
• Lecture nn: Monday, Apr 25, 2011.
• Lecture nn: Wednesday, Apr 27, 2011.
• Lecture nn: Monday, May 2, 2011. (last day of class)