Problems from Chapter 7 of The Colloidal Domain, pg. 398

1. Problem 7.3
2. Problem 7.4

 

3. Using the information presented in class for the persistence length of DNA in an electrolyte solution, and the results from your book for the dependence of persistence length on electrolyte concentration, compute the effective spacing between charges along the DNA molecule's backbone. How does this compare with the 0.34 nm spacing between base pairs and the 3.4 nm helical repeat?

4) Surprisingly, our simple random walk model describes blood-gas transfer in the lungs quite well. (I.e., the oxygenation of the blood and the ridding of the body of carbon dioxide. We will only treat the former problem here.)

The situation found there is the following. O2 molecules in air are inhaled into the lung's tiny air sacs, called alveoli. These molecules must diffuse a distance which can be roughly characterized as one alveolar radius (100 m) to reach the lung's lining. Once at the lining, the oxygen molecules diffuse a tiny distance (a small fraction of a micron) before reaching the interior of capillaries carrying blood which needs to be oxygenated. The radius of a capillary is typically 5 m. Finally, assume that the oxygen must diffuse a distance roughly equal to this capillary radius before encountering and being bound to a molecule of the protein hemoglobin. (Hemoglobin then transports the oxygen to tissues which need it for metabolic processes.)

The process of diffusion (the random walk of a particle in a gas or solution) is described in three dimensions by:

Here C(r,t) is the concentration of the molecules (random walkers) as a function of radial distance, r, from their origin (at r = 0) and time, t. Note: this looks like our random walk result for one dimension, with a slightly altered prefactor, and the radial distance, r, replacing the one dimensional distance, x. The RMS value of the radial distance moved, <r2>, is proportional to time, t--in other words, it's proportional to the number, N, of random walk steps taken and N increases linearly with time.

The constant of proportionality between <r2> and time depends upon a number called the diffusion constant, D. Values relevant to our problem are:

Diffusion constant (O2 in air) D = 0.178 cm2/sec

Diffusion constant (O2 in water) D = 1 ´ 10-5 cm2/sec

1. Show that the three-dimensional diffusion problem can be reduced to a product of three separate x,y, and z diffusion problems. What does the constant A have to equal then? The significance of this is that ANY diffusion problem in any number of N dimensions is just a product of N separate one-dimensional diffusion problems. The scaling exponent of 1/2 holds for any case, therefore.

 

2. Plot the function C(x,t) as a function of x for several different values of t (at least three values of t). What does the area under the curve represent? What relationship should the area follow as a function of time? Explain how your plots describe the spread of molecules from a central point, x = 0, as a function of time.
3. What is the relationship between the RMS distance moved on average, <r2>, and time, t? How much time does it take for the oxygen molecule to diffuse across one alveolar radius? (Compare this to the time between normal breaths. Is there enough time for a typical oxygen molecule to diffuse this distance in one breath?)
4. How much time does it take the oxygen molecule to diffuse across a distance equal to one capillary radius? (Assume the capillary's diffusion constant is roughly that of water.)
5. If a typical capillary hugs an alveolus over a distance of roughly 100 microns, and the blood in a capillary flows at roughly 0.1 cm/sec, how long is a hemoglobin molecule in proximity to the alveolar wall? Is this long enough for a typical oxygen molecule to wander across the alveolus and capillary and encounter the hemoglobin? (Explain why we can ignore the time required for the oxygen molecule to diffuse across a distance only a tenth of a micron or less.)

4. This problem shows that diffusion is very efficient on short distances. Now we will see why our lungs must have such tiny alveoli. How would your computation for a) change if the size of a typical alveolus was 1 cm? 10 cm? Why are alveoli small? Why does your body seldom use purely diffusive transport?

6) Using our two-state (A or B) polymer problem from class, do the following exercises.

Plot the force-extension curve (it's really going to be force as the independent variable and average extension <L> as dependent variable, but you can just plot it on its side!) using your favorite spreadsheet or scientific analysis package. Use the values of the parameters on my plot to recreate this plot, then generate plots to help you answer the following questions

1. How does the shape of the curve change if you vary the energy difference, DE? Try different values of DE, both positive and negative. What is the meaning of different signs here? Do the same exercise, but this time fix the value of DE at the value I used in class, and vary temperature, T, instead. Once again, what changes in this case? Write a short explanation in words of what you think is going on physically.
2. Compute the elastic modulus (effective spring constant) for small force (f _meas), and DE = 0. Since this corresponds to the case where the two states are energetically equivalent, the only thing you are "pulling against" when you exert a force on the polymer to keep it at an average extension <L> is its entropy: the fact that there are far more ways it can configure itself short than long. (You can see this most easily for the limits of smallest length vs. longest length.) You need to do this analytically, rather than graphically, although you can check out your computation using plots or using Mathematica. (Remember Josh's words from last class here, "It seems you are always doing Taylor series in physics, no matter what problem you are trying to solve."

Reread the journal articles "DNA: An extensible molecule" and ""Overstretching B-DNA: the elastic response of individual double-stranded and single-stranded DNA molecules" and answer the following questions:

a) The authors present two different approaches to measuring the force-extension curve for DNA. Explain the differences between their two approaches. Were their molecules the same or different? What about their apparatuses? What about the solutions used? Would you expect them to get comparable results, and if not, what factors would cause them to measure different phenomena? What important simplifications have these authors used?

b) What numerical results do they obtain from their data? Do their results agree? (Answer this question quantitatively, and taking into account uncertainties as much as you can.) Should they? (See question a) above.)

c) Which experimental approach do you think is the best one to use for this problem? Explain your answer. Explain what you see as the major limitations of each technique.

d) What unanswered scientific questions are posed by their work? Name at least one. It can be contained in their papers, from your own imagination, or from other readings you have done. Explain where you got your ideas.

You will be asked to write two papers for this course. The first will be due just after break (the Wednesday after break), and will consist of a 6-8 page report on the use of one of the biophysical techniques covered in class (either experimental, computational or theoretical) to investigate properties of specific biological problems. I will make suggestions about possible topics and references for each as the semester progresses. Here are some initial ideas:

http://expmed.bwh.harvard.edu/projects/polymer/actin_chapter

has a lengthy description of this subject, and many papers exist in the literature

A URL for this topic was included in an earlier email.

Additional systems studied in this fashion include: nacre (the adhesive protein found in mussels and other shellfish); polysaccharides and other systems I may not know about.

Current status of mathematical modeling of biopolymers. Rob Manning could help you find accessible recent work and review articles on this topic.

Polymer reptation. The way in which a polymer moves through a gel or polymer melt (that is, a disorderly tangle of polymers like a wiggling bowl of spaghetti) is described by its snake-like motion through this medium, hence the term "reptation". You can find a description of this process in your text, in De Gennes Scaling Concepts in Polymer Physics (not a very clear one, but a seminal work), and the following URL. The URL also gives a link to one paper describing the theoretical models of this phenomenon. You might be able to find websites with simulations, but you already have access to one simulation in your Random Walk websites for the first problem set. (Remember, the last website mentioned there had a reptation option.) The article is an early contribution to the study of such phenomena directly by imaging individual DNA molecules as they wiggle their way through gels in gel electrophoresis, the most important application of this idea. You can build up a good paper by reading at least four articles in this area, including one recent work you locate yourself.

http://www-cmp.phys.cmu.edu/projects/reptation/repton.html (also has link to theory paper preprint on this topic)

"Observation of Individual DNA molecules Undergoing Gel Electrophoresis", S.B. Smith, P.K. Aldridge, J.B. Callis, Science, Vol. 243 (Jan. 13, 1989) pp. 203-206. and followup work by Smith, Bustamante, et al. (available on-line via JSTOR)

Characteristics of polymer gels These could be biological gels, such as hydrogels formed from synthetic polypeptides (Rob Fairman could help with references on this), biological gels such as hyaluronic acid mentioned in your textbook, or gels for other applications, or the basic properties of gels as a material.

This would have to go beyond the two papers on dendrimers we will read as a class.

1) Given the Lennard-Jones potential between two non-bonded atoms:

a) Compute the force between these two atoms due to VLJ(r).

b) At their equilibrium separation, the force between the two particles is equal to zero. What is their separation, rmin at this point? What is V LJ(r) at this point?

c) Plot VLJ(r) for all r the way we did in class and now label its features using the results of b). Do the same for the force.

d) Near rmin , the equilibrium separation, we can expand the potential VLJ(r) to get the harmonic oscillator approximation. Find K, the effective spring constant, and write down the expression for the harmonic potential. Now, use the equipartition theorem to find the average root-mean-squared displacement r from rmin:

e) Now add to VLJ(r) an electrostatic contribution and compute the new force. (This cannot easily be solved for the ideal separations, etc., but we could find the answer graphically or numerically on a computer--just an example of how fast these problems grow difficult!)

Extra credit: grapple with the question of whether there is a bound solution where force equals zero for finite r for different values of the charges and other parameters in the problem.

1) Write down all terms in Vtotal for glycine-glycine, using equation (6) in Brooks, Karplus and Pettit, Chapter 3, as a guide. Be sure to accurately describe the j,f,w torsional potentials (use either a graph of energy vs. angle or a functional form)! You of course won't know numerical values of the parameters which appear in these equations, but at least denote which bonds, atoms, etc. they refer to.

b) Minimize the conformation using Discover. Use the steepest descents method first, then go to the Newton-Raphson for a final minimization. Do your best to find all local minima for the torsional angles j and f. How different are their energies compared to the global minimum (or the lowest you can find?) Use kBT ~ 1kcal/mole at 300K (room temperature) as a standard to compare these to.)

3) Now, you can find out what the lattices actually look like (I have copies of the patterns). Draw the Bravais lattice and unit cell for each one (there is more than one way to draw the unit cell). Construct the reciprocal lattice for two choices, using the construction we mentioned in class that there is a point in the reciprocal lattice for each plane (line in 2D) in the real lattice, and the k vector associated with such a plane/line has a value of 2p/d, where d is the separation between planes/lines.

1) For the each of following diffraction patterns, find the shape of scatterers in the unit cell and the lattice of a two-dimensional system (if there is a lattice.) These examples are all similar to (and simpler than) those shown on pages 708-709 in Cantor and Schimmel. This exercise requires no computations, just visual inspection of the figures and information from your notes and readings. (The originals are on the door to my office, Stokes 101.)

2) Using the formulas in the lecture notes for the primitive lattice vectors, show that the reciprocal lattice of the simple hexagonal lattice is also simple hexagonal, with lattice constants and , with the directions of the hexagons rotated by 30o about the z axis with respect to the real lattice. The original primitive vectors for the real lattice are

3) For problem 2), draw the diffraction pattern which would result if the hexagonal lattice is oriented in the following fashion. Assume that the lattice dimensions are a=15A and c=20 A, and that the x-rays have wavelength 1.54 A. Give the angles at which two lattice points occur.

4) The diffraction pattern below was collected for a a tetragonal crystal of lysozyme oriented in such a way that the scattering was equivalent to the scattering from a square lattice of protein molecules (see inset.) The diffraction experiment was arranged so that distances on the film correspond directly to angles; the angular distance between the direct beam at the center of the film and the outside edges was 15o. The wavelength of x-rays used was 1.54 A. Compute the dimensions of the square edge of the unit cell derivable from this pattern and the spatial resolution of the measurement.

1)

a) We know that here is no lattice in this sample since the diffraction pattern intensity is not confined to reciprocal lattice points, and hence discrete values of S and q. The intensity is symmetrical in angle for any rotation of any angle, so the molecular form factor (that part of the structure factor due to the Fourier transform of the individual molecule's in the unit cell) is also completely symmetrical. A circular scatterer would satisfy this criterion.

Best guess: a random array of circles

b) The reciprocal lattice can be seen by noting where the intense spots fall on the diffraction pattern. These are arranged in a square lattice, hence the real lattice is also a square. The modulation of intensities can be seen to be symmetrical upon all rotations, just like that in part a) (this is easier to see in the original reproduction.) Hence the individual unit cells are likely filled by circular scatterers:

Best guess: a square lattice of circles.

c) In this case we have the same lattice as in part b), but the modulation of the intensities itself has square symmetry, rather than the symmetry of a circle. Hence, the individual scatterers in the unit cells are themselves squares.

Best guess: a square lattice of squares.

d) Now, inspection of the lattice reveals it to be hexagonal. We know from problem 2)'s statement that the reciprocal lattice of a hexagonal lattice is itself hexagonal. Hence, the real lattice of the sample is hexagonal. The modulation of the intensities is circularly symmetrical, so the individual scatterers in the unit cells are likely circles.

Best guess: a hexagonal array of circles.

2) I asked you to work through the math, which I will do also, but you may also see that the reciprocal lattice spacings are and , by examining layers in the real lattice. I'll do both below:

3) Now the diffraction experiment is specified to be set up so that we probe structures within the plane of the hexagonal layers. The reciprocal lattice will itself be hexagonal, and the diffraction peaks observed will be arranged with hexagonal symmetry. We can compute for specific values of the lattice constants where these points will occur using the following relations:

4) This example comes from data reported in a textbook on crystallography. Let's see whether the numbers add up with the actual values of the lattice constants for lysozyme I find elsewhere. These reported values are a=79.1 A for the square edge of the tetragonal lattice.

The angles on this diffraction pattern can be directly read off using a ruler, since we know that the maximum angle on the film is 15o from the center. The conversion between distances and angle can be derived noticing that there is roughly 2mm distance between two spots, and that a distance of 55mm corresponds to 15o:

The next question is: which angle, q or 2q, corresponds to the distance of 15o reported? I'll address this later using the reported lattice constants--you had no way of knowing these, so you could make either interpretation appropriately! My choice of using 15o as the maximum range of q (left ambiguous in the source) is made using this extra piece of information.

The intensity peaks on the diffraction pattern are placed at regularly-spaced angles in a square array, corresponding to the square reciprocal lattice of the square arrangement of unit cells with lattice constant a in one plane of the tetragonal crystal structure. Our scattering vector is related to values of the scattering angle and lattice constant by the Bragg relation:

Since both directions are equivalent in our lattice, the integer in this equation can be either h or k. We know how far out in angle the first peak (say, h=) is located, and from this we can compute the lattice spacing a. My values give:

This value for the lattice constant, 81 A, agrees to one significant figure (about the validity of my crude angular measurement) with the actual reported value.

The largest values of angle, known to be 15o, also give us the maximum value of the scattering vector S at for which data was collected, and hence the smallest distance at which spatial details are resolved:

It's clear that the peaks at very high angle are still quite intense, so it's likely that data could be collected at perfectly adequate intensity at still larger angles, to reduce the spatial resolution yet further.

Cantor and Schimmel 13-4

Let's do a simple case first, as suggested, to get some ideas. From our definition of the Patterson function we know that whenever two peaks in the electron density are a displacement R apart, then we get a peak in the Patterson with height (# electrons in first peak) X (# electrons in second peak) at position R. Approximating the structure of

a molecule by placing a discrete number of electrons at exact locations, we can use this rule to compute a simple case. Consider the three "atom" "molecule" below:

It has three relevant interatomic vectors, as labeled. Each atom is assumed to have only one electron here. When we move copies of the molecule along the positive or negative direction along these vectors, at least one peak will overlap another. For P(0) [no displacement] the Patterson has value 3 because all three peaks overlap. The other positions in the Patterson have value 1 only because only one set of peaks overlap for each other point. Working using this reasoning we can construct a molecule using the Patterson given, adding one "atom" at a time until we satisfy all of the requirements. The following one-electron "atom", five-atom "molecule" does the job:

"Crystal structure of an antibody to a peptide and its complex with peptide antigen at 2.8 A", Science, vol. 248, pg. 712 (1990) (you can see the full color version by looking through the recent Science stack for the issue with the sea turtle on the cover.)

"Deciphering the message in protein sequences: tolerance to amino acid substitutions", Science, Vol 247, pg. 1306 (1990). (Full color copy on reserve.)