Posted: February 25th, 2015

Biophysics, Quantum Mechanics, Physics

Paper, Order, or Assignment Requirements

Biophysics, physics, calculus, quantum mechanics

BEGR 477 Cellular Biophysics Homework #2. Due 2/26/2015 Please answer the following questions. You may use Mathematica to help you with the calculus, but you must still show your work (such that we can see how you reasoned your way through the problem). 1. Given that the time independent wave function for the particle in a onedimensional box is: ψ (x) = 2 L sin nπ x L ! ” # $ % & a. Determine if this wave function is an eigenfunction of the momentum operator h i ∂ ∂x b. Determine if this wave function is an eigenfunction of the momentum operator squared h i ∂ ( ∂x ) 2 = −h2 ∂2 ∂x2 . c. Could you measure both of these quantities at the same time (i.e. measure one without upsetting the other)? 2. Given the 3 level photophysical-system mentioned during the lecture on Jan 30th, 2014, with a singlet ground state, singlet excited state and triplet excited state and respective rate constants (defined in lecture and shown on the diagram below), determine: a. the fluorescence emission rate, F, dependence on the other parameters in a single equation and graph the fluorescence emission rate dependence on the excitation intensity, I (i.e. fluorescence emission rate, F, on the Y-axis and excitation intensity, I, on the X-axis) b. the maximal fluorescence emission rate when the excitation intensity approaches infinity (i.e. F = ? when I → ∞) c. the saturation intensity, IS3. The following peptide sequence was found to form a stable secondary structural motif and exist in free solution as a dimer: NARVSMKIEAKGDWTGGQMTGDANFRASVDL The monomer is found to associate with membranes, but not the dimer. Comment on the secondary, tertiary, and quaternary structure of this protein using what you learned in lecture as a guide. 4. This problem will explore Blackbody radiation and its relationship with thermodynamics. As you may remember form class, Planck’s law for blackbody radiation is: u(ν,T) = 8πν2 c 3 E = 8πν2 c 3 hν e hν /kBT −1 ” # $ % & ‘ The first term ( 8πν2 c3 ) represents the density of electromagnetic waves (harmonic oscillators) in the cavity and the second term () represents the average energy of per oscillator. a. Show how quantization of energy gives rise to Planck’s Law. Hint: Start with quantized harmonic oscillators of the form En = nhν and calculate the average energy using the Boltzmann equation. You will need the following equations: enx n=0 ∞ ∑ = 1 1− ex nenx n=0 ∞ ∑ = d dx enx n=0 ∞ ∑ = d dx 1 1− ex $ % & ‘ ( ) = ex 1− ex ( ) 2 b. Show that in the limit of high temperatures or longer wavelength (shorter frequency) the average energy reduces to the equipartition theorem and thus Planck’s Law becomes the Rayleigh-Jeans Law. c. Plot the spectral density as a function of wavelength for both the RayleighJeans Law and the Planck Law at 6000K in Mathematica (Hint: To see things most clearly use PlotRange->{{0,2*10^15},{0,5*10^-15}}). Make sure your plot is titled and your axes are labeled (and have units). 5. a. Identify the types and number of the symmetry elements and determine the point groups of the following molecules.b. Use the character tables of the point groups to determine if the 1st and 2nd electronic transitions from the ground state is allowed or not for each of the molecules. Be certain to show and justify your work for each answer. a) H2O b) NH3 c) Ferrocene d) CO2 e) C6H6 (benzene) (You may need to use your chemistry books to determine the structure of the molecules.) 6. The Foerster energy transfer rate, kT, between donor and acceptor fluorophores was determined to be the following in 1946: where τD is the fluorescence decay time of the donor fluorophore, R is the actual distance from the donor absorption and acceptor emission electronic dipoles and R0 is the critical distance (or Foerster Radius) at which the probability is that half of the excited state donor electrons will transfer their energy to the excited state of the acceptor. The critical distance (or Foerster Radius) can be calculated by where κ2 is the orientational alignment factor between the the donor absorption and acceptor emission electronic dipoles, Φ0 is the fluorescence quantum yield of the donor, n is the refractive index of the media and J is the overlap integral or sum given by where FD(λ) is the wavelength dependent donor fluorescence, εA(λ) is the wavelength dependent acceptor absorption coefficients and λ is the wavelength. For this exercise, please calculate the Foerster Radii for the following combinations of fluorophores for FRET pairs: Donor Acceptor 1. Cerulean (CFP) mCitrine (Y pet) (YFP) 2. Rhod TM Cy5 3. Emerald (GFP) mCherry 4. FLaSH ReaSHTo do this you will need to download the absorbance (or excitation) spectra and emission spectra of the fluorophores. You may find your own source, but it is suggested that you go to the Roger Tsien group website to download the fluorophore spectra. You will also need to obtain the fluorescence quantum yields of the donor fluorophores and the maximum absorption coefficients of the acceptor fluorophores. It will be fine to assume that the orientation factor will be random, which sets the κ2 = 2/3 = 0.6667, and that the refractive index will be that of water. You may use Matematica, MatLab or even an MS Excel spreadsheet to calculate these values. Please also note that the integration of FD(λ) d λ is supposed to yield one. Thus, it might be necessary for you to determine the sum or integration of FD(λ) d λ over the spectral range and normalize the overlap integral J to this. It is important that you submit your work of your answers and notebook/workbook files/papers to show how you determined your answer.