Spectroscopy of Macromolecules

1. Spectroscopy of Macromolecules Danger: this is “what you need to know.” One could easily spend a whole semester on this alone. Spectroscopy more often applied than, say, scattering, but…many other courses teach it better. Focus on these two: Fluorescence CD/ORD Reference: VanHolde (newer edition, “Physical Biochemistry” but the older editions are special, if you can still get them)

2. Spectroscopy ↔ Quantum Mechanics A Postulate View of Quantum Mechanics • A system (e.g. molecule) of n particles (electrons, nuclei) is described by wavefunctions (q1,q2…q3n,t) that describe locations (q1,q2,q3) of each particle at time t. • The probability of finding the system in the differential volume element d3V at time t is the squared complex modulus of the wave functions: *d3q

4. The operators for position, momentum, time and energy • are as shown in this table. Position and time pass through unchanged; momentum and energy are filtered through the calculus.

5. The wavefunction is determined from Schrodinger’s equation: Time-dependent Schrodinger’s Eqn. The Hamiltonian operator H follows from a classical mechanics system worked out by Hamilton, where the classical operator was H = K + U, the sum of kinetic and potential energies. This equation ties the action of the energy operator to how the wave function responds. Compare this to Fick’s 2nd law: Dd2c/dx2 = -dc/dt. Like concentration, a wavefunction describes where something is.

6. Why Quantum Mechanics is Hard Because no one except physicists studies enough classical mechanics. Debye is said to have seen very little new in Quantum Mechanics; at least, the math is common to other stuff…if you have studied enough other stuff…which hardly any of us do! So far, we have seen that expectation values are similar to other averages we have computed: sum of probability times thing, divided by sum of probabilities to normalize. The wave function “sandwich” is new…and often associated with a particularly simple matrix mathematics. We also see that Schrodinger’s equation resembles Fick’s equation….which in turn resembles the heat flow equation all engineers learn.

7. Eigenfunctions

8. Basis sets It sometimes works out that a set of eigenfunctions can be used to represent other functions. We say the desired function can be expanded in the set of eigenfunctions. Compare this to writing a vector in terms of unit vectors. Eigenfunctions are most useful when they are orthogonal and complete, meaning that they do not project on to one another and are sufficient to express arbitrary functions (again, compare the traditional unit vectors).

9. A simple example of an operator: kinetic energy only Classically, kinetic energy is Ek=mv2/2 = p2/2m since p = mv. Check the table of Postulate 5 to get the QM analogue: The operator  is discussed in the Math Tuneup, as is its “square”, 2, the Laplacian.

10. Time may not matter Often, the time-dependence of (q,t) is particularly boring; it might just be an oscillation that can be factored out from the positional, q-dependent part. This case is appropriate for stationary operators—ones with no time dependence. If you put the above equation into the full, time-dependent Schrodinger’s equation of Postulate 6, you get (see VanHolde—it is very easy) the simpler, time-independent Shrodinger’s equation, which is an Eigenequation with the particularly interesting and useful Eigenvalue, E. H-sigh equals E-sigh. Another one for permanent memory storage.

11. Simple Systems These notes derive from a modern VanHolde (VanHolde, Johnson & Ho). The authors march through systems you probably saw in PChem already, such as: • Free particle • Particle in box • Hydrogen atom • Approximate solutions (perturbation) • Small molecules (LCAO) In time, I hope this presentation will grow to cover some of those subjects, at least briefly. Meanwhile, I hope the foregoing made QM seem less weird. For now, we need a leap of reasonableness.

12. Leap of Reasonableness QM is the explanation of things we know about atoms, going all the way back to Dalton’s Law of Multiple Proportions. QM explains why it’s CH4, not C1.251H3.785 It’s those stupid waves; together with boundary conditions (like the electron has to be somewhere) they give constructive & destructive interferences—nodes—which makes Chemistry an integer science. Think laser cavity, think wave trough, think booming bass near the corner of a room. For particle existence to be tied to wave amplitude is tantamount to saying: there are discrete energy levels associated with standing waves.

13. U E = h r • Example: nearly harmonic vibration potential. • Weird quantum-mechanical things: • There is a zero-point energy. • Discrete states, separated by near multiples of the zero point energy. • More energy = more nodes in the wave function = fancier dancing by • the electrons. • If this were electronic energy levels, not vibrational levels, that • fancier dancing corresponds to more complex orbitals—e.g., • complicated f orbitals instead of simpler s orbitals.

14. Transitions • Beer-Lambert Law (to be added) • Einstein-Planck Coefficients (to be added; meanwhile, I could weep, these authors make it so simple) • Transition dipole • Orientation of transition dipole }Let’s do these for now

15. Absorption vs. Scattering • Earlier, we talked about how light grabs electrons and shakes them to produce scattering. We used the analogy of a boat floating on rough seas; it produces a little ripple as it bobs up and down. • Molecules that absorb light do the grabbing instead. This stronger interaction is more like the water going over Niagara Falls.

16. Operators Project.(this is a sentence, not a government program) Van Holde (Ch.8, p 373 et seq.) shows that a transition from one quantum state to another (absorption, emission) occurs when: 1) E = h 2) the transition dipole fi is finite and makes a strong projection on the electric field. Transition strength 

17. Transition!

18. Think of the initial state as rolling along the runway. The wings catch some air and—voila!—transition to vertical acceleration. In this case, the initial and final vectors (oops, wavefunctions) are pictured as orthogonal. In QM transitions, this may or may not be allowed. The main thing is: the transition operator (wings) somehow couples one state (horizontal motion) into another (vertical motion).

19. Selection rules: will it fly? Quantum mechanical transitions depend on: • how the transition dipole aligns to the electric field • obscure rules regarding how the field being operated on (initial wavefunction) relates to the new field (final wavefunction). Some transitions are “forbidden”—meaning they happen infrequently.

20. It matters VanHolde Fig. 8.15 UV absorption of crystalline methylthymine

21. Moving right along… Ch. 11 of VanHolde deals with emission spectra. Here is a good chance to observe transition dipoles in…ummm….transit. Emission • Fluorescence = fast emission (allowed transitions) • Phosphorescence = slow emission (forbidden transitions)

22. Franck-Condon Principle, Jablonski Diagram 1st excited state absorption U emission Ground state Evib = h r

23. But it’s not that simple Opposing electron spins: singlet (ground state shown) Aligned electron spins: triplet (excited state shown)

24. Fluorescence solvent effects Example: is that protein aggregating? Put a hydrophobic probe in (pyrene?) and see if it “lights up” to indicate aggregation across the hydrophobic patch. Example: is that arborol self-assembling? Same solution to similar problem as above. +

25. Fluorescence effects not always a tool, sometimes a nuissance. For FPR, the dye can get quenched and not undergo photobleaching. This can happen as a result of variables like pH or salt. Dye can self-quench if too concentrated. That is again a tool: calcein leakage test for vesicles.

26. Fluorescence Machines Can be Simple • Needs picture of simple turner fluorimeter, maybe one from the web, too.

27. Unlike light scattering, Fluorescence is not instantaneous It is a little bit more like preparing radioactive elements: excited molecules spontaneously decide they have had enough of life in the fast lane and return to their ground state. Some tolerate the excited state longer than others.

28. N t Fluorescence Decay: yet another exponential for us to learn. later at first Define: N = number of molecules in the excited state. The number dN decaying back to ground state is proportional to the number available in the excited state. The number N is reported by the proportionate number dN that emit light.

29. Fluorescence decay instruments are much more complex beasts. • Maybe picture from Soper??? Supposed to be picture of flash lamp flashing, With exponential decays atop it.

30. Quantum yield The minimum rate of decay would be the Einstein A value for spontaneous emission, which can be calculated from spectral width (VanHolde Eq. 8.102) The actual decay rate is higher, due to internal conversion, intersystem crossing, nonradiative transfer and any stimulated emission processes from interactions with stray photons. Quantum yield is defined as: q = A/k It is a sensitive indicator of the environment of the dye; q often increases when a dye binds to a molecule. Free dye may be quenched (see previous slides).

31. I absorbed emitted l Another way to think of quantum yield: ratio of (visible) photons in to photons out. Some dyes are very efficient light converters, with quantum yields approaching 100%. Fluorescein (which we often use in FPR) is so efficient that it is hard to know how efficient it is. As I recall, about 90%-100% has been reported. High efficiency is a good thing for FPR: efficient light conversion means little heat production, less damage.

32. Energy Transfer The emitting group does not have to be the same as the absorbing group. Donor Donor hn Acceptor Acceptor Donor Donor + + Acceptor Acceptor + hn

33. I Donor F Acceptor A l Step #2 is the hard part Physics problem: Requires transition dipole interaction between donor and acceptor. Dipole-dipole interactions go like r6. Ro = 6 – 45 Å depending on the DA chemistry and solvent. See VanHolde Table 11.1 for examples. Chemistry problem: Donor’s F-spectrum must significantly overlap acceptor’s A-spectrum.

34. Typical D-A Pairs XXXGet from molecular probes. I will bet acceptors are generally larger molecules than the donors.

35. D-A is a valuable tool for estimating intermolecular distances. A r D ???Does labeling a molecule (twice!) change it? Maybe.

36. Polarization of Fluorescence Light scattering is instantaneous, so draw a donut around the induced dipole and that’s what radiation pattern you will get. Fluorescence and, especially, phosphorescence are slower and the molecule may rotate, taking the dipole with it, before emission. Draw a donut around the dipole at the time of emission.

37. gis in a molecular frame of Reference; this is not the rotation of the molecule but the different transition dipole of emission and absorption; for example, the molecule might distort a bit on absorption, so emission would be in some new direction. Absorption transition dipole Emission transition dipole g Vertical incident light f III I detector

38. If molecule did not rotate (subscript zero)….. You can measure g by freezing out the motion—e.g.,put the molecule in some viscous solvent, cool it, etc.

39. If the molecule does rotate… Then r will deviate from ro, tending more towards zero. (If ro was positive, r goes down). You can use this to estimate rotational diffusion coefficients, or at least to track how they are changing due to binding, adhesion or aggregation. Problem: it’s the rotation of the fluorphore, not necessarily the whole molecule. According to VanHolde, fluorometry is the only spectroscopic method that senses changes in molecular weight (e.g., due to aggregation). What do you think of that assertion?

40. This is called steady state fluorescence polarization anisotropy In this equation, t is the fluorescence decay time and r is a rotational correlation time (proportional to the inverse of rotational diffusion coefficient). You may recall that Dr = kT/8phR3 from our discussion of Hv DLS. So this means that 1/Dr or r represent volume.

41. Perrin plot: r vs T How would you use this? From intercept, get ro, which is related to g. Use slope to estimate t/V. If t is known from fluorescence decay (or maybe even literature) then obtain V. Plot not straight? Could be nonspherical shape, proteins changing with temperature (denaturing), binding to stuff as a function of temperature, etc. T/h

42. r(t) t The well-heeled & talented can do pulsed, polarized fluorescence…