200 Questions

The protein was known to exist in two structural states, A for 25% of the time and

B for the rest of 75%. Each single TEM image of a protein was assigned as being

in one of the two states so as to generate an average image from each subclass.

Sequence data suggested the A–​B transition results in a component of the mol­

ecule displacing by 5 nm.

b

If each TEM images contains a mean of ~12 protein molecules, estimate how

many images one would need to analyze to be able to measure this 5 nm transla­

tion change to at least a 5% accuracy.

5.2

A dehydrated cell membrane was placed in a 200 kV electron microscope normal to

the beam of electrons, resulting in some of the transmitted electrons being diffracted

as they passed through the membrane. A first-​order diffraction peak was measured at

a deflection angle of 3.5°. Estimate the lipid molecule separation in the cell membrane,

stating any assumptions you make.

5.3

Over what range of accelerating voltage do classical and relativistic predictions for the

matter wavelength of an electron agree to better than 1%?

5.4

What is an x-​ray free-​electron laser? Discuss the advantages that this tool has for

determining structures of biological molecules over more traditional x-​ray methods.

5.5

A single cell contains ¹H atomic nuclei, which are mainly associated with water

molecules, but with a significant number associated the –​CH2–​ chemical motif found

in lipids. A population of cells was homogenized and probed using ¹H NMR spectros­

copy in a “400 MHz” machine, which results in a 1.4 kHz difference in the resonance

frequency between H2O and –​CH2–​ protons. Estimate the equivalent chemical shift

in units of ppm.

5.6

Why is NMR spectroscopy often described as the biophysical tool of choice for deter­

mining structures of membrane-​integrated proteins?

5.7

A bespoke solenoid for an NMR spectroscopy device was constructed with a mean

diameter of 50 mm by tightly winding an electrical wire of diameter 0.1 mm with a

total length of 10 m. If a coil current of 5 A is run through the solenoid wire, estimate

the average chemical shift of ¹H atom in ethanol dissolved in TMS.

5.8

With reference to drawing a representative NMR spectrum, explain why the protons

in an ethanol molecule are often categorized as being in one of three types, in terms

of their response to nuclear magnetic resonance.

5.9

What is the “phase problem” in x-​ray crystallography? How can it be overcome?

5.10 If the probability of a radioisotope decaying in small time Δt is equal to the number

N1 of radioisotopes present multiplied by some constant λ1, develop an expression for

the number of radioisotopes present after a general time t.

a

What do we normally call λ1 for a general radioisotope?

b

Derive a relation for the half-​life and mean lifetime for these radioisotopes.

c

This radioisotope was found to decay into a second radioisotope, of number in

the sample denoted by N2, that decayed with a constant λ2 into a nonradioactive

atom, of number N3 in the sample. Derive an expression for N3 as a function

of time.

d

Another radioactive decay chain was found to consist of several more (n −

2) radioactive intermediates from an initial radioisotope, before decaying to

an nth that was not radioactive, with Nn atoms in the sample. Derive a general

expression for Nn as a function of time, and comment on the relevance of one

isotope in the chain having a significantly longer mean lifetime than the other

in the series.

5.11 Electrons can be accelerated to far higher speeds than are currently used in modern

electron microscopes, which therefore would have a smaller Bragg wavelength and

better spatial resolution. Why then not use these to look at biological samples?