Chapter 5 – Detection and Imaging Tools that Use Nonoptical Waves 199
The rate of formation of B from A is negative, that is, the rate of decay of A from
B (since one atom “lost” from A is “gained” by B), or λA
A
N or N
t
A
A
A
exp
0( )
−(
)
λ
λ
.
Rate of decay of B to C is −λB
B
N so rate of change of number of B atoms is:
d
d
exp
A
A
A
B
B
N
t
N
t
N
B =
( )
−(
) −
λ
λ
λ
0
This type of slightly more complicated rate equation can be solved using the
integrating factor method such that in the general case dy/dx + f(x)y = g(x) the
integrating factor I(x) is exp(∫f(x)dx), and by multiplying the original differential
equation by I(x) and integrating, gives a solution y = ∫g(x)I(x)dx/I(x), which can
then be solved given appropriate boundary conditions. After a bit of mental gym
nastics and using the given initial conditions, this therefore simplifies to:
N
N
t
t
B
A
A
B
A
A
B
exp
exp
=
( )
−
−(
) −
−(
)
(
)
λ
λ
λ
λ
λ
0
b
The half-life t1/2 is the decay time taken to reduce the number of atoms twofold, so
from the first part of the answer to (a) above λ = ln 2/t1/2. The half-life of X is several
orders of magnitude higher than the other two isotopes so can be considered
“stable” here, so the formula you derived from part (a) applies, with:
λX=ln 2/(2.4 days × 24 × 60 × 60) = 8.0 × 10-5 counts/s
λY=ln 2/(23.5 min × 60) = 4.9 × 10-4 counts/s
Substituting in these values for 1 hour or t = 60 × 60 = 3600 s indicates NY/NX(0)
≈ 0.11 or 11%.
5.6 SUMMARY POINTS
◾EM is a powerful structural biology tool, but care must be taken to avoid
overinterpretation from sample preparation artifacts.
◾High-energy electrons, x-rays, and neutrons can be used in diffraction experiments
to reveal atomic locations in biomolecules.
◾X-ray crystallography is particularly useful for determining structures of
biomolecules where large crystals with few imperfections can be grown, including
many proteins and nucleic acid complexes.
◾NMR spectroscopy results in a unique molecular signature and is particularly
useful in determining structures of membrane-integrated proteins, which are dif
ficult to crystallize.
◾Diffraction techniques may also be extended to longer length scale investigations
than single atoms, such as biological fibers.
QUESTIONS
5.1
A sample of a protein was prepared for TEM on a 120 keV machine using evaporative
platinum rotary shadowing. A platinum atom is ~0.5 nm in diameter; however, the
smallest observable metal particles were composed of five atoms.
a
What is the practical spatial resolution in this experiment?