Download Spectroscopic methods for biology and medicine

Spectroscopic methods for biology and medicine
Thomas Schultz
Draft date October 28, 2009
2
Chapter 1
Introduction
The lecture offers an overview of spectroscopic and spectrometric methods with
the aim to address questions in the biological sciences.
Biology relies on a vast set of molecular and macromolecular machines, each
performing a well defined task. Our understanding of living systems is therefore
intimately connected to the characterization of structure and function in molecular and macromolecular systems. Spectroscopy can offer the necessary tools to
investigate the relevant structure and function, but the size and complexity of
biological systems is beyond that usually encountered in the physical and chemical sciences and therefore poses a particular challenge. This challenge is met by
extraordinary efforts to extend the sensitivity, specificity, information content,
and in some cases spatial resolution of spectroscopic methods. Goal of this lecture is to review how modern spectroscopy is used in the biological sciences and
can help to shape our understanding of living systems. The lecture also aims
to give an overview about the fundamental limits of spectroscopic and analytic
tools, and to assess how novel developments may promise unprecedented insight
into biological systems.
The subjects covered include:
3
4
CHAPTER 1. INTRODUCTION
1. Absorption and emission of light: Spectroscopic observables
and the physical limits of a measurement.
2. A primer on molecular biology - tools of the trade
3. Measurements with radiowaves: Nuclear Magnetic Resonance spectroscopy and tomography of biological systems.
4. More radiowaves: Electron spins in biology and electron spin
spectroscopy
5. Spectroscopy with light: Probing biology with natural and
artificial chromophores, single-molecule spectroscopy, and
(superresolution) microscopy.
6. X-rays: Crystal structures and ionization events
7. Particle detection: Molecular recognition, radioactive marking, and mass spectrometric methods
8. Theoretical approaches: from molecular dynamics to ab initio calculations
1.1
Spectroscopy of biological systems
A given spectroscopic tool will be suitable to investigate some compounds (or
’analytes’) and answer particular questions, but will not be useful for others.
To gauge the usefulness of a given spectroscopic tool, it is necessary to identify
the fundamental limits in terms of sensitivity, selectivity and resolution. Highly
sensitive methods allow the investigation of small quantities / concentrations
of an analyte. The most sensitive methods nowadays allow the observation of
single molecules (by fluorescence or mass spectroscopy), but offer only a limited
information content for the characterization of the analyte. Selectivity allows
to identify and characterize a given analyte in presence of others, but may seriously limit the scope of a method if only a small set of analytes is accessible
to investigation. Resolution is a fundamental issue determining the information
content in every spectroscopic method and is ultimately limited by Heisenbergs
uncertainty principle. To gauge the usefulness for a given method we therefore
have to discuss the relation between:
Biological
question
⇔
Spectroscopic
method
⇔
physical limits of the
method
The challenge of complexity
As compared to typical systems in chemistry or physics, biological systems are
very large and complex. Furthermore, the complexity spans sizes from meters
to picometers. For a living organism to function properly, every building block
must be in the right place and function properly on both, the meter scale of
1.1. SPECTROSCOPY OF BIOLOGICAL SYSTEMS
5
the human body and the nanometer scale of proteins (Fig. 1.1). The challenge
for spectroscopy is therefore twofold: to identify and characterize the biological
building blocks and to determine their spatial position or distribution.
NH+
human
cell
protein
m
μm
nm
active site
pm
Figure 1.1: The complexity of biological systems spans sizes from meters to picometers. Organs (the heart) must be in the right place of the body, the cell nucleus and
mitochondria must be in the right place of the cell, the peptide strands in a protein
must fold correctly and the active sites or prosthetic groups must have the correct
structure.
It is helpful, however, that a large part of biological molecules is constructed
from only a small set of building blocks. 20 Amino acids form the molecular
building blocks for peptides and proteins and are shown in Fig. 1.2. Bound by
peptide bonds (-CO-NH-), amino acid chains are called peptides (mass < 10
kDa) or proteins (mass ≥ 10 kDa, 1 Da = mass of proton). The proteins, which
sometimes contain additional ”prosthetic groups” (non-amino acid molecules),
perform most catalytic and regulatory functions in the cells and might be called
the molecular machinery of life.
Proteins can be very big (thousands of kDa) and their structure is usually discussed in terms of: primary structure (amino acid sequence), secondary
structure (regular local folding pattern of adjacent amino acids, often α-helix
or β-sheet), tertiary structure (folding of the secondary subunits into the protein shape) and quarternary structure (aggregation of several proteins into a
complex). To perform their function, many proteins must find their way to the
right location in the cell, maybe the cytoplasm or the cell membrane, adding
an additional layer of complexity. To summarize, proteins have a vast structural complexity and an analysis of the structure-function relationship requires
a large amount of experimental analysis.
DNA and RNA bases form the building blocks of DNA and RNA, the second set of biological macromolecules (Fig. 1.3). The bases adenine-thymine,
guanine-cytosine and adenine-uracil (the latter in RNA) can form two or three
complementary hydrogen bonds and thus allow molecular composition between
the correct base pairs. The names allow to distinguish between the pure bases
6
CHAPTER 1. INTRODUCTION
hydrophobic
S
small
COOH
H 2N
Glycine (GLY,G)
Mr 75.07
aromatic
H 2N
H 2N
COOH
H2 N
Alanine (Ala,A)
Mr 89.09
COOH
Valine (Val,V)
Mr 117.15
COOH
H 2N
H 2N
Leucine (Leu,L)
M r 131.17
N
H
COOH
Isoleucine (Ile,I)
Mr 131.17
COOH
Proline (Pro,P)
Mr 115.13
NH
COOH
H2N
Phenylalanine
(Phe,F ) M r 165.19
COOH
Thyrosine (Tyr,Y)
Mr 181.19
H2 N
OH
COO H
H 2N
Serine (Ser,S)
Mr 105.09
COOH
Tryptophan
(Trp,T) Mr 146.19
amide
O
O
H 2N
SH
OH
H2 N
COOH
Threonine (Thr,T)
Mr 119.12
basic
HO
O
COOH
Methionine
(Met,M) Mr 149.21
nucleophilic
HO
acidic
H 2N
O
H 2N
COOH
Cysteine (Cys,C)
Mr 121.16
N
H2 N
NH 2
N
NH
N
HO
H 2N
H 2N
COOH
Aspartic Acid (Asp,D)
Mr 133.10, pKa 3.9
H 2N
COOH
Glutamic Acid (Glu,G)
Mr 147.13 pK a 4.1
H N
2
COOH
H 2N
COOH
Asparagine
Glutamine (Gln,Q)
(Asn,N) Mr 132.18
Mr 146.14
H2 N
COOH
H2N
COOH
H2 N
COOH
Lysine (Lys,K)
Arginine (Arg,R)
Histidine (His,H)
Mr 155.15 pK a 6.0 Mr 146.19 pKa 10.8 Mr 174.20 pKa 12.8
Figure 1.2: The 20 standard amino acids which form the building blocks for proteins.
All natural amino acids with exception of glycine are chiral (L-amino acids).
(nucleic base, e.g. ”adenine”), the base bound to a ribose sugar (nucleoside,
e.g. ”adenosine”) and the phosphorylated base-sugar molecule (nucleotide, e.g.
”adenosine-monophosphate”). In DNA, the nucleotides form macromolecules
bound by the phosphate ”backbone”, the sequence of which contains the genetic information. It would therefore be correct to talk about the bases as
forming the 4 letters of a quarternary alphabet of life.
1.2
Spectroscopic Tools
To perform a measurement, we must observe the interaction of the compound of
interest with another particle. So a basic spectroscopic measurement will consist
of shooting particles with well defined properties at the sample and analyzing
particles which are emitted by the sample as indicated in Fig. 1.4. As a result,
the measurement is due to the properties of the sample, the properties of the
probing particle, and the physical laws governing the interaction between the
two (in many cases called ”selection rules”).
In principle any particle can be used, but in most cases the secondary particle
is a photon. Please be aware, that in some cases photons are conveniently
described as particles with well defined energy / momentum, whereas in other
cases a wave description is more useful. This does not reflect fundamental
differences in the interaction, which always satisfies the same physical laws and
fulfills conditions set by both, particle and wave nature of the photon. Other
particles follow the same physical rules, but are usually described as particles
only, reflecting the more classical nature of heavy particles.
Photons are special particles without any rest mass and without charge,
Therefore the only relevant particle properties are the photon energy, polar-
1.2. SPECTROSCOPIC TOOLS
7
Base pair
Deoxyribonucleic Acid (DNA)
Sugarphosphate
backbone
Sugarphosphate
backbone
Base pairs
A
P
T
S
Hydrogen
bonds
S
G
P
P
C
S
S
P
T
P
A
S
Nucleic acid
S
P
A
P
T
S
S
P
C
P
G
Base pair
S
S
P
G
P
C
S
S
P
Nucleotide
Cytosine
Guanine
H
H
N
H N
O
H N
H
N
O
N
O
3'
Hydroxyl
O
CH2
CH3
N
CH2
O
O
N
O
P
H
H
P
O
O
O
N
H
O
5'
Phosphate
O
Thymine
Adenine
H
CH3
O
H N
N
H
O
CH2
O
N
N
N
O
O
O
O
N
H
CH2
O
P
H
P
O
O
N
H
O
O
3'
Hydroxyl
5'
Phosphate
National
Institutes
of Health
National Human Genome Research Institute
Division of Intramural Research
Figure 1.3: The 4 DNA bases form the letters of the genetic alphabet and specific
hydrogen bonds allow for molecular recognition between the correct complementary
bases.
ization, and coherence properties. According to deBroglie, a particle can be
described as wave with a wavelength inversely proportional to its momentum.
Equation 1.1 gives the relation between energy, momentum, wavelength and frequency relative to the Planck constant of h = 6.62 · 10−34 J · s or ~ = h/(2 · π).
λ wavelength
h, ~ Planck constant
wavelength:
p momentum
ν frequency
c0 speed of light
mass:
energy:
h
p
hν
= 2
c0
λ=
mph
(1.1)
Eph = ~ν
mph photon mass
If we restrict our discussion to photons, then the discussion of measurements
can be tied to the linear scale of the photon energy / frequency (Fig. 1.5).
Most accessible frequency ranges are used for spectroscopy, but some particular
8
CHAPTER 1. INTRODUCTION
sample
Figure 1.4: Spectroscopic measurement: Put the sample in a box, where it is isolated
from the environment. Next, observe/characterize particles which enter or leave the
box to learn about the sample properties.
frequencies and related properties are particularly useful for the analysis of
biological systems.
In the radio wave regime, the very precise control and measurement of frequency and phase allows nuclear magnetic resonance spectroscopy in both, time
and frequency domain. The resulting Fourier transformation (FT) methods allow the measurement of spin correlation (→ ”multidimensional spectroscopy”)
and therefore the structural characterization of macromolecular systems. The
same properties are used for the tomographic imaging of tissue on a macroscopic
scale (>mm). Optical spectroscopy in the visible can be directly combined with
the spatial resolution of a microscope and has large historic and current importance. Diffraction methods are dominating the X-ray regime, because the short
wavelength allows the spatial determination of atomic positions down to sub-Å
precision. In all cases, we can discuss the absorption, emission, or diffraction of
radiowaves
radiation
frequency
MHz
(m)
spectrosc.
observables
nuclear spins
of special
relevance to
biology
FT-methods,
imaging
microwaves
GHz
(mm)
THz
(μm)
rotations
IR Vis UV
X-ray
PHz
(nm)
EHz
(pm)
vibrations
electronic
transitions
microscopy
electronic
diffraction
crystal
structures
Figure 1.5: Qualitative frequency scale for electromagnetic radiation and the
relevant molecular observables.
photons. The following is a short, exemplary discussion of those three types of
interactions to give an overview the fundamental possibilities and limitations of
different spectroscopic methods.
1.2. SPECTROSCOPIC TOOLS
9
E
Efinal
hν
sample
hν
Einitial
Figure 1.6: Absorption measurement: Irradiate the sample with photons of energy
hν. The sample may absorb the photon according to the laws of energy conservation
if hν = Ef inal − Einitial and according to selection rules.
Optical Absorption
All chemical compounds absorb light in the infrared, visible or ultraviolet region.
Hence, absorption spectroscopy is a very general method applicable to almost
all samples. According to the laws of energy conservation, a photon ca only be
absorbed, if the photon energy corresponds to the difference energy between an
initial and final state of the sample: hν = Ef inal − Einitial . The absorbed frequencies therefore depend on the energy levels of the vibrational system (IR) or
electronic system (Vis/UV). As a rule of thumb, delocalized electronic systems
lead to stronger absorption in the visible and near UV, and aromatic compounds
are therefore favorite chromophores for absorption spectroscopy. The strength
of absorption is defined in Beer’s absorption law equation 1.2.
I0 incident intensity
I transmitted intensity
extinction coefficient
absorption:
I = I0 · e−l
z sample length
(1.2)
We can estimate the absorption for a sample of the wild type green fluorescent protein (GFP) with = 7000(M · cm)−1 at 475 nm and a molecular mass
of MR = 30 kDa. If we dissolve 30 µg (= 1 µmol) in a cubic cuvette with 1 cm
length and a volume of 1 cm3 , the resulting concentration is 3 µM. Equation 1.2
then gives a transmission of I = I0 · e−0.007 or an absorption of ≈ 0.7%. Such
a small absorption is difficult to detect, and it is evident that absorption is not
a very sensitive method.
10
CHAPTER 1. INTRODUCTION
Optical Emission
Fluorescence is a widely used method in biochemistry. Photons emitted from a
sample can be detected with a quantum yield approaching 1, hence fluorescence
is a very sensitive method. To continue our example of GFP, we find that this
extraordinary fluorophore has a fluorescence lifetime of τ ≈ 1 ns and a typical
quantum yield of 0.5. If we irradiate a single GFP molecule with a focussed
light source which saturates the absorption, we may collect up to 0.5 photons
per ns, this corresponds to a rate of 5·108 photons/s. A realistic rate may be an
order of magnitude lower, but the fluorescence rate is clearly enough to detect
single molecules.
E
Einitial
sample
hν
hν
Efinal
Figure 1.7: Fluorescence measurement: Irradiate the sample with photons and observe the emission of fluorescence photons. The sample may emit photons according
to the laws of energy conservation and selection rules.
The downside of fluorescence is, that only very few molecules fluoresce with
reasonable quantum yields. Fluorescence is therefore a very sensitive technique,
but not applicable to a large set of samples. Nevertheless, many spectroscopic
studies of biomolecules rely on the fluorescent labelling of compounds with either
artificial fluorophores, or with natural fluorophores such as GFP. The location
of the labelled compound can then be observed in a microscope. Fluorophores
are also attached to antibodies, which bind only to specific molecular targets
and thereby allow the very sensitive identification of the target molecule. Fluorophore markers are also used for sequencing, where they may replace other
biological building blocks for easier detection. Finally, fluorophores found a particular application as probes of local structure by using donor-acceptor pairs,
which can transfer excitation and fluorescent activity if they get spatially close
to one another.
Diffraction
Due to the wave nature of light, scattered photons will lead to interference
patterns which contain information about the relative position of the scattering
centers as shown in Fig. 1.8.
1.2. SPECTROSCOPIC TOOLS
11
A (k=0)
C (k=1)
constructive interf.: sin(αmax) = k ⋅ λ/d
destructive interf.: sin(αmin) = (k+1/2) ⋅ λ/d
B (k=0)
Δx
C
d
α
B
Δx
A
Figure 1.8: Interference from two point sources of light (e.g. pinholes, or scattering
centers). Photons emitted from the two point sources at an angle α have a path
difference ∆x. The path difference can lead to constructive (A,C) or destructive (B)
interference and determines the probability to observe photons at the corresponding
angle
The detected intensity is proportional to the square of the sum of wavefunctions from all waves, (e.g. two waves in Fig. 1.8). The result can be as large as
expected for the classical sum of two waves, but can also be zero if the waves
interfere destructively. The latter occurs whenever two waves have opposite
phase, which is the case if their path differs by k+1/2 values of the wavelength
(k=1,2,...). Equations 1.3 calculate the expected amplitude for a diffraction
angle α.
∆x path difference
α detection angle
d emitter distance
Φn photon wavefunction
t time
λn photon wavelength
I detected intensity
path: ∆x = d · sin(α)
t · c0
∆xn
photon wave: Φn = sin 2π
+ 2π
λn
λn
"
#
2
Z X
2
interference ⇒ I =
Φn
t
n=1
12
CHAPTER 1. INTRODUCTION
∆x0,1 = 0, ∆x:
2
Z t · c0
∆x
t · c0
+ sin 2π
+ 2π
I=
2 · sin 2π
λ
λ
λ
t
trigonometry:
2
Z ∆x
∆x
t · c0
I=
2 · sin 2π
+π
· cos π
λ
λ
λ
t
(1.3)
If more diffraction centers are present, the interference grows more complex.
If the interfering centers are regularly spaced, however, then the pattern becomes
much sharper because only the absolute interference maxima remain. The latter
is the case in crystals and is the reason why X-ray diffraction works best for
crystal structure determination.
Detection of particles other than photons
Spectroscopic or spectrometric information can be gained from the interaction
of the sample with particles other than photons. Electrons have a much shorter
deBroglie wavelength than X-Ray photons and can be used for absorption or
diffraction experiments with very high resolution. The collision of electrons,
ions, or photons with a sample can result in ionic species which may be detected in mass spectrometers. Radioactivity from labelled compounds yield
information about chemical pathways or the spatial distribution of metabolites.
These mostly spectrometric methods (measuring quantities, as opposed to energies) can be extremely sensitive or selective and therefore play a large role in
the analysis of biological systems.
1.3
Physical Limits of Measurements
Sensitivity and selectivity
Biological samples contain large numbers of different compounds in concentrations ranging from single molecules to millimolar concentrations. We therefore
have to consider the sensitivity of spectroscopic methods (i.e. the ability to
detect small quantities of a compound) and the selectivity (i.e. the ability to
distinguish different compounds, or to spatially resolve them). The sensitivity
and selectivity are characteristics of a chosen experimental setup and determine
whether this setup is suited for a desired measurement.
The detection of photons and other particles can be highly efficient. Quantum efficiencies for photon detection with avalanche photodiodes exceeds 50%
and the detection of charged particles is possible with nearly 100% yield. Efficient detection, however, must be coupled to good contrast between the desired
signal and undesired noise to allow sensitive measurements. Fluorescence can
serve as an example for a very sensitive method if scattering and background
1.3. PHYSICAL LIMITS OF MEASUREMENTS
13
fluorescence is low; in this case single molecule detection is possible. Absorption
is also based on photon detection, but the limited absorption quantum yields of
single molecules reduces the sensitivity of this method.
For selective measurements, signals from different species should be distinguishable. High resolution techniques (e.g. some kinds of NMR) can yield
enough information to distinguish multiple compounds, whereas low resolution
spectroscopy only offers the integrated signal of undistinguishable species (e.g.
optical absorption and emission in a homogeneously broadened environment).
The resolution of all spectroscopic techniques is inherently limited by the Heisenberg uncertainty principle
Resolution and the Heisenberg uncertainty principle
The Heisenberg uncertainty principle can be interpreted as effect of the particlewave dualism. The cosine term in equation 1.3 after trigonometric reformulation
is responsible for the zero intensities at all values ∆x/λ = 1/2, 3/2, .... It is
interesting to consider what happens if the distance d becomes similar to the
wavelength: if ∆x/λ is always smaller than 1/2, then the first minimum is no
longer observed. In this case, the distance between the point sources cannot
be determined by investigation of the diffraction pattern. Hence, there is an
uncertainty limit for the determination of position which can be expressed as d ·
sin(α) ≥ λ/2. As shown in figure 1.9, a substitution of the angle by the impulse
components of the photon and the substitution of the deBroglie wavelength
λ = h/p results in the famous Heisenberg uncertainty principle in its impulse
formulation ∆x · ∆p ≥ ~/2; ~ = 1.054 · 10−34 J · s with an additional factor of 2π.
This factor is due to the simplistic assumption, that we must observe exactly
one diffraction minimum within a 90◦ angle. Taking the uncertainty principle
into account, it is obvious why X-rays with a wavelength in the picometer range
are suited for molecular structure determination by diffraction. Photons in the
optical domain with wavelength larger than interatomic distances (∼ Å) on the
other hand cannot be used to resolve molecular structure by diffraction.
The uncertainty principle can therefore be directly deduced from the wave
nature of matter. This is also true for the other formulations of the uncertainty
principle, e.g. if we consider ∆E · ∆t & ~. The term E = hν is well-defined if
the corresponding frequency ν is known. This is clearly the case for a constant,
temporally infinite wave, where the wave can be observed for long times. If, on
the other hand, the wave is truncated after a short time ∆t, then the frequency
is no longer well-defined and the corresponding energy becomes uncertain. Inverting this consideration, we note that the sum of multiple waves with different
frequencies can result in a short pulse as illustrated in Fig. 1.10.
The uncertainty principle limits the energy resolution of photon absorption
/ emission experiments: Rotational and vibrational states in solution have short
lifetimes in the picosecond range due to collisions with the surrounding medium.
1
We can estimate the energy uncertainty using ∆E = h∆ν to get ∆E ≥ 2π·1ps
and calculate a frequency uncertainty of ≈ 160 GHz (5 cm−1 ). The natural
14
CHAPTER 1. INTRODUCTION
First interference minimum observed for:
Δx = d⋅sinα = d⋅(Δp/p) ≥ λ/2
deBroglie: λ = h/p ⇒ d⋅Δp ≥ λ/2
d
α
p
Δx
α
Δp
Figure 1.9: Direct deduction of the Heisenberg uncertainty principle from the requirement that a diffraction minimum must be resolved for the determination of a
distance d. Substitution of the angle α with the corresponding impulse components p
and ∆p and substitution of the deBroglie wavelength does the trick!
linewidth is therefore reasonably narrow as compared to the spacing of vibrational lines, but not as compared to rotational lines. In addition to lifetime
broadening, different particles in the condensed phase see different local environments (e.g. via dipole-charge or dipole-dipole interactions) which shift the
observed transitions and lead to inhomogeneous broadening.
1.4
Physical laws and forces
The properties and the interaction of molecules is primarily due to electrostatic
interactions and to a minor degree to other physical forces.
1.4.1
Fundamental physical laws and postulates
1) Energy Conservation. Energy must be conserved. This law is close to a religious dogma in physics and has survived countless perpetual motion machines
and magic tricks. If energy is not conserved, then the measurement is wrong
or some degree of freedom has been overlooked. This law is the best friend
of everybody in the physical sciences: to understand an experiment you must
always ask where the energy comes from and where it ends up.
Energy can be partitioned between potential energy, kinetic energy, and mass
(Einstein: E = mc2 ). Every degree of freedom in a molecule can absorb or emit
energy in discreet quanta, while the kinetic energy is considered continuous. In
this lecture we will encounter transitions in nuclear and electronic spin states,
rotational and vibrational states and electronic states.
2) Momentum conservation. Momenta must be preserved as religiously as
energies. Unfortunately, it is often much harder to identify all relevant linear and angular momenta in an experiment. In chemistry, the conservation
of momenta are codified in symmetry considerations and, via selection rules,
determine whether transitions are allowed or forbidden.
1.4. PHYSICAL LAWS AND FORCES
15
Figure 1.10: Interference of 11 sine waves with different frequencies (top) results
in a short pulse (bottom) and illustrates the relation between frequency and time in
∆E · ∆t ≥ ~.
3) Postulate of wave particle dualism and the quantization of energy. DeBroglie postulated that impulse (and therefore also the energy) of a photon is
directly related to a wavelength λ = hp . This wave-particle dualism can be extended to other particles and helps to rationalize the quantization of energy and
the uncertainty principle.
1.4.2
Coulomb interaction
The Coulomb law (1.4.2) describes the attractive or repulsive forces between
charged particles. The forces between electrons and nuclei are big and largely
determine molecular structure. The potential energy of an electron at a distance
of one Bohr radius (a0 = 0.529Å) from a proton is easily calculated to be ≈ 27.21
eV. The kinetic energy of the electron in a stable orbit can be determined
by equating the Coulomb and centripetal force and we can directly estimate
the ionization potential of the H atom to be 13.6eV . Please remember that a
Coulomb potential falls off by 1/r and results in long range interactions.
charges q, q 0
electric constant ε0 = 8.8541 · 10−12
qq 0
4πε0 r
qq 0
Force: FC =
4πε0 r2
Energy: V =
F
m
Using the Coulomb law, we can also calculate the interaction between a
charge and a dipole or two dipoles by adding the attractive and repulsive potentials for all particles. The respective interactions falls off with 1/r2 and 1/r3
16
CHAPTER 1. INTRODUCTION
and are therefore of much smaller range than the interaction of charges. The
dipole of water is a considerable 1.8 D with 1 D = 3.335 ·10−30 Cm.
charge q
dipoles µ1 , µ2
angle from dipole axis θ
1.4.3
Magnetic forces
See chapter on NMR.
qµ1 · cosθ
4πε0 r2
µ1 µ2 · (1 − 3cos2 θ)
dipole-dipole: V =
4πε0 r3
dipole-charge: V =
Bibliography
[1] Kuchling Taschenbuch der Physik, 12. Auflage, Horst Kuchling (edt.), Verlag Harri Deutsch, Thun/Frankfurt am Main 1989.
[2] Physical Chemistry for the Life Sciences, Peter Atkins, Julio de Paula
(edts.), Oxford University Press, Oxford 2006.
[3] Catherine A. Royer, ”Approaches to Teaching Fluorescence Spectroscopy”,
Biophysical Journal 68, 1191 (1995).
[4] Shimon Weiss, ”Fluorescence Spectroscopy of Single Biomolecules”, Science
283, 1676 (1999).
[5] Internet Resources from the National Human Genome Research Institute,
Wikipedia, New England Biolabs, and others.
17