r .
Saarinbfi!

v.51... 9
3.3.85.4... inn... 5.3..
-3... 3m. L.

.Afi. #3326» ..

:3! .. . . . . A ,. , . .

.#.§-l(. h“. .. .. . . . . .. . . . . .. . . . . . ..

. . v: m .. . .. .A .. A . . . . . . , A . . .. , _
was} . . . . . .. . . . . _ , . , .. . A , . Wainwrfir... 5.“ ...

«v and haixxat .a u . . . . A . . ._. A _ .. . . , .. . . . I 11...... 1.,

. . 9.5:... 3:! ..I.iA..frl . _ . .. . . . .. . . . .r 51‘. .

. . .Ii Kw.n.li¢tt>!«r.v . , . n A. . A . , A . . . .. .fi:¢lh

. 1.6... in“! . . . H , z! .

é. ...$¢).,.pru.5n§ m ..-.-A. . .

..
u

.5. H. . . . . A . .. .
. . \b‘ I. . . . . , . z . . wild}...
:1 (an; . .n :1 . . . .
.g 4‘35... N .. . .il . .. .
w. 4...»... 1 .533»... .mfi; an t. _,
. . Stu. .mucamw.mmwfimwlxr
. . e... . ,
:, :15. p, ‘5»!
32......«13g. ... .
t. .F)...l.h..

.
. J t. c . ll!
. r. 7.63 a;
7(\ ”vi .3“?!
.1...|.\ . .
.nl. ..

-

. f .
.11“. £50...“ 3.1!.
II. ¢ . I... ts n

. II 2.1.6.112; ( o

. -Qfifinfiafifififzmfi

 

 

fibHE'f-Ug

 

 

 

 

 

LIBRARY 1
Michigan State >
T s 18 to certify that th l
hem '
ELECT s entttl d

 

PLACE IN RETURN BOX to remove this checkout from your record.
To AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6/01 c:/CIRC/DateDuo.p65-p.15

ELECTRON SPIN ECHO ENVELOPE MODULATION STUDIES ON
METHYLMALONYL-COA MUTASE

By

Nadia Anjum Shams-Ahmed

AN ABSTRACT OF A THESIS
Submitted to
Michigan State University
in partial fiilfillment of the requirements
for the degree of
MASTER OF SCIENCE
Department Of Chemistry

1 999

ABSTRACT

ELECTRON SPIN ECHO ENVELOPE MODULATION STUDIES ON
METHYLMALONYL-COA MUTASE

By
Nadia Anjum Shams-Ahmed
Vitamin B”, a cobalt-based cofactor, is involved in many enzymatic
systems, including methylmalonyl-CoA mutase. Methylmalonyl-CoA mutase is an
enzyme that converts L-methylmalonyl-CoA into succinyl-CoA, which is part of
respiratory metabolism. The mechanism involved in this reaction is not yet known. The
most popularly accepted mechanism involves a fi’ee radical pathway.

Ruma Baneijee and Rugmini Padmakumar have tried to identify the radical
intermediates of this enzymatic reaction by using electron paramagnetic resonance (EPR)
spectroscopy. They found inhomogeneous broadening in their spectra, and hyperfine
splittings resulting from coupling of the unpaired electron with the cobalt nucleus. They
also did a power dependence study on the enzyme in the presence of the substrate at two
different temperatures. At 10K they saw two different signals, whereas at 25K they only
saw one. The spectra showed coupling to the cobalt, which indicated that cobalamin, a
cobalt-based radical formed from the B12 cofactor, was one of the radicals. It was not
clear, however, what the other radical species was.

Electron spin echo envelope modulation, a pulsed EPR technique, was performed
on their samples in order to continue their study. ESEEM data showed peaks in the low
frequency region of the spectrum that are characteristic of coupled nitrogen. Isotopic
substitution of protein bound l“N coupled with further ESEEM studies allowed us to

assign these modulations to a histidyl group, coordinated axially to €001).

To Allah, the one God, who blessed me with the ability and energy to comprise this
work, and who helped me in various ways to make this possible.

To Hadhrat Mirza Tahir Ahmad, Khalifatul Masih IV, whose guidance and prayers
provided me with peace of mind.

To my late father, Dr. Salah U. Shams, who taught me about spirituality, dedication, and
hard work through example.

To my husband Tahir, who helped me keep my sanity and never stopped believing in me.

Last but not least, to my mother Kaukab, my sister Sabiha, and my brother Sabah, who
constantly encouraged me and never let me give up.

iii

ACKNOWLEDGMENTS

I would like to thank the following individuals for giving me their time, patience, and
support during the course of this project:
Dr. John McCracken
Dr. Gerald Babcock
Dr. Marilena DiValentin
Pierre Dorlet
Dr. Curt Hoganson
Dr. Steve Siebold
Dr. Hans Schelvis
Gwynne Osaki
Tara Simmons
and

Nasir Bukhari.

iv

TABLE OF CONTENTS

LIST OF TABLES .............................................................................. vi
LIST OF FIGURES ............................................................................ vii
LIST OF ABBREVIATIONS ................................................................. viii
CHAPTER I
INTRODUCTION .............................................................................. 1
References ............................................................................. 13
CHAPTER II
ELECTRON PARAMAGNETIC RESONANCE METHODS .......................... 14
Origin Of the Zeeman Effect ......................................................... l4
Derivation Of the Spherical Harmonics ............................................ 15
Derivation Of the Angular Momentum Operators ................................ 20
Deriving the Zeeman Energy From A Hydrogen-Like Atom (S = ‘/2 ). .. ......24
Deriving the Two Spin Energy States For An Electron .......................... 27
The Nuclear Zeeman Interaction .................................................... 29
Electron Nuclear Hyperfine Interaction: Isotropic and Anisotropic ........... 32
Orientation Of the Lab Axes With Respect To the Hyperfine Axes ............ 40
Data Of Enzyme With Substrate and Product ..................................... 44
Principles Of Transition Rates and Power Saturation ............................ 49
Power and Temperature Dependence Study ....................................... 59
References .............................................................................. 63
CHAPTER III
ELECTRON SPIN ECHO ENVELOPE MODULATION ............................... 64
Two Pulse ESEEM .................................................................... 65
Two Pulse ESEEM For S = 1/2 and I = '/2 Spin System ............................ 69
Two Pulse ESEEM For A System With S = V2 and I = 1 ........................ 73
Modulations Of Two-Pulse ESEEM ................................................ 75
Three-Pulse ESEEM ................................................................... 76
Modulations Of Three-Pulse ESEEM .............................................. 79
The t-Suppression Effect ............................................................. 79
Fourier Transformation ............................................................... 80
References .............................................................................. 83
CHAPTER IV
NUCLEAR QUADRUPOLAR INTERACTION or ”N ................................ 84
Deriving the Nuclear Quadrupolar Interaction Energies For An I = l
Nucleus .............................................................................. 84
Resonant Frequencies Of the Nuclear Quadrupolar Interaction ................. 90

Solving For the Field Gradient q and the Asymmetry Parameter 11...... ...... .91

Deriving the Energy Level Diagram For An S = 1/2 and I = 1 System ......... 92
References .............................................................................. 96

CHAPTER V

ESEEM OF METHYLMALONYL—COA MUT ASE WITH L-METHYL-

MALONYL-COA WITH l“N ................................................................ 97
ESEEM Data Collection .............................................................. 97
Samples Of Methylmalonyl-CoA Mutase .......................................... 97
ESEEM or 1“N Samples .............................................................. 97
Discussion .............................................................................. 99
References .............................................................................. 106

CHAPTER VI

ESEEM OF METHYLMALONYL-COA MUTASE WITH L-METHYL-

MALONYL-COA WITH 1’N ................................................................ 107
ESEEM Data Collection .............................................................. 107
Samples Of Methylmalonyl-CoA Mutase .......................................... 107
ESEEM Of l5N Sample ............................................................... 108
Discussion ............................................................................... 108
Conclusion .............................................................................. l 15
References .............................................................................. 117

LIST OF TABLES

Table 4.1: The solutions of Equations (156), (157), and (158) for the three nuclear spin
states m1= -1, 0, and +1.

vii

Figure1.1:

Figure 1.2:

Figure 1.3:

Figure 1.4:

Figure 1.5:

Figure 1.6:
Figure 1.7:
Figure 1.8:

Figure 1.9:

LIST OF FIGURES

The chemical equation for the production of Acyl-CoA.

The migration of acyl-CoA into the mitochondrion.

(a) At the outer mitochondrial membrane, acyl-CoA react with carnitine to
form acyl-carnitine derivatives, which can cross the inner mitochondrial
membrane.

(b) At the inner mitochondrial membrane, the acyl-carnitine derivatives are

converted back to acyl-CoA and carnitine.

The production of acyl-carnitine derivatives.

Acetyl-CoA, which is a two carbon subunit produced during fatty acid
metabolism or B—oxidation.

Propionyl-CoA, which is produced during B-oxidation of odd-chain-length
saturated fatty acids.

The production of succinyl-CoA fi'om propionyl-CoA.

The structure of 5’-deoxyadenosylcobalamin coenzyme B12 (vitamin B12).
Adenosylcobalamin when recombined with methylmalonyl-CoA mutase.
The postulated reaction mechanism of mmCoA mutase. The two possible

radicals are either the deoxyadenosyl radical or a secondarily generated
protein radical.

Figure 1.10: Banerjee’s and Padmakumar’s EPR spectrum of the holoenzyme with

Figure 2.1:

Figure 2.2:

deuterated substrate, [CD3]methylmalonyl-CoA. The eight small peaks are
due to the hyperfine interaction between the unpaired electron and the
cobalt(II) nucleus.

The rotation of a mass m about a fixed point. The mass is rotating at a
distance r from the center with a velocity v, and has an angular momentum L
perpendicular to the plane of rotation.

The rotation of a two particle system, m1 and m;, about its center of mass.
The particle m1 rotates at a distance r1 from the center of mass, while the
particle m2 rotates at a distance r; from the center of mass. The two particle
system has an angular momentum L perpendicular to the plane of rotation.

viii

Figure 2.3:

Figure 2.4:

Figure 2.5:

Figure 2.6:

Figure 2.7:

Figure 2.8:

Figure 2.9:

Figure 2.10:

Figure 2.11:
Figure 2.12:
Figure 2.13:

Figure 2.14:

Figure 2.15:

The relationship between the cartesian coordinates x, y, and z and the
spherical polar coordinates r, 0, and d).

For an S = ‘/2 system, the two degenerate electronic energy levels split further
and firrther apart as the applied magnetic field is increased.

The energy levels of a proton with I = ‘/2 in a magnetic field.

The vector for the electron magnetic dipole moment He will feel a local field
fiom the vector for the nuclear magnetic dipole moment n... B is the applied
magnetic field and 9 is the angle between the applied field and the axis of the
two magnetic moments. The distance between the two dipoles is r. (The
vector u. represents the state 1118 = - ‘/2 , and the vector 11.. represents the state
m1= + l/2.)

The derivative spectrum of a nucleus with I = V2 interacting with an electron
in an isotropic system. The constant a is the hyperfine splitting constant, and
the resonant field value BM“. = hv/(gB) i a/2.

The derivative spectrum of a nucleus I = '/2 interacting with an electron in an
anisotropic system. It can be seen that A, is not equal to A".

The axis system in terms of the hyperfine. B is the applied magnetic field
and S, is the electron spin in the direction of the applied field.

The axis system in terms of the laboratory field B. The electron is at the
origin interacting with the nearby nucleus.

The EPR spectrum of mmCoA mutase with the substrate L-mmCoA.
The EPR spectrum of mmCoA mutase with [CD3]mmCoA.
The EPR spectrum of mmCoA mutase with the product succinyl-CoA.

For a two level system, with ground state 13 and excited state a, the

microwave-induced upward transition rate mwu equals the microwave-

induced downward transition rate mwd at low power.

(a) When the relaxation rate down is added to the microwave-induced rate
down, they become greater than the upward transition rate.

(b) The microwave power can be increased to overcome the relaxation rate
down, leading to a condition called power saturation.

The two level system of an electron with S = '/2. The transition energy AB =
yhB. The number of spins in the two states are represented by N+ and N..

Figure 2.16: The two level systems of both the electron system, labelled land 2, and the

reservoir, labelled b and a. The diagram in (a) represents an allowed
transition, while the diagram in (b) represents a forbidden transition. The
x’s represent initial states. The populations of the two levels in the electron
system are given by N; and N2, and the populations of the two levels in the
reservoir system are given by N. and Nb.

Figure 2.17: The EPR spectrum of the power dependence study of mmCoA mutase with

L-mmCoA at 10K. There is a shift in g values as the power is varied fi'om
100mW to 0.05 mW. At 100 mW, the g value is 2.14. At 0.05 mW, the g
value is 2.11.

Figure 2.18: The EPR spectrum of the power dependence study of mmCoA mutase with

Figure 3.1:
Figure 3.2:

Figure 3.3:

Figure 3.4:

Figure 3.5:

Figure 3.6:

Figure 3.7:

Figure 3.8:

Figure 3.9:

Figure 4.1:

L-mmCoA at 25K. As the power is varied from 100 mW to 0.05 mW, there
is no shift in the g value. The g value of the lineshape is 2.11.

Microwave pulse sequence of a two-pulse ESEEM experiment.

The modulated echo decay plot of Ni(III)(CN)4(HzO)2.

Classical picture of a two-pulse ESEEM experiment. B0 is the lab field, B 1 is
the magnetic field associated with the microwave pulses, M is the bulk

magnetization, and (no is the microwave pulse frequency.

Energy level diagram describing the quantum mechanical results of an
ESEEM experiment for an S = V2 , I = ‘/2 spin system. The frequencies 0,,
and $5 are coo. = [(0), — N2)2 + 132/41”2 and (Op = [(0), + A/2)2 + 132/41”.

The combination of quantum mechanical results with classical results of an
ESEEM experiment.

The energy level diagram for an S = ‘/z , I = 1 spin system, showing the three
allowed and six semiforbidden transitions.

The microwave pulse sequence of a three-pulse ESEEM experiment.

The classical picture of a three-pulse ESEEM experiment describing the
formation of a stimulated echo.

(a) The free induction decay (FID) spectrum for an S = ‘/2 , I = ‘/2 spin system.
(b) The frequency domain spectrum of the F11) spectrum shown in (a).

Different types of nuclei, varying in charge distributions.

Figure 4.2:

Figure 4.3:
Figure 5.1:

Figure 5.2:

Figure 5.3:
Figure 5.4:

Figure 5.5:

Figure 6.1:

Figure 6.2:

Figure 6.3:

Figure 6.4:

Figure 6.5:

The energy splitting diagram of 14N, showing splittings from the (a) electron
Zeeman energy [-gefleHoms], (b) nuclear Zeeman interaction [-ganHo/h =
-v,.], (c) the electron-nuclear hyperfine interaction [amt], and (d) nuclear
quadrupole interaction. vn is the Larmor fi'equency, which is .94 MHz at
3060 G.

The Fourier transformation spectrum of 1“N.
The field scan of mmCoA mutase with L-mmCoA.

The electron spin echo decay envelope of mmCoA mutase with L-mmCoA at
3060 G.

The Fourier transformation of the spectrum shown in Figure 5.2, at 3060 G.
The Fourier transforrntion of the spectrum shown in Figure 5.2, at 3140 G.

An illustration of cobalamin showing the three possible nitrogens coupled to
the unpaired electron. The four nitrogens around the cobalt center are the
pyrrole nitrogens. The nitrogen at the lower axial position is the nitrogen
from the dimethylbenzimidazole. The dashed line leads to a nitrogen-based
ligand from the protein of the enzyme.

The two-pulse echo-detected EPR experiment of l’N-labelled mmCoA
mutase with L-[CD3]mmCoA.

The Fourier transformation spectrum of the three-pulse ESEEM spectrum of
l’N-labelled mmCoA mutase with L-[CD3]mmCoA at 3060 G.

The Fourier transformation spectrum of the three-pulse ESEEM spectrum of
”N-labelled mmCoA mutase with L-[CD3]mmCoA at 3220 G.

The Fourier transformation spectrum of the three-pulse ESEEM spectrum of
”N-labelled mmCoA mutase with L-[CD3]mmCoA at 4000 G.

An illustration showing cobalamin with the cobalt center coordinated at the
lower axial position to a nitrogen-based ligand from the protein of the

enzyme.

LIST OF ABBREVIATIONS

CoA: Coenzyme A

TCA: Tricarboxylic Acid

ATP: Adenosine Triphosphate
MmCoA: Methylmalonyl-CoA

EPR: Electron Pararnagnetic Resonance
ESEEM: Electron Spin Echo Envelope Modulation
L-mmCoA: L-methylmalonyl-CoA

K: Degrees. Kelvin

mW: MilliWatts

FID: Fourier Induction Decay

l"N: Nitrogen-14

l5N: Nitrogen-15

MHz: MegaHerz

Cm: Centimeters

J: Joules

G: Gauss

Ns: Nanoseconds

2H: Deuterium

xii

INTRODUCTION

Methylmalonyl-CoA mutase is an enzyme that converts L-methylmalonyl-CoA
into succinyl-CoA, which is part of the tricarboxylic acid (TCA) cycle. The TCA cycle is
the component of respiratory metabolism that provides a means for breaking down two
carbon units into carbon dioxide in the presence of oxygen. Succinyl-CoA is also utilized
in the synthesis of heme in animals.1

Succinyl-CoA is produced during fatty acid metabolism. Fatty acids with an odd
number of carbons are rare in many mammalian tissues, but in animals such as cows and
sheep, the oxidation of these fatty acids can account for as much as 25% of their energy
requirements. 1

Inside cells, fatty acids are first reacted with coenzyme A and ATP in the cytosol
to yield fatty acyl-CoA, illustrated in Figure 1.1'. This reaction is catalyzed by acyl-CoA
ligase, also known as thiokinase. At the outer mitochondrial membrane, represented in
‘ Figure 1.2‘, acyl-CoA reacts with carnitine to yield acyl-carnitine derivatives in a
reaction described in Figure 1.3', which can then cross the inner membrane of the
mitochondrion. Once inside the mitochondrion, the acyl-carnitine derivatives are
converted back to acyl-CoA and carnitine, shown in Figure 12‘.1

Inside the mitochondrion, odd-chain-length saturated fatty acids break down by a
process called B-oxidation, to yield a number of acetyl-CoA’s, shown in Figure 1.4', and
one propionyl-CoA, shown in Figure 1.51. It is propionyl-CoA that begins the series of
enzymatic reactions that produces succinyl-CoA, illustrated in Figure 1.6'. Propionyl-

CoA, with the addition of ATP, CO2, and H20, is converted into D-methylmalonyl-CoA

with the enzyme propionyl-CoA carboxylase. Then D-methylmalonyl-CoA is converted

+2

 

M
RCOO' + ATP + CoA g > RCO-CoA + PPi + AMP
14%;“ Acyl-CoA

Figlre 1.1 : The chemical equation for the production of Acyl—CoA.

  
 
  
 

 

 

 

 

 

 

 

 

 
 
  

 

 

Outer Mitochondrial Membrane Inner Mitochondrial Membrane
Cytosol lntermembrane Space Matrix
. . Carrier -
CoA Acyl-Carnltme- '"Prdfe'ihm Haggai“
Carnitine Acyltransferase I CoA
” “ . . arneitin. ‘

Carmtrne A l-

- CY
AcyleCoA Acyltransferase II CoA

Carnitine

 

(b)

Figure 1.2: The migration of acyvaoA into the mitochondrion
(a) At the outer mitochondrial membrane, acyl-CoA react with carnitine to
form acyl-carnitine derivatives, which can cross the inner mitochondrial
membrane.
(b) At the inner mitochondrial membrane, the acyl-carnitine derivatives are
converted back to acyl-CoA and carnitine.

 

RCO-CoA + (CH3)3N+-CH2(IZHCH2COO' ‘——(CH3)3N+-CH2(fHCH2COO' + CoA

Carnitine
0H Acyltransferase ?
Carnitine RC=O
Acyl Carnitine

Figure 1.3: The production of acyl-carnitine derivatives.

O
ll

CH3C-COA

Figure 1.4: Acetyl-CoA, which is a two carbon subunit produced during fatty acid metabolism,
or B-oxidation.

O
I
CH3CH2CJ3-COA

Figure 1.5: Propionyl-CoA, which is produced during B-oxidation of odd-chain-length saturated
fatty acids.

0

ll
CH3CH2C-COA + ATP + C02 + H20
Propionyl-CoA

Propionyl-CoA Carboxylase

C00‘

|
H-C"CH3

l

C
II
o

D—Methy onyl-CoA

—S—CoA

Methylmalonyl-CoA Racemase

(IIOO'
H3CB_ ICE-H

fi—S-COA

O
L-Methylmalonyl-CoA

Methylmalonyl-CoA Mutase

COO'
H2? " CH2

C-S—CoA
ll
0

Succinyl-CoA

Figure 1.6: The production of succinyl-CoA from propionyl-CoA.

into its optical isomer by methylmalonyl-CoA racemase. Finally, methylmalonyl-CoA
mutase converts L-methylmalonyl-CoA into succinyl-CoA.l

The conversion of L-methylmalonyl-CoA into succinyl-CoA is an intramolecular
rearrangement reaction on adjacent carbon atoms (refer to Figure 1.6'). The thioester
group on the or-carbon migrates to the B-carbon in exchange for a hydrogen atom. This
isomerization is catalyzed by methylmalonyl-CoA mutase. The dysfunction of this
enzyme leads to a condition called methylmalonic acidemia, which is an inborn error of
amino acid metabolism. The symptoms and effects of this disease are vomiting,
convulsions, mental retardation, and eventually death.l

MmCoA mutase is a vitamin B12-dependent enzyme. There are a variety of
enzymes that are dependent on the 812 cofactor, such as diol dehydratase, glutamate
mutase, L-B-Lysine aminomutase, ethanolamine ammonia lyase, ribonucleotide
reductase, and methionine synthase.2 The vitamin Br2 coenzymes consist of a cobalt
atom bonded to four pyrrole nitrogens forming a “corrin” ring, as can be seen in Figure
1.7'. The upper axial position on the cobalt can be a methyl, a hydroxide, or a 5’-
deoxyadenosyl group, which is the principle coenzymatic form. In the case of
methylmalonyl-CoA mutase, the 812 cofactor is adenosylcobalamin, where the upper
axial position is a 5’-deoxyadenosine. The lower axial position is occupied by a
dimethylbenzimidazole group.1 However, studies have been done on the methionine
synthase system showing that when the Bu cofactor is bound to the enzyme, the lower
axial position is no longer coordinated to the dimethylbenzimidazole group, but rather to
the side chain of a histidine residue.3 This mode of bonding was also found for

methylmalonyl-CoA mutase, as shown in Figure 1.8.4

 

 

 

 

 

Figure 1.7: The structure of 5’-deoxyadenosylcobalamin coenzyme Bu (vitamin Bu).

Ado

CH2

 

Co

N

c a
N

 

 

Figure 1.8: Adenosylcobalamin when recombined with methylmalonyl-CoA mutase.

The Bl2-dependent enzymes are all involved in rearrangement reactions, but there
has been a lot of ambiguity about the mechanism of the rearrangements. The most
popularly accepted reaction mechanism involves a free radical pathway, which begins at
the metal center of the cofactor.2 This is not a phenomenon that is unique to the Bu-
dependent enzymes. In fact, the B12-dependent enzymes are one of the three families of
the larger metallo-radical class of enzymes, which share similar structural and functional
principles. The three families are the glycyl/thiyl radical enzymes, the B12-dependent
enzymes, and the 02-dependent radical enzymes. Basically, the metal centers of these
enzymes consist of either copper, cobalt, iron, or manganese and act to generate an amino
acid radical. This radical then initiates catalysis by abstracting a hydrogen atom from the
substrate.5 However, it is the actual details of how this mechanism occurs in each system
that is of current interest.

It is believed that the mechanism involved in the production of succinyl-CoA
from methylmalonyl-CoA mutase is comprised of three main steps, illustrated in Figure
1.96. The first step is the homolytic bond cleavage of the cobalt-carbon bond of the B12
cofactor to produce a cobalamin free radical and an adenosyl free radical.6 Upon binding
to the substrate, the 00an ring undergoes conformational changes that lengthens and
weakens the cobalt-carbon bond, allowing for ease of dissociation of the bond. However,
it has not been determined whether the steric interactions alone provide enough energy to
break the bond.2 The second step is hydrogen abstraction fi’om the substrate. After the
cobalt-carbon bond dissociates, the adenosyl radical, either directly or via a protein
radical, abstracts a hydrogen atom from the methyl group of L-methylmalonyl-CoA,

generating a reactive primary radical on the substrate. The third step is the 1,2

 

 

 

 

 

 

 

l
l
l I
J ' I .
I / 9° / l
i l
I K 5 g
i DMB < I HIS '
a / i
L-MmCoA l H 5 Succinyl-CoA
I. K V.
QOSCoA Ado }éc /COSCoA Ado
Hn‘C-CH3m CH2. /C —C;HHn 'CH2
C/OzH/ COZH

 

 

9/ / 9 /
/ /
DMB {Ni/His DMB </::L/H,s

 

 

 

 

 

 

 

 

 

 

 

‘ l
H. IV.
r l
QOSCOA IAdo H\ /COSCOA Ado
H‘i'c— CH _
. 2 CH3 / ' H CH3
cozn m cozrr m

 

 

/’ 9/ m- 9 /

...... (bows m {I

Figure 1.9: The postulated reaction mechanism of mmCoA mutase. The two possible
radicals are either the deoxyadenosyl radical or a secondarily generated
protein radical.

 

 

 

 

 

 

rearrangement reaction. This is the least understood of the three steps. The substrate
rearranges to a more stable secondary radical on the product. Somehow, either via a free
radical, a carbonium ion, a carbanion, or an organocobalt intermediate, the a-carbon
thioester group is switched to the B-carbon. After the substrate leaves, the cobalt atom of
the cobalamin radical and the carbon atom of the adenosyl radical reform their bond.6 A
major question pertaining to this mechanism is whether the adenosyl radical abstracts the
hydrogen atom alone, or if there is a secondary protein radical involved.

Electron paramagnetic spectroscopy provides an ideal tool for characterization of
paramagnetic centers in complex chemical systems. In biological systems containing
large molecules such as proteins, it can become difficult to obtain information about the
paramagnetic center within the molecule. EPR spectroscopy removes this obstacle
because it is only sensitive to the structure directly around the paramagnetic center.
Ruma Banerjee and Rugmini Padmakumar have characterized the EPR properties of
mmCoA mutase under several conditions. No EPR signal was detected for the
apoenzyme, which is the form of the enzyme without the 812 cofactor, nor was an EPR
signal detected for the holoenzyme, which is the form of the enzyme reconstituted with
the 812 cofactor. However, when either the substrate, L-methylmalonyl-CoA, or the
product, succinyl-CoA was added to the holoenzyme and rapidly frozen in liquid
nitrogen, an EPR signal was detected.6 Figure 1.10‘5 illustrates Banerjee’s and
Padmakumar’s spectrum of the holoenzyme rapidly mixed with deuterated substrate,
[CDflmethylmalonyl-CoA. The eight small peaks in the spectrum indicate hyperfine
splittings resulting from coupling of the unpaired electron with the cobalt(II) nucleus,

which has a spin of I = 7/2. This coupling between the unpaired electron and the

10

 

Echo Amplitude

 

2600 3000 3400 3800 4200

Field Strength (G)

Figure 1.10: Banerjee and Padmakumar’s EPR spectrum of the holoenzyme with
deuterated substrate, [CD3]mmCoA. The eight small peaks are due to the
hyperfine interaction between the unpaired electron and the cobalt(II)
nucleus.

11

cobalt(II) nucleus is the result of hyperfine interaction, which is the interaction between
the magnetic moment of the unpaired electron and the magnetic moment of the cobalt(II)
nucleus. Other than the cobalt nucleus, it was not possible to determine any of the other
species present since they were weakly coupled to the unpaired electron.

The purpose of this project was to determine what species were weakly coupled to
the unpaired electron, and then to identify the radicals present in the sample of the
enzyme with the substrate. One way to study weak couplings between an unpaired
electron and surrounding nuclei with nuclear magnetic spins is by the electron spin echo
envelope modulation (ESEEM) technique of pulsed EPR. Once the nuclei have been
identified by ESEEM, they can be assigned by means of isotope labelling. Using both of
these techniques, ESEEM and isotope labelling, more information regarding the identity

of the radical species of mmCoA mutase can be obtained.

12

0‘

References

. Zubay, Geoffrey. Biochemistry. Wm. C. Brown Publishers: Iowa, 1993.

Sigel, H., and A. Sigel. Metal Ions In Biological Systems. Marcel Dekker, Inc., New
York: 1994, Vol. 30, pp. 1-24 and 255-278.

. Matthews, R., et. al., Biochemistry, 1988, Vol. 27, pp. 8458-8465.
. Mancia, et. aI., Structure, 1996, Vol. 4, pp. 339-350.
. Babcock, G., et. 0]., Acta Chemica Scandinavica, 1997, Vol. 51, pp. 533-540.

. Banerjee, R., and R. Padmakumar, The Journal of Biological Chemistry, April 21,

1995, Vol. 270, No. 16, pp. 9295-9300.

l3

Chapter 2: Electron Paramagnetic Resonance Methods

The proposed reaction mechanism of the rearrangement reaction of L-mmCoA to
succinyl-CoA involves radical species. One way to investigate radical systems is by EPR
spectroscopy, or electron paramagnetic resonance spectroscopy. EPR is a spectroscopic
technique allowing the study of molecules containing unpaired electrons by observing the
magnetic fields at which the electron spin energy levels come into resonance with
microwave radiation. An EPR spectrum is obtained by monitoring the microwave
absorption at a fixed frequency as the magnetic field is swept.

In EPR the interaction between the unpaired electron spin moment and the
applied magnetic field is called the Zeeman effect. For the simplest paramagnetic
system, which has a single unpaired electron such that the electronic spin quantum
number S = ‘/2, the isotropic electronic Zeeman harniltonian is

H = 8/3 Bi (1)
where g is the electron g factor, B is the electron Bohr magneton, B is the applied field
strength, and the operator of S; is the electron spin operator in the direction of the field.
Origin of the Zeeman Effect

Atoms have magnetic dipole moments due to the motion of their electrons, and
each electron has its own intrinsic magnetic dipole moment associated with its spin. In
paramagnetic materials, each atom has a permanent magnetic moment, which can be
related to the angular momentum of the atom.1 In order to understand this relationship, it
is necessary to derive the equation for the spherical harmonics, beginning with the

classical model of the rigid rotor. Using the equation for the spherical harmonics, the

14

quantum mechanical operators for angular momentum, L and L2, and spin angular
momentum, S and 82, are obtained.

Derivation Of the Spherical Harmonics

When a particle rotates around a fixed axis as illustrated in Figure 2.12, the

particle has both angular momentum and rotational kinetic energy.2 The kinetic energy is
given by

KE = V; mv2 = p2/2m (2)
where m is the mass of the particle, the momentum p = mv, and v is the linear velocity.
The angular velocity is

v = 27trv = r03 (3)
where v is the fiequency of rotations, r is the radius of the circle of orbit, and the angular
velocity (0 = d0/dt = 27w. Therefore, the kinetic energy of a particle in circular motion
about a fixed point in terms of angular velocity and angular momentum is

KE=‘/2mr’2c02=‘/2Ic02 (4)
where the moment of inertia I = mrz. The angular momentum L is defined as

L = 10). (5)
Thus, the kinetic energy of rotational motion in terms of angular momentum is

K13,m = ‘/210)2 = 13/21. (6)
When two particles rotate about their center of mass, as illustrated in Figure 2.22, they
satisfy the condition

mm = mm (7)
where r; is the distance of m; from the center of mass, and r2 is the distance of m2 fi'om

the center of mass. Since the equilibrium distance R between the particles is

15

 

 

'. I"

Figure 2.1: Therotationofamassmaboutafixedpoint. Themass isrotatingatadistancer

from the center with a velocity v, and has an angular momentum L perpendicular
to the plane of rotation.

 

 

 

 

 

 

Figure 2.2: The rotation of a two particle system, m1 and m2, about its center of mass.
The particle mt rotates at a distance r; from the center of mass, while the
particle m2 rotates at a distance r2 from the center of mass. The two particle
system has an angular momentum L perpendicular to the plane of rotation.

16

R = r; + r2, (3)
then fi'om Equations (7) and (8) the following relationships for the radii can be obtained:
rt = [mz/(mr + m2)] R (9)
and
f2 = [ml/(m1 + m2)] R- (10)
The rotational kinetic energy for this two particle system is
KEN: = ‘/2 mtrlzco2 + ‘/2 m2r22c02 (11)
= 1/2 (mm2 + m2r22) (02
KB... = 1/2 1o2 (12)
where I = [m1m2/(m1 + m2)]R2 after elimination of r1 and r2 by Equations (9) and (10). If
we let m1m2/(m1 + m2) = u, then the moment of inertia can be rewritten as
I = uR2 (13)
where 11, called the reduced mass, is defined as
Up = l/ml + 1/m2 =(m1 + m2)/ m1m2. (14)
By using the reduced mass, the two body problem has been reduced to a one body
problem. Therefore, the rotational kinetic energy of the two body system written in terms
of angular momentum L becomes
K13.m = 1,2/(21) = 1,2/(2.11192).2 (15)
The system of the rigid rotor has kinetic energy but no potential energy. In
quantum chemistry, classical mechanical observables are represented as quantum
mechanical operators. Therefore, the classical kinetic energy corresponds to the

following quantum mechanical harniltonian operator for rotational energy:

17

Ii! .—_ -_’2_‘_v- (16)
2x1
where the Laplacian operator v2 is defined as

2 2 2
V2=;2+;2+:2. (17)

As the energy of the rigid rotor is rotational, it is more convenient to use spherical polar

 

coordinates, which are defined in Figure 2.32. The Laplacian operator in spherical polar

coordinates is

2
V2=—1-2——i(R23-)+ 1 6 + 21 —(s in61—). (18)
R 612 GR R‘sin 05¢ R smear)

In the model of the rigid rotor, R 15 constant because the two particles are at fixed

 

distances from the origin. Therefore, all the derivative terms of R can be ignored, since
the derivative of a constant is zero. Keeping this in mind, when Equation (18) is
substituted into Equation ( 16), the harniltonian operator for the rigid rotor becomes

2 r22 1 a 1 62
H=-——— a — 19
21[sin6166(sn ae)+s 219 are] ( )

where I = 11R2 . Since the rigid rotor wavefunction is a firnction of the angles 0 and d), the

eigenvalue problem to be solved is

’12 l a a l 62
21[stn95‘5 (“ST—“9W1“,2pw2 —2]Y<6’ ¢>= we r» (20)

The solutions Y(0,¢) to this standard differential equation are called spherical harmonics.

 

Since the two quantum numbers 8 (angular momentum quantum number) and m

(magnetic quantum number) arise in the solution of this eigenvalue equation, the

wavefunctions are represented as

Y!” (97 ¢)

18

‘2

z = rcosO

 

 

-------- b---------

,x'x = rsin0cos¢

I
9

y é'rsin'e‘siiii'"=

Figure 2.3: The relationship between the cartesian coordinates x, y, and z and the
spherical polar coordinates r, 0, and :1).

l9

In general, it is found that

RY,"(6,¢)=“—‘§1”—Yr(e,¢) (21)

where I? is the angular momentum quantum number (or azimuthal quantum number).

Therefore, the energies of the rigid rotor are given by

2
E = €(€ + l)h
21
where the angular momentum quantum number (I = 0,1,2,.. .etc.2

(22)

Derivation Of the Angular Momentum Operators

Angular momentum is a vector that has components in the x, y, and 2 directions.
Thus, in order to develop the quantum mechanical operators for angular momentum in
the x, y, and 2 directions, it is necessary to begin with the classical expressions for
angular momentum in all three dimensions.2

As illustrated in Figures 2.12 and 2.22, the angular momentum of a particle or
particles rotating about a fixed point is represented by the vector L in the direction
perpendicular to the plane of the circular motion. Ifa mass m rotates about a fixed point
with linear velocity v, the angular momentum L is given by the cross product of the
radius r and the linear momentum vector p,

L=rxmv=rxp (23)

where the cross product of the vectors r and p is a vector of magnitude lrllplsinO, and 0 is
the angle between r and p. The vectors r and p can be expressed in terms of their

components with the unit vectors

i, j, and k

20

pointing along the x, y, and z axes respectively as follows:

r=xi+yj+zk (24)
and
p=pxi+pyi+pzll (25)
Thus, a determinant can be set up in order to find the cross product of r and p, which

gives the angular momentum L as

A R A

i j k
L=r><p= x y z =(ypz—zp,)i+(zp. -xp.)1'+(xp_v -yp,.)k. (26)
Pa Py Pr

Therefore, the three components of the classical angular momentum of a particle rotating
about a fixed point are

Lx = W2 ‘Zpy (27)

L) = pr — XP2 (28)

and
L; = XPy " YPX- (29)

The square of the angular momentum, L2, is found by taking the scalar product of L with
itself:
LoL = |L||L|cos9 = L’(oos 0) = L2(1) =L2 = If + 1,,2 + L} (30)
where L2 is a scalar quantity.
In order to convert the classical angular momentum into its quantum mechanical
operators, the quantities of L2, Ly, and L2 must be replaced with their corresponding

quantum mechanical operators, namely

21

~_h6_.6 (M)

Px—T~--lh—

16x 0x

~ It 6 6
p =.——=-ih— (32)

y 15y 6y

and

13, =33=4h3 (33)

162 62

Therefore, the angular momentum operators for the x, y, and 2 components are

12, =-ih(y%—z—§l—) (34)

‘ .- 2.- 2.

Ly—-lh(zax x62) (35)
and

* --- i- 31

L2— ”1(an yax). (36)

The quantum mechanical operator for the square of the angular momentum is

if =|12|2 4.12:1“; +122y +121. (37)

As mentioned earlier, since the system is rotational it is more convenient to write the

angular momentum operators in spherical polar coordinates (see Figure 2.32), which are

 

 

~ . . a 6
L = h —+ t0 — 38
x 1(s1n¢ 66 co cos¢ 6¢) ( )
L =ih(-cos¢i+cot9 sin¢—a—) (39)
y 66 a¢
~ . a
L =- h— 40
and
2 1 a a 1 62
L2=-}‘t2 —— ' 6— + . 41
[sinfl 66(Sm 66) sin20 5252)] ( )

It is important to note here that the operators for L, and Ly, Ly and Lz, and L" and L2 do
not commute with each other. However, the operators for Lx, Ly, and L2 each commute
with the operator for L2. This implies that one can measure precisely the square of the
total angular momentum and only one of the components of the angular momentum. For

example, if the magnitude of the total angular momentum, which is defined as

 

|L| = J? =2 J1}, +L3y + L2, (42)
is measured, and L2 is also measured, then it is not possible to measure L,( and Le

precisely. Thus, the eigenfilnction of L2 is an eigenfunction of L2, but not an
eigenfunction of L" or Ly as they do not commute with L2. This is a major difference
between classical and quantum mechanical systems.2

Since the operators for L2 and Lz commute, a firnction can be constructed that is
an eigenfunction of both operators. These wavefunctions are called the spherical
harmonics, and were seen in Equation (21) for the energy of a rigid rotor. For the
classical rigid rotor,

L2 = 21(KE) (43)

while for the quantum mechanical operator

if = 21(r‘1). (44)
Therefore, the eigenvalue equation for the square of the angular momentum becomes

1:2 Y:(9,¢) = m + 1W Y2"(0,¢) (45)
where the angular momentum quantum number 13 = 0,1,2,. . .etc. According to Equation

(45), the total angular momentum squared for a rigid rotor can only have the following

values:

23

I.2 = ((2 +1)h2 (46)
where E = 0,1,2,. . .etc. Similarly, when operating on the spherical harmonics with the 2-

component of the angular momentum operator of L, using the equation

£2Y9”(62¢)=th2"(0,¢) (47)
where the magnetic quantum number m = -£, - £+1,. . ., [-11, the corresponding

eigenvalues for L, are
L, = mi: (48)
where m = -€, - €+1,..., €-1,€.

Thus, since the spherical harmonics are eigenfunctions for the two commuting
operators of L2 and 1,, they can be used to find the corresponding angular momentum
eigenvalues. The spherical harmonics were in terms of the two coordinates 9 and d), 50
only two quantum numbers were seen in the eigenvalues. The eigenvalue of the operator
of L2 was in terms of the angular momentum quantum number If, and the eigenvalue of
the operator for L, was in terms of the magnetic quantum number m. These two quantum

numbers are related in that an angular momentum vector has a value for E, and for each
value of f, the quantum number m has the values -€,. . .,O,. . . ,+€. Therefore, in the absence

of an electric or magnetic field, there is a degeneracy of 2! + 1 orientations for the

angular momentum vector.2
Deriving the Zeeman Energy From A Hydrogen-Like Atom (S = ‘/2)
As seen in Equation (45) of the rigid rotor, the square of the magnitude of the
angular momentum is found by operating on the hydrogenlike wavefimction with the

operator of L2. Since the wavefunction is

24

R(r)Yr"' ((9, 4‘)
and the operator of L2 only operates on 9 and (b, the eigenvalue of L2 was found and

given in Equation (46). Thus, for a hydrogenlike atom, the only values for angular

momentum are

L = e(e +1)h (49)
where I? = 0,1,2,. . .,etc. It was also shown in Equation (48) fiom the rigid rotor that the 2-

component of angular momentum L, is in terms of the magnetic quantum number m,
which describes the orientation of the angular momentum vector. In the absence of a
magnetic field and not including the spin of the electron (which is also a kind of angular
momentum), the energy of the hydrogenlike atom is independent of m. However, in the

presence of a magnetic field, the energy does depend on m.

For each value of (3, there are 26 + 1 different values of m, each corresponding to a

different value of energy. In the presence of a magnetic field the 2K + 1 degeneracy with
respect to m is removed, because when an atom has angular momentum L, the atom acts
like a small magnet. In other words, the atom has a magnetic dipole moment )1, given by
it = 72L (50)
where the gyromagnetic ratio 7, is defined as
72 = -e/(2me) (51)
where me is the mass of the electron and e is the charge of the electron. The z-component

of the dipole moment is
112 = -[e/(2me)] 12. (52)

Substituting the value of L, from Equation (48), Equation (52) becomes

112 = {eh/(21112)] m = 413m (53)

25

where the Bohr magneton 113 (also known as B.) is defined as

#3 = eh/(Zme) (54)

and is the natural unit of magnetic dipole moment for electronic states.2
When a magnetic dipole is placed in a magnetic field oriented along a given

direction, the potential energy is given by

E = 41 e B (55)
where B is the magnetic flux density. If the z direction is chosen to be along B so that
B, = B, then the energy becomes

E = ~112B = [eB/(2m2)1 L2. (56)

The harniltonian operator for a hydrogenlike atom in a magnetic field is found by adding
the potential energy of Equation (56) to the harniltonian in the absence of the magnetic

field:

H2H0+ CB 1“,. (57)

Z

 

2m

C

When the harniltonian of Equation (57) is applied to the eigenfirnctions of a hydrogenlike
atom (the eigenfunctions of the harniltonian in the absence of a magnetic field), it is
found that these functions are also eigenfunctions of the above complete harniltonian with

the following eigenvalues:

m e"z2
= - ° + mB 58
"l" 2(47re:,)2n2h2 ’18 ( )

where z is the atomic number of the atom, 80 is the permittivity of vaccum, the principle

 

quantum number n = 1,2,3,. . .,etc., the angular momentum quantum number I? =

0,1,2,. ..,etc., and the magnetic quantum number m = €,€-1,...,-€. The second term of the

energy in Equation (58) implies that in the presence of a magnetic field, the energy levels

26

have been split into 2! + 1 levels. When spectral lines are split due to a magnetic field, it
is called the Zeeman effect.2
Deriving the Two Spin Energy States For An Electron

It was proposed in 1925 that an electron has an intrinsic angular momentum S.
Since the spin angular momentum of an electron has no analog in classical mechanics,
the spin angular momentum operators cannot be constructed by first writing the classical
hamiltonian. However, the treatment of spin angular momentum is very similar to the
treatment of orbital angular momentum.2

The magnitude of the spin angular momentum S is

J3

S: s(s+1)h=—§—h (59)

where the spin quantum number s (also referred to as the spin) of the electron has the
single value of 1/2. The z-component of the spin angular momentum S, is
S, = msh (60)

where the quantum number m, for the z-component has two possible eigenvalues, +‘/2 and
-‘/2 (also referred to as “spin-up” and “spin-down” respectively). This implies that the
electron spin angular momentum has only two orientations.2

The corresponding spin angular momentum operators of S and S, can be applied
to spin firnctions to yield eigenvalues. However, since the spin eigenfirnctions do not
involve spatial coordinates, the two possible spin functions are represented by or and [3.

Thus, the eigenfunction equations for the spin angular momentum are

Szla)=%(%+l)h2|a)=%h2|a) (61)

27

ézlfl)=%(%+l)h2|fi)=%h2|fl) (62)

Sz|a)=+-;—h|a) (63)

and

. l
S.lfl)=--2—rzlfl). (64)
It should be noted that under the same argument given for the components of the orbital

angular momentum, the components SK and S3, cannot be determined simultaneously with
S, since SK and Sy do not commute with 8,.2
Due to its charge and intrinsic spin angular momentum, an electron has a

magnetic dipole moment 11,, which is proportional to its spin angular momentum 8:

us = {gee/(21112)] S (65)
where the electron g factor ge = 2.002322 for a free electron. The component of the
magnetic moment of the electron in the direction of the applied magnetic field is

112 = -[ gee/(2%)] Sz (66)
where S, is the component of the spin angular momentum in the direction of the field.
Substituting Equation (60) into Equation (66), the magnetic moment in the direction of

the field becomes
112 = -[ geek/(21112)] ms- (67)
From Equation (56) where the potential energy is given as E = -u,B, the energy of the
spin magnetic moment in a magnetic field B is found to be
E = seuamrB. (68)
Since the quantum number m, has the two values +'/2 and -'/2, the electron spin has the

following two energy states in a magnetic field:

28

E = +1682191123 (69)
and
E = -Vzge1taB. (70)
It is the transition between these two levels that is studied in electron spin resonance.2
Figure 2.4 illustrates that as the external magnetic field of the system is increased,
the splitting of the two energy levels (m, = +‘/2, -‘/2) of the unpaired electron increases.
As the microwave radiation is applied at a constant frequency, the resonance condition is
achieved at only one field value at a time. The magnitude of the transition shown in
Figure 2.4 is the energy that must be absorbed from the oscillating magnetic field B to
move from the lower to the upper state.3
The Nuclear Zeeman Interaction
Just as an electron has a spin associated with it, the nucleus of an atom also has a
spin associated with it due to its protons and neutrons. Thenuclear spin interacts with the
magnetic field for the same reasons that the electron spin does, and this interaction is the

nuclear Zeeman effect. The nuclear spin harniltonian is

i1=-g,,6,Bi, (71)
where gn is the nuclear g factor, B, is the nuclear magneton, B is the applied magnetic

field, and the operator of I, is the nuclear spin operator. The nuclear Zeeman energy term
is opposite in sign and smaller in magnitude than the electron Zeeman energy term.
For a nucleus, the total spin angular momentum is I, the spin quantum number is

I, and the component of nuclear spin in the z direction is 1,. Just like S for electrons, I is
an angular momentum, and thus the eigenvalue of I2 is given by 10 + 1)h2. It follows that

the magnitude of I is

29

E ms=ill2 AE=hv=geBeB

 

 

Figure 2.4: For an S = ‘/2 system, the two degenerate electronic energy levels split firrther
and filrther apart as the applied magnetic field is increased.

30

|1| = 1(1+ l) h. (72)

Similarly, the eigenvalues of I, are mth, where m; = -1,-I+1,. . .,O,. . .,I-1,1. Therefore, there
are 21 + 1 values of m, each associated with an eigenstate of I2 and 1,.2

As with the case of the electron, the magnetic dipole moment of the nucleus is
proportional to the spin angular momentum, where

Plane/(2121911 (73)

where g, is the nuclear g factor, rnp is the mass of the proton, and e is the elementary
charge. Depending on the sign of the nuclear g factor, 1.1. and I can either be parallel or
antiparallel. If gn is positive, then u and I are parallel, while a negative g, implies the two
are antiparallel.

Since I is in units of h, the basic unit of nuclear dipole moment, which is the
nuclear magneton me (also known as 13,), can be defined as
it»: = eh/(Zmp) (74)
Using Equation (74), the z-component of the nuclear dipole moment is
112 = [are/(2%)] mth = gnunmr. (75)
Equation (55) defines the energy of a magnetic dipole in an external magnetic
field B as -uoB. If the z direction is chosen to be along the field, then
B = 1123 = '8n14leB (76)
where m; = -I,-I+1,. . .,0,...,I-1,I. Thus a nucleus with spin I has 21 + 1 nondegenerate

energy states in a magnetic field. Transitions among these levels are induced by applying

electromagnetic radiation, with the frequency equal to the energy level spacings. The

31

selection rule for these transitions is Amt =0, and the frequency v of the transitions is
given by
v = AE/h = gnunB/h (77)
which is called the Larmor frequency. There is only one Larmor frequency because the
energy levels are equally spaced. Figure 2.5 illustrates the energy levels of a single
proton in a magnetic field, which is the simplest case of I = V2. Unlike the case of the
electron, the proton has the lower energy when it is parallel to the field (m1 = +‘/::), and
has the higher energy when it is antiparallel to the field (m1: .1/2).2
Electron Nuclear Hyperfine Interaction: Isotropic and Anisotropic

The electron nuclear hyperfine interaction is an interaction between the magnetic
moments of the unpaired electron and the nuclei. The electron spin not only interacts
with the applied magnetic field, but also with the weak local magnetic fields arising fiom
the magnetic moments of nearby nuclei, as illustrated in Figure 2.6. The result of this
interaction on an EPR spectrum is hyperfine structure, which is the splitting of individual
resonance lines into components.4 There are two contributions to the hyperfine
interaction. The first is a Fermi contact interaction, which is isotropic, meaning that the
density of the electron is at the nucleus. Since this component of the hyperfine is
isotropic, changing the sample orientation relative to the magnetic field B does not alter
the EPR spectrum. The second contribution to the hyperfine is a dipole-dipole
interaction, which is anisotropic, meaning that the energy level separations are strongly
dependent on the orientation of the molecule in the applied magnetic field.3

If the electron and nuclear dipoles in Figure 2.6 were to behave classically with

the applied magnetic field B in the z direction, then the classical energy of the dipole-

32

trails—

 

 

Figure 2.5: The energy levels of a proton with I = ‘/2 in a magnetic field.

33

 

Figure 2.6: The vector for the electron magnetic dipole moment u, will feel a local field
from the vector for the nuclear magnetic dipole moment 11,. B is the applied
magnetic field and 0 is the angle between the applied field and the axis of the
two magnetic moments. The distance between the two dipoles is r. (The

vector lle represents the state m, = - V2 , and the vector 11,, represents the state
m; = + 1/:z.

34

dipole interaction between them is given by

3c05261 -1
Emu = - Zia—Ii- r3 flay“ = - Blow/In (78)

where 110 is the permeability of vaccum, and 1,1,, and 1.1,, are the components of the
electron and nuclear dipole moments along the direction of the applied magnetic field.
The two dipoles are separated by a distance r, and 6 is the angle between B and the line
joining the two dipoles. BM, the local field at the electron caused by the nucleus, can

either enhance or diminish the externally applied magnetic field depending on the values

1'!- "m

of9 and r3

The electron is not localized at one position in space since it is in an orbital.
Therefore, the interaction energy must be averaged over the electron probability
distribution function. Ifall values of 9 are possible, such as the case of an electron in an
s orbital, then the average local field at each r can be obtained by first averaging c0529

over a sphere by the following multivariable integration:

<a222>=m°°5295mw=1 (79)
f‘L‘ sin611t9d¢ 3

where sin0d9d¢ is the element of surface area on a sphere in spherical polar coordinates.

 

When the result of Equation (79) is substituted into Equation (78), it can be seen that the
local field Bloc... becomes zero. This is the case of the isotropic hyperfine interaction.3

The harniltonian for the electron nuclear hyperfine interaction is

lit = 5 . A . i (80)
where A is the hyperfine coupling matrix. In an axial system, Axx = A” = A, and A,, =

A“. In an isotropic system such as the case of a hydrogen nucleus interacting with an

35

electron, A, = A", and thus A, = Ay = A, Therefore, the harniltonian of Equation (80)

for an isotropic system becomes

~)

A 0
1745,85,) o A y. (81)
0 0 A z
where A is a scalar. Ifthe electron spin is directed along B in the z direction, then

0 X
0

H) “D

D

Ii=A(oos,)(1) iy . (82)
i

2

Therefore, the hamiltonian for the isotropic hyperfine interaction is

151,, = A s, i, (83)
where A is the isotropic hyperfine coupling constant. This constant measures the
magnetic interaction energy between the electron and nucleus, and is often expressed as
A/h in units of frequency (MHz). It can also be expressed in magnetic field units (mT) as
a = A/(g.B.), which is called the hyperfine splitting constant. For the present system of S
= V2 and I = V2, two peaks would be expected in the absorption spectrum because a spin 1
nucleus splits the spectrum into 2n] + 1 hyperfine lines of equal intensity. Figure 2.7
illustrates what the derivative spectrum would look like for a hydrogen nucleus
interacting with an electron.

In the case of isotropic hyperfine, all values of 6 were equally probable in
Equation (78) for the dipolar energy. However, in systems where the molecule is not flee
to tumble such as in solid samples, all values of 0 are not equally probable, and Bloc.)
does not vanish. In such cases, there is anisotropy in the hyperfine caused by the dipolar

interaction between the electron and nucleus.3 In a system with S = V2 and I = V2, the

36

 

.x..-----.-

BresOnance

 

 

Figure 2.7: The derivative spectrum of a nucleus with I = V2 interacting with an electron
in an isotropic system. The constant a is the hyperfine splitting constant, and
the resonant field value Emma, = hv/(gB) i- a/2.

37

derivative spectrum for an anisotropic system is shown in Figure 2.8. The figure
illustrates that unlike an isotropic system, A, ¢ A“.

The classical expression for the dipolar interaction energy between an electron

and nucleus separated by a distance r is

[11511.9 _§LI'rSX-il°r)] (84)
r

3

Edipolar (r) = h r

47r
where r represents the vector joining the magnetic dipoles of the unpaired electron and
the nucleus, illustrated in Figure 2.6. The vectors u. and 1.1.. are the classical electron and

nuclear magnetic moments, and the superscript T is to indicate the transpose. For the
corresponding quantum mechanical system, the magnetic moment vectors in Equation
(84) must be replaced by their operators. The harniltonian can thus be written as

ST 0 I - 3(ST o rXIT or)] (85)

fidipolnr (r)=--:7;-gflegnfln|: 1'3 r5

By expanding the vectors in Equation (85), the dipolar hanriltonian becomes

 

 

 

r‘-3x2 ~ - r -3y2~ ~ r‘--32 ~ .
r5 lex+ r5 Syly+ r5 S21,-
“ o 3 A “ " " 3 A A A A
”dipolar (r)=-:—”'gflegnfln rfy( ny +Sny)--rT( sz+SzIx)' (86)
3yz ~ ~ .. f‘
'r—5( 3' z+szy)
J

 

 

where g is assumed to be isotropic.3 (For an anisotropic g, g, :1: g", where g, and g"
correspond to A, and All respectively.) Since the hamiltonian is applied to an electron in
an orbital, the quantities in brackets must be averaged over the electron spatial

distribution. The following spin harniltonian in matrix notation becomes

38

 

 

Figure 2.8: The derivative spectrum of a nucleus I = V2 interacting with an electron in an
anisotropic system. It can be seen that A, is not equal to A".

39

9a,1,=-f;;-gflcsnfln><1§, S, S]

 

 

I-t)

)

 

 

< > x
£922) . ,(87,

 

 

r
/I\
U)
«Ml fi
v
/"\
DJ
“MIN
v
A
N
"‘ l
U
DJ
N
h)
v
_)
N

which is abbreviated as

rim, = 8‘ . T oi (88)
where T is the dipolar interaction tensor (in units of Hz) that gauges the anisotropic

nuclear hyperfine interaction.5 Incorporating the isotropic hyperfine term into Equation
(88) gives the complete hamiltonian for the hyperfine interaction as
HM“ =STeAeil (89)
where A, the hyperfine parameter (3x3) matrix, is defined as
A = A13 + T (90)
where A is the isotropic hyperfine coupling constant and 13 is the 3x3 unit matrix.3
Orientation Of the Lab Axes With Respect To the Hyperfine Axes
In an anisotropic system, there are designated hyperfine axes that are separate
from the laboratory axes. Figure 2.9 illustrates the axis system in terms of the hyperfine,
and Figure 2.10 illustrates the laboratory axis system in terms of the lab field B. In order
to establish a common axis system, one set of axes must be rotated with respect to the
other. One way to do this is to hold the hyperfine axes fixed and rotate the laboratory
field about the hyperfine axes.

The complete spin harniltonian is given by

F1
U)
N

 

 

‘-.-----.-b----------

A1

Figure 2.9: The axis system in terms of the hyperfine. B is the applied magnetic field and
S is the electron spin in the direction of the applied field.

13“Z

\
.

 

 

0’ '
I I
0’ I
1' '
0’ '
o I
1' I
0' '
0’ '
o I
L
I ,
Q‘ I
s‘ I y
s I
‘s‘ I
\ I
‘s‘ I
s I
‘s I
\‘ .
‘s I
“ .
‘sr

X

Figure 2.10: The axis system in terms of the laboratory field B. The electron is at the
origin interacting with the nearby nucleus.

41

fizflBogOS+S0AOI-flngnB0I (91)
where g is assumed to be isotropic, the applied field B = Bo(l,,ly,1,), and the operator for

the electron spin, which is directed along the field, is S = S,(l,,ly,1,). The coordinates 1,,

ly, and 12 describe the laboratory field orientation in terms of Bo,and are defined as

, = sinGcosd) (92)

ly = sin0sin¢ (93)
and

l, = cosG. (94)

Writing out the matrices in the harniltonian, Equation (91) becomes

_)

A O O

fi:gflB,Sz+Sz(lx,ly,lz 0 AW 0 iy -p,g,B.(l,i,+l,i,+l,i,). (95)
0 0 A i

22 2

Substituting m for the operator of S,, performing the matrix operations, and regrouping
terms yields
fl = gflB.m,h +(mj1A,x - gnflnB.,)l,Ix + (mjzAyy - gnflnB°)ny +

(m,iiA,, - g, 8,13,),i, (96)

Iquuation (96) is divided by h, then using Equation (77) it can be written as

I“! = gleam, +(m,A,, -v,)l,i, +(m,A,, -v,),iy + (m,A,, -v,)l,i, (97)
where the factor of (27:)1 in the electron Zeeman and the hyperfine terms is incorporated
into the constants. For the case of S = V2 and I = V2, a (4x4) matrix can be constructed

with the z-components along the diagonal. In order to solve for the operators of I, and 1,,

the equations

42

 

1, =1, +ii, (98)
and
i =i,-ti, (99)
are used to find that
i, =-;-(i. +11.) (100)
and
i . .
Iy --§(I, -I_). (101)
Using the following two equations,
A 1
I, m, m,)=[I(I+1)-m,(mI +1)]'2' lms mI +1) (102)
and
.. l
1_|m, m,)=[1(1+1)-m,(rnI -1)]2 |m, rnI —1) (103)

the x and y components of the matrix can be calculated. Using only the top block

diagonal of the full (4x4) matrix, the harniltonian can be rewritten as

IA] =8fl3°ms(l) +

 

 

 

 

o A ll
853° +(A71 -Vn)l—z' 8w +(é_£__vn).l_.’£._(__y_y._yn)._y-
2 2 A 2 ‘1 2 2 2 2 2 (104)
I c
“an“ —v.)'—*+(—’-’--v.>—’- 8” _(A22_,,n,12
2 2 2 2 2 2 2 2

where (1) is the unit matrix indicating that the electron Zeeman energy is a diagonal term.
By letting A, = V2(A,,m,-vn), Ay = V2(Ayyms-vn), and A, = V2(A,,rn,-v,.), the (2x2) matrix

in Equation (104) can be simplified to represent the nuclear part. The resulting matrix is

1 Azl, A,1,—iA,ly
'2'A,l,+iA,ly -Azlz '

43

In order to put the nuclear part of the harniltonian in the same axis system as the
electronic part, the above matrix is diagonalized. For the case of a point dipole-point
dipole model, as illustrated in Figure 2.6, with an axial A tensor so that A,, = Ayy =

A, and A,, = A",

 

21(m,)=i%,/(A,ms —V,)2sin26+(A.m, —-v, )2 cos2 0. (105)

From Equation (105), it can be noted that the rotation of the laboratory axes about the

hyperfine axes is independent of the angle (I). This result is useful for measuring

1" "THE-i

hyperfine splittings by using the following similar equation:

 

v(m,,e)=/1* (m,)-/1'(m,)=\/(A,m, —v,)2sin 26+(A,m, --v,)2 cos2 19 . (106)

Data Of Enzyme With Substrate and Product

Ruma Banerjee and Rugmini Padmakumar wanted to “test” the radical pathway
mechanism by proving, and possibly identifying, the involvement of radicals in the
rearrangement reaction catalyzed by mmCoA mutase. They took recombinant mmCoA
mutase from propionibacterium shermanii and purified it 20-fold to near homogeneity in
a highly active form.6 Then they tested various samples using electron paramagnetic
spectroscopy of the enzyme to see if they would get EPR signals or not. What they found
was that for the apoenzyme, the form of the isolated enzyme, and for the holoenzyme,
which is the enzyme reconstituted with the 812 cofactor, no EPR signal was detected.
However, when either the substrate, L-mmCoA, or the product, succinyl-CoA, was added
to the holoenzyme and rapidly frozen in liquid nitrogen, an EPR signal was detected.6

The EPR signal detected for the sample of the mmCoA mutase with the substrate,

L-mmCoA, is shown in Figure 2.11“. In the high field region, the spectrum shows

 

 

 

 

 

Echo Amplitude

 

L l i j 1 I L I I

2600 3000 3400 3800 4200 I

 

Field Strength (G)

Figure 2.11: The EPR spectrum of mmCoA mutase with the substrate L-mmCoA.

45

hyperfine splittings resulting from coupling of the unpaired electron with the cobalt
nucleus, which has a nuclear spin of I = 7/2. The g, and 81 values are typical for that of
the cobalamin radical, where g, = 2.26, and g" = 2.00.6 Therefore, it is certain that there
is cobalamin in the sample. No other features are resolved due to inhomogeneous
broadening of the lineshape, which is a distribution of resonance frequencies over an
unresolved band caused by many resonance frequencies very close together.3

In the attempt to reduce the inhomogeneous broadening of the lineshape, Banerj ee
and Padmakumar substituted the hydrogen atoms of the methyl group on the L-mmCoA
with deuterium atoms. Hydrogen has a nuclear g value of 5.586, while the nuclear g
value for deuterium is only 0.857. The hyperfine coupling constant A, which is a
measure of the magnetic interaction energy between the electron and nucleus, is
proportional to the nuclear g value.3 Therefore, a smaller nuclear g value will be
associated with smaller hyperfine couplings. By this reasoning, a reduction in the
inhomogeneous linewidth should be detected for the deuterated sample if the substrate is
coupled to the unpaired electron. Figure 2.126 shows the spectrum of the holoenzyme
with deuterated substrate. This spectrum is very similar to the spectrum of the
holoenzyme with protonated substrate shown in Figure 2116, the only difference being
that there is an increase in the signal-to-noise in the spectrum with deuterated substrate.

Banerjee and Padmakumar also tested a sample of the holoenzyme with the
product, succinyl-CoA.6 The EPR spectrum for this sample is shown in Figure 2.136.
The spectrum shows features that are identical to those observed in the spectra of the
holoenzyme with both protonated and deuterated substrate, namely the coupling of the

unpaired electron to the cobalt nucleus in the high field region, and inhomogeneous

 

Echo Amplitude

 

2600 3000 3400 3800 4200

Field Strength (G)

Figure 2.12: The EPR spectrum of mmCoA mutase with [CD3]mmCoA.

47

 

Echo Amplitude

 

l 1 1 1 l I L: l L

3000 3200 3400 3600 3800

Field Strength (G)

Figure 2.13: The EPR spectrum of mmCoA mutase with the product succinyl-CoA.

48

broadening of the lineshape.
Principles Of Transition Rates and Power Saturation

For a two level system with a ground state [3 and an excited state or, illustrated in
Figure 2.14, there are two different phenomenon going on. There are the microwave-
induced transitions up and down, and there is also a relaxation rate down. The
microwave-induced rates up and down are always equal, but when the relaxation rate
down is added to the microwave-induced rate down at low power, the downward
transitions become greater than the one upward transition. Physically then, the excited
state is being depopulated faster than it can be populated. A technique is needed to
suppress the relaxation rate down in order to restore the equilibrium of the upward and
downward transition rates. This can be done by increasing the microwave power.
Therefore, at high power, the upward and downward transition rates are restored to
equilibrium, and theoretically then, the populations of both the ground state and the
excited state are equal. This condition is known as power saturation.3 This concept can
be useful for distinguishing multiple species in EPR, if the different species have unique
saturation properties.

For a macroscopic sample in which a resonance is observed, a two level system
for an electron with S = V2 is represented in Figure 2.157. The number of spins in the
lower and upper states is represented by N- and N, respectively, and the total number of
spins is given by

Nrot = N+ + N.. (107)
When a magnetic field is applied to the system, the number of spins in each level changes

due to induced transitions, but the total number of spins Nu, remains unchanged. The

49

 

 

 

—L_._Y__|[3>

(a) At low power:

qu = de
mwu < mwd + R

(b) At high power:

mwu as mwd +K

2 Saturation

Figure 2.14: For a two level system, with ground state [3 and excited state or, the
microwave-induced upward transition rate mwu equals the microwave-
induced downward transition rate mwd at low power.

(a) When the relaxation rate down is added to the microwave-induced rate
down, they become greater than the upward transition rate.

(b) The microwave power can be increased to overcome the relaxation rate
down, leading to a condition called power saturation.

50

N++ mS=+1/2

AB = yhB

 

N_ + mS=-1/2

Figure 2.15: The two level system of an electron with S = 1/2. The transition energy AB =
yhB. The number of spins in the two states are represented by N + and N,.

51

probability per second of inducing a transition of a spin from the lower state to the upper
state can be represented by W-..+, while the reverse transition can be represented by We.»
Thus, the change of population in terms of N- can be written as the following differential
equation:

%=N,W,+ -N,W__,,. (108)

From time-dependent perturbation theory, the formula for the probability per second

(Pa—.6) that an interaction V(t) induces a transition from a state (a) with energy E, to a

state (b) with energy E1, is

8-9

p b = .2711 (h|v|a]2 6(E, — E, —hw) (109)

Since I (a | V | b) |2 = | (b | V | a) I2, it follows that (P._.b) = (Pim). Therefore, W...+=
We..- 5 W, meaning that the probability per second of inducing a transition of a spin from
the lower state to the upper state is the same as that of the upper state to the lower state.

Thus, Equation (108) can be rewritten as

%=W(N, —N,) (110)
By letting
n = N- - N+ (111)
then from addition and subtraction of Equations (107) and (111) the populations of each
of the two levels can be written as
N-=V2(N+n) (112)
and
N+=V2(N—n). (113)

Substituting Equations (112) and (113) into Equation (110) simplifies the equation to

52

92=-2Wn. (114)

dt
The solution to Equation (114) is

n = n(0)e'2w‘ (115)
where n(0) is the value of n at t = 0. This solution suggests that if there is an initial
population difference it will disappear with time due to the induced transitions.7
The rate of absorption of energy, dE/dt, is obtained by finding the number of spins
per second that go fi'om the lower energy level to the higher energy level, minus the

number that drop back down emitting energy. Thus

5%: Nwrtro - N,th=thn (116)

where to is the frequency of the transition. From Equation (116), it can be seen that in
order for a net absorption of energy, there must be a population difference between the
two energy levels.7

If a magnetic field is applied to an unmagnetized sample, then in order for the
system to be in its most stable configuration, the electron magnetic moments prefer to be
aligned antiparallel to the applied magnetic field. This would require that N. be greater
than N+, which corresponds to a net number of transitions from the upper to the lower
energy state. As this occurs, the spins give up energy, implying that there is a heat
transfer to some other system that is accepting the energy being released. This heat
transfer will continue until the relative populations NJN. correspond to the temperature T
of the reservoir which is receiving the energy. The final equilibrium populations N.° and

N+° are given by

53

——:-=e'kT zen .7 (117)
It can be said then that there exists a mechanism for inducing transitions between
N- and N which is due to the coupling of the spins to some other system, namely the
reservoir. If the probability per second that this coupling induces a spin transition from

the lower to the upper energy level is represented by WT, and the reverse transition is

represented by W1, then the rate equation can be written as

%=+N+W~L-N_WT. (118)

It is important to note here that the upward and downward transition probabilities are no
longer equal. As stated earlier, when magnetizing an unmagnetized sample, a net
downward transition is expected. However, since in the steady-state dNJdt is zero, then

from Equation (118) it is found that

N: _ w T
N: w i '
Using the relationships given in Equations (117) and (119), the ratio of WT to W1 is

 

(119)

given by

WT 13,2
m—e (120)

which illustrates that in this case the transition probabilities are not equal.7

In order to understand the reason why the transition probabilities are unequal for
this case, it is important to realize that the reservoir imposes limitations to the transition
probabilities. Not only does the thermal transition require that the spins couple to some
other system, but this system must also be in an energy state that will allow a transition.
Suppose that the reservoir has two energy levels that are separated by a distance equal to

the two levels of the electron system, illustrated in Figure 2.167. Figure 2.16(a)7

54

Electron Reservoir Electron Reservoir

 

(a) (b)

Figure 2.16: The two level systems of both the electron system, labelled 1 and 2, and the
reservoir, labelled a and b. The diagram in (a) represents an allowed
transition, while the diagram in (b) represents a forbidden transition. The
x’s represent initial states. The populations of the two levels in the electron

system are given by N; and N2, and the populations of the two levels in the
reservoir system are given by N. and N5.

55

represents an allowed transition, where the initial state of the electron system is at the
upper level and the initial state of the reservoir is at the lower level. Therefore, as the
downward transition of the electron releases energy to the reservoir (or lattice), the
reservoir can simultaneously absorb the energy to allow an upward transition, which
satisfies conservation of energy. However, if both the electron system and reservoir have
their initial states in the upper level, shown in Figure 2.16(b)7, a simultaneous transition
is forbidden because energy is not conserved. In other words, both systems will be
simultaneously releasing energy. Thus, this argument illustrates that for the case where
the spins couple to some other system (called the reservoir), the rate of transition of the
electron is dependent on the probability that the reservoir will be in a state that will allow
the transition to occur. Due to this constraint, the transition probabilities W._.+ and W+..-
are not equal.7

According to Figure 2.167, the electron states have populations N1 and N2, and the
lattice states have populations N. and Nb. Thus, the number of transitions per second is
found as

#/s = NleWIbaza (121)
where Wn,_.2. is the probability per second of the transition where the electron is initially
in state 1 and the lattice is initially in state b. By equating the rate of the transition shown
in Figure 2.16(a)7 to its reverse transition, the steady-state condition is represented as
Nlewlb-v2a = NZNaw2a-olb- (122)

According to the quantum theory, W1b_.2, = W232", so that in thermal equilibrium,

 

 

Equation (122) becomes
N N
‘= i. (123)
N2 Nb

56

That implies that at thermal equilibrium, the electron levels will have the same relative
populations as those of the lattice (the two populations will be in thermal equilibrium). It
is also possible now to solve for WT and W1 (the probabilities per second that coupling
of the electron spins to the lattice will induce upward or downward transitions):

WT = N,W2,_.u, (124)

and

Wl = walh-iza = wa2a-olb (125)

where it can be seen that WT and WJ are unequal.7
Referring back to Equation (118), N. and N2 can be substituted by Equations

(112) and (113)to give

i—‘t‘z N(w 9 -w T)-n(w l +w T) (126)

Equation (126) can be rewritten as

 

9’1 = “° ' n (127)
dt Tl
where
w t -w T
no : N —— I
[W Jr +W T] ( 28)
and
l
-_E—=N(wt+w T) (129)
l
The solution of the differential equation given in Equation (127) is
n=no+Ae17 (130)

57

where A is a constant of integration. In the above equations, 110 represents the thermal
equilibrium population difference, and T1 is the characteristic time associated with the
approach to thermal equilibrium, called the “spin-lattice relaxation time”. In other words,
T1 characterizes the time needed to magnetize an unmagnetized sample (where the spins
try to achieve the more stable configuration of aligning with the field). For instance, if a
sample is initially unmagnetized, the magnetization process is described by the following

exponential rise to equilibrium:

1

n=n°+(1—eT‘ (131)
which physically describes a relaxation process.

If the two rate equations for det given in Equations (114) and (127) are

combined, the complete rate equation becomes

(in n - n
—— = —2Wn + ° . 132

The first term in Equation (132) represents the transitions induced by the applied

 

alternating field, while the second term represents the transitions due to thermal

processes. From the steady state condition of Equation (132), it is found that

no
n = —.
1 + 2WT,
From Equation (133) it can be seen that if 2WT;<1, then 11 3 no. That would imply that

(133)
the absorption of energy from the alternating field would not affect the populations much

from their thermal equilibrium values.7

The rate of absorption of energy dE/dt is given by

58

flufznhww = noha) (134)

1+ 2WT '

From Equation (134), it can be seen lthat the power absorbed can be increased by the

electron by increasing the amplitude of the alternating field, as long as 2WT1«1. If,

however, the transition rate W becomes large enough so that W ~ V2 T1, then the power

absorbed levels off even if W is increased. This effect is known as “power saturation”.

In general, systems with long T1 values saturate faster than those with short T; values.7
Power and Temperature Dependence Study

Banerjee and Pamakumar performed a microwave power dependence study on the
sample of the mmCoA mutase with L-mmCoA at both 10K, shown in Figure 2.176, and at
25K, shown in Figure 2186, where the power ranged from 100 to 0.05 mW.6 At 10K, the
g value shifted fiom 2.14 to 2.11. At 25K, there was only one g value at 2.11. One way
to interpret these spectra is by the concept of power saturation.

Looking at the power dependence spectrum at 10K in Figure 2.17", Banerjee and
Padmakumar believed that there were contributions from two different species with
different relaxation properties, one being a fast relaxing species at g = 2.14, and the other
being a slow relaxing species at g = 2.11.6 The species with fast relaxing properties
would be at the higher power, 100 mW, because it would require more power to drive
one-half the population to the excited energy level. The species with slow relaxing
properties would therefore be at the lower power, 0.05 mW. In comparing the two
spectra at 10K and 25K shown in Figures 2.176 and 2.186 respectively, it could be seen

that by increasing the temperature by 15K, the fast relaxing species was lost. Thus, from

their power and temperature dependence study, Banerjee and Padmakumar believed they

59

Echo Amplitude

 

 

 

 

 

l I I d l l l L I l
2600 3000 3400 3800 4200
Field Strength (G)

Figure 2.17: The EPR spectrum of the power dependence study of mmCoA mutase with
L-mmCoA at 10 K. There is a shift in the g value as the power is varied
from 100 mW to 0.05 mW. At 100 mW, the g value is 2.14. At .05 mW,

the g value is 2.11.

 

 

Echo Amplitude

 

 

L 1 1 1 1 1 i r 1 1

2600 3000 3400 3800 4200

 

Field Strength (G)

Figure 2.18: The EPR spectrum of the power dependence study of mmCoA mutase with
L—mmCoA at 25 K. As the power is varied fi'om 100 mW to 0.05 mW,
there is no shift in the g value. The g value of the lineshape is 2.11.

61

had distinguished two radicals which had different relaxation properties. Based on their
EPR studies, Banerjee and Padmakumar concluded that either there were two radical

pairs in the sample, or they had detected two different states of the same radical pair.6

62

fl

2.

References

. Tipler, Paul A. Physics. Worth Publishers: New York, 1991, pp.882-883.

Alberty, Robert A., and Robert J. Silbey. Physical Chemistry. John Wiley and Sons,
Inc.: New York,'1992, pp. 325-331, 350-355.

. Weil, J. A., J. R. Bolton, and J. E. Wertz. Electron Paramagnetic Resonance. John
Wiley & Sons, Inc.: New York, 1994.

Atkins, Peter. Physical Chemistry. W. H. Freeman and Co.: New York, 1994, pp.
553-554.

. Drago, R., Physicau Methods For Chemists. Saunders College Publishers: Fort Worth,
1992.

Banerjee, R., and R. Padmakumar, The Journal Of Biological Chemistry, April 21,
1995, Vol. 270, No. 16, pp. 9295-9300.

Schlichter, C. P. Principles Of Magnetic Resonance. Springer-Verlag: Berlin: 1990,
pp. 4-9. .

63

Chapter 3: Electron Spin Echo Envelope Modulation

Banerjee and Padmakumar's EPR spectra for the samples containing mmCoA
mutase with L-mmCoA showed inhomogeneous broadening of the lineshape.l
Inhomogeneous line broadening can be caused by an inhomogeneous external magnetic
field, by anisotropic interactions in randomly oriented systems, meaning a distribution of
local magnetic fields, or by unresolved hyperfine structure, where many lines are very
close together. Unresolved hyperfine structure occurs when the number of hyperfine
components from nearby nuclei are so large that no structure is observed. What is
detected then is a broad peak, or an "envelope", of a multitude of lines.2 A technique
other than continuous wave EPR spectroscopy must be used to measure hyperfine
couplings.

The conventional EPR spectroscopy is the continuous ane technique, where
monochromatic, continuous microwave radiation is applied to the spin system. A
variation of this technique is a method by which a sequence of short microwave pulses
are applied to the system. This technique is a method of pulsed EPR spectroscopy known
as electron spin echo envelope modulation, or ESEEM. The main advantage of this
technique is that it allows for the measurement of small electron-nuclear hyperfine
couplings that are often masked by inhomogeneous broadening of the EPR lineshape in
continuous wave spectra. The two most common sequences of pulses that can be used

are a two pulse scheme and a three pulse scheme.3

Two Pulse ESEEM

In a two-pulse ESEEM experiment, a 90° microwave pulse is applied to the
sample, as shown in Figure 3.1. After a time t, a 180° microwave pulse is applied to the
sample. After time 1: again, an electron spin echo results. When the time between the
two pulses in the sequence, t, is increased, the spin echo decays exponentially. This
decay is determined by the spin-spin relaxation time. By plotting the integrated intensity
of these echoes as a firnction of the time between the two pulses t, the electron spin echo
decay envelope can be measured. The result is an overall decay of the electron spin
magnetization, which usually shows modulations caused by weak interactions with nuclei
surrounding the paramagnetic centers3, as illustrated in Figure 3.23 .

It is much easier to understand the production of the spin echo fiom the classical
point of view, which is in terms of the bulk magnetization vector M. In the classical
picture, shown in Figure 3.33, initially the bulk magnetization of the sample is directed
along the z-axis (Figure 3.3a3). The laboratory magnetic field, B0, is also directed along
the z-axis. The magnetic field associated with the microwave pulses, B1, is along the
y-axis, perpendicular to the lab field. The first 90° microwave pulse, applied along the
y-axis, rotates the bulk magnetization 900 until it is aligned with the x-axis. During the
time 1 between the first and second pulses, each individual spin packet within the bulk
magnetization vector begins to precess at different angular frequencies because each spin
packet experiences different local magnetic environments. The 180° pulse torques each
spin packet magnetization vector through a 180° angle about the y-axis. This allows
refocusing of the spin packets since some of the packets are rotating with larger angular

frequencies than others. After a time t, the individual magnetization vectors are in phase

65

90O 180° Echo

 

Figure 3.1: The microwave pulse sequence of a two—pulse ESEEM experiment.

Echo Amplitude

 

 

0 l 2 3 4
T (usec)

Figure 3.2: The modulated electron spin echo decay plot of Ni(III)(CN)4(I-I2O)2'
complex.

67

 

 

(a)

C2190

 

 

N

 

 

(c) Free precession period (t) (d) After 1800 pulse

(e) Refocusing of magnetization (21:)
to form spin echo at Zr

 

Figure 3.3: The classical picture of a two-pulse ESEEM experiment. Bo is the lab field,
BI is the magnetic field associated with the microwave pulses, M is the bulk

magnetization, and on is the microwave pulse frequency.

68

along the x-axis, and a primary echo is observed at time 21: after the beginning of the
pulse sequence.3’4’5
Two-Pulse ESEEM For A Spin System With S = V2 and I = V2
For a system with one nucleus with nuclear spin I = V2 coupled to an electron spin
S = V2, the hamiltonian for the case of an isotropic electron g matrix and an axial

hyperfine interaction is

A

H

—-=ru,s, +Anézi, +A,,8,ix -—w,i (135)
h

2.

For spin systems with nuclear spin I 21, there would be a fifth term due to a nuclear
quadrupole interaction, which is more complicated and will thus be discussed later. For
this more simpler case, the first term of the harniltonian given in Equation (135), which is
the electron Zeeman term, describes the interaction of the electron spin with the external
magnetic field B0. The second and third terms of Equation (135) are the electron-nuclear
hyperfine interaction terms, where A,, = A = Ancosze + A,sin20, and A,, = B =
(A”~ ,)cos0sin0. The angle 9 is the angle between the principal axis of the hyperfine
tensor and the laboratory field B0. A" and A, represent the principal values of the axially
symmetric hyperfine tensor, which can be described using two different types of
coupling. The first is a Fermi contact coupling, denoted Aim, as it is isotropic and at the
nucleus, and the second is a dipole-dipole coupling, D = ggnBBn/rs, which depends on the
orientation of the nucleus with respect to the unpaired electron. Using these two coupling
terms, A,l and A, can be defined as

A, = A2,, + 2D (136)
and

A, = Ate - D.3 (137)

69

The fourth term in Equation (135) is the nuclear Zeeman term. The nuclear
Zeeman term describes the interaction of the nuclear spin with the external magnetic field
Bo.

The harniltonian matrix, which is constructed in an uncoupled basis set consisting
of electron and nuclear spin product states Ims,m1>, can be diagonalized for each of the
two electron spin manifolds to give the eigenvalues and eigenvectors of Equation (135).
The only term in the operator that gives rise to off-diagonal elements is the Ix term. The
results can be summarized using an energy level diagram, as shown in Figure 3.43. For
the EPR transitions marked lul and Ivl in Figure 3.43, the normalized probability

amplitudes are

<2|§|3> _ sin[(¢“ ’ Ad] (138)

 

 

WEE-3' ‘T‘
and
1S3 _
M = _<__>_ : cos[M]. (139)
0.5gflBl 2

The angles on and M3 define the axes of quantization for the or and B spin manifolds.
They are defined as sin¢a = B/Zcoa and sindm = B/2w3.3

When combining these quantum mechanical results summarized in Figure 3.43
with the classical picture of echo formation shown in Figure 3.33, the origin of ESEEM
can be understood as the semiclassical picture shown in Figure 3.53. In Figure 3.53, the
microwave frequency coo is equal to a), of Figure 3.43. The focus is on the response of the

packet of spins that makes a transition from l3> to |2> after the first 900 pulse. After the

70

 

|1>

 

|2>

l3>

 

|4>

V2, I = V2 spin system. The frequencies

 

ESEEM experiment for an S

Figure 3.4: The energy level diagram describing the quantum mechanical results of an
(no. and (up are

 

71

 

 

 

(a) (b)

 

 

Figure 3.5: The combination of quantum mechanical results with classical results of an
ESEEM experiment.

first precession period t, the spin packet falls behind the precessional frequency of the
frame, so that it develops a phase ((023 - coo)t with respect to the x-axis. After the 1800
pulse, the spin packet is torqued 1800 about the y-axis, and part of it splits into a smaller
packet that will precess at (1)24, as it has made the transition from |2> to l4>. Since 0324 is
larger than (no, this packet will precess opposite that with the 0123 frequency, and will
interfere with the echo formation at time t after the second pulse. This phenomenon is
known as "branching”. The interference is modulated as the time between the 900 and
1800 pulses is varied with a frequency of |ng - and = (013.3
Two-Pulse ESEEM For A Spin System With S = V2 and I=1

For a system with electron spin S = V2 and nuclear spin I = 1, such as IN, the
energy level diagram becomes more complicated than the S = V2 and I = V2 system shown
in Figure 3.43. Figure 3.66 illustrates the energy level diagram showing the microwave
transitions between the upper and lower electron manifolds. At the beginning of the
two-pulse ESEEM experiment, the electron spin is in state l6>. After the first 900
microwave pulse, the allowed transition |6> to l3>, as well as the semiforbidden
transitions l6> to |2> and l6> to ll> to a lesser degree, are induced. Therefore, after the
first pulse, the wave function for this system represents a superposition of wave functions
corresponding to the three states l1>, |2>, and l3>, but with l3> having the largest
contribution to the sum.‘5

During the time t between the first and second pulses, the states l1>, |2>, and l3>

(-iE1‘l'./h) e(oiE2‘t/n) and e(~iE3'c/A)

evolve with the phase factors e respectively. The second 180°

microwave pulse induces the allowed transitions |1> to l4>, |2> to |5>, and l3> to l6>, as

well as the semiforbidden transitions |l> to |5>, |l> to l6>, |2> to l4>, |2> to l6>, l3> to

73

|1>

|2>

l3>
|4>

|5>

 

 

 

+1/2

|6>

 

V2 , I = 1 spin system, showing the three

allowed and six semiforbidden transitions.

Figure 3.6: The energy level diagram for an S

74

l4>, and l3> to |5>. During the time t between the second pulse and the echo, the
additional phase factors WEN”, coma/n), and €456") become part of the wave function.

Thus, the echo is generated mostly by the allowed transitions, but also in part by the
semiforbidden transitions, and changes in echo amplitude are a result of interference
between the components of the wave fiinction.6
Modulations Of Two-Pulse ESEEM

For a two-pulse ESEEM experiment with S = V2 and I = V2, the modulation

function is given by
Emod (r): |u|4 + Ivl4 + |u|2|v|2 [coswar + 2coscoflr - cos(a)a -w,,)t -cos(a)a + a),3 )r]. (140)

F rom Equation (140), it can be seen that modulations of the two-pulse echo amplitude
occur not only at the fundamental hyperfine frequencies, but also at their sum and
difference frequencies, (coo + cop) and (com - cup). The product of the transition
probabilities of the two individual transitions associated with “branching”, lul2 lvlz,
describes the amplitude of the modulations. The product of the transition probabilities
for the “non-branching” spins, lul4 or Ivl“, describes the non-modulated part of the echo
envelope.3

In an ESEEM experiment, what is experimentally observed is the product of the
modulation function and an exponential decay function, describing the loss of
magnetization as a result of spin relaxation. In a two-pulse experiment, spin-spin
relaxation is generally on the order of one usec. This rapid background decay reduces
the frequency resolution in two-pulse experiments.3

When multiple nuclei contribute to the modulation of a single paramagnetic

center, the modulation function becomes the product of each individual modulation

75

function, given by
1=.(2)=v.,,,,y I]:IE;,,(2) (141)
where N is the number of coupled nuclei. From Equation (141), it can be seen how
complex a two-pulse ESEEM experiment can become if just a few nuclei contribute,
because for each nucleus, there will be fiindamental Am; = i1 frequencies and
combination frequencies, and also there will be new frequencies representing
combinations of the frequencies from different nuclei as well.3
Three-Pulse ESEEM
Reduction of resolution and increase in complexity are two problems with
two-pulse ESEEM that may be avoided by using three-pulse ESEEM. In a three-pulse
ESEEM experiment, the microwave pulse sequence is 90°-t-90°-T-90°, as shown in
Figure 3.7. The first 900 pulse transfers the bulk magnetization vector M along the
y-axis, shown in Figure 3.85 . During the first time t, the bulk magnetization dephases
and the individual spin packets precess with their characteristic angular frequencies. The
second 900 pulse rotates the individual magnetization vectors into the xz plane. During
time T, the transverse magnetization decays, meaning that the individual spin packets
relax back to the z-axis. The third 900 pulse restores the transverse magnetization by
transferring the spin packets along the z-axis onto the y-axis. After time I again, the
individual spin packets dephase about the y-axis, such that the tips of their vectors form
the locus of a circle. At time T+2t from the beginning of the experiment a stimulated
echo is formed along the y-axis.5 In a three-pulse sequence, the background decay is
dependent on electron spin-lattice relaxation, T1, which is much longer than spin-spin

relaxation. This allows for better frequency resolution as compared to the two-pulse

76

90° 900 9o0 Echo

 

Figure 3.7: The microwave pulse sequence of a three-pulse ESEEM experiment.

77

 

 

 

 

 

 

 

(g)

Figure 3.8: The classical picture of a three-pulse ESEEM experiment describing the
formation of a stimulated echo.

78

experiment.3
Modulations Of Three-Pulse ESEEM
For a three-pulse ESEEM experiment, the modulation function for an S = V2 and
I = V2 system is

2

 

Emod (T, T): lul4 + [v]4 +

 

 

 

V

 

u 2{coswar + coswflr + 25in 2(w31)608[wp(7 + T)]} +

(051'

2sin2{—§—]cos[wa(r +T)]. (142)
As shown in Equation (142), the modulations observed in three-pulse ESEEM are those
of the firndamental hyperfine frequencies, and not the combination frequencies.3
In a three-pulse ESEEM experiment with multiple nuclei coupled to a single

paramagnetic center, the overall modulation firnction is

E(r,T)= [Egg—m]: 13;,(2, T)+ I]; E;(2,T)+]. (143)
It is evident from Equation (143) that the products are taken between frequencies of the
same electron manifold, and not combinations of the frequencies between the manifolds,
as is the case in two-pulse ESEEM. Therefore, three-pulse ESEEM removes much of the
complexities that arise in two-pulse ESEEM.3
The t-Suppression Effect
It can be seen from Equation (142) that the 1: values chosen in a three-pulse
experiment will affect the amplitudes of the modulations. Therefore, the value oft can
be varied over a range of values to either enhance or suppress the contribution from one
of the electron spin manifolds. This is known as the "t-suppression effect", and is a
useful technique to use for making spectral assignments. The condition for suppression

of a particular nucleus is that the time t be equal to the inverse of the resonant frequency

79

v of the nucleus. Using 1 to suppress a known nucleus in an experimental sample is
useful in analyzing ESEEM data because it isolates the other peaks, which makes it easier
to assign them.3
Fourier Transformation

The resulting signal of an ESEEM experiment is preceded by a fi'ee induction
decay spectrum, or an FID spectrum, such as the one shown in Figure 3.9(a)2. Once the
spectrum is obtained, the resonant fiequencies present in the F11) spectrum must be
recovered in some way. The FID curve is a sum of oscillating functions, and so the
frequencies can be recovered in terms of the harmonic components of the curve. The FID
curve is analysed using a mathematical technique called Fourier transformation. The
signal, S(t), is originally in the time domain. The total F ID curve is the sum over all

possible contributing frequencies, represented by the integral

S(t) = j I(v)e('m>dv. (144)
In Equation (144), I(v) is the intensity of the contribution of the frequency v, and the

exponential part, cam”

, is the signal oscillating with frequency v. In order to convert the
spectrum into the frequency domain, I(v) must be determined, which can be evaluated by

the integral

1(v) = 2re]28(t)e(2“)dt (145)
where “re” specifies real Csolutions. This integration is carried out over a series of
designated fiequencies v on a computer that is a component of the spectrometer.7 When
the F ID signal is transformed by this method, a frequency-domain spectrum results, as
the one shown in Figure 3.9(b)2. The fiequency-domain spectrum is much more useful

for analysis than the time-domain spectrum because it is easier to assign peaks based on

80

FT

 

 

Figure 3.9: (a) The free induction decay (FID) spectrum for an S = V2 , I = V2 spin
system.
(b) The frequency domain spectrum of the FID spectrum shown in 9(a).

81

characteristic resonant frequencies rather than on the time that separates the microwave

pulses causing the electron spin echo.

82

References

. Banerjee, R, and R. Padmakumar, Ihe Journal Of Biological Chemistry. April
21,1995, Vol. 270, No.16, pp. 9295-9300.

. Weil, J. A, J. R. Bolton, and J. E. Wertz. Electron Paramagnetic Resonance. John
Wiley & Sons, Inc.: New York, 1994.

. McCracken, John, Handbook Of Electron Spin Resonance. Volume II: Electron W
Echo Envelope Modulation. C. P. Poole and H. A. Farach.

. Schweiger, Arthur, Angew. Chem. Int. Engl. Vol. 30, 1991, pp. 269-270.

. Doorslaer, Sabine Van, EPR-Group ETH Zurich Home Page, ”Three-Pulse ESEEM",
1996.

. Berliner, Lawrence J ., and Jacques Reuben, Biological Mmetic Resonance: Volume
3. Plenum Press: New York, pp. 231-233.

. Atkins, P. W. Physical Chemistg: Fourth Edition. W. H. Freeman and Company:
New York, 1990, p. 551.

83

Qhapter 4: Nuclear Quadmmle Integctign Qf l"N

Nitrogen-14 has a nuclear spin of one, and nuclei with I 2 1 have nuclear charge
distributions that are non-spherical. Such nuclei are said to possess an electric
quadrupole moment eQ, where e is the unit of electrostatic charge, and Q is a measure of
the deviation of the nuclear charge distribution from spherical symmetry. For a spherical
nucleus, eQ equals zero, illustrated in Figure 4.11. For a positive value of Q, the charge is
oriented along the direction of the principal axis, which is the axis of the nonbonded
electron pair. For a negative value of Q, the charge accumulation is perpendicular to the
principal axis.l

The quadrupole moment is a property of nuclei with nuclear spin greater than or
equal to one, which arises from a non-spherical charge distribution in the nucleus. At the
same time, there is an electron distribution on the molecule from the valence electrons
which creates an electric field gradient q at the nucleus. When the quadrupole moment
and the field gradient interact at the nucleus, the result is what is called a nuclear
quadrupole interaction. Therefore, the nuclear quadrupole interaction is an electrostatic
interaction between the quadrupole moment of a nucleus and the electric field gradient at
the nucleus due to the surrounding electronic charges in an atom or molecule.1

Deriving the Nuclear Quadrupole Interaction Energies For An I = 1 Nucleus

The nuclear quadrupole interaction harniltonian for an I = l nucleus is

lilo =5'q—S°—9—)-[3i§-2+n(ii-i§)] (146)
where e is the fiindamental charge, q is the field gradient, eQ is the electric quadrupole

moment, and the operator of 12 is the 2 component of the nuclear spin angular momentum

   

(a) (b)
I=O,p=0 I=1/2,u=r=0,cQ=0

 

 

 

(c) (d)

121,u¢0,eQ>0 1219u¢096Q<0

Figure 4.1: Different types of nuclei, varying in charge distributions.

85

operator.3 The x and y components of the nuclear spin angular momentum operator, the

operators of Ix and Iy respectively, can be used to define the step operators of 1+ and I. by

the two equations

i, = i, Hi, (147)
and

i_ = ix —ii,. (148)

The x and y components can be defined in terms of these two step operators as

I = I ' (149)
and
i thL). (150)

y 2i
Squaring both sides of Equation (149) and Equation (150) will give

i: :51: +211 +12] (151)
and
. 1 . . . .
j 2 7h: —21,1_ +1?]. (152)
Therefore
(i: -i§)= éfi‘f +i3) (153)
and the harniltonian of Equation (146) becomes
HQ = flgflki: —2+(%)(ii #3)]. (154)

The Schrodinger equations for the operators of 12, 1+, and I. are

.7254...“
_ r.

EJTCA

A

1,1 m,)=m,|1 m1) (155)

 

A

I, I m,)=[1(1+1)—m,(mI +1)]i|1 (mI +1)) (156)

 

and

. l

1_|1 m,)=[1(1+1)-m,(m,-1)]2|1 (ml-1)). (157)
When Equations (155), (156), and (157) are applied to the three nuclear spin states of an I
= l nucleus, the eigenvalues for the three nuclear spin operators can be determined, as

summarized in Table 4.1. When applying the harniltonian to the diagonal terms, the

 

 

results are
(-1|1§1Q|-1)=(332—Q)(1) (158)
<0|90|0> {3220](4) (159)
and
<+llfiol+1)=[esz)(1) (160)

In each of the solutions for the diagonal terms, the 1') term vanishes. This is due to
orthogonality of the eigenfunctions that arise from the step firnction operators, which are
given in Table 4.1. Along with the three diagonal terms, there are two nonzero

off-diagonal terms, which are

 

(—1|HQ|+1>=[62:Q](U) (161)

and

87

Table 4.1: The solutions of Equations (156), (157), and (158) for the three nuclear spin

states m1= -1, 0, and +1.

 

 

 

'1 m1> i2 i+ i-
ll -1> -111 -1> 21’211 0> 011 -2>
l1 0> 011 o> 2‘011 +1> 21’211 -1>
l1 +l> +1I1 +1> Oll +2> Zmll 0>

 

 

 

 

 

88

 

(+ 1|)?!Q |— 1) = [9:22]“) (162)

The completed matrix in terms of MP is

IO) H) I“)

W [620% 2) o o

1-1) 0 (+011) [9139111)-
111) 0 [1239111) [+0111

Substituting x for (equ/4), the matrix can be written as

 

-2x 0 O
O x xn .
0 x77 x

This matrix will be used to solve for the energies E4, E0, and E41. The "characteristic
equation”4 of this matrix is
-2x-l 0 O
0 A-l A' =0. (163)

0 A' A - ,1
From the above characteristic equation, -2x-7t = 0, and solving for A yields E0 =

-(e2qQ/2). The other two energies, E1 and E1, are found by solving the remaining

determinant within the original characteristic equation, which is

A' A - A.
The two energies are found by solving for A fiom the above determinant. There are two

A-l A'
I [-6 (1641

solutions to the determinant, the first is A. = (equ/4)(1-n), and the second is

89

A. = (equ/4)(l+n). Therefore, the two energies for the m1 = +1 and m; = -1 nuclear states
are E11 = (equ/4)(l in).
Resonant Frequencies Of the Nuclear Quadrupole Interaction

Now that the three energies for the nuclear quadrupole interaction of 14N have
been determined, it is possible to solve for the corresponding resonant frequencies. There
are three possible energy level transitions, E0 —-> E1, E1 —> E“, and E0 —> E).
Calculating these changes in energies will yield the three characteristic resonant
fi'equencies of the nuclear quadrupole interaction.

The first energy transition is Eo —+ EH, which is given by the equation

 

 

15,, —1~:0 = (6220][(1+77)—(‘ 2)]= [6ng

The second transition is E; -—> E+1,

)(3+n)- (165)

 

13,, ‘E-1 = [62%le +n)-(1-n)]= [6ng

The third and final transition is El —2 E0,

J07) (166)

 

c2 Q e2
13-1—15.1 =[ 2 ][(1—n)—(-2)]=[—29](3—n) ‘ (167)
The three changes in energies given in Equations (165), (166), and (167) correspond to
the three characteristic frequencies of the nuclear quadrupole interaction of 1“N. The

three fiequencies are given by

 

 

V+ : [e 2Q](3+n) (168)
v- =[°“2Q](3-n) (169)

and

v0 = [ii—QM (170)
where (ezQ/4) = 1.2071110*5 (cmzC MHz)/J.

Solving For the Field Gradient q and the Asymmetry Parameter 11
Using the nuclear quadrupole resonant frequencies for 14N, v4, 12., and v0, it is
possible to determine the value of q, the constant describing the field gradient. The
equation for V+ is given in Equation (168), and that of v- is given in Equation (169).

Subtracting v- from V.» gives

 

4
which is v0. The electric quadrupole moment eQ for 1“N is 2.0x10'26 cm2,5 and the

(V.-v-)=[equ](2n) (m1

fundamental charge e is 1.6x10"9 0’ Thus (e’Q)/4 is 8.0x10' 4? (e1112 C), which 111 terms
of MHz becomes 1.207x10'6 (cm2 C MI-Iz)/J. Substituting this value into Equation (171)

and solving for qr] gives

cm = Zvo/(ezQ). (172)

Going back to v+, Equation (168) can be rewritten as

2 2
[3.621x10'6[cm gl'flh]]q+[1.207x1o‘[°m Cl“ H6117)“. (173)

by distribution and substitution. From this equation, the field gradient constant q can be

 

 

calculated. Once q has been determined, it can be substituted into Equation (172) to find

the asymmetry parameter 1').

91

Deriving the Energy Level Diagram For An S = V2 and I = 1 System

Knowing the three energies of a nucleus with spin I = 1, the energy level diagram
can be derived for a nucleus such as 1“N coupled to an electron. There are two possible
spin orientations for the unpaired electron, spin up or spin down, are = +V2 and m5 = -V2
respectively. This degenerate electron spin state is first split due to the electron Zeeman
interaction energy, given by

AB = -8eBeB (174)

shown in Figure 4.2(a).‘5'3 For the case where the external magnetic field B is 3060 G, AB
is 5.67S7x10'27 U, or 8.5658x103 MHz.

Each of the electron spin states is firrther split by the nuclear Zeeman interaction.3
With a nuclear spin I = 1, each electron energy level is split into three levels because
there are three nuclear spin states, m1 = +1, m1 = 0, and m; = -1. The nuclear Zeeman
energy is related to the nuclear Larmor frequency by the relationship

21113an = v.1.‘5 (175)

At 3060 G, vn is 0.94 MHz. Therefore, at the upper electron manifold, the three nuclear
Zeeman levels will be m = -1,0, and +1 respectively, and the levels will be split by 0.94
MHz, as shown in Figure 4.2(b). Similarly, at the lower electron manifold, the three
nuclear Zeeman levels will be m = -1,0, and +1 respectively, and again each level will be
split 0.94 MHz apart, shown in Figure 4.2(b).

The nuclear Zeeman energy levels are shifted due to the electron-nuclear
hyperfine interaction energy, given by Am;m,,.6'3 Thus, at the upper electron manifold,
m1 = -1 level is decreased by A/2, A being the hyperfine coupling constant of 14N, and the

m1 = +1 level is increased by A/2, as shown in Figure 4.2 (c). At the lower electron

92

 

 

 

(a) (b) (c) (d)

Figure 4.2: The energy splitting diagram of 14N, showing splittings from the (a) electron
Zeeman energy [-g,B.Bm,], (b) nuclear Zeeman interaction [-gnBuB/h =

-vn], (c) the electron-nuclear hyperfine interaction [Amunx], and ((1) nuclear

quadrupole interaction. vn is the Larmor frequency, which is .94 MHz at
3060 G.

93

manifold, the m1 = -1 level is increased by A/2 while the m1 = +1 level is decreased by
A/2. In both electron manifolds, the m1 = 0 level is neither raised nor lowered.

Each of the electron-nuclear hyperfine levels for 14N is shifted due to the nuclear
quadrupole interaction.3 Solving the nuclear quadrupole interaction harniltonian for l4N,

the quadrupole energy level shifis corresponding to m = 0, :1 are

131.1141= -(equ)/2 (176)

5...-.. = (e2qQ/4)(l+n) (177)
and

12,... = (equ/4)(1-n). (178)

In the upper electron manifold in Figure 4.2, the nuclear Zeeman and
electron-nuclear hyperfine terms almost cancel each other, thereby leaving only the
nuclear quadrupole interaction to determine the energy level splitting. This gives rise to
three transitions corresponding to the three sharp lines in the ESEEM spectra with nuclei
with spin I 2 1, where the frequencies of two add to give the third.3

In the second electron manifold in Figure 4.2, the nuclear Zeeman term is almost
doubled by the electron nuclear coupling. This gives rise to a single broad transition peak
at about four times the nuclear Zeeman frequency, leading to a Am; = 2 transition.2

Thus, in a frequency domain ESEEM spectrum of 1“N coupled to an unpaired
electron, one would expect to see three characteristic sharp peaks between zero and about
two MHz, and one broad peak at about four MHz, such as the spectrum shown in Figure

4333

94

NQI Peaks

m
l
Broad Peak
l

0 1 2 3 4 5 6 7

 

 

 

 

Frequency (MHz)

Figure 4.3: The Fourier transformation spectrum of ]“N.

95

References

. Drago, R., Physical Methods For Chemists. Saunders College Publishers: Fort Worth,
1992.

. Weil, J. A., J. R Bolton, and J. E. Wertz. Electron Paramagnetic Resonance. John
Wiley & So , Inc.: New York, 1994, p. 534.

. J iang, Feng, John McCracken, and Jack Peisach, Journal Of the American Chemical
Society, 1990, Vol. 112, No. 25.

. Boas, Mary. Mathematical Methods In th_e PhyM Sciences: Second Edition. John
Wiley and Sons: New York, 1983, p. 414.

. Lide, David a, PhD. CRC Ha_t_l_db00k OfPhysics and Chemistgg: 72“(1 Edition .
CRC Press Inc.: Boca Raton, 1992.

. Weil, J. A., J. R Bolton, and J. E. Wertz. Electron Paramagnetic Resonance. John
Wiley & Sons, Inc.: New York, 1994, pp. 48-50, 463.

96

Chapter 5: ESEEM Of Methylmalonyl-CoA Mutase With L-Methylmalonyl-CoA
With l“N

ESEEM Data Collection
The ESEEM data were collected on a home built spectrometer. A three-pulse
sequence (900-t-900-T-900) was used. Dead time reconstruction was performed prior to
Fourier transformation. Computer simulations of the ESEEM data were performed on a
Sun Sparcstation 2 computer using FORTRAN software, which is based on the density
matrix formalism of Mims.l
Samples Of Methylmalonyl-CoA Mutase
Three-pulse ESEEM experiments were performed on flow samples of the
enzyme methylmalonyl-CoA mutase with the substrate L-methylmalonyl-CoA. The
samples were prepared by Rugmini Padmakumar.2 The sample used was prepared with
all the nitrogens being l“N.
ESEEM Of l“N Samples
The first experiment was a two-pulse echo-detected EPR experiment at a 1 value
fixed at 500 us. The result is known as a "field scan”, because the external magnetic field
was varied as the echo amplitude was monitored. The resulting spectrum is shown in
Figure 5.1. This experiment was done in order to recognize at which field value the
greatest echo amplitude would occur. The greatest field value was about 3060 G, and a
shoulder was present at about 3140 G. These are the two field values at which the
ESEEM experiments were to be performed.

The next experiment was a three-pulse ESEEM experiment at 3060 G. The first I

97

 

Echo Amplitude
1

 

126

2600 3400
Field Strength (G)

Figure 5.1: The field scan of mmCoA mutase with L-mmCoA, with the fiequency at
8.81 GHz and a I value of 500 ns.

98

value was set at 300 ns, while the time between the second and third pulses, T, was
varied. The echo amplitude was detected as a function of T. The resulting spectrum,
shown in Figure 5.2, was an electron spin echo decay envelope, which is an overall decay
of the magnetization that is modulated by the hyperfine interactions. The spectrum gave
a measure of the echo amplitude as a firnction of the time (t + T) in 11sec. Fourier
transforming this time domain decay spectrum gave the frequency domain spectrum
shown in Figure 5.3. The Fourier transformation spectrum showed frequency peaks at
which the nuclear spins present came into resonance at the given magnetic field values,
which was usefirl for identifying the nuclei coupled to the paramagnetic center. The
spectrum in Figure 5.3 showed prominent peaks at 1.95 MHz, 2.51 MHz, and 4.0 MHz at
3060 G. Figure 5.4 shows the Fourier transformation spectrum that was taken at 3140 G.
The spectrum in Figure 5.4 showed peaks at 2.0 MHz, 2.5 MHz, and 3.9 MHz, which
was in very good agreement with the spectrum taken at 3060 G shown in Figure 5.3.
Discussion

The peaks present in the Fourier transformation spectra of mmCoA mutase with
L-mmCoA, shown in Figure 5.3 and Figure 5.4, were characteristic of the peaks for 1“N.
MN has a nuclear spin of one, and nuclei with I 2 1 have a nuclear quadrupole interaction
associated with them. In the two ESEEM spectra, two sharp peaks were in the region
between zero and two MHz, while one broad peak was at about four MHz. These peaks
could have been assigned to ”N, indicating that “N was interacting with the
paramagnetic center. The one peak absent from the nuclear quadrupole interaction peaks
was at too low of a frequency to be detected.

The two sharp peaks at frequencies of about 2.5 MHz and 2.0 MHz were the

99

 

 

 

 

 

 

IiII [III [III IIII IIII IIII IIII IIIT IIII IIII I7
1703 _ l l I l l l l l I l 1
.,
O l

"D 7 .

:3
0': " "1
in b I W
- -1

O
—=-‘ - 4

0
L1.) 6 4
e la
3 -lllllI[IllIleLlllllllllllLllLlLlLlllllLlllJJlelllLi

 

0 1 2 3 4 5 6 7 8 9 10
tau+T(usec)

Figure 5.2: The electron spin echo decay envelope of mmCoA mutase with L-mmCoA
at 3060 G, with the frequency at 8.81 GHz and a I value of 300 ns.

100

 

 

 

 

 

 

 

 

 

220 — A

D —

1:

a .—

.1:

E:

o \ .' '
.1: _
o 1

LL] 11
-17 ’ E14 . . .' . . . . . .

 

0 2 4 6 8 10 ' 12 14 16
Frequency (MHz)

Figure 5.3: The Fourier transformation of the spectrum shown in Figure 5.2 at 3060 G,
with the frequency at 8.81 GHz and a 1: value of 300 ns.

101

 

 

 

 

 

Echo Amplitude

 

L
DJ
1

 

 

 

Frequency (MHz)

Figure 5.4: The Fourier transformation of the spectrum shown in figure 5.2 at 31406,
with the frequency at 8.81 GHz, and a I value of 300 ns.

102

16

characteristic frequencies w and v., defined in Equations (168) and (169) respectively.
These two frequencies were used to solve for the third characteristic peak, v0. From
Equation (171), v0 was calculated to be about 0.5 MHz. Substituting v0 and the
numerical values of the constants into Equation (172), the value of qr] was found to be
2.07 1x105 J/(cm2 C). The result of Equation (172) was then substituted into Equation
(173) to solve for the field gradient constant q, which was 6.21375x105 V/cmz. The value
of q was then substituted back into Equation (172) to find the value of the asymmetry
parameter 11, which was calculated to be 0.333.

Once the values of q and n were known, Equations (176), (177), and (178) could
be solved for the quadrupole energy level shifts, illustrated in Figure 4.2(d). After

substituting the numerical values for the constants, the energy level shifis in MHz were

EmH) = -15000 MHz (179)
Enu=+1 = 0.9975 MHz (180)
and
Em,__1 = 0.5002 MHz. (181)

Therefore, in both electron manifolds, the m1= 0 energy level decreased by 1.5 MHz,
while the m1 = +1 and m1 = -1 energy levels increased by 0.9975 MHz and 0.5002 MHz
respectively, shown in Figure 4.2(d).

Since the broad peak at about 4.0 MHz was near “exact cancellation”, it was used
to estimate the isotropic hyperfine coupling constant, A1”, for MN. As illustrated in the
energy level diagram in Figure 4.2(c), 2A1,o a 4.0 MHz. Therefore, the approximate

experimental value for Age for 1“'N was about 2.0 MHz.

103

From the ESEEM data, it was evident that the paramagnetic center was
interacting with MN. However, there were three different nitrogens that could couple
with the paramagnetic center, as shown in Figure 5.5. The first possibility was the four
pyrrole nitrogens in the corrin ring. The second possibility was the nitrogen of the lower
axial dimethylbenzimidazole group. Lastly, it was also possible that the nitrogen could
have been fi'om a nitrogen-based ligand on the protein of the enzyme. An experiment
was needed in order to determine which nitrogen was interacting with the paramagnetic

center.

104

 

 

 

   
 

\N-based ligand

Protein

Figure 5.5: An illustration of cobalamin showing the three possible nitrogens coupled to
the unpaired electron. The four nitrogens around the cobalt center are the
pyrrole nitrogens. The nitrogen at the lower axial position is the nitrogen

fi'om the dimethylbenzimidazole. The dashed line leads to a nitrogen-based
ligand from the protein of the enzyme.

105

 

References

1. Mac, Michelle. “Advanced Electron Magnetic Resonance Studies Of Nitrogen

Ligation In Photosynthetic Systems”, Dissertation For the Degree Of Ph.D., 1996, p.
12 1 .

2. Banerj ee, R., and R. Padmakumar, Dre Journal Of Biological Chemistry, April 21,
1995, Vol. 270, No. 16, pp. 9295-9300.

106

 

Charger 6: ESEEM Of Methylmalonyl-CoA Mutase With L-Methylmalonyl-CoA With
1’_N_

A sample was made by Banerjee and Padmakumar of mmCoA mutase with
15N-labelled protein and deuterium-labelled substrate. From this sample it could be
determined if the substrate was interacting with the enzyme, because there would be
hyperfine interaction from deuterium in the ESEEM spectra. This sample would also
assist in the assignment of the nitrogen. Ifthe nitrogen being detected in the ESEEM
data were fiom the Bu cofactor, the pyrrole nitrogens or the nitrogen of the
dimethylbenzimidazole, the quadrupole interaction peaks would reappear in the spectra.
If, however, the nitrogen was from a nitrogen-based ligand fi'orn the enzyme, then peaks
characteristic of 1’N would appear on the ESEEM spectra.

ESEEM Data Collection

The ESEEM data were collected on a home built spectrometer. A three-pulse
sequence (90°-t-90°-T-90°) was used. Dead time reconstruction was performed prior to
Fourier transformation. Computer simulations of the ESEEM data were performed on a
Sun Sparcstation 2 computer using FORTRAN software, which is based on the density
matrix formalism of Mims.l

Samples Of Methylmalonyl-CoA Mutase

Three-pulse ESEEM experiments were performed on frozen samples of the
enzyme methylmalonyl-CoA mutase with the substrate L-methylmalonyl-CoA. The
samples were prepared by Ruma Banerjee and Rugmini Padmakumar.2 The sample used

contained methylmalonyl-CoA mutase with 1’N-labelled protein, deuterated

107

L-methylmalonyl-CoA, and D20 buffer. Banerjee and Padmakumar were
able to separate the B12 cofactor from the enzyme, grow bacteria on media that contained
a nitrogen source labelled with 15N, and then reconstitute the protein with the 131 2
cofactor. In other words, the B12 cofactor consisted of 1“N, while the rest of the protein
was labelled with 15N.

ESEEM or "N Sample

The first experiment done on the 15N-labelled sample was a two-pulse
echo-detected EPR experiment. According to the resulting spectrum, shown in Figure
6.1, the two field values of interest were 3060 G and 3220 G. These were the two field
values at which the ESEEM experiments were to be done.

The next experiments were three-pulse ESEEM experiments at 3060 G and 3220
G, and the Fourier transformation spectra are shown in Figure 6.2 and Figure 6.3
respectively. At 3060 G, the major peaks were at 1.35 MI-Iz,’1.90 MHz, 2.50 MHz, 2.90
MHz, 3.90 MHz, and 13.00 MHz. At 3220 G, the major peaks were at 1.40 MHz, 2.05
MHz, 2.60 MHz, 2.90 MHz, 4.10 MHz, and 13.69 MHz. The two spectra were in close
agreement.

The last ESEEM experiment was a three-pulse ESEEM experiment on the
15N-labelled sample at 4000 G. The Fourier transformation of the three-pulse spectrum is
shown in Figure 6.4. The major peaks were 0.60 MHz, 1.78 MHz, 2.90 MHz, and 5.30
MHz.

Discussion
In the spectrum at 3060 G in Figure 6.2, the peak at 13.00 MHz was the hydrogen

larmor frequency, which is the frequency at which hydrogen comes into resonance at

108

l 423

Echo Amplitude

2500 3500
Field Strength (G)

Figure 6.1: The two-pulse echo-detected EPR experiment of lsN-labelled mmCoA

mutase with L-[CD3]mmCoA at a frequency of 9.00 GHz and a ‘1: value of
500 ns.

109

 

127.....-.........f.

 

 

 

Echo Amplitude

 

 

 

 

r
00
l

 

0‘2 4 6 81012141618

Frequency (MHz)

Figure 6.2: The Fourier transformation spectrum of the three-pulse ESEEM spectrum of
”N-labelled mmCoA mutase with L-[CD3]mmCoA at 30606, with the
frequency at 9.00 GHz and a 1 value of 500 ns.

110

 

107 '-

 

 

 

 

 

Echo Amplitude

 

 

I

Q]!
l

)-
)-
y-
)-
)-

 

 

Frequency (MHz)

Figure 6.3: The Fourier transformation spectrum of the three-pulse ESEEM
spectrum of l’N—labelled mmCoA mutase with L-[CD3]mmCoA at
3220 G, with the frequency at 9.00 GHz and a t value of 450 ns.

111

16

 

$1

 

 

 

 

 

Echo Amplitude

 

-5 - . . 1 1
0 2 4 6 8 10 12 14 16

 

 

Frequency (MHz)

Figure 6.4: The Fourier transformation spectrum of the three-pulse ESEEM spectrum of
”N-labelled mmCoA mutase with L-[CD3]mmCoA at 4000 G, with the
fi'equency at 11.64 GHz and a ‘1: value of 760 ns.

112

 

3060 G. Likewise, the peak at 1.9 MHz was the deuterium larmor fi'equency. The peaks
at 1.35 MHz and 2.5 MHz were both about 0.60 MHz away from the deuterium larmor
frequency. When two peaks are equidistant about the larmor frequency, that is an
indication of hyperfine interaction. Therefore, it was possible that there was hyperfine
interaction from deuterium.

0n the other hand, the peak at 1.35 MHz was also the larmor frequency of 15N,
and the peak at 2.5 MHz, which is 1.15 MHz away from the 15N larmor frequency, may
also have been due to hyperfine from 15N. Ifthis was so, then a peak 1.15 MHz to the
left of the 15N larmor frequency should have also existed, which would have been at
about 0.20 MHz. In order to have detected this peak, if it existed, would have required
that the magnetic field be increased. So the question that needed to be answered was if
there was hyperfine interaction from 15N or 2H? 1

When the magnetic field has been varied, a characteristic peak shift results, which

is given by

Av=———(8“'[::AB). (182)
By increasing the magnetic field to 4000 G and estimating where the peaks for 15N and
2H should be, it became possible to determine which nucleus was coupling to the
paramagnetic center. According to Equation (182), varying the field from 3060 G to
4000 G should have shifted the 15N peaks 0.43 MHkaG, and should have shifted the 2H
peaks 0.65 MHz/kG. Applying these characteristic shifts to the peaks at 3060 G, the
peaks at 4000 G were able to be assigned. In Figure 6.4, the peak at 1.78 MHz was the
larmor frequency for l5N at 4000 G, and the peaks at 0.60 MHz and 2.90 MHz were both

1.18 MHz away from the larmor frequency of 15N. This indicated that there was

113

hyperfine interaction from ”N. The small peak at about 2.60 MHz may have been the 2H
larmor fi'equency, but there was no indication of hyperfine interaction from 2H.
Therefore, the 15N hyperfine peaks were assigned to the nitrogen belonging to the protein
of the enzyme.

Using the data from the ”N experiments, a more accurate hyperfine coupling
constant could be found for 1“N. From the ”N experiments, a hyperfine coupling
constant could be measured for 15N, and then scaled back to find the hyperfine coupling
constant for 1"N. This method would yield a more accurate value for the hyperfine
coupling constant of 1“N because it would not be tainted by the nuclear quadrupole
interaction, as would be the case with the 1“N experiments.

Determining the anisotropy coupling constant of 1“N required scaling back from

the ”N data using the equation

[A] {Sign [A] (183)
2 ("11) 81(611) 2 (”N)

where gn(14N)= 0.403762 and 80(15N)= 0.56638.2 Referring to the experimental ESEEM
spectrum with ”N at 3060 G, shown in Figure 6.2, (A/2)(15N) was measured to be about
1.15 MHz. Substituting these values into Equation (183), (A/2)(14N) was found to be
0.81981 MHz, and thus A0411): 1.63962 MHz. The same procedure was performed on
the ESEEM spectrum at 4000 G, where (A/2)(15N)= 1.18 MHz, shown in Figure 6.4. In
this case, (A/2)(14N) was calculated to be 0.8412 MHz, and thus A(14N)= 1.6824 MHz.
Therefore, the anisotropy coupling constant for 1"N was approximately 1.66 MHz i 0.1

MHz.

114

Conclusion

From the ESEEM experiments, some of the hyperfine lines were assigned to the
nitrogen at the lower axial position of the cobalamin radical, the nitrogen at the lower
axial position being a nitrogen-based ligand from the protein. It was concluded from the
low frequency ESEEM experiments that 15N must be at the lower axial position as
illustrated in Figure 6.5, because the dimethylbenzimidazole group is the only ligand that
was able to exchange with a protein ligand from the enzyme. There was no evidence
from the data that the substrate was part of the paramagnetic center giving rise to the
ESEEM, but rather the 2H was from the solvent. From the data, it was strongly believed
that the enzyme was in its “base-off” form, the “base-off” form being when the lower
axial position is occupied by a nitrogen-based ligand, such as a histidyl group, from the

enzyme.4

115

 

 

 

 

 

    

N-based ligand

on Protein

Figure 6.5: An illustration showing cobalamin with the cobalt center coordinated at the
lower axial position to a nitrogen-based ligand from the protein of the

enzyme.

116

 

References

. Mac, Michelle. “Advanced Electron Magnetic Resonance Studies Of Nitrogen
Ligation In Photosynthetic Systems”, Dissertation For the Degree Of Ph.D., 1996, p.
121.

. Banerjee, R., and R. Padmakumar, The Journal Of Biological Chemistry, April 21,
1995, Vol. 270, No.16, pp. 9295-9300.

. Weil, J. A., J. R. Bolton, and J. E. Wertz. Electron Paramagnetic Resonance. John
Wiley & So , Inc.: New York, 1994, p. 534.

. Frasca, Verna, R. Banerjee, W. Dunham, R. Sands, and R. Matthews. Biochemistry,
1988, Vol. 27, p. 8458.

117

 

IIIIIIIIIIIIIIIIIIIIIIIIIIIIII

111111111 1111111111111111111111

1293 0 369 401