ELECTROSTATIC AND QUANTUM
CHEMICAL INVESTIGATION OF THE
PROTON PUMPING MECHANISM OF
CYTOCHROME cOXIDASE
DISSERTATION
submi ed o
he Facul y o Biology, Chemis y and Geoscience
o he Uni e si y o Bay eu h, Ge many
o ob aining he deg ee o
Doc o o Na u al Sciences
p esen ed by
Punnagai Munusami
Bay eu h 2008
Die o liegende A bei wu de in dem Zei aum on Janua 2005
bis Mai 2008 an de Uni e si ¨
a Bay eu h un e de Lei ung on
P o esso G. Ma hias Ullmann e s ell .
Volls ¨
andige Abd uck de on de Fakul ¨
a Biologie, Chemie
und Geowissenscha en de Uni e si ¨
a Bay eu h genehmig en
Disse a ion zu E langung des akademischen G ades Dok o
de Na u wissenscha en (D . e . na .).
E s e P ¨
u e : P o . D . G. Ma hias Ullmann
Zwei e P ¨
u e : P o . D . Holge Dobbek
D i e P ¨
u e : P o . D . And eas Fe y
P ¨
u ungs o si z: P o . D . J¨
u gen Senke
Tag de Ein eichung: 09.06.2008
Kolloquium: 19.09.2008
ELECTROSTATIC AND QUANTUM
CHEMICAL INVESTIGATION OF THE
PROTON PUMPING MECHANISM OF
CYTOCHROME cOXIDASE

SUMMARY
Cy och ome coxidase is a c ucial enzyme in he espi a o y chain. I ca alyzes he educ ion
o oxygen o wa e and u ilizes he ee ene gy o he educ ion eac ion o p o on pumping
ac oss he inne -mi ochond ial memb ane, a p ocess which esul s in a memb ane elec o-
chemical p o on g adien . Fo each oxygen molecule, eigh p o ons a e aken up om he
ma ix o he mi ochond ia. Fou p o ons oge he wi h ou elec ons a e equi ed o educe
oxygen o wa e a he Fea3-CuBbinuclea cen e and ano he ou p o ons a e ansloca ed
ac oss he memb ane. Al hough se e al high esolu ion s uc u es ha e been sol ed o his
enzyme, he molecula mechanism o he p o on pumping and elec on ans e is no unde -
s ood.
Recen s udies on he cy och ome coxidase CuBcen e sugges ed dep o ona ion o he CuB
bound imidazole ing o his idine (His291 in mammalian cy och ome coxidase o His334
in Rhodobac e sphae oides cy och ome coxidase) as a key elemen in he p o on pumping
mechanism [1–6]. The cen al ea u e o his p oposed mechanism is ha he pKa alue o
he imidazole signi ican ly lowe ed depending on he edox s a e o he me als in he binuclea
cen e . The ene ge ic easibili y o his mechanism is es ed in his wo k.
To ind a eliable me hod o calcula e e ec i e cha ges o he pKacalcula ions, he cha ge
dis ibu ion o he ipep ide Ala-Asn-Ala wi h di e en con o ma ions has been analyzed us-
ing he hyb id densi y unc ional me hod (B3LYP) wi h 6-31G* basis se . Popula ion analysis
me hods (such as Mulliken and Na u al Popula ion Analysis) and elec os a ic po en ial (ESP)
me hods (such as CHELPG, MK and RESP) a e used o analyze he cha ge dis ibu ion o he
ipep ide Ala-Asn-Ala. This ex ensi e s udy p o ided a be e unde s anding o each me hod
and he pa ame e s which in luence he pa ial a omic cha ges. The esul s show ha ESP
me hods like CHELPG and MK gi e eliable cha ges when p ope sampling poin s a e used o
he po en ial i .
To comp ehend he ole o he CuBbound his idines in he eac ion mechanism o cy och ome
coxidase, densi y unc ional heo y is used in combina ion wi h con inuum elec os a ics o
calcula e he pKa alues o hese imidazole ings in he aqueous solu ion as well as in he
p o ein. The pKa alues o His334, His333 and H2O molecule a e calcula ed bo h in oxidized
and educed s a e o CuBcen e . The Fini e Di e ence Poisson Bol zmann (FDPB) me hod and
he conduc o -like pola izable con inuum model (C-PCM) a e used o de e mine he sol a ion
ee ene gies in aqueous solu ion.
All possible p o ona ion equilib ium eac ions in he CuBcen e a e s udied o unde s and he
dep o ona ion eac ions o he bound H2O molecule, His333 and His334. In aqueous solu ion,
pKa alues o 15.2, 15.9 and 7.4 we e ob ained o dep o ona ion o His334, His333 and H2O
7
espec i ely. These pKa alues in aqueous solu ion show ha His334 and His333 a e likely
o be p o ona ed a physiological pH.
The p o ein en i onmen shi s he pKa alues o he CuBligands o e en highe alues in he
ange be ween 15 o 60. These pKa alues o CuBligands a e signi ican ly highe compa ed o
aqueous solu ion. The high pKa alues show ha His334 is p o ona ed du ing all s eps o he
ca aly ic cycle and demons a e ha he Fe and Cu ion oxida ion s a es do no lowe he pKa
alues o CuBligands and in ol ed in shi ing he pKa alues o CuBligands o highe alues.
These esul s a e incompa ible wi h he p oposed ole o His334 as a key elemen in he
pumping mechanism. Acco ding o he pKa alues, he p o on pumping model as sugges ed
by S ucheb ukho [1] migh no be possible wi h he in ol emen o His334. The pKa alues
o he His333 in he CuBcen e a e always shi ed o highe alues bo h in he educed and in
he oxidized s a e o he CuBcen e . The pKa alues o His333 show ha his esidue is likely
o be p o ona ed in he p o ein and an in ol emen in he eac ion mechanism o cy och ome c
oxidase can he e o e be uled ou .
ZUSAMMENFASSUNG
Cy och om cOxidase is ein wich iges Enzym in de A mungske e. Es ka alysie die Re-
duk ion on Saue s o zu Wasse und nu z die eie Ene gie de Reduk ion, um P o onen
du ch die inne e mi ochond iale Memb an zu pumpen, ein Vo gang, de zu einem elek oche-
mischen P o oneng adien en ¨
ube de Memb an ¨
uh . F¨
u jedes Saue s o molek¨
ul we den
ach P o onen on de mi ochond ialen Ma ix au genommen. Vie P o onen zusammen mi
ie Elek onen sind n¨
o ig, um Saue s o am Binuklea zen um (Heme-Fea3—CuB) zu Wasse
zu eduzie en und wei e e ie P o onen we den du ch die Memb an anspo ie . Obwohl
¨
u dieses Enzym einige hochau gel¨
os en S uk u en bes imm wu den, is de molekula e
Mechanismus des P o onenpumpens und des Elek onen ans e s nich e s anden.
Neue e S udien am CuB-Zen um de Cy och om cOxidase legen nahe, dass die Dep o o-
nie ung eines am CuBgebundenen Imidazol ings ein Schl¨
usselelemen im P o onenpump-
mechanismus da s ell (His291 in de S¨
auge ie Cy och om cOxidase ode His334 in de
Rhodobac e sphae oides Cy och om cOxidase) [1–6]. De zen ale Punk dieses o geschla-
genen Mechanismuses is eine e hebliche Ve schiebung des pKa-We es eines Imidazols im
CuB-Zen um zu nied ige en We en in Abh¨
angigkei om Redoxzus and de Me alle im bi-
nuklea en Zen um. Die ene ge ische M¨
oglichkei dieses Mechanismus wi d in diese A bei
gep ¨
u .
Um eine e l¨
assliche Me hode ¨
u die Bes immung e ek i e Ladungen ¨
u die pKa-Be ech-
nung zu inden, wu de die Ladungs e eilung im T ipep ids Ala-Asn-Ala in e schiedenen
Kon o ma ionen mi els eine Hyb iddich e unk ionsme hode (B3LYP) mi 6-31G* als Basis-
sa z analysie . Popula ionsanalyseme hoden (Mulliken-Analyse und Na u al Popula ion Ana-
lysis) und Me hoden, die das elek os a ischen Po en ial (ESP) e wenden (CHELP, MK und
RESP), wu den benu z , um die Ladungs e eilung des T ipep ids Ala-Asn-Ala zu analysie en.
Diese aus ¨
uh lichen Un e suchung de Me hoden zu Be echnung de Pa ialladungen lie e
ein besse es Ve s ¨
andnis jede Me hode und de Pa ame e , die die pa ielle A omladung be-
ein lussen. Die E gebnisse zeigen, dass die ESP-Me hoden, wie CHELP und MK, e l¨
assliche
Ladungen e geben, wenn geeigne e P obenpunk e ¨
u den Po en ial i benu z we den.
Um die Rolle de an CuBgebundenen His idine im Reak ionsmechanismus de Cy och om c
Oxidase zu e s ehen, wu den Dich e unk ions heo ie-Me hoden in Ve bindung mi Kon inu-
umselek os a ik-Rechnungen e wende , um die pKa-We e de Imidazol inge in w¨
ass ige
L¨
osung und auch im P o ein zu be echnen. Die pKa-We e on His334, His333 und des ge-
bundenen Wasse molek¨
uls wu den im oxidie en und eduzie en Zus and des CuBZen ums
be echne . Sol a a ionsene gien in w¨
ass ige L¨
osung wu den mi Hil e on ini e-di e ence-
9
16 CONTENTS
3.1.1 Wa e unc ion based me hods . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.1.2 Po en ial based me hods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.2 S uc u e p epa a ion and Compu a ional Me hods . . . . . . . . . . . . . . . . . 72
3.3 Resul sandDiscussion.................................. 75
3.3.1 Mullikencha ges.................................. 76
3.3.2 NPAcha ges .................................... 76
3.3.3 CHELPGcha ges.................................. 76
3.3.4 MKcha ges..................................... 76
3.3.5 RESPcha ges.................................... 78
3.4 Conclusions......................................... 80
4 P o ona ion and edox po en ials o cy och ome cni i e educ ase 87
4.1 Hemes and he calcium binding si e in cy och ome cni i e educ ase . . . . . . 89
4.2 P epa a ion o he c ys al s uc u e o cy och ome cni i e educ ase . . . . . . . 90
4.2.1 Densi y Func ional calcula ions . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.2.2 Con inuum elec os a ic calcula ions . . . . . . . . . . . . . . . . . . . . . . 91
4.3 Resul sandDiscussion.................................. 93
4.4 Conclusions......................................... 99
5 pKacalcula ions o binuclea cen e o cy och ome coxidase using DFT calcula-
ions 103
5.1 pKacalcula ions...................................... 104
5.2 Compu a ionalMe hods ................................. 105
5.2.1 Densi y Func ional calcula ions . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.2.2 Cha ge i ing.................................... 108
5.2.3 Sol a ion ee ene gy calcula ions . . . . . . . . . . . . . . . . . . . . . . . . 109
5.3 Resul sandDiscussion.................................. 111
5.3.1 pKa alues o he CuBcen e ........................... 112
5.3.2 pKa alues o he heme a3cen e ........................ 120
5.4 Conclusions......................................... 121
6 pKacalcula ions o CuBligands in cy och ome coxidase 125
6.1 P e ious compu a ional wo k on cy och ome coxidase................ 126
6.2 S uc u e p epa a ion and models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.2.1 P epa a ion o X- ay s uc u e o p o ein . . . . . . . . . . . . . . . . . . . . 129
6.2.2 Redoxcen e models................................ 130

CONTENTS 17
6.3 Densi y Func ional calcula ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.4 Elec os a ic calcula ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.4.1 Calcula ion o a e age pKainp o ein...................... 135
6.5 Resul sandDiscussion.................................. 137
6.6 Conclusions......................................... 149
7 Concluding Rema ks and Ou look 151
Bibliog aphy 153
18 CONTENTS
CHAPTER 1
INTRODUCTION
1.1 INTRODUCTION
1.1.1 CYTOCHROME cOXIDASE - A REDOX-DRIVEN MOLECULAR MACHINE
Cy och ome coxidase is he e minal membe o he elec on anspo sys em o mi ochond ia
and many bac e ia. I ca alyzes he educ ion o molecula oxygen o wa e and pumps p o ons
ac oss he memb ane [7, 8]. In his p ocess he memb ane elec ochemical p o on g adien is
gene a ed; he ene gy s o ed by he p o on g adien is subsequen ly u ilized o ATP syn hesis
[9].
Cy och ome coxidase is esponsible o ca alyzing he educ ion o mo e han 95% o he oxy-
gen aken up by ae obically g owing highe o ganisms. Cy och ome coxidase is s uc u ally
classi ied as a membe o he supe amily o heme-coppe con aining e minal oxidases com-
posed o 4 o 13 subuni s whose la ges and mos hyd ophobic subuni s, I, II and III, a e
encoded by mi ochond ial DNA. These subuni s collec i ely ha e 18 hyd ophobic segmen s
o ming memb ane-spanning helices simila o hose o cy och ome b[10]. The p o ein sub-
uni I o mos cy och ome coxidases con ain wo heme amolecules, called heme aand heme
a3and a coppe B cen e (CuB). Heme a3and CuB o m a binuclea cen e whe e molecula
oxygen is educed in o wa e . Elec ons om cy och ome ca e i s ans e ed o he cop-
pe A cen e (CuA), which is loca ed in he subuni II. The coppe A cen e passes hem o
heme a, which ans e s he elec ons o he binuclea cen e , whe e he molecula oxygen is
educed in o wa e (see Eq. (1.1)). The educ ion o oxygen o wa e in he ca aly ic cen e o
cy och ome coxidases gene a es ene gy ha is necessa y o p o on ansloca ion om he
mi ochond ial ma ix.
4 cy c ed + 8 H+
(in)+ O2−→ 4 cy cox + 2 H2O + 4 H+
(ou )(1.1)
1.1.2 STRUCTURE AND FUNCTION OF CYTOCHROME cOXIDASE
C ys alliza ion and X- ay di ac ion analysis o he mi ochond ial and o se e al bac e ial
cyc och ome coxidases ha e se he s age o unde s anding o he complex unc ions o his
enzyme [7].
19
20 In oduc ion
Q
+
+
+
+
ATP syn hase
Complex IV
ADP ATP
In e memb ane
Inne Mi ochond ial
Memb ane
H
H
H
O
2H O
2
Complex III
2
Succina e
Complex II
Succina e Fuma a e
NADH NAD+
H
Complex I
NADH
Cy c
dehyd ogenase dehyd ogenase oxidase
Cy och ome bc1Cy och ome c
space
ma ix
QH2QH2
Fe Fe Cu
Cu
Cu
Cu
Q
QH
Q
2
Q/
Figu e 1.1. The mi ochond ial elec on- anspo chain. The elec ons a e ans e ed
om complex I o complex IV. Elec ons a e ans e ed be ween complexes I and III by he
mobile elec on ca ie coenzyme Q (Q) and om complexes III o VI by he pe iphe al mem-
b ane p o ein cy och ome c(Cy c). Complex II also ans e s elec ons o Q. The pa hways
o elec on ans e ( ed) and p o on pumping (blue) a e indica ed.
CYTOCHROME cOXIDASE IN MITOCHONDRIAL RESPIRATION. The mi ochond ion is he si e o
euka yo ic oxida i e me abolism. In oxida i e phospho yla ion, elec ons a e ans e ed om
NADH (Nico inamide Adenine Dinucleo ide educed o m) o FADH2(Fla in Adenine Di nu-
cleo ide, educed o m) o O2 ia memb ane-bound p o ein complexes. The ee ene gy o
he elec on ans e eac ion is coupled o ATP syn hesis [10]. A schema ic ep esen a ion
o elec on anspo chain is shown in Figu e 1.1. P o ein complexes embedded in he in-
ne mi ochond ial memb ane ca alyze he elec on ans e om NADH o oxygen. These
p o ein complexes a e commonly e e ed as espi a o y chain. Some o he compounds a e
highly mobile (coenzyme Q (Q) and cy och ome c) which shu le elec ons be ween he ans-
memb ane p o ein complexes. The ee ene gy eleased in he edox eac ions is s o ed as
elec ochemical g adien o p o ons ac oss he memb ane. This p o on g adien is u ilized o
gene a e ATP.
2 NADH + 2 H++ O2−→ 2 H2O + 2 NAD+(1.2)
The espi a o y chain in he inne mi ochond ial memb ane is commonly o ganized in o ou
ansmemb ane complexes, namely NADH dehyd ogenase (complex I), succina e dehyd oge-
nase (complex II), cy och ome bc1(complex III) and cy och ome coxidase (complex VI). Two
mobile elec on ca ie s Q and cy och ome cshu le elec ons be ween he ansmemb ane
p o ein complexes.
Complex I passes he elec ons om NADH o Q. The elec on ans e akes place along i on-
sul u (FeS) clus e s. Du ing he elec on ans e , complex I pumps p o ons om he ma ix
o he in e memb ane space.
1.1. In oduc ion 21
Complex II also ans e s elec ons o Q, howe e in he case o complex II, he sou ce o he
elec ons is FADH2p oduced in he ci ic acid cycle. Since he s anda d edox po en ial o
FAD is sligh ly lowe han ha o Q, complex II does no pump p o ons om he ma ix o he
in e memb ane space. The elec ons om complex I and complex II a e shu led o complex
III by educed Q molecules.
Complex III (cy och ome bc1) passes he elec ons o he nex mobile ca ie cy och ome c
which shu les he elec ons o complex IV (cy och ome coxidase). Complex IV passes he
elec ons o dia omic oxygen and he eby educing i in o wo wa e molecules ( inal elec on
accep o ). Bo h cy och ome bc1and cy och ome coxidase ansloca e p o ons om he ma ix
o he in e memb ane space.
THE THREE-DIMENSIONAL STRUCTURE OF HEME-COPPER OXIDASES
Heme-coppe oxidases a e memb ane p o eins ound in he espi a o y chain o ae obic o gan-
isms. They a e he e minal elec on accep o s coupling he ansloca ion o p o ons ac oss
he memb ane wi h he educ ion o oxygen o wa e . The heme-coppe oxidase supe amily
is di ided in o wo main b anches based on he iden i ies o he elec on dona ing subs a es:
cy och ome coxidases use a wa e soluble p o ein, cy och ome c, as elec on dono whe eas
ubiquinol oxidases use a memb ane soluble ubiquinol molecule as hei elec on dono . Bo h
he membe s o he supe amily sha e impo an s uc u al and unc ional ea u es.
STRUCTURE OF THE CYTOCHROME cOXIDASE FROM Pa acoccus deni i icans. The p o ein was
o iginally isola ed as a wo-subuni enzyme complex om he cy oplasmic memb ane o he
soil bac e ium P. deni i icans [11]. Michel and co-wo ke s de e mined he s uc u es o he
educed and oxidized P. deni i icans enzyme and ound no s uc u al di e ence [12]. The ou
subuni s o oxidase has been c ys allized in he p esence o dodecyl mal oside as a complex
wi h a monoclonal an ibody agmen (F ) di ec ed agains an epi ope on he hyd ophilic do-
main o subuni II and i s s uc u e was de e mined a 2.8 ˚
A [12, 13]. La e , he wo-subuni
complex s uc u e was sol ed a 2.7 ˚
A again using he F app oach o inc ease he pola
su aces o he p o ein complex and undecyl mal oside as de e gen [14].
STRUCTURE OF THE CYTOCHROME cOXIDASE FROM Rhodobac e sphae oides. The s uc u e
o cy och ome coxidase om R. sphae oides has been sol ed by Iwa a e al. [15]. The c ys al
s uc u es we e de e mined o he wild ype and a mu an by eplacing he glu ama e esidue
286 o subuni I by glu amine (see Figu e 1.2).
STRUCTURE OF THE CYTOCHROME cOXIDASE FROM BOVINE HEART. The c ys al s uc u e o
bo ine hea cy och ome coxidase a 2.8 ˚
A was de e mined by Yoshikawa e al. [16–18] in
1995. The c ys al s uc u es o he bo ine hea cy och ome coxidase in he ully oxidized
and he ully educed s a es we e de e mined by he same g oup in 1998 [16]. The p o ein is
composed o 13 di e en subuni s.
UBIQUINOL OXIDASE. Ubiquinol oxidases ake pa in he bac e ial elec on anspo chain
oxidizing ubiquinol in o ubiquinone and educing oxygen o wa e . These enzymes a e one

22 In oduc ion
se o he many al e na i e e minal oxidases in he b anched p oka yo ic elec on anspo
chain. The o e all s uc u e o he ubiquinol oxidase is simila o ha o he mammalian
cy och ome coxidase, wi h he addi ion o a pola ubiquinol-binding si e embedded in he
memb ane.
The cy och ome coxidase con ains ou edox cen e s: CuA, heme a, heme a3and CuB. The
oxygen educ ion akes place in he binuclea cen e Fea3-CuBand u ilizes cy och ome cas
an elec on dono . A second majo b anch o he aa3-cy och ome coxidase uses ubiquinol
o menaquinol as he educing subs a e, and in hese cases he CuAcen e is absen . Such
enzymes a e he cy och ome bo3o Esche ichia coli [19, 20] and cy och ome aa3-600 o Bacillus
sub ilis [21–23]. The cy och ome aa3- ype e minal quinol oxidase o B. sub ilis ca alyzes he
ou -elec on educ ion o oxygen in o wa e .
STRUCTURE OF THE ba3-CYTOCHROME cOXIDASE FROM The mus he mophilus. The s uc u e
o he ba3-cy och ome coxidase om T. he mophilus has been epo ed a a esolu ion o 2.4
˚
A [24]. The c ys al s uc u es o ecombinan cy och ome ba3-cy och ome coxidase om T.
he mophilus was epo ed a 2.3 ˚
A by Hunsicke -Wang and co-wo ke s [25]. The model o he
ba3-cy och ome coxidase is composed o h ee p o ein subuni s I, II and IIa. The main pa
o he complex is o med by subuni I wi h 13 ansmemb ane helices, which binds he heme
b, heme as3as well as CuB. The a- ype heme o he ba3-oxidase co esponds o he heme as
p esen in he SoxB- ype e minal oxidases [24]. Heme bis he simples p o oheme con aining
a low-spin i on wi h wo his idine esidues as axial ligands. The binuclea cen e is o med
be ween heme as3and CuBas in he cy och ome coxidase.
STRUCTURE OF THE bo3-CYTOCHROME cOXIDASE FROM Esche ichia coli. The s uc u e o cy-
och ome bo3oxidase om E. coli was epo ed a 3.5 ˚
A by Ab amson e al. [26]. The cy-
och ome bo3ubiquinol oxidase is a ou -subuni heme-coppe oxidase ha ca alyzes he
ou -elec on educ ion o O2 o wa e and unc ions as a p o on pump. All he edox cen e s
a e loca ed in subuni I, wi h a low spin p o oheme bac ing as an elec on dono o a binu-
clea cen e ha is composed o an o- ype heme, heme o3, and a coppe ion CuB. Subuni s o
I, II and III o ubiquinol oxidase a e homologous o he co esponding subuni s in he aa3- ype
cy och ome coxidase, and he ligands o he wo heme g oups and he CuBha e been iden i-
ied as his idine esidues. In con as o he cy och ome coxidase, subuni II o he ubiquinol
oxidase has nei he CuAcen e , no a cy och ome cbinding si e. Ins ead, he heme b ecei es
elec ons di ec ly om a memb ane solubilized ubiquinol molecule.
SUBUNITS OF CYTOCHROME cOXIDASE
The subuni s o cy och ome coxidase om R. sphae oides a e discussed in he ollowing
sec ion.
Subuni I (see Figu e 1.2) o bac e ial cy och ome coxidase is la gely embedded in he mem-
b ane, wi h i s 12 ansmemb ane helices shaped in a h ee-winged p opelle a angemen
[7, 12]. The N- e minus and he long, exposed C- e minus o he polypep ide ace he cy o-
plasmic side. The h ee edox cen e s, he wo a- ype hemes and he coppe B cen e , a e
liga ed by amino acid side chains o his subuni . His idines a e he axial ligands o he low-
spin heme a, whe eas a his idine and a p esumed hyd oxyl o a wa e molecule a e ligands
1.1. In oduc ion 23
2
H O
O2
SU II
SU III
SU I
+
CuA
heme a 3CuB
4H
+
4H 4H
+
Cyc c4
ma ix
memb ane
In e memb ane space
SU IV
heme a
Figu e 1.2. The X- ay s uc u e o cy och ome coxidase om Rhodobac e
sphae oides.The elec ons a e ans e ed om cy och ome c o coppe A cen e . P o-
ons a e pumped om ma ix o in e memb ane space and ou mo e p o ons a e deli e ed
o he binuclea cen e . Oxygen is educed o wa e in he binuclea cen e (heme a3...CuB).
o he high-spin heme a3moie y. Bo h hemes a e o ien ed pe pendicula o he memb ane
plane. Heme a3, oge he wi h a coppe ion (CuB)in i s immedia e icini y, o ms he binuclea
cen e whe e oxygen binding and educ ion akes place.
Subuni II (see Figu e 1.2) o bac e ial cy och ome coxidase has a bipa i e s uc u e. The
N- e minal has wo ansmemb ane helices ollowed by a hyd ophilic, 10-s anded β-ba el
domain ex ending in o he pe iplasm, comp ising he CuAcen e . The CuAcen e con ains wo
coppe ions in a mixed- alence (CuI.CuII)s a e and 2.6 ˚
A apa , gi ing ise o a cha ac e is ic
EPR (elec on pa amagne ic esonance) signal in he oxidized s a e which was obse ed in he
o he enzymes as well [27]. The wo coppe ions a e b idged by wo cys ein hiola es.
Subuni III (see Figu e 1.2) o bac e ial cy och ome coxidase is ully embedded in he mem-
b ane. No edox co ac o s a e associa ed wi h his subuni . When bo h subuni s III and IV
a e emo ed om he P. deni i icans cy och ome coxidase he e is no loss in he ca aly ic
unc ions. I has been sugges ed ha subuni III s abilizes he in eg i y o he binuclea cen-
e in subuni I [28]. The cle o he subuni III may be a binding si e o o he memb ane
p o eins. Subuni III may be in ol ed in assembly o he oxidase o o m he en ance o a
oxygen channel leading o he ac i e si e [29]. I possesses se en ansmemb ane helices ha
a e di ided by a la ge V-shaped cle in o wo bundles, one o med by he i s wo helices, and
he o he by helices III o VII. In his cle , lipid molecules a e ound o be i mly bound o he
24 In oduc ion
conse ed esidues. In mammalian oxidase h ee hyd ophobic channels we e p oposed and
hese hyd ophobic channels a e sugges ed o be he po en ial pa hways o oxygen o each
he binuclea cen e [18]. The oxygen channels s a a he p o ein-memb ane in e ace nea
he cen e o he lipid bilaye , whe e oxygen solubili y is much highe han in he aqueous
phase. One o such channel has also been iden i ied in he bac e ial oxidase [29]. I s a s in
he V-shaped cle o he subuni III di ec ly abo e a igh ly bound lipid molecule and leads
h ough subuni I in o he binuclea si e.
Subuni IV (see Figu e 1.2) o he bac e ial enzyme o bo h P. deni i icans and R. sphae oides
consis s o a single ansmemb ane helix in con ac wi h subuni I and III. The unc ion o
his small subuni is unknown [12].
THE REDOX CENTERS OF CYTOCHROME cOXIDASE
C256
H217
H260
E254
CuA
M263
C252
Figu e 1.3. The coppe A (CuA) cen e o cy och ome coxidase om R. sphae oides.
The coppe A (CuA) cen e wi h i s liga ing esidues is shown. The elec ons a e ans e ed
om cy och ome c o he coppe A cen e . The amino acid numbe ing o R. sphae oides is
used h oughou his hesis.
COPPER A(CuA)CENTER. The CuAcen e is loca ed 8 ˚
A abo e he memb ane su ace. The
CuAcen e con ains wo coppe ions (see Figu e 1.3). These coppe ions a e b idged by wo
cys eins sul u a oms and ha e addi ional p o ein ligands. The wo coppe ions o he CuA
cen e a e coo dina ed by wo His, one Me , a backbone ca bonyl oxygen o a Glu and wo
b idging Cys esidues. The spec oscopic measu emen s indica e ha in he educed o m o
he CuAcen e bo h coppe ions a e in hei Cu(I) s a e whe eas in he ully oxidized o m, he
newly acqui ed elec on appea s o be delocalized be ween he wo coppe ions such ha hey
assume he mixed [Cu1.5+...Cu1.5+]s a e.
1.1. In oduc ion 25
H102
heme a
H421
Figu e 1.4. The heme ao cy och ome coxidase om R. sphae oides.The heme a
po phy in ing sys em is shown wi h wo his idine esidues as axial Fe-ligands. The amino
acid numbe ing o R. sphae oides is used h oughou his hesis.
HEME aCENTER. The heme acen e consis s o wo his idine esidues as axial i on-ligands
(see Figu e 1.4). I ans e s he elec ons om CuA o he binuclea cen e . The heme is
non-co alen ly bound o he p o ein and heme acon ains a o myl g oup and a hyd ophobic
hyd oxye hyl- a nesyl g oup. The heme en i onmen s in P. deni i icans and bo ine hea
cy och ome coxidase a e e y simila .
HEME a3AND COPPER B(CuB)CENTER: THE BINUCLEAR CENTER. The binuclea cen e is
o med by heme a3and CuBwhich is he ca aly ic cen e o O2 educ ion (see Figu e 1.5). The
heme a3i on appea s o be i e old coo dina ed wi h a his idine o subuni I. The coppe ion in
he CuBcen e is coo dina ed by Na oms o His334 and His333 and Nδa om o His284 and
he his idines a e a anged in an equila e al iangle, cen e ed on CuB. Molecula oxygen is
supposed o bind be ween he heme a3i on and CuB. The i on o heme a3is 0.36 ˚
A ou o he
heme plane in P. deni i icans bu almos wi hin he plane in he bo ine hea . C ys allog aphic
s udies on cy och ome coxidase ha e e ealed an unique and unexpec ed pos ansla ional
modi ica ion in he enzyme ac i e si e [16]. A y osine (Y288) which is loca ed e y nea o he
binuclea cen e is co alen ly linked o he No his idine esidue (H284) which also se es
as a ligand o he CuBcen e (see Figu e 1.5). The c oss-linked y osine and his idine was
obse ed in bo h bac e ial and mammalian cy och ome coxidases.
AMINO ACIDS AS REDOX CENTERS. In addi ional o ou me al edox cen e s, he e a e o he
amino acids nea he ac i e si e o he enzyme ha could concei ably o m adicals and ac
as a edox-ac i e cen e s in cy och ome coxidase. The y osine (Y288) is he mos p oxi-
ma e amino acid which is c oss-linked o he CuBhis idine (H284) [30]. The EPR e idence
32 In oduc ion
ASTATE. The p oposed ca aly ic cycle o he molecula s eps aking place in he ac i e si e
o cy och ome coxidase du ing ca alysis is gi en in Figu e 1.8. Oxygen binds o he educed
binuclea cen e and o ming he A s a e (adduc a e oxygen binding).
PSTATE. A s a e leads o he so-called pe oxy s a e P. The P s a e appea s o exis in wo
o ms, Pm, a wo elec on educed and he P s a e. The Pms a e esul ed om he in e nal
elec on edis ibu ion and he y osine is p oposed o be neu al adical (see Figu e 1.8).
Weng and Bake [55] we e he i s o sugges ha he P s a e is no a pe oxy s a e bu
he oxo- e yl s a e wi h hyd oxyl g oup bound o he CuBcen e . They sugges ed ha a
yp ophan is he sou ce o he missing elec on in analogy o cy och ome cpe oxidase. La e
Ki agawa and co-wo ke s [56] p o ided e idence by Raman spec oscopy ha P s a e is a
hyd ogen-bonded oxo- e yl s a e. The s uc u e o he Pms a e shown in Figu e 1.8 has
been p oposed se e al imes [57]. The exis ence o he co alen y osine-his idine c oss-link
should be aken as he e idence ha a y osine adical is o med du ing he ca aly ic cycle
o cy och ome coxidase, because he c oss-linking o y osines is ypical o a adical eac ion
ca alyzed by pe oxidase. The P s a e o he binuclea cen e may be conside ed as a “high-
ene gy” s a e, he decomposi ion o which will d i e p o on ansloca ion [58]. The subsequen
ans o ma ion o he P s a e in o F s a e is coupled o ansloca ion o he p o on ac oss he
memb ane, ne up ake o ano he p o on in o he binuclea cen e and an accompanying
gene a ion o ansmemb ane elec ic po en ial [59].
FSTATE TO OSTATE. The F s a e esul s om he hi d elec on ans e om cy och ome c
oge he wi h he acquisi ion o wo p o ons which con e he y osine adical o phenola e
s a e. The e a e o e whelming spec oscopic da a ha heme a3is in he oxy e yl o m in
F s a e (see Figu e 1.8). Howe e , he e appea s o be mul iple o ms and ambigui y wi h
espec o he edox s a us and he p o ona ion s a e o g oups in he immedia e icini y o
he ac i e si e, depending on how F s a e is gene a ed. A ou h and inal elec on ans e and
p o on acquisi ion yields he oxidized O s a e ia H s a e. The H s a e in e media e possesses
hyd oxyl g oup a he heme a3.
OAND RSTATE. In O s a e, all he ou edox cen e s a e ully oxidized and he heme a3i on
is bound wi h he wa e ligand and CuBcen e is bound wi h hyd oxyl g oup. The y osine is
co alen ly c oss-linked o he CuBligand o His284. The oxidized binuclea complex is educed
o R s a e ia he o ma ion o he one-elec on educed E s a e. P o ons a e ansloca ed om
ma ix du ing his p ocess.
Wiks ¨
om and co-wo ke s wo ked o es ablish much o he concep ual and an expe imen al
amewo k o s udying he oxidase mechanism, including impo an expe imen al e idence
ha p o on pump is coupled only o he P→F and F→O s a e ansi ions and ha each o
hese one elec on edox s eps esul s in he pumping o wo p o ons [8, 60]. This pa adigm
has been gene ally accep ed o he pas decade, hough some p oblems ha e been poin ed ou
and discussed [61]. Now he pa adigm ha pumping is coupled only o he las wo s eps in
he ca aly ic cycle is being challenged by Michel [53] on he basis o a e-e alua ion o se e al
key expe imen s. The conclusion o he e-e alua ion is ha he F →O s a e ansi ion is
coupled o he pumping o only one p o on and no wo. Al hough conside able p og ess has

1.3. Ou line o he hesis 33
been made conce ning he p o on inpu pa hways in cy och ome coxidase, e y li le is known
abou he mechanism and how he p o on pump ac ually wo ks.
PROTON–COUPLED ELECTRON TRANSFER
The cy och ome oxidase ene ge ically couple he elec on ans e eac ions associa ed wi h
educ ion o oxygen o wa e and pumps p o on ac oss he memb ane. E en hough a as
amoun o s uc u al and unc ional in o ma ion o cy och ome coxidase a e a ailable om
expe imen al and heo e ical da a [8, 14, 16, 50–52], ac ual s ep o coupling he he edox
eac ions o he p o on ansloca ion is poo ly unde s ood. How he p o ons a e deli e ed
exac ly o he binuclea cen e , he p o on ansloca ion and exi pa hways and he chemical
in e media es in ol ed in he ca aly ic cycle a e s ill emain unclea .
1.3 OUTLINE OF THE THESIS
The aim o he hesis is o unde s and he p o on pumping mechanism o cy och ome coxi-
dase. Al hough he s uc u es o cy och ome coxidase has been sol ed o se e al o ganism,
he molecula mechanism o p o on pumping emains unclea . In his hesis, he eac ion
mechanism o cy och ome coxidase is analyzed by combining he densi y unc ional heo y
(DFT) and con inuum elec os a ic calcula ions.
The heo y behind he elec os a ic calcula ions, Poisson-Bol zmann equa ion, i a ion be-
ha io calcula ions and DFT a e desc ibed in chap e 2 and di e en cha ge me hods a e
epo ed in chap e 3. P o ona ion p obabili ies and edox po en ials o cy och ome cni i e
educ ase which se es as a simple elec on ans e sys em o s udy cy och ome coxidase
a e discussed in chap e 4.
To in es iga e he eac ion mechanism and he ole o he his idines bound wi h he CuB
cen e and he his idine coo dina ing he heme a3, he pKacalcula ions we e pe o med on
he CuBand heme a3using DFT in combina ion wi h con inuum elec os a ic models. The
pKa alues a e calcula ed using di e en basis se s and di e en sol a ion models. The Fini e
Di e ence Poisson Bol zmann (FDPB) me hod and he conduc o -like pola izable con inuum
model (C-PCM) a e used o de e mine he sol a ion ee ene gies in aqueous solu ion. The
in luence o di e en cha ges, basis se s and sol a ion models on he pKacalcula ions o he
CuBha e been s udied (Chap e 5). The pKa alues o he CuBligands and heme a3cen e in
aqueous solu ion a e epo ed in Chap e 5.
In chap e 6 he a e age pKa alues o he CuBligands in he p esence o di e en heme a3
edox s a es in cy och ome coxidase a e epo ed.
This hesis p o ides insigh s in o he ole o CuBligands in he eac ion mechanism o cy-
och ome coxidase. The mos accu a e me hods a e epo ed o he pKacalcula ions o he
CuBand heme a3cen e . This hesis is hus a s ep owa ds a be e unde s anding o he ole
o he CuBligand in he eac ion mechanism o cy och ome coxidase.
34 In oduc ion
CHAPTER 2
COMPUTATIONAL METHODS
2.1 ACID-BASE AND REDOX REACTIONS EQUILIBRIA
2.1.1 FUNDAMENTAL DESCRIPTION OF ACID-BASE AND REDOX EQUILIBRIA
Biological molecules, such as p o eins and nucleic acids bea nume ous unc ional g oups
such as ca boxyl and amino g oups ha can unde go acid-base eac ions. The dissocia ion
o a p o on om a monop o ic acid is gene ally gi en by:
HA A−+ H+(2.1)
The ee ene gy change (∆Ga) o his eac ion can be ela ed o he equilib ium cons an (Ka),
∆Ga=−RT ln Ka(2.2)
Ka=[H+][A−]
[HA] (2.3)
The pH o he solu ion is de ined as he nega i e decadic loga i hm o he hyd ogen ion con-
cen a ion and he pKao an acid is de ined as he nega i e decadic loga i hm o he Ka
alues.
pH = −log[H+](2.4)
pKa=−logKa(2.5)
The Hende son-Hasselbalch equa ion combines Eq. (2.4) and (2.5) o
pH = pKa+ log [A−]
[HA] (2.6)
The p o ona ion p obabili y is gi en by [62]:
pp o =[HA]
[HA] + [A−]=10pKa−pH
1 + 10pKa−pH (2.7)
35
36 Compu a ional me hods
The pKa alue o an acid is he pH alue a which he concen a ion o he p o ona ed and
dep o ona ed o ms o he acid equal. Simila o he abo e equilib ium p o ona ion eac ion,
he equilib ium be ween he edox eac ion is,
Aox +e−A−
ed (2.8)
The educ ion equilib ium cons an KET o his eac ion is,
KET =[A−
ed]
[Aox][e−](2.9)
Eq. (2.9) is analogous Eq. (2.6). The s anda d edox po en ial o he abo e edox eac ion is
gi en by:
E0=RT
Fln KET (2.10)
The edox po en ial o he solu ion is gi en by,
E=−RT
Fln[e−](2.11)
The Ne ns equa ion combines he s anda d edox po en ial E0and he edox po en ial o he
solu ion E
E=E0+RT
Fln [A]
[A−](2.12)
whe e Fis he Fa aday cons an , Ris he gas cons an and Tis he empe a u e.
The p obabili y o inding g oup A in he educed s a e is he e o e gi en by [62]:
p ed =[A−]
[A−] + [A] =exp RT
F(E0−E)
1 + exp RT
F(E0−E)(2.13)
2.1.2 COMPUTATION OF ACID-BASE AND REDOX EQUILIBRIA
The pKais di ec ly ela ed o he ee ene gy o he dep o ona ion eac ion in aqueous solu ion
∆Gdep o
wa e by he ollowing Eq. (2.14):
pKa=1
ln 10kBT∆Gdep o
wa e (2.14)
The ∆Gdep o
wa e can be exp essed as a sum o wo con ibu ions: he sol a ion ene gy di e ence
∆∆Gdep o
sol be ween he associa ed and he dissocia ed sys em and he gas phase dep o ona ion
ene gy ∆Gdep o
ac . These e ms can be ob ained om he he modynamic cycle (see Figu e 2.1).
pKa=1
ln 10kBT(∆Gdep o
ac + ∆∆Gdep o
sol )(2.15)
2.1. Acid-base and edox eac ions equilib ia 37
(A ) (H )
+
wa e
acuum
AH
AH
AH+
∆G
∆G∆G∆G(AH)
sol
dep o
ac
sol sol
AH+
G∆dep o
wa e
Figu e 2.1. The modynamic cycle o calcula e absolu e pKa alues. The ee ene gy
o dissocia ion is calcula ed in acuum and he eac an and p oduc s a e hen ans e ed
om acuum o wa e . The ee ene gy o dissocia ion o a p o on om an acid in wa e
(∆Gdep o
sol ) is calcula ed indi ec ly.
The sol a ion ene gy di e ence ∆∆Gdep o
sol is ob ained om Eq. (2.16)
∆∆Gdep o
sol = ∆Gsol (A−)+∆Gsol (H+)−∆Gsol (AH)(2.16)
The sol a ion ene gy o he p o ona ed and dep o ona ed s a es, ∆Gsol (A−)and ∆Gsol (AH)
can be calcula ed by sol ing he Poisson-Bol zmann equa ion Eq. (2.19). The sol a ion ene gy
o a p o on is measu ed expe imen ally om he po en ial o he s anda d hyd ogen elec ode.
The sol a ion ene gy o –264.6 kcal/mol [63] is used o he p o on in he p esen s udy.
The gas phase p o ona ion ene gy ∆Gdep o
ac is gi en by Eq. (2.17)
∆Gdep o
ac = ∆Hdep o
ac + ∆Hdep o
ib +H ans(H+) + ∆(pV )−T[S(H+)] (2.17)
whe e ∆Hdep o
ac is he di e ence in he gas phase ene gy o he associa ed (p o ona ed) and
dissocia ed (dep o ona ed and hyd ogen ion) sys em which can be ob ained om quan um
chemical calcula ions. The ∆Hdep o
ib is he change in he ib a ional ene gy be ween he p o-
ona ed and dep o ona ed s a es and his can be ob ained om no mal mode analysis. The
H ans(H+) = 3
2RT is he ansla ion ene gy o a p o on and ∆(pV )is he ene gy change due
o he olume change in he gas phase eac ion which is es ima ed o be kBT om he ideal
gas app oxima ion. The T[S(H+)] is he en opic con ibu ion o he gas phase ee ene gy o
p o on and is se o 7.8 kcal/mol [62] as de i ed om he Sacku -Te ode equa ion. The edox
po en ial E edox
0can also be compu ed by a simila app oach.
The edox po en ial E edox
0can be calcula ed om Eq. (2.18)
E edox
0=1
F(∆H edox
ac + ∆∆G edox
sol )+∆SHE (2.18)

38 Compu a ional me hods
whe e ∆H edox
ac is he di e ence in he gas phase ene gy be ween he oxidized and educed
s a es. The ∆∆ edox
sol is he sol a ion ene gy di e ence be ween he oxidized and educed s a es
which can be calcula ed by sol ing he Poisson-Bol zmann equa ion. Fis he Fa aday con-
s an (23.06 kcal/mol) and ∆SHE is he s anda d po en ial o he hyd ogen elec ode (–4.43
V) [62].
2.2 CONTINUUM ELECTROSTATICS
Elec os a ic e ec s a e e y impo an in many phenomena in biology and chemis y. In
biology, elec os a ics is a key componen in de e mining he s uc u e o p o eins, nucleic
acids and memb anes. Elec os a ics o p o eins can be s udied by di e en models. The
quan um chemical calcula ions can gi e accu a e esul s, bu i is compu a ionally expensi e
and ime consuming me hod. An al e na i e app oach in ol es con inuum o mac oscopic
models, in which he sol en p ope ies a e desc ibed in e ms o a e age alues. By his
model i is possible o desc ibe solu e molecules in a omic de ails and ea ing he sol en
in e ms o a e age p ope ies [64]. Following app oxima ions will be used in he con inuum
elec os a ics model. The sol en wa e is assumed as a con inuum liquid wi h high dielec ic
cons an (= 80) (see Figu e 2.2). The p o ein sys em will be assigned a low dielec ic cons an
(= 2 −4). The sol en which con ains mobile ions a e modeled as mobile cha ges. The
p o ein sys em is ep esen ed as a low dielec ic medium con aining ixed poin cha ges and
he mobile ions a e assumed o be in ini ely small in size [62, 65, 66] (see Figu e 2.2).
+
+
−
+
+
+
+
−
−
−
−
+
++
+
+
−
−
−
−
−
mobile ions
Sol en wi h
wi h ionic s eng h I
Fixed cha ges
wi hin he
molecule
(low )
ε
(high )ε
Ion exclusion
laye
Figu e 2.2. A molecule in a he e ogeneous dielec ic medium. A molecule o low
dielec ic wi h ixed cha ges in a high dielec ic sol en wi h mobile cha ges (ion).
2.2. Con inuum elec os a ics 39
2.2.1 THE POISSON-BOLTZMANN EQUATION
The ea men o cha ges in acuum is he simples o m o he classical elec os a ics. Pois-
son equa ion can be applied o ea such elec os a ic in e ac ions.
The elec os a ic po en ial φ(~ )a posi ion (~ ), a ising om cha ge densi y ρ(~ )in acuum is
gi en by,
~
∇2φ(~ ) = −ρ(~ )
ε0
(2.19)
whe e ε0is he dielec ic cons an in acuum and ~ is gi en in Ca esian coo dina es. The ∇is
he Nabla-ope a o (~
∇= (∂/∂x, ∂/∂y, ∂/∂z)) and ~
∇2=~
∇ · ~
∇is Laplace-ope a o . The Solu ions
o Eq. (2.19) yields he amilia coulombs law.
Fo inhomogeneous dielec ic medium, he dielec ium a ies wi h posi ion (~ ), and Poisson
equa ion is modi ied o
~
∇ · [ε(~ )~
∇φ(~ )] = −ρ(~ )
ε0
(2.20)
The p o ein sys em is assumed o ha e mobile ions. The cha ge dis ibu ion o he mobile
ions ρions(~ )can be desc ibed by Debye-H¨
uckel heo y. The heo y desc ibes he dis ibu ion
o ions by combining elec os a ics wi h s a is ical he modynamics and uses he Bol zmann
s a is ics based on he elec os a ic in e ac ion ene gy Wibe ween ion iand he solu e [67].
The cha ge dis ibu ion esul ing om he dis ibu ion o ions ci(~ )and o he ions a e
ρions(~ ) =
K
X
i=1
ci(~ )·Zi·e0=
K
X
i=1
cbulk
iZie0·exp −Wi(~ )
RT (2.21)
whe e cbulk
iis he concen a ion o ion iin he bulk, Kis he numbe o di e en ype o ions,
Ziis he uni less o mal cha ge o ion ype i,e0is he elemen a y cha ge, Ris he uni e sal
gas cons an and Tis he absolu e empe a u e. The summa ion uns o e all K ypes o ion
i.
The in e ac ion ene gy Wiis app oxima ed by he po en ial ene gy Epo o ion iin he elec-
os a ic po en ial φ(~ )due o cha ge dis ibu ions o solu e ρsolu e(~ )and mobile ions ρions(~ ).
This app oxima ion Wi≈Epo =Zie·φ(~ )yields
ρions(~ ) =
K
X
i=1
cbulk
iZie0·exp −Zie0·φ(~ )
RT (2.22)
Wi h ρ(~ ) = ρsolu e(~ ) + ρions(~ ), Eq. (2.19) can be e-w i en o gi e he Poisson-Bol zmann
equa ion (PBE)
~
∇ · [ε(~ )~
∇φ(~ )] = 1
ε0 ρsolu e(~ ) +
K
X
i
cbulk
iZie0·exp −Zie0·φ(~ )
RT !(2.23)
40 Compu a ional me hods
Using he linea app oxima ion exp(−x)≈1−x alid o small x1, o Zie0·φ(~ )/RT 1
K
X
i
cbulk
iZie0·exp −Zie0·φ(~ )
RT =
K
X
i
cbulk
iZie0−
K
X
i
cbulk
iZie0·Zie0·φ(~ )
RT (2.24)
As he o al cha ge o he ions in solu ion is equal o ze o, he i s e m on he igh side o
Eq. (2.24) is ze o.
~
∇ · [ε(~ )~
∇φ(~ )] = 1
ε0 ρsolu e(~ ) +
K
X
icbulk
iZ2
ie2
0
RT φ(~ )!(2.25)
The linea Poisson-Bol zmann equa ion (LPBE) can be ew i en by in oducing he ionic
s eng h and he Debye-H¨
uckel in e se, κ
I=1
2
K
X
i=1
cbulk
iZ2
i(2.26)
κ2(~ ) = 8πe2
0I(~ )
RT (2.27)
~
∇ · [ε(~ )~
∇φ(~ )] = −4πρsolu e(~ ) + κ2φ(~ )(2.28)
Eq. (2.28) is sol ed nume ically o yield φ(~ ).
The Poisson-Bol zmann equa ion is qui e good o he mobile ions (mono alen and mul i a-
len ). The Linea Poisson-Bol zmann equa ion b eakdowns o sol en wi h high concen a ion
bu hese limi a ions will no a ec esul s o he biological sys em since bo h ionic s eng hs
and he elec os a ic in e ac ions a e mode a e in hese sys ems.
NUMERICAL SOLUTION TO LINEAR POISSON-BOLTZMANN EQUATION. Analy ical solu ions a e
known only o simple geome ies. Fo complex sys ems, nume ical me hods a e applied o
sol e he Poisson-Bol zmann equa ion. A wide a ie y o p oblems ha e been s udied using he
ini e di e ence me hod. Howe e he e a e also o he nume ical me hods such as bounda y
elemen me hod which essella es he dielec ic bounda y be ween he in e io and ex e io
o he molecule. A ini e di e ence me hod is applied in he p esen wo k o sol e he PBE.
In ini e di e ence me hod (see Figu e 2.3), he p o ein (solu e) is mapped in o a cubic g id
along wi h he sol en . The alues o he elec os a ic po en ial, cha ge densi y, g id cons an
h, dielec ic cons an and ionic s eng h a e assigned o each g id poin [68]. The a omic
cha ges usually do no coincide wi h he g id poin s and so he cha ge is alloca ed o he
eigh su ounding g id poin s in such a way ha he close he cha ges o he g id poin ,
he g ea e he p opo ion o i s o al cha ge ha is alloca ed. The linea Poisson-Bol zmann
2.2. Con inuum elec os a ics 41
φ0
φ3
5
ε
1
ε6
φ2φ4
0
0
1
1
0
0
0
1
1
1
0
0
1
1
0
0
0
1
1
1
0
0
1
1
00
00
11
11
0
0
1
1
h
φ5
φ1
φ6
ε2ε
ε3
ε4
q0
0
Ι
Figu e 2.3. The cube used in he ini e di e ence me hod o sol ing he Poisson-
Bol zmann equa ion.
equa ion Eq. (2.28) is in eg a ed o e he olume Vo he cubic g id:
Z~
∇ · [ε(~ )~
∇φ(~ )]d −Zκ2
0(~ )φ(~ )+4πZρsolu e(~ )d~ = 0 (2.29)
In he i s s ep, he in eg al is ans o med in o a su ace in eg al using Gauss’s heo em:
The olume in eg al is gi en by:
Z~
∇ · [ε(~ )~
∇φ(~ )]dA −h3κ2
0φ0+ 4πq0= 0 (2.30)
The su ace in eg al is calcula ed sepa a ely o all six sides o he cubic g id elemen .
Z~
∇ · [ε(~ )~
∇φ(~ )]dA =
6
X
i=1
h2εi(φi−φ0)
h(2.31)
6
X
i=1
hεi(φi−φ0)−h3κ2
0φ0+ 4πq0= 0 (2.32)
he ini e di e ence exp ession o φ0:
φ0=(P6
i=1 hεiφi)+4πq0
P6
i=1 hεi+h3κ2
0
(2.33)
48 Compu a ional me hods
pKp
=1
RTln10(Go
p−Go
)(2.45)
The mac oscopic p ¯
Kk alues cha ac e ize he equilib ia be ween wo mac oscopic p o ona ion
s a es o he p o ein i.e., be ween he wo mac os a es wi h kand k−1p o ons bound. The
k h mac oscopic p ¯
Kk alue is gi en by
p¯
Kk=−log PM
mP2N
xδ(k−1)e−βGo
x,m
PM
mP2N
xδ(k)e−βGo
x,m !(2.46)
A mac oscopic p ¯
Kk alue is he he modynamic a e age o e all mic oscopic equilib ia in-
ol ed in he elease o he k h p o on. The p oduc o e he i s o he k h mac oscopic ¯
Kk
alues equals o he k h polynomial coe icien PN
mP2N
xδ(k)e−βGo
x,m o he pa i ion unc ion
in Eq. (2.42). The mac oscopic p ¯
Kk alues can consequen ly also be used o o mula e he
pa i ion unc ion. Mo eo e , he o al i a ion cu e o a polyp o ic acid is o en in e p e ed
in e ms o mac oscopic p ¯
Kk alues. The pKa alues o h ee p o on binding si es a e also
calcula ed in simila way. The sys em wi h h ee p o on binding si es a e shown in Figu e 2.7
[75].
CALCULATION OF TITRATION CURVES
The p o ona ion s a e o a p o ein wi h N i a able g oup is 2N. The i a ion cu e o a single
si e in a p o ein is gi en by he he modynamic a e age o e all possible p o ona ion s a es a
each pH- alue
hxii=
2N
X
n=0
x(n)
iexp −G(n)(pH)
RT 
2N
X
n=0
exp −G(n)(pH)
RT 
(2.47)
To ob ain he p o ona ion s a e ene gy Gn, he Eq. (2.47) should be e alua ed o 2N imes.
Since cy och ome coxidase and o he model p o eins a e huge, he he modynamic a e age
hxiig ows exponen ially wi h he numbe o si es. The a e age beha io o such sys em can
be ob ained by Me opolis Mon e Ca lo app oach which is used o de e mine he p o ona ion
o many in e ac ing si es as a unc ion o pH.
THE METROPOLIS MONTE CARLO APPROACH. The Me opolis Mon e Ca lo me hod is used o
ob ain he i a ion cu es o amino acids wi hin he p o ein [76]. Ini ially a andom p o ona-
ion s a e nis chosen, he ene gy o he co esponding p o ona ion s a e is calcula ed. Each
Mon e Ca lo (MC) s ep chooses he p o ona ion s a e andomly and he ene gy o he cu en
p o ona ion s a e is calcula ed and he change in ene gy is compu ed, i.e., he ene gy di e -
ence be ween cu en p o ona ion s a e and he p e ious p o ona ion s a e ∆G = Gold−Gcu en .

2.2. Con inuum elec os a ics 49
The cu en s a e is accep ed wi h a p obabili y pacco ding o Me opolis c i e ion
p=


1 i ∆G≤0,
exp (−∆G
RT ) i ∆G > 0(2.48)
The abo e p ocess is epea ed o each MC s ep. The collec ion o p o ona ion s a es s a
a e he equilib a ion s ep. The s eps in ol ed in MC a e shown in Figu e 2.8. The Me opolis
Mon e Ca lo app oach also e ec i ely sample he p o ona ion s a es o he s ongly coupled
si es. The s ongly coupled g oups ha e he ollowing s a es: he singly p o ona ed s a e
(1,0) and (0,1) he low ene gy s a es and s a e (0,0) and (1,1) high ene gy s a es. The e o e
wo ansi ions a e needed o change om (1,0) o (0,1) o a oid he high- ene gy in e media e
s a es (0,0) and (1,1). To sol e his sampling p oblem, double and iple mo es a e in oduced,
which simul aneously change he p o ona ion s a es o wo o h ee si es in MC s ep.
CORRELATION CURVES. The coupling be ween he p o ona ion o m o i a able g oup iand
g oup jis gi en by:
c=hxi,ji−hxii·hxji(2.49)
whe e he hxiiand hxjia e he p obabili y o he i a able g oup iand g oup jwhich a e
p o ona ed espec i ely and hxi,j iis he p obabili y o bo h a e p o ona ed in he same ime.
The co ela ion be ween wo si es can be ob ained om abo e Eq. (2.49).
TANFORD-ROXBY PKaVALUES
The Tan o d-Roxby app oxima ion assumes ha he a e age p o ona ion o a i a able esidue
depends on he a e age cha ges o all o he i a able g oups. The a e age pKa alue o he
esidue iinside he p o ein [77, 78] hpKin
a,i ip o is ob ained om Eq (2.50).
pKa,i =pKin
a,i +
N
X
j=1
(hxji − x(0)
j)Wij (2.50)
whe e he in insic pKa alue (pKin
a,i ) is he pKa alue ha he pa icula i a able g oup
would ha e, i all o he i a able g oups a e in hei e e ence o m. This e m includes he
sol a ion ene gy and he in e ac ion wi h non- i a ing esidues and he p o ein backbone.
The e m Wij ep esen s he in e ac ion ene gy be ween he i a able g oups iand jin hei
cha ged om; hxji ep esen s he p o ona ion p obabili y o he g oup jwhich is ob ained by
a he modynamic a e age o e all possible p o ona ion s a es and x0
jis he e e ence p o o-
na ion o m o si e j. The hxjia e ob ained om MC calcula ions. These a e age pKa alues
do no ep esen an equilib ium si ua ion, bu hey a e a good app oxima ion o he eal pKa
alue o he si e.
50 Compu a ional me hods
∆ G= G − G
∆G
S a e
Choose andom
Ini ial P o ona ion
P o ona ion o
∆old
Change
Check 0
Choose andom
1
0
Accep new
S a e
∆G
Yes No
∆
Keep old
S a e
Sa e S a e
Mo e MC
Calcula e a e age
P obabili y
Exi
Yes
No
No
Yes
Si e i
Si e i
new
Check
e
/k T
B
Figu e 2.8. The Me opolis Mon e Ca lo (MC) app oach. Ini ial andom p o ona ion
s a e iis chosen and he p o ona ion is changed. The ene gy di e ence be ween he new
s a e and old s a e is calcula ed. The new s a e is accep ed i he ene gy is smalle han ze o.
I he ene gy is g ea e han ze o, he Bol zmann ac o o he change in ene gy is checked.
I he change in ene gy is la ge han a numbe andomly chosen be ween 0 and 1 hen
he s a e is accep ed. The s eps a e epea ed un il con e gence o he sampled p ope ies is
achie ed.
2.3. Quan um Chemis y 51
2.3 QUANTUM CHEMISTRY
The applica ion o quan um-mechanical p inciples o chemical p oblems has gained momen-
um in pas ew decades. The quan um mechanical me hods can be o g ea use in he
simula ion o p o eins. They can be used o ob ain pa ame e s o molecula mechanics ype
po en ial ene gy unc ions. In he p esen wo k, he quan um chemical (QC) me hods a e
used o calcula e he ee ene gies o he eac ions, ib a ional ene gies, sol a ion ene gies
and pa ial a omic cha ges. This chap e desc ibed some o he basic p inciples, me hods o
quan um chemis y ha lead o unde s and he elec onic s uc u e beha io .
SCHR ¨
ODINGER EQUATION
The p inciples o densi y unc ional heo y a e con enien ly expounded by making e e ence
o con en ional wa e unc ion heo y. The ime independen , non- ela i is ic Sch ¨
odinge
equa ion o a many elec on sys em can be w i en as:
HΨ = EΨ(2.51)
His he Hamil onian ope a o o a sys em o nuclei and elec ons desc ibed by posi ion
ec o s ~
RAand ~ i, espec i ely. Ψis he wa e unc ion o he many pa icle sys em and E is
he ene gy o he eigen unc ion. In SI uni s he Sch ¨
odinge equa ion o he hyd ogen a om
can be w i en as:
−~2
2m∇2
iΨi(~ )−e2
4πε0 Ψi(~ ) = EiΨi(~ )(2.52)
In Eq. (2.52), i s e m in he le hand side (LHS) is he kine ic ene gy o he elec on sys em
and he second e m is he po en ial ene gy e m and Eiis he ene gy o he sys em, whe e
eand ma e he elec on cha ge and mass espec i ely, ~is Plank’s cons an di ided by 2π,
is he dis ance be ween elec on and p o on and ε0is he pe mi i i y o ee space. In he
SI sys em o uni s he abo e equa ion is a he unwieldy o deal wi h because o he iny
numbe s associa ed wi h quan i ies as m,e2and ~2. The incon enience can be emo ed by
ew i ing he equa ion in e ms o ‘na u al’ a omic uni s.
The Sch ¨
odinge equa ion can be exp essed in a omic uni s whe e me=|e|=~= 4πε0= 1 and
Eq. (2.52) becomes Eq. (2.53)
−1
2∇2
i−1
Ψi(~ ) = EiΨi(~ )(2.53)
52 Compu a ional me hods
The Hamil onian o Nelec ons and Mnuclei can be exp essed as
H=−
N
X
i=1
−1
2∇2
i−
M
X
A=1
1
2MA
∇2
A−
N
X
i=1
M
X
A=1
ZA
iA
+
N
X
i=1
N
X
j>i
1
ij
+
N
X
A=1
N
X
B>A
ZAZB
RAB
(2.54)
In Eq. (2.54) he dis ance be ween he i- h elec on and he A- h nucleus is ~ iA=|~ i-~
RiA|, he
dis ance be ween he i- h and j- h elec on is ~ ij=|~ i-~ j|and he dis ance be ween he A- h
and B- h nucleus is ~
Rij=|~
Ri-~
Rj|.MAis he a io o he mass o nucleus A o he mass o
an elec on and ZAis he a omic numbe o nucleus A. The Laplacian ope a o s ∇2
iand ∇2
A
in ol e di e en ia ion wi h espec o he coo dina es o he i- h elec on and A- h nucleus,
espec i ely. The i s e m in he abo e equa ion is he ope a o o he kine ic ene gy o
he elec ons; he second is he ope a o o he kine ic ene gy o he nucleus; he hi d e m
ep esen s he Coulomb a ac ion be ween elec ons and nuclei; he ou h and i h e m
ep esen s he epulsion be ween elec ons and be ween nuclei, espec i ely.
The Bo n-Oppenheime app oxima ion [79, 80] conside s elec ons as mo ing pa icles in he
ield o ixed nuclei. Due o signi ican di e ences in mass be ween elec ons (mel∼10−31
kg) and he nucleus (mnuc∼10−27 kg), he elec onic deg ees o eedom can be conside ed o
espond ins an aneously o any change in he nuclea con igu a ion i.e., hei wa e unc ions
always co espond o a s a iona y s a e. Hence, he in e ac ion be ween di e en s a iona y
s a es. Applica ion o he Bo n-Oppenheime app oxima ion enables one o o mula e he
elec onic Hamil onian. The ime-independen elec onic Sch ¨
odinge equa ion desc ibing he
mo ion o elec ons in he elec os a ic ield o he s a iona y nuclei comp ises only e m
one, h ee and ou o Eq. (2.54), whe e he elec onic Hamil onian and he elec onic wa e
unc ions depend only pa ame ically on he coo dina es o he nuclei.
2.3.1 DENSITY FUNCTIONAL THEORY
The densi y unc ional heo y (DFT) is a ema kable heo y ha allows one o eplace he
complica ed N-elec on wa e unc ion by a much simple elec on densi y ρ(~ ). The basis
o DFT is he p oo by Hohenbe g and Kohn [81] ha he g ound-s a e elec onic ene gy is
de e mined comple ely by he elec on densi y ρ(~ )[79, 80, 82, 83]. The Hohenbe g-Kohn
says ha he molecula ene gy, he wa e unc ion and all o he molecula p ope ies can be
compu ed om he g ound s a e elec on p obabili y densi y. E en wi h he p oo ha e e y
speci ic elec on densi y gi es a speci ic ene gy ( o he g ound s a e), he connec ion be ween
he ene gy and he unc ional has no been disco e ed. The hypo he ical unc ional can be
di ided in o h ee pa s: he kine ic ene gy T[ρ(~ )], he co e-elec on a ac ion Ene[ρ(~ )], and
he elec on-elec on epulsion Eee[ρ(~ )]. The elec on-elec on epulsion is usually di ided
in o an exchange pa K[ρ(~ )] and a Coulomb pa J[ρ(~ )]. The exac kine ic ene gy (T) o a
g ound s a e is gi en by
T=
∞
X
i
nihΨi| −1
2∇2
i|Ψii(2.55)
2.3. Quan um Chemis y 53
whe e Ψia e na u al spin o bi als and ni hei occupa ion numbe , 0 ≤ni≥1. Fo an
in e ac ing sys em, Eq. (2.55) con ains an in ini e numbe o e ms, making an exac solu ion
una ainable. Kohn and Sham [81] p esen ed a o malism o ea ing he kine ic pa o he
ene gy.
In Ha ee’s model [79, 80], he elec ons mo e in an e ec i e po en ial υH(~ )c ea ed by he
a omic co es and a mean ield c ea ed by he o he elec ons is gi en as:
υH(~ ) = −
nuclei
X
a
Za
|~
Ra−~ |+Zρ(~ 0)
|~ −~ 0|d~ 0(2.56)
whe e a uns o e all he nuclei, Zais he nuclea cha ge on a om a,~
Radeno es he posi ion
o he nuclei and ρ(~ )is he elec on densi y.
In his app oxima ion, a one-pa icle Sch ¨
odinge equa ion can be de ined as:
−1
2∇2
i+υH(~ )Ψi=EiΨi(2.57)
whe e Eiis he ene gy o he sys em.
The mean densi y is gi en by
ρ(~ ) =
N
X
i
|Ψi(~ )|2(2.58)
Fo he g ound s a e, a summa ion o e he Nspin o bi als wi h lowes eigen alues is pe -
o med.
Eq. (2.56-2.58) a e sol ed sel -consis en ly. F om a ial guess o he elec on densi y, a
po en ial is ob ained by Eq. (2.56). This po en ial is used in Eq. (2.57). The one-elec on
o bi als ob ained a e pu in o Eq. (2.58). The p ocedu e is epea ed un il a se o con e ged
one-elec on o bi als a e ob ained. Fo a sys em o independen non-in e ac ing elec ons, he
common one-body po en ial, υKS is assumed.
−1
2∇2
i+υKS(~ )Ψi=EiΨi(2.59)
The local one-body po en ial υKS gi es, by de ini ion, a non-in e ac ing elec on densi y ha
has he same o m as Eq. (2.57).
On compa ing Eq. (2.55) and Eq. (2.59), he exp ession o he kine ic ene gy o he non-
in e ac ing elec ons (Ts[ρ]), becomes

54 Compu a ional me hods
Ts[ρ] =
N
X
i=1
hΨi| −1
2∇2
i|Ψii(2.60)
whe e he sum iagain uns o e all o bi als. The lowe index sin Ts[ρ], deno es he single-
elec on equa ions. As he elec ons do in e ac in eali y, he exp ession is no exac , and he
ollowing inequali y holds whe e T[ρ]is he exac kine ic ene gy
Ts[ρ]≤T[ρ](2.61)
The emaining pa o he kine ic ene gy de ines he co ela ion con ibu ion:
Tc[ρ] = T[ρ]−Ts[ρ]≥0(2.62)
whe e Tcis included in a exchange/co ela ion e m Exc. The Kohn-Sham equa ions can
be sol ed analogously o he Ha ee equa ions, wi h he di e ence ha he po en ial in
Eq. (2.57), υH, is eplaced by υe :
υe (~ ) = υ(~ ) + Zρ(~ 0)
|~ −~ 0|d~ 0+υxc(~ )(2.63)
whe e υxc(~ )deno es he local exchange/co ela ion po en ial and υ(~ )co esponds o he
ex e nal po en ial.
Wi hin he KS o malism, he ene gy o he g ound s a e can be ob ained om
EDF T [ρ] = X
i
Ei+Exc [ρ]−Zυxc(~ )ρ(~ )d −1
2Zρ(~ )ρ(~ 0)
|~ −~ 0|(2.64)
o mo e gene ally, di ided in o i s componen s:
EDF T [ρ(~ )] = Ts[ρ(~ )] + Ene [ρ(~ )] + J[ρ(~ )] + Exc [ρ(~ )] (2.65)
An exac ene gy exp ession has been ob ained om Eq. (2.65). The i s e m is he kine ic
ene gy o he non-in e ac ing elec ons and he second e m is he co e-elec on a ac ion
and he hi d e m ep esen s he Coulombic epulsion unc ional, he elec os a ic elec on-
elec on epulsion and he inal e m is he exchange/co ela ion e m. Excep he las e m
exchange/co ela ion ene gy, all o he e ms a e sol ed exac ly. The new challenge was o
ind a solu ion o Exc.
2.3. Quan um Chemis y 55
DIFFERENT DFT MODELS
Mos DFT models di e in hei exp ession o he exchange/co ela ion ene gy Exc. I is
always spli in o sepa a e exchange and co ela ion e ms, Exand Ec. The exchange ene gy is
“by de ini ion” gi en as a sum o con ibu ions om αand βspin densi ies, as exchange ene gy
only in ol es elec ons o he same spin. The kine ic ene gy, he nuclea -elec on a ac ion
and Coulomb e ms a e i ially sepa able. The co ela ion ene gy con ains con ibu ions
om he in e ac ions be ween all elec ons:
Ex(ρ) = Eα
x(ρα) + Eβ
x(ρβ)(2.66)
Ec(ρ) = Eαα
c(ρα) + Eββ
c(ρβ) + Eαβ
c(ρα, ρβ)(2.67)
Whe e ραand ρβa e densi ies o he αand βspin. The o al densi y is he sum o he αand β
con ibu ions, ρ=ρα+ρβ he la ges con ibu ion o Exc comes om he exchange pa Ex.
THE LOCAL DENSITY APPROXIMATION
In he Local Densi y App oxima ion (LDA) i is assumed ha he densi y can be ea ed locally
as a uni o m elec on gas o he elec on densi y is assumed o be slowly a ying in space
ELDA
xc [ρ] = Zρ(~ )εuni
xc (ρ)(2.68)
εuni
xc gi es he exchange/co ela ion ene gy pe elec on in a uni o m elec on gas. The ana-
ly ical exp ession o he exchange ene gy can be ob ained om he Di ac o mula
ELDA
xc [ρ] = −CxZρ4/3(~ )d~ (2.69)
whe e Cxis gi en as:
Cx=3
43
π1/3
In he mo e gene al case, whe e αand βdensi ies a e no equal, LDA has been eplaced by
he Local Spin Densi y App oxima ion (LSDA).
ELSDA
xc [ρ] = −21/3CxZρ4/3
α(~ ) + ρ4/3
β(~ )d~ (2.70)
56 Compu a ional me hods
Fo he LDA co ela ion ene gy, no exac solu ion is known bu i has been de e mined by
Mon e Ca lo me hods o a numbe o di e en densi ies. In o de o use hese esul s in DFT
calcula ions, a sui able analy ic in e pola ion o mula is desi able. Vosko, Wilk and Nusa i
[84] i ed a ious da a o ob ain eliable analy ic exp essions o he LDA co ela ion ene gy
and in gene al conside ed o be a e y accu a e i . This exchange unc ional is used in he
calcula ions.
THE GENERALIZED GRADIENT APPROXIMATION
The Gene alized G adien App oxima ion (GGA) [79, 80] is he imp o emen o e he LSAD
app oach whe e he elec on dis ibu ion is conside ed as a non-uni o m elec on gas. The
idea behind his is no only conside ing he elec on densi y, bu also he g adien o he
densi y, ha is he change in he densi y a a gi en poin .
The e m GGA was p obably i s in oduced in connec ion wi h he PW86 unc ional, de el-
oped by Pe dew and Wang in 1986 [85]. Pe dew and Wang p oposed modi ying he LSDA
exchange exp ession o ha shown in Eq. (2.71)
EP W 86
x=εLDA
x(1 + ax2+bx4+cx6)1/15 (2.71)
x=| ∇ρ|
ρ4/3(2.72)
whe e xis a dimensionless g adien a iable, and a,band cbeing sui able cons an whe e he
summa ion o e equi alen exp ession o αand βdensi ies is implici ly assumed.
A gene al GGA unc ional has he o m:
εGGA =εGGA(ρα, ρβ,∇ρα,∇ρβ)(2.73)
Se e al GGA unc ionals, o bo h exchange and co ela ion, ha e been p oposed in he li e -
a u e [85–90].
Becke [87] p oposed a widely used co ela ion (B o B88) o he LSDA exchange ene gy:
εB88
x=εLDA
x+ ∆εB88
x(2.74)
∆εB88
x=−βρ1/3x2
1+6βx sinh−1x(2.75)
The βpa ame e is de e mined by i ing o known a omic da a and xis de ined in Eq. (2.72).
2.3. Quan um Chemis y 57
Pe dew and Wang ha e p oposed an exchange unc ional simila o B88 o be used in connec-
ion wi h he PW91 co ela ion unc ional gi en below.
εP W 91
x=εLDA
x 1 + xa1sinh−1(xa2)+(a3+a4e−bx2)x2
1 + xa1sinh−1(xa2) + a5x2!(2.76)
whe e aiand b a e sui able cons an s and xis de ined in Eq. (2.72). whe e xis gi en in
Eq. (2.72) This GGA unc ional is used o he calcula ions.
HYBRID METHODS
The Kohn-Sham ex ensions e ealed ha imp o ed exp essions o elec on exchange and
co ela ion can lead o be e wa e unc ions and elec onic ene gy. The e o e majo a emp s
we e ca ied ou in he pas o ind imp o ed and be e elec on exchange and co ela ion
exp essions.
Wa e unc ion heo y (Ha ee-Fock) can in p inciple p o ide he exac exchange ene gy.
EHF
x=−1
2
n
X
i
n
X
jZ Z Ψi(~ 1)Ψj(~ 1)Ψi(~ 2)Ψj(~ 2)
|~ 1−~ 2|d~ 1~ 2(2.77)
I could be assumed ha he ideal exchange/co ela ion ene gy would hen be ob ained om
Exc =EHF
x+EDF T
c(2.78)
In hyb id DFT me hods in gene al, he exchange e m o DFT is co ec ed by a con ibu-
ion om exac exchange ene gy. In he abo e, he DFT exchange has been eplaced by HF
exchange.
BECKE3 The mos popula hyb id me hods a e based on Becke’s h ee unc ional B3 [91]. I
con ains h ee adjus able pa ame e s a0,axand ac:
EB3
xc =a0EHF
x+ (1 −a0)ELSDA
x+ax∆EB88
x+ELSDA
c+ac∆EGGA
c(2.79)
Becke used PW91 o he co ela ion pa [86, 88, 92]. The name o he unc ional , B3PW91,
implies i s use o a h ee-pa ame e scheme, as well as GGA exchange and co ela ion unc-
ionals B and PW91, espec i ely.
EB3LY P
xc =a0EHF
x+ (1 −a0)ELSDA
x+ax∆EB88
x+ELSDA
c+ac∆EP W 91
c(2.80)
64 Cha ge dis ibu ion in pep ide con o ma ions
e ec o hese me hods on dipole momen s. They checked he abili y o hese cha ges by
ep oducing he expe imen al dipole momen s o 89 molecules. Va ious me hods like RESP
and CHELP-Bow we e used o d i e pa ial a omic cha ges o he disaccha ides om he
quan um chemical elec os a ic po en ial and momen s [137].
The a omic cha ges o a ious con o ma ions was s udied by Richa ds and co-wo ke s [138]
who epo ed wo ela ed me hods, bo h o hem employ a cons ained minimiza ion o he
di e ence be ween he quan um mechanical and classical MEP (Molecula Elec os a ic Po-
en ial) wi h espec o he a omic cha ges. The i s me hod in ol es de e mining he MEP
and cons aining he cha ges o ep oduce he dipole a an al e na i e geome y. The second
me hod in ol es de e mining he MEP o each con o ma ion o in e es and and weigh ing
he MEP o each con o ma ion acco ding o he app op ia e Bol zmann ac o .
The cha ge dis ibu ion o he ipep ide Ala-Asn-Ala wi h di e en con o ma ions ha e been
analyzed using he hyb id densi y unc ional me hod (B3LYP) wi h 6-31G* basis se . The
popula ion analysis me hods such as Mulliken, NPA and he elec os a ic po en ial me hods
CHELPG, MK and RESP a e used o analyze he cha ge dis ibu ion o he ipep ide Ala-Asn-
Ala. The ex ensi e s udies on he cha ge me hods can p o ide a be e unde s anding o each
me hod and he pa ame e s which in luence he pa ial a omic cha ges. The di e en cha ge
me hods discussed in his chap e a e used o ob ain he poin cha ges o he elec os a ic
calcula ions and o he calcula ion o pKa alues o CuBligands in aqueous solu ion.
3.1 THEORETICAL BACKGROUND
In his sec ion he heo y behind cha ge me hods, he ad an age and disad an ages o each
cha ge me hods a e explained.
3.1.1 WAVE FUNCTION BASED METHODS
MULLIKEN POPULATION ANALYSIS
In Mulliken popula ion analysis, he elec ons a e di ided up amongs he a oms acco ding
o he deg ee o which di e en a omic o bi al (AO) basis unc ions con ibu e o he o e all
wa e unc ion [79, 80, 139, 140]. The o al numbe o elec ons a e ob ained by expanding
he wa e unc ion in i s AO basis se ,
N=
elec ons
X
jZψj(~ j)ψj(~ j)d~ j(3.1)
=
elec ons
X
jX
,s Zcj ϕ (~ j)cjsϕs(~ j)d~ j(3.2)

3.1. Theo e ical backg ound 65
=
elec ons
X
jX
c2
j +X
6=s
cj cjsS s(3.3)
whe e ψis he wa e unc ion o molecule j, and sa e he index o AO basis unc ion ϕ,cj is
he coe icien o he basis unc ion in molecula o bi al j,cjs is he coe icien o he basis
unc ion sin molecula o bi al j.S s is he usual o e lap ma ix elemen [79]. F om Eq. (3.3)
i s clea ha he o al numbe o elec ons a e di ided in o wo sums, one including only he
squa es o single AO basis unc ions and he o he including p oduc s o wo di e en AO basis
unc ions. The i s e m in he pa en heses in Eq. (3.3), is he sum o elec ons associa ed
wi h single basis unc ion only and he second e m, which ep esen s he elec ons ‘sha ed’
be ween basis unc ions. The o al numbe o elec ons pe a om Nkis gi en by
Nk=
elec ons
X
jX
∈k
c2
j +X
,s∈k, 6=s
cj cjsS s +X
∈k,s/∈k
cj cjsS s(3.4)
The o hono mali y o basis unc ions o di e en angula momen um bo h esiding on he
same a om kcauses many e ms in he second sum o Eq. (3.4) o be ze o. The Mulliken
pa ial a omic cha ge is de ines as
qk=Zk−Nk(3.5)
whe e Zkis he nuclea cha ge o a om kand Nkis compu ed acco ding o he Eq. (3.4).
Wi h minimal o small spli - alence basis se s, he Mulliken cha ges end o be easonably
in ui i e. The use o non-o hogonal basis se in he Mulliken analysis, howe e , can lead o
some undesi able esul s. In Mulliken analysis whe e he o al numbe o elec ons a e di-
ided o e AO basis unc ions, i is possible o indi idual basis unc ions o ha e occupa ion
numbe s g ea e han 1 o less han 0. Such si ua ions ob iously can ha e no physical mean-
ing. The elec ons a e also di ided equally in he case a oms wi h di e en elec onega i i ies.
Mulliken pa ial cha ges p o e o be e y sensi i e o basis-se size, so ha compa isons o
pa ial a omic cha ges om di e en le els o heo y a e in no way possible. Mo eo e , wi h
e y comple e basis se s, he Mulliken cha ges ha e a endency o become unphysical.
NATURAL POPULATION ANALYSIS
Na u al A omic O bi als (NAOs) we e de ined by P. O. L¨
owdin [141] as he eigen unc ions o
he i s -o de educed densi y ma ix [142]. They a e in insic o bi als {θk} ha a ise as
eigen unc ions o he i s -o de educed densi y ope a o ˆ
Γ,
ˆ
Γθk=qkθk(3.6)
66 Cha ge dis ibu ion in pep ide con o ma ions
which is o med by ‘ educing’ he wa e unc ion p obabili y dis ibu ion o he single-pa icle
le el,
ˆ
Γ = NZψ(1,2, ..., N)ψ∗(10,2...N)dτ2...dτN(3.7)
and whose eigeno bi als a e hence ‘na u al’ o he ψi sel . The subsys ems o he N-pa icle
sys em a e bes o mula ed in e ms o educed densi y ope a o s. In pa icula , he squa ed
ampli ude |hψ(1,2,...,N)|φ(1)i|2 ha an elec on o ψ(1,2,...,N) is ‘in’ o bi al φ(i.e., he popula-
ion o φ(1) in he wa e unc ion) is igo ously exp essed, o any possible o bi al φ, as
qφ=hφ|ˆ
Γ|φi(3.8)
The occupancies qφa e in insically non-nega i e and limi ed by he Pauli exclusion p inciple
e.g., o he spa ial o bi al φ( ),
0≤qφ≤2(3.9)
The sum o occupancies qko e any comple e o hono mal se {θk}accoun s o all Nelec-
ons,
X
k
hφk|ˆ
Γ|φki=X
k
qk=N(3.10)
Mulliken popula ion commonly ail o sa is y he physical cons ains (Eq. 3.9 and 3.10). The
na u al o bi als {θk}o Eq. (3.6) a e in insically bes possible o conside ing maximum oc-
cupancy in he smalles possible numbe o o bi als.
The s a ing poin o he Na u al Popula ion Analysis (NPA) is he de e mina ion o an op imal
se o e ec i e a omic o bi als o he molecula en i onmen .
To ind he maximum-occupancy o bi als o a om A, he local eigen ec o s ˜
θ(A)
io he one-
cen e a omic block ˆ
Γ(A) associa ed wi h basis AOs on a om A is ob ained i s .
ˆ
Γ(A) ˜
θ(A)
i= ˜q(A)
i˜
θ(A)
i(3.11)
A e he maximum-occupancy o bi als {˜
θ(A)
i}ha e been ob ained o each a om A, i is nec-
essa y o es o e ull o hogonali y. The ‘p e-o hogonal’ o bi als {˜
θ(A)
i}a e hen subjec ed
o an occupancy-weigh ed symme ic o hogonaliza ion o ind inal NAOs {θi(A)}. Once he
inal o hono mal NAOs θi(A) a e ob ained, he Eq. (3.8) a e used o igo ously compu e hei
‘na u al’ popula ions qi(A),
3.1. Theo e ical backg ound 67
qi(A) =hθ(A)
i|ˆ
Γ|θ(A)
ii(3.12)
which may be summed o gi e he o al numbe o elec ons NA
NA=X
i
qi(A) (3.13)
and ‘na u al cha ge’ QAon a om A wi h a omic numbe ZAis gi en as
QA=ZA−NA(3.14)
In acco d wi h Eq. (3.9 and 3.10), he na u al popula ions qi(A) necessa ily sa is y he Pauli
p inciple (0 ≤qi(A) ≤2) and sum co ec ly o he o al numbe o elec ons. The Eq. (3.12-3.14)
summa ize ‘na u al popula ion analysis’.
The “Na u al Popula ion Analysis” is an al e na i e o he con en ional Mulliken popula ion
analysis [79, 80, 143, 144]. I seems o exhibi imp o ed nume ical s abili y and be e in
desc ibing he elec on dis ibu ion in compounds o high ionic cha ac e . The mos appealing
ea u e o he NPA scheme is ha each a omic pa ial cha ge e ec i ely con e ges o a s able
alue wi h inc easing basis-se size.
3.1.2 POTENTIAL BASED METHODS
Widely dis ibu ed me hods o gene a e a omic pa ial cha ges om QC compu a ions a e
called po en ial-based me hods such as CHELP, CHELPG, MK and RESP me hod. The cha ges
a e calcula ed by ma ching he elec os a ic po en ial (ESP) om he ins an aneous a omic
pa ial cha ge dis ibu ion wi h he QC ESP compu ed on op imized geome ies in acuum.
These me hods ha e in common ha hey use alues o he elec os a ic po en ial on spa ial
posi ions ha a e dis ibu ed a ound he a oms o he molecule. All po en ial-based me hods
ake in o accoun only he po en ial poin s ou side he dW adii o he a oms o de e mine
cha ges. As hey canno be de e mined undoub edly, e ec i e dW adii a e a ma e o deba e
in he li e a u e and a e ea ed di e en ly in he po en ial-based me hods. Di e en schemes
a e cu en ly in use o selec he poin s ou side he dW adii o he a oms o he molecule.
MOLECULAR ELECTROSTATIC POTENTIAL
A unca ed mul ipole expansion is a compu a ionally con enien single-cen e o malism ha
allows one o quan i a i ely compu e he deg ee o which a posi i e o nega i e es cha ge is
a ac ed o epelled by he molecule ha is being ep esen ed by he mul ipole expansion
[79]. The MEP can be compu ed exac ly on any gi en posi ion ~
68 Cha ge dis ibu ion in pep ide con o ma ions
VMEP (~ ) =
nuclei
X
k
Zk
|~ −~ k|−ZΨ(~ 0)1
|~ −~ 0|Ψ(~ 0)d~ 0(3.15)
Whe e Zkis he nuclea cha ge and he sum un o e s all he a om k. The MEP is an obse -
able, al hough in p ac ice i is a he di icul o design app op ia e expe imen s o measu e
i . Compu a ionally, he molecula elec os a ic po en ial is usually e alua ed as
VMEP (~ ) =
nuclei
X
k
Zk
|~ −~ k|−Zρ(~ 0)
|~ −~ 0 |d~ 0(3.16)
whe e ρis he elec on densi y. The pa ial a omic cha ges a e ypically de i ed om MEP.
The common no a ion MEP is eplaced wi h ESP o , ‘elec os a ic po en ial’. The ESP is
he mos ob ious p ope y o ob ain he pa ial a omic cha ges ha will be use ul o model
molecula -molecula in e ac ions a sho and long ange. All ESP cha ge- i ing schemes
in ol e de e mining a omic pa ial cha ges qk,
VESP (~ ) =
nuclei
X
k
qk
|~ −~ k|(3.17)
Al hough he e a e many posi i e aspec s o he ESP de i ed cha ges, o en hey a e ound o
be e y sensi i e o small changes o he molecula s uc u e o in he sampling o he ESP.
CHELP METHOD
In CHELP (CHa ges om ELec os a ic Po en ials) me hod [79, 145] he elec os a ic po en ial
is de e mined a a selec ed numbe o poin s a ound he molecule chosen in sphe ical shells,
1˚
A apa , o ou een nea ly symme ically placed poin s a ound each a om. Poin which all
wi hin he dW adius o any o he a oms a e disca ded due o he la ge dis o ions caused
by close p oximi y o he nuclei. Typically 100-300 poin s a e selec ed in he egion ex ending
o 3 ˚
A om he dWs su ace o he molecule is conside ed.
F om Eq. (3.16) he ESP is calcula ed quan um chemically and a leas squa e p ocedu e is
used o i he cha ge qj o each a omic cen e jin he molecule. The calcula ed classical ESP
is gi en by Eq. (3.17). The χ2
esp o be minimized by leas -squa es p ocedu e is de ined as in
Eq. (3.18).
χ2
esp =X
i
(Vi−Ei)2(3.18)
A he minimum, o all j
3.1. Theo e ical backg ound 69
∂(χ2
esp)
∂qj
= 0 (3.19)
whe e
∂(χ2
esp)
∂qj
=−2X
i
Vi−Ei
ij
= 0 (3.20)
and hus a sys em o equa ions can be o med and sol ed in ma ix o m. The bes leas
squa e i o cha ges o he elec os a ic po en ial is ob ained by inding he minimum o y,
y(q1, q2, ...qn) =
m
X
i=1
[Vi−Ei(q1, q2, ...qn)]2(3.21)
whe e mis he o al numbe o poin s o i , Viis om QC wa e unc ions as gi en in Eq. (3.16)
and Eiis he elec os a ic po en ial in he monopole app oxima ion and gi en in Eq. (3.17).
The minimum o ycan be ob ained by inding he s a iona y poin s o he Lag angian unc ion
z:
z(q1, q2, ...qn) = y(q1, q2, ...qn) + λg(q1, q2, ...qn)(3.22)
whe e gis he cons ain imposed on he i . Fo example, ollowing cons ain s can be used:
cons aining he o al cha ge, molecula dipole, he cha ges o a g oup o a oms o a pa icula
alue and ow o mo e a oms ha ing equi alen cha ges.
The linea leas squa e p oblem wi h imposed cons ain s can be sol ed using he Lag angian
mul iplie me hod. The unc ion co esponding o o al cha ge cons ain is:
g(q1, q2, ...qn) = Xqi−q o = 0 (3.23)
The ex emum poin s o za e hen ob ained by sol ing ∂z/∂λ = 0 and ∂z/∂qk= 0. A se o n+1
equa ions a e ob ained whe e nis he numbe o a oms. The solu ion o co esponding ma ix
equa ion yields he cha ges.
The CHELP me hod has o a ional in a iance p oblems due o he spa se poin sampling and
dependency upon he molecula coo dina e sys em o choosing es poin s.
CHELPG METHOD
The p inciple di e ence be ween he CHELPG (CHa ges om ELec os a ic Po en ials using a
G id based me hod) and CHELP is ha CHELPG [79, 146] p ocedu e employs a poin selec ion

70 Cha ge dis ibu ion in pep ide con o ma ions
algo i hm based upon egula ly spaced poin s. The algo i hm in ol es de ining a cube o
poin s spaced 0.3-0.8 ˚
A apa con aining he molecule and including 2.8 ˚
A o head space
on all sides. Figu e 3.1 desc ibes he si ua ion o wa e as a simple example. All poin s
ou side he 2.8 ˚
A adius a e also elimina ed om he i ing p ocedu e. The emaining poin s
o m a ela i ely homogeneous laye a ound he molecule. The elec os a ic po en ial a each
sample poin s is calcula ed analy ically om he wa e unc ion and geome y da a. These
da a a e hen used as inpu o he Lag ange mul iplie leas -squa es ou ine, which has
been cons ained o i he exac molecula cha ge, bu no o he molecula dipole momen .
CHELPG ypically u ilize 5000-6000 poin s. The CHELPG cha ges a e ob ained o all he i e
con o ma ions by a ying he g id poin s pe squa e angs om. In he p esen s udy he g id
poin s in he CHELPG me hod is a ied om 1 o 6 (poin s pe squa e angs om) in ESP i .
O
HH
28 pm 28 pm
Figu e 3.1. CHa ges om ELec os a ic Po en ials using a G id based me hod. The
dimensions o he cube a e chosen such ha he molecule is loca ed a he cen e o he
cube, adding 28.0 pm head space be ween he molecule and he end o he box in all h ee
dimensions.
One o he weak poin s o CHELPG is he ea men o la ge sys ems, in which some o he
inne mos a oms a e loca ed a away om he poin s a which he ESP is e alua ed. In such
si ua ion, a ia ions o he inne mos a omic cha ges will no lead o signi ican changes in
he ESP ou side he molecule and i ing o hese a omic cha ges will he e o e no esul in
meaning ul esul s. Rep esen a i e a omic cha ges o lexible molecules should he e o e be
de i ed as a e age alues o e se e al con o me s.
MK METHOD
The Me z Kollman (MK) me hod [79, 147, 148] uses he Connolly su ace algo i hm [149] o
calcula e a shell o a numbe o shells o poin s a a speci ied dis ance om he molecule
and hen calcula e he elec os a ic po en ial a hese poin s. The use o successi e shells
o a ious mul iples o he dWs adii, anging om 1.4 o 2.0 imes he adius (smalle
adii lead o posi i e po en ials in he lone-pai egion). The shells o 1.4, 1.6, 1.8, and 2.0
3.1. Theo e ical backg ound 71
imes he dWs adius a e used o a oid a i ac s om being oo close o he a oms. The
densi y o poin s in hese su aces depended on he size o he molecule and ypically 200-
300 poin s/molecules a e gene a ed by a ying 1-5 poin s pe squa e angs om. The Me z
Kollman me hod is desc ibed o wa e in Figu e 3.2. The wa e molecules is co e ed wi h
ou shells wi h 1.4, 1.6, 1.8, 2.0 and 2.2 imes he dWs adius. In he p esen s udy he MK
cha ges a e ob ained o all he i e con o ma ions by a ying he g id poin s and he shells.
The g id poin s a ied om 1 o 6 and he shell is a ied om 2 o 9 o he maximum limi
wi h 1.4, 1.6, 1.8, 2.0, 2.2, 2.4, 2.6, 2.8 and 3.0 imes he dWs adius.
Vdw en elope
HH
O
Shell 1 (scaling 1.4)
Shell 2 (scaling 1.6)
Shell 3 (scaling 1.8)
Shell 4 (scaling 2.0)
Shell 5 (scaling 2.2)
Figu e 3.2. Me z-Singh-Kollman me hod. The ESP is calcula ed a a numbe o g id
poin s loca ed on se e al shells a ound he molecule. The shells a e cons uc ed as an
o e lay o an de Waals sphe es a ound each a om. The shells o 1.4, 1.6, 1.8,2.0 and 2.2
imes he dWs adius a e shown.
The majo s eng h o he elec os a ic po en ial cha ges is ha hey op imally ep oduce
he in e molecula in e ac ion p ope ies o he molecules wi h a simple wo-body addi i e
po en ial, i sui ably accu a e le el o quan um chemical calcula ion is used o de i e he ESP
a ound he molecule. The majo weakness o hese cha ges is hey a e no easily ans e able
be ween common unc ional g oups in ela ed molecules [133].
RESTRAINED ELECTROSTATIC POTENTIAL METHOD (RESP)
In RESP me hod [150], he es ain s a e in oduced in he o m o penal y unc ion in o he
i ing p ocedu e conside ably o o e come he limi a ions o o he ESP cha ge me hods.
Wi h he cha ge i ing p ocedu e χ2
esp (see sec ion 3.1.2), an addi ion penal y unc ion χ s is
added. So χ2is o be minimized now
χ2=χ2
esp +χ2
s (3.24)
72 Cha ge dis ibu ion in pep ide con o ma ions
and he leas squa es minimum is de ined as
∂(χ2)
∂qn
=∂(χ2
esp)
∂qn
+∂(χ2
s )
∂qn
(3.25)
The simple ha monic Eq. (3.26) o hype bolic Eq. (3.27) unc ion is used as a penal y unc ion.
χ2
s =aX
m
(q0n−qn)2(3.26)
χ2
s =aX
m
((q2
n+b2)1/2−b)(3.27)
whe e ais he scaling ac o de e mining he s eng h o he es ain and q0is he a ge
o he es ain . Mulliken cha ges a e used as a a ge cha ges [150]. Gene ally he RESP
cha ges a e used in molecula mechanics o m any o ganic and bio-o ganic sys ems.
In he p esen s udy, he RESP cha ges a e ob ained by using bo h Mulliken and NPA cha ges
as ini ial cha ges o he ha monic es ain . The RESP cha ges a e ob ained om bo h elec-
os a ic po en ials calcula ed om CHELPG and MK me hods.
MULTIPOLE DERIVED CHARGES
Ano he se o cha ges used in he wo k is mul ipole de i ed cha ges. The mul ipole de i ed
cha ges (MDC) a e used o ob ain he sol a ion ene gies based on FDPB me hod combina-
ion wi h DFT le el o heo y (see sec ion 5.3.1). The MDC analysis is based on he ideas
used o he Dipole P ese ing Cha ge analysis [151] bu o mula ed by using mo e accu a e
a omic dipoles. The e a e h ee s ages in ol ed in he me hod: Fi s he molecula elec on
densi y is w i en as a sum o a omic densi ies. Second, om hese a omic densi ies a se
o a omic mul ipoles a e de ined, which is used o ge he elec os a ic po en ial ou side he
cha ge dis ibu ion. Thi d, hese a omic mul ipoles a e econs uc ed by using a scheme ha
dis ibu es cha ges o e all a oms o ep oduce hese mul ipoles exac ly. I is claimed ha
his me hod can o e come he d awbacks o he ESP me hods because his me hod uses he
a omic mul ipole expansion o e molecula mul ipole expansion. A de ailed desc ip ion o
a omic mul ipole expansion me hod is gi en in Re [152]. The MDC used in he p esen wo k
a e cha ges de i ed om monopole (MDCm), dipole (MDCd) and quad upole (MDCq) momen s.
3.2 STRUCTURE PREPARATION AND COMPUTATIONAL METHODS
The ini ial s uc u e o he ipep ide Ala-Asn-Ala was builded using CHARMM p og am [153]
package. The C- e minal alanine and he N- e minal alanine a e capped wi h ace ela e and
me hyla e g oups espec i ely. Fi e di e en con o ma ions a e gene a ed by changing he
3.2. S uc u e p epa a ion and Compu a ional Me hods 73
dihed al angles o a oms 6, 7, 17 and 18 om 120◦ o 180◦by s ep o 15◦. The ipep ide
conside ed o he cha ge calcula ion is shown in Figu e 3.3
1Ala − 2Asn − 3Ala
N
CA
O
N
CA
CG OD1
C
N
CA
NT
C
CAT
CB
ND2
1
2
3
4
5
6
7
8
9
11
12
13
15
10
16
17
18
19
21
22 23
CAY
CY
CB
C
OY CB
O
O
14
20
Figu e 3.3. The ipep ide Ala-Asn-Ala conside ed o he cha ge calcula ion is shown.
The ipep ide Ala-Asn-Ala is shown wi h he PDB a om ype.
Each con o ma ion was op imized by ixing he dihed al angle. All he con o ma ions we e
op imized by hyb id densi y unc ional heo y B3LYP wi h 6-31G* basis se . The geome y op-
imiza ions we e pe o med using GAUSSIAN 03 p og am [154]. The CHELPG and MK cha ges
we e also ob ained using he same p og am package. The RESP cha ges we e ob ained using
RESP p og am [155].
The comple e se o calcula ions pe o med on ipep ide Ala-Asn-Ala a e shown in Figu e 3.4.
Popula ion analysis we e pe o med o all he con o ma ions. The choice o he popula ion
analysis me hod includes he adi ional Mulliken popula ion analysis [139, 140] and NPA
[143, 144], bo h o hese me hods a e based on he molecula wa e unc ion, bu NPA uses he
na u al a omic basis unc ion o o e come he d awback o he Mulliken popula ion analysis.
Va ious me hods a e a ailable o i he molecula elec os a ic po en ial o he g id poin s
wi h i ing p ocedu es. The h ee me hods conside ed in ou s udies a e he CHELPG [145] ,
MK [147, 148], and RESP [156] me hods. The CHELPG me hod is a g id based me hod whe e
poin s a e selec ed in a egula ly spaced cubic g id. To ob ain cha ges independen o he
con o ma ions o he ipep ide, he sampling poin s we e inc eased by a ying he g id poin s
in CHELPG and MK me hod and he cha ges we e calcula ed o each con o ma ion.
In CHELPG me hod, he g id poin s pe squa e angs om a e a ied om 1 o 6 in he ESP i
by in oking he GAUSSIAN 03 op ion IOp(6/42=N) whe e N is he poin s pe squa e angs om.
MK me hod uses he Connolly su ace algo i hm [149] o calcula e a shell o a numbe o shells
80 Cha ge dis ibu ion in pep ide con o ma ions
MODEL 3
In model 3, he ESP ob ained om MK me hod by a ying g id poin s om 1-6 a e used o
calcula e RESP cha ges and he Mulliken cha ges a e used as ini ial cha ges o ha monic
es ain . As in model 1, when less g id poin s and s ong es ain 0.5 o 0.1 a e used in
he i s s age i , he RESP cha ges app oach he Mulliken cha ges. When he g id poin s a e
inc eased om 3 o 4, he e in no change in he cha ge and inc easing he sampling poin s
do no ha e any e ec on he cha ges. The calcula ions show ha a leas 3 poin pe squa e
angs om should be used o ge a s able cha ges and he es ain ac o o 0.0005 should be
used o ge a easonable cha ges.
MODEL 4
In Model 4, he NPA cha ges a e used as ini ial cha ges wi h he combina ion o ESP om
MK me hod. The ends a e almos same. Analysis o he RESP cha ges also show ha mo e
han 3 poin s pe squa e angs om should be used o ge s able cha ge wi h combina ion o
es ain ac o 0.0005.
MODEL 5
In Model 5, he RESP cha ges a e calcula ed by using he ESP ob ained om MK me hod by
a ying he shells om 2 o 9. The e is an i egula end in he RESP cha ges wi h espec o
he con o ma ions. The analysis shows ha a leas mo e han 4 shells should be used o ge
a s able cha ges o each con o ma ions.
All he models show ha he RESP cha ges app oach he ini ial cha ges (Mulliken and NPA)
when s ong es ain is used and he RESP cha ges app oach he ESP cha ges (CHELPG and
MK) when weak es ain is used. The es ain beyond 0.00005 does no make any change
in he cha ges.
3.4 CONCLUSIONS
The Mulliken, NPA, CHELPG, MK and RESP a omic cha ges a e de i ed o di e en con o -
ma ion o he ipep ide Ala-Asn-Ala. All he pa ial a omic cha ges we e ob ained om hyb id
densi y unc ion heo y B3LYP wi h 6-31G* basis se . The sensi i i y o cha ges wi h di e en
con o ma ions and he in luence o wa e- unc ion based (Mulliken and NPA) and elec os a ic
po en ial based cha ge me hods (CHELPG, MK and RESP) ha e been explo ed in de ail. The
ac o s in luencing he cha ges a e a ied which includes inc easing he poin s pe squa e
angs om bo h in CHELPG and MK me hod and he shells a e a ied om 2-9 he maximum
limi in MK me hod. The RESP cha ges we e calcula ed by using he elec os a ic po en ial
om CHELPG and MK me hod using Mulliken and NPA cha ges as ini ial cha ges wi h ha -
monic unc ion as penal y unc ion. The RESP cha ges we e also ob ained by using di e en
scaling ac o om a s ong o weak es ain .
The a omic cha ges ob ained om bo h CHELPG and MK me hod a e dependen on he con-
o ma ions. The Mulliken and NPA cha ges show less con o ma ional dependency compa e o

3.4. Conclusions 81
he CHELPG and MK cha ges. Di e ence in poin s selec ion o he ESP i s can d ama ically
a ec he pa ial cha ges. The magni udes o he calcula ed poin cha ges we e ound o be
somewha dependen on he choice o poin s pe squa e angs om and he shells. The mag-
ni ude o he cha ges a e mo e o less same o he bo h CHELPG and MK me hod when g id
poin s o 3 o 4 pe squa e angs om we e used. The esul s show ha minimum 2 o 3 g id
poin s pe squa e angs om should be used in bo h me hods and inc easing he g id poin s o
maximum limi 6 does no change he magni ude o cha ge much. The MK cha ges show less
dependen on con o ma ion when minimum 3 o 4 shells we e used and he cha ges emains
almos same when shells a e inc eased o maximum limi 9. The esul s show ha minimum
3 o 4 shells should be used in MK me hod.
The RESP cha ges a e mainly dependen on he s ong o weak es ain used. When s ong
es ain s a e used he cha ges a e close o he Mulliken o NPA cha ges and i weak e-
s ain s a e used he cha ges a e close o he ESP cha ges. The magni ude o he cha ges
exhibi much smalle luc ua ions when es ain ac o o 0.0005 was used in he i s s age
i . The es ain weigh o 0.0005 appea s o be op imal o ob aining RESP cha ges o ipep-
ide Ala-Asn-Ala. The model 3 combina ion wo ks ine o ob ain RESP cha ges o ipep ide
Ala-Asn-Ala. The cha ges p edic ed by he Mulliken popula ion analysis and NPA schemes
a e la ge han hose p edic ed by he ESP schemes.
This conclusions should be ca e ully econside ed o o he sys ems wi h pola bonds in which
elec ons co ela ion e ec s may be mo e signi ican .
82 Cha ge dis ibu ion in pep ide con o ma ions
A.1
A.2
A.3
B.1
B.2
B.3
Figu e 3.6. A omic cha ges o hea y a oms in he backbone and side chains o di -
e en con o ma ions o ipep ide Ala-Asn-Ala ob ained om CHELPG and MK me h-
ods a e shown in he plo s. (A.1). CHELPG cha ges o hea y a oms in he backbone
and side chain whe e 1 poin pe squa e angs om is used in he CHELPG me hod. (A.2).
CHELPG cha ges o hea y a oms in he backbone and side chain whe e 2 poin s pe squa e
angs om a e used in he CHELPG me hod. (A.3). CHELPG cha ges o hea y a oms in he
backbone and side chain whe e 3 poin s pe squa e angs om a e used in he CHELPG
me hod. (B.1). MK cha ges o hea y a oms in he backbone and side chain whe e 1 poin
pe squa e angs om is used in he MK me hod. (B.2). MK cha ges o hea y a oms in he
backbone and side chain whe e 2 poin s pe squa e angs om a e used in he MK me hod.
(B.3). MK cha ges o hea y a oms in he backbone and side chain whe e 3 poin s pe
squa e angs om a e used in he MK me hod. The black ci cle (◦) co esponds o con o -
ma ion 1 (1-20), ed squa e (2) co esponds o con o ma ion 2 (2-135), g een diamond ()
co esponds o con o ma ion 3 (3-150), blue iangle up (M) co esponds o con o ma ion 4
(4-165), magen a s a (∗) co esponds o con o ma ion 5 (5-180)
3.4. Conclusions 83
A.4
A.5
A.6
B.4
B.5
B.6
Figu e 3.7. A omic cha ges o hea y a oms in he backbone and side chains o di -
e en con o ma ions o ipep ide Ala-Asn-Ala ob ained om CHELPG and MK me h-
ods a e shown in he plo s. (A.4). CHELPG cha ges o hea y a oms in he backbone
and side chains whe e 4 poin s pe squa e angs om a e used in he CHELPG me hod.
(A.5). CHELPG cha ges o hea y a oms in he backbone and side chains whe e 5 poin s
pe squa e angs om a e used in he CHELPG me hod. (A.6). CHELPG cha ges o hea y
a oms in he backbone and side chains whe e 6 poin s pe squa e angs om a e used in he
CHELPG me hod. (B.4). MK cha ges o hea y a oms in he backbone and side chain whe e
4 poin s pe squa e angs om a e used in he MK me hod. (B.5). MK cha ges o hea y
a oms in he backbone and side chains whe e 5 poin s pe squa e angs om a e used in he
MK me hod. (B.6). MK cha ges o hea y a oms in he backbone and side chains whe e 6
poin s pe squa e angs om a e used in he MK me hod. The black ci cle (◦) co esponds o
con o ma ion 1 (1-20), ed squa e (2) co esponds o con o ma ion 2 (2-135), g een diamond
() co esponds o con o ma ion 3 (3-150), blue iangle up (M) co esponds o con o ma ion
4 (4-165), magen a s a (∗) co esponds o con o ma ion 5 (5-180)
84 Cha ge dis ibu ion in pep ide con o ma ions
C.1
C.2
C.3
C.4
C.5
Figu e 3.8. A omic cha ges o hea y a oms in he backbone and side chains o di e -
en con o ma ions o ipep ide Ala-Asn-Ala ob ained om CHELPG and MK me hods
is shown in he plo s. (C.1). MK cha ges o hea y a oms in he backbone and side chains
o con o ma ion 1 (1-120) whe e shells a ied om 2-9 and 4 poin s pe squa e angs om is
used in he MK me hod. (C.2). MK cha ges o hea y a oms in he backbone and side chains
o con o ma ion 2 (1-135) whe e shells a ied om 2-9 and 4 poin s pe squa e angs om is
used in he MK me hod. (C.3). MK cha ges o hea y a oms in he backbone and side chains
o con o ma ion 3 (1-150) whe e shells a ied om 2-9 and 4 poin s pe squa e angs om is
used in he MK me hod. (C.4). MK cha ges o hea y a oms in he backbone and side chains
o con o ma ion 4 (1-165) whe e shells a ied om 2-9 and 4 poin s pe squa e angs om is
used in he MK me hod. (C.5). MK cha ges o hea y a oms in he backbone and side chains
o con o ma ion 5 (1-180) whe e shells a ied om 2-9 and 4 poin s pe squa e angs om is
used in he MK me hod.
3.4. Conclusions 85
D.1
D.2
D.3
D.4
D.5
Figu e 3.9. A omic cha ges o hea y a oms in he backbone and side chains o di -
e en con o ma ions o ipep ide Ala-Asn-Ala ob ained om RESP me hods is shown
in he plo s. (D.1). RESP cha ges o hea y a oms in he backbone and side chains o con-
o ma ion 1 (1-120). (D.2). RESP cha ges o hea y a oms in he backbone and side chains
o con o ma ion 2 (1-135). (D.3). RESP cha ges o hea y a oms in he backbone and side
chains o con o ma ion 3 (1-150). (D.4). RESP cha ges o hea y a oms in he backbone and
side chains o con o ma ion 3 (1-165). (D.5). RESP cha ges o hea y a oms in he backbone
and side chains o con o ma ion 3 (1-180). The black ci cle (◦) co esponds o RESP cha ges
whe e ESP om CHELPG me hod is used wi h combina ion o Mulliken cha ges as ini ial
cha ges. The ed squa e (2) co esponds o o RESP cha ges whe e ESP om CHELPG
me hod is used wi h combina ion o NPA cha ges as ini ial cha ges. The g een diamond ()
co esponds o RESP cha ges whe e ESP cha ges MK me hod is used wi h combina ion o
Mulliken cha ges as ini ial cha ges. The blue iangle up (M) co esponds o RESP cha ges
whe e ESP om MK me hod wi h combina ion o NPA cha ges as ini ial cha ges.

86 Cha ge dis ibu ion in pep ide con o ma ions
CHAPTER 4
PROTONATION AND REDOX POTENTIALS OF
CYTOCHROME cNITRITE REDUCTASE
Cy och ome coxidase is a la ge memb ane p o ein designed o u ilize he ene gy o elec on
ans e and oxygen educ ion o pump p o ons ac oss he memb ane. The molecula mech-
anism o he ene gy con e sion p ocess is no been well unde s ood [8, 40–42]. Elec os a ic
calcula ions on o he p o eins wi h simple , be e esol ed s uc u es can help o unde s and
he possible mechanism o elec on ans e in cy och ome coxidase.
Cy och ome cni i e educ ase can se e as such a simple sys em. Cy och ome cni i e
educ ase ca alyzes he six elec on educ ion o ni i e o ammonia. This second pa o he
espi a o y pa hway o ni a e ammoni ica ion is he key s ep in he biological ni ogen cycle
[158–161].
Cy och ome cni i e educ ase is a mul iheme cenzyme and i s ac i e si e is a p o opo phy in
IX which is co alen ly linked o he p o ein backbone. A lysine was ound o eplace he usual
his idine as a p oximal ligand o he heme i on [162–164]. The i e hemes in he monome
o cy och ome cni i e educ ase a e in close con ac wi h Fe-Fe dis ance o 9 o 12.8 ˚
A. The
cy och ome cni i e educ ase dime is shown in Figu e 4.1 and he a angemen o hemes in
he dime is shown in Figu e 4.2. The Heme I o ms he ac i e si e and he o he hemes, Hemes
II, III and IV a e almos coplana wi h he ca aly ic Heme I. Hemes II and V a e a he apa
and a e no coplana wi h Hemes I, III and IV. All he hemes excep Heme I a e bis-his idinyl-
coo dina ed and linked o he pep ide backbond by hioe he bonds o he cys eine esidues o
a classical heme-binding mo i o pe iplasmic p o eins, Cys-X1-X2-Cys-His. Heme I howe e
has he binding mo i Cys-X1-X2-Cys-Lys, whe e he ni ogen a om o he lysine eplacing ha
o his idine.
The cy och ome cni i e educ ase ca alyzes he educ ion o bo h ni i e and sul ide wi h
high speci ic ac i i y and ge s elec ons om he memb anous quinol pool, he eby gene a ing
a p o on mo i e o ce. Heme I is clea ly he subs a e-binding si e and p e ious spec oscopic
s udies [165, 166] in a simila ni i e educ ase had p o ed ha one o he heme was high-
spin and i was he ac i e si e o he enzyme. The cy och ome cni i e educ ase can be ound
in he pe iplasm and o ms a s able, memb ane associa ed complex wi h i s elec on dono
N H, a membe o he NapC/Ni T amily o he e aheme cy och omes [167, 168].
Cy och ome cni i e educ ase possesses mo e han 180 p o ona able g oups, o which en
o hem a e he heme p opiona es in each monome . In o al, cy och ome cni i e educ ase
87
88 P o ona ion and edox po en ials o cy och ome cni i e educ ase
con ains 10 c- ype hemes. In en hemes, eigh hemes a e c- ype heme i.e., wo his idines
coo dina ing o he i on and in o he hemes he lysine is co alen ly linked as a p oximal
ligand o he heme i on o he c- ype heme. Al hough all c- ype hemes a e o he same chemical
na u e, hei midpoin po en ials di e .
The cy och ome cni i e educ ase is a homodime ic enzyme wi h 10 c- ype hemes which
a e a anged such ha he nea es neighbo s a e in close p oximi y. I has been epo ed
[164, 169] ha he educ ion po en ial o he 10 heme cen e s anges om ca. -30 o -320 mV
in cy och ome cni i e educ ase om Esche ichia coli. The p o ein ilm ol amme ic expe i-
men s e ealed ha he heme oxida ion s a e has a p o ound, and o en unan icipa ed e ec
on he in e ac ions wi h subs a e molecules, ni i e, hyd oxyl amine and inhibi o cyanide
[169]. The oxida ion p obabili ies o he hemes and p o ona ion p obabili ies o he heme p o-
piona es and edox po en ials o he hemes in cy och ome cni i e educ ase a e calcula ed
o s udy he edox po en ial p o iles. The elec os a ic in e ac ions be ween he p o ona able
and edox-g oups a e s udied by sol ing he Poisson-Bol zmann equa ion using ini e di e -
ence me hod. The elec os a ic calcula ions a e pe o med in cy och ome cni i e educ ase, a
less complex p o ein be o e pe o ming calcula ions on he complex p o ein like cy och ome c
oxidase which con ains ou edox cen e s CuA, heme a, heme a3and CuB. The aim o he
p esen s udy is o ob ain he edox mid-poin po en ials o he hemes o unde s and he
edox po en ial p o iles o hemes when he ca aly ic Heme I was blocked wi h he inhibi o
cyanide.
Figu e 4.1. The ni i e educ ase dime and he heme a angemen . A on iew wi h
he dime axis o ien a ed e ically, he i e hemes in each monome (s ick model) and he
Ca2+ (yellow) a e shown.
4.1. Hemes and he calcium binding si e in cy och ome cni i e educ ase 89
Heme I
Heme III
Heme II
Heme IV
Heme V
Figu e 4.2. The a angemen o hemes in ni i e educ ase dime is shown. The
o e all o ien a ion o hemes co esponds o ni i e educ ase dime is shown. Hemes in
he le monome a e labeled acco ding o hei a achmen o he p o ein chain. The same
labels a e used h oughou he s udy.
4.1 HEMES AND THE CALCIUM BINDING SITE IN CYTOCHROME c
NITRITE REDUCTASE
Hemes I, III and IV a e close enough o allow di ec π-elec on in e ac ion o he po phy in
ings. The p opiona e side chains o heme I o m pa o he ac i e-si e ca i y, while hose
o Heme IV a e exposed o he sol en and he Heme III p opiona es a e hyd ogen-bonded
inside he p o ein. All he hemes show sligh dis o ion om ideal plana i y and i is mo e
p onounced in Heme II and less in Heme I. I is sugges ed [162] ha Heme II could unc ion as
he en y poin o elec ons (see Figu e 4.2). The Ca2+ binding si e appea s o be an essen ial
s uc u al ea u e in he o e all a chi ec u e o he enzyme and he egion su ounding he
calcium binding si e is one o he mos highly conse ed pa s o he whole sequence [163].
The calcium binding si e o cy och ome cni i e educ ase wi h he impo an esidues a e
shown in Figu e 4.3. The esidues Ty 281, Lys274, Gln276 and His277 ac as ligands o
he calcium binding si e. I was p oposed ha he calcium binding si e hold he key esidues
needed o ca alysis [162]. The se o y osine esidues nea he calcium binding si e and
ac i e si e migh play a ole in he eac ion mechanism o cy och ome cni i e educ ase by
o ming possible adical in e media es o he s epwise educ ion o ni i e o ammonia [170].
96 P o ona ion and edox po en ials o cy och ome cni i e educ ase
Hemes
Mid-poin edox po en ials
in [mV]
Oxd.aRed.bExp.c
Heme II -210 -210 -323 (Heme IV)
Heme III -30 -50 -37 (Heme II)
Heme IV -270 -270 -107 (Heme III)
Heme V -210 -210 -323 (Heme V)
Table 4.3. The calcula ed mid-poin edox po en ials o Hemes II o V a pH=7. The
mid-poin edox po en ials ob ained o he Hemes II-V in he educed and oxidized s a es
o Heme I a e gi en. The expe imen al mid-poin edox po en ials a e a anged whe e he
expe imen al alues ma ch closly wi h he calcula ed mid-poin po en ials. The o de o
assignmen o edox-po en ial o a pa icula heme is e y con o e sial [169].
aThe mid-poin edox po en ials we e calcula ed o Hemes II-V by ixing he ca aly ic Heme
I in he oxidised s a e.
bThe mid-poin edox po en ials we e calcula ed o Hemes II-V by ixing he ca aly ic Heme
I in he educed s a e.
cThe expe imen al mid-poin edox po en ials o Hemes II-V om Re [169].
Figu e 4.6. Oxida ion p obabili ies o di e en hemes in cy och ome cni i e educ-
ase om W. succinogenes a pH=7. The hemes do no show a s anda d Ne ns i a ion
beha io because o he in e ac ion wi h each o he and wi h he i a able g oups in hei
icini y.
o hese in e ac ions, i a ion cu es o i a able g oups in p o eins can de ia e conside -
ably om sigmoidal Ne ns o Hende son-Hasselbalch i a ion cu es o isola ed i a able
g oups. Thus, i is no always possible o assign a unique mid-poin po en ial o pKa alue
o a speci ic i a able g oup. The e o e, he pH alue a which he p o ona ion p obabili y o
he p o ona able g oup is 0.5 is o en used o desc ibe he i a ion beha io . Likewise, he

4.3. Resul s and Discussion 97
solu ion edox po en ial E◦
1/2a which he oxida ion p obabili y o he edox-ac i e g oup is
0.5 is used o desc ibe he i a ion beha io . The oxida ion p obabili ies o di e en hemes in
cy och ome cni i e educ ase a pH=7 a e gi en in Figu e 4.6. The i a ion cu es de ia es
om he s anda d Ne ns i a ion cu es. This de ia ion is expec ed because o he in e -
ac ion be ween he hemes and he in e ac ion o hemes wi h o he p o ona able g oups and
heme p opiona es.
Heme II Heme III
Heme IV Heme V
Figu e 4.7. Oxida ion p obabili ies o he Hemes II-V. Calcula ed oxida ion p obabili ies
o heme II-V o cy och ome cni i e educ ase om W. succinogenes. The p obabili ies a e
colo coded as indica ed by he scale nex o he diag am. The g een colo indica es ha he
oxida ion p obabili y o he hemes is abou 0.5.
The compu ed edox po en ials o hemes ange om –270 o –30 mV. Howe e he mid-poin s
canno be compa ed wi h he expe imen al mid-poin po en ial because he expe imen al mid-
poin po en ials we e ob ained using ni i e as inhibi o . The edox po en ials in he p esen
s udies we e compu ed by blocking he ca aly ic Heme I by a cyanide whe e he blocking
p e en s he elec on ans e om he ca aly ic heme o he inhibi o . Bu e al. [169]
ob ained edox po en ial o –107 mV o Heme III. The edox po en ial o –30 mV was ob ained
o Heme III om p esen calcula ions. On compa ing he edox po en ials o Hemes II-V, in
98 P o ona ion and edox po en ials o cy och ome cni i e educ ase
he oxidized and in he educed s a e o Heme I, he edox po en ials emain almos same
expec o Heme III whe e he di e ence is –20 mV. Hemes II and V ha e same mid-poin
po en ial e en hough he dis ance be ween he hemes a e mo e compa e o heme IV and V.
The Heme III has high edox po en ial compa ed o o he hemes.
The oxida ion p obabili ies o he Hemes II-V a e shown in Figu e 4.7. Wi hin he p o ein,
he heme mid-poin edox po en ials a e a ec ed by cha ges on he ionizable amino acids,
pola g oups and o he hemes. The mid-poin po en ial o he Hemes II-V dec eases wi h he
inc ease in pH. The oxida ion p obabili ies o he Hemes II-V (see Figu e 4.7) show ha edox
s a es o he hemes a e s ongly depend on he pH. The pH dependence is less p onounced in
Hemes II and V which a e no coplana wi h o he hemes and mo e p onounced in Heme III
and IV.
PROTONATION PROBABILITIES OF HEMES I-V PROPIONATES
a.1. b.1.
Figu e 4.8. P o ona ion p obabili ies o he hemes p opiona es A and D o Heme I.
(a.1)-(b.1). Calcula ed p o ona ion p obabili ies o heme p opiona es A and D o Heme I. The
p obabili ies a e colo coded as indica ed by he scale nex o he diag am. The g een colo
indica es ha he p o ona ion p obabili y o he esidues is abou 0.5.
The p o ona ion p obabili ies o heme p opiona es (A and D) om Heme I and Heme II-V a e
shown in Figu e 4.8 and 4.10 espec i ely. The in e ac ion ene gies be ween he hemes and
heme p opiona es a e gi en in Figu e 4.9. The p o ona ion p obabili ies o Heme I p opiona es
a e s ongly depend he edox s a es o o he hemes. The p o ona ion p obabili ies o Heme
I p opiona e A (plo a.1) s ongly depend on bo h he pH and edox po en ial compa ed o
Heme I p opiona e D (plo b.1). Compa ing he p o ona ion p obabili ies o heme p opiona es
A o Hemes II-V, Heme IV and Heme III p opiona es show s ong dependence on bo h he
pH and edox po en ial whe e as Heme II and V show dependence o he pH only. In heme
p opiona es D o Hemes II-V, only Heme IV and V show s ong dependence on pH and Heme II
and III p opiona es a e comple ely dep o ona ed and p o ona ed espec i ely. The in e ac ion
be ween he heme cha ges depend on how deeply he hemes a e bu ied in he p o ein as well
as he dis ance be ween he hemes. The s abiliza ion o he hemes by p opionic acids can
4.4. Conclusions 99
Figu e 4.9. In e ac ion ene gies o Hemes (I-V) wi h heme p opiona es A and D in
cy och ome cni i e educ ase. The plo shows he in e ac ion ene gies o Hemes (I-V)
wi h heme p opiona es A and D in cy och ome cni i e educ ase.
also ha e in luence on he edox po en ial o he hemes. The in e ac ion ene gies be ween he
hemes a e highe compa ed o he heme p opiona es.
The in e ac ion ene gies be ween Heme III and IV is a ound 3 kcal/mol due o he s aking
posi ion o Heme III and IV wi h espec o Heme I and V. The analysis o he in e ac ion
be ween hemes show ha hey ha e in luence on he heme mid-poin po en ials. The coupling
o he pKashi o p opiona es wi h edox po en ials o he hemes a e suppo ed om he
s uc u e because o he p oximi y o hemes o hese p opiona es.
4.4 CONCLUSIONS
Elec os a ic calcula ions we e pe o med on a simple , be e esol ed c ys al s uc u e o
cy och ome cni i e educ ase be o e doing calcula ions on he complex sys em cy och ome c
oxidase. The edox po en ials we e calcula ed and in he ange be ween –270 o –30 mV. The
edox po en ials a e almos same o he oxidized and educed enzyme excep o Heme III. The
oxida ion and p o ona ion p obabili ies o he hemes and heme p opiona es we e calcula ed.
The edox po en ials o he hemes a e sensi i e o hei loca ion and a e also in luenced by he
p o ona ion p obabili ies o he su ounding i a able g oups and as well as he p opiona es
in he p o ein. Heme I, III and IV a e close enough o allow di ec π-elec on in e ac ions o
he po phy in ing.
The mid-poin po en ial o he Hemes II-V dec eases wi h inc ease in pH. The oxida ion p ob-
abili ies o he Hemes II-V show less dependence on pH and Heme III and IV shows s ong
dependence. The oxida ion p obabili ies show ha he edox s a es o he hemes a e s ongly
100 P o ona ion and edox po en ials o cy och ome cni i e educ ase
depend on he pH. The p o ona ion p obabili ies o Heme I p opiona es a e s ongly depend on
he edox s a es o o he hemes. On compa ing he p o ona ion p obabili ies o p opiona es A
o Hemes II-V, p opiona es o Heme IV and Heme III show s ong dependence on bo h he pH
and edox po en ial and whe e as Heme II and V show dependence on he pH only. In heme
p opiona es D o Hemes II-V, only Heme IV and V show s ong dependence on pH and Heme II
and III p opiona es a e comple ely dep o ona ed and p o ona ed espec i ely. The in e ac ion
be ween he hemes and heme p opiona es play a majo ole in uning he edox po en ials o
he hemes.
4.4. Conclusions 101
a.2.
a.3.
a.4.
a.5.
b.2.
b.3.
b.4.
b.5.
Figu e 4.10. P o ona ion p obabili ies o he heme p opiona es A and D o Hemes
II-V. (a.2)-(a.5) Calcula ed p o ona ion p obabili ies o heme p opiona es A o Hemes II-V.
(b.2)-(b.5) Calcula ed p o ona ion p obabili ies o heme p opiona es D o Hemes II-V. The
p obabili ies a e colo coded as indica ed by he scale nex o he diag am. The g een colo
indica es ha he p o ona ion p obabili y o he esidues is abou 0.5.

102 P o ona ion and edox po en ials o cy och ome cni i e educ ase
CHAPTER 5
PKaCALCULATIONS OF BINUCLEAR CENTER OF
CYTOCHROME cOXIDASE USING DFT
CALCULATIONS
Knowledge o he pKa alues o ionizable g oups a e impo an o an unde s anding o many
phenomena o chemis y o bo h he gas phase and he solu ion. The pKa alues a e o
pa icula in e es in elucida ing he eac ion mechanisms, especially hose in ol ing p o on
ans e s and in e p e ing he binding o subs a es o inhibi o s o enzymes. Expe imen al
de e mina ions o indi idual pKa alues a e di icul in complex sys ems. One case in poin
conce ns he di ec measu emen o pKa alues o i a ing g oups o ca aly ically impo an
esidue o subs a e in he enzyme-subs a e complexes [101].
Cy och ome coxidase con ains ou edox cen e s CuA, heme a, heme a3and CuB. I ca alyzes
he educ ion o oxygen o wa e . In he cou se o i s ca aly ic ac i i y, elec ons deli e ed
by cy och ome ca e ans e ed ia CuAand heme a o he Fea3-CuB he binuclea cen e ,
whe e he educ ion o oxygen akes place. Fo e e y molecula oxygen a o al o eigh p o ons
a e consumed om he ma ix side o he mi ochond ion: ou p o ons a e deli e ed o he
binuclea cen e o wa e o ma ion and ano he ou p o ons a e ansloca ed ac oss he
memb ane (see Eq. (5.1)).
4 cy c ed + 8 H+
(in)+ O2−→ 4 cy cox + 2 H2O + 4 H+
(ou )(5.1)
A de ailed pic u e o he unde lying mechanism o how he p o ons a e ansloca ed o he
binuclea cen e and how hey each he opposi e side o he enzyme is a ma e o deba e
[57, 179–181]. The analysis o he many a ailable c ys allog aphic s uc u es o cy och ome c
oxidase does no show any speci ic pa hways connec ing he D-channel o he binuclea cen e
o leading p o ons o he ex e io side o he memb ane. Recen s udies ha e p oposed he
di ec in ol emen o he one o he CuBhis idine ligands His291 in bo ine hea o His334 in
R. sphae oides, a p o on loading si e in he pumping mechanism [1, 2, 8, 12].
Based on elec os a ic calcula ions S ucheb ukho e al. p oposed ha CuBbound imidazole
ing o His334 play a key ole in he p o on pumping mechanism [1, 2]. The cen al ea u e o
he p oposed mechanism is ha he pKa alues o he imidazole a y signi ican ly depending
on he edox s a e o he me als in he binuclea cen e [5, 6]. The p oposed model s a es
103
104 pKacalcula ions o binuclea cen e o cy och ome coxidase using DFT calcula ions
ha upon educ ion o he binuclea cen e , His334 dep o ona es a i s Nδ1[5]. F om DFT
calcula ions on he models o CuBin aqueous solu ion, hese au ho s ound ha he pKa alue
o His334 is 8.6 o he oxidized CuBcen e and 13.2 o he educed CuBcen e . Ano he DFT
s udy [182] on he CuBcen e ob ained a pKa alue o abo e 13.5 o His334 in he oxidized
s a e making i unlikely ha a edox-coupled p o ona ion s a e change o His334 di ec ly
in ol ed in he p o on pumping mechanism. The disc epancy be ween hese wo s udies
lead o some deba e in he ield [4, 183]. Howe e in he wo s udies [5, 182], he c ys al
coo dina es we e no ully elaxed leading o un ealis ic bond leng hs. Mo eo e in he la e
s udy [183], i was no conside ed ha some esidues o he cy och ome coxidase can adop
a p o ona ion ha de ia es om hei usual p o ona ion in aqueous solu ion. Siegbahn and
co-wo ke s, pe o med hyb id densi y unc ional heo y calcula ions using B3LYP unc ional
o s udy he ene ge ics o he p o on ansloca ion in cy och ome coxidase [184]. They also
compu ed he edox po en ials o he me al cen e s and he y osyl adicals, as well as pKa
alues o impo an g oups along he ansloca ion pa hs [184, 185]. Ryde and co-wo ke s
sys ema ically in es iga ed how he axial ligand in heme p o eins in luences he geome y,
elec onic s uc u e and spin s a es o he ac i e si e using densi y unc ional B3LYP me hod
and medium-sized basis se s [186].
To in es iga e he eac ion mechanism and as well as he ole o he his idines bound wi h
he CuBcen e and he his idine coo dina ing he heme a3, we calcula ed he pKa alues o
he Cu-bound imidazole in a ious CuBand Fea3complexes using densi y unc ional heo y
in combina ion wi h classical con inuum elec os a ics models. In he p e ious s udies only,
he pKa alue o His334 was de e mined [5, 6, 182] and no all possible mic oscopic pKa
alues we e calcula ed in o de o check o in e nal consis ency. In he p esen s udy, we
calcula ed all mic oscopic pKa alues o he CuBand heme a3cen e bo h in he educed and
in he oxidized s a es [187]. The Fini e Di e ence Poisson Bol zmann (FDPB) me hod and
conduc o -like pola izable con inuum model (C-PCM) a e used o de e mine he sol a ion ee
ene gies in aqueous solu ion. The dependence o he sol a ion ee ene gies on he pa ame e s
a e explo ed by compa ing hese wo sol a ion models. The p esen s udy a emp s o ind
eliable me hod o ob ain p ope pKa alues o he CuBcen e in aqueous solu ion.
5.1 PKaCALCULATIONS
pKacalcula ions a e used mainly in s udies o enzyme mechanisms. B ie in oduc ion o
acid-base equilib ia eac ions a e gi en in sec ion 2.1.1 and 2.1.2 [62]. I is equi ed o
ha e eliable and accu a e means o calcula ing ela i e and/o absolu e pKa alues and
p ope unde s anding o he ac o s in ol ed in pKacalcula ions. The pKa alues a e e y
sensi i e o he cha ges used o he sol a ion ee ene gy calcula ions. In his chap e , he
pKacalcula ions on CuBand heme a3cen e in he aqueous solu ion a e epo ed. The pKa
calcula ions in cy och ome coxidase a e epo ed and discussed in chap e 6. The accu acy
o he pKacalcula ions depend on di e en ac o s such as he sol a ion ene gies o he
eac an and p oduc , he sol a ion ene gy used o he p o on and he model used o he pKa
calcula ions. The calcula ion o absolu e pKa alues o species in solu ion is a delica e ask,
which necessi a es he use o high-le el QM me hods capable o a aining chemical accu acy
[63]. The ea men o sol a ion ep esen s he leas eliable aspec s o pKacalcula ions [182].
The hea o he pKaes ima ion me hods a e he calcula ed a omic cha ges in case o FDPB
5.2. Compu a ional Me hods 105
me hod. To gauge he dependence o pKa alues on he le el o heo y used o desc ibe
he sol a ion, FDPB me hod and C-PCM [121, 188] a e used o de e mine he sol a ion ee
ene gies in bulk wa e .
5.2 COMPUTATIONAL METHODS
5.2.1 DENSITY FUNCTIONAL CALCULATIONS
The s a ing s uc u es o he CuBand heme a3cen e s we e aken om he X- ay c ys al
s uc u e o bo ine hea cy och ome coxidase ob ained by Yoshika a e al. a 2.3 ˚
A esolu ion
(2OCC in he p o ein da a bank) [16]. The minimal model equi ed o calcula e he pKao a
CuBcen e consis s o he Cu ion, me hylimidazole ep esen ing he coo dina ing his idines
284, 333 and 334, a me hyl g oup ep esen ing Ty 288 (which is c oss-linked o His284)
and he bound H2O molecule as CuBligand. The me hyl g oups a ached o he imidazole
ings ep esen he Cβo his idines. A ull desc ip ion o Ty 288 is no included because
he calcula ions a e pe o med only o ob ained he pKacalcula ions o his idines and H2O
coo dina ing o CuBcen e . The model compound conside ed o he CuBcen e is gi en in
Figu e 5.1. The model o he heme a3cen e consis s o he heme a3and an axial his idine
which is modeled by me hylimidazole. The six h coo dina ion is modeled wi h H2O molecule
as in he CuBcen e . The hyd ophobic hyd oxye hyl- a nesyl g oup is unca ed nex o he
hyd oxyl g oup. The model compound conside ed o he heme a3cen e is gi en in Figu e 5.2.
CuB
H284 Y288
+2/+1
H333
H334
Figu e 5.1. The model compound o CuBcen e . The his idines coo dina ing he coppe
ion a e modeled by me hyl imidazoles. The c oss-linked y osine 288 is eplaced by me hyl
g oup. The ou h coo dina ion is he H2O molecule. The p o on binding si es a e indica ed
by whi e sphe es.
The CuBcen e has h ee p o on binding si es which a e he H2O molecule, Nδ1o His333, and
Nδ1o His334. The h ee p o on binding si es lead o eigh possible mic oscopic s a es and
wel e p o on equilib ium eac ions [73]. In heme a3cen e , he e a e wo p o on binding si es
he H2O molecule and he Nδ1o His419. The wo p o on binding si es lead o ou possible
mic oscopic s a es and ou p o on equilib ium eac ions [73]. All possible p o ona ion s a es
112 pKacalcula ions o binuclea cen e o cy och ome coxidase using DFT calcula ions
S ep Rea. S a e P od. S a e TZP TZ2P+axc-BLYPbuncons ain c
∆∆Gsol pKa∆∆Gsol pKa∆∆Gsol pKa∆∆Gsol pKa
1Cu+2
B-111 Cu+2
B-011 98.77 6.9 99.00 7.9 97.56 6.5 80.93 6.5
Cu+2
B-111 Cu+2
B-101 96.19 16.1 96.76 17.6 96.38 13.4 80.31 15.8
Cu+2
B-111 Cu+2
B-110 97.60 15.1 96.02 14.9 96.33 11.5 80.11 13.3
2a Cu+2
B-110 Cu+2
B-010 17.47 6.4 22.21 7.3 19.32 7.6 24.10 8.4
Cu+2
B-101 Cu+2
B-001 21.45 4.4 19.22 5.5 21.55 5.0 25.13 3.3
2b Cu+2
B-110 Cu+2
B-100 20.42 10.9 23.64 13.9 21.09 10.0 27.71 12.9
Cu+2
B-011 Cu+2
B-001 18.87 13.6 16.98 15.2 20.37 11.9 24.52 12.7
2c Cu+2
B-011 Cu+2
B-010 16.30 14.6 19.23 14.4 18.09 12.6 23.28 15.2
Cu+2
B-101 Cu+2
B-100 21.83 9.9 22.90 11.2 21.04 8.1 27.50 10.3
3Cu+2
B-100 Cu+2
B-000 -48.81 4.6 -49.54 4.4 -47.95 5.6 -26.60 3.9
Cu+2
B-010 Cu+2
B-000 -45.86 9.1 -48.11 11.0 -46.18 8.0 -23.00 8.4
Cu+2
B-001 Cu+2
B-000 -48.43 10.1 -45.86 10.2 -48.46 8.7 -24.23 11.0
Table 5.2. The sol a ion ene gy di e ence be ween dep o ona ed and p o ona ed
o ms and he pKa alues o all possible p o ona ion equlib ium eac ions in he ox-
idized s a e o CuBcen e . The sol a ion ene gy di e ence be ween dep o ona ed and
p o ona ed (∆∆Gsol ) o ms and he pKa alues o all possible p o on equlib ium eac ions
a e gi en. All sol a ion ene gies we e ob ained by FDPB me hod using CHELPG cha ges.
The sol a ion ene gies a e gi en in kcal/mol. aPW91 wi h TZ2P+ basis se (di usion unc-
ions o i on and coppe ). bPW91 wi h BLYP exchage-co ela ion unc ionals and TZP basis
se . cPW91 wi h TZP basis se whe e he geome y is comple ely elaxed.
His333 and His334. The second p o on is emo ed om H2O in ‘s ep 2a’ and in ‘s ep 2b’
om His333 and om His334 in ‘s ep 2c’. In ‘S ep 3’ he hi d p o on is emo ed om H2O,
His333 and His334. All he Tables epo ed in his sec ion ollow he same no a ion. The
nomencla u e used a e shown in Figu es 5.3, 5.4, 5.5 and 5.6 espec i ely. The pKa alues o
all possible p o on equilib ium eac ions a e calcula ed o check he in e nal consis ency i.e.,
he dep o ona ion o he same p o on binding si e becomes mo e di icul when less p o ons
a e bound o he cen e . The sol a ion ene gies we e calcula ed wi h di e en cha ges o
ob ain app op ia e model pKa o he CuBand heme a3cen e which is equi ed o calcula e
he a e age pKa alues in he p o ein.
5.3.1 PKaVALUES OF THE CUBCENTER
The en halpy di e ence be ween he dep o ona ed and he p o ona ed o m (∆Hdep o
ac ) we e
calcula ed o all mic oscopic s a es o CuBcen e in he oxidized and in he educed s a e us-
ing PW91 me hod wi h TZP, TZ2P+ basis se and also using BLYP as exchange and co ela ion
unc ionals. Table 5.2 lis s he sol a ion ene gy di e ence (∆∆Gsol ) be ween dep o ona ed
and he p o ona ed o ms. The pKa alues a e calcula ed based on Eq. (2.14). The pKa alues
a e e y sensi i e o he basis se used and also wi h di e en exchange co ela ion unc ion-

5.3. Resul s and Discussion 113
S ep Rea. S a e P od. S a e
PW91/TZP
MDCmMDCdMDCq
∆∆Gsol pKa∆∆Gsol pKa∆∆Gsol pKa
1Cu+2
B-111 Cu+2
B-011 112.57 17.1 97.55 6.1 97.40 5.9
Cu+2
B-111 Cu+2
B-101 117.00 31.4 95.40 15.5 93.70 14.3
Cu+2
B-111 Cu+2
B-110 115.00 27.9 95.60 13.7 96.60 14.4
2a Cu+2
B-110 Cu+2
B-010 23.85 11.1 20.49 8.6 15.54 5.0
Cu+2
B-101 Cu+2
B-001 29.28 10.2 22.14 5.0 20.30 3.6
2b Cu+2
B-110 Cu+2
B-100 36.45 22.6 22.79 12.6 17.40 8.7
Cu+2
B-011 Cu+2
B-001 33.71 24.5 19.99 14.4 16.60 11.9
2c Cu+2
B-011 Cu+2
B-010 26.28 21.9 18.54 16.2 14.74 13.4
Cu+2
B-101 Cu+2
B-100 34.45 19.1 22.99 10.7 20.30 8.8
3Cu+2
B-100 Cu+2
B-000 -46.06 6.6 -46.12 6.6 -49.58 4.1
Cu+2
B-010 Cu+2
B-000 -33.46 18.2 -43.82 10.6 -47.72 7.7
Cu+2
B-001 Cu+2
B-000 -40.89 15.6 -45.27 12.4 -49.58 9.2
Table 5.3. The sol a ion ene gy di e ence be ween dep o ona ed and p o ona ed
o ms and he pKa alues o all possible p o on equlib ium eac ions in he oxidized
s a e o CuBcen e . The sol a ion ene gy di e ence be ween p o ona ed and dep o ona ed
o ms (∆∆Gsol ) and he pKa alues o all possible p o on equlib ium eac ions in he ox-
idized s a e o CuBcen e a PW91 me hod wi h TZP Basis se a e gi en. The sol a ion
ene gies a e ob ained om FDPB me hod using mul ipole de i ed cha ges. The sol a ion
ene gies a e gi en in kcal/mol.
als. Compa ing he pKa alues ob ained om he di e en basis se and exchange unc ionals,
he a ia ion be ween he pKa alues a e ound o be i egula . The majo p oblem obse ed
in he pKa alues end is ha he pKa alues co espond o ‘s ep 3’ whe e a hi d p o on
is emo ed om he doubly dep o ona ed s a e is 4.6 o by TZP basis se and 4.4 o TZP
wi h di used unc ions (TZ2P+) and 5.6 wi h BLYP exchange and co ela ion unc ionals. The
ends in he pKa alues show ha he emo al o p o on om he double dep o ona ed s a es
is a o ed which is no ene ge ically plausible. The p o ona ion equilib ium should show he
co ec end i.e., he dep o ona ion o he same p o on binding si e becomes mo e di icul
when less p o ons a e bound o he cen e .
On compa ing he pKa alues o eac ions which co espond o he emo al o p o on om
H2O, Cu+2
B-111 →Cu+2
B-011, Cu+2
B-110 →Cu+2
B-010 and Cu+2
B-100 →Cu+2
B-000, he emo al o
hi d p o on has he lowe pKa alue compa ed o ‘s ep 1’ and ‘s ep 2a’. The sol a ion ene gy
di e ence be ween he eac ions a e also di e e y much. The sol a ion ene gy di e ence
a e e y high o he eac ions in ‘s ep 1’ whe e a single p o on is emo ed om he ully
p o ona ed s a e. The o al cha ges o he eac an s a e +2 and he p oduc is +1 in ‘s ep
114 pKacalcula ions o binuclea cen e o cy och ome coxidase using DFT calcula ions
Mic o. S a e TZP TZ2Paxc-BLYPbUncons ain cMDCmdMDCddMDCqd
Cu+2
B-111 -150.80 -150.69 -150.17 -150.16 -175.60 -149.7 -154.90
Cu+2
B-011 -52.03 -51.69 -52.61 -53.90 -63.03 -52.15 -57.50
Cu+2
B-101 -54.61 -53.93 -53.79 -54.74 -58.60 -54.30 -61.20
Cu+2
B-110 -53.20 -54.67 -53.84 -55.02 -60.60 -54.10 -58.30
Cu+2
B-001 -33.16 -34.71 -32.24 -34.50 -29.32 -32.16 -40.90
Cu+2
B-010 -35.73 -32.46 -34.52 -36.18 -36.75 -33.61 -42.76
Cu+2
B-100 -32.78 -31.03 -32.75 -31.27 -24.15 -31.31 -40.90
Cu+2
B-000 -81.59 -80.57 -80.70 -81.51 -70.21 -77.43 -90.48
Table 5.4. Sol a ion ene gies o each mic oscopic s a e o CuBcomplex in oxidized
s a e. The sol a ion ene gies o he mic oscopic s a e o CuBcomplex ob ained by FDPB
me hod. aPW91 wi h TZ2P+ basis se (di usion unc ions o i on and coppe ). bPW91
wi h BLYP exchage-co ela ion unc ionals and TZP basis se . cPW91 wi h TZP basis se
whe e he geome y is comple ely elaxed du ing op imiza ion. dPW91 wi h TZP basis se
and MDCs a e used o sol a ion ene gy calcula ions. F om a−cCHELPG cha ges a e used
o sol a ion enegy calcula ions. All he sol a ion ene gies a e gi en in kcal/mol.
1’ eac ions. Use o highe basis se and di e en exchange co ela ion unc ionals does no
help e y much o imp o ed he ends in he pKa alues. To ind he p ope eason o such
ends in he pKa he sol a ion ene gies we e calcula ed wi h di e en cha ges. The cha ges
ob ained om he mul ipole momen a e used ins ead o CHELPG cha ges. Table 5.3 show he
sol a ion ene gy di e ence be ween dep o ona ed and p o ona ed o ms whe e he sol a ion
ene gies we e ob ained om FDPB me hod using mul ipole de i ed cha ges (MDC) (see sec ion
3.1.2).
The pKa alues a e e y sensi i e o he cha ges used. The sol a ion ene gies ob ained om
monopole de i ed cha ges a e e y high compa ed o he sol a ion ene gies ob ained om
dipole and quad upole cha ges. Ob iously, he pKa alues a e e y high wi h MDCm. On
compa ing he Tables 5.2 and 5.3, he pKa alues show almos he same end.
The sol a ion ene gies ob ained om each mic oscopic s a es o he CuBcen e using FDPB
me hods wi h di e en basis se , exchange co ela ion unc ionals, di e en cha ges a e gi en
in Table 5.4. The sol a ion ene gies ob ained om he he CHELPG cha ges a e simila wi h
∼1-2 kcal/mol di e ence wi h di e en basis se and exchange co ela ion unc ionals. The
sol a ion ene gies om he MDCmcha ges a e much highe compa ed o he sol a ion ene -
gies ob ained om CHELPG cha ges. The pKa alues we e calcula ed o he educed s a e o
he CuBcen e o ob ain he pKa alues o H2O, His333 and His334.
The gas phase ee ene gies a e ob ained o all mic oscopic s a e o CuBcen e in educed
s a e using PW91/TZP basis se . The en halpy di e ence be ween dep o ona ed and p o-
ona ed o m (∆Hdep o
ac ), he sol a ion ene gies o he dep o ona ed (∆GA−
sol ) and p o ona ed
(∆GAH
sol ), and he calcula ed pKa alues a e gi en in Table 5.5. The pKa alues o he educed
s a e o he CuBcen e also show he same end like oxidized s a e o CuBcen e . The pKa al-
ues co esponding o ‘s ep 2a’ a e lowe han he eac ions in ‘s ep 1’. The sol a ion ene gies
5.3. Resul s and Discussion 115
S ep Rea. S a e P od. S a e PW91/TZP
∆GAH
sol ∆GA−
sol ∆Hdep o
ac ∆∆Gsol pKa
1Cu+1
B-111 Cu+1
B-011 -49.55 -34.35 -2.03 15.20 19.9
Cu+1
B-111 Cu+1
B-101 -49.55 -36.06 -8.59 13.59 14.7
Cu+1
B-111 Cu+1
B-110 -49.55 -33.54 -9.14 16.01 15.3
2a Cu+1
B-110 Cu+1
B-010 -34.35 -77.56 67.82 -49.60 9.0
Cu+1
B-101 Cu+1
B-001 -33.54 -78.32 47.80 -42.29 13.0
2b Cu+1
B-110 Cu+1
B-100 -77.56 -197.25 48.48 -44.78 27.5
Cu+1
B-011 Cu+1
B-001 -82.98 -197.25 41.24 -43.90 16.7
2c Cu+1
B-011 Cu+1
B-010 -78.32 -197.25 60.71 -48.79 28.1
Cu+1
B-101 Cu+1
B-100 -34.35 -82.98 47.93 -42.36 19.8
3Cu+1
B-100 Cu+1
B-000 -36.06 -77.56 143.20 -118.93 14.2
Cu+1
B-010 Cu+1
B-000 -33.54 -82.98 123.86 -114.11 24.4
Cu+1
B-001 Cu+1
B-000 -36.06 -78.32 143.33 -119.00 13.6
Table 5.5. The en halphy (∆Hdep o
ac ) and he sol a ion ene gy di e ence (∆∆Gsol ) be-
ween p o ona ed and dep o ona ed o ms, he sol a ion ene gies o he p o ona ed
(∆GAH
sol ) and dep o ona ed (∆GA−
sol ) o m, and he calcula ed pKa alues o CuBcen e
in he educed s a e. The en halphy o di e endce (∆Hdep o
ac ) be ween p o ona ed and de-
p o ona ed o m, he sol a ion ene gies o he p o ona ed (∆GAH
sol ) and dep o ona ed (∆GA−
sol )
o m, he sol a ion ene gy di e ence be ween dep o ona ed and p o ona ed o m (∆∆Gsol ),
he calcula ed pKa alues o CuBcen e in he educed s a e o m a DFT me hod wi h TVP
basis se a e gi en. The sol a ion ene gies a e ob ined by FDPB me hod using CHELPG
cha ges. The en halphies and sol a ion ee ene gies a e gi en in kcal/mol.
a e e y low o eac ions co espond o ‘s ep 3’. The ‘s ep 1’ eac ions show posi i e sol a ion
ene gies. The en halpy di e ence be ween he dep o ona ed and he p o ona ed o ms a e
posi i e o he eac ions co esponding o ‘s ep 3’ whe e he hi d p o on is emo ed om he
doubly dep o ona ed s a e. The esul s show ha he pKa alues depend on he basis se s,
di e en exchange and co ela ion unc ionals and also he mode o he op imiza ion. Bu
he lowe pKa alues ob ained o he iply dep o ona ed eac ions may be because o he
sol a ion model used o he sol a ion ene gy calcula ions.
All he sol a ion ene gies om Table 5.2 o 5.4 we e calcula ed using he FDPB me hod.
Though his me hod was success ul in many cases, [101–104] o he calcula ion o he pKa
alues o CuBcomplexes, his me hod seems o be insu icien . To ha e be e unde s anding
o he ac o s in luencing he pKa alues, he pKacalcula ions a e also pe o med using he
B3LYP me hod wi h 6-31+G* and 6-31G* basis se s. The sol a ion ene gies we e calcula ed by
C-PCM and FDPB me hod using di e en cha ges like Mulliken, NPA, CHELPG, MK and RESP.
The pKa alues ob ained a B3LYP le el o heo y wi h 6-31+G* basis se and he sol a ion
ene gy di e ence be ween dep o ona ed and p o ona ed o ms o he oxidized s a e o CuB
116 pKacalcula ions o binuclea cen e o cy och ome coxidase using DFT calcula ions
S ep Rea. S a e P od. S a e
B3LYP/6-31+G*
CHELPG MK RESP
∆∆Gsol pKa∆∆Gsol pKa∆∆Gsol pKa
1Cu+2
B-111 Cu+2
B-011 102.49 6.9 101.64 6.3 101.25 6.0
Cu+2
B-111 Cu+2
B-101 105.21 20.4 105.32 20.5 105.39 20.5
Cu+2
B-111 Cu+2
B-110 104.95 19.0 104.94 19.0 105.16 19.1
2a Cu+2
B-110 Cu+2
B-010 18.65 2.1 18.31 1.9 18.05 1.7
Cu+2
B-101 Cu+2
B-001 22.20 -0.5 20.14 -2.0 20.47 -1.8
2b Cu+2
B-011 Cu+2
B-001 24.92 13.0 23.82 12.2 24.61 12.7
2c Cu+2
B-011 Cu+2
B-010 21.11 14.2 21.61 14.5 21.96 14.8
3Cu+2
B-010 Cu+2
B-000 -37.33 9.5 -38.06 8.9 -38.24 8.8
Cu+2
B-001 Cu+2
B-000 -41.14 10.7 -40.27 11.3 -40.89 10.9
Table 5.6. The sol a ion ene gy di e ence be ween dep o ona ed and p o ona ed
o ms (∆∆Gsol ) and pKa alues o all possible p o on equlib ium eac ions in he
oxidized s a e o CuBcen e . The pKa alues a e ob ained a B3LYP/6-31+G* basis se .
The sol a ion ene gies a e calcula ed wi h CHELPG, MK and RESP cha ges. The sol a ion
ene gies a e calcula ed using FDPB me hod. The sol a ion ene gies a e gi en in kcal/mol.
cen e a e gi en in Table 5.6. Ini ially he sol a ion ene gies we e calcula ed wi h CHELPG,
MK and RESP cha ges. The RESP cha ges a e ob ained by using hype bolic es ain and
ESP om MK me hod. The sol a ion ene gies we e calcula ed using FDPB me hod. All he
mic oscopic s a es o he CuBcen e we e op imized wi h B3LYP/6-31+G* basis se . The ESP
cha ges CHELPG and MK cha ges we e ob ained by i ing he ESP ob ained om B3LYP/6-
31+G* by CHELPG and MK me hod. The mic oscopic s a e Cu+2
B-100 was uns able and did no
con e ge. The eac ions in ol ing Cu+2
B-100 mic oscopic s a e is no epo ed in he Table 5.6.
On compa ing he sol a ion ene gy di e ence (∆∆Gsol ) be ween he eac ion and p oduc
s a es, he sol a ion ene gies co espond o ‘s ep 1’ a e mo e posi i e compa ed o o he s eps.
The pKa alues o ‘s ep 2a’ a e e y low and e en nega i e o he eac ion Cu+2
B-101 →Cu+2
B-
001. Acco ding o he pKa alues, he emo al o second p o on om he singly dep o ona ed
s a e is a o ed compa ed o he emo al o single p o on om he ully p o ona ed s a e. The
pKa alues ob ained using CHELPG and MK a e almos simila and in some cases hey a y
up o 1 pKauni . The pKa alues ob ained om RESP cha ges a e close o he pKa alues
ob ained om MK cha ges. Ei he he le el o heo y o he basis se wi h di usion and po-
la iza ion unc ion does no help o imp o e he pKa ends. To check he e ec o di usion
unc ion on he pKacalcula ions, he calcula ions we e done a B3LYP le el o heo y wi h
6-31G* basis se wi hou di usion unc ions. The sol a ion ene gy di e ence be ween dep o-
ona ed and p o ona ed o ms, he pKa alues o he oxidized s a e o CuBcen e a e gi en
in Table 5.7. The sol a ion ene gies we e calcula ed wi h Mulliken, NPA, CHELPG and MK
5.3. Resul s and Discussion 117
S ep Rea. S a e P od. S a e
B3LYP/6-31G*
Mulliken NPA CHELPG MK
∆∆Gsol pKa∆∆Gsol pKa∆∆Gsol pKa∆∆Gsol pKa
1Cu+2
B-111 Cu+2
B-011 104.54 11.3 103.52 10.6 105.71 12.1 105.91 12.3
Cu+2
B-111 Cu+2
B-101 93.22 20.0 86.91 15.5 94.38 20.9 94.36 20.9
Cu+2
B-111 Cu+2
B-110 94.94 19.4 87.92 14.3 94.92 19.4 95.71 20.0
2a Cu+2
B-110 Cu+2
B-010 33.24 13.3 32.15 12.5 31.66 12.1 30.73 11.5
Cu+2
B-101 Cu+2
B-001 32.22 8.7 28.38 5.9 32.89 9.2 32.52 8.9
2b Cu+2
B-110 Cu+2
B-100 27.62 15.8 22.07 11.7 30.33 17.7 27.73 15.8
Cu+2
B-011 Cu+2
B-001 20.90 17.4 11.77 10.8 21.56 17.9 20.97 17.5
2c Cu+2
B-011 Cu+2
B-010 23.64 21.4 16.55 16.2 20.87 19.4 20.53 19.1
Cu+2
B-101 Cu+2
B-100 29.34 15.1 23.08 10.6 30.87 16.2 29.08 14.9
3Cu+2
B-100 Cu+2
B-000 -32.74 13.0 -35.64 10.9 -35.53 11.0 -33.71 12.3
Cu+2
B-010 Cu+2
B-000 -38.36 15.5 -45.72 10.2 -36.86 16.6 -36.71 16.7
Cu+2
B-001 Cu+2
B-000 -35.62 19.5 -40.94 15.7 -37.55 18.1 -37.15 18.4
Table 5.7. The sol a ion ene gy di e ence be ween dep o ona ed and p o ona ed
o ms (∆∆Gsol ) and pKa alues o all possible p o on equlib ium eac ions in he
oxidized s a e o CuBcen e The pKa alues a e ob ained a B3LYP/6-31G* basis se .
The sol a ion ene gies a e calcula ed wi h Mulliken, NPA, CHELPG and MK cha ges. The
sol a ion ene gies a e calcula ed using FDPB me hod. The sol a ion ene gies a e gi en in
kcal/mol.
cha ges. The sol a ion ene gies we e also calcula ed wi h he RESP cha ges wi h wo di e en
es ain s hype bolic and ha monic es ain using Mulliken cha ges as ini ial cha ges. The
pKa alues ob ained a e much be e compa ed o he pKa alues ob ained a B3LYP/6-31+G*
basis se . Bu he pKa alues do no show co ec end i.e., he dep o ona ion o he same
p o on binding si e becomes mo e di icul when less p o ons a e bound o he cen e .
The pKa alues ob ained using he CHELPG and MK cha ges a e compa able. The sol a ion
ene gy is mo e posi i e o he single dep o ona ed s ep compa ed o he o he s s eps. The
sol a ion ene gies a e nega i e o he eac ions whe e he hi d p o ons a e emo ed om he
doubly p o ona ed s a e. The pKa alues ob ained wi h ei he Mulliken o NPA cha ges show
he same end in he pKa alues as in pKa alues ob ained om ESP based cha ges.
The sol a ion ene gy di e ence be ween dep o ona ed and dep o ona ed o ms, pKa alues
o all possible p o on equilib ium eac ions in he oxidized s a e, and he sol a ion ene gies
calcula ed wi h RESP cha ges a e gi en in Table 5.8. The sol a ion ene gies we e calcula ed
using FDPB me hod. The RESP cha ges a e ob ained by using he Mulliken and NPA cha ges
as ini ial cha ges o ha monic and hype bolic es ain . The elec os a ic po en ial om MK
cha ges we e used. The pKa alues ob ained by using RESP cha ges also show he same
end as o he ESP cha ge me hods. To ge he p ope unde s anding o he in luence o he
sol a ion ene gy on he pKacalcula ions, he C-PCM was used o calcula e he sol a ion ee

118 pKacalcula ions o binuclea cen e o cy och ome coxidase using DFT calcula ions
S ep Rea. S a e P od. S a e
B3LYP/6-31G*
RESPaRESPb
qw =0.005 qw =0.01 qw =0.005 qw =0.01
∆∆Gsol pKa∆∆Gsol pKa∆∆Gsol pKa∆∆Gsol pKa
1Cu+2
B-111 Cu+2
B-011 104.91 11.6 104.59 11.3 105.83 12.2 105.64 12.1
Cu+2
B-111 Cu+2
B-101 94.85 21.2 94.23 20.7 94.50 21.0 94.52 21.0
Cu+2
B-111 Cu+2
B-110 96.89 20.8 96.76 20.7 95.68 19.9 95.70 20.0
2a Cu+2
B-110 Cu+2
B-010 30.06 11.0 30.36 11.2 31.02 11.7 31.12 11.7
Cu+2
B-101 Cu+2
B-001 33.14 9.3 33.38 9.5 33.37 9.5 33.81 9.8
2b Cu+2
B-110 Cu+2
B-100 29.36 17.0 29.65 17.2 28.95 16.7 29.16 16.9
Cu+2
B-011 Cu+2
B-001 23.08 19.0 23.02 18.9 22.04 18.2 22.69 18.7
2c Cu+2
B-011 Cu+2
B-010 22.04 20.2 22.53 20.6 20.87 19.4 21.18 19.6
Cu+2
B-101 Cu+2
B-100 31.40 16.6 32.18 17.2 30.13 15.7 30.34 15.8
3Cu+2
B-100 Cu+2
B-000 -36.33 10.4 -36.60 10.3 -34.31 11.9 -34.46 11.8
Cu+2
B-010 Cu+2
B-000 -37.03 16.5 -37.31 16.3 -36.38 17.0 -36.42 16.9
Cu+2
B-001 Cu+2
B-000 -38.07 17.7 -37.80 17.9 -37.55 18.1 -37.93 17.8
Table 5.8. The sol a ion ene gy di e ence be ween dep o ona ed and p o ona ed
o ms (∆∆Gsol ) and pKa alues o all possible p o on equlib ium eac ions in he ox-
idized s a e o CuBcen e . The pKa alues a e ob ained a B3LYP/6-31G* basis se . The
sol a ion ene gies a e calcula ed wi h RESP cha ges. The sol a ion ene gies a e calcula ed
using FDPB me hod. The sol a ion ene gies a e gi en in kcal/mol ahype bolic es ain
wi ih ini ial mulliken cha ges and elec os a ic po en ial om MK me hod. bha monic e-
s ain wi h mulliken cha ges as ini ial cha ges and elec os a ic po en ial om MK me hod.
Res ain ac o (qw ) o 0.005 and 0.01 a e used o RESP calcula ions.
ene gies. The sol a ion ene gy di e ence be ween dep o ona ed and p o ona ed o ms, pKa
alues ob ained a B3LYP wi h TZVP, 6-31+G* and 6-31G* basis se a e gi en in Table 5.9.
The pKacalcula ions wi h C-PCM a B3LYP le el wi h TZVP and 6-31+G* also show lowe pKa
alues o he dep o ona ion eac ion which a e no ene ge ically plausible. The pKa alues
ob ained a B3LYP wi h 6-31G* show he p ope end. The p o ona ion equilib ium show he
co ec end, i.e., he dep o ona ion o he same p o on binding si e becomes mo e di icul
when less p o ons a e bound o he cen e . The en halpy o di e ence (∆Hdep o
ac ), he sol a ion
ene gy di e ence (∆∆Gsol ) and he calcula ed pKa alues o CuBcen e in he educed s a e
a B3LYP le el wi h 6-31G* a e gi en in Table 5.10. The sol a ion ene gies a e ob ained by
using C-PCM. The pKain he educed CuBcen e also show co ec end. The sol a ion
ene gies ob ained o all he eigh mic oscopic s a es a e gi en in Table 5.12 o compa ison.
The pKa alues ob ained om B3LYP/6-31G* le el o heo y show ha he sol a ion ene gies
should be calcula ed in a sel -consis ency manne as in C-PCM. The FDPB model ea s he
in e ac ions in a simple way. The solu e cha ge dis ibu ion is conside ed as classical en i y
and he elec onic pola iza ion due o sol en e ec s is disca ded. The FDPB model su e s
5.3. Resul s and Discussion 119
S ep Rea. S a e P od. S a e
C-PCM
B3LYP/TZVP B3LYP/6-31+G* B3LYP/6-31G*
∆∆Gsol pKa∆∆Gsol pKa∆∆Gsol pKa
1Cu+2
B-111 Cu+2
B-011 98.47 7.0 97.20 3.1 98.55 7.4
Cu+2
B-111 Cu+2
B-101 96.45 16.4 96.85 14.3 86.72 15.9
Cu+2
B-111 Cu+2
B-110 95.96 14.9 96.24 12.7 90.25 15.2
2a Cu+2
B-110 Cu+2
B-010 23.96 8.9 27.01 8.1 29.96 13.0
Cu+2
B-101 Cu+2
B-001 27.60 6.0 26.52 2.6 35.84 10.6
2b Cu+2
B-110 Cu+2
B-100 -a- -a- -a- -a- 27.84 18.1
Cu+2
B-011 Cu+2
B-001 25.58 15.4 26.17 13.9 24.01 19.1
2c Cu+2
B-011 Cu+2
B-010 21.45 16.8 26.05 17.7 21.66 20.8
Cu+2
B-101 Cu+2
B-100 -a- -a- -a- -a- 31.37 17.4
3Cu+2
B-100 Cu+2
B-000 -a- -a- -a- -a- -33.22 14.3
Cu+2
B-010 Cu+2
B-000 -32.48 16.3 -35.40 10.9 -35.34 19.4
Cu+2
B-001 Cu+2
B-000 -36.61 17.7 -35.52 14.8 -37.69 21.0
Table 5.9. The sol a ion ene gy di e ence be ween dep o ona ed and p o ona ed
o ms (∆∆Gsol ) and pKa alues o all possible p o on equlib ium eac ions in he
oxidized s a e o CuBcen e . The pKa alues a e ob ained a B3LYP wi h TZVP, 6-31+G*
and 6-31G* basis se . The sol a ion ene gies a e calcula ed wi h C-PCM.
aThe s uc u e o Cu+2
B-100 was uns able and did no con e ge a he gi en le el o heo y
and he basis se used.
by se e e limi a ions o ob ain he sol a ion ene gy o he CuBcomplexes. The p esen esul s
show ha in he sol a ion ene gy calcula ions he elec onic pola iza ion due o he sol en
should be included explici ly. In C-PCM, he in e ac ion ene gies be ween he sol en and he
solu e a e included in he Hamil onian i sel . In C-PCM he elec on dis ibu ion o he solu e
is ob ained wi h he ixed nuclei in he p esence o he eac ion ield o he sol en .
Using he C-PCM o he oxidized s a e o he CuBmodel in he aqueous solu ion, a pKa alue
o 15.9 and 15.2 a e ob ained o he i s dep o ona ion eac ion o His333 and His334 and
he pKa alue o 7.4 o he H2O coo dina ing he coppe . Ob iously, he pKa alues o he wo
his idines a e oo high o allow hei dep o ona ion in aqueous solu ion. The pKa alues o
he educed s a e is e en highe because he nega i e cha ge o he elec on is s abilizing he
p o on. The pKa end is ob ained co ec ly only in he combina ion o B3LYP/6-31G*/C-PCM
le el.
120 pKacalcula ions o binuclea cen e o cy och ome coxidase using DFT calcula ions
S ep Rea. S a e P od. S a e B3LYP/6-31G*
∆GAH
sol ∆GA−
sol ∆Hdep o
ac ∆∆Gsol pKa
1Cu+1
B-111 Cu+1
B-011 -36.17 -15.89 304.45 20.28 32.5
Cu+1
B-111 Cu+1
B-101 -36.17 -12.90 280.73 23.27 17.7
Cu+1
B-111 Cu+1
B-110 -36.17 -12.37 277.77 23.80 15.9
2a Cu+1
B-110 Cu+1
B-010 -a- -a- -a- -a- -a-
Cu+1
B-101 Cu+1
B-001 -12.90 -63.16 379.80 -50.26 36.2
2b Cu+1
B-110 Cu+1
B-100 -12.37 -47.50 344.09 -35.13 21.3
Cu+1
B-011 Cu+1
B-001 -15.89 -63.16 356.08 -47.27 21.1
2c Cu+1
B-011 Cu+1
B-010 -a- -a- -a- -a- -a-
Cu+1
B-101 Cu+1
B-100 -12.90 -47.50 341.13 -34.60 19.5
3Cu+1
B-100 Cu+1
B-000 -47.50 -162.00 460.90 -114.50 48.4
Cu+1
B-010 Cu+1
B-000 -a- -a- -a- -a- -a-
Cu+1
B-001 Cu+1
B-000 -63.16 -162.00 422.23 -98.84 31.7
Table 5.10. The en halphy (∆Hdep o
ac ) and he sol a ion ene gy di e ence (∆∆Gsol ) be-
ween p o ona ed and dep o ona ed o m and he calcula ed pKa alues o CuBcen e
in he educed s a e. The en halphy di e ence ∆Hdep o
ac be ween p o ona ed and dep o o-
na ed, sol a ion ene gy di e ence be ween dep o ona ed and p o ona ed o m ∆∆Gsol and
he calcula ed pKa alues o CuBcen e in he educed s a e a B3LYP/6-31G* le el a e
gi en. The sol a ion ene gies a e ob ined by using C-PCM.
aThe s uc u e o Cu+1
B-010 was uns able and did no con e ge a he gi en le el o heo y
and he basis se used.
5.3.2 PKaVALUES OF THE HEME a3CENTER
The en halpy o di e ence ∆Hdep o
ac be ween dep o ona ed (HAH
ac ) and p o ona ed (HA−
ac ), he
sol a ion ene gies o he dep o ona ed (∆GA−
sol ) and p o ona ed (∆GAH
sol ), sol a ion ene gy di -
e ence be ween dep o ona ed and p o ona ed o m ∆∆Gsol and he calcula ed pKa alues o
CuBcen e in he educed and oxidized o m o heme a3cen e a e gi en in Table 5.11. The
sol a ion ene gies we e calcula ed using CHELPG cha ges in case o PW91 me hod and C-PCM
sol a ion model was used in B3LYP/6-31G* le el. The pKa alues ob ained a B3LYP/6-31G*
le el a e highe han he pKa alues ob ained a PW91/TZP basis se . The pKa alues 19.3
and 10.7 co espond o he dep o ona ion o His419 and H2O espec i ely. The pKa alue o
he doubly dep o ona ed and educed heme a3cen e a e e en oo high. The pKa alues o
he his idine a e oo high o allow hei dep o ona ion in aqueous solu ion.
5.4. Conclusions 121
Rea. S a e P od. S a e PW91/TZPaB3LYP/6-31G*
∆Hdep o
ac ∆∆Gsol pKa∆Hdep o
ac ∆∆Gsol pKa
Fea3+3-11 Fea3+3-01 268.65 17.33 7.8 268.65 21.47 10.7
Fea3+3-11 Fea3+3-10 277.26 19.7 10.4 277.26 24.56 19.3
Fea3+3-01 Fea3+3-00 343.72 -43 16.5 343.72 -33.52 25.4
Fea3+3-10 Fea3+3-00 335.11 -45.37 13.8 335.11 -36.61 16.8
Fea3+2-11 Fea3+2-01 364.65 -43.78 18.3 364.65 -45.85 31.7
Fea3+2-11 Fea3+2-10 349.27 -38.39 13.5 349.27 -36.00 27.7
Fea3+2-01 Fea3+2-00 402.75 -93.08 19.9 402.75 -79.27 35.2
Fea3+2-10 Fea3+2-00 418.13 -98.47 24.7 418.13 -89.12 39.3
Table 5.11. The en halphy (∆Hdep o
ac ) and he sol a ion ene gy di e ence be ween de-
p o ona ed and p o ona ed o m ∆∆Gsol and he calcula ed pKa alues o CuBcen e
in he educed and oxidized s a es o heme a3cen e . The pKa alues a e calcula ed
using PW91/TZP and B3LYP/6-31G* basis se s. The sol a ion ene gies a e calcula ed us-
ing CHELPG cha ges in case o PW91 me hod and C-PCM sol a ion model was used in
B3LYP/6-31G* le el. The sol a ion ene gies and en halpies a e gi en in kcal/mol.
5.4 CONCLUSIONS
In he p esen s udy, we calcula ed all mic oscopic pKa alues o he CuBcen e combining
PW91 and B3LYP wi h bo h FDPB and C-PCM sol a ion models, wi h he pu pose o assess-
ing he alidi y o he ecen ly p oposed pumping mechanism o cy och ome coxidase, based
on he dep o ona ion o he His334 [1, 2]. The pKa alues o H2O, His333 and His334 we e
calcula ed in he aqueous solu ion o ind he possibili y o hese ligands ole in he p o on
pumping mechanism o cy och ome coxidase. Ex ensi e s udies we e done o unde s and he
in luence o a ious ac o s on he pKa alues o he CuBcen e . The pKacalcula ions pe -
o med by wo di e en densi y unc ional me hods PW91 and B3LYP and sol a ion models
(FDPB and C-PCM) show ha in sol a ion ene gy calcula ions he elec onic pola iza ion due
o he sol en should be included explici ly, which is lacking in FDPB me hod. Due o his lim-
i a ions, he pKa alues calcula ed using he sol a ion ene gies ob ained om FDPB me hod
do no show he end co ec ly i.e., he emo al o he hi d p o on is a o ed compa ed o
o he eac ions [193, 194] which is ene ge ically no plausible. The B3LYP/6-31G*/C-PCM
le el should be used o ge co ec pKa end o CuBwhe e sol a ion ene gies we e calcula ed
by C-PCM in which he elec onic pola iza ion o he solu e due o he sol en eac ion ield is
ob ain by sel -consis en manne .
The expe imen al pKa alue o he dep o ona ion o imidazole is app oxima ely 14 [193, 194].
Ou calcula ions, in aqueous solu ion on small CuBmodels do no show any signi ican change
o he pKadue o he liga ion o he imidazoles o Cu ion. Fu he mo e, he esul s ob ained
om educed CuBcen e e els a small inc ease in he pKa alues o imidazoles in aqueous
solu ion. The pKa alues o 15.9 and 15.2 o he i s dep o ona ion eac ion o His333 and
128 pKacalcula ions o CuBligands in cy och ome coxidase
In he ansi ion be ween he F→Fps a es, a chemical p o on (p o on in ol ed in oxygen
educ ion) en e s he binuclea cen e and con e s he hyd oxyl CuBligand in o wa e he eby
inc easing he o al cha ge o he si e om +1 o +2. This s a e o he ca aly ic cycle is
supposedly connec ed o he i s dep o ona ion o he His334 imidazole ing.
The coupled elec on-p o on ans e eac ion leads o H0s a e. I was p oposed [1] ha H0is
likely in dynamic equilib ium wi h he H s a e in which i on is in he +3 s a e and coppe ion
is in he +2 s a e and each o he me als is liga ed o he hyd oxyl g oup.
The ansi ion be ween he H and O s a es is connec ed o a second dep o ona ion o His334.
The p o ona ion o he hyd oxyl g oup bound o heme a3leads o he o ma ion o O s a e,
which is associa ed wi h he pumping e en . The O s a e can e ol e in wo di e en ways. The
p o on en e ing he binuclea cen e can p o ona e he hyd oxyl g oup o CuBcen e which
is ene ge ically leas a o able (O∼s a e). The ano he possible pa hway (OH) esul s in he
o ma ion o E s a e.
The E s a e was p oposed [1] o be he modynamically uns able and con e ed o Ep. In E
s a e, he edox s a e o heme a3changes om +3 o +2. In Eps a e, he chemical p o on en e s
he ac i e si e and con e s he hyd oxyl g oup o wa e , igge ing he hi d dep o ona ion
o His334. The nex incoming p o on eloads he His334 si e, closing he ca aly ic cycle wi h
emo al o wa e molecule.
In he wo s udies [1–3, 5, 6, 182], he c ys al coo dina es o he CuBcen e was no ully e-
laxed du ing he geome y op imiza ions which can lead o un ealis ic bond leng hs. Mo eo e
in la e s udies [182], i was no conside ed ha some esidues o he cy och ome coxidase
can adop a p o ona ion de ia ing om hei usual p o ona ion in aqueous solu ion. In he
wo s udies, only he dep o ona ion o His334 was de e mined and no all possible mic oscopic
p o on equilib ium eac ions in he p o ein a e conside ed. The e is a possibili y ha His333,
ano he ligand o CuBcen e can se e as a p o on loading si e o subsequen pumping which
was no examined.
To unde s and he coupling o he elec on ans e and p o on ansloca ion, DFT and con-
inuum elec os a ic calcula ions we e pe o med on R. sphae oides cy och ome coxidase.
Ini ially all he mic oscopic pKa alues o he bound H2O, His333 and His334 in CuBcen-
e we e calcula ed in aqueous solu ion by combining DFT calcula ions wi h PCM model (see
chap e 5). In o de o ake he in luence o he p o ein en i onmen in o accoun , he a e age
pKa alues o H2O, His333 and His334 ligands bound o CuBwe e calcula ed in he p o ein
by conside ing he p o ona ion o he p o ein a pH=7. The CuBcen e was conside ed in
bo h educed and oxidized s a e. The heme a3cen e was conside ed wi h all possible ligand
s a es and ligand unbound s a es. The elec os a ic calcula ions we e pe o med by sol ing
he Poisson Bol zmann Equa ion by ini e di e ence me hod.
6.2 STRUCTURE PREPARATION AND MODELS
O e he pas yea s, subs an ial p og ess has been made in he unde s anding o he s uc u e
and unc ion o cy och ome coxidase. A landma k in he ield o cy och ome coxidase esea ch
was he de e mina ion o he h ee-dimensional s uc u es o he bac e ial cy och ome cox-
idase om he soil bac e ium P. deni i icans [12, 14] and i s mammalian coun e pa om

6.2. S uc u e p epa a ion and models 129
bo ine hea mi ochond ia [16–18]. The i s c ys als o he bac e ial enzyme we e ob ained
using he ou subuni cy och ome coxidase [12]. La e , an imp o ed c ys al o m o he
unc ionally ac i e wo-subuni cy och ome coxidase complex wi h he F an ibody agmen
as he ou -subuni cy och ome coxidase was obse ed and he s uc u e was de e mined
a 2.7 ˚
A esolu ion [14]. The s uc u es o he bac e ial and he mi ochond ial enzymes a e
su p isingly simila . The co e pa s (subuni s I, II, and III) o he wo c ys al s uc u es look
nea ly iden ical a he a omic le el.
The ollowing sec ion deals wi h he s uc u es and necessa y p epa a ions o he elec o-
s a ic calcula ions o cy och ome coxidase.
6.2.1 PREPARATION OF X-RAY STRUCTURE OF PROTEIN
The elec os a ic calcula ions we e pe o med on he X- ay s uc u e o R. sphae oides (PDB
ID:1M56). The s uc u e o cy och ome coxidase ha e been de e mined a 2.8 ˚
A esolu ion
by Iwa a e al. [15]. The c ys al s uc u es we e de e mined o wild ype and mu an by
he eplacemen o glu ama e-286 o subuni I by glu amine. The elec os a ic calcula ions
we e pe o med on he wild ype enzyme. The o e all s uc u e o he subuni I-IV o R.
sphae oides is e y simila o hose o he P. deni i icans while subuni I-III a e simila o
he co esponding subuni s o he bo ine hea cy och ome coxidase. The en i onmen s
a ound he edox-me al cen e s ( he binuclea cen e , heme aand CuA) and he non- edox-
ac i e cen e s (Mg2+ and Ca2+) a e s uc u ally e y simila o hose o bo h he bo ine hea
and P. deni i icans. Six phospholipid molecules we e iden i ied in he X- ay s uc u e o R.
sphae oides and a e assigned as phospha idyle hanolamines om he shapes o he elec on
densi y. These phospholipid molecules a e also included in he elec os a ic calcula ions. The
elec os a ic calcula ions we e pe o med in he monome o cy och ome coxidase.
S uc u es we e p epa ed o elec os a ic calcula ions using he CHARMM [153] molecula mod-
eling package. Hyd ogen a om posi ions we e gene a ed using he HBUILD algo i hm imple-
men ed in he CHARMM p og am. All he hea y a oms we e ixed and ene gy was minimized
using he CHARMM o ce ield. The ene gy minimiza ions we e pe o med by 500 s eepes decen
(SD) s eps, ollowed by 500 conjuga e g adien (CG) s eps.
Pa ial a omic cha ges o a oms o he s anda d amino acids we e aken om he CHARMM
pa ame e se . The pa ial a omic cha ges o he edox-me al cen e s ( he binuclea cen-
e , heme aand CuA) and he non- edox-ac i e cen e s (Mg2+ and Ca2+) we e ob ained om
quan um chemical calcula ions (see sec ion 6.3). The p o ein s uc u es we e minimized wi h
he c ys allog aphic wa e molecules. In elec os a ic calcula ions he c ys allog aphic wa e
molecules we e emo ed since he cons uc ion o wa e hyd ogens would a bi a ily assign
a ce ain o ien a ion o he wa e molecules ha would a ec elec os a ic calcula ions. The
C- e minus o subuni s I-IV o cy och ome coxidase we e no esol ed and a neu al blocking
g oup (ace ly g oup) was added o he C- e minus.
CONSTRUCTION OF A MEMBRANE MODEL. In o de o include he e ec o he memb ane en i-
onmen in o he elec os a ic calcula ions, he p o ein complex was embedded in o a cylind i-
cal shaped bel o uncha ged dummy a oms whe e he dummy a oms model he hyd ophobic
memb ane co e (see Figu e 6.2). The dummy a oms which mimics he memb ane en i on-
130 pKacalcula ions o CuBligands in cy och ome coxidase
men we e assigned a low dielec ic cons an in he elec os a ic calcula ions. The Bondi adii
we e used o he p o ein a oms [175]. The ca i ies wi hin he p o ein ha show no connec ion
o he memb ane en i onmen we e no be illed wi h he dummy a oms. These ca i ies a e
conside ed o con ain dis o ed wa e molecules and assigned a high dielec ic cons an .
The s uc u e o cy och ome coxidase wi h memb ane was ob ained om he o ien a ions o
p o ein in memb anes (OPM) da abase [205–207]. The OPM da abase cu en ly includes all
unique s uc u es o ansmemb ane p o ein complexes and selec ed mono opic, pe iphe al
p o eins and memb ane-bound pep ides om PDB wi h hei calcula ed memb ane bound-
a ies. Coo dina e iles o he p o eins wi h calcula ed memb ane bounda ies a e a ailable
o download. The coo dina es o cy och ome coxidase wi h calcula ed memb ane bound-
a ies we e supe imposed wi h he c ys al s uc u e o R. sphae oides using he p og am
supe impose. The memb ane was cons uc ed a ound his supe imposed s uc u es. The
Vol p og am [208] was used o cons uc a memb ane a ound he p o ein. The hickness o
he memb ane model is 30 ˚
A . The adius o he con ac ed memb ane is 50 ˚
A . A p obe adius
o 1.4 ˚
A was used o gene a ing he p o ein su ace. The memb ane model was cons uc ed
o all s uc u es conside ed in he elec os a ic calcula ions.
(A) (B)
Figu e 6.2. Memb ane model. (A): View o cy och ome coxidase om R. sphae oides
along he memb ane plane. The dummy a oms which mimics he memb ane en i onmen
we e assigned a low dielec ic cons an . (B): View om he cy oplasm.
6.2.2 REDOX CENTER MODELS
In he eac ion mechanism o cy och ome coxidase, ou edox cen e s a e in ol ed. The e-
duc ion o molecula oxygen o wa e akes place in he binuclea cen e whe e he elec ons
and p o ons a e deli e ed o oxygen educ ion and ou p o ons a e ansloca ed ac oss he
inne -mi ochond ial memb ane, a p ocess which esul s in a memb ane elec ochemical p o-
6.2. S uc u e p epa a ion and models 131
on g adien . The p o on ansloca ion is coupled wi h he elec on ans e which makes he
eac ion di icul o s udy. To inco po a e he edox cen e s o he p o ein, he edox cen e s
we e conside ed o he quan um chemical calcula ions o ge he cha ges wi h di e en ox-
idized and educed s a es as well hese models we e used o he DFT calcula ions o ob ain
he gas phase and sol a ion ene gies. The models used o he edox cen e s a e desc ibed
below. DFT calcula ions we e also pe o med in non- edox-ac i e cen e s (Mg2+ and Ca2+) o
ob ain pa ial a omic cha ges.
MODEL FOR CuACENTER. The CuAcen e con ains wo coppe ions (see Figu e 6.3) and a e
b idged by wo cys eins sul u a oms. The wo coppe ions o he CuAcen e a e coo dina ed
by wo His, one Me , a backbone ca bonyl oxygen o Glu, and wo b idging Cys esidues. The
liga ed cys eins we e simpli ied as me hyl hiola es. The his idines we e modeled by he me hyl
imidazoles. The backbone ca bonyl oxygen o Glu was included in he model compound. The
amino acid side chains we e cu a he Cαa om and hei Cβa oms we e ixed in hei c ys al
s uc u e posi ions. The ini ial coo dina es we e ob ained om he c ys al s uc u e o R.
sphae oides. All he hyd ogens we e added using babel p og am. The CuAcen e model was
used o ob ain he cha ges o he calcula ions. The cha ges we e ob ained o bo h educed
and oxidized s a es o CuAcen e .
CuA
M263
C256
C252
H260
E254
H217
Figu e 6.3. The model compound o CuAcen e The cys eins a e simpli ied as me hyl
hiola es. The his idines a e modeled by me hyl imidazole.
MODEL FOR HEME aHeme acen e is wi h wo his idine esidues as axial i on-ligands (see
Figu e 6.4). The his idine esidues we e modeled by me hyl imidazole. The heme p opiona es
we e cu o and subs i u ed by hyd ogen a oms. The hyd ophobic hyd oxye hyl- a nesyl
g oup was unca ed nex o he hyd oxyl g oup. The calcula ions we e pe o med bo h in he
educed and in he oxidized s a e.
132 pKacalcula ions o CuBligands in cy och ome coxidase
heme a
H421
H102
Figu e 6.4. The model compound o heme a.The his idines coo dina ing he i on a e
modeled by me hyl imidazole. The hyd ophobic hyd oxye hyl- a nesyl g oup was unca ed
nex o he hyd oxy g oup. The heme p opiona es we e cu o and subs i u ed by hyd ogen
a oms.
MODEL FOR CuBCENTER. The model o he CuBcen e consis s o Cu ion, me hylimidazole
model o coo dina ing his idines 284, 333, and 334 and he y osine 288 was modeled by
me hyl g oup (see Figu e 6.5). The ou h coo dina ion was modeled wi h H2O molecule and a
hyd oxyl g oup. The calcula ions we e pe o med bo h in he educed and he in he oxidized
s a e.
CuB
H284 Y288
+2/+1
H333
H334
Figu e 6.5. The model compound o CuBcen e . The his idines coo dina ing he coppe
ion a e modeled by me hyl imidazole. The c oss linked y osine 288 is eplaced by me hyl
g oup. The ou h coo dina ion is he H2O molecule. The p o on binding si es a e indica ed
by whi e sphe es.
6.3. Densi y Func ional calcula ions 133
MODEL FOR HEME a3CENTER. The model o he heme a3cen e consis s o he heme a3and
a axial his idine which was modeled by me hylimidazole (see Figu e 6.6). The six h coo di-
na ion was modeled wi h H2O molecule (FeIII-H2O and FeII-H2O), hyd oxyl g oup (FeIII-OH
and FeII-OH), oxo- e yl (FeIV=O) and ligand unbound s a e (FeIII and FeII). The hyd ophobic
hyd oxye hyl- a nesyl g oup was unca ed nex o he hyd oxyl g oup. The heme a3cen e
was op imized bo h in oxidized and educed s a es wi h all ligands men ioned abo e.
His419
heme a3
Figu e 6.6. The model compound o heme a3cen e . The his idines coo dina ing he
i on a e modeled by me hyl imidazole. The hyd ophobic hyd oxye hyl- a nesyl g oup was
unca ed nex o he hyd oxyl g oup. The heme p opiona es we e cu o and subs i u ed by
hyd ogen a oms. The H2O molecule is conside ed in he six coo dina ion posi ion is shown.
The p o on binding si es a e indica ed by whi e sphe es.
6.3 DENSITY FUNCTIONAL CALCULATIONS
The DFT calcula ions [81, 172] we e pe o med o ob ain pa ial a omic cha ges o he e-
dox cen e s CuA, heme a, heme a3and CuB espec i ely. Fo edox cen e s CuA, heme a
and heme a3 he Pe d ew-Wang 91 (PW91) calcula ions we e pe o med wi h ADF 2004.01
[173] p og am. The pa ial a omic cha ges o CuBcen e we e ob ained by B3LYP me hod
using 6-31G* basis se s using GAUSSIAN 03 p og am. The DFT calcula ions we e pe o med
on he oxidized s a es o CuAand heme a. Following se en s a es we e conside ed o heme
a3: aqua e ic s a e (FeIII-H2O, cha ge=+1, S=5/2), aqua e ous s a e (FeII-H2O, cha ge=0,
S=2), hyd oxyl s a e (FeIII-OH, cha ge=0, S=5/2 and FeII-OH, cha ge=-1, S=2), oxo- e yl
s a e (FeIV=O, cha ge=0, S=2) and ligand unbound s a es (FeIII, cha ge=+1, S=5/2 and FeII,
cha ge=0, S=2). The local densi y app oxima ion (LDA) o exchange and co ela ion a e based
on he pa ame iza ion o Vosko, Wilk and Nausai [84]. The Pe d ew-Wang 91 (PW91) [86]
exchange and co ela ion unc ionals we e used o he gene alized g adien app oxima ion
(GGA). The nume ical in eg a ion scheme used in his calcula ion was Vo onoi polyhed on
me hod de eloped by e Velde e al. wi h he accu acy pa ame e ACCINT se o i s de aul
alue. A se o iple ζSla e ype o bi al (STO) was employed wi h single pola iza ion unc ion.
The inne co e shells we e ea ed by he ozen co e app oxima ion. All he calcula ions we e

134 pKacalcula ions o CuBligands in cy och ome coxidase
done wi h a spin-un es ic ed scheme. The op imiza ion we e pe o med by he quasi New on
me hod and he Hessian was upda ed wi h he B oyden-Fle che -Gold a b-Shanno s a egy.
The dis ibu ions o pa ial a omic poin cha ges we e compu ed by i ing he molecula elec-
os a ic po en ials calcula ed by ADF 2004.01 p og am [173]. The p og am cha ge i was
used o ob ain poin cha ges which is based on CHELPG algo i hm [174]. The ne cha ge o he
molecule and he h ee Ca esian dipole momen componen s om PW91 calcula ions we e
adop ed as cons ain s o he cha ge i . Poin cha ges we e hen compu ed by de e mining
he elec os a ic po en ial, how well he poin cha ges e ec i ely ep oduce he elec os a ic
po en ial. The ESP cha ges we e calcula ed on he cubic g id wi h uni o m spacing o 0.2 and
3˚
A ou e bounda y a ound each a om o he molecule. The a oms we e assigned he Bondi
alues o 1.7 o ca bon, 1.2 o hyd ogen, 1.55 o ni ogen, 1.5 o oxygen, 1.3 o i on and
1.8 o sul u . To minimize he unce ain ies in he i ing p ocedu e he single alue decom-
posi ion (SVD) me hod [209] was used o ob ain a model wi h s able a omic cha ges. Pa ial
a omic cha ges o he amino acids a e aken om he CHARMM22 pa ame e se . The pa ial
a omic cha ges o all he edox cen e s we e de i ed om PW91 calcula ions. In addi ion
o a omic coo dina es and a omic cha ges, elec os a ic calcula ions equi e adii. The adii
used we e aken om Bondi [175].
All mic oscopic s a es o CuBcen e we e op imized wi h hyb id densi y unc ional calcula ions
(B3LYP) using 6-31G* basis se s. The geome y op imiza ions we e pe o med using GAUSSIAN
03 p og am. The single poin calcula ions we e pe o med on he op imized s uc u es o he
CuBcen e using B3LYP/6-31G*/C-PCM le el. The elec os a ic po en ial ob ained om C-
PCM sel -consis en eac ion ield we e i ed by Me z-Kollman me hod o ob ain he poin
cha ges.
6.4 ELECTROSTATIC CALCULATIONS
The elec os a ic calcula ions we e pe o med on Fea3-CuBcomplexes (see Figu e 6.7). The
CuBcen e was allowed o adop each o he eigh s a es shown in Figu e 5.3 and 5.4 espec-
i ely. Following se en possible s a es o heme a3cen e we e conside ed: FeIV=O, FeIII-H2O,
FeII-H2O, FeIII-OH, FeII-OH, FeIII and FeII. These se en s a es wi h di e en CuBcen e a e
shown in Figu e 6.7. The CuAcen e and heme awe e conside ed oxidized h oughou he
s udies. Con inuum elec os a ic calcula ions we e pe o med wi h he QMPB which sol e he
linea Poisson-Bol zmann equa ion by nume ical ini e di e ence me hod. The whole sys em
is di ided in o h ee egions wi h h ee di e en dielec ic cons an o i= 1 o he ac i e si e
(quan um egion) in his case he CuBcen e , s= 80 o he sol en egion, and p= 4 o he
p o ein. Compa ed o he pu ely elec os a ic dielec ic cons an = 2,p= 4 was adop ed
o he p o ein, which allows some mobili y o he p o ein dipoles and accoun s o some e-
o ien a ional elaxa ion o he p o ein in an app oxima e way. The ionic s eng h was se o
0.1 mol/l and he absolu e empe a u e is se o 300 K. The bounda y be ween he in e io
and he ex e io is de ined as he sol en con ac and e-en an su aces o 1.4 ˚
A sphe ical
sol en p obe olling o e he an de Waals su ace o he p o ein. The a omic adii o he
p o ein’s a om and he cha ges o he p o ein’s non- i a ing a oms we e aken om he pola
hyd ogen pa ame e se o CHARMM22. Elec os a ic po en ial was calcula ed by i s ocusing
he g id on he p o ein model compound and he second g id was cen e ed on he i a able
6.4. Elec os a ic calcula ions 135
Fe =O
IV
HH
O
HH
O
Fe −H O
II
Fe −H O
III
O
H
Fe −OH
III
O
H
Fe −OH
II Fe
III
Fe
II
heme a3heme a3
heme a3
heme a3
CuBCuBCuB
CuB
CuB
heme a3
heme a3
heme a3CuBCuB
Cu
His280
His333
His334
O
HH
1+/2+
Fe Cu
His280
His333
His334
O
HH
1+/2+
Fe
Cu
His280
His333
His334
O
HH
1+/2+
Fe Cu
His280
His333
His334
O
HH
1+/2+
Fe
Cu
His280
His333
His334
O
HH
1+/2+
Fe
Cu
His280
His333
His334
O
HH
1+/2+
Fe
Cu
His280
His333
His334
O
HH
1+/2+
OFe
4+ 3+ 2+
3+ 2+ 3+
2+
2 2
Figu e 6.7. The Fea3-CuBcomplexes conside ed in he p esen s udy. The Fea3-CuB
complexes conside ed o he p esen s udy a e shown. The nomencla u e o he co e-
sponding complexes a e gi en in he bo om.
g oup. A coa se g id wi h 1 ˚
A g id spacing and a ine g id wi h 0.25 ˚
A g id spacing we e used.
Aspa a es, glu ama es, lysine, his idines ( wo si es o each his idines) cys ein, y osine and
N- and C- e mini a e ea ed as i a able g oups. pKmodel
a alues o he i a able g oups a e
gi en in Table 6.1. The pKa alues calcula ed o he CuBcen e bo h in he educed and in
he oxidized s a es in he aqueous solu ion a e aken as a model pKa alues o he CuBcen e
in p o ein.
6.4.1 CALCULATION OF AVERAGE PKaIN PROTEIN
The elec os a ic calcula ions we e pe o med on he p o ein s uc u e o R. sphae oides.
The elec os a ic ene gies ob ained om he QMPB calcula ions we e decomposed in o Bo n,
backg ound and he in e ac ion ene gies. The di e ence in he Bo n ene gy (∆∆GBo n) (see
Eq. 2.36) and he backg ound ene gy (∆∆Gback) (see Eq. 2.37) o a p o ona ion eac ion o
136 pKacalcula ions o CuBligands in cy och ome coxidase
Ti a able g oup pKmodel
a
aspa a e 4.0
glu ama e 4.4
a ginine 10.4
lysine 10.4
his idine Nδ6.6
his idine N7.0
y osine 9.6
cys eine 9.5
C- e minus 3.8
N- e minus 7.5
Table 6.1. pKmodel
a alues o he i a able g oups. The pKmodel
a alues a e aken om
Re . [70] and Re . [71].
he si e can be di ec ly ob ained om he elec os a ic calcula ions (see sec ion 2.2.2). The
a e age pKa alues o CuBligand in p o ein we e ob ained om Eq. (6.1).
pKa,i =pKin
a,i +
N
X
j=1
(hxji − x(0)
j)Wij (6.1)
whe e he in insic pKa alue (pKin
a,i ) is he pKa alue ha he pa icula i a able g oup
would ha e, i all o he i a able g oups a e in hei e e ence o m. This e m includes he
sol a ion ene gy and he in e ac ion wi h non- i a ing esidues and he p o ein backbone.
The e m Wij ep esen s he in e ac ion ene gy be ween he i a able g oups iand jin hei
cha ged om; hxji ep esen s he p o ona ion p obabili y o he g oup jwhich is ob ained by
a he modynamic a e age o e all possible p o ona ion s a es and x0
jis he e e ence p o o-
na ion o m o si e j. The hxjia e ob ained om MC calcula ions. These a e age pKa alues
do no ep esen an equilib ium si ua ion, bu hey a e a good app oxima ion o he eal pKa
alue o he si e.
The p o ona ion p obabili ies we e ob ained by Me opolis Mon e Ca lo calcula ions. The
i a ion cu es we e calcula ed by Mon e Ca lo p og am, GMCT de eloped in ou g oup. The
p o ona ion p obabili ies we e compu ed a pH 7. The empe a u e was se o 300 K. The
double lip and iple lips we e se o 2 and 3 pH uni s espec i ely. The numbe o ull MC
scans we e se o 30,000 and 100 equilib ium MC scans we e pe o med be o e he MC ull
scan.
6.5. Resul s and Discussion 137
6.5 RESULTS AND DISCUSSION
The pKa alues calcula ed o he Cu-bound ligands in a ious CuBand heme a3complexes
in cy och ome coxidase a e discussed in his sec ion. The pKa alues discussed in his
sec ion a e a e age pKa alues o he dep o ona ion eac ions a pH=7. The e ec o he
p o ein en i onmen on he CuBligands we e s udied by combining he DFT and con inuum
elec os a ic calcula ions. The pKa alues ob ained o he CuBcen e bo h in oxidized and
educed s a e in aqueous solu ion a e discussed in de ail in chap e 5.
PKaVALUES OF MICROSCOPIC STATES OF CuBLIGANDS IN CYTOCHROME cOXIDASE. The pKa
alues o all possible p o on equilib ium eac ions o oxidized (Cu+2
B) and educed (Cu+1
B)CuB
cen e in he p esence o di e en heme a3s a es o cy och ome coxidase a e gi en in Table 6.2
and 6.3 espec i ely.
The pKa alue o H2O ligand was calcula ed bo h in oxidized and educed s a es o he CuB
cen e . The e is no expe imen al pKa alue a ailable o H2O ligand in he CuBcen e . The
pKa alues o H2O ligand ob ained in he aqueous solu ion a e used as model pKa alues.
Compa ed o he pKa alues o he H2O in aqueous solu ion, he p o ein en i onmen shi s
he pKa alues o he H2O bo h in oxidized and educed s a e o CuBcen e o highe alues.
The pKa alues a e e y high o he educed s a e o he CuBcen e compa ed o he oxidized
s a e. The pKa alues a e in he ange o 51-60 which show ha he dep o ona ion o he
H2O in he educed s a e is highly un a o able. The pKa alues calcula ed in he FeII-OH
s a e a e ob iously high due o -1 cha ge on he heme a3cen e . The pKa alues inc ease
wi h he change in cha ge s a e o heme a3cen e om +1→0→-1. The high pKa alues o
dep o ona ion o H2O show ha i is p o ona ed wi hin he physiological pH.
The e is a possibili y ha His333 ano he ligand o CuBcen e , can se e as a p o on loading
si e o subsequen pumping. The pKa alues o His333 ligand ob ained in he aqueous
solu ion is used as model pKa alues. The e is no expe imen al pKa alue a ailable o His333
ligand in he CuBcen e . The low dielec ic p o ein en i onmen shi s he pKa alues o he
His333 bo h in oxidized and educed s a e o CuBcen e o highe alues compa ed o esul s
in aqueous solu ion. The expe imen al pKa alue o he dep o ona ion o imidazole, which
p oduces an anionic imidazole, is app oxima ely 14 [193, 194]. In p esen calcula ions, he
pKa alues o His333 dep o ona ion a e in he ange o 37-48 in he oxidized s a e and 44.5-
53.8 in he educed s a e. The high pKa alues in he p o ein show ha His333 is likely o
be p o ona ed wi hin he physiological pH bo h in he oxidized and in he educed s a e o he
CuBcen e . The p o ona ed Nδ1o His333 is hyd ogen bonded o Th 352. As a esul , he
dep o ona ion o His333 may be signi ican ly hinde ed.
The pKa alue o His334 ob ained bo h in aqueous solu ion and p o ein (hpKa,iip o ) wi h di -
e en heme a3ligand a e gi en in Table 6.2 and 6.3 espec i ely. The pKa alues o His334
ligand ob ained in he aqueous solu ion a e used as model pKa alues. The e is no expe imen-
al pKa alue a ailable o His334 ligand in he CuBcen e . The p o ein en i onmen shi s
mos o he pKa alues o he His334 bo h in oxidized and educed s a e o CuBcen e . The
pKa alues a e shi ed o highe alues in he oxidized s a e o he CuBcen e and a e in he
ange o 27.5-39. The pKa alues a e e en shi ed o highe alues in he educed s a e o he
CuBcen e . The pKa alue o 33.5 is ob ained in he case o oxo- e yl s a e (FeIV=O) whe eas
144 pKacalcula ions o CuBligands in cy och ome coxidase
Subuni Residue
hxia pH=7
FeIII-H2O FeIII FeII FeII-H2O FeIII-OH FeIV=O FeII-OH
[+1] [+1] [0] [0] [0] [0] [-1]
GLU-533 0.002 0.001 0.002 0.002 0.002 0.002 0.002
HIS-534 0.411 0.409 0.436 0.437 0.426 0.434 0.461
ASP-536 0.001 0.003 0.003 0.002 0.003 0.004 0.004
GLU-539 0.001 0.001 0.001 0.002 0.002 0.002 0.003
TRP-540 1.000 1.000 1.000 1.000 1.000 1.000 1.000
GLU-548 0.004 0.003 0.003 0.003 0.003 0.003 0.003
HIS-549 0.107 0.118 0.123 0.119 0.122 0.117 0.128
GLU-552 0.002 0.003 0.004 0.004 0.003 0.004 0.003
LYS-556 0.997 0.998 0.999 0.997 0.998 0.997 0.999
ARG-557 0.999 1.000 0.999 0.999 0.999 0.999 1.000
GLU-558 0.003 0.003 0.003 0.003 0.002 0.002 0.002
ASP-559 0.002 0.002 0.002 0.001 0.002 0.001 0.002
TRP-560 1.000 1.000 1.000 1.000 1.000 1.000 1.000
HEA a-PROPA 0.000 0.000 0.000 0.000 0.000 0.000 0.000
HEA a-PROPD 0.000 0.000 0.000 0.000 0.000 0.000 0.000
HEA a3-PROPA 0.000 0.000 0.000 0.000 0.000 0.000 0.000
HEA a3-PROPD 0.000 0.000 0.000 0.000 0.000 0.000 0.000
II GLU-31 0.019 0.015 0.016 0.015 0.021 0.016 0.015
ARG-35 1.000 1.000 1.000 1.000 1.000 1.000 1.000
HIS-55 0.004 0.003 0.005 0.003 0.005 0.004 0.007
TRP-56 1.000 1.000 1.000 1.000 1.000 1.000 1.000
ASP-58 0.000 0.000 0.000 0.001 0.001 0.000 0.005
TYR-78 1.000 1.000 1.000 1.000 1.000 1.000 1.000
TRP-81 1.000 1.000 1.000 1.000 1.000 1.000 1.000
ARG-82 1.000 1.000 1.000 1.000 1.000 1.000 1.000
HIS-84 0.000 0.000 0.000 0.000 0.000 0.000 0.000
GLU-85 0.000 0.000 0.000 0.000 0.000 0.000 0.000
LYS-86 0.924 0.921 0.923 0.925 0.920 0.925 0.924
ARG-87 1.000 1.000 1.000 1.000 1.000 1.000 1.000
LYS-89 0.366 0.355 0.363 0.362 0.356 0.354 0.371
ARG-93 1.000 1.000 1.000 1.000 1.000 1.000 1.000
HIS-96 0.045 0.046 0.048 0.051 0.048 0.053 0.052
con inued on nex page

6.5. Resul s and Discussion 145
Subuni Residue
hxia pH=7
FeIII-H2O FeIII FeII FeII-H2O FeIII-OH FeIV=O FeII-OH
[+1] [+1] [0] [0] [0] [0] [-1]
GLU-101 1.000 0.999 1.000 1.000 0.999 1.000 1.000
TRP-104 1.000 1.000 1.000 1.000 1.000 1.000 1.000
GLU-128 0.011 0.013 0.012 0.012 0.014 0.014 0.016
GLU-131 0.011 0.011 0.011 0.014 0.009 0.012 0.009
ASP-133 0.001 0.001 0.001 0.001 0.001 0.001 0.001
LYS-137 0.999 1.000 1.000 1.000 1.000 1.000 1.000
TYR-141 1.000 1.000 1.000 1.000 1.000 1.000 1.000
TRP-143 1.000 1.000 1.000 1.000 1.000 1.000 1.000
TYR-144 1.000 1.000 1.000 1.000 1.000 1.000 1.000
TRP-145 1.000 1.000 1.000 1.000 1.000 1.000 1.000
TYR-147 1.000 1.000 1.000 1.000 1.000 1.000 1.000
GLU-148 0.030 0.027 0.031 0.034 0.029 0.031 0.031
TYR-149 1.000 1.000 1.000 1.000 1.000 1.000 1.000
ASP-151 0.006 0.008 0.006 0.009 0.009 0.009 0.008
GLU-152 0.005 0.005 0.006 0.005 0.004 0.007 0.006
GLU-153 0.010 0.010 0.009 0.008 0.010 0.010 0.012
GLU-157 0.080 0.080 0.080 0.086 0.075 0.078 0.081
TYR-159 1.000 1.000 1.000 1.000 1.000 1.000 1.000
ASP-169 0.004 0.004 0.005 0.006 0.005 0.005 0.005
ARG-171 1.000 1.000 1.000 1.000 1.000 1.000 1.000
GLU-175 0.002 0.003 0.002 0.002 0.002 0.001 0.002
GLU-177 0.011 0.009 0.009 0.010 0.011 0.012 0.007
GLU-182 0.013 0.013 0.011 0.012 0.015 0.017 0.015
TYR-185 1.000 1.000 1.000 1.000 1.000 1.000 1.000
ARG-187 1.000 1.000 1.000 1.000 1.000 1.000 1.000
ASP-188 0.001 0.001 0.001 0.001 0.001 0.001 0.001
GLU-189 0.272 0.271 0.262 0.263 0.254 0.268 0.274
ASP-195 0.001 0.000 0.000 0.001 0.001 0.001 0.001
LYS-204 1.000 1.000 1.000 1.000 1.000 1.000 1.000
ASP-214 0.000 0.000 0.000 0.000 0.000 0.000 0.000
TRP-219 1.000 1.000 1.000 1.000 1.000 1.000 1.000
LYS-227 0.984 0.984 0.996 0.997 0.997 0.997 1.000
con inued on nex page
146 pKacalcula ions o CuBligands in cy och ome coxidase
Subuni Residue
hxia pH=7
FeIII-H2O FeIII FeII FeII-H2O FeIII-OH FeIV=O FeII-OH
[+1] [+1] [0] [0] [0] [0] [-1]
ASP-229 0.000 0.000 0.000 0.000 0.000 0.000 0.000
ARG-234 1.000 1.000 1.000 1.000 1.000 1.000 1.000
TRP-239 1.000 1.000 1.000 1.000 1.000 1.000 1.000
ARG-241 1.000 1.000 1.000 1.000 1.000 1.000 1.000
GLU-243 0.000 0.000 0.000 0.000 0.000 0.000 0.000
ARG-244 0.998 0.999 0.998 0.999 0.999 0.998 0.999
GLU-245 0.040 0.043 0.040 0.039 0.042 0.041 0.042
TYR-262 1.000 1.000 1.000 1.000 1.000 1.000 1.000
LYS-268 0.975 0.981 0.979 0.977 0.980 0.975 0.977
GLU-272 0.013 0.016 0.017 0.015 0.014 0.014 0.017
GLU-273 0.007 0.005 0.004 0.005 0.006 0.004 0.007
TYR-275 1.000 1.000 1.000 1.000 1.000 1.000 1.000
TRP-278 1.000 1.000 1.000 1.000 1.000 1.000 1.000
GLU-280 0.016 0.012 0.015 0.014 0.015 0.015 0.014
ARG-283 1.000 1.000 1.000 1.000 1.000 1.000 1.000
TYR-287 1.000 1.000 1.000 1.000 1.000 1.000 1.000
GLU-288 0.054 0.055 0.053 0.052 0.053 0.057 0.052
III HIS-3 0.245 0.241 0.236 0.233 0.232 0.241 0.239
LYS-5 0.997 0.998 0.998 0.998 0.998 0.997 0.997
HIS-7 0.000 0.000 0.000 0.000 0.000 0.000 0.000
ASP-8 0.000 0.000 0.000 0.000 0.000 0.000 0.000
TYR-9 1.000 1.000 1.000 1.000 1.000 1.000 1.000
HIS-10 0.001 0.001 0.001 0.001 0.002 0.002 0.002
TRP-17 1.000 1.000 1.000 1.000 1.000 1.000 1.000
TRP-35 1.000 1.000 1.000 1.000 1.000 1.000 1.000
HIS-37 0.460 0.462 0.486 0.493 0.491 0.490 0.510
TRP-42 1.000 1.000 1.000 1.000 1.000 1.000 1.000
TYR-53 1.000 1.000 1.000 1.000 1.000 1.000 1.000
TRP-58 1.000 1.000 1.000 1.000 1.000 1.000 1.000
TRP-59 1.000 1.000 1.000 1.000 1.000 1.000 1.000
ASP-61 0.004 0.004 0.004 0.003 0.005 0.005 0.004
GLU-65 0.000 0.000 0.000 0.000 0.000 0.000 0.000
con inued on nex page
6.5. Resul s and Discussion 147
Subuni Residue
hxia pH=7
FeIII-H2O FeIII FeII FeII-H2O FeIII-OH FeIV=O FeII-OH
[+1] [+1] [0] [0] [0] [0] [-1]
GLU-68 0.046 0.045 0.042 0.040 0.041 0.043 0.042
ASP-70 0.837 0.836 0.847 0.848 0.848 0.845 0.845
HIS-71 0.000 0.000 0.000 0.000 0.000 0.000 0.000
ARG-76 1.000 1.000 1.000 1.000 1.000 1.000 1.000
ARG-80 1.000 0.999 0.999 0.999 1.000 0.999 1.000
TRP-81 1.000 1.000 1.000 1.000 1.000 1.000 1.000
GLU-90 0.969 0.975 0.979 0.977 0.981 0.975 0.984
TRP-97 1.000 1.000 1.000 1.000 1.000 1.000 1.000
TRP-99 1.000 1.000 1.000 1.000 1.000 1.000 1.000
LYS-103 1.000 1.000 1.000 1.000 1.000 1.000 1.000
HIS-104 0.000 0.000 0.000 0.000 0.000 0.000 0.000
TYR-107 1.000 1.000 1.000 1.000 1.000 1.000 1.000
GLU-112 0.018 0.018 0.019 0.017 0.017 0.021 0.020
ASP-117 0.020 0.021 0.025 0.025 0.022 0.024 0.031
GLU-123 0.057 0.060 0.057 0.059 0.055 0.060 0.057
ASP-129 0.000 0.001 0.000 0.001 0.000 0.001 0.000
TRP-131 1.000 1.000 1.000 1.000 1.000 1.000 1.000
HIS-132 0.013 0.012 0.016 0.015 0.014 0.013 0.017
CYS-143 1.000 1.000 1.000 1.000 1.000 1.000 1.000
CYS-146 1.000 1.000 1.000 1.000 1.000 1.000 1.000
TRP-150 1.000 1.000 1.000 1.000 1.000 1.000 1.000
HIS-152 0.000 0.000 0.000 0.001 0.000 0.000 0.000
HIS-153 0.035 0.035 0.031 0.034 0.035 0.037 0.034
HIS-157 0.417 0.433 0.430 0.408 0.413 0.416 0.419
GLU-158 0.072 0.069 0.068 0.073 0.073 0.065 0.071
ARG-161 0.998 0.997 0.998 0.997 0.998 0.998 0.997
ARG-162 1.000 1.000 1.000 1.000 1.000 1.000 1.000
ASP-163 0.000 0.001 0.000 0.001 0.001 0.001 0.000
TRP-166 1.000 1.000 1.000 1.000 1.000 1.000 1.000
TYR-184 1.000 1.000 1.000 1.000 1.000 1.000 1.000
GLU-185 1.000 1.000 1.000 1.000 1.000 1.000 1.000
TYR-186 1.000 1.000 1.000 1.000 1.000 1.000 1.000
con inued on nex page
148 pKacalcula ions o CuBligands in cy och ome coxidase
Subuni Residue
hxia pH=7
FeIII-H2O FeIII FeII FeII-H2O FeIII-OH FeIV=O FeII-OH
[+1] [+1] [0] [0] [0] [0] [-1]
HIS-188 0.982 0.983 0.978 0.985 0.981 0.983 0.982
TYR-198 1.000 1.000 1.000 1.000 1.000 1.000 1.000
HIS-209 0.000 0.000 0.000 0.000 0.000 0.000 0.000
HIS-212 0.029 0.025 0.021 0.022 0.019 0.025 0.016
CYS-223 1.000 1.000 1.000 1.000 1.000 1.000 1.000
ARG-226 0.321 0.332 0.342 0.250 0.256 0.254 0.361
ARG-229 0.997 0.996 0.997 0.997 0.997 0.997 0.996
HIS-231 0.001 0.001 0.001 0.001 0.000 0.001 0.001
GLU-235 0.002 0.002 0.002 0.001 0.001 0.002 0.001
LYS-236 0.290 0.275 0.275 0.299 0.298 0.302 0.282
HIS-237 0.521 0.512 0.505 0.562 0.548 0.561 0.512
GLU-241 0.140 0.138 0.150 0.156 0.156 0.150 0.148
TRP-245 1.000 1.000 1.000 1.000 1.000 1.000 1.000
TYR-246 1.000 1.000 1.000 1.000 1.000 1.000 1.000
TRP-247 1.000 1.000 1.000 1.000 1.000 1.000 1.000
HIS-248 0.000 0.000 0.000 0.000 0.000 0.000 0.000
ASP-251 1.000 1.000 1.000 1.000 1.000 1.000 1.000
TRP-254 1.000 1.000 1.000 1.000 1.000 1.000 1.000
TYR-262 1.000 1.000 1.000 1.000 1.000 1.000 1.000
TRP-264 1.000 1.000 1.000 1.000 1.000 1.000 1.000
IV HIS-11 0.316 0.318 0.320 0.320 0.317 0.316 0.319
ASP-17 0.000 0.000 0.000 0.000 0.000 0.000 0.000
GLU-22 0.002 0.002 0.002 0.003 0.001 0.002 0.003
LYS-23 0.993 0.996 0.994 0.995 0.995 0.994 0.995
ARG-30 0.999 1.000 0.999 0.999 0.999 0.999 1.000
TRP-34 1.000 1.000 1.000 1.000 1.000 1.000 1.000
The pKa alue o 46.4 and 46.2 a e ob ained o His334 in H0and H s a es espec i ely. The
high pKa alues show ha His334 emains p o ona ed in hese s a es. The nex dep o ona ion
is expec ed in connec ion wi h he o ma ion o he O s a e and a pKao 40.7 is ob ained o
His334 which is associa ed wi h p o on pumping e en . The high pKa o he His334 show
ha i is p o ona ed in O s a e. The O s a e was p oposed [1] o e ol e in wo di e en ways.
The p o ona ion o he hyd oxyl g oup o he CuBcen e leads o O∼s a e and a pKa alue o
27.5 is ob ained o he dep o ona ion o His334 and in nex possible s ep (OH)apKa alue o
6.6. Conclusions 149
40.4 is ob ained o His334. The pKa alues o 45.4, 32.7 and 45.9 a e ob ained o His334
in E, Epand R s a es espec i ely. All hese high pKa alues show ha His334 is p o ona ed
in O, E, Epand R s a es.
In he p esen s udy, high pKa alues we e ob ained compa ed o he pKa alues epo ed
by S ucheb ukho and Fadda e al. This di e ence may be due o how he pKacalcula ions
we e done. Fo example he model pKa alues used o he CuBligands and he eliabili y o
he cha ges used o he elec os a ic calcula ions. In p esen s udy he a e age pKa alues
o he CuBligands we e calcula ed by Tan o d-Roxby app oxima ion [77]. S ucheb ukho and
Fadda e al. calcula ed he pKa alues by aking only he wo ins ances like p o ona ed and
dep o ona ed s a es o he i a able g oups and no all ins ances we e conside ed by hem.
In ou calcula ions, we conside ed all ins ances o he i a able g oups, o example ou in-
s ances we e conside ed o he y osines and yp ophanes. An ex ensi e bench ma k s udies
we e pe o med o ob ain mo e accu a e cha ges and sol a ion ene gies. In he wo s udies
[1–3, 5, 6, 182], he c ys al coo dina es o he CuBcen e was no ully elaxed du ing he
geome y op imiza ion which can lead o un ealis ic bond leng hs. Fadda e al. only op imized
he p o ona ed s uc u es, while he dep o ona ed was no op imized i.e., same geome y was
used o bo h p o ona ed and dep o ona ed s a es. The homogeneous dielec ic model o he
p o ein was used in he calcula ions. The cha ges o he CuBcen e used in he calcula ions
we e ound wi hou ega d o he eac ion ield i.e., no in a sel -consis en manne . The p o-
ona ion s a e o he p o ein is no p ope ly de ined in hei calcula ions whe e he cha ges
o he i a able g oups a e c i ical o elec os a ic calcula ions. The p o ona ion depends on
he edox s a es o he p o ein. This issues we e comple ely igno ed in Fadda e al. wo k.
6.6 CONCLUSIONS
The DFT in combina ion wi h con inuum elec os a ic calcula ions we e used o calcula e
he pKa alues o he CuBligands, H2O, His333 and His334 in a ious CuBand Fea3-CuB
complexes o ind he ole o CuBligands in he eac ion mechanism o cy och ome coxidase.
The His334 which ac s as ligand o he CuBcen e was p oposed [1] o be in ol ed in he
eac ion mechanism o cy och ome coxidase.
In p esen s udy all he possible edox combina ion o CuB-heme a3cen e o cy och ome c
oxidase a e conside ed and all mic oscopic pKa alues o H2O, His333 and His334 we e cal-
cula ed o di e en s a es o he CuB-heme a3cen e . The pKa alues o His334 in F→O,
E→Ep ansi ions o he ca aly ic cycle we e s udied. Ou calcula ions a e based on he de-
ailed s udy o all he pKa alues o he ligands in he CuBcen e conside ing all edox s a es
o CuBand heme a3cen e , which allows us o analyze he e ec o he edox s a e o CuB-
heme a3cen e . Acco ding o he p oposed scheme [3, 5, 6], His334 unde goes dep o ona ion
whene e he chemical p o on en e s he binuclea cen e and con e he hyd oxyl ligand o
wa e molecule. This model was p oposed based on he pKa alue o His334 whe e he pKa
alue o His334 was shi ed o lowe pKa alue a ound 5 pKauni s.
In he p esen s udy, he pKa alues o CuBligands a e signi ican ly shi ed o highe alues
in p o ein compa ed o aqueous solu ion. The pKa alues o CuBligands a e in he ange o
15-60. The high pKa alues o His334 show ha His334 is p o ona ed du ing all s eps o he
ca aly ic cycle. The pKa alues o His334 in p o ein inc ease signi ican ly compa ed o he

150 pKacalcula ions o CuBligands in cy och ome coxidase
aqueous solu ion when he CuBcen e is p esen in he oxidized s a es which is inconsis en
wi h he pKa alues epo ed by he S ucheb ukho e al. The p o ein en i onmen shi s
he pKa alues o all complexes o highe alues, independen ly o he edox s a e o he
me als. Acco ding o he pKa alues o His334 he p o on pumping model as sugges ed by
S ucheb ukho [1] migh no be possible.
CHAPTER 7
CONCLUDING REMARKS AND OUTLOOK
This hesis is ocused on he pKacalcula ions o CuBand heme a3cen e o cy och ome c
oxidase using densi y unc ional heo y and con inuum elec os a ic models o unde s and
he ole o he CuB-bound his idines in eac ion mechanism o cy och ome coxidase. The
cy och ome coxidase ene ge ically couples he elec on ans e eac ions associa ed wi h
he educ ion o oxygen o wa e , o pump p o ons ac oss he memb ane. Al hough a as
amoun o s uc u al and unc ional in o ma ion o cy och ome coxidase is a ailable om
expe imen al and heo e ical da a, he ac ual s ep o coupling he edox eac ions o he p o on
ansloca ion is poo ly unde s ood.
Recen ly S ucheb ukho e al. [1–3, 5, 6] sugges ed ha he dep o ona ion o he CuBligand
His334 plays a cen al ole in he p o on pumping mechanism o cy och ome coxidase. Ac-
co ding o his sugges ion, His334 dep o ona es a i s Nδ1when he CuBcen e ge s oxidized.
F om ou esul s, His334 is p o ona ed du ing all s eps o he ca aly ic cycle in cy och ome c
oxidase and concluding ha a simple dep o ona ion o His334 in he p o ein en i onmen is
impossible in an equilib ium si ua ion due o he high pKa alue ha his g oup shows when
i is bound o CuBcen e . A ole o his esidue in he mechanism o p o on pumping migh
no be possible.
The pKacalcula ions on CuB-bound His333 and His334 in aqueous solu ion (Chap e 5) lead
o he ollowing conclusions:
The pKa alues o he i s dep o ona ion eac ions o His333 and His334 a e oo high (∼15)
o allow hei dep o ona ion in aqueous solu ion. The expe imen al pKa alue o he de-
p o ona ion o imidazole is app oxima ely 14 [193, 194]. The double and iple dep o ona ion
eac ions a e no ene ge ically plausible in aqueous solu ion due o hei high pKa alues. The
pKa alues in he he ange o 15.2-21 demons a e ha His334 and His334 a e p o ona ed
a physiological pH.
The pKa alues o CuBbound His333 and His334 in oxidized s a e do no show any signi ican
change o he pKa alue due o hei coo dina ion o Cu ion in aqueous solu ion. The esul s
ob ained om educed CuBcen e e els a small inc ease in he pKa alues o imidazoles. The
pKa alues o His333 and His334 in he educed s a e o CuBcen e a e e en highe o allow
hei dep o ona ion in aqueous solu ion. The esul s show ha he sol a ion ene gies needed
o pKacalcula ions should be calcula ed by including he elec onic pola iza ion due o he
sol en explici ly i.e., in sel -consis en manne o ob ain p ope end in he pKa alues o
he CuBcen e .
151
152 Concluding Rema ks and Ou look
The pKacalcula ions on CuB-bound His333 and His334 in cy och ome coxidase (Chap e 6)
a pH=7 lead o he ollowing conclusions:
The pKa alues o CuBligands a e signi ican ly shi ed o highe alues in p o ein compa ed
o aqueous solu ion. The pKa alues o CuBligands a e in he ange o 15-60. The CuB
bound His334 is p o ona ed du ing all s eps o he ca aly ic cycle p o ing ha he Fe and
Cu ion oxida ion s a es do no lowe he pKa alues o CuBligands in he p o ein. The pKa
alues o CuBligands a e signi ican ly shi ed o highe alues dependen on he edox s a es
o he CuBand heme a3cen e . The p o ein en i onmen inc eases he pKa alues o all
complexes o highe alues independen o he edox s a e o he me als. The His333 and
His334 a e likely o be p o ona ed a physiological pH. The double dep o ona ion and iple
dep o ona ion eac ions in p o ein a e ene ge ically un a o able. The pKa alues o His333
show ha his esidue is likely o be p o ona ed in he p o ein and an in ol emen o his
esidue as p o on loading si e in he eac ion mechanism o cy och ome coxidase can he e o e
be uled ou . These esul s a e inconsis en wi h he p oposed ole o His334 as a key elemen
in he pumping mechanism. The p o on pumping model as sugges ed by S ucheb ukho [1]
wi h he in ol emen o His334 migh no be possible.
A ew issues could no be comple ely sol ed in he amewo k o his s udy and should be he
opic o u u e esea ch:
In he p esen wo k, Ty 280 which is c oss-linked wi h His284 is eplaced by me hyl g oup in
he CuBmodel. This c oss-linked His-T y esidue can be ep esen by c oss-linked imidazole-
phenol model and he pKa alues and edox po en ials o he Ty 280 can be calcula ed o
unde s and he ole o c oss-linked T y280 in he eac ion mechanism o cy och ome coxi-
dases.
To unde s and he edox coupled p o ona ion eac ions in cy och ome coxidases, he mid-
poin po en ial o he edox cen e s CuA, heme a, heme a3and CuBcan be calcula ed using
accu a e heo y which can be used o s udy edox beha io o he cy och ome coxidase.
In he p esen wo k, minimal models a e used o he CuBand heme a3cen e s. The in-
e ac ions be ween he CuBand heme a3cen e s in he p o ein a e ea ed as elec os a ic
in e ac ions. La ge models can be used o he binuclea cen e including he CuBcen e wi h
he wo his idine ligands and he c oss-linked His-Ty esidues and heme a3cen e wi h sub-
s i u ed po phy in. High le el quan um chemical me hods can be pe o med on his comple e
binuclea cen e , o s udy he in luence o he heme a3 edox s a es on CuBligands.
The c ys al s uc u es o bo h mammalian and bac e ial cy och ome coxidases do no con ain
all wa e molecules, ha could be in ol ed in he p o on ans e pa hways. Compu e algo-
i hms can be used o de e mine he likely posi ions and model in e nal wa e molecules in
he p o ein ca i ies o unde s and he hyd ogen bond ne wo ks in p o on ans e pa hways.
We plan o use Vol p og am (de eloped in ou g oup) o place wa e molecules and o ind he
hyd ogen bond ne wo ks in cy och ome coxidase.
The complex coupling o elec on and p o on ans e eac ions in cy och ome coxidases mus
be elucida ed in de ail. In he u u e, we plan o use he DMC (Dynamic Mon e Ca lo) p og am
o s udy he cha ge ans e and p o on ans e a es in cy och ome coxidase.
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