Antibiotics: Protein Synthesis, Nucleic Acid Synthesis and Metabolism.
in the Analysis of Protein and Nucleic Acid Secondary ...
2
Figure 1. The backbone path of a protein (PDB code 1ag6) is simplified by ignoring R-group appendages and by defining the orientation of the residue based on generic backbone features. In this case, each peptide bond is idealized as a planar shape whose orientation is based on the orientation of local backbone atoms. Novel Use of Novel Use of Quaternions Quaternions in the Analysis of in the Analysis of Protein and Nucleic Acid Secondary Structure Protein and Nucleic Acid Secondary Structure Daniel D. Kohler and Robert M. Hanson St. Olaf College, Northfield, Minnesota 237 th National Meeting of the American Chemical Society, Salt Lake City, Utah, March 24, 2009 Quaternionshave been known for some time to be expedient with problems involving rigid body rotations, but despite heavy usage of these highly visual objects, meaningful visualization within applications such as molecular modeling has been largely devoid. To address this disparity, this study applies quaternionsto parameterize protein backbone structure, focusing on the identification of helix parameters through visual and metric means. Quaternion maps are introduced as a tool for analysis of backbone character, anda measure of secondary structure backbone volatility, which we define as quaternion straightness, is introduced. We find that this quaternion analysis provides a simple way to identify structural motifs in secondary structure, as well as a metric means to define helix parameters. Straightness is further applied as a statistical tool for gross evaluation of peptide structural integrity. A direct relation between relative quaternion differences and Ramachandrantorsion angles is found. • A unit quaternion is a normalized 4- dimensional vector. A quaternion is frequently separated into a scalar part and a vector part. • Rotational operators in R 3 are homomorphic to the unit quaternions. • Quaternions can be parameterized by Rodriguez parameters: for a rotation of θ radians about normal vector n, Abstract Abstract Rigid Body Idealization Rigid Body Idealization Let a peptide's amino acid residues be indexed following the order of peptide bonding, from N- terminus to C-terminus. We wish to characterize the structural difference from residue i to residue i+1. As described by Quine, 1 one can think of a protein's secondary structure as a “discrete curve” in R 3 , where, as shown in Figure 1, backbone atoms of a given residue are defined on the curve as points in space. It is asserted that structural parameters can be simplified by ignoring the minor fluctuations in bond angles and lengths ofthe backbone atoms, so that all backbone moieties are equivalent. The coordinates of residue atoms can then be simplified analytically by reducing the expression of points to a common geometric feature. Not only does this reduce the number of independent structures in the modelingprocedure (individual points in the moiety can now be considered a single collection of dependently linked points), but it also invites a characterization of the backbone structure analogous to motion of a rigid body. Any positional change of a rigid body can be described through a single rotation and a single translation. 2 Quaternions Quaternions and Quaternion Maps and Quaternion Maps Quaternion Differences Quaternion Differences Helices Helices Quaternion Straightness Quaternion Straightness Structural Integrity Structural Integrity Conclusions Conclusions Acknowledgements Acknowledgements Works Cited Works Cited Peptide Frames Peptide Frames To describe the orientation of peptide residues we define special bases, or frames, for each residue. A frame is a triplet of orthonormal vectors associatedwith a point in space. Frames can be defined according to the generic bonds of a residue. Consequently, frames can fail to represent the residue accurately when the residue does not obey generic bond angles sufficiently. Frames should be chosen based on the more consistent bonding characteristics of the backbone. Figure 2. C-, P-, and N-based frames. The P and N frames are examples of plane-based frames, where the plane is colored green. The C frame is based on the tetrahedral geometry of the alpha carbon. A reference frame is given adjacent to each model. To view the multitude of quaternionsmeasured, we use quaternion maps, 3 which plot the vector component of quaternions(Figure 3). Figure 3. The protein crambin(PDB 1crn) with frames and a quaternion map associated with those frames. Coloring scheme is from N-terminus (blue) to C-terminus (red). Jmolis used for visualization. 4 Quaternionsprovide a novel way to express both differences and similarities between individual pairs of residues as well among entire strands. The mathematicalbasis of structural integrity and straightness correlates well with simple measures using Ramachandranangles and has the advantage of easily being extended to non-protein systems. While this study focuses primarily on sequential residues in peptides, another study in our group (poster BIOL 149) examined a broader use of quaternions to look at the flexibility of residues over time in a moleculardynamics calculation. Statistical approaches might be used to correlate straightness values and more qualitative visual depictions of secondary structure. Figure 7. Relative distributions of straightness values for various molecules. Models with a relatively higher proportion of straightness near 1.00 have relatively more order in their secondary structure. Models 1ppk and 1a6g have very different straightness distributions, which can be seen visually by inspection and measured quantitatively using quaternions. Quaternion differences can be measured two different ways depending upon the intended application. One option is to measure the difference between two quaternions with respect to the reference frame, RF (the x, y, and z coordinates of a pdbfile). Additionally, one can consider the quaternion that rotates one peptide frame, PF i , to the next, PF i+1 with respect to PF i (Figure 4). Figure 4. Rotations are a non-commutative algebra, and pathways to a given basis can yield entirely different results with different information. The frame pf i is a pseudo-frame, not physically present in the molecule. 1ppk 1a6g • Absolute difference (dq). The quaternion that rotates PF i to PF i+1 . This preserves information of global orientation. • Relative difference (dq*). The quaternion that rotates PF i to PF i+1 as if PF i were the reference frame. This relativizesthe domains of each transformation. Figure 5. Structural measurements of an ideal helix. The structure was generated by iterating a rigid body transformation of an idealized alanineresidue 16 times using Ramachandranangles (phi, psi, omega) = (-65,-40,0), and obtaining quaternion maps of the C frame. Figures (a),(b), and (c) have the same reference frame perspective, as shown by the global axes assignments. In (b), the observed point is between the viewer and the center when viewed down the axis of the helix. Quaternion map (d) displays (c) from a different perspective. Figure 6. The M2 membrane H+ channel of influenza (1mp6) and its absolute quaternion difference map. The cluster of points is between the viewer and the center when viewed down the axis of the helix. (a) (b) (c) (d) • Prof. Andrew J. Hanson, University of Indiana-Bloomington • St. Olaf College • HHMI [1] Quine, J. Journal of Molecular Structure 1999, 460, 53-66. [2] Kneller, G. R.; Calligari, P. ActaCryst. 2006, 62, 302. [3] Hanson, A. “Visualizing Quaternions;”Morgan Kaufmann, 2006. [4] Jmol, http://jmol.sourceforge.net/ Shown on the right are two depictions of structure 1a6g. The model on the left is colored by structure, as defined in the PDB file. On the right, coloring is based on “straightness”as defined by the four-dimensional dot product of consecutive quaternion differences, which are displayed below the structures. Several regions of high straightness where no PDB structure is defined indicate short motifs not found by the DSSP algorithm. Shown on the left is a correlation between straightness, as defined above, and a function of Ramachandranangles defined as: f(i) = phi i+1 + psi i –phi i –psi i-1 Cis-prolinesare anomalous, as expected. This Ramachandranfunction could be used as an alternative measure of straightness if access to quaternion information is limited.
Transcript of in the Analysis of Protein and Nucleic Acid Secondary ...
Fig
ure
1.
Th
e b
ack
bo
ne
pat
h o
f a
pro
tein
(P
DB
co
de
1ag
6) i
s si
mp
lifi
ed b
y i
gn
ori
ng
R-g
rou
p a
pp
end
ag
es a
nd
by
d
efin
ing
th
e o
rien
tati
on
of
the
resi
du
e b
ased
on
gen
eric
bac
kb
on
e fe
atu
res.
In
th
is c
ase,
eac
h p
epti
de
bo
nd
is
idea
lize
d a
s a
pla
nar
sh
ape
wh
ose
ori
enta
tio
n i
s b
ased
on
th
e o
rien
tati
on
of
loca
l b
ack
bo
ne
ato
ms.
No
vel
Use
of
No
vel
Use
of
Qu
ater
nio
ns
Qu
ater
nio
ns
in t
he
An
aly
sis
of
in t
he
An
aly
sis
of
Pro
tein
an
d N
ucl
eic
Aci
d S
eco
nd
ary
Str
uct
ure
Pro
tein
an
d N
ucl
eic
Aci
d S
eco
nd
ary
Str
uct
ure
Da
nie
l D
. K
oh
ler
an
d R
ob
ert
M. H
an
son
St.
Ola
f C
oll
ege,
No
rth
fiel
d,
Min
nes
ota
23
7th
Na
tio
na
l M
eeti
ng
of
the
Am
eric
an
Ch
emic
al
So
ciet
y, S
alt
La
ke
Cit
y, U
tah
, Ma
rch
24
, 200
9
Qu
ater
nio
ns
hav
e b
een
kn
ow
n f
or
som
e ti
me
to b
e ex
ped
ien
t w
ith
pro
ble
ms
inv
olv
ing
rig
id
bo
dy
ro
tati
on
s, b
ut
des
pit
e h
eav
y u
sag
e o
f th
ese
hig
hly
vis
ual
ob
ject
s, m
ean
ing
ful
vis
ual
izat
ion
w
ith
in a
pp
lica
tio
ns
such
as
mo
lecu
lar
mo
del
ing
has
bee
n l
arg
ely
dev
oid
. T
o a
dd
ress
th
is
dis
par
ity
, th
is s
tud
y a
pp
lies
qu
ater
nio
ns
to p
aram
eter
ize
pro
tein
bac
kb
on
e st
ruct
ure
, fo
cusi
ng
o
n t
he
iden
tifi
cati
on
of
hel
ix p
aram
eter
s th
rou
gh
vis
ual
an
d m
etri
c m
ean
s.
Qu
ater
nio
n m
aps
are
intr
od
uce
d a
s a
too
l fo
r an
aly
sis
of
bac
kb
on
e ch
arac
ter,
an
da
mea
sure
of
seco
nd
ary
st
ruct
ure
bac
kb
on
e v
ola
tili
ty, w
hic
h w
e d
efin
e as
qu
ater
nio
n s
trai
gh
tnes
s, i
s in
tro
du
ced
. W
e fi
nd
th
at t
his
qu
ater
nio
n a
nal
ysi
s p
rov
ides
a s
imp
le w
ay t
o i
den
tify
str
uct
ura
l m
oti
fs i
n
seco
nd
ary
str
uct
ure
, as
wel
l as
a m
etri
c m
ean
s to
def
ine
hel
ix p
aram
eter
s.
Str
aig
htn
ess
is
furt
her
ap
pli
ed a
s a
stat
isti
cal
too
l fo
r g
ross
ev
alu
atio
n o
f p
epti
de
stru
ctu
ral
inte
gri
ty. A
dir
ect
rela
tio
n b
etw
een
rel
ativ
e q
uat
ern
ion
dif
fere
nce
s an
d R
amac
han
dra
nto
rsio
n a
ng
les
is f
ou
nd
.
•A
un
it q
uat
ern
ion
is
a n
orm
aliz
ed 4
-d
imen
sio
nal
vec
tor.
A q
uat
ern
ion
is
freq
uen
tly
sep
arat
ed i
nto
a s
cala
r p
art
and
a
vec
tor
par
t.
•R
ota
tio
nal
op
erat
ors
in
R3
are
ho
mo
mo
rph
icto
th
e u
nit
qu
ater
nio
ns.
•
Qu
ater
nio
ns
can
be
par
amet
eriz
ed b
y
Ro
dri
gu
ez p
aram
eter
s: f
or
a ro
tati
on
of
θra
dia
ns
abo
ut
no
rmal
vec
tor
n,
Ab
stra
ctA
bst
ract
Rig
id B
od
y I
deali
zati
on
Rig
id B
od
y I
deali
zati
on
Let
a p
epti
de'
s am
ino
aci
d r
esid
ues
be
ind
exed
fo
llo
win
g t
he
ord
er o
f p
epti
de
bo
nd
ing
, fro
m N
-te
rmin
us
to C
-ter
min
us.
W
e w
ish
to
ch
arac
teri
ze t
he
stru
ctu
ral
dif
fere
nce
fro
m r
esid
ue
ito
re
sid
ue
i+1.
As
des
crib
ed b
y Q
uin
e,1
on
e ca
n t
hin
k o
f a
pro
tein
's s
eco
nd
ary
str
uct
ure
as
a “d
iscr
ete
curv
e”in
R3, w
her
e, a
s sh
ow
n i
n F
igu
re 1
, b
ack
bo
ne
ato
ms
of
a g
iven
res
idu
e ar
e d
efin
ed o
n t
he
curv
e as
po
ints
in
sp
ace.
It
is a
sser
ted
th
at s
tru
ctu
ral
par
amet
ers
can
be
sim
pli
fied
b
y i
gn
ori
ng
th
e m
ino
r fl
uct
uat
ion
s in
bo
nd
an
gle
s an
d l
eng
ths
of
the
bac
kb
on
e at
om
s, s
o t
hat
all
b
ack
bo
ne
mo
ieti
es a
re e
qu
ival
ent.
Th
e co
ord
inat
es o
f re
sid
ue
ato
ms
can
th
en b
e si
mp
lifi
ed
anal
yti
call
y b
y r
edu
cin
g t
he
exp
ress
ion
of
po
ints
to
a c
om
mo
n g
eom
etri
c fe
atu
re. N
ot
on
ly d
oes
th
is r
edu
ce t
he
nu
mb
er o
f in
dep
end
ent
stru
ctu
res
in t
he
mo
del
ing
pro
ced
ure
(in
div
idu
al p
oin
ts
in t
he
mo
iety
can
no
w b
e co
nsi
der
ed a
sin
gle
co
llec
tio
n o
f d
epen
den
tly
lin
ked
po
ints
), b
ut
it a
lso
in
vit
es a
ch
arac
teri
zati
on
of
the
bac
kb
on
e st
ruct
ure
an
alo
go
us
to m
oti
on
of
a ri
gid
bo
dy
. A
ny
p
osi
tio
nal
ch
ang
e o
f a
rig
id b
od
y c
an b
e d
escr
ibed
th
rou
gh
a s
ing
le r
ota
tio
n a
nd
a s
ing
le
tran
slat
ion
.2
Qu
ate
rnio
ns
Qu
ate
rnio
ns
an
d Q
uate
rnio
n M
ap
san
d Q
uate
rnio
n M
ap
s
Qu
ate
rnio
n D
iffe
ren
ces
Qu
ate
rnio
n D
iffe
ren
ces
Heli
ces
Heli
ces
Qu
ate
rnio
n S
traig
htn
ess
Qu
ate
rnio
n S
traig
htn
ess
Str
uct
ura
l In
teg
rity
Str
uct
ura
l In
teg
rity
Co
ncl
usi
on
sC
on
clu
sio
ns
Ack
no
wle
dg
em
en
tsA
ckn
ow
led
ge
me
nts
Wo
rks
Cit
ed
Wo
rks
Cit
ed
Pep
tid
e F
ram
es
Pep
tid
e F
ram
es
To
des
crib
e th
e o
rien
tati
on
of
pep
tid
e re
sid
ues
we
def
ine
spec
ial
bas
es, o
r fr
ames
, fo
r ea
ch
resi
du
e. A
fra
me
is a
tri
ple
t o
f o
rth
on
orm
al v
ecto
rs a
sso
ciat
edw
ith
a p
oin
t in
sp
ace.
Fra
mes
can
b
e d
efin
ed a
cco
rdin
g t
o t
he
gen
eric
bo
nd
s o
f a
resi
du
e. C
on
seq
uen
tly
, fra
mes
can
fai
l to
re
pre
sen
t th
e re
sid
ue
accu
rate
ly w
hen
th
e re
sid
ue
do
es n
ot
ob
ey g
ener
ic b
on
d a
ng
les
suff
icie
ntl
y.
Fra
mes
sh
ou
ld b
e ch
ose
n b
ased
on
th
e m
ore
co
nsi
sten
t b
on
din
g c
har
acte
rist
ics
of
the
bac
kb
on
e.
Fig
ure
2.
C-,
P-,
an
d N
-bas
ed f
ram
es.
Th
e P
an
d N
fram
es a
re e
xam
ple
s o
f p
lan
e-b
ased
fra
mes
, w
her
e th
e p
lan
e is
co
lore
d g
reen
. T
he
C f
ram
e is
bas
ed o
n t
he
tetr
ah
edra
l g
eom
etry
of
the
alp
ha
carb
on
. A
ref
eren
ce f
ram
e is
giv
en
adja
cen
t to
eac
h m
od
el.
To
vie
w t
he
mu
ltit
ud
e o
f q
uat
ern
ion
sm
easu
red
, w
e u
se q
uat
ern
ion
map
s,3
wh
ich
plo
t th
e v
ecto
r co
mp
on
ent
of
qu
ater
nio
ns
(Fig
ure
3).
Fig
ure
3.
Th
e p
rote
in c
ram
bin
(PD
B 1
crn
) w
ith
fr
ames
an
d a
qu
ater
nio
n m
ap a
sso
ciat
ed w
ith
th
ose
fr
ames
. C
olo
rin
g s
chem
e is
fro
m N
-ter
min
us
(blu
e)
to C
-ter
min
us
(red
).
Jmo
lis
use
d f
or
vis
ual
izat
ion
.4
Qu
ater
nio
ns
pro
vid
e a
no
vel
way
to
exp
ress
bo
th d
iffe
ren
ces
and
sim
ilar
itie
s b
etw
een
in
div
idu
al
pai
rs o
f re
sid
ues
as
wel
l am
on
g e
nti
re s
tran
ds.
Th
e m
ath
emat
ical
bas
is o
f st
ruct
ura
l in
teg
rity
an
d
stra
igh
tnes
s co
rrel
ates
wel
l w
ith
sim
ple
mea
sure
s u
sin
g R
amac
han
dra
nan
gle
s an
d h
as t
he
adv
anta
ge
of
easi
ly b
ein
g e
xte
nd
ed t
o n
on
-pro
tein
sy
stem
s. W
hil
e th
is s
tud
y f
ocu
ses
pri
mar
ily
on
se
qu
enti
al r
esid
ues
in
pep
tid
es, a
no
ther
stu
dy
in
ou
r g
rou
p (
po
ster
BIO
L 1
49)
exam
ined
a b
road
er
use
of
qu
ater
nio
ns
to l
oo
k a
t th
e fl
exib
ilit
y o
f re
sid
ues
ov
er t
ime
in a
mo
lecu
lar
dy
nam
ics
calc
ula
tio
n.
Sta
tist
ical
ap
pro
ach
es m
igh
t b
e u
sed
to
co
rrel
ate
stra
igh
tnes
s v
alu
es a
nd
mo
re
qu
alit
ativ
e v
isu
al d
epic
tio
ns
of
seco
nd
ary
str
uct
ure
.
Fig
ure
7.
Rel
ativ
e d
istr
ibu
tio
ns
of
stra
igh
tnes
s v
alu
es f
or
va
rio
us
mo
lecu
les.
Mo
del
s w
ith
a r
elat
ivel
y h
igh
er
pro
po
rtio
n o
f st
raig
htn
ess
nea
r 1.
00 h
av
e re
lati
vel
y m
ore
ord
er i
n t
hei
r se
con
dar
y s
tru
ctu
re.
Mo
del
s 1p
pk
an
d 1
a6g
h
av
e v
ery
dif
fere
nt
stra
igh
tnes
s d
istr
ibu
tio
ns,
wh
ich
ca
n b
e se
en v
isu
all
y b
y i
nsp
ecti
on
an
d m
easu
red
qu
an
tita
tiv
ely
u
sin
g q
uat
ern
ion
s.
Qu
ater
nio
n d
iffe
ren
ces
can
be
mea
sure
d t
wo
d
iffe
ren
t w
ays
dep
end
ing
up
on
th
e in
ten
ded
ap
pli
cati
on
. O
ne
op
tio
n i
s to
mea
sure
th
e d
iffe
ren
ce b
etw
een
tw
o q
uat
ern
ion
sw
ith
res
pec
t to
th
e re
fere
nce
fra
me,
RF
(th
e x,
y,
and
zco
ord
inat
es o
f a
pd
bfi
le).
Ad
dit
ion
ally
, on
e ca
n
con
sid
er t
he
qu
ater
nio
n t
hat
ro
tate
s o
ne
pep
tid
e fr
ame,
PF
i, to
th
e n
ext,
PF
i+1
wit
h r
esp
ect
to P
Fi
(Fig
ure
4).
Fig
ure
4.
Ro
tati
on
s ar
e a
no
n-c
om
mu
tati
ve
alg
ebra
, a
nd
p
ath
way
s to
a g
iven
bas
is c
an y
ield
en
tire
ly d
iffe
ren
t re
sult
s w
ith
dif
fere
nt
info
rmat
ion
. T
he
fra
me
pfiis
a
pse
ud
o-f
ram
e, n
ot
ph
ysi
call
y p
rese
nt
in t
he
mo
lecu
le.
1ppk
1a6g
•A
bso
lute
dif
fere
nce
(d
q).
T
he
qu
ater
nio
n t
hat
ro
tate
s P
Fito
PF
i+1.
Th
is p
rese
rves
in
form
atio
n
of
glo
bal
ori
enta
tio
n.
•R
ela
tiv
e d
iffe
ren
ce (
dq
*).
Th
e q
uat
ern
ion
th
at
rota
tes
PF
ito
PF
i+1
as i
fP
Fiw
ere
the
refe
ren
ce
fram
e.
Th
is r
elat
iviz
esth
e d
om
ain
s o
f ea
ch
tran
sfo
rmat
ion
.
Fig
ure
5.
Str
uct
ura
l m
easu
rem
ents
of
an
id
eal
hel
ix.
Th
e st
ruct
ure
was
gen
erat
ed b
y i
tera
tin
g a
rig
id b
od
y
tran
sfo
rmat
ion
of
an
id
eali
zed
ala
nin
ere
sid
ue
16 t
imes
usi
ng
Ram
ach
an
dra
na
ng
les
(ph
i, p
si, o
meg
a) =
(-6
5,-4
0,0)
, an
d
ob
tain
ing
qu
ater
nio
n m
aps
of
the
Cfr
ame.
F
igu
res
(a),
(b),
an
d (
c) h
ave
the
sam
e re
fere
nce
fra
me
per
spec
tiv
e, a
s sh
ow
n
by
th
e g
lob
al a
xes
ass
ign
men
ts.
In
(b
), t
he
ob
serv
ed p
oin
t is
bet
wee
n t
he
vie
wer
an
d t
he
cen
ter
wh
en v
iew
ed d
ow
n t
he
axis
of
the
hel
ix.
Qu
ater
nio
n m
ap (
d)
dis
pla
ys
(c)
fro
m a
dif
fere
nt
per
spec
tiv
e.
Fig
ure
6.
Th
e M
2 m
emb
ran
e H
+ ch
an
nel
of
infl
uen
za
(1m
p6)
an
d i
ts a
bso
lute
qu
ater
nio
n d
iffe
ren
ce m
ap.
T
he
clu
ster
of
po
ints
is
bet
wee
n t
he
vie
wer
an
d t
he
cen
ter
wh
en v
iew
ed d
ow
n t
he
axis
of
the
hel
ix.
(a)
(b)
(c)
(d)
•P
rof.
An
dre
w J
. H
anso
n,
Un
iver
sity
of
Ind
ian
a-B
loo
min
gto
n
•S
t. O
laf
Co
lleg
e
•H
HM
I
[1]
Qu
ine,
J.
Jou
rnal
of
Mol
ecu
lar
Str
uct
ure
1999,
460,
53-
66.
[2]
Kn
elle
r, G
. R
.; C
alli
gar
i, P
. A
cta
Cry
st. 2
006,
62,
302.
[3]
Han
son
, A
. “V
isu
aliz
ing
Qu
ater
nio
ns;
”M
org
an
Ka
ufm
an
n,
2006
.[4
] Jm
ol,
htt
p:/
/jm
ol.
sou
rcef
org
e.n
et/
Sh
ow
n o
n t
he
rig
ht
are
two
d
epic
tio
ns
of
stru
ctu
re 1
a6g
. T
he
mo
del
on
th
e le
ft i
s co
lore
d b
y
stru
ctu
re, a
s d
efin
ed i
n t
he
PD
B f
ile.
O
n t
he
rig
ht,
co
lori
ng
is
bas
ed o
n
“str
aig
htn
ess”
as d
efin
ed b
y t
he
fou
r-d
imen
sio
nal
do
t p
rod
uct
of
con
secu
tiv
e q
uat
ern
ion
dif
fere
nce
s,
wh
ich
are
dis
pla
yed
bel
ow
th
e st
ruct
ure
s. S
ever
al r
egio
ns
of
hig
h
stra
igh
tnes
s w
her
e n
o P
DB
str
uct
ure
is
def
ined
in
dic
ate
sho
rt m
oti
fs n
ot
fou
nd
by
th
e D
SS
P a
lgo
rith
m.
Sh
ow
n o
n t
he
left
is
a co
rrel
atio
n b
etw
een
st
raig
htn
ess,
as
def
ined
ab
ov
e, a
nd
a f
un
ctio
n
of
Ram
ach
and
ran
ang
les
def
ined
as:
f(i)
= p
hi i+
1+
psi
i–
ph
i i–
psi
i-1
Cis
-pro
lin
esar
e an
om
alo
us,
as
exp
ecte
d. T
his
R
amac
han
dra
nfu
nct
ion
co
uld
be
use
d a
s an
al
tern
ativ
e m
easu
re o
f st
raig
htn
ess
if a
cces
s to
q
uat
ern
ion
in
form
atio
n i
s li
mit
ed.
Usi
ng
Qu
ate
rnio
ns
and
Mo
lecu
lar
Dy
na
mic
s S
imu
lati
on
s in
Dru
g D
esU
sin
g Q
uat
ern
ion
s an
d M
ole
cula
r D
yn
am
ics
Sim
ula
tio
ns
in D
rug
Des
ign
ign
Sea
n B
. Jo
hn
sto
n a
nd
Ro
ber
t M
. Ha
nso
nS
t. O
laf
Co
lleg
e, N
ort
hfi
eld
, M
inn
eso
ta23
7thN
ati
on
al
Mee
tin
g o
f th
e A
mer
ica
n C
hem
ica
l S
oci
ety
, Sa
lt L
ak
e C
ity
, Uta
h, M
arc
h 2
4, 2
009
•Q
uat
ern
ion
s ar
e an
ex
ten
sio
n o
f th
e co
mp
lex
nu
mb
ers
(wh
ere
i2= j
2= k
2= i
jk=
-1).
2
•U
nit
qu
ater
nio
ns
(qu
ater
nio
ns
of
len
gth
on
e) h
ave
bee
n i
nv
esti
gat
ed f
or
thei
r u
niq
ue
app
lica
tio
n t
o t
he
mea
sure
men
t an
d v
isu
aliz
atio
n o
f o
rien
tati
on
s.3
•A
ny
fra
me
of
refe
ren
ce (
incl
ud
ing
th
ose
rep
rese
nti
ng
mo
lecu
lar
ori
enta
tio
ns)
can
be
rep
rese
nte
d b
y a
un
it q
uat
ern
ion
.
•U
nit
qu
ater
nio
n d
iffe
ren
ces
(or
qu
ater
nio
n d
eriv
ativ
es)
rep
rese
nt
rota
tio
ns,
an
d c
an b
e ex
pre
ssed
in
ter
ms
of
an a
bso
lute
fra
me
of
refe
ren
ce (
q2q
1-1
) o
r a
rela
tiv
e fr
ame
of
the
rota
tin
g o
bje
ct (
q1
-1q
2)
.
Mo
lecu
lar
dy
nam
ics
sim
ula
tio
ns
are
rou
tin
ely
car
ried
ou
t w
ith
bio
log
ical
mac
rom
ole
cule
s to
m
od
el t
he
way
th
ey i
nte
ract
wit
h s
olv
ent.
1F
rom
su
ch s
imu
lati
on
s, a
pp
rox
imat
e b
ind
ing
fre
e en
erg
y o
f a
pro
tein
to
a l
igan
dca
n b
e ca
lcu
late
d.
Mu
ch o
f m
edic
inal
ch
emis
try
is
con
cern
ed
wit
h i
nd
ivid
ual
in
tera
ctio
ns
bet
wee
n s
pec
ific
am
ino
aci
ds
and
a d
rug
, h
ow
ever
. T
hro
ug
h a
m
ole
cula
r d
yn
amic
s tr
ajec
tory
, an
am
ino
aci
d p
arti
cip
atin
g i
n b
ind
ing
a l
igan
dm
ay h
ave
dec
reas
ed f
lex
ibil
ity
. U
nit
qu
ater
nio
ns
off
er a
un
iqu
e w
ay t
o q
uan
tify
th
is f
lexi
bil
ity
. W
e es
tab
lish
qu
ater
nio
ns
for
resi
du
es i
n H
IV-1
pro
teas
e, a
nd
ex
amin
e th
e sa
me
resi
du
es w
hen
th
e en
zym
e is
bo
un
d t
o t
he
inh
ibit
or
lop
inav
ir(L
PV
).
Qu
ater
nio
ns
are
des
ign
ated
by
a c
oo
rdin
ate
fram
e u
sin
g t
hre
e at
om
s.
•T
he
firs
t at
om
is
the
cen
ter
(in
th
is e
xam
ple
, th
e al
ph
a-ca
rbo
n).
•T
he
seco
nd
ato
m (
carb
oxy
l ca
rbo
n)
giv
es t
he
dir
ecti
on
of
the
x-a
xis
(re
d
arro
w).
•T
he
y-a
xis
(g
reen
arr
ow
) is
per
pen
dic
ula
r to
th
e x
-ax
is a
nd
in
th
e p
lan
e fo
rmed
by
th
e al
ph
a ca
rbo
n, c
arb
on
yl
carb
on
, an
d b
ack
bo
ne
nit
rog
en.
•T
he
z-ax
is i
s p
erp
end
icu
lar
to b
oth
x a
nd
y a
xes
, an
d f
ou
nd
usi
ng
th
e ri
gh
t-h
and
-ru
le.
Ab
stra
ctA
bst
ract
Qu
ater
nio
ns
in 3
DQ
uat
ern
ion
s in
3D
Qu
ater
nio
n M
ap
sQ
uat
ern
ion
Ma
ps
•U
nit
qu
ater
nio
ns
can
be
dis
pla
yed
in
a g
rap
hic
al m
ann
er.3
(Fig
. 1)
•Q
uat
ern
ion
der
ivat
ives
(re
pre
sen
tin
g t
he
rota
tio
n r
equ
ired
to
pro
ceed
fro
m o
ne
ori
enta
tio
n t
o
ano
ther
) ca
n b
e d
isp
lay
ed a
s w
ell.
•B
oth
ver
sio
ns
giv
e in
form
atio
n a
bo
ut
the
stru
ctu
re.
•H
IV-1
pro
teas
e is
th
e su
bje
ct o
f m
uch
res
earc
h a
nd
sim
ula
tio
n.4
-6
•V
ario
us
inh
ibit
ors
hav
e b
een
stu
die
d a
nd
sim
ula
ted
.
•T
he
enzy
me
affo
rds
an o
pp
ort
un
ity
to
see
if
add
itio
n o
f lo
pin
avir
affe
cts
the
flex
ibil
ity
of
amin
o a
cid
s.
•C
on
tin
uin
g r
esea
rch
co
uld
co
mp
are
inh
ibit
ors
fo
r w
hic
h b
ind
ing
fre
e en
erg
y d
ata,
bo
th
com
pu
tati
on
al a
nd
ex
per
imen
tal,
is
kn
ow
n.
HIV
HIV
-- 1 P
rote
ase
1 P
rote
ase
Qu
ater
nio
n d
esig
nat
ion
sQ
uat
ern
ion
des
ign
atio
ns
Qu
ater
nio
n d
esig
nat
ion
s ar
e se
nsi
tiv
e to
glo
bal
pro
tein
ro
tati
on
or
"tu
mb
lin
g"
du
rin
g t
he
mo
lecu
lar
dy
nam
ics
calc
ula
tio
n.
Tu
mb
lin
g i
s o
bse
rvab
le i
n t
he
qu
ater
nio
n m
ap, w
her
e a
syst
emat
ic s
pre
adin
g
of
qu
ater
nio
ns
wit
hin
th
e m
ap i
nd
icat
es t
um
bli
ng
of
the
pro
tein
. T
his
is
seen
in
th
e fi
rst
two
q
uat
ern
ion
map
s in
Fig
ure
2.
Th
e re
lati
ve
qu
ater
nio
n d
iffe
ren
ce o
f tw
o q
uat
ern
ion
s (F
ig. 2
c) r
emo
ves
th
is a
rtif
act
of
the
calc
ula
tio
n.
Sim
ilar
ly, t
he
rela
tiv
e d
iffe
ren
ce b
etw
een
tw
o r
efer
ence
fra
mes
lo
cate
d
on
a s
ing
le r
esid
ue
can
rem
ov
e th
e ef
fect
of
tum
bli
ng
.
Th
e T
um
bli
ng
Pro
ble
mT
he
Tu
mb
lin
g P
rob
lem
Sam
ple
qu
ater
nio
n m
aps
Sam
ple
qu
ater
nio
n m
aps
In p
ure
wat
er,
thre
e si
mu
lati
on
s u
sin
g t
he
AM
BE
R m
ole
cula
r d
yn
amic
s su
ite7
wer
e ca
rrie
d o
ut,
o
ne
wit
h t
he
A c
on
form
atio
n o
f lo
pin
avir
, on
e w
ith
th
e B
co
nfo
rmat
ion
, an
d o
ne
un
bo
un
d:
•ea
ch s
olu
te w
as s
olv
ated
exp
lici
tly
usi
ng
TIP
3P w
ater
s
•th
e i
nit
ial
stru
ctu
re’s
en
erg
y w
as m
inim
ized
, th
e sy
stem
was
hea
ted
fro
m 0
K t
o 3
00K
fo
r 50
p
s, t
he
syst
em e
qu
ilib
rate
d a
t co
nst
ant
pre
ssu
re f
or
50 p
s, a
nd
a 5
00 p
seq
uil
ibra
tio
n w
as r
un
to
ch
eck
fo
r tr
end
s in
tem
per
atu
re, d
ensi
ty,
and
to
tal
ener
gy
to
ensu
re s
yst
em s
tab
ilit
y a
nd
b
ack
bo
ne
roo
t-m
ean
-sq
uar
e d
evia
tio
n t
o c
hec
k s
tru
ctu
re c
han
ges
Th
e sy
stem
s w
ere
then
sim
ula
ted
fo
r ~1
.5 n
s, o
utp
utt
ing
str
uct
ure
dat
a ev
ery
10
ps.
Usi
ng
Jm
ol,
8q
uat
ern
ion
s an
d q
uat
ern
ion
dif
fere
nce
s w
ere
calc
ula
ted
fo
r ea
ch s
nap
sho
t in
th
e tr
ajec
tory
. A
spar
tate
resi
du
es 2
5, 2
9, a
nd
60
on
bo
th c
hai
ns
wer
e st
ud
ied
pri
mar
ily
.
•“a
sp”
fram
es a
re d
esig
nat
ed i
n t
he
man
ner
des
crib
ed i
n p
ost
er B
IOL
110
(D
an K
oh
ler)
.
•“a
spO
”fr
ames
are
bas
ed o
n t
he
pla
ne
of
the
carb
ox
yl
gro
up
.
•“b
ase”
qu
ater
nio
ns
serv
e as
a r
efer
ence
po
int
for
com
par
ing
bo
un
d a
nd
un
bo
un
d p
rote
ase.
•A
sp25
is
the
cata
lyti
c re
sid
ue,
asp
29 i
s an
im
po
rtan
t b
ind
ing
res
idu
e, a
nd
asp
60 i
s o
n t
he
ou
tsid
e o
f th
e en
zym
e, s
erv
ing
as
a co
ntr
ol.
Met
ho
d d
eta
ils
Met
ho
d d
eta
ils
Tes
tin
g b
ou
nd
/un
bo
un
d h
yp
oth
esi
sT
esti
ng
bo
un
d/u
nb
ou
nd
hy
po
the
sis
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
124-base
124O-base
25-base
25O-base
60-base
60O-base
29-base
29O-base
128-base
128O-base
quaternion difference label
unbound
A conform
er
B conform
er
Exa
min
ing
E
xam
inin
g l
igan
dli
gan
db
ind
ing
bin
din
g
Th
e m
easu
rem
ent
and
vis
ual
izat
ion
of
qu
ater
nio
n d
iffe
ren
ces
len
ds
insi
gh
t in
to t
he
ran
ge
of
mo
tio
n a
nd
fle
xib
ilit
y o
f p
rote
in r
esid
ues
an
d b
ou
nd
lig
and
sd
uri
ng
a m
ole
cula
r d
yn
amic
s ca
lcu
lati
on
. Jm
ol
11.7
in
clu
des
a v
arie
ty o
f q
uat
ern
ion
-bas
ed f
un
ctio
nal
ity
and
all
ow
s a
bro
ad
ran
ge
of
inv
esti
gat
ion
s u
sin
g q
uat
ern
ion
s in
way
s th
at h
ave
no
t b
een
av
aila
ble
pre
vio
usl
y.
Co
ncl
usi
on
sC
on
clu
sio
ns
I w
ou
ld l
ike
to a
ckn
ow
led
ge
the
Un
iver
sity
of
Min
nes
ota
Su
per
com
pu
tin
g I
nst
itu
te f
or
the
use
of
the
AM
BE
R s
oft
wa
re s
uit
e o
n C
alh
ou
n.
Th
an
ks
to
Dr.
Jef
f S
chw
inef
us
for
intr
od
uci
ng
me
to s
uch
an
in
tere
stin
g a
rea
. Th
is w
ork
wa
s su
pp
ort
ed b
y a
St.
O
laf
Co
lleg
e Magnus the Good c
oll
ab
ora
tiv
e re
sea
rch
g
ran
t.
Ack
no
wle
dg
em
en
tsA
ckn
ow
led
ge
me
nts
1.
Pat
rick
, G
. L
. A
n i
ntr
odu
ctio
n t
o m
edic
ina
l ch
emis
try
, 3
rd e
d.;
Ox
ford
Un
iver
sity
Pre
ss:
Ox
ford
; N
ew Y
ork
, 2
005
.2
. N
ich
ols
on
, W
. K
. In
trod
uct
ion
to
abs
trac
t a
lgeb
ra,
3rd
ed
.; W
iley
-In
ters
cien
ce:
Ho
bo
ke
n,
N.J
., 2
007
.3
. H
anso
n,
A.
Vis
ua
lizi
ng
qu
ate
rnio
ns;
Mo
rgan
Ka
ufm
ann
: S
an
Fra
nci
sco
, C
A,
20
06
.4
. K
oy
ano
, K
.; N
akan
o,
T. J
ourn
al
of S
yn
chro
tron
Rad
iati
on20
08, 1
5, 2
39-2
42.
5.
Wit
tay
anar
aku
l, K
.; H
ann
on
gb
ua,
S.;
Fei
g,
M.
Jou
rna
l of
Com
pu
tati
ona
l Ch
emis
try
2008
, 29
, 673
-68
5.6
. P
uro
hit
, R
.; R
ajas
ekar
an,
R.;
Su
dan
dir
ad
oss
, C
.; D
oss
, C
. G
. P
.; R
aman
ath
an,
K.;
Rao
, S
. In
tern
ati
ona
l Jo
urn
al
of B
iolo
gic
al M
acr
omol
ecu
les
20
08,
42, 3
86-3
91.
7.
Cas
e, D
. A
.; D
ard
en,
T.
A.;
T.E
. C
he
ath
am,
I.;
Sim
mer
lin
g,
C.
L.;
Wan
g,
J.;
Du
ke,
R.
E.;
Lu
o,
R.;
Mer
z,
K.
M.;
Pea
rlm
an,
D.
A.;
Cro
wle
y,
M.;
Wal
ke
r, R
. C
.; Z
han
g,
W.;
Wa
ng
, B
.; H
ayik
, S
.; R
oit
ber
g,
A.;
Sea
bra
, G
.; W
on
g,
K.
F.;
Pae
san
i, F
.; W
u,
X.;
Bro
zel
l, S
.; T
sui,
V.;
G
oh
lke,
H.;
Yan
g,
L.;
Tan
, C
.; M
on
gan
, J.
; H
orn
ak
, V
.; C
ui,
G.;
Ber
oz
a, P
.; M
ath
ews,
D.
H.;
Sch
afm
eist
er,
C.;
Ro
ss,
W.
S.;
P.A
. K
oll
ma
n(2
00
6)
AM
BE
R 9
, U
niv
ersi
ty o
f C
ali
forn
ia,
San
Fra
nci
sco
.8
. J
mo
l. h
ttp
://J
mo
l.so
urc
efo
rge.
net
Wo
rks
Cit
edW
ork
s C
ited
Fig
ure
1.
a. A
n i
dea
lize
d h
elix
. b
. R
ota
tin
g t
o l
oo
k d
ow
n t
he
hel
ix.
c.
Sw
itch
ing
to
a q
uat
ern
ion
map
dir
ectl
y f
rom
b.
(no
tice
th
e ax
es a
re t
he
sam
e),
we
see
that
th
e q
uat
ern
ion
s fo
rma
per
fect
cir
cle.
d.
In t
he
qu
ater
nio
n d
eriv
ativ
e m
ap,
all
po
ints
map
to
th
e ti
p o
f th
e ar
row
bec
au
se e
ach
dif
fere
nce
fro
m o
ne
fram
e to
th
e n
ext
is t
he
sam
e.
Vis
ual
izat
ion
s w
ere
ma
de
usi
ng
th
e Jm
ol
sig
ned
ap
ple
t.8
a.b
.c.
d.
Fig
ure
2.
a. Q
uat
ern
ion
map
fo
r th
e p
lan
e o
f th
e ca
rbo
ny
l o
f as
p12
4. b
. T
he
qu
ater
nio
n m
ap f
or
the
hy
dro
xy
gro
up
of
lop
inav
ir.
c. T
he
rela
tiv
e q
uat
ern
ion
dif
fere
nce
bet
wee
n t
he
qu
ater
nio
ns.
d.
Th
e fr
ames
rep
rese
nti
ng
th
e q
uat
ern
ion
s st
ud
ied
in
th
is s
ecti
on
. V
isu
aliz
atio
ns
wer
e m
ade
usi
ng
th
e Jm
ol
sig
ned
ap
ple
t.8
a.b
.c.
d.
Fig
ure
3.
a. Q
uat
ern
ion
dif
fere
nce
map
fo
r as
p12
8O a
nd
asp
128
bas
e. b
Q
uat
ern
ion
dif
fere
nce
map
fo
r as
p29
O a
nd
an
un
rela
ted
qu
ater
nio
n o
n l
op
inav
ir.
c. Q
uat
ern
ion
dif
fere
nce
map
fo
r as
p60
on
th
e o
uts
ide
of
the
pro
tein
an
d t
he
hy
dro
xy
gro
up
of
lop
inav
ir.
Vis
ual
izat
ion
s w
ere
ma
de
usi
ng
th
e Jm
ol
sig
ned
ap
ple
t.8
Fig
ure
4.
Rel
ativ
e q
uat
ern
ion
dif
fere
nce
s w
ith
in s
pec
ific
res
idu
es g
ive
an
ind
icat
ion
of
the
ran
ge
of
flex
ibil
ity
of
the
resi
du
e si
de
chai
n d
uri
ng
th
e si
mu
lati
on
. N
ote
th
at b
ind
ing
of
lop
ina
vir
is n
ot
nec
essa
rily
a p
red
icto
r o
f re
sid
ue
flex
ibil
ity
, ev
en i
n t
he
acti
ve
site
. T
he
larg
e d
evia
tio
ns
in b
oth
ou
r co
ntr
ol
(asp
60)
and
th
eb
ind
ing
sit
e re
sid
ue
asp
29
are
evid
ence
th
at t
hes
e tw
o c
arb
ox
yla
teg
rou
ps
flip
ped
ori
enta
tio
n d
uri
ng
th
e si
mu
lati
on
.
Fig
ure
5.
Co
mp
aris
on
s o
f q
uat
ern
ion
dif
fere
nce
s fo
r a
var
iety
of
inte
ract
ing
an
d n
on
-in
tera
ctin
g l
op
inav
ir/p
rote
in
inte
ract
ion
s. O
ver
all,
it
seem
s th
at t
he
B c
on
form
atio
n i
s sl
igh
tly
mo
re “
sett
led
.”A
s in
Fig
ure
4,
thes
e re
sult
s in
dic
ate
that
in
th
e A
co
nfo
rmat
ion
sim
ula
tio
n a
sp60
an
d a
sp29
fli
pp
ed.
(No
te t
hat
on
ly q
uat
ern
ion
s b
ased
on
ca
rbo
ny
l o
xy
gen
ssh
ow
th
is e
ffec
t.)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
LPV1-asp25
LPV1-asp25O
LPV1-asp124
LPV1-asp124O
LPV1-asp60
LPV1-asp60O
LPV2-asp25
LPV2-asp25O
LPV2-asp124
LPV2-asp124O
LPV2-asp60
LPV2-asp60O
LPV3-asp29
LPV3-asp29O
LPV4-asp128
LPV4-asp128O
average deviation
A conform
ation
B conform
ation
