Ka
lma
n F
ilte
r E
sti
ma
tes
of
the
Co
nto
ur
Le
ng
th o
f a
n U
nfo
ldin
g P
rote
in
in S
ing
le-M
ole
cu
le F
orc
e S
pe
ctr
os
co
py
Ex
pe
rim
en
tsV
ice
nte
I.
Fe
rna
nd
ez
1,
Pa
lla
v K
os
uri
2,
Vic
en
te P
aro
t4a
nd
Ju
lio
M.
Fe
rná
nd
ez
3
1D
epart
ment
of M
echan
ica
l E
ng
ineeri
ng,
Massach
usett
s I
nstitu
te o
fT
echnolo
gy,
Bosto
n 0
213
92D
ept
of B
iochem
istr
y a
nd 3
Dep
art
ment of B
iolo
gic
al S
cie
nces,
Co
lum
bia
Un
ivers
ity,
New
York
1002
7,
4P
ontificia
Un
ivers
idad C
ató
lica
de C
hile
, S
antiag
o
Ab
str
act
Fo
rce s
pectr
osco
py m
easu
rem
en
ts o
f sin
gle
mo
lecu
les u
sin
g A
FM
have e
nab
led
th
e s
tud
y o
f a
ran
ge o
f m
ole
cu
lar
pro
pert
ies n
ot
accessib
le w
ith
b
ulk
m
eth
od
s.
Th
ese p
rop
ert
ies o
f in
tere
st
mu
st
typ
icall
y b
e i
nfe
rred
by m
an
uall
y f
itti
ng
mo
dels
to
sele
cte
d p
ort
ion
s o
f m
easu
red
data
. A
s
man
ual
inte
rven
tio
n i
n t
he f
itti
ng
pro
cess e
asil
y i
ntr
od
uces a
bia
s i
n t
he a
naly
sis
, th
ere
is a
need
fo
r m
ore
so
ph
isti
cate
d
an
aly
sis
m
eth
od
s
cap
ab
le
of
inte
rpre
tin
g
data
in
an
u
nb
iased
an
d
rep
eata
ble
“h
an
ds-o
ff”
man
ner.
Here
we a
pp
ly a
n exte
nd
ed
K
alm
an
fi
lter
to th
e e
sti
mati
on
of
pro
tein
co
nto
ur
len
gth
(L
c)
du
rin
g m
ech
an
ical
un
fold
ing
, b
ased
on
fo
rce a
nd
exte
nsio
n d
ata
fro
m
an
AF
M e
xp
eri
men
t. T
his
filte
r p
rovid
es a
n o
nlin
e a
nd
fu
lly a
uto
mate
d e
sti
mate
of
Lc
based
on
a
syste
m
mo
del,
the
exp
eri
men
tal
measu
rem
en
ts,
an
d
no
ise
sta
tisti
cs.
Th
e
syste
m
mo
del
co
mp
rises a
ph
ysic
al
mo
del
of
the c
an
tile
ver
an
d a
no
nlin
ear
WL
Cap
pro
xim
ati
on
of
the e
xte
nd
ed
p
rote
in.
Wh
en
man
uall
y f
itti
ng
th
e W
LC
mo
del
to f
orc
e-e
xte
nsio
n d
ata
fro
m u
biq
uit
in p
rote
ins,
the
esti
mate
of
the c
han
ge i
n c
on
tou
r le
ng
th d
uri
ng
un
fold
ing
is d
istr
ibu
ted
no
rmall
y w
ith
mean
23.3
n
m a
nd
vari
an
ce 1
0.2
nm
2.
Testi
ng
th
e K
alm
an
filte
r o
n t
he s
am
e p
rote
in y
ield
s ∆ ∆∆∆
Lc
wit
h a
24.8
nm
m
ean
an
d 2
.0 n
m2
vari
an
ce.
As t
he v
ari
an
ce l
imit
s r
eso
luti
on
in
esti
mati
ng
th
e n
um
ber
of
am
ino
acid
s r
ele
ased
by u
nfo
ldin
g,
it i
s c
lear
that
the K
alm
an
filte
r p
resen
ts a
su
bsta
nti
al
imp
rovem
en
t o
ver
the c
on
ven
tio
nal
meth
od
. W
e t
here
by d
em
on
str
ate
th
at
the K
alm
an
filte
r p
rovid
es a
po
werf
ul
un
bia
sed
ap
pro
ach
to
in
terp
reti
ng
fo
rce sp
ectr
osco
py d
ata
, cap
ab
le o
f in
cre
asin
g re
so
luti
on
b
eyo
nd
th
e t
rad
itio
nal exp
eri
men
tal li
mit
. D
ue t
o t
he f
lexib
ilit
y o
f th
is a
pp
roach
, it
can
be e
xte
nd
ed
to
mo
nit
ori
ng
oth
er
sta
te v
ari
ab
les o
f m
ole
cu
lar
sys
tem
s o
bserv
ed
b
y vari
ou
s f
orm
s o
f fo
rce
sp
ectr
osco
py,
inclu
din
g o
pti
cal an
d m
ag
neti
c t
weezers
.
Fig
ure
1M
ech
an
ical str
etc
hin
g o
f a p
oly
pro
tein
usin
g s
ing
le-m
ole
cu
le a
tom
ic f
orc
e m
icro
sco
py.
(A)
(i)
A s
ingle
poly
pro
tein
mole
cule
is h
eld
betw
een t
he c
antile
ver
tip
and t
he c
overs
lip,
whose p
ositio
n c
an b
e
contr
olle
d w
ith h
igh p
recis
ion u
sin
g a
pie
zoele
ctr
ic p
ositio
ner
(pie
zo).
(ii)
Movin
g t
he c
overs
lipaw
ay f
rom
the
tip exert
s a str
etc
hin
g fo
rce on th
e poly
pro
tein
, w
hic
h in
tu
rn bends th
e cantile
ver.
T
he bendin
g of
the
cantile
ver
changes t
he p
ositio
n o
f th
e laser
beam
on t
he s
plit
photo
dio
de (
PD
), r
egis
tering t
he p
ulli
ng f
orc
e.
The a
pplie
d f
orc
e c
an b
e d
ete
rmin
ed f
rom
the s
pring c
onsta
nt
of
the c
antile
ver
and t
he d
egre
e o
f cantile
ver
bendin
g.
At
this
hig
h pulli
ng fo
rce,
a pro
tein
dom
ain
unfo
lds.
(iii)
The unfo
lded dom
ain
can now
re
adily
exte
nd,
rela
xin
g t
he c
antile
ver.
(iv
)T
he p
iezo c
ontinues t
o m
ove,
str
etc
hin
g t
he p
oly
pro
tein
to a
new
hig
h
forc
e p
eak,
repeating t
he s
equence u
ntil
the w
hole
poly
pro
tein
has u
nfo
lded.
This
pro
cess r
esults i
n a
forc
e
exte
nsio
n c
urv
e w
ith a
chara
cte
ristic s
aw
tooth
pattern
shape.
(B)
A t
ypic
al
saw
tooth
pattern
curv
e o
bta
ined
by s
tretc
hin
g a
n I27 p
oly
pro
tein
. T
he labels
i–iv
repre
sent th
e s
equence o
f events
show
n in A
.
Fig
ure
2
Sch
em
ati
c
dep
icti
on
o
f th
e
Exte
nd
ed
K
alm
an
F
ilte
r (E
KF
) im
ple
men
tati
on
fo
r sin
gle
p
rote
in
forc
e
sp
ectr
osco
py
wit
h
an
A
FM
.T
he
EK
F
estim
ate
s t
he c
urr
ent
conto
ur
length
of
the
pro
tein
based u
pon t
he f
orc
e m
easure
ments
(F
t) u
p t
o t
he p
resent
tim
e.
It
is a
n e
xte
nsio
n
of
the
Kalm
an
filter
for
syste
ms
with
nonlin
ear
dynam
ics.
The K
alm
an f
ilter
is a
n
optim
al
estim
ation a
lgorith
m g
iven a
know
n
linear
syste
m w
ith G
aussia
n n
ois
e.
B
In g
ene
ral, t
he
exte
nded
Ka
lman
filt
er
(EK
F)
tra
cks a
n e
stim
ate
of
the
sta
te
ve
cto
r q
nT
he
sta
te v
ecto
r is
co
mpo
sed
of
the
con
tou
r le
ngth
of
the p
rote
in
an
d t
he
po
sitio
n o
f th
e c
an
tile
ve
r, w
hic
h a
re t
he
hid
den
va
riab
les t
ha
t fu
lly
de
term
ine
the
syste
m.
The
alg
orith
m u
se
s a
mode
l of
the
syste
m t
o p
red
ict
the
me
asu
rem
en
t a
t tim
este
pn
ba
se
d o
n t
he
inpu
t a
t n
-1an
d t
he
estim
ate
a
t n
-1.
The
err
or
be
twee
n t
he
pre
dic
ted
va
lue
and
th
e t
rue
me
asu
red
va
lue
at
tim
e
nis
th
en
u
sed
to
upda
te
the
e
stim
ate
w
ith
a
gain
K
tha
t is
de
term
ined
by t
he
EK
F a
lgo
rith
m.
In o
rde
r to
op
tim
ally
de
term
ine
Kfo
r
ea
ch
tim
este
p,
an
estim
ate
of
the
co
va
rian
ce
ma
trix
is a
lso
tra
cked
by t
he
alg
orith
m.
Dis
cu
ssed
in
Fig
ure
3,
the
pro
tein
is m
od
ele
d u
sin
g t
he
Worm
-Lik
e C
ha
in
(WLC
) m
ode
l of
po
lym
er
ela
sticity,
giv
en
by e
qua
tion
(1
).
The
WLC
mode
l p
rovid
es t
he
ten
sio
n f
orc
e i
n t
he
pro
tein
(F
p)
giv
en
th
e e
xte
nsio
n o
f th
e
pro
tein
(y-u
) a
nd
the
con
tou
r le
ngth
(L
c).
T
he
can
tile
ve
r m
od
el
in t
urn
is
spe
cifie
d b
y t
he
tra
nsfe
r fu
nction
be
twe
en
the
inpu
t p
rote
in t
en
sio
n a
nd
the
ou
tpu
t can
tile
ve
r fo
rce (
2).
The
co
rre
spond
ing c
oeff
icie
nts
a
re g
ive
n
in
equa
tion
s
(3)
and
(4
).
As
men
tioned
in
Fig
ure
3,
the
re
su
lt is a
filt
er
tha
t de
pend
s o
n the
pre
vio
us t
wo
tim
este
ps.
Equa
tio
n
(5)
sh
ow
s
the
sta
te
ve
cto
r u
sed
in
this
im
ple
men
tation
. It
is
co
mpo
sed
of
the
ca
ntile
ve
r de
fle
ction
(y)
an
d t
he
co
nto
ur
len
gth
of
the
pro
tein
be
ing s
tre
tche
d (
Lc).
T
he
cu
rren
t an
d p
revio
us v
alu
es o
f y a
nd
Lc
are
bo
th i
nclu
de
d i
n t
he
sta
te v
ecto
r, a
s r
equ
ired
by t
he
can
tile
ve
r m
ode
l. T
he
fu
ll syste
m m
ode
l u
se
d b
y t
he
EK
F a
lgo
rith
m i
s d
escrib
ed
by
equa
tion
s (
6)
and
(7
).
Equa
tion
(6
)de
scribe
s t
he p
rogre
ssio
n o
f th
e s
tate
ve
cto
r.
Both
th
e c
an
tile
ve
r d
eflection
and
the
con
tou
r le
ngth
are
assu
med
to
ha
ve
indepe
nden
t w
hite
Gau
ssia
n p
roce
ss n
ois
e (
wi,n)
so
urc
es.
Apa
rt f
rom
th
e n
ois
e,
the
co
nto
ur
len
gth
is m
ode
led
as c
on
sta
nt.
Althou
gh
th
is i
s c
lea
rly i
nco
rre
ct
glo
ba
lly,
it m
ode
ls t
he
pe
riod
be
twe
en
un
fold
ing e
ven
ts a
ccu
rate
ly.
Th
e c
an
tile
ve
r d
efle
ction
is u
pda
ted
by t
he
co
mb
ina
tion
of
can
tile
ve
r d
yn
am
ics a
nd
WLC
mode
ls.
The
non
linea
rity
in
th
e W
LC
mode
l is
the
re
ason
tha
t an
exte
nded
Ka
lman
filt
er
mu
st
be
use
d a
s o
ppo
se
d t
o a
re
gu
lar
Ka
lman
filt
er.
E
qua
tio
n (
7)
is t
he
mea
su
rem
en
t e
qua
tion
, lin
kin
g t
he
sta
te v
ariab
les t
o t
he
exp
erim
en
tally
me
asu
red
qu
an
tity
. I
n t
he
mea
sure
men
t, t
he
re is a
n a
dd
itio
na
l sou
rce
of
Gaussia
n n
ois
e (
vn).
T
he
se
equa
tio
ns f
ully
define
the
pro
ble
m f
or
the
app
lica
tion
of
the
EK
F a
lgo
rith
m.
The
alg
orith
m its
elf is s
tand
ard
and
can
be
fo
und
in
te
xts
su
ch
as [
RE
F 1
].
The
EK
F a
lgo
rith
m o
nly
utiliz
es t
he
mea
su
rem
en
ts th
at ha
ve
alrea
dy b
een
made
in
ord
er
to c
rea
te its
estim
ate
of
the
sta
te v
ariab
les.
As a
re
su
lt,
it
can
be
run
con
cu
rren
tly w
ith
th
e e
xp
erim
en
t itse
lf.
In
the
cu
rren
t ana
lysis
, th
e e
stim
ation
wa
s d
one
aft
er
the
exp
erim
en
t on
a b
atc
h o
f tr
ace
s.
The
cau
sal e
stim
ation
re
su
lts a
re s
ho
wn
in
Fig
ure
5.
The
se
re
sults s
ho
w a
sub
sta
ntial im
pro
vem
en
t o
ve
r th
e c
om
monly
use
d h
an
d-f
ittin
g m
eth
od
s.
(1)
(2)
(3)
(4)
(5)
(6)
2
2
1
1
2
2
1
1
1)
(
)(
−−
−−
++
+=
za
za
zb
zb
zF
zF
p
•
+
−
−
=
=
∑ =
−+
−
−−
+
+
+
+
nn
nnn
j
jn
j
jn
jn
jn
j
n
nn
n
nww
Lc
Lcy
ya
Lc
uy
Wb
k
Lc
Lcyy
q,
2
,1
1
0
11
1
1
1
00
10
00
01
1
[]T
nn
nn
nL
cL
cy
yq
11
−−
=
[]
nn
nv
qk
F+
⋅=
00
0
−
+−
−−
=
−
=
−
Lcu
y
Lcu
y
pTk
Lcu
yW
FB
p41
141
2
[]
0
.19
20
.33
4
=b
[]
0.1
96
0
.66
9-
=a
(7)
(1)
(2)
(3)
(4)
(5)
(6)
2
2
1
1
2
2
1
1
1)
(
)(
−−
−−
++
+=
za
za
zb
zb
zF
zF
p
•
+
−
−
=
=
∑ =
−+
−
−−
+
+
+
+
nn
nnn
j
jn
j
jn
jn
jn
j
n
nn
n
nww
Lc
Lcy
ya
Lc
uy
Wb
k
Lc
Lcyy
q,
2
,1
1
0
11
1
1
1
00
10
00
01
1
[]T
nn
nn
nL
cL
cy
yq
11
−−
=
[]
nn
nv
qk
F+
⋅=
00
0
−
+−
−−
=
−
=
−
Lcu
y
Lcu
y
pTk
Lcu
yW
FB
p41
141
2
[]
0
.19
20
.33
4
=b
[]
0.1
96
0
.66
9-
=a
(7)
Fig
ure
5.
Resu
lts
of
EK
F
imp
lem
en
tati
on
o
n
exp
eri
-m
en
tal
data
.E
xperim
ents
w
ith
ubiq
uitin
poly
pro
tein
s
at
an
exte
nsio
n
rate
of
400
nm
/s
pro
duced 1
90 s
aw
tooth
tra
ces f
or
analy
sis
.
(A)
A
sam
ple
tr
ace
from
the d
ata
set.
The f
inal
peak
is a re
sult of
the dis
socia
tion of
the p
rote
in f
rom
the c
antile
ver
tip.
These p
eaks w
ere
exclu
ded f
rom
th
e a
naly
sis
in L
cste
p s
izes.
(B)
The
resultin
g
estim
ate
of
the
conto
ur
length
of
the
pro
tein
,
corr
espondin
g to
th
e data
in
part
(A
). A
pers
iste
nce le
ngth
of
0.4
nm
w
as
chosen f
or
the p
rote
in m
odel, w
hic
h i
s t
he c
om
monly
used v
alu
e f
or
ubiq
uitin
.
The inset
enla
rges o
ne o
f th
e s
teps in L
c,
show
ing t
hat
the c
onverg
ence b
ehavio
r does
not
matc
h
any
of
those
from
F
igure
3.
T
he
overs
hoot
that
occurs
im
media
tely
after
the s
tep i
mplie
s a
n e
rror
in t
he p
rote
in m
odel
and c
onfirm
s t
he
expecte
d fa
ilure
of
the W
LC
m
odel
at
low
fo
rces.
(C
) T
he dis
trib
ution and
sta
tistics o
f th
e e
stim
ate
d c
hanges in c
onto
ur
length
s d
uring u
nfo
ldin
g c
om
pare
d
with e
arlie
r re
sults f
it b
y h
and w
ith t
he W
LC
model, a
s i
n f
igure
2.
The d
ata
for
the
hand-f
itte
d
ste
ps
is
from
[R
EF
2].
T
he
spre
ad
of
EK
F
estim
ate
s
is
substa
ntially
narr
ow
er,
with a
sta
ndard
devia
tion l
ess t
han h
alf
that
of
the h
and-
fitted d
istr
ibution.
A n
oticeable
skew
is o
bserv
ed in t
he E
KF
ste
p s
ize h
isto
gra
m.
There
fore
the f
it p
lotted i
s n
ot
a
Gaussia
n
dis
trib
ution,
but
a
genera
lized
extr
em
e
valu
e
dis
trib
ution.
T
he p
ara
mete
rs o
f th
e
dis
trib
ution
are
: ξ
=
-0.1
5
σ=
1.3
and µ
= 2
4.6
. ξ,
σ,
and µ
are
th
e
shape,
scale
, and
location p
ara
mete
rs r
espectively
.
Though p
relim
inary
, th
is m
ay b
e
linked
to
the
underlyin
g
pro
tein
m
echanic
s in w
hic
h larg
e c
onto
ur
length
s a
re p
refe
rentially
chosen
am
ong a
vaila
ble
configura
tions.
Fig
ure
3M
od
els
descri
bin
g t
he e
xp
eri
men
tal
syste
m f
or
the e
xte
nd
ed
Kalm
an
fi
lter
imp
lem
en
tati
on
.T
he s
yste
m m
odel
is d
ivid
ed in
to t
wo sequential
part
s:
A
lin
ear
model
of
the c
antile
ver
dynam
ics d
riven b
y t
he o
utp
ut
of
a n
onlin
ear
pro
tein
m
odel.
(A)
The c
antile
ver
model
consis
ts o
f a s
econd o
rder
LT
I syste
m f
it t
o t
he
nois
e s
pectr
um
of
a f
ree c
antile
ver.
This
assum
es t
he n
ois
e i
s w
hite a
nd a
cts
sole
ly
on t
he t
ip o
f th
e c
antile
ver.
It
is im
port
ant
to d
istinguis
h t
he
heavily
dam
ped c
antile
ver
spectr
um
near
a s
urf
ace (
blu
e lin
e)
from
the s
pectr
um
far
from
asurf
ace (
bla
ck l
ine).
(B
)T
he p
rote
in f
orc
es a
re m
odele
d b
y t
he W
orm
-Lik
e C
hain
(W
LC
) m
odel
of
pro
tein
ela
sticity.
This
model
is k
now
n t
o f
it t
he p
rote
in f
orc
e-e
xte
nsio
n c
urv
e w
ell
for
hig
h
forc
es
only
. P
revio
usly
, data
w
as
fit
with th
e W
LC
m
odel
as show
n,
where
th
e
unfo
ldin
g s
tep s
ize a
nd p
ers
iste
nce l
ength
are
chosen m
anually
to o
bta
in t
he b
est
fit
over
the e
ntire
tra
ce. F
or
this
exam
ple
, ∆
Lc
= 2
3.1
nm
, and p
= 0
.37 n
m.
Fig
ure
4K
alm
an
filte
r esti
mati
on
ap
plied
to
sim
ula
ted
d
ata
.(A
)S
imula
ted
data
genera
ted
directly
from
th
e
cantile
ver
and W
LC
models
with a
pers
iste
nce length
of
0.4
nm
and a
pre
dete
rmin
ed s
tepw
ise c
onsta
nt
conto
ur
length
(L
c).
W
hite G
aussia
n m
easure
ment
nois
e w
ith a
sta
ndard
devia
tion o
f 10 p
Nhas b
een a
dded t
o t
he s
imula
tion.
(B
)E
KF
estim
ate
s
of
the
Lc
based
on
various
pers
iste
nce
length
s,
again
st
the t
rue L
c(d
otted l
ine).
The c
onverg
ence
behavio
r is
str
ongly
dependent
on
the
valu
e
of
the
pers
iste
nce l
ength
. F
or
the t
rue v
alu
e o
f th
e p
ers
iste
nce
length
, th
e e
stim
ate
quic
kly
converg
es t
o t
he t
rue
Lc.
If
the
pers
iste
nce l
ength
is o
ff,
slo
w c
onverg
ence r
esults,
with a
slo
pe that in
dic
ate
s the s
ign o
f th
e e
rror.
Co
nclu
sio
ns
•Im
ple
men
tin
g
a
Kalm
an
fi
lter
en
han
ces
the
esti
mate
o
f th
e
ste
pw
ise i
ncre
ases i
n c
on
tou
r le
ng
th o
f u
nfo
ldin
g p
rote
ins b
y
pro
vid
ing
mo
re c
on
sis
ten
t an
d a
ccu
rate
resu
lts
•T
his
ap
pro
ach
p
rovid
es
an
u
nb
iased
“h
an
ds
off
”w
ay
of
extr
acti
ng
in
form
ati
on
in
real ti
me o
n m
ole
cu
lar
pro
pert
ies
•T
he
ch
oic
e
of
pers
iste
nce
len
gth
m
ay
be
mad
e
befo
re
the
exp
eri
men
t, o
r it
can
be a
uto
mati
call
y c
ho
sen
in
po
st-
an
aly
sis
, m
akin
g t
he p
roced
ure
fu
lly i
nd
ep
en
den
t o
f th
e e
xp
eri
men
ter
•T
he o
n-l
ine o
pera
tio
n o
f th
e K
alm
an
fi
lter
op
en
s th
e d
oo
r to
n
um
ero
us a
pp
licati
on
s s
uch
as im
pro
ved
feed
-back s
ys
tem
s•
Th
e K
alm
an
filte
r sh
ou
ld b
e u
tilized
mo
re c
om
mo
nly
in
pro
vid
ing
u
nb
iased
esti
mati
on
s
of
hid
den
sta
te
vari
ab
les
in
sin
gle
m
ole
cu
le e
xp
eri
men
ts
Refe
ren
ces:
[1]
Mo
hin
der
S. G
rew
al an
d A
ng
us P
. A
nd
rew
s, K
alm
an
Filte
rin
g –
Th
eo
ry a
nd
Pra
cti
se u
sin
g M
AT
LA
B, 2n
d e
dit
ion
,
Wiley-I
nte
rscie
nce, 2001
[2]
Mari
an
o C
arr
ion
-Vazq
uez e
t al., T
he m
ech
an
ical sta
bilit
y o
f u
biq
uit
in is lin
kag
e d
ep
en
den
t, N
at
Str
uct
Bio
l, 2
007
