Single-Pa icle O bi T acking
Se up, Cha ac e isa ion and Applica ion
De Uni e si ä Bay eu h
zu E langung des G ades eines
Dok o s de Na u wissenscha en (D . e . na .)
genehmig e Abhandlung
on
Dominique E ns
gebo en am 28. Ap il 1982
in Ti schen eu h
1. Gu ach e : P o . D . J. Köhle
2. Gu ach e : P o . D . M. Weiss
Tag de Ein eichung: 27.09.2012
Tag des Kolloquiums: 14.12.2012
Abs ac
T acking o single nanoobjec s (e.g. beads, p o eins o molecules) is o undamen al
in e es in many esea ch ields, pa icula ly in he ields o biophysics and ma e ial
sciences. In o ma ion such as he local iscosi y o he s uc u e a ound he ace
pa icle can be ga he ed on he nanome e scale. Fu he , he pho ophysical p op-
e ies o con o ma ional dynamics o he ace can be s udied wi hou he need o
immobilising hem. Mo eo e , undamen al esea ch wi h espec o di usion p o-
cesses like he no mal B ownian mo ion o anomalous di usion can be examined
wi h he acquisi ion o single-pa icle ajec o ies.
In his hesis, he de elopmen and expe imen al ealisa ion o an op ical se up
which eco ds he 2-dimensional ajec o ies o single luo escen ly labeled poly-
s y ene beads, ei he 20 nm o 50 nm in diame e , wi h a high spa ial and empo al
esolu ion is in oduced. Combining single molecule luo escence echniques wi h
a new me hod called single-pa icle o bi acking he spa ial posi ion o he beads
could be de e mined wi h an accu acy o less han 10 nm a a ime esolu ion o
4 ms. The idea is o manipula e he exci a ion ligh spa ially and empo ally o
loca e a pa icle. In o de o do so, special op ics which de lec a lase beam and
guide i on a ci cula pa h we e used. Subsequen ly, his o a ing beam is p o-
jec ed by a mic oscope in o he sample wi h he di using pa icles. Due o he
spa ially and pe iodically modula ed exci a ion ligh , he emission signal o he
bead is modula ed wi h he equency o he o a ion o he lase ocus. The ampli-
ude o he modula ed emission signal depends on he posi ion o he pa icle wi hin
he exci a ion o bi . An ingeniously de eloped algo i hm calcula es he posi ion o
he pa icle wi h espec o he cen e o he o bi by demodula ing he emission
signal and es o es he pa icle back o he o bi cen e. Applying his me hod
successi ely, he ajec o y o he di using bead can be econs uc ed. Besides he
expe imen al ealisa ion, he cha ac e isa ion o he se up in e ms o he spa ial
and empo al accu acy as well as he expe imen al sho comings ha in luences
he measu ed ajec o ies and hence, he in e p e a ion o he da a, we e also he
main opics o his wo k. Fo his pu pose a e e ence sample o 20 nm sized beads
in glyce ol was used. The accu acies we e s udied mainly by compu e simula ions
and he a i ac s by expe imen s. The echnical de ails o he se up and he cha -
ac e isa ion esul s we e published (publica ion P1).
The eco ded ajec o ies we e analysed wi h a ious me hods, among which he
commonly used mean squa ed displacemen (MSD) yields he esul s wi h highes
in o ma ion. The di usion coe icien as well as he di usion beha iou could be
quan i ied. Wi h his me hod he ob ainable accu acy in measu ing he di usion
coe icien by he acquisi ion o single-pa icle ajec o ies was s udied as a unc ion
i
o he leng h o he ajec o ies and as a unc ion o he numbe o i ing poin s
ha we e used o a linea i o he expe imen ally de e mined MSD-cu es. As
expec ed, he ela i e e o o he de e mined di usion coe icien ge s be e o
longe ajec o ies. Fu he , an op imal numbe o i ing poin s o he linea ap-
p oxima ion o he MSD-cu es was ound, which yields he mos exac alues o
he di usion coe icien s and which is independen o he ajec o y leng h. Fo
he i s ime, expe imen al esul s on ha issue we e compa ed wi h heo e ical
p edic ions, whe e a good ag eemen was ound. These indings we e published
(publica ion P2). By he use o he S okes-Eins ein ela ion he di usion coe i-
cien s could u he be con e ed o pa icle adii. A close examina ion o hese
adii emphasises he in luence o he a o e men ioned numbe o i ing poin s. Fo
he op imal alue, signi ican ly p ecise adii could be de e mined.
Finally, an applica ion o he new se up is p esen ed. In coope a ion wi h he
chai o expe imen al physics I (g oup o P o . D . M. Weiss) o he Uni e si y
o Bay eu h, he di usion beha iou o single nanopa icles in a complex luid was
s udied. Backg ound he e o is he in es iga ion o biochemical eac ions in a biolog-
ical cell, whose kine ic is gi en by he di usion o he co esponding eac ion pa -
ne s. Due o he high c owding o he cell compa men s he di usion is hinde ed.
The di usion beha iou in hese sys ems is called anomalous and mo e exac ly
subdi usi e. Se e al heo e ical models ha e been de eloped o explain his phe-
nomenon, bu ye wi hou expe imen al e i ica ions. He e, he di usion o 50 nm
sized polyme beads in he model sys em dex an (a highly b anched biopolysac-
ca ide) is in es iga ed expe imen ally wi h high spa ial and empo al esolu ion.
The da a we e analysed in he g oup o he coope a ion pa ne which yields a e y
good ag eemen wi h he model o “ ac ional B ownian mo ion”. These esul s
we e also published (publica ion P3).
A inal ou look conce ns possible echnical ad ancemen s o he expe imen al se up,
in pa icula o measu e 3-dimensional ajec o ies, and se e al applica ions a
which he acking o single pa icles migh be help ul o a be e unde s anding
o he sys em o in e es .
ii
Ku zda s ellung
Die äumliche Ve olgung einzelne Nanopa ikel (z.B. Polys y olkolloide, P o ei-
ne ode Moleküle) is ü iele Fo schungsgebie e, o allem in de Biophysik und
den Ma e ialwissenscha en, on g oßem In e esse. So können un e ande em In-
o ma ionen übe die lokale Viskosi ä ode übe S uk u en in de Umgebung
des Teilchens au eine Nanome e skala gewonnen we den. Wei e hin können die
pho ophysikalischen Eigenscha en ode die Kon o ma ionsdynamik de e olg en
Teilchen selbs s udie we den, ohne sie zu immobilisie en. Auch die G undlagen
on Di usionsp ozessen, wie zum Beispiel die B ownsche Bewegung ode anomales
Di usions e hal en, können du ch die Messungen on T ajek o ien einzelne Teil-
chen un e such we den.
In diese Disse a ion wi d die En wicklung und expe imen elle Umse zung eines
op ischen Au baus zu Messung on zweidimensionalen T ajek o ien einzelne luo-
eszenzma kie e Polys y olbeads mi einem Du chmesse on 20 nm bzw. 50 nm
mi hohe äumliche und zei liche Au lösung o ges ell . Du ch die Kombina ion
on Einzelmolekül luo eszenz echniken mi eine neua igen Me hode mi de engli-
schen Bezeichnung „single-pa icle o bi acking“ konn e eine äumliche Au lösung
in de Posi ionsbes immung de Beads on wenige als 10 nm bei eine Zei au lösung
on 4 ms e ziel we den. Die Idee dabei is , das An egungslich äumlich und zei -
lich zu manipulie en, um die Posi ion eines Teilchens zu bes immen. Dazu we den
spezielle op ische Bauelemen e e wende die einen Lase s ahl au einen Kegel-
man el ablenken. Nach olgend wi d diese o ie ende Lase s ahl mi Hil e eines
Mik oskops in die P obe mi den di undie enden Teilchen p ojizie . Au g und de
äumlichen und pe iodischen Modula ion des An egungslich s is auch das Emissi-
onssignal des Teilchens mi de F equenz des o ie enden Lase okus modulie . Die
Ampli ude des modulie en Emissionssignals is on de Teilchenposi ion inne halb
des O bi s abhängig, welche du ch das okussie e An egungslich e zeug wi d. Ein
speziell en wickel e Algo i hmus be echne die Teilchenposi ion bezüglich des Mi -
elpunk es des O bi s indem das Emissionssignal demodulie wi d. Anschließend
wi d das Teilchen um den be echne en We zu ück in die Mi e des O bi s e scho-
ben. Sukzessi e Anwendung diese Be echnungsme hode lie e die ekons uie e
T ajek o ie des Teilchens. Schwe punk diese A bei wa neben de echnischen
Realisie ung, die Cha ak e isie ung des Au baus in Bezug au die äumliche und
zei liche Au lösung de T ajek o ien, sowie au expe imen elle Unzulänglichkei en,
welche die gemessenen T ajek o ien und dami auch die In e p e a ion de Messda-
en, beein lussen. Dazu wu de die Re e enzp obe on 20 nm g oßen Teilchen in Gly-
ce in e wende . Die e eichba en Au lösungsg enzen wu den haup sächlich du ch
den Einsa z compu e ges ü z e Simula ionen e i izie , wohingegen die A e ak e
iii
expe imen ell un e such wu den. Die diesbezüglich e ziel en E gebnisse sowie die
echnischen De ails des Au baus wu den e ö en lich (Publika ion P1).
Die au genommenen T ajek o ien wu den au e schiedene Weise analysie , wo-
bei die wei e b ei e e Me hode des mi le en Ve schiebungsquad a s (engl.: mean
squa ed displacemen , MSD), E gebnisse mi dem höchs en In o ma ionsgehal lie-
e e. Sowohl de Di usionskoe izien als auch das Di usions e hal en konn e quan-
i izie we den. Mi Hil e diese Analyse wu de die e eichba e Genauigkei on
Di usionskoe izien en du ch Messungen on Einzel eilchen ajek o ien in Abhän-
gigkei de T ajek o ienlänge und de Anzahl an Fi punk en, die ü eine linea e
Ku enanpassung an die expe imen ell bes imm en MSD-Ku en e wende wu de,
un e such . Die Analyse de Messda en zeig e e wa ungsgemäß, dass de ela i e
Fehle des Di usionskoe izien en ü länge e T ajek o ien kleine is . Wei e hin
wu de eine op imale Anzahl an Fi punk en ü die MSD-Ku enanpassung ge un-
den, die unabhängig on de T ajek o ienlänge is und die genaues en We e ü die
Di usionskoe izien en lie e . Die expe imen ellen E gebnisse diese Un e suchung
wu den e s mals mi heo e ischen Vo he sagen e glichen, wobei eine gu e Übe ein-
s immung ge unden wu de. Die Resul a e wu den e ö en lich (Publika ion P2).
Mi Hil e de S okes-Eins ein Beziehung konn en wei e hin die Di usionskoe izien-
en in Teilchen adien umge echne we den. Eine genaue Be ach ung de Radien
e deu lich den Ein luss de Anzahl an Fi punk en. Fü die op imale Anzahl an
Fi punk en wu den wesen lich p äzise e We e e mi el .
Als Anwendung des neuen Au baus wu de in Koope a ion mi dem Leh s uhl Expe-
imen alphysik I (A bei sg uppe on P o . D . M. Weiss) de Uni e si ä Bay eu h
das Di usions e hal en on einzelnen Polys y olbeads in eine komplexen Flüssig-
kei s udie . Hin e g und hie bei is die Un e suchung biochemische Reak ionen
inne halb eine biologischen Zelle, de en Kine ik du ch die Di usion de en sp e-
chenden Reak ionspa ne gegeben is . Diese Di usion is du ch die hohe Dich e an
Zellkompa imen en s a k eingesch änk . Man sp ich deshalb on einem anoma-
lem Di usion e hal en, genaue gesag on Subdi usion. Ve schiedene heo e ische
Modelle zu Besch eibung dieses Phänomens wu den en wickel , wobei eine expe-
imen elle Ve i ika ion noch nich möglich wa . In diese A bei wi d die Di usion
on 50 nm g oßen Polys y olbeads in dem Modellsys em Dex an (hoch e zweig es
Biopolysaccha id) mi hohe äumliche und zei liche Au lösung un e such . Die
Messda en wu den in de A bei sg uppe des Koope a ionspa ne s analysie und
zeig en eine seh gu e Übe eins immung mi dem Modell „ ac ional B ownian mo-
ion“. Die E gebnisse wu den eben alls e ö en lich (Publika ion P3).
Ein abschließende Ausblick be ass sich mi de echnischen Wei e en wicklung
des expe imen ellen Au baus, speziell mi de Messung 3-dimensionale T ajek o-
ien, und mi e schiedenen Anwendungsmöglichkei en, bei denen die Bewegung
einzelne Teilchen au schluss eiche E kenn nisse lie e n können.
i
Con en s
I In oduc ion 1
1 Mo i a ion 3
2 Theo e ical & expe imen al backg ound 5
2.1 Theo y.................................. 5
2.1.1 Di usion equa ion . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.2 B ownianmo ion........................ 6
2.1.3 T ajec o y analysis . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 T acking echniques........................... 10
2.2.1 CCD-T acking.......................... 11
2.2.2 O bi -T acking . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.3 Al e na i e app oaches . . . . . . . . . . . . . . . . . . . . . 13
3 Ma e ials 15
3.1 Beadsandsamples ........................... 15
3.2 Samplep epa a ion........................... 17
4 Simula ions 19
4.1 Gene alp ocedu e ........................... 19
4.2 Spa io- empo al accu acy . . . . . . . . . . . . . . . . . . . . . . . . 22
4.3 Loosingapa icle............................ 25
5 The acking se up 29
5.1 Op ics&Ha dwa e........................... 29
5.2 As a icpa icle............................. 30
5.2.1 Simula ion............................ 31
5.2.2 Expe imen ........................... 33
5.3 A i ac s & Co ec ions . . . . . . . . . . . . . . . . . . . . . . . . . 34
6 Single-pa icle acking: esul s & discussion 37
6.1 Analysing di usion based on single-pa icle ajec o ies . . . . . . . 38
6.1.1 Mean squa ed displacemen analysis . . . . . . . . . . . . . . 38
6.1.2 Al e na i e app oaches . . . . . . . . . . . . . . . . . . . . . 42
6.2 Accu acy o di usion coe icien s . . . . . . . . . . . . . . . . . . . . 47
6.3 Pa iclesizes .............................. 48
Con en s
6.4 C owdedFluids............................. 52
7 Ou look 57
Appendix 59
A P og amcode.............................. 59
B Ha monic app oxima ion . . . . . . . . . . . . . . . . . . . . . . . . 61
Bibliog aphy 63
Lis o publica ions 69
Danksagung 71
E klä ung 73
II Publica ions 75
P1. Se up o single-pa icle o bi acking: a i ac s and co ec ions 77
P2. Measu ing a di usion coe icien by single-pa icle acking: S a is-
ical analysis o expe imen al mean-squa ed-displacemen cu es 90
P3. F ac ional B ownian Mo ion in C owded Fluids 102
i
Lis o abb e ia ions
The ollowing lis o abb e ia ions summa izes all used pa ame e s and gi es an
explana ion o he symbols.
R adius o he o bi
ν equency o he o a ing ocus
ωangula eloci y o he o a ing ocus (ω= 2πν)
w ull wid h a hal maximum o he ocussed lase spo
I0maximum emission o a luo escen pa icle placed di ec ly in he
ocal spo
Ibbackg ound emission
S0maximum numbe o emi ed pho ons o a luo escen pa icle
placed di ec ly in he ocal spo
Sbnumbe o backg ound pho ons
Sn heo e ical numbe o emi ed pho ons du ing he sampling
in e all δ
Sn,poiss simula ed numbe o emi ed pho ons by he use o a poisson
dis ibu ion
Imean emission in ensi y o he luo escen ace
xp, yp eal x and y coo dina es o he pa icle
xc, yccalcula ed x and y coo dina es o he pa icle
xs, ysx and y coo dina es o he piezo s age
a adius o he pa icle
Ddi usion coe icien
kBBol zmann cons an
T empe a u e, a which he expe imen s we e pe o med
η isosi y o he used luids
αanomaly pa ame e
nnumbe o i ing poin s used o a linea i o he MSD cu es
ime
τlag ime
∆ ime esolu ion o he expe imen s
∆ spa ial esolu ion
δ sampling ime o he expe imen s and he simula ions
NSnumbe o sampling da a poin s
N o al numbe o da a poin s o a ajec o y
Nseg numbe o da a poin s o a cu segmen
Nens numbe o ajec o ies an ensemble o segmen s consis s o
ii
2 Theo e ical & expe imen al backg ound
cu en densi y j, which a ises due o a concen a ion g adien ∇c.
j=−D∇c(2.1)
This equa ion is called he i s Fickian law. Beside he concen a ion g adien , he
pa icle cu en densi y depends u he on he di usion coe icien D. The la e
was de eloped in he wo ks o S okes and Eins ein [4] and is de ined as:
D=kBT
6πηa (2.2)
He e, kBis he Bol zmann cons an , T he empe a u e o he sys em, η he is-
cosi y and a he hyd odynamic adius o he di using pa icles. Wi h he use o
he con inui y equa ion (2.3)
d
d c+∇j= 0,(2.3)
which has i s o igin in he law o conse a ion o pa icles, he di usion equa ion is
ob ained: ∂
∂ c=D∇2c(2.4)
In his o m he di usion coe icien is ega ded as cons an . The di usion equa ion
desc ibes he dynamics o he concen a ion o pa icles o molecules. S ic ly
speaking, his equa ion holds ue o a con inuum o pa icles in an in ini e space.
To s udy single pa icle phenomena, he concen a ion has o be in e p e ed as a
p obabili y densi y o ind a pa icle in space. Howe e , in his wo k he di usion
o single pa icles is s udied by measu ing he ajec o y o he pa icle. Hence, in
he ollowing he heo e ical desc ip ion o his s ochas ic mo ion is in oduced.
2.1.2 B ownian mo ion
A his poin i is wo h men ioning, ha he ollowing heo e ical de i a ions
can be ound in g ea de ail in he book An in oduc ion o Dynamics o Colloids
w i en by J. Dhon [33]. The di usion o a pa icle in a s a ic iscous luid can
be unde s ood as a andom walk, also called B ownian mo ion. The ma hema ical
desc ip ion o such a s ochas ic p ocess is based on he Lange in equa ion (eq.
(2.5)).
m¨ ( ) = −γ˙ ( ) + Fs( )(2.5)
He e, he ec o ( ) ep esen s he posi ion o a pa icle wi h mass ma he ime
. The pa icle, ha mo es wi h espec o he liquid, expe iences an accele a ing
o ce Fs( )and a ic ion o ce −γ˙ ( )whe e γdeno es he ic ion coe icien . Fo
sphe ical pa icles wi h adius ain a luid wi h iscosi y η he ic ion coe icien is
gi en by:
γ= 6πηa (2.6)
6
2.1 Theo y
The o igin o he accele a ing o ce a e he mal luc ua ions o he liquid molecules
and he concomi an collisions wi h he pa icle. The o ce Fs( )in equa ion (2.5)
can be sepa a ed in a s eng h pa ame e K, and a s ochas ic a iable ( ), ep e-
sen ing he andom o ien a ion.
Fs( ) = K( )(2.7)
The s ochas ic a iable ( ), also known as whi e noise, ull ils wo condi ions.
Fi s , i is iso opic in space and second, wo consecu i e alues ( o ces) in ime a e
unco ela ed, i.e.
h( )i= 0 (2.8)
h( )( 0)i=δ( − 0)(2.9)
( )2= 1 (2.10)
whe e h·i deno es a e aging o e ime o an ensemble. A dis inc a e aging me hod
will be w i en as h·iT o ime a e aging and h·iE o ensemble a e aging, espec-
i ely. The s eng h can be calcula ed om he luc ua ion-dissipa ion- heo em:
hFs( )Fs( 0)i=K2h( )( 0)i(2.11)
= 2nγkBTδ( − 0)(2.12)
The luc ua ion s eng h depends on he ic ion coe icien and he empe a u e.
nis he numbe o dimensions. He e a 2-dimensional andom walk is analysed, i.e.
2n= 4. Combining equa ions (2.7) and (2.12) he s ochas ic o ce esul s o:
Fs( ) = p4γkBT( )(2.13)
The o al o ce (m¨ ( )) is apidly luc ua ing on ime scales o 10−14 s. Due o he
no mally ela i e la ge mass o he pa icle, he B ownian mo ion co e s a ypical
ime scale o 10−9s. The sys em is highly o e damped and we can neglec he
le -hand side o equa ion (2.5). The Lange in equa ion educes o:
˙ ( ) = 1
γFs( )(2.14)
Because o he andom na u e o he o ce, e e y ealisa ion o equa ion 2.14 leads
o a new ajec o y o he pa icle.
The Lange in equa ion p o ides disc e e s eps o he pa icle mo emen wi h an-
dom o ien a ion, ha make up he ajec o y. Such a ajec o y can be simula ed
wi h an i e a i e Eule me hod. Fo a small ime s ep τ he posi ion o he pa icle
a he ime +τcan be calcula ed om i s posi ion ( )a ime by
( +τ) = ( ) + ˙ ( )τ. (2.15)
7
2 Theo e ical & expe imen al backg ound
Using equa ions (2.13) and (2.14), his yields
( +τ) = ( ) + s4kBT
γτ ( )τ(2.16)
which inco po a es an addi ional ac o 1/√τ o a p ope desc ip ion o he mo e-
men . This equa ion desc ibes he B ownian mo ion o a pa icle and was applied
in his hesis o simula e ajec o ies using a home-w i en Ma lab p og am.
2.1.3 T ajec o y analysis
In he ollowing I will gi e an o e iew on possible me hods o analyse a single-
pa icle ajec o y. In pa icula , hese a e i) he mean squa ed displacemen as
a unc ion o a lag ime τ(MSD(τ)) [34], ii) he spa ial ex end and shape, also
e med asphe ici y [35, 36] and iii) he cumula i e dis ibu ion unc ion (CDF) o
squa ed displacemen s [37, 38]. All o hese me hods will be used in sec ion 6 o
he analysis o expe imen al single-pa icle ajec o ies.
i) Mean squa ed displacemen . The MSD can be calcula ed ei he ime-
a e aged o ensemble-a e aged, i.e. h∆ ( )2iTo h∆ ( )2iE. The la e one equi es
a s a is ical ele an ensemble o ajec o ies, while he i s one is commonly used
o a ew single ajec o ies wi h a high numbe o x,y-posi ion pai s and is hence
mo e sui ed o single-pa icle acking expe imen s. The MSD o a pa icle a e
a ime s ep τis de e mined acco ding o:
h∆ ( )2iT=( ( +τ)− ( ))2T=4kBT
γτ(2.17)
He e he equa ions (2.16) and (2.10) we e used o calcula e he MSD. Wi h he
S okes-Eins ein equa ion (2.2) and equa ion (2.6) he di usion coe icien can be
de e mined.
h∆ ( )2iT= 4Dτ (2.18)
The linea dependence in ime holds ue o no mal (B ownian) di usion. Bu ,
o sys ems showing anomalous di usion, he linea dependence b eaks down and
a powe law wi h a scaling exponen αis in oduced [39].
h∆ ( )2iT= 4 ˜
Dτα(2.19)
He e, he di usion coe icien has o be in e p e ed as a gene alized di usion coe i-
cien ˜
D, ha explains he di usion in he sys em unde in es iga ion. The anomaly
pa ame e αis ega ded as a s eng h o he anomaly and can be used o g oup
he di usion beha iou . P ocesses wi h an exponen α > 1a e called supe di usi e
8
2.1 Theo y
and hose wi h α < 1subdi usi e. Only i α= 1 B ownian mo ion is ob ained.
This alue can be de e mined easily wi h loga i hmic calculus o equa ion (2.19).
logh∆ ( )2iT=αlog τ+ log 4 ˜
D(2.20)
The slope ep esen s he scaling exponen , which is ob ained by a linea i o he
MSD da a poin s, plo ed in a loga i hmic scale.
In an expe imen only disc e e posi ions a e a ailable, i.e. he ime-a e aged MSD o
a single ajec o y ( )cons i u ing Nposi ion de e mina ions, has o be calcula ed
o consecu i e lag imes τ=k∆ (k= 1..(N−1)) acco ding o
MSDT(τ) = ∆ (k∆ )2T=1
N−k
N−k
X
n=0
[ (n∆ )− ((n+k)∆ )]2(2.21)
He e h·iTsymbolizes ime-a e aging o e he espec i e lag ime. The al e na i e
ensemble-a e age calcula ion is gi en as ollows:
MSDE(τ) = ∆ (k∆ )2E=1
Nens
Nens
X
m=1
[ m(k∆ )− m(0)]2(2.22)
whe e Nens deno es he numbe o ajec o ies he ensemble consis s o and m(0)
ep esen s he s a ing posi ion o each ajec o y m.
ii) Shape o a ajec o y. The a e age spa ial ex end o a ajec o y can be
es ima ed by he adii o gy a ion Rxand Ry. Hence, he gy a ion enso To a
2-dimensional ace has o be calcula ed [35, 36].
Tij =1
N
N
X
n=1
( i(n∆ )−h ii) ( j(n∆ )−h ji)(2.23)
He e, he indices i, j deno e he x- and y-componen o a posi ion ec o ( )and
he b acke s h ii ep esen he co esponding cen e o masses o he x- ace and
he y- ace.
h i,ji=1
N
N
X
n=1
i,j(n∆ )(2.24)
Diagonalisa ion o Tyields he eigen alues, i.e. he squa ed adii o gy a ion.
T=R2
x0
0R2
y(2.25)
The eigen ec o s de ines he o ien a ion o he espec i e gy a ion ellipse. The spa-
ial ex end o a ajec o y changes wi h he numbe o da a poin s and he mobili y
9
2 Theo e ical & expe imen al backg ound
o he ace pa icle. An unambiguous c i e ion ega ding he di usion beha iou
emains elusi e.
A mo e sui able alue is he asphe ici y A, p o iding a single pa ame e ha de-
e mines he shape o a andom walk [36].
A=DR2
y−R2
x2E
DR2
y+R2
x2E(2.26)
The calcula ion o A equi es a e aging o e a sub-ensemble (h·i) ha can be ob-
ained om cu ing he ajec o y in o consecu i e segmen s wi h an equal numbe
o posi ions Nseg. Fo each o he segmen s, he adii o gy a ion we e calcula ed (c .
eqn. 2.23 - 2.25) and he asphe ici y was de e mined acco ding o equa ion (2.26).
T i ial alues o he asphe ici y a e gi en o a pe ec od-like shape, whe e one
o he adii o gy a ion is 0, leading o A= 1, and o a pe ec sphe ic shape wi h
equal adii, i.e. A= 0. Fo a andom walk one inds A= 4/7[35].
iii) Cumula i e dis ibu ion unc ion. An al e na i e me hod o in es iga e
he di usion p ocess o a single pa icle is o calcula e he (disc e e) cumula i e
dis ibu ion unc ion o he squa ed displacemen s ∆ 2a a ce ain lag ime τ, i.e.
CDF(∆ 2, τ)[37, 38]. In o de o do so, he numbe o squa ed displacemen s
smalle o equal o a gi en ∆ 2is coun ed acco ding o
CDF ∆ 2, τ=X
∆ 2(τ)≤∆ 2
P∆ 2, τ(2.27)
whe e P(∆ 2, τ)deno es he empi ical dis ibu ion o he squa ed displacemen s
o a lag ime τ. This is done consecu i ely h oughou a ajec o y un il he highes
∆ 2is eached.
2.2 T acking echniques
Nume ous kinds o se ups ha e been in en ed o ollow he 2- and e en 3-dimen-
sional mo ion o single pa icles [13, 17, 18, 20, 21, 23, 24, 29, 40]. Mos ly, luo escen
echniques we e used, which equi es luo escen pa icles. The e o e, ei he dye
molecules, polyme beads ha a e loaded wi h dyes, o labeled p o eins a e possible
ace s. Fo non- luo escing pa icles, o he mic oscopy echniques, e.g. da k ield
mic oscopy is a possible me hod. In he ea ly yea s, simple ideo mic oscopy [17]
was used o ollow he 2-dimensional mo ion o single lipids and bigge molecules.
Fu he de elopmen s leads o high s anda d CCD- acking echniques ha eco d
he mo ion o a single pa icle wi h a high spa ial and empo al esolu ion [18].
Besides he common CCD- acking echnique, o he me hods eme ged, whe e he
10
2.2 T acking echniques
spa ial and empo al esolu ion as well as he o al obse a ion ime ha e been
imp o ed u he . In his sec ion I will gi e a b ie in oduc ion o he widely
used CCD- acking me hod, explain he echnique o single-pa icle o bi acking,
which was exploi ed in his hesis and gi e inally an o e iew o some al e na i e
expe imen al app oaches.
2.2.1 CCD-T acking
T acking echniques, ha use a cha ged coupled de ice (CCD) as a de ec ion uni
a e called CCD- acking, which is he mos wide-sp ead me hod o single-pa icle
acking [13, 17, 18, 20, 23, 29]. The p inciple is o eco d successi e images o
he sample wi h he mo ing pa icles. Each pa icle in a CCD-image is displayed
as a di ac ion limi ed spo , which is ypically sp ead o e some 10 pixels on he
CCD-chip. A wo-dimensional Gaussian i o he spo is applied, whe e he cen-
e o his i gi es he ac ual pa icle posi ion wi h an accu acy o be e han
he classical di ac ion limi o ligh . By doing so successi ely wi h all eco ded
ames, he ajec o ies o all he pa icles wi hin he CCD-images a e ob ained.
The ad an age o his echnique is he ela i ely simple expe imen al se up and
he high posi ion accu acy ha can be ob ained. Fu he , mul iple single pa icles
can be acked simul aneously. Ye , his expe imen al app oach o en lacks a high
empo al esolu ion, because he da a s o age o he images wi h a high in o ma ion
densi y is ime consuming. A leas sophis ica ed imp o emen s ha e o be applied
o a oid his sho coming. No only he ime esolu ion is es ic ed, also he o-
al obse a ion ime is, because he CCD-images need la ge compu e memo y. A
ideo o abou en minu es can easily exceed 100 Gigaby e o ha d d i e space. To
ci cum en hese limi a ions, o he acking echniques ha e been in en ed.
2.2.2 O bi -T acking
The basic idea o his me hod is o ocus a lase beam in he ocal plane o he
sample and le his ocal poin o a es a ound a luo escen pa icle. By acqui ing
he emission signal, which is modula ed by he equency o he o a ing lase beam,
he posi ion o he pa icle can be aced. By demodula ion o his signal he
ac ual pa icle posi ion can be calcula ed. Theo e ically his app oach was s udied
by Ende lein [41, 42] and success ully implemen ed by he g oups o G a on [40],
Mabuchi [21] and Lamb [43]. In igu e 2.1 he p inciple is shown schema ically
o wo di e en pa icle posi ions (le hand side) wi h he co esponding emission
signals ( igh hand side). Fo pa icles, ha mo e be ween he cen e and he im
o he o bi , he emission signal is pe iodically modula ed as i is depic ed in igu e
2.1 ( igh hand side). A high modula ion occu s o pa icles ha a e apa om
he cen e o he o bi (see ig. 2.1a), while i ge s weak o pa icles ha a e close
o he cen e (see ig. 2.1b). The ampli ude o he modula ion changes as a unc ion
11
2 Theo e ical & expe imen al backg ound
Figu e 2.1: Schema ic explana ion o he o bi acking echnique. All necessa y pa am-
e e s, i.e. he adius o he o bi R, he angula eloci y ω, he wid h o he ocus wand
he posi ion o he pa icle xpand yp, d awn as a ed sphe e, a e displayed. This scheme is
no d awn o scale. In a) he pa icle is a o -cen ed, whe eas i is close o he cen e o
he o bi in b). On he igh hand side o he igu e he co esponding emission in ensi ies
o he pa icle a e ske ched. Adap ed om publica ion P1.
o he posi ion. Hence, om he demodula ion o he emission, he di ec ion as well
as he absolu e dis ance om he cen e o he o bi can be calcula ed.
The emission signal I( )depends on he ela i e posi ion o he pa icle (xpand yp)
and he o a ing lase ocus and can be w i en as ollows [41, 42]:
I( ) = I0exp −2
w2(xp−Rcos(ω ))2exp −2
w2(yp−Rsin(ω ))2+Ib(2.28)
He e, I0is he maximum emission in ensi y, i.e. he pa icle is a he posi ion o he
ocal spo , R he adius o he o bi , w he 1/e2-wid h o he lase ocus, ω= 2πν
he angula eloci y o he o a ing ocus and Ib he backg ound in ensi y. By using
lock-in echniques, he posi ion o he pa icle can be calcula ed om his emission
12
2.2 T acking echniques
signal acco ding o he equa ions
xp( ) = w2
2RRT
0I( ) cos(ω )d
RT
0I( )d , yp( ) = w2
2RRT
0I( ) sin(ω )d
RT
0I( )d .(2.29)
The in eg a ion bounda ies a e om 0 (begin o he posi ion de e mina ion) o
T, which is a ime ha co esponds o a mul iple o he cycling ime. Once he
posi ion o a pa icle wi h espec o he cen e o he o bi is calcula ed, a eedback
mechanism has o be implemen ed, ha es o es he pa icle back in he cen e o
he o bi . Gene ally, wo possibili ies exis o do so. On he one hand, his is beam
scanning, whe e he whole ligh o bi is mo ed acco ding o he new posi ions o he
pa icle by applying he co esponding eedback signals o he op ical elemen s ha
a e esponsible o he gene a ion o he o bi . This a e ypically scanning mi o s
o acous o op ical de lec o s. On he o he hand, his is sample scanning, whe e
he eedback signals a e applied o ha dwa e elemen s, ha mo e he whole sample,
which is mos ly ealised by a piezos age. The eedback loop is as ollows: acqui e
emission signal, calcula e posi ion, es o e pa icle in he cen e o he o bi , and so
on. Doing so successi ely, he whole ajec o y o a pa icle can be econs uc ed.
This new se up o he eco ding o single-pa icle ajec o ies is qui e powe ul as
i combines he spa ial accu acy o a CCD came a wi h he empo al esolu ion o
a single pho on de ec o .
2.2.3 Al e na i e app oaches
In he ollowing some al e na i e app oaches and new de elopmen s in he esea ch
ield o single-pa icle acking a e in oduced. All o hem ha e in common, ha
hey wan o push o wa d he empo al and spa ial accu acy o he posi ion de-
e mina ions.
In he esea ch g oup o W. E. Moe ne wo ingenious me hods ha e been in en ed.
The i s one is called he ABEL (An i-B ownian ELec okine ic) ap [15, 44, 45].
He e, a pa icle is apped be ween ou elec odes. As soon as he pa icle mo es
owa ds one o hem a ol age o he co esponding elec ode is applied ha in-
duces a low ield in he medium whe e he pa icle is di using and which pushes
he pa icle back o he cen e o he ap. The eedback mechanism is implemen ed
by eco ding he mo ion o he pa icle wi h a CCD-came a. In a newe e sion o
his se up, he a o e explained o bi acking is used o eco d he posi ion o he
pa icle [46], which is as e .
The second se up, ha was de eloped in he g oup o Moe ne modula es he poin
sp ead unc ion (PSF) [24, 47]. By a spa ial ligh modula o a double-helix PSF
is gene a ed, whe e he pa icle o in e es is loca ed wi hin ha double helix. In
o he wo ds, a spa ial ca i y o ligh su ounds he pa icle. A mo emen in any
di ec ion is de ec ed by he emission signal o he luo escen pa icle. Compu e
13
2 Theo e ical & expe imen al backg ound
so wa e is hen able o econs uc he posi ion in h ee dimensions.
Ano he me hod was pu o wa d by he g oup o H. Yang, in which he emi ed
ligh o a pa icle was spli ou imes by p ism mi o s [22]. Each pa o he emis-
sion was acqui ed by a sepa a e a alanche pho o diode (APD). F om he in ensi y
a ios be ween he ou de ec o s he posi ion o he pa icle can be ex ac ed. By
a 3-dimensional piezos age his mo ion is compensa ed. The eedback signal o he
piezo is used o de e mine he ajec o y.
In p inciple all echniques ha e in common, ha hey p obe he space a ound a
pa icle, ei he by apping he pa icle, by modula ion o he exci a ion ligh , o
by spli ing he emission signal. F om he acqui ed signal o he ace pa icles,
eedback mechanisms es o e he ini ial posi ion. Successi e calcula ions and he
use o sophis ica ed algo i hms p o ide he econs uc ed ajec o y.
14
The good hing abou science is ha i ’s
ue whe he o no you belie e in i .
Neil deG asse Tyson
3 Ma e ials
Fo he cha ac e isa ion and acking expe imen s, wo sizes o beads (20 nm and
50 nm) loaded wi h wo di e en luo escen dye molecules (nile ed and hodamine)
and ou kinds o sample subs ances (poly- inyl-alcohol (PVA), glyce ol, suc ose
and dex an) we e used. Among hese, he combina ion o 20 nm sized beads in
PVA is used o s a ic expe imen s, whe e he beads a e immobile. Fu he , ano he
i e combina ions o beads/samples we e used o he cha ac e isa ion o he se up
including he de e mina ion o he dynamic accu acies and o he in es iga ion o
anomalous di usion. In he ollowing pa ag aphs I will desc ibe he cha ac e is ics
o he beads as well as he p epa a ion o he sample. The de ailed p epa a ion
o he bead/sample-mix u es o he espec i e expe imen s can be ound in he
co esponding publica ions (publica ion P1: 20 nm beads in PVA and in glyce ol,
publica ion P2: 20 nm beads in glyce ol, publica ion P3: 50 nm beads in suc ose
and dex an).
3.1 Beads and samples
In o de o cha ac e ise he acking pe o mance o he expe imen al se up and
o measu e di usion p ocesses, wo di e en sizes o dye labeled beads we e used.
On he one hand hese a e 20 nm (diame e ) sized polys y ene beads (Molecula
P obes) ha a e loaded wi h he dye nile ed and a e u he s abilized wi h ca -
boxyla e g oups a ached o he su ace o a oid agg ega ion. Acco ding o he
manu ac u e , he beads a e suspended in wa e a a concen a ion o 20 mg/ml.
On he o he hand hese a e polys y ene-based la ex mic osphe es (Polysciences)
wi h a diame e o 50 nm, ha a e labeled wi h he dye hodamine. He e, he con-
cen a ion o he s ock solu ion is 200 mg/ml wi h no addi ional s abilize s. The
molecula s uc u e as well as he no malized luo escence exci a ion and emission
spec a o he wo highly luo escen molecules a e displayed in ig. 3.1a,b. The
spec a ha e been eco ded wi h a comme cial luo escence spec ome e (Ca y
Eclipse, Va ian). The e o e he espec i e beads we e dissol ed in millipo e wa e
and his solu ion was hen illed in cu e es. The luo escence exci a ion and emis-
15
4 Simula ions
was se o w= 270 nm, he adius o he o bi o R= 190 nm, which is he op imal
alue acco ding o [50] whe e R=w/21/2was de e mined, and he equency o
he ocus o a ion was se o ν=ω/2π= 1 kHz. The sampling ime was δ = 2 µs,
which esul s o a numbe o sampling in e als o NS= ∆ /δ = 2000. This co e-
sponds o 4 pe iods o o a ion o he ocus. (La e in his hesis we will see, ha
his pa ame e s ma ch he expe imen al condi ions.) F om he numbe o simula ed
pho ons, he mean emission in ensi y o he ace esul s o I= 57kcps and he
backg ound in ensi y o Ib= 0.5 kcps (kcps: kilo coun s pe second).
To calcula e he coo dina es xcand ycacco ding o he equa ions (4.3), he simu-
la ed signal Sn,poiss as well as he alues Sn,poiss cos(ωnδ )and Sn,poiss sin(ωnδ )
a e accumula ed du ing each 4 ms pe iod (Snin eqn. (4.3) was eplaced by
Sn,poiss). By successi ely epea ing his p ocedu e, he ajec o y o he pa i-
cle, i.e. ( )=(x( ), y( )) can be econs uc ed. Figu e 4.1c displays bo h, he
ajec o y gene a ed wi h he Lange in equa ion (black) and he ajec o y, ha is
econs uc ed. Beside some sligh de ia ions acco ding o he posi ion unce ain-
ies (noise and mo emen du ing signal acquisi ion), he “simula ed econs uc ed”
ajec o y ma ches he “simula ed eal” one.
To summa ize, i s a “ as ” ajec o y on a ime scale o δ = 2 µsis gene a ed, ha
co esponds o he mo emen o he pa icle du ing he in eg a ion ime ∆ . Ou
o he posi ions o his ajec o y, he signal Sn,poiss is calcula ed, accumula ed and
mul iplied wi h a cosine and sin unc ion, necessa y o he calcula ion o he pa -
icle posi ion. Finally, he posi ions xcand yca e calcula ed acco ding o 4.3. All
simula ions we e pe o med using home-w i en Ma lab p og ams. The sou ce code
o he gene a ion o he “simula ed eal” and “simula ed econs uc ed” ajec o ies
is gi en in he appendix A. This simula ions a e an e ec i e ool
• o scan he pa ame e ange (e.g. o a ion equency ν, ime esolu ion ∆ ,
o bi adius R, e c.), which is app op ia e o he expe imen o achie e he
bes acking pe o mance,
• o compa e he esul s o simula ions and expe imen s,
•and o s udy he in luences o se e al expe imen al a i ac s (e.g. noise and
posi ion a e aging) o he in luences o di usion pa ame e s (e.g. iscosi y
ηo bead size a) on he analysis o he mean squa ed displacemen and he
concomi an in e p e a ions.
4.2 Spa io- empo al accu acy
Wi h he use o he a o e desc ibed simula ions o he econs uc ed ajec o ies
he acking pe o mance by means o he spa ial and empo al accu acy is s udied
22
4.2 Spa io- empo al accu acy
as a unc ion o he emission in ensi y and as a unc ion o he di usion coe i-
cien . S a ing poin is he simula ion o he numbe o pho ons Sn,poiss ha is
used by he eedback algo i hm o de e mine he posi ion. The heo e ical and
Poisson dis ibu ed numbe o pho ons, i.e. Snand Sn,poiss, du ing a ime in e -
al ha co esponds o he ime esolu ion o ∆ = 4 ms o a mo ing bead is
displayed in ig. 4.2a, ep esen ed in ed and black, espec i ely. Fo a be e i-
sualiza ion a low emission signal o I= 30 kcps was chosen, o he wise he Poisson
dis ibu ed numbe o pho ons a e oo c owded. The di usion coe icien was se
o D= 43 ×10−3µm2/s. The signal o he heo e ical numbe o pho ons Sn
is pe iodic, bu hea ily luc ua ing. The eason is he mo emen o he pa icle,
du ing he in eg a ion ime ∆ . As al eady men ioned, he emi ed pho ons depend
on he posi ion o he pa icle wi hin he o bi . As soon as he mo emen happens
on a as e ime scale han he in eg a ion ime, he pe iodic emission luc ua es
acco dingly. Because he Poisson dis ibu ed numbe o pho ons is calcula ed om
he heo e ical numbe o pho ons, he desc ibed beha iou (pe iodic and luc ua -
ing emission) is ca ied o wa d o Sn,poiss. The densi y o pho ons (Sn,poiss) is high
when Snis high.
Howe e , despi e he luc ua ions o he emi ed pho ons, he eedback algo i hm
is s ill able o calcula e he posi ion o he pa icle, wi h an accu acy, ha is de e -
mined by pho on s a is ics (noise) and posi ion a e aging du ing da a acquisi ion.
Figu e 4.2b shows wo 1-dimensional “simula ed econs uc ed” ajec o ies wi h
N= 5000 da a poin s o he ime esolu ions ∆ = 0.5 ms (blue) and ∆ = 9.0 ms
( ed), bo h wi h a mean emission in ensi y o I= 157 kcps and a di usion coe i-
cien o D= 400 ×10−3µm2/s. I is i ial, ha he pa icle simula ed wi h he
highe ime esolu ion o ∆ = 9.0 ms can mo e a la ge dis ance, because he o al
obse a ion ime o he mo emen is =N∆ and hence, longe han o a pa -
icle simula ed wi h ∆ = 0.5 ms. Howe e , a he han he co e ed dis ance, he
posi ion e o in he ajec o y is conside ed. Despi e a la ge numbe o pho ons,
which a ou s he posi ion de e mina ion, he “simula ed econs uc ed” ajec o y
de e mined wi h he ime esolu ion o ∆ = 9.0 ms appea s mo e noisy han he
ajec o y de e mined wi h ∆ = 0.5 ms. Ob iously, he mo ion du ing he acqui-
si ion ime is dominan . Dependen on he emission in ensi y and he mobili y o
he pa icle (di usion coe icien ), bo h e ec s con ibu e o he spa ial accu acy.
To de e mine his accu acy, he posi ional e o o a ajec o y has o be calcula ed.
In o de o do so, he oo -mean-squa e ( ms) e o be ween he posi ions o he
“simula ed eal” ajec o y, xc, and he posi ions o he “simula ed econs uc ed”
ajec o y, xp, has o be de e mined.
σx=σ ms =
u
u
1
N
N
X
i=1
(xci −xpi)2(4.4)
23
4 Simula ions
Figu e 4.2: Simula ion o he dynamic posi ion accu acy. a) heo e ical ( ed) and Pois-
son dis ibu ed (black) numbe o emi ed pho ons wi hin one ime bin o ∆ b) calcula ed
posi ion as a unc ion o he numbe o posi ion de e mina ions N o a ime esolu ion
o ∆ = 0.5 ms (blue) and ∆ = 9.0 ms c),d) dynamic posi ion accu acy o a high mean
emission in ensi y (c) and a low mean emission in ensi y (d) as a unc ion o he ime
esolu ion o a ious di usion coe icien s be ween he s a ic case o D= 0 µm2/sand a
as pa icle mo ion o D= 6.4µm2/s. The wo ajec o ies, simula ed in b) a e indica ed
in d) by he numbe s 1 and 2. Mo e de ails see ex .
In ig. 4.2c,d he 1-dimensional spa ial accu acy σxas a unc ion o he ime esolu-
ion ∆ o di usion coe icien s be ween D= 0 µm2/s(s a ic) and D= 6.4µm2/s
a e shown o a low ( ig. 4.2c) and a high ( ig. 4.2d) mean emission in ensi y o
I= 30 kcps and I= 157 kcps, espec i ely. The low emission signal co esponds
o a ypical emission o a single molecule, while he high signal co esponds o he
ypical emission o a luo escen bead. Quali a i ely, bo h g aphs show he same
esul s. Wi h an inc easing alue o he ime esolu ion (in e e yday language: he
ime esolu ion ge s wo se) he spa ial accu acy apidly ge s be e un il a mini-
mum alue is eached, om which i g ows slowly. To explain his beha iou le
us conside a gi en di usion coe icien . Fo e y low ime esolu ions he pa icle
can be ega ded as s a ic and he posi ion accu acy is ge ing be e acco ding o
σx∝1/√∆ . This is e iden o he spa ial accu acy o he s a ic pa icle (black
da a poin s in ig. 4.2c,d). Fo highe alues o ∆ he posi ion o he pa icle ge s
blu ed due o i s mo emen du ing one ime esolu ion s ep, i.e. he posi ional e -
24
4.3 Loosing a pa icle
o is g owing. Hence, i exis s a minimum, whe e he bes spa ial accu acy can be
achie ed. Fo as pa icles, i.e. high di usion coe icien s, he posi ion a e aging
e ec is mo e p onounced han o slow pa icles. A common compa ison is he di -
usion o a 20 nm bead in wa e , which has a di usion coe icien o D= 21.5µm2/s.
Acco ding o he p esen ed simula ions, he pa icle can no be acked o a leas
a e y high ime esolu ion is equi ed. Fo a success ul acking expe imen , he
pa icle has o be la ge and/o he iscosi y o he su ounding luid has o be
highe .
Howe e , his wo g aphs ( ig. 4.2c,d) se e as an o ien a ion o se ing sui able
expe imen al pa ame e s and o know he heo e ical limi s o he acking pe o -
mance. In ad ance, he expe imen alis can check i he pa icles in he sys em
he wan s o in es iga e can be acked and i so, which spa ial accu acy can be
expec ed. A p oblem occu s, i he simula ed accu acies a e compa ed wi h expe -
imen al ones. In an expe imen he “ eal” posi ion, which was necessa y o his
calcula ions, is no accessible and o he me hods o he de e mina ion o he spa-
ial accu acy has o be used. One possibili y is discussed in de ail in publica ion
P1, whe e he o se o he MSD-cu e was used as an indica o o he spa ial
accu acy.
4.3 Loosing a pa icle
As long as he pa icle can be acked, i s ays inside he ligh o bi , gene a ed by
he o a ing ocus, and is es o ed o he cen e o he o bi a e e e y ime pe iod
∆ . Hence, his means, ha he emission in ensi y o he pa icle is kep cons an
du ing acking. As soon as he pa icle ge s los , he emission in ensi y dec eases
immedia ely o he backg ound in ensi y. Possible easons o loose a pa icle a e a
weak emission signal o a high mobili y.
The si ua ions o a success ul and a ailed acking we e simula ed o a bead wi h
adius a= 10 nm a a empe a u e o T= 294 K. The op g aph o ig. 4.3a
displays he 1-dimensional “simula ed eal” (black) and “simula ed econs uc ed”
( ed) ajec o y o a slow di using bead wi h a high emission in ensi y. The pa am-
e e s we e se o I= 157 kcps and η= 1.2 Pa s, which co esponds o a di usion
coe icien o D= 17.95 ×10−3µm2/s. The econs uc ed posi ions ollow nicely
he eal ace. A de ailed iew is shown in he inse o ig. 4.3. The co esponding
in ensi y ace (bo om g aph o ig. 4.3a) yields he expec ed cons an emission
in ensi y, i.e. he pa icle is acked con inuously.
To simula e a acking expe imen , whe e he pa icle ge s los , a as bead wi h
a low emission in ensi y is used. The ajec o ies ( eal and econs uc ed) a e
shown in he uppe pa o ig. 4.3b. He e, I= 21 kcps and η= 0.1 Pa s, i.e.
D= 215.3×10−3µm2/s, we e chosen. The di usion is oo as and he emission is
oo low, o ollow he mo emen o he pa icle. A a ime o abou 2.5 s he posi-
25
4 Simula ions
Figu e 4.3: Resul s o he simula ions o a as and high emi ing pa icle (a) and a slow
and low emi ing pa icle (b). The op g aphs display he ime aces o he ue (black) and
he calcula ed posi ions ( ed) and he bo om g aphs o he in ensi y ime ace. The inse
in (a) is an enla ged iew o he wo ajec o ies. The blue dashed line in (b) indica es he
ime, whe e he pa icle is los .
ion can no be calcula ed anymo e and as men ioned abo e, he emission in ensi y
d ops ins an ly o he backg ound le el (indica ed by he blue dashed line in ig.
4.3b). This happens o example, i he bead mo es ha as , ha i exceeds he
im o he o bi o i he emission is no high enough o calcula e he posi ion. This
example explains nicely he c i e ion o a e mina ion o a acking expe imen .
As soon as he emission signal d ops o he backg ound le el he measu emen is
s opped.
Wi h a ough es ima ion, a limi ing case o he 2-dimensional acking pe o mance
can be de e mined o a high emission signal. The pa icle can de ini ely no be
acked, i he mean displacemen be ween wo consecu i e posi ions is la ge han
he alue o he adius o he o bi R, i.e. i
ph∆ (k∆ )2i ≥ R o k = 1 (4.5)
⇔4D∆ ≥R2
⇒D≥R2
4∆
By chance i could happen, ha he pa icle can be acked pa ially, bu wi h
a mean displacemen la ge han he adius o he o bi , a pe manen acking is
no possible. Fo he pa ame e s ∆ = 4 ms and an o bi adius o R= 190 nm,
he limi ing di usion coe icien is de e mined o D= 2.26 µm2/s. This can also
be compa ed wi h he alues gi en in ig. 4.2d. Sys ems, ha exhibi s a la ge
di usion coe icien han he de e mined one, can no be acked wi h his se ings.
Coming back o he example o a 20 nm bead in wa e , success ul acking would
26
4.3 Loosing a pa icle
equi e a ime esolu ion o abou ∆ = 0.4 ms, acco ding o equa ion 4.6.
In he ollowing sec ion he pa ame e limi a ions o he expe imen al se up a e
in oduced. A his poin I wan o men ion, ha a he p esen s age o he se up
i is no possible o ack a 20 nm bead in wa e . The empo al accu acy has o be
imp o ed by a ac o o 10.
27
A lea ning expe ience is one o hose hings
ha say, “You know ha hing you jus
did? Don’ do ha .”
Douglas Adams
5 The acking se up
The main wo k o his hesis was he de elopmen o a new expe imen al se up
ha is capable o measu ing single-pa icle ajec o ies wi h a high spa io- empo al
esolu ion. I s a ed om an emp y op ical able and ended wi h he success ul im-
plemen a ion o single-pa icle acking expe imen s. Among he a o e men ioned
a ious echniques, he o bi acking me hod wi h sample scanning was chosen.
This enabled he eco ding o successi e posi ions o a pa icle o mo e han 10
minu es wi h a ime esolu ion o 4 ms, esul ing in ajec o ies wi h mo e han
1.5×105posi ions. A spa ial esolu ion o be e han 10 nm was achie ed. How-
e e , p io o he acquisi ion o e aluable ajec o ies, nume ous cha ac e isa ion
expe imen s ha e o be pe o med, o iden i y possible sho comings ha dis u b
he measu emen s. This wo k was subs an ial, so ha i was sui ed o a publica-
ion in he Jou nal o he op ical socie y o Ame ica A (JOSA A) wi h he i el
“Se up o single-pa icle o bi acking: a i ac s and co ec ions”, which can be
ound in pa II (publica ion P1) and which is he majo publica ion o his hesis.
I includes he desc ip ion o he expe imen al se up, i s ull cha ac e isa ion and
he i s success ul acking expe imen s wi h his se up. In he ollowing sec ions
he con en o his publica ion is summa ized and addi ional esul s on he spa ial
acking accu acy o a s a ic pa icle a e gi en.
5.1 Op ics & Ha dwa e
In p inciple, he expe imen al se up consis s o a home-build con ocal luo escence
mic oscope, a de ec ion uni , and a lase beam de lec ion uni , which is he co e pa
o he se up. Two acous o-op ical de lec o s (AOD) guide he lase beam (514 nm)
on a ci cula pa h. This ligh o bi is p ojec ed in o he mic oscope and is u he
e lec ed owa ds a wa e -imme sion objec i e, ha ocuses he o bi in o he plane
o he sample, whe e he dye loaded pa icles a e exci ed. The emission signal o
he pa icles is hen collec ed by he same objec i e and di ec ed o ei he a CCD
came a o wide ield imaging o o an a alanche pho o diode (APD) o acking
wi h high ime esolu ion. The op ical pa hway was calcula ed wi h he aid o ay
29
5 The acking se up
ans e ma ix analysis, because he mu ual dis ances be ween all op ical elemen s
ha e o be as exac as possible. To ope a e he se up, an ingenious home-w i en
p og am was used o implemen he acking algo i hm o he calcula ion o he
pa icle posi ions. This was done wi h a p og ammable measu ing ins umen ,
ha is also esponsible o he gene a ion o all ou pu signals, e.g. he piezo and
AOD signals, and he acquisi ion o he emission signal, eco ded by he APD. The
use o only one single communica ion ins umen be ween he PC and he se up
made i possible o a oid ha dwa e, like equency gene a o s, lock-in ampli ie s o
coun e ca ds o he PC, and o ha e a pe ec ime synch onisa ion be ween all
ou pu and inpu signals, which is a p e equisi e o his acking me hod. Only i
he ou pu signals o he gene a ion o he ligh o bi a e synch onized wi h he
da a acquisi ion o he emission signal, a meaning ul posi ion o he pa icle can
be calcula ed. The p esen ed o bi acking echnique is able o econs uc he
2-dimensional mo ion o he pa icle as long as his mo ion akes place in he plane
o he o bi , which is also he xy-plane. The p oblem is now o keep he pa icle
in his plane, e.g. by compensa ing o he pa icle mo ion in he 3 d dimension (z-
axis). This was done by implemen ing a z- acking algo i hm, which was de eloped
oge he wi h he diploma s uden S e an Hain. The piezo wi h he moun ed sample
is wobbled up and down along he z-axis, i.e. also he pa icle is mo ed up and down
and pene a es he o bi once in each di ec ion. Fo e e y up (down) mo emen he
z-posi ion a which he pa icle ea u es he highes emission is s o ed and se es as
he new o igin o he subsequen down (up) mo emen . This enabled he acking
o e an ex ended pe iod o ime. Typically, acking imes o 10 minu es we e
used o analyse he di usion p ocesses, bu i is possible o ollow he mo ion o
he ace o mo e han 30 minu es. A he p esen s age o he se up he o al
eco ding ime depends only on bleaching o he ace beads and on he maximum
scan ange o he piezo (100 µmpe axis). The impo an pa ame e s o he se up
a e he o bi adius R, he equency o he ocus o a ion νand he numbe o
o a ion pe iods P, ha we e used o se ing he ime esolu ion (∆ =P/ν) and
he pa ame e s o he z- acking algo i hm. The la e ones a e a numbe o s eps
wi h a de ined s ep size o he up and down mo emen s o he piezo. These alues
ha e o be op imized sepa a ely o e e y new sample. Combining simula ions,
li e a u e esea ch and cha ac e isa ion measu emen s allowed o de ine op imal
se ings o he emaining pa ame e s. The alues we e ound o be R= 190 nm,
ν= 1 kHz and P= 4. Ha ing a sui able se o pa ame e s, a cha ac e isa ion is
necessa y o lea n he capabili y o he se up.
5.2 A s a ic pa icle
The mos basic cha ac e isa ion is he in es iga ion o a s a ic pa icle. I is use ul
o s udying he posi ion accu acy wi hou any dis u bing e ec s due o he mo ion
30
5.2 A s a ic pa icle
o he pa icle. Fi s his si ua ion is again simula ed o know he heo e ical
p edic ions, which a e hen compa ed wi h expe imen al da a. The e o e, in he
ollowing wo subsec ions I will explain he simula ed as well as he expe imen al
de e mined posi ion accu acy o a s a ic pa icle, ollowed by a compa ison.
5.2.1 Simula ion
The p ocedu e o he simula ion is simila o he al eady desc ibed one in chap e
4, wi h he di e ence, ha he “simula ed eal” ajec o y is now a ixed posi ion.
He e, i was se o (xp, yp) = (0,0), which co esponds o he cen e o he o bi .
Wi h he eedback mechanism swi ched on, i does no ma e a which s a ing
posi ion wi hin he o bi he pa icle is placed. Because a e he i s posi ion
calcula ion, he eedback loop es o es he pa icle back in he cen e o he o bi
wi h an accu acy, ha is de e mined by acquisi ion limi s (noise and posi ion a e -
aging du ing da a acquisi ion). The simula ion o he heo e ical and he Poisson
dis ibu ed numbe o pho ons, i.e. Snand Sn,poiss a e shown in ig. 5.1a du ing
he acquisi ion ime o ∆ = 4 ms. A pa icle a he (0,0) posi ion should esul in
a cons an alue o he heo e ical numbe o emi ed pho ons, because in an ideal
case, also xsis ze o and he exp ession xp−xsin equa ion (4.2) anishes.
Sn=δ S0exp
−2
w2(−Rcos(ωnδ ))2+ (−Rsin(ωnδ ))2
| {z }
=R2
+Sbδ (5.1)
=δ S0exp −2R2
w2+Sbδ =cons .
The eason o he pe iodic signal a e he pho on s a is ics and he eedback mecha-
nism. Due o he Poisson dis ibu ed emission, he calcula ed posi ion is no exac ly
ze o. Via xs his non-ze o posi ion is ed back o he gene a ion o he numbe o
pho ons. Hence, he exp ession xp−xsin equa ion 4.2 is non-ze o, leading o a
weak pe iodic signal Sn. The emission o pho ons, i.e. Sn,poiss s ays mo e o less
cons an , because he absolu e alues o Sni sel as well as he ampli ude a e qui e
small.
F om he simula ed numbe o pho ons, he posi ion o he pa icle is econs uc ed.
Figu e 5.1b displays he x-posi ion ime- ace o N= 5000 posi ion de e mina-
ions, o a ime esolu ion o ∆ = 0.5 ms (blue) and ∆ = 9.0 ms ( ed). Bo h
simula ions we e pe o med o a mean emission in ensi y o I= 157 kcps. The
high luc ua ions in he posi ion a a ime esolu ion o ∆ = 0.5 ms compa ed o
∆ = 9.0 ms a e asc ibed o he educed numbe o pho ons, ha can be acqui ed
du ing he ime esolu ion o ∆ = 0.5 ms. A highe pho on coun a e leads o a
mo e accu a e posi ion de e mina ion. To quan i y his posi ional noise, he e o
in calcula ing a posi ion in one dimension, which also de ines he spa ial accu acy,
31
6 Single-pa icle acking: esul s & discussion
Figu e 6.1: a) Example o a ajec o y o a 20 nm sized bead in pu e glyce ol, eco ded o
= 608 s (N= 1.52 ×105da a poin s). b) Ex ended iew o a sequence wi h Nseg = 2000
posi ions ( = 8 s).
6.1 Analysing di usion based on single-pa icle
ajec o ies
F om se en di e en beads each wi h a nominal diame e o 20 nm di using in
pu e glyce ol, ajec o ies o abou 1.5×105posi ions we e eco ded. Fo e e y
un a pa icle was selec ed om he wide ield image ha was su icien ly sepa a ed
om o he pa icles and om he su ace o he co e slips. An example o a ypical
ajec o y is displayed in ig. 6.1a. I ep esen s 1.52×105da a poin s eco ded wi h
a ime esolu ion o ∆ = 4 ms and co esponds o an elapsed ime o 608 s which is
indica ed by he colou code whe e blue co esponds o he begin o he ajec o y
and ed o i s end. In he ollowing he analysis o he mean squa ed displacemen s
and h ee al e na i e app oaches we e discussed on one ep esen a i e ajec o y.
6.1.1 Mean squa ed displacemen analysis
F om he eco ded ajec o y ( ) = (xp( ), yp( )), consis ing o Nposi ions, he
ime-a e aged MSD a e a ime lag τ=k∆ is calcula ed acco ding o equa ion
2.21 [34]. Fo he ajec o y displayed in ig. 6.1a he inse in ig. 6.2a shows he
ull MSD as a unc ion o he lag ime. Fo lag imes τ < 50 s he MSD ea u es
a linea inc ease ollowed by s ong luc ua ions a longe imes ha e lec he
p og essi ely dec easing a e aging. The e o e he analysis o he MSD cu es is
usually limi ed o he i s ew da a poin s. Ha ing mo e han 105da a poin s a
38
6.1 Analysing di usion based on single-pa icle ajec o ies
Figu e 6.2: a) Time-a e aged MSD o he ull ajec o y shown in ig. 6.1a. The inse
displays he ull MSD, while he main igu e shows he i s 50 da a poin s. The ed
line co esponds o a linea i o he da a poin s. b) Double-loga i hmic plo o he da a
shown in a). The ed line co esponds o a linea i wi h a slope o 1.05 c) MSDs o 76
sub ajec o ies ha esul om cu ing he ull ajec o y in o pieces o 2000 da a poin s
each. Simila ly o pa a) o he igu e, he inse shows he ull MSDs and he main igu e
displays he i s 50 da a poin s o each MSD. The do ed g een line co esponds o he
MSD cu e shown in a). d) Dis ibu ions o he di usion coe icien and he anomaly
pa ame e as ob ained om he MSDs shown in c).
hand, he MSD shown in ig. 6.2a was es ic ed o he i s 50 da a poin s co e-
sponding o τ= 0.2 s. I is wo h o no e ha e en he 50 h da a poin ep esen s
an a e age o e mo e han 150000 en ies. The ull line in ig. 6.2a co esponds
o a linea i , which is in e y good ag eemen wi h he da a and om which a
di usion coe icien o hDiT= 17.19 ×10−3µm2/sis ex ac ed. Fo his analysis,
he i s da a poin was neglec ed due o emaining expe imen al a i ac s (oscil-
la ions o he piezo, de ails see publica ion P1). The no a ion hDiTwas used o
indica e ha his alue s ems om a ime-a e aged MSD acco ding o equa ion
2.21. This numbe can be compa ed wi h he p edic ion om he S okes-Eins ein
ela ion DSE =kBT/6πηa [5]. Fo T= 294 K,η= 1.2 Pa ·s[39], and a mean bead
adius o a= 10 nm as gi en by he manu ac u e , DSE = 17.94 ×10−3µm2/swas
ound. Al hough he ag eemen be ween hDiTand DSE is be e han 5%, one can
39
6 Single-pa icle acking: esul s & discussion
a gue ha he disc epancy migh e lec de ia ions o he adius o he bead om
he alue o 10 nm. Changing he poin o iew and aking he heo e ical (S okes-
Eins ein) di usion coe icien as “ eal”, a= 10.4 nm o he ac ual size o he bead
is ound. Figu e 6.2b shows he same da a on a double loga i hmic plo ha can be
i ed wi h a linea slope o hαiT= 1.05, which is in line wi h he expec ed α= 1
o B ownian mo ion and jus i ies he linea i o he MSD da a as explained in
he heo y pa (sec ion 2.1.3).
O en i is no possible o measu e ha long ajec o ies due o bleaching p o-
cesses o he ace pa icle, due o weak emission signals o simply, because he
pa icle mo es ou o he de ec able a ea. In o de o mimic o ha e only sho e
ajec o ies wi h less da a poin s he long ajec o y was chopped in o 76 sub a-
jec o ies, also called segmen s, o 2000 da a poin s each (co esponding o a ime
ange o 8 s), see igu e 6.1b. F om hese sub ajec o ies he ime-a e aged MSDs
we e calcula ed. The esul o all 76 da a se s is shown in he inse o igu e 6.2c
and e eals e y la ge a ia ions o he MSDs wi h espec o each o he . Fo he
same easons as de ailed abo e, igu e 6.2c displays only he i s 50 da a poin s o
he MSDs om all 76 sub ajec o ies. A bundle o linea MSDs ha a y in slope
was ound. Since hese da a ha e been eco ded om he same indi idual polyme
bead i can be excluded ha his sp ead e lec s a ia ions in he bead size. The
di e en slopes a he esul om he educ ion o he s a is ical weigh o he
ime-a e aged MSDs o he sub ajec o ies. This is in line wi h he ac ha he
ime-a e aged MSD om he o al ajec o y o 1.52 ×105da a poin s uns nicely
h ough he cen e o he bundle o pa ial MSDs, as shown in igu e 6.2c by he
do ed g een line o compa ison. Analysing he slopes o he indi idual MSDs in
e ms o a di usion coe icien and an anomaly pa ame e yields he dis ibu ions
shown in igu e 6.2d. The mean (empi ical s anda d de ia ion) o he di usion
coe icien is hhDiTiE= 17.15 ×10−3µm2/s(sD= 2.39 ×10−3µm2/s) and o he
anomaly pa ame e hhαiTiE= 1.01 (sα= 0.05). The no a ion hh·iTiEsymbolizes
ha he da a ha e been e alua ed om (sho ) ime-a e aged MSDs ha ha e been
a e aged o e an ensemble (he e 76 sub aces).
The his og ams p esen ed in ig. 6.2d can be in e p e ed as he empi ical p obabili y
densi y unc ions (PDFs) o measu e a dis inc ange o alues o hese pa ame e s.
Ma hema ical s a is ics ells us, ha he bes es ima o o he “ eal” alue o he
di usion coe icien is p o ided om he i s momen o his p obabili y densi y
which co esponds o hhDiTiE. The good ag eemen be ween he nume ical alues
ound o hDiT, i.e. he esul om he ull ajec o y o 1.5×105da a poin s, and
hhDiTiEcon i ms ha he s a is ical sho comings o he ajec o ies wi h only
2000 da a poin s a e a e aged ou in he long ajec o y. Howe e , i shows as
well ha an accu a e de e mina ion o he di usion coe icien equi es ei he a
e y long ajec o y o knowledge abou he (empi ical) PDF o he ou come o
an expe imen on a sho e ajec o y. A single expe imen on a sho ajec o y is
no su icien and he esul o he di usion coe icien can di e by up o a ac o
40
6.1 Analysing di usion based on single-pa icle ajec o ies
Figu e 6.3: Ensemble-a e aged mean squa ed displacemen o he cu sequences o he
single ajec o y shown in ig. 6.1a. The inse displays he double-loga i hmic plo ed
da a, whe eas he main igu e shows he linea ones. The ed solid lines a e linea i s o
he da a poin s.
o 2.
Assuming ha e godici y is p ese ed, he cu sequences o he sub-ensemble can
also be used o calcula e he ensemble-a e aged MSD acco ding o he equa ion
2.22. This me hod is an al e na i e way o in es iga e a di usion p ocess, bu
is less sui ed o single ajec o y expe imen s, because he numbe o aces o
a s a is ical ele an ensemble, eco ded unde iden ical expe imen al condi ions
is mos ly oo small o yield p ope esul s. Howe e , o a compa ison wi h he
p esen ed esul s also Dand αwe e de e mined wi h his me hod. F om line
i s ( ed solid lines in ig. 6.3) o he linea as well as he double-loga i hmic da a
hDiE= 17.06×10−3µm2/sand hαiE= 0.97 was achie ed (c . ig. 6.3). Again bo h
alues i qui e well wi h he alues de e mined by he abo e men ioned analysing
me hods. Sligh di e ences s em om he a he weak ensemble. No iola ion o
he e godic p inciple is obse ed.
The p esen ed MSD analysis was inally pe o med o all he se en measu ed a-
jec o ies. The esul s om he ime-a e aged MSDs (hDiT,hαiT), he mean alues
o he ensemble o sub ajec o ies (hhDiTiE,hhαiTiE) as well as he esul s om
he ensemble-a e aged MSDs (hDiE,hαiE) a e summa ized in able 6.1. Beside he
alues ob ained om he pu e ensemble-a e age analysis, which a e no mally no
used because o i s weak s a is ics, all he alues om he ime-a e aged analysis
ag ee qui e well wi h espec o each o he . Fo all he ollowing discussions ega d-
ing his sample (20 nm bead in glyce ol), he measu ed no mal B ownian mo ion
wi h α= 1 is used.
41
6 Single-pa icle acking: esul s & discussion
Exp. n . hDiT/hhDiTiE/hDiE/hαiThhαiTiEhαiE
10−3µm2/s
1 17.19 17.15 17.06 1.04 1.01 0.97
2 21.58 21.64 21.03 1.02 1.02 0.99
3 16.70 16.60 17.71 1.03 1.01 0.96
4 17.49 17.47 18.18 1.04 1.01 1.15
5 15.01 14.98 14.95 1.05 0.94 0.91
6 24.14 24.10 24.57 1.04 1.01 1.02
7 12.32 12.34 13.30 1.01 1.04 9.94
Table 6.1: Summa y o he di usion coe icien s and he anomaly pa ame e s ob ained
om ime-a e aged as well as ensemble-a e aged analysis. The en ies a e he ime-
a e aged di usion coe icien (anomaly pa ame e ) hDiT(hαiT), he mean alue de e -
mined om he ensemble o sub ajec o ies hhDiTiE(hhαiTiE) and he ensemble-a e aged
di usion coe icien (anomaly pa ame e ) hDiE(hαiE).
The analysis o single-pa icle ajec o ies wi h he use o he mean squa ed dis-
placemen s (MSD) is he mos common me hod. Among he he e p esen ed ech-
niques o ajec o y analysis, his is he only one, whe e he di usion coe icien as
well as he anomaly pa ame e can be de e mined. The mos in o ma ions ega ding
a di usion p ocess can be ex ac ed.
6.1.2 Al e na i e app oaches
Cumula i e dis ibu ion unc ion
This me hod supplies an al e na i e app oach o analyse a ajec o y [37, 38]. Ac-
co ding o he explana ion in sec ion 2.1.3 he empi ical cumula i e dis ibu ion
unc ion (CDF) o he squa e displacemen s a he lag imes τ= 4 ms −0.2 s
(τ= (1−50)∆ ) was de e mined consecu i ely o a single ajec o y. An example,
calcula ed a a lag ime o τ= 40 ms is shown in ig. 6.4a (black squa es). The
da a poin s can be modeled wi h a double-exponen ial unc ion as gi en by
CDF ∆ 2, τ= 1 −βexp −∆ 2
∆ 2
1+ (1 −β)exp −∆ 2
∆ 2
2 (6.1)
He e, he ∆ 2
ia e ela ed o a as (1) and slow (2) componen o he di usion
coe icien , i.e. ∆ 2
i= 4Diτ, weigh ed by a ac ion β. Fo sys ems, whe e pu e
B ownian mo ion is p esen , βequals 1 and he CDF educes o a single-exponen ial
wi h a single di usion coe icien D1=Dcd . Ye , o sys ems showing subdi usi e
beha iou βclea ly di e s om 1 and a double-exponen ial wi h a so called slow
componen o he di usion coe icien (∆ 2
2)is be e sui ed o desc ibe he da a.
Hence, his me hod is powe ul in dis inguishing no mal om anomalous di usion,
jus by de e mining β. The a o e men ioned i unc ion (eq. 6.1) was applied o
42
6.1 Analysing di usion based on single-pa icle ajec o ies
Figu e 6.4: a) Example o a cumula i e densi y unc ion a a lag ime o τ= 40 ms.
b) di usion coe icien and ac ion β aken om double-exponen ial i s o he empi ical
CDF. The a ow ma ks he alues ha co espond o he CDF shown in a). Due o emain-
ing expe imen al sho comings he i s da a poin was neglec ed o he in e p e a ion.
he da a poin s (c . ig. 6.4a, ed solid line). The shown example (τ= 40 ms)
yields β= 1.002 which is in nea ly pe ec ag eemen wi h B ownian mo ion. The
alue o he ac ion as a unc ion o he lag ime can be seen in ig. 6.4b, uppe
pa . Fo his single ajec o y on a e age a ac ion o hβi= 0.99 was ound, which
is close o uni y and esembles B ownian mo ion wi h a single-exponen ial o he
CDF. Fu he , he mean di usion coe icien was calcula ed wi h his me hod o
hDcd i= 17.44 ×10−3µm2/s, which is shown g aphically in ig. 6.4b, lowe pa .
The calcula ed alue o he example CDF, shown in ig. 6.4a a e ma ked wi h an
a ow. Again, he i s da a poin s o τ= 4 ms was neglec ed due o expe imen al
sho comings. Compa ed o he alue ob ained om he ime-a e aged MSD (c .
able 6.1, exp. n . 1), a good ag eemen is achie ed. This p ocedu e was done o
all he 7 eco ded ajec o ies. The esul s can be ound in able 6.2. F om he
o e all a e age o he se en ajec o ies o he ac ion hβi= 0.96 one can conclude
ha he he e used sys em e lec s B ownian mo ion. Addi ionally, he di usion
coe icien i s qui e well wi h he al eady de e mined ones om he mean squa ed
displacemen analysis. The sligh disc epancy o heo e ical/expec ed alues can
be add essed o he usage o pa ly unco ec ed ajec o y da a (only he se up
induced a i ac s we e co ec ed, i.e. he sys ema ic e o in he acking algo i hm
and he (mis)alignmen o he piezo s age) which was necessa y o his e alua ion.
The ad an age o his echnique is he capabili y o s udy mul i-componen di u-
sion phenomena, e.g. a wo-componen di usion can be in e p e ed as a slow and
a as mobili y o he acked pa icle. The ac ion be ween hese wo mobili ies
can hen be used o dis inguish no mal om anomalous di usion. One has o be
ca e ul, ha his ac ion is no ela ed o he anomaly pa ame e α. Ano he big
ad an age is, ha his me hod can also be applied o ela i e sho ajec o ies
43
6 Single-pa icle acking: esul s & discussion
Exp. n . hβi hDicd /
10−3µm2/s
1 0.99 17.44
2 0.90 22.15
3 0.98 16.81
4 0.92 18.19
5 0.97 16.46
6 1.01 24.05
7 0.97 11.98
Table 6.2: Summa y o he esul s om he analysis acco ding o he cumula i e dis i-
bu ion unc ion om 7 indi idual polyme beads. The en ies deno e he expe imen n .,
he ac ion hβiT, and he di usion coe icien s hDicd .
wi h a ew posi ion de e mina ions [37].
Spa ial ex end & shape
Ano he me hod o ex ac in o ma ion om a acking expe imen is o de e mine
he spa ial ex end by means o a gy a ion ellipse a ound a ajec o y and i s shape,
i.e. he asphe ici y. The me hod, how o calcula e his alue is discussed in de ail in
sec ion 2.1.3. In ig. 6.5a he gy a ion ellipse wi h i s adii o gy a ion is o e layed
o a ep esen a i e ajec o y wi h N= 1.52 ×105da a poin s. The o e all spa ial
ex end o he ajec o y is a he elonga ed han ci cula . The la e one would be
expec ed o a pe ec andom walk and holds ue o an ensemble o aces. Bu ,
o a single ajec o y he elonga ed e sion is ound [35]. This beha iou is be e
desc ibed by he shape o he ace ins ead o he spa ial ex end, which can a y
(mainly) due o he mobili y o he pa icle and he leng h o he ajec o y. The
shape is de e mined by he asphe ici y, which is a measu e o he de ia ion om
a sphe ic (3D) o a ci cula (2D) shape. Acco ding o eqn. 2.26 he asphe ici y A
o an indi idual ajec o y is calcula ed o a ce ain leng h (numbe o posi ions
Nseg) o cu segmen s. (As a eminde , he calcula ion o A equi es a e aging o e
an ensemble o aces.) I is qui e ha d o decide, a which segmen leng h he alue
o Ais eliable. The e o e he asphe ici y o a ace was calcula ed as a unc ion
o he leng h Nseg, i.e. A(Nseg). The esul o he 7 indi idual measu ed aces is
shown in ig. 6.5b. Fo a segmen leng h o Nseg = 2 he shape o he ajec o y
is pe ec od-like and leads a alue o A(2) = 1. Wi hin he i s 15 da a poin s
he asphe ici y con e ges owa ds a alue o A= 4/7(g ey line in ig. 6.5b), ha
is he analy ical alue o a 2-dimensional andom walk [35, 36]. To e i y his
beha iou o he he e p esen ed measu emen s in a mo e quan i a i e way, a s a-
is ical ele an window (g ey shaded a ea in ig. 6.5b) o he calcula ion o a mean
asphe ici y hAiwas chosen. This egion was es ic ed o 20 ≤Nseg ≤250 whe e
a he one end (Nseg >20)A(Nseg)is ega ded as ully con e ged and on he o he
44
6.1 Analysing di usion based on single-pa icle ajec o ies
Figu e 6.5: Asphe ici y o all ajec o ies plo ed agains he sequence leng h Nseg. The
g ey shaded egion is he window, whe e he asphe ici y is a e aged. The g ey solid line
ep esen s he analy ical alue o a 2-dimensional andom walk, i.e. A=4/7.
hand (Nseg <250) he a e aging in calcula ing he asphe ici y is high enough. The
alues o he 7 expe imen s a e summa ized in able 6.3. On a e age, he mean
asphe ici y was de e mined o hAi7= 0.566 which is in e y good ag eemen wi h
he analy ical alue o A= 4/7≈0.571 o a B ownian mo ion. The index 7 indi-
ca es he a e age o e all se en expe imen s. The nea ly pe ec coincidence e lec s
u he mo e, ha no signi ican signs o d i o low is o e layed o he ajec o ies
ha would lead o a dis o ion and concomi an ly in luence he asphe ici y. Fo
he in es iga ion o sys ems showing anomalous di usion, his me hod can be used
o cla i y he ype o sub-di usion, which is desc ibed in publica ion P3 (see pa
II).
Wi h his me hod, nei he he di usion coe icien no he anomaly pa ame e can
be de e mined. Hence, i yields minimal in o ma ions abou a di usion p ocess.
Bu , in addi ion o he ime-dependen pa ame e s (D,α) a s uc u al pa ame e
Exp. n . hAi
1 0.555
2 0.575
3 0.564
4 0.563
5 0.540
6 0.569
7 0.595
Table 6.3: Summa y o he de e mined mean asphe ici ies hAio he se en indi idual
ajec o ies.
45
6 Single-pa icle acking: esul s & discussion
is ob ained, ha desc ibes he de ia ions om a ci cula shape (in case o a 2D
ajec o y). In combina ion wi h he anomaly his echnique is powe ul in in es-
iga ing subi usi e beha iou , e.g. o c owded luids like i is discussed in sec ion
6.4 o his hesis.
E godici y b eaking pa ame e
Recen ly a echnical pa ame e o de e mining he deg ee o e godici y b eaking
has been published [51, 52].
E(τ) = Dh∆ (τ)2i2
TEE−hh∆ (τ)2iTi2
E
hh∆ (τ)2iTi2
E
(6.2)
The calcula ion o his e godici y b eaking pa ame e E(τ) equi es again an en-
semble o sho ajec o ies, ha can be ob ained by spli ing a long ace in o sho
segmen s wi h equal leng h. F om hese sho segmen s he ime-a e aged MSDs
a e calcula ed. Acco ding o equa ion 6.2 he ime-a e aged MSDs, i.e. h∆ (τ)2iT,
a e on he one hand squa ed and hen ensemble-a e aged, symbolized by h·iE, and
on he o he hand i s ensemble-a e aged and hen squa ed. Fo alues o Eclose
o 0 he sys em is ega ded as non e godici y b eaking [51, 52].
In an analog way o he de e mina ion o he asphe ici y he e godici y b eaking
pa ame e was calcula ed. Fo a e aging E(τ)we used a ime window ha co -
esponds o he analysis o he di usion coe icien and he anomaly pa ame e
(2-50 da a poin s). The esul s o he a e aged e godici y b eaking pa ame e hEi
o e e y single ajec o y is lis ed in able 6.4. All alues a e smalle han 0.020
which is in e p e ed as non-e godici y b eaking, compa ed o alues ound in he
li e a u e o weak e godici y b eaking sys em. The e, E= 0.57 [51] and hence a
ac o o mo e han 20 imes la ge han he he e ob ained alues. This addi ional
pa ame e e lec s he expec ed B ownian mo ion, whe e e godici y is p ese ed.
Exp. n . hEi
1 0.017
2 0.019
3 0.020
4 0.020
5 0.018
6 0.017
7 0.020
Table 6.4: Summa y o he de e mined mean e godici y b eaking pa ame e hEio he
se en indi idual ajec o ies.
46
6.2 Accu acy o di usion coe icien s
6.2 Accu acy o di usion coe icien s
How well a di usion coe icien can be de e mined by a mean squa ed displacemen
analysis o expe imen al single-pa icle ajec o ies is an impo an issue. In he
p e ious chap e i was explained, ha long ajec o ies esul s in accu a e di usion
coe icien s. Howe e , one can imagine, ha no only he leng h o he ajec o y is
impo an , also unce ain ies in he posi ion de e mina ion o he numbe o i ing
poin s used in he MSD analysis ha e an e ec on he accu acy o he measu ed
di usion coe icien . To wha ex end hese pa ame e s in luence he accu acy is
discussed in g ea de ail in he li e a u e on he basis o simula ions and nume ical
calcula ions [53–55]. Fi s i was s udied by Qian and Sax on and was u he de-
eloped and sub ilized by Michale who ook also expe imen al localisa ion e o s
in o accoun . Up o now a compa ison wi h expe imen al da a emained elusi e.
In his sec ion he p oblem is ea ed om he expe imen al poin o iew and he
esul s we e compa ed wi h he p edic ions made in he a o e men ioned li e a u e.
The indings we e published in Physical Chemis y Chemical Physics (see publica-
ion P2). In he ollowing I will gi e a summa y o his wo k.
Many s udies ha use single-pa icle acking o in es iga e di usion p ocesses, es-
pecially in he ield o biophysics, o en lack o su icien ly long ajec o ies, ei he
because he pa icle di uses ou o he de ec able a ea o due o pho obleaching
o he luo escen ace pa icles. Hence, he MSD poin s o a ajec o y a e less
a e aged which esul s in a less accu a e di usion coe icien . In o de o in es iga e
his accu acy, many ajec o ies a e equi ed o pe o m a s a is ical analysis. Usu-
ally, i is no possible o acqui e a s a is ically ele an numbe o ajec o ies wi h
iden ical expe imen al condi ions. While i is a he simple o keep he iscosi y
and empe a u e be ween se e al independen acking expe imen s cons an , he
bead size de e mined om each eco ded ajec o y can a y d as ically. The e-
o e, in he li e a u e simula ions wi h a high numbe o ic i ious ajec o ies wi h
iden ical pa ame e s ( ajec o y leng h N, empe a u e T, iscosi y η, bead adius
a) we e used o cla i y his issue [53–55]. In his wo k an expe imen al app oach o
measu e he accu acy o a di usion coe icien is p esen ed.
Exploi ing single-pa icle o bi acking, e y long ajec o ies comp ising mo e
han N= 1.5×105da a poin s we e eco ded om indi idual 20 nm sized pa icles
in glyce ol ha unde go B ownian mo ion. Each o hese la ge da a se s can be
decomposed in o an ensemble o segmen s wi h a a ious numbe o da a poin s
Nseg, p o iding he same ensemble o mean squa ed displacemen cu es om he
same indi idual pa icle. By a linea i o each MSD cu e o an ensemble wi h
a gi en alue o Nseg, he slopes D∗as a unc ion o he numbe o i ing poin s
nwe e de e mined (ns a s om he MSD poin 2 and ends a n+ 1, de ails see
publica ion P2). Subsequen ly o each alue o n, he ela i e e o sD∗/D∗was
calcula ed om he his og ams o he slopes and plo ed agains each o he . He e,
he (empi ical) mean D∗and s anda d de ia ion sD∗we e de e mined. Figu e 6.6
47
6 Single-pa icle acking: esul s & discussion
Figu e 6.9: Resul s o he anomaly pa ame e αas a unc ion o he de e mined bead
adius a. The ull blue ( ed) squa es a e he nominal 20 nm sized beads in he sample
suc ose (dex an), while he open blue ( ed) squa es a e he nominal 50 nm beads in suc ose
(dex an). The g ey line se es as a guide o he eye.
i.e. he iscosi y s ays cons an (see equa ion 6.4).
1
D=6πη
kbTa=γa (6.4)
This p esen a ion is sui able enough, because only he dependence in he pa icle
sizes and no he absolu e alues o he adii a e impo an . The expe imen s we e
pe o med in wo luids wi h di e en iscosi ies, i.e. o a p ope isualiza ion wo
x-axis we e in oduced, whe e he bo om (blue) axis in igu e 6.9 co esponds o
he measu emen s in suc ose and he op ( ed) axis o he measu emen s in dex an.
To compa e he wo da a se s, he axis we e u he scaled acco ding o he smalles
measu ed alues o 1/D o he nominal 20 nm sized beads. Despi e he nominal
alues o 20 nm and 50 nm o he bead sizes a dis ibu ion was de e mined, which
was discussed in de ail in sec ion 6.3. Fo his measu emen , a b oad dis ibu ion
was help ul, because he size e ec on he di usion beha iou was s udied. The
measu emen s in suc ose showed independen o he size o he beads an anomaly
alue close o 1, while he measu emen s in dex an showed a clea dependence
on he size o he beads. Howe e , all measu emen s o he 50 nm sized beads
in dex an showed an unambiguous subdi usi e beha iou . The mean anomaly
alues o he 50 nm sized beads we e de e mined o hαsuci= 0.98 o suc ose and
54
6.4 C owded Fluids
o hαdexi= 0.82 o he c owded dex an solu ion. The clea sepa a ion be ween
hese wo alues is e iden , i.e. he subdi usion is no jus an a i ac . A plausible
explana ion is he size a io o he beads and he mesh size o he dex an ne wo k.
Small beads can di use nea ly eely h ough he ne wo k s uc u e, while la ge
pa icles a e hinde ed. Hence, in he ollowing only he da a ob ained o he mea-
su emen s wi h bead sizes o 50 nm a e discussed.
The expe imen ally de e mined ajec o ies o he 50 nm sized beads in dex an
we e compa ed wi h simula ed ajec o ies based on he h ee heo e ical models
men ioned a o e. These models ha e in common, ha hey can desc ibe subdi u-
sion. Bu , while he CTRW model shows sligh e godici y b eaking acco ding o
he ecen ly de eloped e godici y b eaking pa ame e [51, 52] (see eq. (6.2), he
o he wo models do no . Also he analysis o he expe imen ally measu ed a-
jec o ies do no show any signs o e godici y b eaking. This ules ou he CTRW
model o a p ope desc ip ion o he c owding induced subdi usion in dex an.
The emaining wo models (OD and FBM) show e godic beha iou and he e o e
ano he c i e ium has o be ound o deciphe be ween hose wo heo ies.
The idea is o calcula e he asphe ici y Aand he anomaly αo expe imen ally
de e mined ajec o ies and compa e hose esul s wi h he esul s ob ained om
simula ed ajec o ies ha epose on he heo ies o OD and FBM. Fo hese sim-
ula ions, ajec o ies wi h α alues be ween 0.5and 1.0we e gene a ed and he
asphe ici y was calcula ed subsequen ly, yielding he asphe ici y as a unc ion o
he anomaly, i.e. A(α). This analysis was pe o med in he g oup o he coope a-
ion pa ne . The de ails a e desc ibed in he publica ion P3. The mean alues o
he expe imen ally de e mined anomaly and asphe ici y a e hαi= 0.82 ( ide sup a)
and hAi= 0.46, espec i ely. Fo he compa ison, he α alue om he expe imen
(0.82) was inse ed in bo h unc ions A(α), ob ained by he simula ions wi h OD
and FBM. This yields AOD = 0.56 and AFBM = 0.47. A nea ly pe ec ag eemen
be ween he expe imen al da a and he model o ac ional B ownian mo ion was
ob ained.
Wi h his wo k a clea s a emen ega ding he heo e ical desc ip ion o a sub-
di usi e p ocess in a complex luid was made, which helps o be e unde s and
he eac ion kine ics in li ing cells. I is wo h o no e, ha i is o c ucial impo -
ance o he analysis o he expe imen al da a o measu e e y long ajec o ies
wi h a high spa ial and empo al accu acy. Ha ing hose da a a compa ison wi h
simula ions was possible.
55
The u u e belongs o hose who belie e in
he beau y o hei d eams
Eleano Roose el
7 Ou look
The desc ibed expe imen al se up o single-pa icle o bi acking, he da a anal-
ysis and he applica ions a e on a le el we e ascina ing expe imen s and in es i-
ga ions can be pe o med. The se up is well cha ac e ised and i s esul s showed
i s capabili y. Bu some imp o emen s in all o he men ioned ields a e help ul o
push his esea ch o wa d and o ge a highly de eloped se up. In he ollowing a
sho ou look o possible applica ions and se up imp o emen s a e gi en.
Ins ead o concen a ing on acking expe imen s in he esea ch sec o o bio-
physics, whe e mos o he g oups a e wo king, he ield o ma e ial sciences can be
in es iga ed, whe e acking o single nanoobjec s can help o be e unde s and
p ocesses on a nanome e leng h scale. Fo example he swi ching beha iou o a
liquid c ys al is challenging when wo king on a submic ome e scale. I aceable
molecules o pa icles can be a ached o he molecules o a liquid c ys al, he ime-
dependen mo ion o hese molecules can be eco ded wi h a high p ecision. This
would help o be e unde s and he phase ansi ion o liquid c ys als. Fu he ,
i molecules wi h a de ined ansi ion dipole momen a e used, also he o a ional
di usion can be in es iga ed.
Ano he possibili y o use pa icle acking in ma e ial sciences is he examina ion
o di usion p ocesses h ough nanopo ous memb anes. Recen ly success ul expe i-
men s we e published, whe e he di usion h ough swi chable nanopo es ha a y
in size we e measu ed [64]. Wi h he single-pa icle acking expe imen s, he s uc-
u e o he po es can be de e mined, wi h a esolu ion a beyond he di ac ion
limi o ligh . This helps o cons uc il e sys ems on a mic o- and e en nanome e
scale.
Wi h he he e p esen ed se up, a he momen only 2-dimensional acking expe i-
men s can be pe o med wi h a high spa io- empo al esolu ion. Bu wi h a lowe
ime esolu ion also 3-dimensional ajec o ies can be eco ded. The 3 d dimension
becomes accessible h ough he z- acking algo i hm. He e i was only used o keep
he pa icle o in e es in he ocal plane, bu he algo i hm s o es he z-posi ion
o he pa icle wi h a lowe ime esolu ion. In igu e 7.1 he 3-dimensional ace
wi h a numbe o N= 15000 posi ions o a 20 nm sized bead, di using in glyce ol
is shown o a ime esolu ion o ∆ = 40 ms, which is a ac o o 10 slowe han
57
7 Ou look
Figu e 7.1: 3-dimensional ajec o y o a single pa icle wi h a size o 20 nm, ha di uses
in he luid glyce ol. The ace consis s o N= 15000 da a poin s wi h a ime esolu ion
o ∆ = 40 ms. The g een, blue and ed ajec o y co esponds o he p ojec ions o he xz,
yz, and xy plane, espec i ely.
he es o he esul s. Rega ding he posi ion accu acies o he z-posi ion, his has
o be in es iga ed in a u u e wo k, he 2-dimensional accu acy (xy plane) is he
same like in he p esen ed wo k (≈10 nm). To imp o e also he empo al eso-
lu ion o he z dimension, a as scanning in his di ec ion has o implemen ed.
One op ion is o ins all a hi d AOD, ha is esponsible o he gene a ion o a
second ligh o bi . The op ics has o be aligned in a manne , ha one o he ligh
o bi s is sligh ly abo e he pa icle and he o he is sligh ly below. By swi ching
he exci a ion in ensi ies o hese wo o bi s pe iodically on and o wi h a di e en
equency han he one used o he ocus o a ion, in an analog way he z-posi ion
o he pa icle can be calcula ed be demodula ing he emission signal.
The empo al esolu ion o all o he axis can be imp o ed by implemen ing a beam
scanning echnique, a he han sample scanning wi h he piezo. By mo ing he
lase beam wi h he AODs a much highe empo al esolu ion should be ob ained,
because no mechanical elemen s es ic he scanning mechanism.
These we e jus some imp o emen s ha can be done o b ing he se up o a new
le el. Especially wi h he highe ime esolu ion, di usion p ocesses on a as e
ime scale becomes accessible.
58
Appendix
A P og am code
In he ollowing he Ma lab sou ce code o he gene a ion o he “simula ed eal”
and “simula ed econs uc ed” ajec o ies is displayed.
1
2% Simula ion o he " eal " and he " econs uc ed " ajec o y
3% ***********************************************************
4
5clc;
6clea ;
7 o ma long
8n _ ace =1;
9
10 %**** acking pa ame e ************************************
11 d =2 E -6; % sampling ime [s]
12 =1000; % o bi equency [ Hz]
13 o_x =190E -9; % o bi adius x [ nm ]
14 o_y =190E -9; % o bi adius y [ nm ]
15 w =270E -9; %1/e^2 o ocus [nm]
16 gb =500; % backg ound emission [ cps]
17 g0 =450000; % emission in he ocus
18 omega =2* pi* *d ; % o bi equency
19 s =100; % samples ( leng h o ajec o y )
20
21
22 o i2 =0: n _ ace -1
23
24 % ***** di usion pa ame e ** ********** ********** *******
25 e a =1.2; % iscosi y [ Pas]
26 =10 E -9; % adius o sphe e [m]
27 kb =1.3806504 E -23; % Bol zmann cons an [J/K]
28 T =294; % Tempe a u e [K]
29 g =6* pi* e a* ; % ic ion coe icien ( S okes )
30 c= sq (2* kb *T/g/d ); % p e ac o s ochas ic o ce
31 D=kb*T/g; % di usion coe icien
32 P=4; % o bi pe iods
33 a=P /( * d ); % sampling in e als
34 =P/ ; % ime esolu ion [s]
35
36
37 % ***** ini ial alues ******* ************** *************
38 x=0E -9; %x o " eal" ace
39 y=0E -9; %y o " eal" ace
40 dx =0; %x s ep o " eal " ace
41 dy =0; %y s ep o " eal " ace
42 xc =0; %x o " econs uc ed " ace
43 yc =0; %y o " econs uc ed " ace
44 dxc =0; %x s ep o " econs uc ed " ace
45 dyc =0; %y s ep o " econs uc ed " ace
46 xs =0; %x o piezos age
47 ys =0; %y o piezos age
48 x_a = ze os (1,s); % a ay o " eal " x alues
49 y_a = ze os (1,s); % a ay o " eal " y alues
59
7 Ou look
50 xc_a=ze os (1,s); % a ay o " econs uc ed " x alues
51 yc_a=ze os (1,s); % a ay o " econs uc ed " y alues
52 ime=ze os (1,s); % a ay o ime
53 in = ze os (1,s); % a ay o in ensi y
54 andn (’s a e ’,sum (1000*clock ));
55
56
57 %** gene a ion o he " eal " and " econs uc ed " ace *
58 o i3 =1:s
59
60 S= ze os (1 ,a); % heo e ical n . o pho ons
61 Sp= ze os(1,a); % poisson dis . n . o pho ons
62 Scos=ze os (1,a);
63 Ssin=ze os (1,a);
64 cos_ = ze os (1,a);
65 sin_ = ze os (1,a);
66
67
68 o n=1: a
69 % ********* " eal " ajec o y **********
70 s_x= andn ;
71 s_y= andn ;
72
73 dx= s_x * c*d ;
74 dy= s_y * c*d ;
75
76 x=x+dx ;
77 y=y+dy ;
78
79 % ***** " econs uc ed " ajec o y *****
80
81 S(n)=d * gb+d *g0*exp ( -2/w ^2*(( x-xs - o_x * cos( omega *n)) ^2+
82 (y-ys - o_y* sin( omega *n)) ^2) );
83 cos_ (n) =cos( omega * n);
84 sin_ (n) =sin( omega * n);
85 end
86
87 x_a (i3)=x *1 E9;
88 y_a (i3)=y *1 E9;
89
90 Sp= poiss nd (S);
91 Scos = Sp .* cos_ ;
92 Ssin = Sp .* sin_ ;
93
94 % ***** calcula ion o he posi ion ********
95
96 i (sum(Sp)==0)
97 dxc =0;
98 dyc =0;
99 else
100 dxc =w ^2/(2* o_x )* sum( Scos ) /sum( Sp);
101 dyc =w ^2/(2* o_x )* sum( Ssin ) /sum( Sp);
102
103 xc=xc+ dxc;
104 yc=yc+ dyc;
105
106 xc_a ( i3)= xc *1 E9 ;
107 yc_a ( i3)= yc *1 E9 ;
108
109 % ******** eedback mechanism **************
110 xs=xc;
111 ys=yc;
112
60
B Ha monic app oxima ion
113 % ******** ime and in ensi y **************
114
115 ime (i3)=i3* ;
116 in ( i3)= sum(Sp)/ ;
117
118 end
119 end
B Ha monic app oxima ion
In his sec ion he de i a ion o he ha monic modula ion o he emission signal o
pa icle posi ions close o he cen e o he ligh o bi is shown. The e o e equa ion
(2.28) om he sec ion 2.2.2 is used.
I( ) = I0exp −2
w2(xp−Rcos(ω ))2exp −2
w2(yp−Rsin(ω ))2+Ib(B.1)
A 2-dimensional Taylo se ies o his equa ion in he neighbo hood o he posi ion
(x0, y0) = (0,0) gi es:
I( )≈I( )|x0,y0+∂
∂xp
I( )x0,y0
xp+∂
∂yp
I( )x0,y0
yp+O(x2, y2)
=Ib+I0e2
w2R2cos2(ω )e2
w2R2sin2(ω )
+I0e−2
w2(xp−Rcos(ω ))2e−2
w2(yp−Rsin(ω ))2·−4
w2(xp−Rcos(ω ))x0,y0
xp
+I0e−2
w2(xp−Rcos(ω ))2e−2
w2(yp−Rsin(ω ))2·−4
w2(yp−Rsin(ω ))x0,y0
yp
+O(x2, y2)
=Ib+I0e2R2
w2+I0e2R2
w2·4R
w2cos(ω )xp+I0e2R2
w2·4R
w2sin(ω )yp
+O(x2, y2)
=Ib+I0e2R2
w21 + 4R
w2(xpcos(ω ) + ypsin(ω ))+O(x2, y2)
The emission signal is modula ed wi h a cos- and a sin- unc ion.
61
Bibliog aphy
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molecules in solu ion using he abel ap,” Appl. Phys. B 99, 23–30 (2010).
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63
Danksagung
Zum Schluss möch e ich mich noch bei einigen Leu en bedanken, die au iel äl ige
Weise zum Gelingen diese A bei beige agen haben.
Als E s es möch e meinem Dok o a e P o . D . Jü gen Köhle einen Dank aus-
sp echen. Du ch sein Ve auen in meine A bei ha e es mi e möglich , einen
neuen Au bau zu en wickeln, de nach einigen Hochs und Tie s le z endlich aus-
ge ei wa und neue Pe spek i en ü die wei e e Fo schung e ö ne . E ha mi
s e s die nö ige F eihei gegeben meine Ideen umzuse zen und ha mi imme du ch
zahl eiche Diskussionen übe meine Messe gebnisse wei e gehol en, wenn ich seine
Un e s ü zung b auch e.
Ein wei e e g oße Dank geh an unse en Koope a ionspa ne P o . D . Ma hias
Weiss. Von Ma hias konn e ich iel übe anomale Di usionsp ozesse und neue
Auswe eme hoden le nen. E wa imme diskussionsbe ei und ha sich ü meine
P obleme iel Zei genommen. Wei e hin möch e ich D . Ma cel Hellmann danken,
de mich in p og amm echnischen F agen zu Auswe ung de Messda en un e -
s ü z ha .
De Se iceab eilung de Fi ma Jäge Mess echnik GmbH möch e ich ü den he -
o agenden Suppo danken. Insbesonde e is hie S e an Plappe zu e wähnen,
de mi des ö e en die nö igen Tipps zu So wa eans eue ung des Au baus gab
und auch o wäh end ein eges In e esse am Fo sch i meine A bei zeig e.
Danke auch an P o . D . We ne Köhle , de mi bei g undlegenden F agen zu
Di usionsp ozessen wei e hal . Unse e echnischen Anges ell en, ohne die ein Leh -
s uhl meine Meinung nach nie ich ig unk ionie en kann, wa en imme hil sbe ei
und so g en an ielen S ellen ü eine E leich e ung de A bei . S e an Schlich wa
bei elek o echnischen F agen s e s ein gu e Ansp echpa ne . Wal aud Joy is
mi bei chemischen F agen bezüglich de P obenp äpa a ion mi Ra und Ta zu
Sei e ges anden. We ne „Wö ni“ Reichs ein is imme ü einen zu S elle und ha
mi o Allem in de An angsphase du ch einige, eils g öße e, Umbau en im La-
bo gehol en. Wei e hin möch e ich de Mechanikwe ks a und insbesonde e F ank
Neumann und Olli ein Lob aussp echen. Ohne sie wä en einige Kons uk ionen
wohl nich zus ande gekommen. Ein g oße Dank geh an die „Ve wal ungszen a-
le“ des Leh s uhls, das Sek e a ia . Besonde s e wähnen möch e ich hie E elyn
Hülsmann, die sich um säm liche Ve wal ungsangelegenhei en gekümme ha und
imme ein o enes Oh ü die „P oblemchen“ de Mi a bei e ha .
Vielen Dank auch an unse e Pos Docs Richa d Hildne und Abey Issac ü iele in-
e essan e Diskussionen und das Ko ek u lesen meine A bei . Ich möch e an diese
S elle auch „meinem“ e s en Bachelo Daniel Zalami ü neue Ideen zu Wei e en -
wicklung des Au baus und ü die gu e Zusammena bei danken. Ein Dankeschön
71
geh auch an die Mi s ei e de Dok o anden wäh end meine Zei am Leh s uhl,
Tobias P lock und Flo ian Sp ei le , die den Sp ung in die „ech e“ Wel be ei s hin-
e sich geb ach haben. Vo allem Flo ian möch e ich ü die g oße Un e s ü zung
du ch iele Diskussionen übe PC-P og amme, angewand e Op ik und In e p e-
a ion on Messungen danken. Nich zu e gessen sind na ü lich Ral Kunz und
Paul Böhm, die mi mi die Zei am Leh s uhl begonnen haben und langsam auch
dem Ende de P omo ion en gegensehen. Au geh ’s Jungs! Ma c Jend ny danke
ich ü eine coole Banz-Tagung. Besonde s he o heben möch e ich S e en Ha -
mann, Ma hias Gebha , Axel He mann, Vanessa Wei h, Lau a Win e ling, Lisa
K ap und Flo ian Schwaige . Ein Teil da on is zwa schon lange nich meh am
Leh s uhl, sie haben abe en scheidend dazu beige agen, dass ich heu e de bin,
de ich bin. Ich danke euch ü eine Menge schöne E lebnisse in Bay eu h, di e se
Wande ungen, exzessi e Pa ies, lus ige Kochabende, den Ka eeecken a sch und
ieles meh . Was wä en die Miamiam-Besuche ohne euch gewesen. Vo allem Flo-
ian möch e ich g oßen Dank aussp echen. Die unzähligen Diskussionen mi ihm
wa en ü mich nich nu physikalisch seh we oll, auch das Humo ni eau wu -
de au ungeahn e Ebenen gelenk . Mann, mann, mann. Highligh s wa en London
(Baby!) und das Pokal inale in Do mund (Heja BVB!). Es wa eine geniale Zei .
Alle wei e en Kollegen, die hie nich namen lich au ge üh sind (bi e nich bö-
se sein), möch e ich na ü lich auch ü das angenehme A bei sklima danken. De
Leh s uhl is ein bun e „Hau en“ aus chao ischen und o ganisie en, uhigen und
au b ausenden, engagie en und wenige engagie en, abe imme gu au geleg en
Kollegen. Ich möch e mich bei Allen ü die in e essan en achlichen und (zum Teil
seh abs usen) nich achlichen Gesp äche in de Ka eeecke, bei di e sen G ill ei-
e n und abendlichen Kneipen- und Bie ga enbesuchen he zlichs bedanken. Das
A bei sum eld wa du chwegs posi i und es ha seh iel Spaß gemach in diese
G uppe zu a bei en.
Auch meinen F eunden auße halb de Uni e si ä gebüh ein Dank ü die Ablen-
kung om eilweise s essigen All ag. Auße dem möch e ich auch de Familie meine
F eundin ü iele schöne S unden und Aus lüge danken.
Schließlich möch e ich mich bei meine Familie und insbesonde e bei meinen El-
e n ü die jah elange und uneingesch änk e Un e s ü zung bedanken. Ih hab
mi imme den nö igen Rückhal gegeben, was mi ein so gen eies S udium und
eine anschließende P omo ion e möglich e. Ih wa imme ü da und da ü möch e
ein ach nu DANKE sagen. Ohne euch wä e ieles nich möglich gewesen. Zu gu e
Le z möch e ich mich bei meine F eundin Nadine bedanken. Die le z en Jah e mi
Di wa en wunde schön und ho en lich olgen noch iele wei e e. Du wa s imme
ü mich da, has mich bei meine A bei imme un e s ü z und mein Leben au
iel äl ige Weise be eiche . Danke, ü all die schönen Momen e und dass du imme
ü mich da bis . ILD
72
E klä ung
Hie mi e siche e ich an Eides s a , dass ich die o liegende A bei selbs s ändig
e ass und keine ande en als die on mi angegebenen Quellen und Hil smi el
benu z habe.
Ich e klä e, dass ich keine ühe en P omo ions e suche un e nommen habe. Die
o geleg e Abhandlung wu de wede in gleiche noch in ähnliche Fo m eine an-
de en P ü ungsbehö de zu E langung eines akademischen G ades o geleg .
Deswei e en e klä e ich, dass ich Hil e on gewe blichen P omo ionsbe a e n bzw.
- e mi le n ode ähnlichen Diens leis e n wede bishe in Ansp uch genommen ha-
be noch kün ig in Ansp uch nehmen we de.
Bay eu h, den 27.09.2012
Dominique E ns
73
Pa II
Publica ions
Publica ion P1
Se up o single-pa icle o bi acking:
a i ac s and co ec ions
Dominique E ns , S e an Hain, and Jü gen Köhle
published in:
J. Op . Soc. Am. A 29, 1277-1287, (2012)
c
2012 Op ical Socie y o Ame ica
h p://dx.doi.o g/10.1364/JOSAA.29.001277
Se up o single-pa icle o bi acking:
a i ac s and co ec ions
Dominique E ns , S e an Hain, and Jü gen Köhle *
Expe imen al Physics IV and Bay eu h Ins i u e o Mac omolecula Resea ch (BIMF),
Uni e si y o Bay eu h, 95440 Bay eu h, Ge many
*Co esponding au ho : jue gen.koehle @uni‐bay eu h.de
Recei ed Ma ch 16, 2012; accep ed Ma ch 30, 2012;
pos ed Ap il 3, 2012 (Doc. ID 164958); published June 7, 2012
We epo on an expe imen al se up o single-pa icle o bi acking, which allows ollowing luo escen
nanopa icles o mo e han 10 min wi h a empo al esolu ion o 4 ms and a dynamic posi ion accu acy o be e
han 10 nm. On a model sample—20 nm sized luo escen polyme beads in glyce ol—we will illus a e how
a i ac s caused by una oidable expe imen al sho comings (migh ) obscu e he expe imen al esul and how
misin e p e a ions can be p e en ed. © 2012 Op ical Socie y o Ame ica
OCIS codes: 180.2520, 180.5810, 300.6280.
1. INTRODUCTION
The s udy o anspo p ocesses on molecula leng h scales is
o g ea impo ance in many ields o esea ch [1–9]. In he li e
sciences, eac ion kine ics may depend c ucially on he di u-
sion o he eac an s [10–12], o in ma e ial enginee ing he
low o ma e ial h ough an in e ace is o g ea impo ance
o il e ing and ca alysis [6,13]. Many me hods o s udying
such p ocesses ely on luo escence mic oscopy. Ini ially, his
was la gely es ic ed o luo escence eco e y a e pho o-
bleaching [14], which is an ensemble echnique ha p ohibi s
p obing o anspo p ocesses beyond he classical di ac-
ion limi o ligh mic oscopy. Mo eo e , due o ensemble
a e aging, complex di usion beha io migh ge masked in
pho obleaching expe imen s.
The si ua ion changed d as ically wi h he ad en o single-
molecule echniques. Con ocal mic oscopy lies a he hea o
luo escence-co ela ion spec oscopy (FCS) and i s a ian s
[9,15]. The disad an age o his me hod is ha he a e age
ansi ion ime o a pa icle h ough he de ec ion olume
o abou 1μm3is e y sho . Hence, one is ei he limi ed o
sho obse a ion imes o one a e ages sequen ially o e
many pa icles, making i di icul o ex ac in o ma ion abou
empo al o spa ial inhomogenei ies. Al e na i ely, wide- ield
luo escence mic oscopy has been employed o ollow he
di usion o indi idual nano-objec s. The gene al idea o
single-pa icle acking (SPT) is o de e mine he posi ion
o an indi idual pa icle by i ing i s di ac ion-limi ed image
on a CCD came a o he known poin -sp ead unc ion o
he mic oscope o , mo e p agma ically, by i ing i o a wo-
dimensional Gaussian. Associa ing he cen e o he i wi h
he posi ion o he pa icle allows de e mina ion o he spa ial
posi ion wi h an accu acy a beyond he classical di ac ion
limi o ligh mic oscopy and o ollow he di usion o his
pa icle wi h high p ecision [1,3,16–21]. Fascina ing expe i-
men s ha e been epo ed using SPT in combina ion wi h
luo escen ly labeled pa icles. Examples a e he di usion
o indi idual lipids in memb anes [4,22], he mo emen o
p o eins o quan um do s in cells [5,21,23], o he s udy o
biomolecula mo o s [24]. I e en has become possible o
ace he in ec ion pa hways o indi idual i uses [25].
While SPT allows o de e mining he spa ial posi ion o he
ace pa icle wi h excep ional accu acy, i p o ides only
limi ed empo al esolu ion, mainly gi en by he eadou ime
o he CCD came a. Ins ead o ollowing he spa ial posi ion o
a pa icle by eco ding a sequence o images in epi luo es-
cence mic oscopy, Ende lein p oposed a me hod nowadays
e med single-pa icle o bi acking in [26,27]. The e, he
exci a ion ligh is ocused in o he plane o he sample and
o a es on a ci cle ha encloses he pa icle o in e es ; see
Fig. 1. As long as he pa icle is loca ed p ecisely a he cen e
o he o bi , i expe iences a cons an exci a ion in ensi y, de-
spi e he a ia ion o he posi ion o he ocus. Upon any
mo emen o he pa icle away om he cen e , his si ua ion
changes d as ically and he exci a ion in ensi y, and concomi-
an ly he in ensi y o he emi ed luo escence I om he
pa icle, becomes modula ed wi h he o a ion equency o
he ligh o bi , acco ding o
I I0exp −2
w2xp−Rcosω 2
· exp −2
w2yp−Rsinω 2IB:(1)
He e I0co esponds o he maximum emission in ensi y, R o
he adius o he ligh o bi , w o he wais o he ocused
exci a ion beam, xPand yP o he coo dina es o he pa icle
in he plane o he ligh o bi , ω o he cycle equency o he
ligh o bi , and IB o he backg ound in ensi y. Demodula ion
o he emission signal yields he coo dina es o he pa icle
xpw2
2RRT
0I cosω d
RT
0I d ;y
pw2
2RRT
0I sinω d
RT
0I d ;(2)
and a eedback loop can be implemen ed ha ollows he ajec-
o y o he pa icle xp ;y
p as a unc ion o ime.
E ns e al. Vol. 29, No. 7 / July 2012 / J. Op . Soc. Am. A 1277
1084-7529/12/071277-11$15.00/0 © 2012 Op ical Socie y o Ame ica
ha he co ec ions (3) and (4) a e signi ican only a sho
ime scales o small mo emen s, espec i ely. As soon as τ≫
Δ o MSD ≫σ2is ul illed, he in luence o hese co ec ions
on he esul s is negligible.
4. EXAMPLE
As an example, we p esen an expe imen whe e we s udied
he di usion o a 20 nm sized ace pa icle in pu e glyce ol.
This sample was chosen, because glyce ol does no o m
ne wo k s uc u es and no mal, B ownian di usion can be ex-
pec ed [42], i.e., a linea dependence o he MSD o he aces
as a unc ion o he lag ime. O mo e o mally MSDτ∼τα,
wi h α1.
We eco ded ajec o ies wi h 150,000 posi ions, co e-
sponding o an expe imen al ime o 10 min. In o de o com-
pensa e o he dec ease o he emission signal in he cou se
o ime, o example due o bleaching e ec s, he exci a ion
powe was a ied be ween 130 nW–1.5 μW, which ensu ed a
a he cons an emission o abou 1.5×105cps (coun s pe
second) du ing he expe imen . In mos o he expe imen s,
he mean exci a ion powe was abou Pex 560 nW, which
co esponds o an exci a ion in ensi y o Iex 250 W∕cm2
in he ocus o he mic oscope objec i e. The o a ion
equency o he ligh o bi was se o ν1kHz wi h P4
pe iods o o a ion, esul ing in a bin ime o Δ 4ms. The
pa ame e s o he z acking we e Nz10 and s5
(≈15 nm). We eco ded ajec o ies om se en di e en pa -
icles in o de o compensa e o he (sligh ) dispe sion in he
diame e s o he beads and o possible a ia ions o hei
shapes.
In Fig. 8 he i s 50 da a poin s o he “ensemble”a e age o
ime-a e aged MSDs a e shown o di e en s ages o he co -
ec ions. The MSD ha has been calcula ed a e ca ying ou
co ec ions (1) and (2) o he coo dina es is shown by he
black colo code (squa es), he one ha esul s a e s ep
(3) is gi en by he blue colo code (ci cles), and inally he
ully co ec ed MSD [a e s ep (4)], is indica ed by he ed
colo code ( iangles). In o de o show he di e ences be-
ween he h ee MSDs mo e clea ly, some da a poin s a e
shown on an expanded scale in he op igh inse o Fig. 8(a).
The solid lines in Fig. 8(a) e e o linea i s, which di e only
by hei o se ; see inse , op le . F om he slope we ob ain
he di usion coe icien Dexp 0.0178 μm2∕s ha can be
compa ed wi h he p edic ion acco ding o he S okes–
Eins ein ela ion D heo kBT∕6πηa. He e a e e s o he a-
dius o he pa icle, η o he iscosi y o he medium, T o
he empe a u e, and kB o he Bol zmann cons an . Using
T294 K, η1.2Pa · s [42], and a10 nm, we ob ain
D heo 0.0179 μm2∕s in nea ly pe ec ag eemen wi h he ex-
pe imen al alue. The di e ences in he o se s o he h ee
MSD cu es become clea in he op le inse o Fig. 8(a),
which shows he ex apola ion o he h ee i ed cu es
s a
end
x
y
-47 nm
-38 nm
Fig. 7. (Colo online) Measu ing he low in he sample chambe . (a) Sequence o 1000 da a poin s (4 s) aken om a long ajec o y o 150,000 da a
poin s. The g ay a ow co esponds o he displacemen ec o o he pa icle du ing he 4 s. (b) Sca e plo o 1050 displacemen ec o s om
se en independen ajec o ies. The ec o s ha e been shi ed wi h hei s a ing poin o a common o igin. (c) His og am o he xcomponen o he
displacemen ec o s. The ull line co esponds o a Gaussian i cen e ed a a mean o −47 nm. (d) His og am o he ycomponen o he dis-
placemen ec o s. The ull line co esponds o a Gaussian i cen e ed a a mean o −38 nm.
1284 J. Op . Soc. Am. A / Vol. 29, No. 7 / July 2012 E ns e al.
owa d τ0. Once he di usion coe icien is ob ained, he
o he wo co ec ions, i.e., adding 4∕3DΔ and sub ac ing
he emaining o se a e i ial. F om he las co ec ion,
we ob ain 2σ2, which p o ides he dynamic posi ion accu acy
o σ7.5nm o he example shown he e.
In o de o enhance he isibili y o any de ia ion om
α1, Fig. 8(b) displays he same da a on a double loga i h-
mic scale. The lines a e linea i s wi h slopes α12 1.00
(black squa es), α30.98 (blue ci cles), and α41.03 ( ed
iangles), which all ag ee e y well wi h bo h he da a and
he expec a ion. He e he indices e e o he le el o co ec-
ions. Howe e , i should be no ed ha only he ully co ec ed
MSD obeys log MSDταlog τlog 4D, which allows ex-
ac ing he scaling exponen αby linea i ing. The o he wo
exponen s (α12 and α3) should be ega ded only as appa en
scaling pa ame e s α ollowing he no a ion in [40]. Ne e he-
less, o any le el o co ec ion, he de ia ions o hese pa am-
e e s om α1a e wi hin he expe imen al accu acy,
es i ying ha he expec ed B ownian mo ion is e ealed,
and ha he expe imen s we e pe o med in a egime whe e
co ec ions (3) and (4) we e o mino impo ance.
An al e na i e me hod o analyze single-pa icle ajec o ies
wi h espec o he di usion beha io , elies on he cumula-
i e dis ibu ion unc ion [CDF( 2τ)] o he squa ed displace-
men s 2 o a ce ain lag ime τ[43,44]. The analysis o
ou da a acco ding o his p o ocol is gi en in de ail in
Appendix A.4 and ep oduces B ownian mo ion.
5. CONCLUSION
We ha e desc ibed he de ails o an expe imen al se up ha
exploi s o bi acking o ollow he mo emen o an indi i-
dual luo escen pa icle wi h a posi ion accu acy a beyond
he classical di ac ion limi . The igu es o me i depend on
he dynamics o he sys em and on he numbe o de ec ed
pho ons pe uni ime. Typical alues o di usion in iscous
media ha we achie ed a e a posi ion accu acy be e han
10 nm, a empo al esolu ion o 4 ms, and a o al obse a ion
ime o mo e han 600 s. This allowed eco ding o ajec o ies
o he pa icle mo emen consis ing o mo e han 105da a
poin s p o iding excellen s a is ics o da a e alua ion. How-
e e , he expe imen al ealiza ion o o bi acking was hin-
de ed by se e al sho comings leading o a i ac s in he
da a ha migh misleadingly be in e p e ed as an unde lying
anomalous di usion p ocess. We ha e p esen ed a de ailed
cha ac e iza ion o hese p oblems, elucida ed he o igin o
se e al a i ac s, and showed how o co ec he da a acco d-
ingly. The ope a ion o he se up and he in luence o he a -
i ac s on he da a we e illus a ed o a model sys em om
which i is known ha i ea u es no mal di usion. We ha e
demons a ed ha he no mal di usion can be e ealed on all
expe imen ally accessible ime scales, i he co ec ions o he
a i ac s a e p ope ly aken in o accoun . The ag eemen be-
ween he measu ed and calcula ed di usion coe icien is
be e han 1%.
APPENDIX A
1. Beam Pa h Calcula ion
The op ical sys em o he exci a ion ligh was designed using
ay ans e ma ix analysis [45], which has been p o en o be
a powe ul ool o ace a Gaussian beam h ough a complex
op ical se up. The beam is ep esen ed by a wo-componen
ec o bb; φwhe e bco esponds o he dis ance o
he beam om he op ical axis and φ o he angle be ween
he p opaga ion di ec ion o he beam and he op ical axis.
Each op ical elemen can be ep esen ed by a ans e ma ix
ha ela es a gi en inpu ec o bin o a dis inc ou pu ec o
bou . The op ical pa h o a se ies o op ical elemen s is hen
simply calcula ed by mul iplying he espec i e ans e
ma ices. Fo he se up desc ibed in he ex , we ind o
he ans e ma ix o he ull sys em
bou
φou − 7 5
6 −1
2
7 5
6
2 6
7 50·bin
φin ;(A1)
whe e he mic oscope objec i e has been app oxima ed as a
single lens wi h 73mm. All o he ocal dis ances a e gi en
in he ex . This simple se o linea equa ions allows o cal-
cula e he adius o he ligh o bi (Rou bou ) and i s o ien-
a ion wi h espec o he op ical axis (φou ) as a unc ion o
he incoming beam adius bin and de lec ion angle φin. He e
we used bin 0;0.76 m ads a ing a he cen e o he i s
AOD and ob ain bin 190 nm;0, i.e., a ocused lase beam
ha o a es on a cylinde ha ing he op ical axis as symme y
axis; see Fig. 9.
Fig. 8. (Colo online) Ensemble a e age o ime-a e aged MSDs
om se en ajec o ies aken om di e en 20 nm sized beads in gly-
ce ol a di e en s ages o he co ec ion. (a) Linea plo : aw da a
(black squa es), da a co ec ed o posi ion a e aging (blue ci cles),
and ully co ec ed da a ( ed iangles). All lines co espond o linea
i s. The op igh inse shows he same da a a a scale ha has been
expanded by a ac o o 100. The op le inse shows he ex apola ion
o he i s owa d τ0and he espec i e in e cep s wi h he MSD
axis. Fo his example, we ob ain Dand σas gi en in he igu e.
(b) Same da a on double loga i hmic axis. The lines a e linea i s
o he da a wi h slopes o 1.00 (black squa es), 0.98 (blue ci cles),
and 1.03 ( ed iangles). Fo mo e de ails, see he ex .
E ns e al. Vol. 29, No. 7 / July 2012 / J. Op . Soc. Am. A 1285
2. Radia ion P essu e
The o ce exe ed on a luo escen pa icle due o he inciden
pho ons is gi en by FPPex∕c, whe e Pex deno es he exci-
a ion powe and c he speed o ligh . Fo Pex ≈1μW, which is
ypical o ou expe imen s, and he ex eme assump ion ha
all pho ons would be abso bed, his yields a o ce o abou
10−15 N. This igu e has o be compa ed wi h he o ce exe ed
on he pa icle ( adius a) due o ic ion in a medium ( iscosi y
η), which is gi en by FS okes 6πηa. E en o a low- iscosi y
medium like wa e (η10−3Pa · s), his yields o a pa icle
wi h a adius o 10 nm a o ce o FS okes ≈10−10 N, exceeding
he o ce induced by he adia ion p essu e by se e al o de s
o magni ude.
3. Op ical T apping
The po en ial o an op ical ap is gi en by
U ap 4Pexn4
ma3
w2cm2−1
m22;(A2)
whe e Pex e e s o he exci a ion powe , nm o he index o
e ac ion o he medium, a o he adius o he pa icle, w o
he beam wais , and mnp∕nm o he a io o he indices o
e ac ion o he pa icle and he medium, espec i ely
[46,47]. Using he nume ical alues om ou expe imen s,
i.e., Pex 1μW, nm1.33,np1.65,a10 nm, and w
270 nm, we ob ain a apping po en ial o abou U ap ≈
10−26 J, which is o de s o magni ude smalle han he he mal
ene gy a oom empe a u e, which amoun s o U he m
kBT≈10−21 J.
4. Cumula i e Dis ibu ion Func ion
The empi ical CDF ( 2,τ) o a ce ain lag ime τ(da a no
shown) is de e mined by coun ing he numbe o squa ed
displacemen s smalle o equal o 2. Fo no mal di usion
a single-exponen ial cu e is expec ed, while in sys ems wi h
anomalous di usion de ia ions om his unc ion a e aken
in o accoun by a double-exponen ial cu e gi en as
CDF 2;τ1−βexp − 2
2
11−βexp − 2
2
2:(A3)
He e, he 2
ia e i pa ame e s ha a e ela ed o he di usion
coe icien s, commonly in e p e ed as a as and a slow com-
ponen , weigh ed wi h a pa ame e β. Fo B ownian mo ion,
his educes o he single-exponen ial; i.e., β1[43,44].
F om ou da a we calcula ed he CDFs o lag imes
τ2–50Δ , and we de e mined βacco dingly. Figu e 10 dis-
plays he CDF o τ40 ms as a ypical example, which
shows a nea ly pe ec ag eemen be ween he da a and
he i . A e aging o e all lag imes τ2–50Δ and se en
ajec o ies yields β0.96. I should be no ed ha he
CDF analysis does no include he co ec ions o he posi ion
a e aging and he ini e signal- o-noise a io [co ec ion s eps
(3) and (4)]. This analyzing me hod gi es only limi ed
in o ma ion abou he deg ee o subdi usion, bu is sui ed
o dis inguish no mal om anomalous di usion.
ACKNOWLEDGMENTS
We hank We ne Köhle , Flo ian Schwaige , and Flo ian
Sp ei le o ui ul discussions and g a e ully acknowledge
inancial suppo om he Ge man Science Founda ion
(DFG) wi hin he amewo k o he Resea ch Uni “Nich li-
nea e Dynamik komplexe Kon inua”(FOR 608).
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Publica ion P2
Measu ing a di usion coe icien by single-pa icle
acking: S a is ical analysis o expe imen al
mean-squa ed-displacemen cu es
Dominique E ns and Jü gen Köhle
published in:
Phys. Chem. Chem. Phys. 15, 845-849, (2013)
Elec onic supplemen a y in o ma ion (ESI) a ailable
c
Royal Socie y o Chemis y 2013
h p://dx.doi.o g/10.1039/c2cp43433d
This jou nal is c he Owne Socie ies 2013 Phys. Chem. Chem. Phys., 2013, 15, 845--849 845
Ci e his: Phys.Chem. Chem.Phys., 2013,
15, 845
Measu ing a diffusion coefficien by single-pa icle
acking: s a is ical analysis o expe imen al mean
squa ed displacemen cu es†
Dominique E ns and Ju
¨ gen Ko
¨hle *
We p o ide expe imen al esul s on he accu acy o diffusion coefficien s ob ained by a mean squa ed
displacemen (MSD) analysis o single-pa icle ajec o ies. We ha e eco ded e y long ajec o ies
comp ising mo e han 1.5 10
5
da a poin s and decomposed hese long ajec o ies in o sho e
segmen s p o iding us wi h ensembles o ajec o ies o a iable leng hs. This enabled a s a is ical
analysis o he esul ing MSD cu es as a unc ion o he leng hs o he segmen s. We ind ha he
ela i e e o o he diffusion coefficien can be minimized by aking an op imum numbe o poin s in o
accoun o i ing he MSD cu es, and ha his op imum does no depend on he segmen leng h.
Ye , he magni ude o he ela i e e o o he diffusion coefficien does, and achie ing an accu acy in
he o de o 10% equi es he eco ding o ajec o ies wi h abou 1000 da a poin s. Finally, we
compa e ou esul s wi h heo e ical p edic ions and ind e y good quali a i e and quan i a i e
ag eemen be ween expe imen and heo y.
1 In oduc ion
Diffusion is o en exploi ed o examine in e ac ions and mo e-
men s o indi idual nanoscopic objec s in so ma e and/o
biological en i onmen s on a molecula leng h scale.
1–10
Ea ly
expe imen al wo k was ca ied ou using luo escence eco e y
a e pho obleaching (FRAP)
11
which yields he ensemble a e age
o he diffusing pa icles and which is, dic a ed by he diff ac ion
o ligh , es ic ed in spa ial esolu ion o leng h scales o abou
200–300 nm. Al e na i ely, esea che s employed luo escence
co ela ion spec oscopy (FCS),
12,13
which p o ides he a e age
o e a numbe o indi idual objec s ha a e egis e ed sequen-
ially and om which i is assumed ha hey beha e uni o mly.
Since abou wo decades single-pa icle acking (SPT) became a
aluable ool o map ou he mo emen o an indi idual pa icle
wi h high spa ial and empo al esolu ion.
1–4,14–21
The de eloped
me hodology co e s echniques whe e he mo emen o an
indi idual pa icle can be ollowed by eco ding i s diff ac ion-
limi ed image on a sequence o CCD ames,
1
sophis ica ed
app oaches ha compensa e he B ownian mo ion using
elec okine ic o ces,
14
echniques ha use s uc u ed illumina-
ion by ac i ely designing he poin -sp ead unc ion o he
mic oscope,
21,22
as well as me hods ha ely on a spa ial modula-
ion o he ligh ha a els o o comes om he pa icle.
19
Fascina ing esul s ha e been ob ained, o example in biophysics
he mo emen o molecules, i uses, o mo o p o eins could be
made isible,
4–6
and in he ma e ials science anspo p ocesses
h ough nanopo ous s uc u es
7,8
o he mani es a ion o diffusion
anomalies in liquid c ys als and mesopo ous s uc u es could be
ollowed.
9,23,24
Typically he luo escence o a pa icle is moni o ed as a
unc ion o ime and he posi ion o he pa icle is ex ac ed
om he da a wi h sub-diff ac ion limi ed accu acy. This p o ides
he ajec o y ( ) o he pa icle ha is commonly analysed in
e ms o he mean squa ed displacemen (MSD). Fo a 2-dimen-
sional diffusion p ocess he MSD gene ally scales wi h a powe law
acco ding o MSD( )=4D
˜
a
,whe eD
˜is he gene alized diffusion
coefficien , and a he anomaly pa ame e . Fo a= 1 he unde lying
p ocess co esponds o no mal diffusion (B ownian mo ion)
25
and
D
˜ educes o he diffusion coefficien Dknown om Eins ein.
26
O he wise he p ocess is called subdiffusi e (ao1) o supe -
diffusi e (a>1).
Fo ob ious easons an expe imen al ajec o y can only be
eco ded wi h a dis inc empo al esolu ion, i suffe s om
localisa ion e o s due o he mo emen o he pa icle du ing
da a acquisi ion,
27
i is affec ed by una oidable signal- o-noise
Expe imen al Physics IV and Bay eu h Ins i u e o Mac omolecula Resea ch (BIMF),
Uni e si y o Bay eu h, 95440 Bay eu h, Ge many. E-mail: jue gen.koehle @uni-
bay eu h.de; Fax: +49 921 55 4002; Tel: +49 921 55 4000
† Elec onic supplemen a y in o ma ion (ESI) a ailable: Analysis o all measu ed
ajec o ies. See DOI: 10.1039/c2cp43433d
Recei ed 28 h Sep embe 2012,
Accep ed 13 h No embe 2012
DOI: 10.1039/c2cp43433d
www. sc.o g/pccp
PCCP
PAPER
846 Phys. Chem. Chem. Phys., 2013, 15, 845--849 This jou nal is c he Owne Socie ies 2013
limi a ions
28
and las bu no leas i is inhe en ly o ini e
leng h. Hence, i is o c ucial impo ance o unde s and how
accu a e he diffusion coefficien can be ex ac ed om a eal
expe imen al MSD cu e.
25,29,30
Al hough he ma hema ical
amewo k o he MSD analysis is known o many yea s, he
implica ions o he expe imen al limi a ions on he accu acy o
he measu ed pa ame e s ha e been conside ed in de ail only
ecen ly.
29,31
These s udies add ess he achie able p ecision o
he diffusion coefficien ha can be ob ained om a gi en MSD
cu e as a unc ion o he expe imen al pa ame e s men ioned
abo e. In o de o es hei heo e ical esul s he au ho s had
o ely on ic i ious acking expe imen s based on simula ions
a he han on expe imen al da a. The eason is ha a sophis-
ica ed s a is ical analysis o he acking da a equi es a e y
la ge da a se which is difficul o ob ain, because he pa icle
migh ge los o acking due o diffusion ou o he ocal
olume o due o pho obleaching. O en i is al eady a g ea
challenge o egis e ajec o ies consis ing o some hund ed
da a poin s. Nai ely speaking, a ajec o y o a bi a y leng h
could be acqui ed by epea ing a acking expe imen unde
exac ly he same expe imen al condi ions on se e al nano-
pa icles. Howe e , since e en nominally iden ical nanopa icles
a e sligh ly diffe en in shape and size, he s a is ics o he
(unknown) size dis ibu ion o he nanopa icles will be supe -
imposed on he s a is ics o he diffusion coefficien . Mo eo e ,
he p ecision wi h which he diffusion coefficien can be de e -
mined om a MSD cu e depends on he accu acy o he MSD
da a poin s and on he numbe o i ing poin s ha a e aken
in o accoun .
29,30
The e o e, he nume ous heo e ical and
nume ical de elopmen s ha allow assessmen o he expe i-
men al sho comings s ill awai expe imen al e i ica ion.
In his wo k we use single-pa icle o bi acking, which allows
us o ob ain single-pa icle ajec o ies ha consis o mo e han
1.5 10
5
da a poin s wi h a empo al esolu ion o 4 ms and a
spa ial accu acy o be e han 10 nm.
32
Such a long ajec o y can
be di ided in o a sequence o segmen s, whe e each segmen can
be conside ed as an indi idual ajec o y ha , by de ini ion, has
been eco ded on exac ly he same pa icle unde iden ical expe i-
men al condi ions. This enables us o e alua e he s a is ics o he
diffusion coefficien ex ac ed om he segmen s as a unc ion o
he leng h o he segmen s and o compa e he esul s wi h he
heo e ical p edic ions made in he li e a u e.
25,29
2 Expe imen al sec ion
2.1 Sample p epa a ion
Fo he single-pa icle acking expe imen s we use luo escen
beads wi h a diame e o 20 nm ha a e loaded wi h nile ed
(Molecula P obes, 20 mg ml
1
dissol ed in wa e ). This
solu ion is u he dilu ed in wa e o a concen a ion o 0.1 nM
and subsequen ly mixed wi h pu e glyce ol (Sigma) esul ing in a
concen a ion o 2 pM o he ace s. F om ha solu ion a d op o
abou 25 ml is sandwiched be ween wo mic oscope co e slips ha
a e cleaned wi h ace one. In o de o p e en e apo a ion o he
sol en (and he esul ing low ield in he sample) he edges o he
co e slips a e sealed wi h g ease (High-Vacuum G ease, Wacke ).
This cons uc is moun ed on op o a 3-axis piezo s age (T i o
102, piezosys em Jena) p o iding a scan ange o 100 mm o
each axis. All expe imen s a e pe o med a oom empe a u e,
i.e. (21 0.5) 1C.
2.2 Expe imen al se up
The home-buil se up o single-pa icle o bi acking has been
desc ibed in g ea de ail in a sepa a e pape .
32
B ie ly, he
ou pu om an A /K -ion lase (Inno a 70C Spec um, Cohe en )
ope a ed a 514 nm is guided h ough a de lec ion uni consis ing
o wo mu ually pe pendicula a anged acous o op ical de lec o s
(AOD, DTSX-400-532, Pegasus) ha gene a e a o a ing ligh o bi .
This o bi is p ojec ed ia a dich oic beam spli e (z532RDC, AHF)
owa ds an in ini y-co ec ed wa e -imme sion objec i e
(UPLSAPO, 60, NA = 1.2, Olympus). This esul s in a ocussed
lase beam wi h a wais o w= 270 nm ha o a es on an o bi wi h
a adiuso R= 190 nm in he ocal plane o he objec i e. The
equency o he o a ion can be adjus ed by he AODs and is se
o 1 kHz.
The emission o he luo escen nanopa icles is collec ed
wi h he same objec i e, passes he dich oic and is ocussed
ei he on o he chip o a CCD (sensicam qe, PCO) o an
a alanche pho o diode (SPCM-AQR-14, Pe kin Elme ). Residual
lase ligh ha passes he dich oic is supp essed by a dielec ic
op ical il e (HQ545LP, OD = 6 a 514 nm, AHF). To spo he
loca ion o he ace s we ope a e he se up in wide ield mode.
The e o e he de lec ion uni is swi ched off and an addi ional
lens in he exci a ion pa h de ocusses he lase ligh o an a ea
o 80 80 mm
2
. Wi h he aid o he piezo s age an app op ia e
pa icle is mo ed close o he posi ion whe e he ligh o bi will
appea (cen e o he ield o iew). Subsequen ly, he op ics a e
swi ched o con ocal mode, he ligh o bi is gene a ed and he
algo i hm o au oma ed acking is s a ed.
We eco d he emission in ensi y o he luo escen pa icle
which is modula ed by he equency o he o a ing lase ocus.
By demodula ing his emission signal we a e able o calcula e
he x-, y-posi ion o he pa icle wi h espec o he cen e o he
o bi . The posi ion p o ides a eedback signal o he piezo and
he pa icle is mo ed ( oge he wi h he sample) back o he
cen e o he o bi . These s eps (collec emission – calcula e
posi ion – mo e piezo) a e epea ed con inuously, which allows
us o econs uc he mo emen o a luo escen ace pa icle
o mo e han 10 minu es wi h a spa ial esolu ion o be e
han 10 nm. The empo al esolu ion o he expe imen s is
D = 4 ms which esul s in ajec o ies o N= 1.5 10
5
x-,
y-posi ion pai s.
3 Resul s and discussion
An example o a ypical ajec o y measu ed wi h ou se up is
displayed in Fig. 1. I ep esen s 1.52 10
5
da a poin s and
co esponds o an elapsed ime o 608 s which is indica ed by
he colou code, whe e blue co esponds o he s a o he
ajec o y and ed o i s end. In o de o mimic o ha e only
sho e ajec o ies wi h less da a poin s we cu he long
ajec o y in o segmen s ha we e ea ed as independen
Pape PCCP
This jou nal is c he Owne Socie ies 2013 Phys. Chem. Chem. Phys., 2013, 15, 845--849 847
ajec o ies o sho e leng h. In he ollowing we deno e he
leng h o he ull ajec o y as N
T
(he e N
T
= 1.52 10
5
) and he
leng h o a segmen as N
seg
. Fo ou s udy we choose N
seg
= 10,
20, 40, 60, 80, 100, 200, 400, 600, 800, and 1000, which yields
ensembles o N
T
/N
seg
sho ajec o ies o equal leng h. The
idea is now o de e mine he diffusion coefficien D om he
slope o he ime-a e aged MSDs o each segmen and o
examine he s a is ical a ia ion o Dwi hin each ensemble o
ajec o ies. Ye , acco ding o e . 29 he e exis s an op imum
numbe o da a poin s o he MSD ha should be conside ed o
ob ain he bes esul o he diffusion coefficien . This can be
unde s ood as ollows. Fo inc easing lag imes he accu acy o
he da a poin s in he MSD dec eases due o he p og essi ely
dec easing a e aging o he a ailable da a. Fo example, he
i s da a poin o he MSD ep esen s an a e age o e (N
seg
1)
posi ions o he pa icle whe eas he las da a poin has no
been a e aged a all. Hence, i ing he slope o he MSD cu e
by aking oo many da a poin s in o accoun leads o a
de e io a ion a he han an imp o emen o he esul . On
he o he hand, he e y i s poin s o he MSD a e s onge
subjec ed o localisa ion e o s, ei he due o noise (s a ic e o )
o due o blu ing o he posi ion o he pa icle du ing da a
acquisi ion (dynamic e o ). Bo h effec s a e age ou o MSD
poin s a longe lag imes.
As a consequence o his, we i s ha e o ind ou he
op imum numbe o da a poin s ha should be conside ed
o i ing he slope o he MSD. In he ollowing, he p o ocol
o doing so will be explained on he example o N
seg
= 1000
which yields an ensemble o 152 ajec o ies o equal leng h
and he same numbe o MSD cu es. Fo his ensemble we
i ed he slope, D*, o each MSD cu e by an unweigh ed linea
i o he i s nda a poin s. Mo e p ecisely, we ha e skipped he
e y i s da a poin o he MSDs, because i u ned ou ha i is
s ongly affec ed by esidual oscilla ions o he piezo. These
oscilla ions affec he posi ion de e mina ion and educe he
accu acy o he i s poin o he MSD cu e, whe eas he
in luence o hese oscilla ions on he accu acy o he succeeding
MSD poin s le el off ( o de ails see Expe imen al sec ion and
e . 32). The e o e he i was applied o he da a poin s om 2 o
(n+ 1) and he slope D* o he MSD cu es was de e mined as a
unc ion o n. In o de o be compa ible wi h he exis ing
li e a u e we p e e he slope D* o he MSD cu es a he han
he diffusion coefficien D=D*/4.
29
An example o he dis ibu-
ion o D* is shown in he op igh inse o Fig. 2 o n=4,i.e.
aking only he da a poin s 2–5 o i ing he MSDs in o accoun
as indica ed schema ically in he op le inse o Fig. 2. Sub-
sequen ly, we de e mined om each his og am he i s and he
second momen p o iding he empi ical mean Dand he
empi ical s anda d de ia ion s
D
*
o his pa ame e , and plo ed
he a io sD=Das a unc ion o he numbe o i ing poin s n.
The esul o his p ocedu e is shown in Fig. 2 o he examples
o N
seg
= 100 and N
seg
= 1000. Fo bo h samples, he ela i e e o
sD=D i s dec eases o g owing nand hen apidly inc eases i
mo e i ing poin s a e aken in o accoun . He e we ind an
op imum o he accu acy o he slope o he MSDs o n=4.
While he ela i e accu acy ha can be achie ed o D*(abou 8%
o N
seg
= 1000, and abou 25% o N
seg
= 100) clea ly depends on
he leng hs o he segmen s, i is in e es ing o no e ha he
numbe o i ing poin s n ha yield he op imum esul
does no .
In o de o acili a e a quan i a i e compa ison o he
da a shown in Fig. 2 wi h he heo e ical p edic ions in he
li e a u e
29
we ha e o eso o he educed localisa ion e o
x=s
2
/DD , whe e sis he localisa ion e o , D he diffusion
Fig. 1 Example o a ajec o y o a 20 nm sized bead in pu e glyce ol. The colou
code e e s o he elapsed ime o 608 s (N= 1.52 10
5
da a poin s; blue
co esponds o he s a and ed co esponds o he end).
Fig. 2 Rela i e e o o he slope D* ob ained om unweigh ed linea i s o he
MSD cu e as a unc ion o he numbe o i ing poin s n o he segmen leng h
N
seg
= 100 (open symbols) and N
seg
= 1000 ( ull symbols). The inse op le
displays schema ically a MSD cu e as a unc ion o he lag ime and he da a
poin s ha a e conside ed o he linea i ( ed) o ob ain D*. Fo all i s he i s
da a poin o he MSD (b acke s) is igno ed ( o de ails see ex ). The inse op
igh shows as an example o he dis ibu ion o he slopes wi hin he ensemble
o N
T
/N
seg
ajec o ies o n= 4 and N
seg
= 100, om which D( i s momen ;
empi ical mean alue) and s
D*
(second momen ; empi ical s anda d de ia ion)
can be calcula ed.
PCCP Pape
Publica ion P3
F ac ional B ownian Mo ion in C owded Fluids
Dominique E ns , Ma cell Hellmann, Jü gen Köhle and Ma hias Weiss
published in:
So Ma e 8, 4886-4889, (2012)
Elec onic supplemen a y in o ma ion (ESI) a ailable
c
Royal Socie y o Chemis y 2012
h p://dx.doi.o g/10.1039/C2SM25220A
F ac ional B ownian mo ion in c owded luids†
Dominique E ns ,‡
a
Ma cel Hellmann,‡
b
J€
u gen K€
ohle *
a
and Ma hias Weiss*
b
Recei ed 30 h Janua y 2012, Accep ed 22nd Ma ch 2012
DOI: 10.1039/c2sm25220a
Di usion in c owded luids, e.g. in he cy oplasm o li ing cells, has
equen ly been epo ed o show anomalous cha ac e is ics (so-
called ‘subdi usion’). Se e al andom walk models ha e been
p oposed o explain hese obse a ions, ye so a an expe imen ally
suppo ed decision in a o o one o hese models has been lacking.
He e, we show ha expe imen ally ob ained ajec o ies in a p o o-
ypical c owded luid show an asphe ici y ha is mos consis en
wi h he p edic ions o ac ional B ownian mo ion, i.e. an an i-
co ela ed, an i-pe sis en gene aliza ion o no mal B ownian
mo ion ha is ela ed o he luid’s iscoelas ici y.
Mac omolecula c owding, i.e. a o al concen a ion o a a ie y o
mac omolecules up o 400 mg ml
1
, is a common phenomenon in
in acellula luids.
1
C owding can ha e a conside able impac on
(bio)chemical eac ions,
2
hence challenging insigh s de i ed om
biochemical assays in dilu e aqueous solu ions. The phospho yla ion
pa e n o he mi ogen-ac i a ed p o ein kinase (MAPK), o
example, has been shown o a y g ea ly wi h he deg ee o cy o-
plasmic c owding:
3
In dilu e solu ions, MAPK was wice phospho -
yla ed by i s kinase in a dis ibu i e manne , whe eas adding a i icial
c owding agen s esul ed in a p ocessi e phospho yla ion and hence
a mo e e icien ac i a ion o MAPK. Recen ly, a heo e ical expla-
na ion o hese esul s has been gi en in e ms o c owding-induced
anomalous di usion.
4
Indeed, c owding is known o s ongly al e
he di usional mobili y o mac omolecules.
5
Apa om a me e
educ ion o he di usion coe icien , i.e. an inc eased iscosi y o he
luid, anomalous di usion has also been equen ly obse ed in
c owded luids in i o
6–10
and in i o.
11–15
He e, he mean squa e
displacemen (MSD) o a di using pa icle was shown o scale o e
se e al decades as h ( )
2
i
a
wi h a< 1 (‘subdi usion’).
In spi e o he equen obse a ion o subdi usion, e en in ai ly
uns uc u ed luids in i o, an expe imen ally suppo ed and unam-
biguous explana ion o he e ec in e ms o a andom walk model
has emained elusi e. So a , h ee ypes o andom walks ha e been
conside ed as an explana ion o c owding-induced subdi usion: (1)
Obs uc ed di usion (OD), i.e. he mo ion o a ace pa icle in
a maze o immobile obs acles,
16
(2) ac ional B ownian mo ion
(FBM) due o he iscoelas ici y o he c owded luid,
15
and (3)
a con inuous ime andom walk (CTRW) in which he di using
ace akes powe -law dis ibu ed es s be ween pe iods o ee
di usion. The CTRW model is special since i shows weak e godici y
b eaking
17,18
whe eas OD and FBM a e e godic andom p ocesses
wi h s a iona y inc emen s. Recen expe imen al da a ha e indica ed
ha CTRW may be less well sui ed o explain c owding-induced
subdi usion
15,19
a leas on sho and in e media e ime scales.
20
The main p oblem in ela ing expe imen al da a o he abo e
models is a lack o de ailed in o ma ion on he di usion p ocess:
se e al echniques, e.g. luo escence co ela ion spec oscopy, only
epo he MSD and lea e all highe momen s o he di usion
p opaga o unde e mined. Single-pa icle acking (SPT) echniques
allow one o eco d indi idual ajec o ies and hence can o e come
his limi a ion.
21,22
Howe e , p ecise posi ion de e mina ion in SPT
equi es he collec ion o many pho ons o he mo ing ace which
se s limi a ions o he empo al esolu ion and he o e all leng h o
he eco ded ajec o y (due o bleaching o he dye). Ye , an
unambiguous deciphe ing o he andom walk model om ai ly
sho SPT ajec o ies, o en accompanied by an un a o able spa ial
and empo al esolu ion, is challenging.
He e, we ha e u ilized a as and p ecise single-pa icle acking
echnique o eco d pa icle ajec o ies wi h a leng h o 10
5
posi ions
and a spa io- empo al esolu ion o 10nm and 4ms. F om ajec o ies
in p o o ypical c owded and pu ely iscous luids, we ha e de e -
mined he ime- and ensemble-a e aged MSD o he di using pa icle
as well as he andom walk’s asphe ici y. As a esul , we ha e ound
ha a ansien , ye long-las ing subdi usion eme ged in a c owded
bu no in a pu ely iscous luid. The anomaly was associa ed wi h an
e godic mode o mo ion as e idenced by a ecen ly in oduced
e godic y b eaking pa ame e . Compa ing he andom walks’
asphe ici y wi h hose p edic ed by compu e simula ions o no mal
B ownian mo ion, FBM, CTRW, and OD, we ha e ound ha ou
expe imen al da a in c owded luids a e bes desc ibed by he FBM
model. Since FBM is closely ela ed o iscoelas ici y, we pu o wa d
he hypo hesis ha mac omolecula c owding equips luids wi h
iscoelas ic p ope ies ha en o ce a ac ional B ownian mo ion o
di using ace pa icles.
Single-pa icle acking (SPT) is equen ly limi ed by a poo
empo al and/o spa ial esolu ion as well as ai ly sho ajec o ies.
These limi a ions can be o e comeusinga acking echnique ha
has been de eloped wi hin he las ew yea s:
23–26
AGaussian ocus
ci cles a high speed a ound a luo escen pa icle wi h he pa icle
a
Expe imen al Physics IV, Uni e si y o Bay eu h, 95440 Bay eu h,
Ge many. E-mail: [email p o ec ed]
b
Expe imen al Physics I, Uni e si y o Bay eu h, 95440 Bay eu h,
Ge many. E-mail: [email p o ec ed]
† Elec onic supplemen a y in o ma ion (ESI) a ailable: Expe imen al
and Nume ical Me hods. See DOI: 10.1039/c2sm25220a
‡ These au ho s con ibu ed equally o his wo k.
4886 | So Ma e , 2012, 8, 4886–4889 This jou nal is ªThe Royal Socie y o Chemis y 2012
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loca ed a he posi ion o he s eepes g adien o he exci a ion
in ensi y. Taking a di usion s ep o escape his posi ion is compen-
sa ed by mo ing he sample s age ia a nega i e eedback loop.
Hence, he wo-dimensional cen e -o -mass mo ion can be acked
wi h a high spa ial and empo al esolu ion.
Using his app oach (see he ESI† o he schema ic se up and
echnical de ails), we we e able o ack luo escen beads (diame e
50nm) o up o en minu es wi h a empo al esolu ion o D ¼4ms
and a spa ial accu acy o D ¼10 nm. We ha e acked pa icles in
wo p o o ypical luids: (i) in a pu ely iscous solu ion ob ained by
mixing 60% suc ose (pe weigh ) in o wa e , and (ii) in a c owded
luid, whe e 30% dex an (500 kDa) was dissol ed in wa e . Fo he
la e , anomalous di usion has al eady been epo ed p e iously.
6,12
F om a sligh ly highe numbe o acqui ed ajec o ies, we ha e
e ained o each luid only hose 21 SPT ajec o ies o u he
analysis ha con ained 4.5 10
4
–1.5 10
5
posi ions wi hou blanks,
i.e. we disca ded hose ew ajec o ies in which a weak emission
signal lead o a ansien loss o he acked bead. The chosen
ajec o ies did no show any signs o d i . Rep esen a i e ajec o ies
o bo h luids a e shown in he ESI.†
As a i s s ep in he analysis, we calcula ed o each o he selec ed
ime aces
i
h ( ¼iD ) he ime-a e aged MSD,
D ð Þ2ET¼1
NkX
Nk
i¼1
ð i iþkÞ2
:(1)
Rep esen a i e ime-a e aged MSDs o suc ose and dex an
solu ions a e shown in Fig. 1a. To highligh he eme gence o
a di usion anomaly, we ha e di ided ou he leading o de o he
MSD, i.e. we ha e plo ed D( )¼h ( )
2
i
T
/ as a unc ion o . While
he pu ely iscous suc ose solu ion yielded a ho izon al line, D( )¼
cons ., a ansien powe -law decay eme ged o he c owded luid.
F om he ansien scaling D( )1/
0.2
(ob ained wi hin he g ey
shaded egion), we in e ed h ( )
2
i
T
0.8
o small and in e media e
ime scales. This obse a ion is in quan i a i e ag eemen wi h
p e ious epo s
15
on simila p obes. Beyond z1 s a c osso e
owa ds no mal di usion eme ges, i.e. D( ) ends owa ds a ho i-
zon al line. Indeed, his beha io is expec ed o all o he abo e
men ioned andom walk models o subdi usion since adap ing hem
o a physical sample equi es speci ica ion o a minimum and
maximum leng h/ ime scale.
To de e mine he anomaly o each ajec o y, we es ic ed he
i ing p ocess o he empo al ange 50 ms # #500 ms which is no
a ec ed by some emaining ine ia aces o he se up ( <50ms;see
discussion in he ESI†) bu also does no su e om he eme ging
c osso e o no mal di usion a la ge ime scales. The esul ing
anomaly alues, a, o all ajec o ies a e summa ized in Fig. 1b. A
clea sepa a ion o he da a o he pu ely iscous suc ose solu ion
(a e age: hai¼0.98) and he esul s o a c owded dex an solu ion
(a e age: hai¼0.82) can be seen.
F om he obse a ion h ( )
2
i
T
a
we can al eady in e ha he
CTRW model wi h i s dis inc weak e godici y b eaking canno
desc ibe he expe imen al da a since i p edic s
17,18
h ( )
2
i
T
. Indeed,
e en o a unca ed CTRW model wi h only a ansien scaling p(s)
s
(1 + a)
o he dis ibu ion o wai ing imes one obse es h ( )
2
i
T
(c . ESI†). Hence, e en a mo e ealis ic adap a ion o he CTRW
model appea s incompa ible wi h ou expe imen al da a.
Nex , we calcula ed o all ajec o ies an e godici y pa ame e
17
ha anishes i e godici y is p ese ed:
Eð Þ¼ D ð Þ2E2
T
E
D ð Þ2ET2
E
D ð Þ2ET2
E
(2)
To his end, we ha e cu each ajec o y in o segmen s o N¼3000
ime s eps and used hese segmen s o he ensemble a e aging h.i
E
.
As a esul , we obse ed ha o all ajec o ies hEi#0.03 (Fig. 1c).
He e, he a e age o E( ) was aken in he same empo al window in
which awas also de e mined. This esul s ongly suppo s he no ion
ha all ajec o ies we e e godic. In pa icula , ou da a sepa a es well
om he p edic ions o a non- unca ed CTRW model ha yields
a lowe bound E(a#0.9) $0.1.
17
Howe e , o he unca ed
CTRW model (c . ESI†) we also ob ained Ez0.03 on he expe i-
men ally ele an ime scale. We a ibu e his e ec o he unca ion
o p(s) which na ows he dis ibu ion o appa en di usion cons an s
in h ( )
2
i
T
.
17,18
Hence, based only on E, a clea -cu decision ha ou
expe imen al da a is inconsis en wi h a unca ed CTRW model is
no possible.
We nex inspec ed he ajec o ies’ shape o gain deepe insigh s
in o he unde lying ype o andom walk. The asphe ici y p o ides
a simple ye powe ul pa ame e o quan i y he shape o ac al
objec s like andom walks.
27
Diagonalizing he andom walk’s
gy a ion enso T
ij
(c . ESI†) yields he p incipal axes o gy a ion and
Fig. 1 (a) Rep esen a i e ime-a e aged MSD, shown as D( )¼h ( )
2
i
T
/
o highligh he asymp o ic scaling. Da a o suc ose solu ions (blue
ci cles) ollows he an icipa ed scaling o no mal di usion (D( )¼
cons .). In con as , da a o dex an solu ions ( ed squa es) shows
a ansien subdi usion (dashed line, D( )1/
0.2
). Fo > 1 s a c osso e
o he asymp o ic scaling (a¼1, D( )¼cons .) is isible. The g ey shaded
egion indica es he empo al window in which he cu es we e i ed o
ex ac he anomaly a. (b) Anomaly alues a o each ajec o y as
ob ained om i ing he ime-a e aged MSD in he indica ed ime
window. A clea sepa a ion be ween a suc ose solu ion (blue ci cles,
hai¼0.98) and a c owded dex an solu ion ( ed squa es, hai¼0.82) is
e iden . (c) The e godici y b eaking pa ame e hEi[eqn (2)] o all
ajec o ies was e y small, indica ing e godici y.
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he co esponding eigen alues, i.e. he squa ed p incipal adii o
gy a ion, R
2
i
. Res ic ing ou sel es o wo dimensions ( he expe i-
men al ajec o ies a e wo-dimensional objec s), he asphe ici y is
de ined as
A¼h(R
2
1
R
2
2
)
2
i/h(R
2
1
+R
2
2
)
2
i. (3)
We no e ha Ain ol es an a e aging o e he ensemble o walks
(indica ed by h.i). The limi ing cases A¼0andA¼1 esemble
a pe ec sphe e and a simple od, espec i ely. Fo B ownian mo ion
in wo dimensions an exac alue is a ailable:
27
A¼4/7. Hence, e en
an indi idual ajec o y o a wo-dimensional B ownian andom
walk di e s d as ically om a ci cula shape a each ins an o ime.
The ime-a e aged o ien a ion o he longes p incipal axis o gy a-
ion, howe e , is iso opic. Mo eo e , he iso opy o di usion is also
eco e ed in an ensemble o pa icles due o he unco ela ed andom
o ien a ions o he gy a ion axes.
We ha e de e mined ia simula ions he alues o A o FBM, OD,
and a unca ed CTRW a a ying anomaly alues a(see ESI† o
de ails). Ou simula ion esul s e ealed ha o 0.5 #a#1, which is
he expe imen ally ele an egime, he asphe ici y changes almos
linea ly wi h a,i.e. A ¼m
1
a+b
1
. Fo OD we ound m
1
¼0.120
0.006, b
1
¼0.458 0.004 whe eas o FBM we ob ained m
1
¼
0.638 0.009,b
1
¼0.057 0.006. Fo he unca ed CTRW model
we ound Az4/7 i espec i e o a(c . ESI†). This esul can be
a ionalized by bea ing in mind ha a CTRW ajec o y a any
ins ance o ime looks simila o he pa h o no mal B ownian mo ion.
To compa e ou expe imen al ajec o ies o hese p edic ions, we
assigned he p e iously de e mined anomaly a o each ajec o y (c .
Fig. 1b). Then, we calcula ed he accompanying asphe ici y: Since he
anomaly e lec s a scaling o sho and in e media e imes, a consis-
en es ima e o he andom walk’s asphe ici y mus ela e o he same
ime scale. The e o e, each ajec o y was cu in o sequences o N¼
3000 ime s eps o leng h D , and he a e age o e hese sub- ajec-
o ies yielded he (mean) asphe ici y [eqn (3)] o he en i e ajec o y
on he leng h and ime scales du ing which anomalous di usion was
obse ed. As can be seen om Fig. 2a, he cloud o da a poin s o
suc ose solu ions o e laps well wi h he an icipa ed esul o no mal
B ownian mo ion, i.e. he mean o all 21 da a poin s (hai¼0.98 and
hAi¼0.58) ag ees quan i a i ely wi h he expec a ion a¼1andA¼
4/7 z0.57. Hence, suc ose solu ions indeed ea u e no mal B ownian
ajec o ies also om he geome ic pe spec i e. In dex an solu ions,
howe e , we ob ained hai¼0.82 and hAi¼0.46 which is mos
consis en wi h he simula ion esul s o he FBM model ha
p edic s locally a mo e sphe ical shape o he ajec o y due o he
an i-pe sis ence o he andom walk.
Gi en ha FBM is closely ela ed o he iscoelas ici y o non-
New onian and c owded luids,
13,28
he eme gence o subdi usion
may be aced back o ansien es o ing o ces on sho leng h and
ime scales. I is hence meaning ul o ansla e he SPT ajec o ies
in o he luid’s complex shea modulus,
28
G(u)¼G0(u)+iG00(u).
He e, he eal (imagina y) pa o G(u) ep esen s he elas ic ( iscous)
modulus o he luid. Employing a semi-analy ical app oach, we ha e
i ed he ime-a e aged MSD o each ajec o y by an empi ical
exp ession w( )¼a
0
a
+a
1
o cap u e he ansien anomaly and he
asymp o ic no mal di usion. The esul ing i pa ame e s we e hen
used o de e mine he complex shea modulus as desc ibed ea lie .
13
F om he ensemble o complex shea moduli o each luid, we ha e
de e mined he minimum and maximum alues o G0and G00.As
expec ed, suc ose showed a anishing elas ic con ibu ion whe eas he
c owded dex an solu ion showed a signi ican iscoelas ici y o la ge
equencies (Fig. 2b). Since high equencies a e ela ed o small imes,
his iscoelas ic beha io is in ima ely linked o he ansien sub-
di usion obse ed o small and in e media e imes. A simila
iscoelas ic beha io ( ela ed o subdi usion) has been obse ed o
he cy oplasm and nucleoplasm o li ing cells.
13,29
In conclusion, we ha e shown wi h an ad anced SPT app oach
ha a pu ely iscous suc ose solu ion ea u es no mal B ownian
mo ion o ace pa icles wi h an asphe ici y o he andom walk ha
ag ees e y well wi h analy ical p edic ions. In con as , di usion in
a c owded dex an solu ion was anomalous (‘subdi usion’). T ajec-
o ies showed no signs o e godici y b eaking and hei asphe ici y
was in quan i a i e ag eemen wi h p edic ions o he FBM model. In
con as , obs uc ed di usion (i.e., a s anda d andom si e pe cola-
ion model) and CTRW we e incompa ible wi h he expe imen al
da a. This esul is co obo a ed by he associa ed complex shea
modulus: A s ong iscoelas ic beha io o he c owded dex an
solu ion was seen a high equencies as expec ed due o he ela ion
o FBM wi h iscoelas ic media.
I is emp ing o specula e abou he easons and consequences o
ou inding in he con ex o li ing ma e . Since he deg ee o
Fig. 2 (a) Asphe ici y Aas a unc ion o he anomaly a(da a o suc ose
and dex an shown as blue ci cles and ed squa es, espec i ely). Mean
alues (s anda d de ia ion) a e indica ed by c oss hai s. Dashed lines
indica e simula ion esul s o OD and FBM. Da a o suc ose solu ions
a e in e y good ag eemen wi h he asymp o ic alue A¼4/7 o a¼1
(dash-do ed ma k), whe eas da a o dex an compa e a o ably o he
p edic ions o FBM. (b) Elas ic ( ed) and iscous (g ey) moduli, G0and
G00, as ob ained om he ensemble o ajec o ies in a c owded dex an
solu ion. Shown a e he minimum and maximum alues o G0and G00 a
each equency u,i.e. all ajec o ies lie wi hin he indica ed bands. Fo
low equencies he luid is almos comple ely iscous whe eas o u>
100 s
1
, a clea iscoelas ic beha io eme ges.
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cy oplasmic c owding appea s o be conse ed,
29
cells migh ha e
adap ed o highly c owded condi ions and aim a main aining his
s a e (c . also discussion in e . 2 and 29). Indeed, a po en ial bene i o
FBM-like subdi usion in cells is he inc eased e u n p obabili y o
a posi ion in h ee-dimensional space. In pa icula , FBM wi h a<2/
3 yields a bulk- illing andom walk ha can massi ely inc ease he
cap u e p obabili y o a a ge as compa ed o no mal di usion.
30
Mo eo e , an enhanced ebinding due o FBM mos likely is he
explana ion o he ecen ly obse ed phospho yla ion enhancemen
o MAPK unde c owded condi ions.
3,4
As an enhanced ecu ence is
a gene ic ea u e o FBM-like subdi usion, we expec ha he
beha io o a mul i ude o biochemical pa hways in cells will ha e o
be e isi ed and in e p e ed in ligh o ou indings.
Acknowledgemen s
DE and JK g a e ully acknowledge inancial suppo by Resea ch
Uni FOR608. MH was pa ly inanced by he Ge man-Is aeli
P ojec Coope a ion GA309/10. We would like o hank S e an Hain
o echnical suppo .
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F ac ional B ownian Mo ion in C owded Fluids – Supplemen
Dominique E ns 1, Ma cel Hellmann 2, J¨u gen K¨ohle 1, and Ma hias Weiss 2
1Expe imen al Physics IV, Uni e si y o Bay eu h, D-95440 Bay eu h, Ge many and
2Expe imen al Physics I, Uni e si y o Bay eu h, D-95440 Bay eu h, Ge many
I. EXPERIMENTAL APPROACH
Fluids o SPT expe imen s we e ob ained by dissol -
ing dex an (500 kDa, Sigma) and suc ose (342 Da, Ro h)
in millipo e wa e a concen a ions o abou 430 mg/ml
(30% w/w) and 1500 mg/ml (60% w/w), espec i ely.
Rhodamine- agged ace beads (50 nm, Polysciences)
we e added om a p edissol ed solu ion, esul ing in a
ypical concen a ion o abou 2 pM. Hence, 3-5 beads
we e obse ed in he ocal plane o he mic oscope’s wide-
ield image (80 ×80 µm2).
Abou 35 µl o each sample was placed be ween
ace one-cleaned co e slips and sealed on he edges wi h
highly iscous g ease o p e en e apo a ion o adhesion
o ces ha would induce a low ield.
T acking expe imen s we e pe o med wi h a home-
buil single-pa icle acking se up (Fig. 1) using a no el
acking echnique [1–5]. He e we only desc ibe he basic
concep , echnical de ails will be p esen ed elsewhe e.
The ou pu o an A /K -Ion lase (Inno a 70C Spec-
um, Cohe en ) a a wa eleng h o 514nm wi h a ci -
cula pola isa ion (due o a λ/4-wa epla e) was used as
an exci a ion ligh sou ce. The lase beam was di ec ed
h ough a se ies o wo pe pendicula ly a anged acous o
op ical de lec o s (AOD, DTSX-400-532, Pegasus), e-
sponsible o he gene a ion o an o bi ing lase beam
wi h o a ion equency . The o a ing lase beam was
hen passed h ough a elecen ic lens sys em and di-
ec ed in o a home-buil con ocal mic oscope. The lase
ligh was e lec ed by a dich oic beamspli e (z532RDC,
AHF) owa ds an in ini y co ec ed wa e -imme sion ob-
jec i e (UPLSAPO, 60x, NA=1.2, Olympus). The sam-
ple wi h he diffusing ace pa icles was moun ed on op
o a h ee-dimensional piezo s age. This se up allowed us
o c ea e an o bi adius Rin he ange o 0 o 5 µm
in he ocal plane o he objec i e. Sui able dye-labeled
pa icles in he icini y o he o a ing ocal spo we e
exci ed. The emi ed ligh was collec ed by he same
objec i e, passed he dic oic beamspli e and a u he
dielec ic il e (HQ525LP, OD=6 @ 514nm, AHF) o
supp ess emaining lase ligh . Finally i was ocussed
ei he on o he chip o a CCD (sensicam qe, PCO) o
an a alanche pho odiode (APD, SPCM-AQR-14, Pe kin
Elme ) wi h a sensi i e a ea o 180 µm in diame e .
The se up was capable o wo king in a wide ield and
a con ocal ope a ion mode. Fo he wide ield mode an
op ional lens in on o he mic oscope was lipped in o
he op ical pa h o de ocus he exci a ion o an a ea o
abou 80 ×80 µm2in he plane o he sample. In his
mode he de lec ion uni is se o a neu al s a e (no de-
lec ion). The diffusing pa icles a e loca ed wi hin he
0s
400s
2 m
suc ose
2 m
dex an
μ μ
A /K -Ion lase
AOD AOD
Piezo
APD
CCD
Mic oscope & sample s age
L4L3L2L1LpLp
L5L6
L8
L7
F
l/4
Calcula ion uni
A cos(w )
A sin(w )
emission
x
y
LWF
Piezo s age
(a)
(b)
FIG. 1: (a) Ske ch o he expe imen al se up wi h lenses o
beam p o ile op imiza ion (Lp), a wide ield lense (LWF), a di-
elec ic il e (F), and an acous o op ical de lec o (AOD). Sig-
nals we e collec ed ei he wi h a CCD came a o an a alanche
pho odiode (APD). The calcula ion uni p o ided he d i e
signals o he AODs, ga he ed he emission in ensi y, cal-
cula ed he posi ions xand y, and ed he nega ed posi ion
o a piezo s age. (b) Rep esen a i e ajec o ies in a pu ely
iscous suc ose solu ion (le ) and a c owded dex an luid
( igh ); colo -coding blue o ed highligh s he empo al di-
ec ion o he ajec o y. The gy a ion ellipsoids which e lec
he andom walks’ asphe ici y a e supe imposed in g ey. Due
o a highe mobili y in he suc ose solu ion, he ellipsoids di -
e in size.
CCD image and a e mo ed by he piezo s age o a p ope
posi ion nea he cen e o he lase o bi . The con ocal
mode is subsequen ly used o pe o m he measu emen s.
By lipping he op ional lens back, he emission is now de-
ec ed wi h he APD and he de lec ion uni is swi ched
on. The emission in ensi y o he mo ing pa icle is mod-
ula ed wi h he known equency o he lase o bi .
Using a acking so wa e based on a lock-in echnique
[1, 3] we we e able o econs uc he wo-dimensional
mo ion o pa icles om he equency-modula ed luo-
escence signal: F om he de ec ed pho ons he posi ion
wi h espec o he o bi cen e was calcula ed, and he
piezo s age was ed wi h a signal co esponding o he
Elec onic Supplemen a y Ma e ial (ESI) o So Ma e
This jou nal is © The Royal Socie y o Chemis y 2012
2
nega ed posi ion. The whole ajec o y o he ace can
be econs uc ed by moni o ing he eedback signal o he
piezo.
Expe imen s we e done wi h an o bi equency o =
1kHz. E e y ou pe iods o o a ion he posi ion was
calcula ed esul ing in a ime esolu ion o ∆ = 4 ms.
The adius o bes acking pe omance depends on he
beam wais wo he ocal spo and was ound o be R=
w/√2 [6]. A ypical wid h o w= 270 nm lead o a adius
o R= 190 nm.
II. EVALUATION WINDOW OF
EXPERIMENTAL DATA
We ha e chosen o e alua e he scaling p ope ies o
he expe imen al da a in he empo al ange 50-500ms,
e.g. when inspec ing he MSD. Enla ging he window in
which he anomaly αand he co esponding asphe ic y A
we e de e mined will al e he s a ed numbe s. Ex end-
ing he i ange o la ge imes will include mo e o he
c osso e owa ds he asymp o ic no mal diffusion and
hence α→1 and A→4/7. This unde lines he ansien
na u e o he anomaly. In be e wo ds, signi ican elas-
ic es o ing o ces a e only p esen in c owded luids o
la ge equencies, i.e. o apid mo ion on sho leng h
and ime scales.
Ex ending he i ange o smalle imes will include
pa icula ea u es o he measu emen p ocess. The ac-
cu acy o he posi ion measu emen in SPT depends on
he numbe o pho ons acqui ed. I oo ew pho ons a e
de ec ed, small diffusion s eps a e masked by noise and
he MSD appea s o con e ge o a cons an o →0
[7, 8]. This beha io may mimic a subdiffusi e cha ac-
e is ics a small imes (’ alse posi i es’). In ou expe i-
men s pho on s a is ics was sufficien ly high o make his
effec negligible, i.e. he scaling exponen o a pu ely is-
cous luid de ia ed om uni y only by less han 2%. Bu
e en when ha ing enough pho on s a is ics and no mal
diffusion, an appa en ly anomalous cha ac e is ics may
eme ge: Since diffusion does no s op du ing he acquisi-
ion p ocess (gi en by he acquisi ion ime ∆ ), he MSD
will ake on a o m ⟨ ( )2⟩= 4D( −∆ /3) [9]. Due
o he sub ac ion o a cons an , he MSD hence may
mimic a supe diffusi e scaling o sho imes. To a oid
all hese con ibu ions, we ha e es ic ed ou sel es o
he indica ed i window which is leas affec ed by he
abo e men ioned p ocesses.
F om he wo-dimensional ajec o y wi h Nposi ion
and a ime esolu ion o ∆ , we ob ained he gy a ion
enso ia
Tij =1
N
N
∑
n=1
( i(n∆ )−⟨ i⟩) ( j(n∆ )−⟨ j⟩).(1)
He e, ⟨ i⟩deno es he i- h componen o he cen e o
mass. Diagonalizing Tij yields he p incipal axes o gy a-
ion and he co esponding eigen alues, i.e. he squa ed
p incipal adii o gy a ion, R2
i.
III. SIMULATIONS
We ha e conside ed wo diffe en models o anomalous
diffusion, namely diffusion in a pe cola ion sys em (ob-
s uc ed diffusion, OD) and ac ional B ownian mo ion
(FBM). Compu e simula ions o he espec i e p ocess
p o ide nume ical alues o he shape pa ame e s ha
can be compa ed o expe imen al da a.
Obs uc ed diffusion was simula ed on a squa e la ice
(350 ×350 si es) wi h pe iodic bounda y condi ions. A
ixed ac ion o andomly chosen si es we e occupied
by s a ic obs acles and ace pa icles we e allowed o
mo e on he emaining ee si es acco ding o he blind
an algo i hm (see, e.g. [10]). Depending on he occupied
olume ac ion , he suppo becomes a ac al [11], and
diffusion can become ( ansien ly) anomalous. Fo a c i -
ical concen a ion o obs acles p= 0.40726 [12], he pe -
cola ion h eshold in wo dimensions, subdiffusion wi h
α≈0.69 is obse ed on all ime scales whe eas o < p
a ansien , ye long-las ing subdiffusion wi h a ini e-size
co ec ed anomaly αeme ges. Indeed, o < pno mal
diffusion is asymp o ically es o ed. Fo > p, ace s
a e con ined o ini e domains, i.e. an ini ial subdiffusion
is obse ed bu asymp o ically he pa icle is bound o a
ce ain egion in space. In ou simula ions, we a ied he
occupied olume ac ion in he ange 0.33 ≤ ≤0.42
which esul ed in s aigh powe laws o he pa icles’
MSD wi hin he simula ion pe iod. Fo e e y alue o
, we simula ed 1.5×106 andom walks, whe e o e e y
1000 h un a new en i onmen was c ea ed. Each an-
dom walk was s a ed a a andomly chosen acan si e.
Occasionally, pa icles we e apped in a small sub olume
o he la ice due o he andom placemen o obs acles.
We iden i ied such si ua ions and emo ed apped a-
jec o ies om he analysis.
Fo he simula ion o FBM we used he ci culan
me hod [13] which is in p inciple exac , i.e. he de ia-
ions be ween ’ ue’and simula ed FBM a e due o com-
pu a ional limi a ions like ini e nume ical accu acy. The
me hod elies on he embedding o he co a iance ma ix
o FBM in o a ci culan ma ix ha is diagonalized by
a disc e e Fou ie ans o m. Using a as Fou ie ans-
o m (FFT), he simula ion ime o a ajec o y o leng h
Nscales as Nlog N. We gene a ed 106independen a-
jec o ies, each ha ing N= 213 posi ions. The anomaly
was a ied in he ange 0.5≤α≤0.9. Fo no mal di -
usion (α= 1), we elied on B ownian Dynamics simu-
la ions [14] ha a e based on he o e damped Lange in
equa ion, ( +∆ ) = ( )+ξ(∆ ) wi h ξbeing a andom
a iable wi h whi e noise cha ac e is ics.
IV. RESULTS ON A TRUNCATED CTRW
MODEL
To o e come he somewha a i icial ea u es o he
CTRW model due o i s asymp o ic scaling o he dis-
ibu ion o wai ing imes, p(τ)∼1/τ1+α, we ha e con-
Elec onic Supplemen a y Ma e ial (ESI) o So Ma e
This jou nal is © The Royal Socie y o Chemis y 2012
3
(a)
(b) (c)
10-2 10-1 100101
[s]
100
101
D( )
~1/ 0.2
10-3 10-2 10-1 100
10-5
10-3
10-1
101
τ
p(τ)
~1/τ1.8
0.7 0.8 0.9 1.0 1.1
α
0.3
0.4
0.5
0.6
0.7 A
OD
FBM
FIG. 2: (a) Rep esen a i e ime-a e aged MSD o unca ed
CTRW model (α= 0.8), shown as D( ) = ⟨ ( )2⟩/ o
highligh he asymp o ic scaling. Da a o he ime- and
ensemble-a e aged MSD a e shown in blue and ed, espec-
i ely. Fo he ensemble-a e aged quan i y a ansien scal-
ing D( )∼1/ 0.2in he expe imen ally ele an in e al is
obse ed be o e asymp o ically eaching he no mal diffusion
limi (D→cons .). Hence, he unca ed CTRW model only
shows ansien ly a weak e godici y b eaking. (b) The p oba-
bili y dis ibu ion unc ion o wai ing imes, p(τ) used o he
unca ed CTRW model. A e a ansien powe -law scaling,
he dis ibu ion displays an exponen ial ail ha en o ces an
asym o ic con e gence o he MSD o no mal diffusion. (c)
The asphe ici y o he unca ed CTRW model o α= 0.8
( ed do ) de ia es conside ably om he p edic ions o he OD
and FBM models (dashed lines). I is mos consis en wi h
he limi ing alue A= 4/7 o no mal diffusion and hence
incompa ible wi h he expe imen al da a ound o a c owded
dex an solu ion.
s uc ed a unca ed CTRW model. In pa icula , we
ollowed p e ious epo s ha had implied exponen ially
unca ed powe -law dis ibu ions [15]. A unca ed p(τ)
is expec ed o yield a long-las ing ansien subdiffusion
which asymp o ically con e ges o no mal diffusion.
We he e o e ha e simula ed a wo-dimensional CTRW
wi h α= 0.8 and a unca ed dis ibu ion o wai ing
imes wi h pa ame e s ha yielded a close ma ch wi h
he expe imen al MSD da a. The chosen wai ing ime
dis ibu ion and he esul ing beha io o he MSD (again
shown as D( ) = ⟨ ( )2⟩/ ) a e epo ed in Fig. 2. As
can be seen in Fig. 2a, he ime-a e aged D( ) is app oxi-
ma ely cons an whe eas he ensemble-a e aged quan i y
shows a ansien scaling ∼1/ 0.2(i.e. ⟨ ( )2⟩E∼ 0.8)
be o e con e ging o he asymp o ic limi o no mal di -
usion. Hence, he CTRW’s ea u e o a linea scaling o
he ime-a e aged MSD pe sis s e en o he unca ed
model. The associa ed wa ing ime dis ibu ion (Fig. 2b)
shows a powe -law decay o e se e al o de s o magni ude
be o e being exponen ially unca ed.
We nex de e mined he asphe ici y o he unca ed
CTRW model. Since he ajec o y o a CTRW a any
ins ance o ime geome ically looks like he pa h o no -
mal B ownian mo ion, we expec ed a alue A≈4/7
o he unca ed CTRW model. Indeed, ou expec a-
ion u ned ou o be co ec (Fig. 2c). Fu he mo e, A
did no change signi ican ly wi h he imposed anomaly
α(da a no shown). Shi ing he unca ion o la ge
and la ge imes esul ed in a sligh inc ease o A a he
han educing he alue. The e o e, based on he scal-
ing o ⟨ ( )2⟩Tand he asphe ici y, we can no only ule
ou OD and he ull CTRW model bu also a ( unca ed)
CTRW model as an explana ion o he expe imen ally
obse ed anomalous diffusion.
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Elec onic Supplemen a y Ma e ial (ESI) o So Ma e
This jou nal is © The Royal Socie y o Chemis y 2012
