Ped o Ângelo Pe ei a da Sil a
Disse a ion p esen ed o ob ain he Ph.D deg ee in Biology
Ins i u o de Tecnologia Química e Biológica An ónio Xa ie | Uni e sidade No a de Lisboa
Inse he e a colo
image wi h ounded
co ne s
Quan i a i e image analysis o cells
using mo phodynamical models:
Sea u chin spe ma ozoa as case s udy
Oei as,
Ap il, 2017
Ped o Ângelo Pe ei a da Sil a
Disse a ion p esen ed o ob ain he Ph.D deg ee in Biology
Ins i u o de Tecnologia Química e Biológica An ónio Xa ie | Uni e sidade No a de Lisboa
Oei as, Ap il, 2017
Quan i a i e image analysis o cells
using mo phodynamical models:
Sea u chin spe ma ozoa as case s udy
Summa y
Digi iza ion and obo iza ion o labo a o y equipmen has ecen ly con ibu ed o
he gene a ion o high con en o da a and i s me ada a. While his seems like an
ad an age o science’s cele i y, he analysis o such da a became he limi ing s ep
– a e y na ow bo leneck. Such is he case o imaging da a acquisi ion and i s
analysis. A e collec ing Gigaby es o images, esea che s spend se e al o de s o
magni ude o mo e ime o de e mine he egions o in e es (ROIs) (e.g. cell) and o
measu e ele an a ibu es (e.g. mean luo escence in ensi y). This manual cu a ion
o da a p omo es ano he issue ha is ela ed wi h he ep oducibili y o he analysis,
e.g., he same esea che will ha dly selec he exac same ROIs in he same da a se .
Fu he mo e, he e is also he possibili y o bias in he selec ion o which cells o use
in he analysis by biased de e mina ion o he ROIs. All o hese conside a ions can be
sol ed by au oma ion o imaging da a analysis. Gi en he same ini ial pa ame e s, he
analysis p og am will selec he same ROIs, measu e and p ocess hose measu emen s
in he same, epe i i e way, hus p oducing ep oducible analysis. By checking i s
inpu pa ame e s, one can also analyze i he e is a bias in he au oma ic selec ion
o in he analysis pipeline. Finally, he p ocessing speed o cu en cen al p ocessing
uni s (CPUs) and g aphic p ocessing uni s (GPUs) allow o as analysis, which is
c ucial o close he gap be ween da a acquisi ion and analysis.
The idea o au oma ion o imaging analysis is a om new and he e a e
many comme cial and open sou ced p og ams a ailable o ei he simple o complex
analysis p ocedu es. Howe e , mos o hem can only be applied o speci ic model
o ganism ypes and, some imes, only i hey a e used unde a speci ic expe imen al
p o ocol. One majo aspec hinde ing his p og ess is ha mos app oaches ely on
segmen a ion algo i hms (i.e. classi ica ion o pixels as ROI, backg ound o deb is)
and il e ing (i.e. ejec ion/accep ance o ROIs acco ding o hei cha ac e is ics).
In p ac ice, he compu e algo i hm does no know wha i is looking o in he
i

da a and o en includes di e en objec s (including deb is) o misses pa s o he
objec s a s udy. This aspec will p omo e bias in he subsequen analysis s eps,
po en ially c ea ing a i ac ual da a. We p opose o use a ma hema ical model as
a p io i knowledge o he a ge model cell o be i ed di ec ly o imaging da a.
Ou hypo hesis is ha , by es ima ing he numbe and o m o he pa icula objec s
o in e es ha bes desc ibe an image (i.e. maximum likelihood), we can ob ain
he mos p obable ea u es and cha ac e is ics o hose objec s, hus imp o ing hei
de ec ion, acking and cha ac e iza ion.
Image analysis o spe ma ozoa is an example whe e image segmen a ion is he
p ima y me hod used, and he e o e hese cells we e chosen as he biological case
s udy o explo e ou hypo hesis. Spe ma ozoa a e highly specialised mo ile cells
whose unc ion is o ind and e ilize he conspeci ic eggs du ing ep oduc ion.
Fo his eason hey a e cells o he u mos undamen al impo ance and also o
majo medical ele ance, conside ing ha spe m mo ili y abno mali ies a e a majo
cause o human couple in e ili y. No su p isingly Compu e Assis ed Spe m Analysis
(CASA) sys ems, implemen ing image segmen a ion ha e been de eloped ea ly and
made hei way in o ou ine clinical p ac ice. In mammals, spe m cells mus swim a
pa h housands o imes hei own body leng h h ough a complex in e io geome y,
o en illed wi h highly iscous liquids and po en ially hos ile immune cells. The
o e whelming majo i y do no e en each he allopian ubes, le alone he si e o
e iliza ion. In ma ine in e eb a es, e iliza ion occu s du ing b oadcas spawning
e en s in which he spe m ha e o ind hei conspeci ic eggs li e ally in a sea o
eggs o mul iple species. In he labo a o y s udies, sea u chin spe ma ozoa end o
accumula e and swim con ined o he liquid-solid bounda y plane which made hem
pa icula ly sui able o imaging mo ili y and chemo axis in esponse o molecula cues
eleased by he egg. Fo his eason sea u chin spe m became an impo an biological
pa adigm. Se e al s udies ha e shown ha spe m espond o spe m ac i a ing
pep ides eleased om he egg jelly laye . The ansduc ion o hese signals esul s
in se ies o cy osolic calcium spikes ha a e concomi an wi h ansien pe iods o
g ea e asymme y o he lagella bending wa es ha eo ien he cellula ajec o y.
In spe m o some species o sea u chin, such as Ly echinus pic us he calcium spike
ains a e coo dina ed in space and ime o p oduce chemo ac ic ajec o ies owa ds
he sou ce o he pep ides, while in spe m o S ongylocen o us pu pu a us he
calcium spikes p oduce diso ien ed beha io . Howe e , he ac ha he na u al
ii
en i onmen o he spe ma ozoa is a h ee dimensional olume has aised conce ns
on he ele ance and gene ali y o he knowledge de i ed om s udies in he plane.
Since spe ma ozoa a e a class o e y as mo ing cells hey a e pa icula ly demanding
on ime-lapse and h ee-dimensional mic oscopy imaging ins umen s. Pa icula ly
challenging is he measu emen o luo escen epo e s on bea ing lagella in h ee
dimensions ha canno be imaged wi h cu en ly a ailable high-pe o mance con ocal
mic oscopes. To his end, imaging sys ems ailo ed o image spe m cells ha e
been de eloped in he las wo decades. One o he sys ems, he 2D+Z( )sys em
(i.e. epo ed in he li e a u e as 3D+ ), uses a piezoelec ic de ice o oscilla e
an objec i e a high equencies, allowing us o ake wo-dimensional (2D) images
( ames) a di e en dep hs (Zaxis) as a unc ion o ime. Using his mic oscopy
ins umen a ion i was shown ha S. pu pu a us spe m displayed di e en a e age
pa h eloci y and cu a u e when con ined o when ee-swimming. Un o una ely,
he dep h o each ame is no epo ed by he sys em and an algo i hm based on
image co ela ion and on i ing he cha ac e is ic Z( ) unc ion o he piezoelec ic
was used o de e mine i . Al hough his me hod pe o med adequa ely in a ious
expe imen al da a se s, i pe o med poo ly in many o he s in which he eal and he
in e ed dep h unc ions became ou -o -phase. This has been impai ing compa a i e
s udies o he ee-swimming mo ili y and chemo axis o spe ma ozoa om he wo
sea u chin species.
The objec i e o his hesis was wo- old. The i s was o de elop and apply
image analysis me hods based on a p io i knowledge by i ing a ma hema ical model
o he objec o be de ec ed o acked di ec ly o imaging da a. The second objec i e
was o de elop and apply me hods o compa e he 3D mo ili y o spe ma ozoa o L.
pic us and S. pu pu a us.
Chap e 2 is a p oo -o -concep ha a cellula model can be used o 2D
imaging da a analysis. A de ailed ma hema ical model was de eloped desc ibing he
spe ma ozoon mo phodynamics and how i swims in a luid, gi en he changes o i s
mo phology. Thus, his mechanis ic model includes wo modules: he shape, whe e
he head is a e olu ion ellipsoid and he lagella bea ing is gi en by de ining he local
cu a u e as a eling wa e unc ion and he local o sion as a cons an alue; and
he mechanics, whe e physics a e modeled by Resis i e Fo ce Theo y. This model
was compa ed o imaging da a using he ollowing p ocedu e. Fi s we ake he
cu en model s a e and ende a model image co esponding o ha s a e, hen we
iii
escale i o he imaging da a spa ial and empo al esolu ions and las ly con ol e i
wi h he poin -sp ead unc ion cha ac e is ic o luo escence mic oscopy, i ha is he
case. The model image is compa ed o he co esponding ime-poin o imaging da a
by co ela ion. I was shown ha , unde ce ain condi ions, maximizing he c oss-
co ela ion is equi alen o maximizing he likelihood (i.e. how likely he pa ame e s
a e gi en by he da a). As ou model is non-linea , we maximized he likelihood
o ou model pa ame e s by a simple e olu iona y Mon e Ca lo simula ion, which
selec s and p opaga es he pa ame e s wi h highes co ela ion coe icien (single
ime-poin ) o highes co ela ion sum (mul iple ime-poin s, i.e. he summa ion o
he all co ela ion coe icien s) o e some i e a ions. The p opaga ion s ep allows
o small changes in he pa ame e s o explo e he op imiza ion landscape bu hese
changes become smalle a each i e a ion in o de o p omo e exploi a ion.
We i ed he mo phodynamical model o a spe m cell o di e en imaging da a
se s. The i s imaging da a se was gene a ed using he model i sel (in silico)
o p o e ha he i ing algo i hm is able o eco e he pa ame e s he o iginal
pa ame e s. We show we can dec ease he dis ance be ween he ini ial pa ame e se
(i.e. andomized) and he pa ame e selec ed a e i ing o h ee independen ini ial
se s. A second da a se was used o compa e he pe o mance o ou me hod and
ha o a human. Fo his, an imaging da a o L. pic us spe m was acqui ed wi h high
spa ial and empo al esolu ion. A cell in his da a se was acked semi-au oma ically
by a collabo a o econs i u ing bo h posi ion and lagella con o ma ions. The same
cell was i ed by ou model o show ha he solu ions ob ained also minimize he
dis ance be ween he human-de e mined and model- i ed con o ma ions. No e he
disc epancies can o igina e on he assump ion o ou model ha he lagella shape
pa ame e s a e cons an , which is mo e likely o b eak down as ime elapses. To
assess whe he ou me hod can ack cells in images wi h lowe spa ial and empo al
esolu ion, we i ed ou model o imaging da a o S. pu pu a us in hose condi ions
and he esul s we e simila o he ones wi h high esolu ion da a. Finally, we
hypo hesized ha we should be able o es ima e ea u es o he cells ha a e modeled
bu no isible in he imaging da a. To es i we used imaging da a ob ained om
spe ma ozoa labelled wi h bo h a luo escen memb ana ma ke and a luo escen
Ca2+ indica o . The o me labels he whole cell and he la e labels only he head
in an uns imula ed cell. On he same cell and a he same ime, we can image bo h
ma ke s using a ligh spli e o p oduce he co esponding wo images in he same
i
ame. Fi ing he model independen ly o he whole cell o head-only imaging da a
we a i ed o e y close pa ame e se s and lagella posi ions. Al oge he , hese
esul s show we can i a model di ec ly o da a and e en es ima e ea u es ha
canno be di ec ly measu ed in he da a, an ad an age o e segmen a ion me hods
ha ely on he da a i sel .
Chap e 3 ackles he p oblem o analysing and compa ing he mo ili y o he
spe m o he wo sea u chin species while con ined o plana swimming o while
mo ing eely in a olume. Analysing spe ma ozoa swimming in a olume equi es
special imaging sys ems. We used he sys em p oposed by Co kidi e al., (2008)
which implies in e ing he dep h o each ame om i s ime-poin be o e ob aining
he 3D coo dina es o he cells. To deal wi h his in e ence p oblem we s a ed
by es ima ing he imes a which he piezoelec ic was a ei he a maximum o
a minimum dep h posi ion. Based on he assump ion ha cells do no displace
signi ican ly wi hin a piezoelec ic pe iod, meaning ames a he same dep h posi ion
should be highly co ela ed, we used he collec i e in o ma ion o he co ela ion o
all ames and hei subsequen images. A e we i ed he canonical, empi ically
de e mined piezoelec ic Z( ) unc ion be ween hose ex emes. The compu a ional
speed o he implemen a ion o he me hod de eloped in Chap e 2 was low and i s
applica ion o da a con aining housands o ames (i.e. as he ones p oduced by he
2D+Z( )mic oscopy sys em) was no compu a ionally easible. As he op imiza ion
o such algo i hms is no he ocus o his hesis, we used a mixed app oach o de ec
cells and econs i u e hei 3D ajec o ies om 2D+ da a. To de ec cells we
used a mo e adi ional 3D empla e o he spe ma ozoon composed by 2D ames,
each co esponding o a di e en di ac ion pa e n esul ed om he ela i e o se
be ween he cen oid o he cell and he objec i e. Each di ac ion pa e n was
c oss-co ela ed o each ame and he posi ion o he maximum co ela ion posi ion
was sa ed, om which we calcula ed he cen oid posi ion a e clus e ing (i.e. o
know which sa ed posi ions belong o he same cell). These de ec ions we e hen
a ibu ed o cells by clus e ing acco ding o hei dis ances in space and ime. To
in e he a e age pa h o he cells (neglec ing he ine wiggling o he heads), we
i ed he minimal se o helical segmen s ha explained he ajec o y poin s o
each cell using Bayesian in o ma ion c i e ia as he sco ing c i e ia o a Dynamical
P og amming p oblem, a me hod we called piecewise helix i ing.
S. pu pu a us spe ma ozoa we e s udied bo h in con ined and ee-swimming
suas al e a¸c˜oes mo ol´ogicas. Assim, es e modelo mecˆanico ´e compos o po dois
m´odulos: o ma, onde assumimos que a cabe¸ca ´e um elipsoide de e olu¸c˜ao, que
a cu a u a local do lagelo ´e dada pela onda p og essi a e que a o ¸c˜ao lagela
´e cons an e, de inindo assim o ba imen o lagela ; e mecˆanica, onde a ´ısica ´e
ap oximada pela Teo ia de Fo ¸ca Resis i a. Es e modelo oi compa ado di e amen e
com uma imagem usando o p ocedimen o seguin e. P imei o peg´amos no es ado
a ual do modelo e cons u´ımos a imagem-modelo co esponden e a esse es ado,
depois eescal´amos a imagem-modelo pa a a esolu¸c˜ao espacial e empo al da imagem
expe imen al e, po im, iz´emos a sua con olu¸c˜ao com a un¸c˜ao de p opaga¸c˜ao de
pon o ´o ico (poin -sp ead unc ion) ca ac e ´ıs ica da mic oscopia de luo escˆencia,
se o o caso. A imagem-modelo oi compa ada com a imagem co esponden e
ao mesmo ins an e a a ´es de co ela¸c˜ao. Foi epo ado que, em ce as condi¸c˜oes,
maximiza a co ela¸c˜ao-c uzada ´e equi alen e a maximiza a e osimilhan¸ca (i.e. uma
medi¸c˜ao de quan o os conjun os de pa ˆame os s˜ao explicados pelos dados). De ido
ao nosso modelo n˜ao se linea , maximiz´amos a e osimilhan¸ca dos pa ˆame os do
nosso modelo usando um simples modelo e olu i o baseado em simula¸c˜oes de Mon e
Ca lo, selecionando e p opagando os conjun os de pa ˆame os com maio coe icien e
de co ela¸c˜ao (uma imagem) ou com a maio soma de co ela¸c˜oes ( ´a ias imagens, i.e.
a soma de odos os coe icien es de co ela¸c˜ao) du an e algumas i e a¸c˜oes. A e apa de
p opaga¸c˜ao pe mi e pequenas al e a¸c˜oes aos pa ˆame os de o ma a pe mi i explo a
o espa¸co de pa ˆame os sendo as al e a¸c˜oes o nam-se cada ez meno es a cada no a
i e a¸c˜ao.
Ajus ´amos o nosso modelo mo odinˆamico de um espe ma ozoide a di e en es
conjun os de imagens. O p imei o conjun o oi c iado a pa i do p ´op io modelo (in
silico) com o in ui o de p o a que conseguimos ob e os pa ˆame os usados pa a
ge a as imagens. Mos ´amos que a dis ˆancia en e o conjun o de pa ˆame os eal
e inal (i.e. ajus ado) diminui a cada i e a¸c˜ao, ap´os o ajus e independen e de ˆes
condi¸c˜oes iniciais alea ´o ias. O segundo conjun o de imagens e e como obje i o
a e igua a e ic´acia do nosso m´e odo compa ando-a `a de um humano. Pa a es e
im, ob i emos images de al a esolu¸c˜ao espacial e empo al de L. pic us. Uma
cel´ula des e conjun o de dados oi as eada de o ma semi-au om´a ica po um
colabo ado , econs uindo a posi¸c˜ao e o ma do lagelo. Ajus ´amos o nosso modelo
`a mesma c´elula e mos ´amos que solu¸c˜oes ob idas diminuem a dis ˆancia en e o
modelo humano e modelo mo odinˆamico. No e-se que as pequenas di e en¸cas en e
xii

os dois modelos pode ˜ao e o igem na suposi¸c˜ao do modelo de que os pa ˆame os
s˜ao cons an es, o que ´e cada ez mais imp o ´a el quan o mais empo passa. Pa a
de e mina se o nosso m´e odo consegue as ea c´elulas em imagens com baixa
esolu¸c˜ao espacial e empo al, ajus ´amos o nosso modelo a imagens de S. pu pu a us
com essas ca ac e ´ıs icas, endo ob ido esul ados semelhan es aos de al a esolu¸c˜ao.
Po ´ul imo, le an ´amos a hip´o ese de se poss´ı el es ima ca ac e ´ıs icas modeladas
mas que n˜ao s˜ao is´ı eis nos dados expe imen ais. De o ma a es ´a-la, us´amos
imagens ob idas de espe ma ozoides a ados com ma cado es de luo escˆencia de
memb ana (c´elula comple a) e de c´alcio (Ca2+, apenas a cabe¸ca du an e condi¸c˜oes
n˜ao-quimio ´a icas). ´
E poss´ı el ob e a in o ma¸c˜ao de ambos os ma cado es na
mesma c´elula no mesmo ins an e usando um sepa ado de eixes. Conseguimos
ob e pa ˆame os e posi¸c˜oes lagela es semelhan es depois do ajus e independen e aos
dados onde a c´elula comple a ou apenas a cabe¸ca es ˜ao is´ı eis. No seu conjun o,
os nossos esul ados mos am que ´e poss´ı el ajus a modelos di e amen e a imagens
com a an agem de consegui es ima ca ac e ´ıs icas que n˜ao podem se medidas
di e amen e a a ´es desses dados, o que n˜ao ´e az´ı el a a ´es dos m´e odos adicionais
de segmen a¸c˜ao, que dependem mui o dos p ´op ios dados.
O cap´ı ulo 3 a a do p oblema de analisa e compa a a mo ilidade dos
espe ma ozoides de duas esp´ecies de ou i¸co-do-ma quando es ˜ao con inados ou
nadando li emen e num olume. A an´alise de espe ma ozoides nadando num olume
necessi a de sis emas de aquisi¸c˜ao de imagens especiais. Us´amos o sis ema p opos o
po Co kidi e al., (2008) que implica in e i a p o undidade de cada imagem a a ´es
do empo a que oi adqui ida an es de ob e as coo denadas 3D das c´elulas. Pa a
isso, come¸c´amos po es ima os ins an es em que o piezoel´e ico se posicionou
a p o undidades m´aximas e m´ınimas. Apoiados na suposi¸c˜ao de que as c´elulas
n˜ao se mo em signi ica i amen e du an e um pe ´ıodo do piezoel´e ico (imagens
adqui idas `a mesma p o undidade de e ˜ao es a posi i amen e co elacionadas),
us´amos a in o ma¸c˜ao cole i a da co ela¸c˜ao en e odas as imagens e as suas izinhas
subsequen es (den o de um pe ´ıodo do piezoel´e ico). Depois ajus ´amos a un¸c˜ao
can´onica do mo imen o do piezoel´e ico en e esses ex emos. Pa a de e a as c´elulas
us´amos um modelo 3D do espe ma ozoide m´edio compos o po imagens 2D, cada
uma co esponden e a um pad ˜ao de di a¸c˜ao esul an e da dis ˆancia en e a obje i a e
o cen oide da c´elula. Cada pad ˜ao de di a¸c˜ao oi usado pa a aze co ela¸c˜ao-c uzada
com cada imagem, sendo g a adas as posi¸c˜oes de co ela¸c˜ao m´axima a a ´es das quais
xiii
o cen oide da c´elula oi calculado, ap´os o seu ag upamen o (i.e. de o ma a iden i ica
que posi¸c˜oes pe encem `a mesma c´elula, den o de um pe ´ıodo do piezoel´e ico).
Es as de e¸c˜oes o am a ibu´ıdas a di e en es c´elulas a a ´es do ag upamen o pela sua
dis ˆancia no espa¸co e no empo. De o ma a a e igua os ´a ios compo amen os
na a ´o ios, ajus ´amos o meno conjun o de h´elices que explicasse um conjun o de
pon os de uma aje ´o ia. Pa a isso, us´amos o c i ´e io de in o ma¸c˜ao Bayesiano
como c i ´e io de pon ua¸c˜ao de um p oblema de p og ama¸c˜ao dinˆamica, um m´e odo
que apelid´amos como ajus e po peda¸cos helicoidais (piecewise helix i ing).
Os compo amen os de na a¸c˜ao em modo li e ou con inado dos espe ma ozoides
de S. pu pu a us o am es udados p e iamen e, mas nada ´e conhecido ace ca
da na a¸c˜ao li e dos espe ma ozoides de L. pic us. Que endo amb´em sabe o
desempenho do nosso m´e odo de as eio, us´amo-lo pa a de e mina a elocidade,
cu a u a e o ¸c˜ao das aje ´o ias m´edias de ambas as esp´ecies pa a os dois
modos de na a¸c˜ao. Os nossos esul ados o am consis en es com o que es ´a
epo ado na li e a u a. Mos ´amos, pela p imei a ez, que as aje ´o ias de na a¸c˜ao
li e dos espe ma ozoides de L. pic us ˆem elocidades meno es do que a ou a
esp´ecie, ao con ´a io do que acon ece no modo con inado. A o ¸c˜ao m´edia das
aje ´o ias espe m´a icas des a esp´ecie ´e amb´em meno do que a de S. pu pu a us.
In e essan emen e, a cu a u a da aje ´o ia dos espe ma ozoides de L. pic us n˜ao se
al e a en e os modos de na a¸c˜ao li e e con inado, enquan o es a ˜
Al’ di e en e pa a
os de S. pu p ua us. Es udos ecen es com espe ma ozoides de A bacia punc ula a
mos am que as suas aje ´o ias ˆem ca ac e ´ıs icas semelhan es `as de L. pic us. De
o ma a pode explica odos es es dados, adap ´amos o nosso modelo mo odinˆamico
pa a consegui simula a na a¸c˜ao con inada quando a o ¸c˜ao lagela ´e di e en e
de ze o e p ocu ´amos os conjun os de pa ˆame os que p oduzissem aje ´o ias
semelhan es `as des as esp´ecies. Os nossos esul ados suge em que os espe ma ozoides
de S. pu pu a us ˆem maio cu a u a lagela m´edia (e, p o a elmen e, amb´em
maio o ¸c˜ao lagela ) quando nadam em modo li e, compa a i amen e `a na a¸c˜ao
con inada. Es e e ei o pode ´a de e -se ao aumen o de iscosidade pe o da supe ´ıcie
que a e a mais o ba imen o lagela dos espe ma ozoides nes a esp´ecie do que nas
ou as duas.
Usando o p ocedimen o de an´alise de imagens do sis ema 2D+Z( )que
desen ol emos no cap´ı ulo 3, in es ig´amos o compo amen o quimio ´a ico dos
espe ma ozoides de L. pic us e de S. pu pu a us no cap´ı ulo 4. Em es udos de
xi
quimio axia des es espe ma ozoides em na a¸c˜ao con inada, oi c iado um g adien e
de um p´ep ido quimi ´a ico libe ado pelo o o (Spe ac ). Pa a esse e ei o, usou-se
uma o ma bloqueada (enjaulada) do p´ep ido cuja a inidade pa a o seu ece o ´e
1000 ezes meno e um g adien e de concen a¸c˜ao da o ma a i a oi modulada po
i adia¸c˜ao ul a iole a (UV) (desenjaulamen o). Es e p ocesso he desenjaulamen o
oi usado pa a p oduzi um g adien e quimio ´a ico no sis ema de mic oscopia 3D.
Duas condi¸c˜oes o am es adas: com e sem quimioa a o enjaulado, endo i adiando
ambas com UV du an e dois segundos, de inindo assim os in e alos empo ais an es,
du an e e depois da i adia¸c˜ao. As ajec ´o ias das c´elulas o am econs i u´ıdas
po ajus e po peda¸cos helicoidais e ob i ´emos s´e ies empo ais dos pa ˆame os dos
segmen os helicoidais. De o ma a e em conside a¸c˜ao a dependˆencia dos dados
den o de cada in e alo empo al, ajus ´amos modelos linea es mis os aos dados.
Apesa des a es a ´egia e sido capaz de dis ingui os pa ˆame os ob idos pa a as
duas esp´ecies, n˜ao conseguimos encon a nenhuma al e a¸c˜ao nos pa ˆame os que
osse consis en e com quimio axia p o ocada pelo desenjaulamen o do quimioa a o .
In e p e ´amos es e esul ado como sendo consequˆencia de uma c ia¸c˜ao de icien e do
g adien e que n˜ao oi, p esumidamen e, de e ado pelas c´elulas. Depois suge imos
´a ias al e a¸c˜oes aos p o ocolos expe imen ais de o ma a a e igua se exis em ou as
condi¸c˜oes que p omo am quimio axia. Nou o es udo, a quimio axia em na a¸c˜ao
li e dos espe ma ozoides de A. punc ula a oi conseguida usando i adia¸c˜ao UV
sus en ada. Uma es a ´egia semelhan e pode ´a se e icien e pa a os espe ma ozoides
das duas esp´ecies es udadas aqui. Al e na i amen e, pode emos c ia g adien es
di e en es modulando o pe il de i adia¸c˜ao a a ´es do uso de ib as ´o icas di e en es.
Finalmen e, discu imos a nossa ese de que o uso do conhecimen o a p io i sob a
o ma de modelo ma em´a ico da c´elula- ou o ganismo-al o pa a ajus a dados pode
aumen a a esolu¸c˜ao e pe mi i es ima ca ac e ´ıs icas que n˜ao o am nem podem
se medidas di e amen e a a ´es desses dados. ´
E poss´ı el expandi o algo i mo
desen ol ido pa a o modelo biol´ogico al o azendo as al e a¸c˜oes ap op iadas. No
caso ap esen ado, podemos adiciona um m´odulo de sinaliza¸c˜ao e es uda a na a¸c˜ao
espe ma oz´oide em ambien es quimio ´a icos. Ac edi amos que is o se ´a c ucial
pa a descob i as causas e mecanismos dos casos de in e ilidade em humanos
que ainda n˜ao comp eendemos. Conseguimos amb´em melho a a de e mina¸c˜ao
da p o undidade das imagens ge adas pelo sis ema 2D+Z( )e ainda a an´alise
desses dados. Al´em disso, melho ´amos a an´alise desses dados de o ma a consegui
x
di e encia a na a¸c˜ao e compo amen o de con inamen o en e as duas esp´ecies. A
aplica¸c˜ao do modelo mo odinˆamico pe mi iu-nos suge i que os espe ma ozoides de
S. pu pu a us ˆem maio o ¸c˜ao lagela aquando a na a¸c˜ao li e, quando compa ado
com o modo con inado. In elizmen e, n˜ao conseguimos de e a compo amen o
quimio ´a ico nas condi¸c˜oes es adas, pelo que ou os g adien es de e ˜ao se es ados.
No ge al, os m´e odos desen ol idos aqui, em conjun o com ou as ´ecnicas ecen es,
se ˜ao essenciais pa a a an´alise de imagem e, mais especi icamen e, pa a comp eende
a quimio axia dos espe ma ozoides.
x i
Acknowledgemen s
Ha ing changed om molecula o compu a ional biology I knew my pa h would no
be an easy one. Howe e , i was o su e a mo e challenging, ewa ding and un
ha I would ha e imagined. Fo all his, i s and o emos , I would like o hank my
supe iso , Jo ge Ca nei o, o he inc edible oppo uni y ha he ga e me. Wi h his
guidance and company, I was able o g ow mo e as a scien is and as a pe son. Those
momen s o long discussions, ei he us a ing o enligh ening, will simul aneously
haun and mo i a e me as p ecious lessons ha will hope ully guide me in he igh
di ec ion, whiche e i may be. Be i nea a blackboa d o beside a ma ga i a, I hope
o con inue o ake bo h his scien i ic and pe sonal ad ices as a p o essional and a
iend in he nea and a u u e.
To he Quan i a i e O ganism Biology g oup, I eally app ecia e he g ow h,
guidance and e en he occasional silly alks. Thank you Thiago Guzella, Tom Webe ,
Danesh Ta apo e, Delphine Pessoa, Eleono a Tulumello and Diogo San os (you know
you a e like pa o ou g oup). Special hanks go o Tiago Macˆedo o he many
discussions and much ime spen du ing b eaks, lunches and coding ad ice. I also
would like o hank Nuno Sep´ul eda, a o me g oup membe , o ad ice on s a is ics.
I hank all he people om he Ins i u o Gulbekian de Ciˆencia o making e e y
wo king day ligh e and un. I hank Claudine Chaouiya, M´onica Dias and And eas
Bohn o accep ing o be pa o my hesis commi ee and o he use ul insigh s
and c i icism. I also hank he many PhD s uden s commi ees ha o ganized he
AMeeGuS e ea s which allowed me o ain p esen a ions, discuss my wo k and
enjoy he beau i ul enues. I also hank he IGC band o allowing me o play music
wi h hem.
This wo k would no be possible wi hou he collabo a ion wi h he conso ium
Conso cio de Fisiolog´ıa del Espe ma ozoide and he Depa amen o de Ingenie ´ıa
Celula y Bioca ´alisis o heIns i u o de Bio ecnolog´ıa, Uni e sidad Nacional
x ii

Au ´onoma de M´exico (UNAM). I hank Albe o Da szon o he pa ience, ad ice,
insigh , c i icism and iendship. Fo all o hem, I hank he amazing hospi ali y ha
hey ecei ed me wi h and o making home o seem less dis an . Speci icaly, I would
like o hank he ollowing pe sons:
•Albe o Da szon and Cl´audia T e i˜no, o making some o my ips o Mexico
possible and o ecei ing me in open a ms.
•A u o Pimen el, Ad´an Gue e o, Ta iana Luna and Gab iel Co kidi o he
discussions, c i icism and o p o iding he imaging da a, which was essen ial
o his hesis, and o he pa ience in aining me o ope a e he 2D+Z( )
mic oscopy sys em.
•Oma Jose, Ta iana Luna, Albe o Vicens, A u o Pimen el, Ad´an Gue e o,
Ana Lau a, Ca mem San ana, Ana Ru h and F ancisco Balde as o showing
me he bo h he un and he delicious sides o Mexico. You made i easie o
a oid home sickness.
•Ana Ru h o sha ing he house, meals, bi hdays and p o iding a second home.
•Many o he g oup membe s who, one way o ano he , con ibu ed ei he
academically o cul u ally.
Also om UNAM, I would like o hank Gus a o Mekkel o suppo and Daniel P iego
o he discussions abou spe m channels and company du ing lunches.
To he amazing Co A g oup, I hank And ´e San os, Sa a Sil a, Diana Macedo
and Jo˜ao Ba is a o he mos unp oduc i e mo nings, a e noons and nigh s ha I
could ha e only imagined. I was only o a sho ime ha we managed o wo k
oge he bu i was un!
On a mo e pe sonal no e, I would like o add ess my pa en s who suppo ed me
in e e y imaginable way du ing all di e en aspec s o my li e. Wi hou you, I simply
would no ge his a . Thank you o being he e... always... unwea y... ime a e
ime!
Finally, I would like o acknowledge he pe son who sac i iced mo e han anyone
du ing my PhD. Thank you Aud ey Lopes o hose lonely nigh s I could no make
you company because o my wo k, o ge ing my spi i s up when I hough only ime
could do i and o being he amazing pe son I can coun on o a laugh , a ge away
and o sha ing a li e. No wo ds can e e epay you.
x iii
Con en s
Summa y i
Sum´a io ix
Acknowledgemen s x ii
Lis o Tables 3
Lis o Figu es 5
Ac onyms 9
1 In oduc ion 11
1.1 Modeling and Modeling Rela ion . . . . . . . . . . . . . . . . . . . . 12
1.1.1 Ma hema ical modeling . . . . . . . . . . . . . . . . . . . . . 15
1.2 Imagingda a............................... 16
1.2.1 Imaging da a acquisi ion: om scene o image . . . . . . . . 16
1.2.2 Imaging da a analysis: om image o scene . . . . . . . . . . 18
1.3 Sea u chin spe ma ozoa . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.3.1 Spe m cell mo phology . . . . . . . . . . . . . . . . . . . . . 22
1.3.2 Spe m mo ili y . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.3.3 Spe m chemo axis . . . . . . . . . . . . . . . . . . . . . . . . 25
1.3.4 3D imaging o spe m cells . . . . . . . . . . . . . . . . . . . 27
1.3.5 Ma hema ical modeling o spe ma ozoa . . . . . . . . . . . . 28
1.4 In his hesis............................... 31
1.5 Ma hema ical no a ion . . . . . . . . . . . . . . . . . . . . . . . . . 33
1
CONTENTS
2 Mo phodynamical image analysis o spe ma ozoa swimming in he
plane 41
2.1 In oduc ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.2 Ma e ials and Me hods . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.2.1 Mo phodynamical model . . . . . . . . . . . . . . . . . . . . 44
2.2.2 Compa ing and i ing he model o imaging da a . . . . . . . 47
2.2.3 Implemen a ion de ails . . . . . . . . . . . . . . . . . . . . . 48
2.2.4 Imaging da a . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.3 Resul s.................................. 51
2.3.1 Co ela ion sco es sensi i i y o Resis i e Fo ce Theo y (RFT)
pa ame e s............................ 51
2.3.2 P ecise es ima ion o model pa ame e s using ei he in silico
o L. pic us da a ........................ 55
2.3.3 In e ence o lagella con o ma ions by acking only he head . 56
2.4 Discussion ................................ 59
2.4.1 Expanding ou knowledge-based model . . . . . . . . . . . . . 63
2.5 Conclusion................................ 65
3 Compa a i e s udy o sea u chin spe m mo ili y – con ined and ee
swimming 69
3.1 In oduc ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.2 Ma e ials and Me hods . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.2.1 Biological ma e ials and image acquisi ion . . . . . . . . . . . 73
3.2.2 3D ajec o ies o ee swimming spe ma ozoa . . . . . . . . . 74
3.2.3 2D imaging da a o spe ma ozoa and ajec o ies . . . . . . . 80
3.2.4 Da a............................... 80
3.2.5 Mo phodynamical model . . . . . . . . . . . . . . . . . . . . 81
3.2.6 Compa ing and i ing he model o ajec o ies . . . . . . . . 83
3.3 Resul s.................................. 84
3.3.1 Accu a e and p ecise econs i u ion o h ee dimensions (3D)
spe m ajec o ies . . . . . . . . . . . . . . . . . . . . . . . . 84
3.3.2 Piecewise helix i ing allows disc imina ion o bo h species by
hei ajec o y pa ame e s in ee swimming . . . . . . . . . . 85
3.3.3 S. pu pu a us adius o oscula ing ci cle is di e en be ween
ee and con ined swimming . . . . . . . . . . . . . . . . . . 88
2
CONTENTS
3.3.4 Highe asymme y o lagella bea ing accoun s o he
obse ed cu a u e a io . . . . . . . . . . . . . . . . . . . . 88
3.4 Discussion ................................ 93
3.4.1 F om 2D+Z( ) o 3D ajec o ies . . . . . . . . . . . . . . . 95
3.4.2 F ee swimming ajec o ies . . . . . . . . . . . . . . . . . . . 97
3.4.3 Con ined s ee swimming . . . . . . . . . . . . . . . . . . . 97
3.4.4 Conclusion and u u e wo k . . . . . . . . . . . . . . . . . . . 99
4 Analysis o spe m chemo axis in h ee dimensions 103
4.1 In oduc ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.2 Me hods.................................106
4.2.1 Spe m imaging da a . . . . . . . . . . . . . . . . . . . . . . . 106
4.2.2 Imaging da a analysis . . . . . . . . . . . . . . . . . . . . . . 106
4.2.3 S a is ical analysis . . . . . . . . . . . . . . . . . . . . . . . . 107
4.3 Resul s..................................108
4.3.1 3D expe imen s wi h caged Spe ac . . . . . . . . . . . . . . 108
4.4 Discussion ................................109
4.5 Conclusion................................112
5 Gene al Discussion 115
5.1 Modeling o spe ma ozoa . . . . . . . . . . . . . . . . . . . . . . . . 118
5.2 Ad ances in Imaging analysis . . . . . . . . . . . . . . . . . . . . . . 120
5.3 B inging i oge he . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
3
ACRONYMS
HCN hype pola iza ion-ac i a ed cyclic nucleo ide-ga ed
KCNG K+-selec i e cyclic nucleo ide-ga ed
MC Mon e Ca lo
MLE maximum likelihood es ima ion
NCE Na+-Ca2+-K+exchange
NHE Na+-H+exchange
ODE o dina y di e en ial equa ion
PCA P incipal Componen Analysis
PDF p obabili y densi y unc ion
pHiin acellula pH
PSF poin -sp ead unc ion
RFT Resis i e Fo ce Theo y
ROI egion o in e es
SAP spe m ac i a ing pep ide
SBT Slende Body Theo y
SNR signal- o-noise a io
TIRF o al in e nal e lec ion luo escence
UV ul a iole
10

Chap e 1
In oduc ion
Today biological sciences a e gene a ing imaging da a a a huge pace. This is usually
ollowed by se e al weeks o pains aking analysis by esea che s who usually selec
he egions o ( hei ) in e es . As such, imaging da a analysis is cu en ly one o he
majo bo lenecks in scien i ic p oduc i i y. Also, he analysis made by a pe son is
seldom ep oducible, e en i he same pe son we e o eanalyze he same da a (e.g. by
selec ing di e en egions o in e es ). This has led o an e o o au oma e imaging
analysis bu many me hods canno be gene ally applied o all cell ypes, ma ke s,
e c., because he compu e does no know wha i is looking o and il e s like size
and ma ke in ensi y a e no enough o de ec o ack he cell e icien ly. Fo hese
easons many scien is s s ill eso o manual anno a ion and analysis o hei imaging
da a.
Spe m analysis is one such example. These as cells equi e high empo al
esolu ion om he mic oscopy se up i one hopes o s udy hei mo ili y; his is
he eason why mo e da a pe second is gene a ed in his kind o se up. I you couple
his wi h h ee dimensions (3D) s ack imaging, a ew Gigaby es a e easily gene a ed
and s o ed in a ew seconds. Au oma ic image analysis is c i ical o deal wi h such
p oblem. This aspec has hinde ed compa a i e s udies on how spe ma ozoa om
di e en species swim and, consequen ly, on how hey eac o di e en chemo ac ic
g adien s. These kind o s udies can gi e insigh s on human e ili y p oblems which
causes a e cu en ly unknown.
In his hesis we explo ed au oma ic image analysis based on a p io i knowledge
o de ec and ack spe ma ozoa in bo h 2D+ and 2D+Z( )imaging da a. We hen
used his me hodology o make a compa a i e s udy o he mo ili y o spe ma ozoa
11
CHAPTER 1. INTRODUCTION
om wo di e en sea u chin species and o assess hei chemo axis beha io in 3D.
As his wo k in ol ed di e se scien i ic ields, which made a con inuous in oduc o y
low challenging, we will i s in oduce some o hese ields independen ly. They will
be in eg a ed and expanded as we in oduce new ones. In mo e de ail, we will s a
by making an in oduc ion o modeling and Modeling Rela ion, hen we will mo e on
o cu en imaging analysis me hods and how models ha e been used o do i and,
inally, we will in oduce ou biological case s udy – he sea u chin spe ma ozoon,
expanding i s s a e-o -a in he ligh o modeling and image analysis. The ea e ,
he objec i e and ou line o his hesis will be p esen ed.
1.1 Modeling and Modeling Rela ion
Making suppo ed s a emen s abou he wo ld equi es e idence. This e idence is
usually he esul o i ing a model o da a ob ained somewhe e and somehow. I
can be as simple as he case o assuming he model o people heigh in a class o
be Gaussian dis ibu ed and o assess i hose dis ibu ions a e di e en as we i
he model o samples o di e en classes, using he app op ia e s a is ical model.
Howe e , a lo can go w ong in his simple p ocess. Is he dis ibu ion o heigh s
on each class co ec ly modeled by a Gaussian one? Is he sample o each class
ep esen a i e o hei eal dis ibu ion? Is he ule well calib a ed and sui able o
measu e he expec ed di e ences? Wha is he e o o noise o he measu emen s?
A e all he assump ions o he s a is ical es me , e.g., a S uden ’s - es o es o
di e ences o he mean heigh assumes independen da a bu a e we su e he e is no a
s uden aking bo h classes p esen in bo h samples? A good expe imen al design will
deal wi h mos o hese issues bu complex ques ions, sampling me hods, measu ing
de ices o models migh ha e cha ac e is ics which, in he pa icula ensemble used,
a e incompa ible o answe he ques ion. Many imes, he incompa ible cha ac e is ic
is no ob ious, i is di icul o iden i y o simply is no e en no iced o exis . A he
end, he e can be ouble in ei he o bo h model and da a.
In o de o be e unde s and whe he we should belie e mo e in he da a o he
model, we will i s in oduce he Modeling Rela ion de eloped by Rosen, (1991)
( ig. 1.1). In a nu shell, his is a amewo k ha ela es he na u al sys em, how we
pe cei e i and how we can in e and alida e he mechanisms occu ing in i using
o mal (i.e. ma hema ical) language. Thus, we ind use ul o desc ibe he di e en
12
1.1. MODELING AND MODELING RELATION
Model Na u al
Sys em 1
Cause
2
Measu emen
3
In e ence
4
Measu emen , In e p e a ion, P edic ion
Figu e 1.1: Rosen’s Modeling Rela ion. Adap ed om (Rosen, 1991).
ypes o in e en ial p ocesses.
A na u al sys em is a se o p ope ies and he e en s ha change hose p ope ies.
To unde s and he na u al sys em we mus know and unde s and he cause o such
p ope ies and e en s (mapping 1). Howe e , we canno make di ec assessmen o
he causes in he na u al sys em. Thus, we mus use ou senses o pe cei e i so
we ake measu emen s o he p ope ies and e en s – an encoded ep esen a ion o
a subse o he na u al sys em (mapping 2). We mus also de elop a model ha
in e s some hing abou he encoded da a (mapping 3). This, howe e , does no say
any hing abou he na u al sys em. In o de o achie e co espondence be ween he
in e en ial (mapping 3) and he causal (mapping 1) p ocesses, he in e enced esul
mus be decoded back o he na u al sys em (mapping 4), hence making a guess o
expec a ion o how he na u al sys em changed. I his las s ep canno be made,
hen ou model does no desc ibe he causal p ocess.
In he end, we can choose o belie e he da a, he model o a mix u e o bo h.
Belie ing he da a is he mos popula app oach. Resea che s o en collec da a om
samples and hen i di e en models o i un il one is ound, usually he one mos
signi ican ly suppo ed by he da a. This can lead o da a-d i en models ha me ely
desc ibe he da a bu ha do no p o ide new knowledge. New samples o mo e da a
poin s may need al e a ions o he p e ious model o accommoda e hem. Hence,
his is no he case o a phenomenological model.
Belie ing he model is he mos in amous app oach as he e is he case o
belie ing he al e na i e hypo hesis o a ques ion and i ing di e en dis ibu ions
13
CHAPTER 1. INTRODUCTION
and s a is ical models un il he esea che is p o ed igh . This is a ypical case
when people wi h poo s a is ics knowledge“ o u e he da a un il he da a con ess”.
As his is conside ed bad scien i ic p ac ice, we do no ind i use ul o discuss his
u he . The e is, howe e , a easonable p e e ence o model-d i en belie s i i is
impossible o pe cei e he ex e nal wo ld wi h cu en echnology i measu emen s
(mapping 4) a e belie ed o su e om he issues e e ed abo e (o o he s). The
o me was he case o he p edic ion o he Higgs boson (ATLAS Collabo a ion, 2012;
CMS Collabo a ion, 2012) and also o he g a i a ional wa es p oposed by Eins ein’s
Gene al Rela i i y which could only be measu ed ecen ly (Abbo e al., 2016).
Simila ly, in biology he e a e many heo e ical p edic ions awai ing o echnology
o be able o measu e hem. Many me abolic and signaling pa hways o memb ane
channel unc ions and mechanisms, p oposed by a ailable da a and knowledge, a e
no easy o be es ed in i o due o lack o speci ic inhibi o s and blocke s. Also,
measu emen o all componen s o he pa hway is usually imp ac ical as he a ailable
me hods can lead o a i ac ual beha io .
In my opinion, mixed-belie is whe e mos scien is wan o be. By o mula ing
a hypo hesis and designing an expe imen o es i , which is able o join bo h he
app op ia e p ocedu e o da a collec ion and he co ec s a is ical es , he scien is
o malizes a model (mapping 3) o which hey will eed da a (mapping 2) and om
which hey ge a p edic ion which is alida ed (mapping 4). Then (s)he is happy o
say ha he cause o he na u al sys em has co espondence o he in e ence p ocess
o he model (mapping 3) and a new disco e y (o a eplica e assay) is con i med.
In his case, he scien is is bo h con iden on he p ope ies o he collec ed da a
cha ac e is ics and on he model de ails. No e ha all model p edic ions om e e y
meaning ul inpu da a mus be e i ied o e ec i e co espondence o he in e en ial
and causal p ocesses. Fo example, New on’s law o g a i y allowed Le Ve ie and
Adams o hypo hesize he exis ence o an unknown plane a ec ing he p edic ed o bi
o U anus, and hei calcula ions p edic ed he posi ion o Nep une, la e con i med
by Galle. Bo h da a and g a i a ional model we e belie ed o be co ec ed and,
consequen ially, he sola sys em model was modi ied acco dingly. Al hough he
g a i a ion model was o mula ed and calib a ed om ea hly objec s and co ec ly
p edic ed he posi ion and exis ence o many ’ou -o - his-wo ld’ objec s, i was no
success ul in modeling Me cu y’s o bi . Only wi h Eins ein’s Gene al Rela i i y was
able o p edic he co ec amoun o p ecession o he o bi ’s eccen ici y.
14
1.1. MODELING AND MODELING RELATION
Taking all his in o conside a ion, we should be looking o build a model whe e,
independen ly o he measu ing p ocess, he e is enough de ail in he in e en ial
p ocess ha p edic s se e al di e en aspec s o he na u al sys em (i.e. ou biological
model). As an e o o con i m all p edic ions (mapping 4) should be made, om
all meaning ul measu emen s (mapping 2), he model should also be simple enough
ha he co espondence be ween in e en ial and causal p ocesses can be a ained.
1.1.1 Ma hema ical modeling
Looking a he Modeling Rela ion om he ma hema ics poin -o - iew, we de ine a
ma hema ical model which is a mapping ha will ake some inpu (as he o m o
da a) and gene a e some p edic ions (also measu ed in he o m o da a). This model
will be desc ibed by a se o assump ions and hei consequen pa ame e se (θ).
Taking measu emen s o he na u al wo ld (i.e. da a) is o en a ec ed by noise o
o he ans o ma ions (see sec ion 1.2 o mo e de ails) which can also be desc ibed
ma hema ically. How should we choose be ween wo di e en ep esen a ions o
he da a? I we go o a mo e complex model (e.g. he ex eme case o he
da a i sel ), no o li le in o ma ion may be gained, and, i we go o he simple
model, he abs ac ion o imp ecision may be so g ea i could be ep esen ing ei he
his o comple ely di e en da a. Bo h in s a is ics and o he a eas, he e is he
adi ion o choosing he mos pa simonious model, meaning he simple one ha
can model he da a well enough. Modeling he da a well enough is usually gi en by
he likelihood (L), a measu e ha a se o pa ame e s a e suppo ed by he da a, and
he complexi y is usually gi en by he numbe o pa ame e s (k). Bo h F equen is
and Bayesian schools de i ed he Akaike’s in o ma ion c i e ia (AIC) and Bayesian
in o ma ion c i e ia (BIC), espec i ely, which a e sco es ha gi e di e en weigh s o
he likelihood and o he pa ame e numbe (Has ie e al., 2009). The i s is de i ed
om in o ma ion heo y and bo h can be de i ed om he Bayesian amewo k. When
o use one o he o he is a deba e las ing o decades bu i is gene ally conside ed
BIC gi es mo e penal y o he numbe o pa ame e s (Bu nham and Ande son, 2002).
Wha e e he c i e ia chosen, i is i s ela i e di e ence ha holds some meaning –
he model wi h lowe sco e is he mos pa simonious.
Conside now ha we ha e wo di e en in e en ial models (i.e. he measu emen
models a e he same) ha may o may no ha e he same pa ame e se . How should
we choose be ween hese compe ing models when hey a e applied o he same da a?
15

CHAPTER 1. INTRODUCTION
A lo o e o and deba e has also been pu o answe his ques ion. As ou in e en ial
model ans o ms measu ed da a in o p edic i e da a, we can also use AIC o BIC o
compa e models (Has ie e al., 2009). In he special case whe e he models a e nes ed
(i.e. one o hem is a sub-model o he o he , e.g., wi h one o he pa ame e s ixed),
we can use he likelihood a io es : using he a io o each model’s likelihood, we
can build a s a is ic ha is chi-squa ed dis ibu ed and compu e a con idence in e al
o he di e ence. I he e is no di e ence, he simple model should be chosen.
How a e Model Rela ions and model cons uc ion a ec ed in imaging da a
p ocessing and analysis? To answe his ques ion we mus i s hink abou wha
is an image and how i is o med.
1.2 Imaging da a
1.2.1 Imaging da a acquisi ion: om scene o image
In a mode n op ical o elec on mic oscopy se up, a sample is i adia ed wi h pho ons
o elec ons, espec i ely, and hese al e hei pa h by in e ac ing wi h i . The
i adia ed media is hen cap u ed by a de ec o which ansla es he in o ma ion o
elec ic impulses, e.g. by a cha ged coupling de ice (CCD) o a complimen a y me al
oxide semiconduc o (CMOS) de ice, which a e in u n sa ed as digi al in o ma ion.
The inal imaging da a (I,i.e. he measu emen ) will be a dis o ed ep esen a ion
o he scene (S,i.e. he na u al sys em) and his dis o ion can be he esul o
ou di e en ans o ma ions on he uns ans o med ep esen a ion o he scene, he
image (Iu) (Knill and Richa ds, 1996). These ans o ma ions a e lis ed below (no e
’A→B’ means objec Ais mapped o objec B):
•Noise and blu (S→(Iu∗β)i+i): he image is he esul o he con olu ion
o he scene by he blu ing ke nel (β), o which backg ound noise (b) and
sampling e o (i∈∼ N(b, σ)) a e added o he pixel i. This is he
ans o ma ion mos commonly add essed in imaging analysis. The noise may
no be Gaussian, i.e. mos imes i is no , al hough we can use i as a
ai app oxima ion in many o hose cases. No e he symbol ’∗’ ep esen s
con olu ion.
•Supe posi ion (S∼(S1+· · · +Sn)o σ(S1,...,Sn)): i deals wi h he ac
ha a complex scene o signal can usually be decomposed in o simple elemen s.
Wa ele unc ions and Fou ie analysis a e examples whe e linea combina ions
16
1.2. IMAGING DATA
o wa ele o sinusoidal unc ions, espec i ely, a e added o model a complex
signal. Al e na i ely, o he ypes o ules (σ) migh be applied o he se o
he simple indi idual componen s. These me hods can be used o dec ease
he numbe o pa ame e s necessa y o desc ibe a scene. The use o a subse
o componen s esul ing om he P incipal Componen Analysis (PCA) o he
signal is also a popula app oach.
•Domain wa ping (S→(Iu◦ψ)): di e en empo al and spa ial con ac ions
o expansions (ψ) o he scene’s domain migh occu when acqui ing he signal
o he same objec . An example is aking a pho og aph o a pe son’s ace om
wo di e en poin -o - iews – he objec is he same bu he esul ing images
a e di e en . No e he symbol ’◦’ ep esen s unc ion composi ion.
•In e up ions (S={O1, O2} → Iu={I1|D0,I2|D−D0}): many imes a
scene is a composi ion o se e al objec s (Oi), om which we only obse e a
subdomain (D, D0), e.g., due o occlusions o missing da a.
Because any combina ion o hese ans o ma ions is possible, imaging da a o en
needs o be p e-p ocessed and analyzed o ex ac he in o ma ion he esea che
desi es. P e-p ocessing usually in ol es low-le el ope a o s, e.g. backg ound
co ec ion, con as enhancemen o decon olu ion, while image analysis ies o
de ec and measu e ea u es on he images, e.g. ack cells and measu e hei mean
luo escence o size.
An impo an p e-p ocessing me hod o luo escen imaging da a is decon olu ion,
which is ela ed o he i s ans o ma ion e e ed abo e. In mo e de ail, he non-
uni o m gene a ion o i adia ion (i.e. exci a ion) sou ce, i s a el h ough he op ical
componen s un il i eaches he sample, he andom emission in space and ime by he
luo escen ma ke and i s cap ion h ough mo e op ical componen s un il i eaches
one o he se e al uni s o he de ec o causes he inal image o be a dis o ed
ep esen a ion o he o iginal sample – a con olu ed one. The dis o ion ope a o is
called poin -sp ead unc ion (PSF) and, i we know his unc ion, we can decon olu e
he inal image and ob ain a sha pe image which is a close ep esen a ion o he
objec (Aga d and Seda , 1983; Zhang e al., 2007). Bo h 3D luo escence mic oscopy
and supe - esolu ion mic oscopy need o es ima e accu a e PSFs so he decon olu ion
does no c ea e a i ac s o abe a ions, he eason why di e en me hods o calcula e
he e ec i e PSF o each sys em a e cu en ly being de eloped (Pa wa y and P eza,
2015). On he o he hand, o mos uses he PSF can be app oxima ed by a Gaussian
17
CHAPTER 1. INTRODUCTION
unc ion using he in o ma ion ega ding he mic oscopy se up used (e.g. objec i e
nume ical ape u e and he exci a ion and he luo opho e’s emission wa eleng hs)
(Zhang e al., 2007). A e eco e ing he decon olu ed image, and assuming his
p ocess does no c ea e a i ac s, i is easie o analyze i , in o de o es ima e he
scene i encodes.
1.2.2 Imaging da a analysis: om image o scene
The objec i e o imaging da a analysis is o decode he scene encoded in images. A e
he imaged objec s cells? Whe e a e hey loca ed? Wha is hei shape? Wha is hei
size and wha is he concen a ion o a speci ic ma ke inside hem? These a e he kind
o ques ions a esea che o en does when analyzing imaging da a and hei espec i e
answe s will p o ide he da a o e i y he hypo heses a hand. Al hough a human
can unde s and a scene ep esen ed by an image, a compu e p og am needs o be
coded wi h he ope a ions o pe o m such ask. We can use a model wi h pa ame e s
θ o ep esen a scene encoded by an image (I). Due o he signal ans o ma ions
in oduced be o e, speci ically he andom sou ces, he e is a p obabili y (P(I|θ))
ha a gi en scene/pa ame e s esul s in a pa icula image. When we speci y a
model, he se o possible images will ollow a gi en dis ibu ion wi h o al p obabili y
summing o one. In e sely, we also ha e he likelihood o he pa ame e s gi en a
pa icula image (L(θ|I)), meaning ha we a e measu ing how likely i is o a
se o pa ame e s (i.e. scene) o ha e o med ha speci ic imaging da a (i.e. he
sum o e he pa ame e space can be di e en han one). I is no an unusual
p ac ice o es ima e he model’s pa ame e s by maximizing he likelihood, usually
by sol ing o when he pa ial de i a i es a e ze o. The pa ame e s can also be
es ima ed by Bayesian in e ence. Bayes heo y de ines ha he pos e io p obabili y
(P(θ|I)) is he likelihood (P(I|θ))1 imes he p io p obabili y (P(θ)) o e he
expec edness o he image (P(I)): P(θ|I) = P(I|θ)P(θ)/P(I)(Knill and
Richa ds, 1996). Assuming he same imaging da a, he expec edness is cons an (i.e.
a no malizing ac o ) so he pos e io is p opo ional o he likelihood imes he p io
(P(θ|I)∝P(I|θ)P(θ)). While bo h he F equen is and Bayesian app oaches
make use o a model o gene a e he p obabili y mass unc ion (o p obabili y densi y
unc ion, in he case o a con inuous model) only he la e akes p io knowledge in o
accoun , i.e. in he shape o he p obabili y o he pa ame e s. Ei he case p esen
1No e hey de ine he likelihood di e en ly han he F equen is s.
18
1.2. IMAGING DATA
he same challenge when dealing wi h complex o nume ical models, i is some imes
imp ac ical o ge an algeb aic unc ion o he likelihood so di e en op imiza ion
algo i hms can be used o ind he pa ame e s which maximize P(θ|I).
A ypical expe imen al pipeline whe e imaging analysis is used is summa ized in
Du ou e al., (2015). Following a s a ing expe imen al design, mic oscopy da a is
ob ained and cells a e de ec ed. Thei desc ip ions a e hen ex ac ed and ed o a
machine lea ning algo i hm which allow us o selec he mos ele an ea u es o make
biological in e ences. Finally, hese allow us o p opose new hypo hesis and design
new expe imen s, closing he cycle. We will now de ail some o he p ocesses and
cha ac e is ics en ailed by he cu en implemen a ion o his expe imen al pipeline.
One o he c ucial s eps in image analysis is i s segmen a ion in o de o de e mine
he egions o in e es (ROIs) (e.g. cells) in each image. We can conside wo di e en
s a egies o do his. The i s is image-based and ies o assign each pixel o a gi en
class, e.g. cell, backg ound, luo escen ma ke o deb is. This is he main me hod
used o s udy cell mo phology (Smi h e al., 2009b), p o ein colocaliza ion, emo e
signaling, magne ic esonance images (Ahmed and Mohamad, 2011), angiogenesis
and s em cell (Rabbani and Ja anma d, 2011), o name a ew ep esen a i e s udies.
The e a e wo main app oaches: pixel-based, whe e he mul idimensional in o ma ion
o each pixel is used by supe ised and unsupe ised me hods o classi y i ; and objec -
o ien ed classi ica ion, a bo om-up app oach we e neighbo ing pixels a e sequen ially
clus e ed acco ding o some homogenei y c i e ia and he di e en g oups a e hen
classi ied using, e.g., pixel in ensi y, shape o ex u e ea u es (Inglis e al., 2010). The
homogenei y c i e ia is usually a sco e o an ene gy unc ion ha ep esen s simila i y
o dissimila i y o be maximized o minimized, espec i ely, and i s o mula ion is
c ucial o sol e he p oblem a hand wi hou c ea ing a i ac s. Co ela ion o leas
squa ed dis ance a e popula as simila i y measu es (Has ie e al., 2009). Pixel-based
me hods a e pe haps he mos widely used ones and can be as simple as de ining
an in ensi y h eshold. Then, e e y pixel which in ensi y is abo e ha h eshold is
e ained while hose which do no a e se o ze o. A body o wo k was done in
o de o de ine he h eshold alue(s) au oma ically (Ren e al., 2010). The second
segmen a ion s a egy is model-based and ies o es ima e he da a ha p oduced
an image o o i some kind o model di ec ly o he image. Gi en an explici o
implici pa ame ic model i is possible o maximize a sco e unc ion o e.g. ob ain
he bounda y egions o an objec , e.g. using ac i e con ou s (Xu and P ince, 1998).
19
CHAPTER 1. INTRODUCTION
he a e o change o i s concen a ion con ols chemo axis (Wood e al., 2003;
Al a ez e al., 2012). The mechanism by which calcium a ec s he lagella shape is
s ill unknown.
The e is a conside able body o wo k done o es ablish he signalling cascade
om he spe m ac i a ing pep ide (SAP) binding o guanylyl cyclase (GC, i.e.
he ecep o ) o changes in [Ca2+]i(Da szon e al., 2011; Kaupp, 2012; Sei e
e al., 2015; Gonz´alez-Co a e al., 2015). B ie ly ( ig. 1.3), he binding elici s
syn hesis o cyclic guanosine monophospha e (cGMP) which will ac i a e he K+-
selec i e cyclic nucleo ide-ga ed (KCNG) channels. The exi o po assium ions will
hype pola ize he cell memb ane and hus ac i a e he Na+-H+exchange (NHE) and
he hype pola iza ion-ac i a ed cyclic nucleo ide-ga ed (HCN) channels which will
alkalinize he cy osol and depola ize he memb ane, espec i ely. The depola iza ion
ac i a es he ol age-dependen Ca2+ (CaV) channels which will inc ease he [Ca2+]i.
The inc ease in in acellula pH (pHi) will p esumably ac i a e Ca Spe , a pHiand
mildly ol age dependen Ca2+ channel, hus also inc easing he [Ca2+]i. Re u ning o
basal [Ca2+]ile els a e s imula ion is done by he Na+-Ca2+-K+exchange (NCE)
and a phospodies e ase (PDE) which hyd olyzes cGMP. Also, a calcium-dependen
K+(BK) channel hype pola izes de memb ane o he basal memb ane po en ial.
While some o hese componen s ha e been iden i ied in sea u chin spe m and hei
unc ion has been es ablished, o he s emain elusi e. The iden i y o he CaVchannel
is s ill unknown and he unc ion o NHE has no been i mly p o ed. The e is
also e idence o a di e en calcium-dependen K+channel which can modula e
he calcium spike ain (Espinal e al., 2011; Espinal-En ´ıquez e al., 2014). Also,
he ensemble and p opo ions o calcium channels p esen in he sea u chin spe m
lagellum a e ye o be de e mined.
Con a y o L. pic us, o he same chemoa ac an molecule and g adien , S.
pu pu a us spe m eac by elici ing u n-and- un episodes bu hey do i in andom
di ec ions (Gue e o e al., 2010), a beha io ha can ha dly be called chemo axis. As
hese s udies we e pe o med when he cells a e con ined o he wa e -glass in e ace,
we canno exclude he hypo hesis ha a om he bounda y, in 3D, cells would be
able o u n in a chemo ac ic way. The chemo axis o A. punc ula a ee swimming
spe m has been cha ac e ized in 3D (Jikeli e al., 2015) bu he ques ion whe he
all sea u chin species spe m eac he same way emains o be answe ed. Because
small local changes o he ajec o y cu a u e can ha e an huge impac on he global
26

1.3. SEA URCHIN SPERMATOZOA
Hype . Depo.
Cy osol
Ou side
Plasma memb ane
SAP
GC
cGTP
cGMP
PDE
KCNG
K+
KCNG
NHE
H+
Na+
NHE
HCN
Na+
HCN
CaV
Ca2+
CaV
Ca Spe
Ca2+
Ca Spe
NCE
Na+
Ca2+
K+
NCE
BK
K+
BK
Figu e 1.3: Chemo ac ic signaling cascade o sea u chin spe m. The plasma ic
memb ane can be hype pola ized (Hype .) o depola ized (Depo.). Blue boxes
ep esen di e en memb ana channels (see sec ion 1.3.3 o de ails). Black a ows
ep esen ion anspo , s a e ans e ences and eac ions, depending on he con ex .
G een and ed solid a ows ep esen ac i a ion and inhibi ion, espec i ely, o
channels and p ocesses. Dashed a ows ep esen inhibi ion by low pHi( ed) and
ac i a ion by high pHi(g een).
ajec o y (Gue e o e al., 2011), u he s udies on spe m mo ili y a e equi ed o
unde s and how spe m cells swim and eo ien in space in o de o ha e mo e insigh s
on spe ma ozoan chemo axis.
1.3.4 3D imaging o spe m cells
Mul idimensional eco ding o biological p ocesses is a s anda d app oach in biological
esea ch and i can encompass spa ial dimensions, ime and colo (i.e. usually by
di e en luo escen labels). This scaling in dimensions has been ins umen al o s udy
aspec s and de ails (e.g. by co ela ion) ha we e no possible be o e, including in
he s udy o spe m mo ili y and chemo axis. Using luo escen p obes and a con ocal
mic oscope we a e now able o ge 3D spa ial econs uc ion o cells and hei labelled
s uc u es by ocusing a di e en Zposi ions (slice). Howe e , mos o hese assays
equi e immobile o ixed ma e ial. To make 3D empo al sc eenings, di e en cells
a e ixed a di e en ime-poin s e.g. a e ea men . Imaging mo ile shape-changing
cells in 3D plus ime, howe e , has been challenging, specially when hey swim a
200-300 µm.s-1.
Some 3D imaging se ups ha e p e iously been de eloped and can help o add ess
27
CHAPTER 1. INTRODUCTION
his issue bu hey can only de ec he head o he cell. Examples include using wo
pe pendicula came as o obse e a 3D olume (C enshaw, 1991) o using an objec i e
coupled o a piezoelec ic de ice in o de o ake XY slices a di e en Zposi ions
a an eno mous a e (Co kidi e al., 2008). The majo disad an age o he la e is
he non-negligible e o on he es ima ion o he Zposi ion o he cell while scanning
conside able olumes (i.e. de e mined by he Zampli ude o he piezoelec ic de ice).
Su e al., (2012) de eloped di e en sys em based on holog aphy whe e o e 1,500
cells can be acked wi h submic on p ecision on a olume as big as 17 mm3. The
same sys em was la e used o s udy he 3D chemo axis o A bacia punc ula a spe m
(Jikeli e al., 2015). Ne e heless, combining his sys em wi h luo escen labeling is
no s aigh o wa d (Rosen and B ooke , 2008; Nadeau e al., 2016), a ea u e ha
is essen ial o s udy spe ma ozoan chemo axis, namely, how does he [Ca2+]iand
he pHichange. Also, he lagella canno be ye esol ed in any o hese se ups and
we know ha he cell’s ajec o ies a e no su icien o unde s and 3D chemo axis
(C enshaw, 1989). On he o he hand, he 2D+Z( )sys em o Co kidi e al., (2008)
has al eady been shown o be able o segmen he human spe m lagellum (Sil a-
Villalobos e al., 2014) and has he po en ial o use luo escen ma ke s. In ha
s udy, hey scanned only 16 µm in he Z-axis wi h a piezoelec ic equency o 90
Hz and a ame a e o 5000 Hz, co esponding o a mean spacing be ween slices is
∼0.6µm and he Z-e o men ioned becomes negligible. Howe e , his is no he
case i we ack mul iple cells in a conside able la ge olume, as in Pimen el e al.,
(2012), so an imp o emen o he cell’s coo dina e p ecision is equi ed. Al hough i s
p omising po en ial o elucida e chemo axis, no signi ican ou pu has been p oduced
wi h he 2D+Z( )sys em. Sol ing i s ew issues will be in umen al o add ano he
dimension o 3D spe m chemo axis.
1.3.5 Ma hema ical modeling o spe ma ozoa
Spe ma ozoa do no ha e he machine y necessa y o syn hesize p o eins so i is no
possible o use a gene ic manipula ion app oach o s udy hem. Hence, much o
he esea ch on spe ma ozoa ha e been made using di e en ma ke s and d ugs such
as inhibi o s. Fo he case o spe m mo ili y, he heo e ical app oach has been a
p ecious ins umen . G ay and Hancock, (1955) we e he i s o success ully model
in e eb a e spe m mo ili y. They p oposed he Resis i e Fo ce Theo y (RFT) whe e
he lagellum is app oxima ed as a se o in ini ely small ods. As he axoneme mo o s
28
1.3. SEA URCHIN SPERMATOZOA
elici mo emen o he cell, each lagella piece mo es, c ea ing a o ce on he luid,
which will in u n exe an opposi e o ce on ha lagella piece (i.e. New on’s ac ion-
eac ion law). They came o he conclusion ha , in o de o he cell o mo e o wa d,
he lagella d ag coe icien s pe pendicula and pa allel o he lagella cen eline need
o ha e a a io highe han one. Hal a cen u y la e i was shown ha his heo y can
model bull (F ied ich e al., 2010) and sea u chin (Jikeli e al., 2015) spe m wi h high
p ecision, in ei he con ined o ee-swimming assays. Fo con ined swimming, ad hoc
inc eases o lagella a ios a e usually conside ed o measu ed in o de o compensa e
he neglec o long- ange hyd odynamic o ces (Smi h e al., 2009a; F ied ich e al.,
2010).
RFT is in ac an app oxima ion (i.e. neglec s long- ange hyd odynamic o ces) o
he mo e gene al amewo k – Slende Body Theo y (SBT), which explici ly desc ibes
he luid lows gene a ed by he cell(s) and how he luid also a ec s he spe ma ozoan
con o ma ion using Na ie -S okes equa ions (Johnson and B okaw, 1979). La e
i was es ablished ha one could implemen his heo y using s okele s, a as e
nume ical app oxima ion (Gillies e al., 2009; Co ez, 2001). These and o he simila
amewo ks allow o s udy he con ining p ocess (Smi h e al., 2009a; Elge i e al.,
2010), i.e. how cells become apped in he wa e -glass in e ace, as hey can model
he o ces he bounda y exe s bo h on he luid and on he cell. As con ined swimming
is mo e p e alen in in e nal e ilizing species, mos o hese s udies a e pe o med
assuming a mammalian spe ma ozoan model. Al hough no o -plane componen s o
he lagella bea ing we e equi ed o accumula ion in su aces, i was also possible
o con ine cells wi h helicoidal lagella bea ing (Smi h e al., 2009a). Fo he sea
u chin spe m model, i was shown ha a ac ion o he cell su ace is he esul o
hyd odynamic in e ac ions be ween he lagellum, he su ace and he o wa d h us
o he cell. Those o ces p omo e bo h a o que on he swimme ha aligns i pa allel
o he bounda y and a o ce ha app oxima es i o he wall, including a ail epulsion
( hus, a head a ac ion) a sho dis ances om he su ace (Elge i e al., 2010).
On he o he hand, i was also shown o he ac i e lagella model (i.e. euka yo ic)
ha he con ining beha io depends on many ac o s and canno be known a p io i
(E ans and Lauga, 2010).
These and o he heo e ical amewo ks ha e been de eloped o s udy o he
pe spec i es o he spe m cell. In in e species compa a i e s udies, he op imal
mo phology o mo ili y o uni lagella ed cells was shown o be dependen on he a io
29
CHAPTER 1. INTRODUCTION
be ween head and lagella leng h, a he han hei absolu e alues (Humph ies e al.,
2008), and ha o each head leng h he e is a ini e op imal lagella leng h (Tam,
2008). The op imal lagella s oke cha ac e is ics (displacemen gi en expended
ene gy) we e ound o be e y simila o he biological cases (Tam, 2008; Spagnolie
and Lauga, 2010; Lauga and Eloy, 2013). These include he exis ence o hal -
in ege wa eleng h, which is p oposed o educe o a ion and inc ease he ansla ional
eloci y, simila wa e ampli ude o lagella leng h a io and he dec ease in cu a u e
along he p opaga ing wa e. Lauga and Eloy, (2013) e en sugges ha euka yo ic
lagella a e mechanically op imal.
No e he e a e also s udies which model he sliding ubules o he axoneme
explici ly (Camale and J¨
uliche , 2000; Cibe , 2002; Riedel-K use e al., 2007) and
some e en explici ly include he dynein machine y (Hines and Blum, 1978; Hines and
Blum, 1979; B okaw, 2014).
Spe ma ozoan chemo axis has also been he objec o many heo e ical
s udies. Simple models o mo ili y and chemo axis which also neglec long-
ange hyd odynamical o ces o he lagellum al oge he . One example using
o dina y di e en ial equa ions (ODEs) also measu ed chemo axis in popula ion by
assuming s ochas ic p ocesses o agglome a ion (i.e. as an in e se p ocess o
di usion) wi h di e en deg ees o beha io al and en i onmen al assump ions (Kelle
and Segel, 1970; Ho s mann, 2003). O he app oaches use ei he he s imulus
(chemoa ac an ) o he de i a i e o [Ca2+]i o a ec he a e age pa h cu a u e
di ec ly (F ied ich and J¨
uliche , 2007; Al a ez e al., 2012). In a consequen s udy
whe e bo h s imulus and calcium de i a i e model chemo axis by a ec ing he mean
lagella cu a u e, wo di e en beha io s we e iden i ied: an ’on esponse’, whe e he
cell s eadily bu slowly edi ec s i s ajec o y o maximize he mean s imula ion (i.e.
owa ds he cen e o a chemoa ac an sou ce-poin ); and an ’o esponse’, whe e
he cell pe o ms an ab up change in di ec ion when i swims down he g adien (Jikeli
e al., 2015). No wi hs anding, we canno gi e meaning o he simple adap a ion
module (i.e. signaling module) pa ame e s in e ms o he componen s desc ibed o
he biological signaling cascade. The e a e some ma hema ical models ha ansla e
he signaling cascade o boolean ne wo ks (Espinal e al., 2011) and hose ha
use ODEs a e unde de elopmen (Daniel Espinosa, manusc ip unde p epa a ion,
pe sonal communica ion). Howe e , we ha e ye o see an in eg a i e model whe e
shape, mechanics and signaling cascade a e p esen (also unde de elopemen ).
30
1.4. IN THIS THESIS
1.4 In his hesis
As in oduced be o e, he e is a bo leneck be ween imaging da a acquisi ion and i s
analysis. This occu s a se e al le els, including speed, objec i i y, ep oducibili y and
obus ness. Au oma ic analysis by compu e s can add ess hese issues bu pains aking
wo k is gene ally needed o adap an algo i hm o he sys em in s udy, as hey a e
no gene aly appliable o all o hem. One example whe e his occu s is he lack o
compa a i e s udy o sea u chin spe m ee-swimming mo ili y and chemo axis, which
we will use he e as a case s udy.
As a pa o he specia ia ion p ocess, one expec s spe m om di e en species o
eac di e en ly o he chemoa ac an s eleased by he homologous eggs. Howe e ,
he na u e o such di e ence is unknown. I has been specula ed ha species-speci ic
chemo axis plays a key ole in inc easing he chances o e iliza ion du ing b oadcas
spawning. Mo ili y and i s coo dina ion a e he co ne s ones o chemo axis which
jus i ies he demand o compa a i e s udies o spe m o di e en species. Since sea
u chin spe ma ozoa a e eleased and p esumably e ilize he eggs in a 3D se ing,
i is essen ial o unde s and how spe ma ozoa om di e en species swim eely in
he olume. These 3D compa a i e s udies ha e been hinde ed by he limi a ions
o 3D imaging me hods o as cells and he lack o eliable and obus imaging
analysis me hods ha can deal wi h his da a ype and amoun . We p opose ha
me hods using a p io i knowledge, in he o m o a ma hema ical model ha desc ibes
bo h he o m and de o ma ion o he cell and he mechanics ha p opel i o wa d
wi hin a luid, will imp o e he de ec ion and acking quali y, wi h he ad an age
o easy change o he o m and physic desc ip o s. Due o i s po en ial o s udy
chemo axis, we decided o use he 2D+Z( )mic oscopy se up (Co kidi e al., 2008)
bu an imp o emen o he accu acy and p ecision o he cell’s posi ion in 3D is
o pa amoun impo ance o ob ain eliable da a. Thus, he objec i e o he wo k
p esen ed in his hesis is wo- old:
1. To de elop and apply an au oma ic me hod o spe ma ozoan de ec ion and
acking in imaging da a using a mechanis ic model o hese cells as a p io i
in o ma ion;
2. To de elop and es ools o he analysis o spe ma ozoan mo ili y and
chemo axis ha enable compa a i e s udies ac oss species and plana and 3D
modes o swimming.
To accomplish he i s objec i e we hypo hesize ha we can use a mechanis ic
31

CHAPTER 1. INTRODUCTION
model o he spe ma ozoon as a p io i knowledge o be i ed di ec ly o imaging
da a by maximum likelihood. Due o i s physical na u e and high le el o de ail, he
model should impose mo phological and kinema ic cons ains ha accu a ely desc ibe
he objec (s) depic ed in he imaging da a. I his me hodological hypo hesis holds
ue, hen hese me hods is expec ed o allow us o dis inguish di e en ea men s
o condi ions as hey will esul in dispa a e pa ame e iza ion o he model.
Fo he second objec i e we will add ess se e al issues, mos ly ela ed o he
2D+Z( )mic oscopy sys em and he da a i gene a es. Can we imp o e he
de ec ion’s p ecision and accu acy o he cell’s posi ion? Fu he mo e, can we use
he app oach de eloped o he i s objec i e o desc ibe spe ma ozoan mo ili y and
chemo axis? I so, is he e a di e ence in 3D spe m mo ili y be ween L. pic us and
S. pu pu a us?
In Chap e 2 we de eloped a simple amewo k o biological model-based image
analysis by compa ing in silico imaging da a p oduced by a mo phodynamical model
o mic oscopy images in o de o es ima e he posi ion, o ien a ion, o m and physical
pa ame e s o a spe ma ozoon and i s su ounding media. We p o ed ou amewo k
allows us o ack shape-shi ing cells qui e p ecisely (as well as a human does i ).
Be e ye , we show we can in e he in isible lagella bending pa e ns and posi ions
by acking only he head. The s udy in his chap e is he co e o a manusc ip in
p epa a ion.
The 3D compa a i e s udy is desc ibed in Chap e 3. Using he mechanis ic
models app oach u ned ou o be compu a ionally no easible due he dimension
o po en ial pa ame e space and da a se s. We had o educe he combina o ial
and compu a ional p oblem using a mixed app oach. In a mo e adi ional app oach,
ha in ol ed he de elopmen o an accu a e es ima ion o he Zposi ion associa ed
wi h each ame, we used p io knowledge on he de ocused appea ance o he cells
o es ima e cell cen oids in space and ime. Then, a simpli ied kinema ic model
o he helical swimming pa hs was used o econs i u e he ajec o ies by piecewise
helical segmen i ing. Using his me hod we we e able o econ i m he ajec o y
pa ame e s o p e ious manual analysis (Co kidi e al., 2008; Gue e o e al., 2010)
and, mo e impo an ly, o e eal a di e ence in con ining beha io be ween he
wo species conside ed. We hen used he mo phodynamical model o in e ha
S. pu pu a us spe m ha e highe mean lagella cu a u e in ee as compa ed o
con ined swimming.
32
1.5. MATHEMATICAL NOTATION
In Chap e 4 we used he de ec ion me hod de eloped in he hi d chap e o
assess he ee-swimming chemo ac ic beha io o L. pic us and S. pu pu a us. By
i ing linea mixed models o he da a, we concluded ha no chemo axis was de ec ed
in he expe imen al condi ions used.
Finally, we make a gene al discussion and conclusion connec ing he p e ious
chap e s, highligh ing he hesis con ibu ions and u u e pe spec i es in Chap e 5.
1.5 Ma hema ical no a ion
Fo con enience o he eade , we p esen a ew de ails o he ma hema ical no a ion
used h oughou his hesis.
Gene ally, bold symbols ep esen enso s (ei he column ec o s o ma ices) and
no mal ype ace ep esen scala s. Pa ame ic unc ions a e ep esen ed no mally by
(x)bu some imes, abusing he no a ion, hey can be ep esen ed wi hou hei
a iable(s) ( ). No e ha , due o he high numbe o pa ame e s and a iables used
h oughou his hesis, he same symbol can ha e di e en meanings in di e en
chap e s.
Re e ences
Abbo , B. P. e al. (2016). “Obse a ion o g a i a ional wa es om a bina y black hole
me ge ”. Physical Re iew Le e s 116.(6), pp. 1–16.
A zelius, B. A. (1955).“The ine s uc u e o he sea u chin spe ma ozoa as e ealed by he
elec on mic oscope.”Zei sch i u Zell o schung und mik oskopische Ana omie (Vienna,
Aus ia : 1948) 42.(1-2), pp. 134–148.
Aga d, D. A. and Seda , J. W. (1983).“Th ee-dimensional a chi ec u e o a poly ene nucleus”.
Na u e 302, pp. 676–681.
Ahmed, M. M. and Mohamad, D. B. (2011). “Segmen a ion o b ain magne ic esonance
images (MRIs): a e iew”. In e na ional Jou nal o Ad ances in So Compu ing and i s
Applica ions 3.(3), pp. 1–24.
Al a ez, L., Dai, L., F ied ich, B. M., Kashika , N. D., G ego , I., Pascal, R., and Kaupp, U. B.
(2012).“The a e o change in Ca 2+ concen a ion con ols spe m chemo axis”. Jou nal
o Cell Biology 196.(5), pp. 653–663.
ATLAS Collabo a ion (2012).“Obse a ion o a new pa icle in he sea ch o he S anda d
Model Higgs boson wi h he ATLAS de ec o a he LHC”. Physics Le e s B 716.(1),
pp. 1–29.
33
CHAPTER 1. INTRODUCTION
Baba, S. A. and Mogami, Y. (1985).“An app oach o digi al image analysis o bending shapes
o euka yo ic lagella and cilia”. Cell Mo ili y 5.(6), pp. 475–489.
Bishop, D. W. (1962). “Spe m Mo ili y”. The Ame ican Physiological Socie y 42.(1),
pp. 338–360.
B¨
ohme , M. e al. (2005). “Ca2+ spikes in he lagellum con ol chemo ac ic beha io o
spe m.”The EMBO jou nal 24.(15), pp. 2741–2752.
B okaw, C. J. (1965). “Non-sinusoidal bending wa es o spe m lagella.” The Jou nal o
expe imen al biology 43, pp. 155–169.
— (1966). “E ec s o inc eased iscosi y on he mo emen s o some in e eb a e
spe ma ozoa”. Con ol 45, pp. 113–139.
— (1971). “Bend p opaga ion by a sliding ilamen model o lagella”. Jou nal o
Expe imen al Biology 55, pp. 289–304.
— (1975).“Molecula mechanism o oscilla ion in lagella and muscle”. P oceedings o he
Na ional Academy o Sciences 72.(8), pp. 3102–3106.
— (1979). “Calcium-induced asymme ical bea ing o i on-dememb ana ed sea u chin
spe m lagella.”The Jou nal o cell biology 82.(2), pp. 401–11.
— (1984).“Au oma ed me hods o es ima ion o spe m lagella bending pa ame e s.”Cell
mo ili y 4.(6), pp. 417–430.
— (2003).“Swimming wi h h ee-dimensional lagella bending wa es”. Second In e na ional
Symposium on Aqua-Biomechanics, Honolulu, HI.
— (2014). “Compu e simula ion o lagella mo emen X: Double pai spli ing and bend
p opaga ion modeled using s ochas ic dynein kine ics”. Cy oskele on 71.(4), pp. 273–284.
Bu nham, K. P. and Ande son, D. R. (2002). “Model Selec ion and Mul imodel In e ence”.
Second. Sp inge -Ve lag.
Camale , S. and J¨
uliche , F. (2000).“Gene ic aspec s o axonemal bea ing”. New Jou nal o
Physics 2, pp. 24.1–24.23.
Chen, D. T. N., Heymann, M., F aden, S., Nicas o, D., and Dogic, Z. (2015). “ATP
consump ion o euka yo ic lagella measu ed a a single-cell le el.” Biophysical jou nal
109.(12), pp. 2562–73.
Cibe , C. (2002). “Axonemal ac i i y ela i e o he 2D/3D-wa e o m con e sion o he
lagellum”. Cell mo ili y and he cy oskele on 51.(No embe 2001), pp. 89–111.
CMS Collabo a ion (2012).“Obse a ion o a new boson a a mass o 125GeV wi h he CMS
expe imen a he LHC”. Physics Le e B 716.(1), pp. 30–61.
Co kidi, G., Taboada, B., Wood, C. D., Gue e o, A., and Da szon, A. (2008).“T acking spe m
in h ee-dimensions.” Biochemical and Biophysical Resea ch Communica ions 373.(1),
pp. 125–129.
Co ez, R. (2001). “The me hod o egula ized S okesle s”. SIAM Jou nal on Scien i ic
Compu ing 23.(4), pp. 1204–1225.
34
REFERENCES
Cosson, J., G oison, A.-L., Suque , M., Fau el, C., D eanno, C., and Billa d, R. (2008).
“Ma ine ish spe ma ozoa: acing epheme al swimme s.” Rep oduc ion (Camb idge,
England) 136.(3), pp. 277–94.
Cosson, J., Hui o el, P., and Gagnon, C. (2003).“How spe ma ozoa come o be con ined o
su aces.”Cell mo ili y and he cy oskele on 54.(1), pp. 56–63.
C enshaw, H. C. (1989).“Kinema ics o helical mo ion o mic oo ganisms capable o mo ion
wi h ou deg ees o eedom”. Biophysical jou nal 56.(No embe ), pp. 1029–1035.
— (1991). “A echnique o acking spe ma ozoa in h ee dimensions wi hou iscous wall
e ec s”. Compa a i e Spe ma ology 20, pp. 353–357.
C enshaw, H. C., Ciampaglio, C. N., and McHen y, M. (2000). “Analysis o he h ee-
dimensional ajec o ies o o ganisms: es ima es o eloci y, cu a u e and o sion om
posi ional in o ma ion”. Jou nal o Expe imen al Biology 203, pp. 961–982.
Da szon, A., Gue e o, A., Galindo, B. E., Nishigaki, T., and Wood, C. D. (2008). “Spe m-
ac i a ing pep ides in he egula ion o ion luxes, signal ansduc ion and mo ili y.”The
In e na ional jou nal o de elopmen al biology 52.(5-6), pp. 595–606.
Da szon, A., Nishigaki, T., Bel an, C., and T e i˜no, C. L. (2011).“Calcium channels in he
de elopmen , ma u a ion, and unc ion o spe ma ozoa.” Physiological e iews 91.(4),
pp. 1305–55.
de La Go ce, M., Pa agios, N., and Flee , D. J. (2008). “Model-based hand acking wi h
ex u e, shading and sel -occlusions”. 26 h IEEE Con e ence on Compu e Vision and
Pa e n Recogni ion, CVPR (June), pp. 1–8.
Du ou , A., Liu, T. Y., Duc oz, C., Tou nemenne, R., Cummings, B., Thibeaux, R.,
Guillen, N., He o, A., and Oli o-Ma in, J. C. (2015). “Signal p ocessing challenges in
quan i a i e 3-D cell mo phology: Mo e han mee s he eye”. IEEE Signal P ocessing
Magazine 32.(1), pp. 30–40.
Dupon , S., Do ey, N., S umpp, M., Melzne , F., and Tho ndyke, M. (2013). “Long- e m
and ans-li e-cycle e ec s o exposu e o ocean acidi ica ion in he g een sea u chin
S ongylocen o us d oebachiensis”. Ma ine Biology 160.(8), pp. 1835–1843.
Elge i, J., Kaupp, U. B., and Gomppe , G. (2010). “Hyd odynamics o spe m cells nea
su aces.”Biophysical jou nal 99.(4), pp. 1018–1026.
Espinal, J., Aldana, M., Gue e o, A., Wood, C. D., Da szon, A., and Ma ´ınez-Mekle , G.
(2011). “Disc e e dynamics model o he spe ac -ac i a ed Ca2+ signaling ne wo k
ele an o spe m mo ili y.”PloS one 6.(8), e22619.
Espinal-En ´ıquez, J., Da szon, A., Gue e o, A., and Ma ´ınez-Mekle , G. (2014). “In silico
de e mina ion o he e ec o mul i- a ge d ugs on calcium dynamics signaling ne wo k
unde lying sea u chin spe ma ozoa mo ili y”. PLOS one 9.(8), pp. 1–11.
E ans, A. and Lauga, E. (2010). “P opulsion by passi e ilamen s and ac i e lagella nea
bounda ies”. Physical Re iew E 82.(4), pp. 1–12.
35
CHAPTER 2. MORPHODYNAMICAL IMAGE ANALYSIS OF SPERMATOZOA
SWIMMING IN THE PLANE
he ue pa ame e s used o gene a e he syn he ic da a and ha we can ack a
spe ma ozoon as good as a human can. Using his amewo k we we e e en able o
in e he o m and posi ion o he lagellum by acking only he head, i.e., in he
images in which he lagella whe e no isible. Using such mo phodynamical models
as a p io i knowledge migh gi e he necessa y in o ma ion ha au oma ed imaging
analysis me hods demand in o de o become mo e eliable, independen and obus .
Acknowledgmen s
Jo ge Ca nei o and Ped o ˆ
Angelo Sil a designed he esea ch. Ped o ˆ
Angelo Sil a
pe o med he wo k and Jo ge Ca nei o supe ised. Ad´an Gue e o and Albe o
Da szon p o ided he 2D+ imaging da a. Ped o ˆ
Angelo Sil a, Albe o Da szon
and Jo ge Ca nei o con ibu ed o w i ing he manusc ip . This wo k was unded by
Funda¸c˜ao pa a a Ciˆencia e Tecnologia, Po ugal, (SFRH/BD/79261/2011), Ins i u o
Gulbenkian de Ciˆencia, Po ugal, and Ins i u o de Bio ecnolog´ıa, UNAM, Mexico.
The wo k de eloped in his chap e is he objec o a manusc ip in p epa a ion, o
be submi ed o an in e na ional pee - e iewed jou nal.
42

2.1. INTRODUCTION
2.1 In oduc ion
The au oma ion o mic oscopy sys ems and he eme gence o mul idimensional
measu emen o biological da a has ecen ly been p oducing da a a a o e whelming
pace, one ha cu en imaging analysis p ocedu es a e unable o cope wi h. The e
has been an e o o au oma e image analysis bu he speci ics o each s udy makes i
di icul o ha e gene al ools ha allow o ex ac in o ma ion om he imaging da a.
The e o e, scien is s o en wande on empi ical es ing o hese ools pa ame e s. This
c ea es no only highly biased esul s bu also i ep oducible ones, as many o hese
pa ame e s a e a ely epo ed in many s udies. As such, au oma ic me hods o
image analysis which do no equi e pa ame e weaking and can be gene ally used
in an wide a ay o s udies a e desi ed.
The majo di e ence be ween a compu e and a human when ex ac ing
in o ma ion om imaging da a is ha he la e is ich in a p io i, implici knowledge.
Based on pas expe iences, a pe son uses implici models o dis inguish be ween
backg ound, cell, nucleus, and so on. In compa ison, he compu e algo i hms a e
e y limi ed in he esou ces hey can deploy, which a e es ic ed o aw spec al,
spa ial and empo al da a, and a se o s a is ical ools bu no model o connec
hem all. Wha i he compu e was endowed wi h a p io i knowledge on he objec
o s udy, including i s mo phology and dynamics, and o he mic oscopy se up such as
he poin -sp ead unc ion and ligh ing condi ions? I should hen be possible o he
compu e algo i hms o ex ac in o ma ion as good as any human. Be e , in ac , i
we conside he ep oducibili y o he de e minis ic me hods. No only ha , wha i
he compu e can ex ac in o ma ion ha a human canno , e.g. a ea u e non-linea ly
co ela ed wi h mo phological dynamics which is no di ec ly measu able/no iceable
in he imaging da a?
When one is dealing wi h imaging da a, he e is a dis inc ion be ween image
p ocessing (i.e. low-le el image manipula ion such as educ ion o noise and
backg ound sub ac ion) and image analysis – he ex ac ion o in o ma ion om
imaging da a. Fo his end, he e is a long adi ion o segmen ing he image
o i ing pa ame ic models o each ime-poin independen ly and only hen he
ex ac ed in o ma ion is used o i mo phodynamical models (B okaw, 1984; Baba
and Mogami, 1985; F ied ich e al., 2010; Su e al., 2012; Su e al., 2013; Jikeli
e al., 2015). In he compu e ision ield, ace- ecogni ion and acking usually
eso s o he Lucas-Kanade algo i hm as i allows o de o m empla es in o de o
43
CHAPTER 2. MORPHODYNAMICAL IMAGE ANALYSIS OF SPERMATOZOA
SWIMMING IN THE PLANE
iden i y hem in he nex ame (Bake and Ma hews, 2004). O he me hods use
maximum likelihood o es ima e he da a ha p oduced a gi en imaging da a se o
immobile s uc u es (Vulo i´c e al., 2013; Ke ann e al., 2016). To he bes o ou
knowledge, me hods whe e he o m and de o ma ion o he objec , i s in e ac ion
wi h su ounding media and ans o ma ion o he imaging p ocess a e all combined
as pu ely ma hema ical and heo e ical desc ip o s ha e no been implemen ed o
es ima e he mo phology and posi ion o mo ile cells ha change hei shape in ime.
Ou objec i e is o de elop a amewo k o de ec and ack cells using a
mo phodynamical model. We will gene a e syn he ic imaging da a based on he model
and compa e he syn he ic images ende ed om his model di ec ly o expe imen al
imaging da a by co ela ion, a p oxy o likelihood (Zucke , 2003). We will show he e
ha his me hod is able o de ec and ack sea u chin spe ma ozoa as good as
a human and e en ou pe o m he human abili y o ecognize pa e ns by in e ing
lagella posi ions and con o ma ions when hese s uc u es a e no isible in he
images. As such, he use o mechanis ic models wi hin he image analysis p ocedu es
is ins umen al by inc easing he esolu ion o he analysis and by allowing o in e
s uc u es ha a e missing om he imaging da a.
2.2 Ma e ials and Me hods
2.2.1 Mo phodynamical model
Conside a ma hema ical model o a cell in which he mo phology changes a e de ined
wi hin he model i sel – his is a mo phodynamical model. In ou pa icula case,
he spe ma ozoon cell model (S) is de ined by he shape (Σ) and mechanics (Φ)
modules.
Shape
We can de ine he mo phology o a spe ma ozoon by de ining wo egions, he
head and he lagellum. We assume he head is a e olu ion ellipsoid wi h hal
axes aand b=c. The lagellum is composed o N ods wi h o al leng h L
(µm). The con o ma ion o he lagellum a a pa icula ime is gi en by i s
cu a u e κalong lagella a cleng h s, a i s o de a eling wa e: κ(s, ) =
K0+ (A0+A1e−A1s) cos (ωκ −λκs+φ), which is de ined by he mean lagella
cu a u e (K0), basal cu a u e ampli ude (A0), exponen ial e m o cu a u e
44
2.2. MATERIALS AND METHODS
ampli ude (A1), angula eloci y o bea ing (ωκ) and cu a u e wa eleng h (λκ).
The phase (φ) nea he apical pa o he lagellum ( i s 0.5 µm) is φ1while o he
emaining lagellum is φ0. Fo a plana bea ing, we assume he lagella o sion along
a cleng h o be ze o (τ(s, ) = T0= 0 ad.µm-1). The in insic lagella posi ion
( (s, )) is calcula ed by a cleng h in eg a ion o he Cosse a ame (Cao e al.,
2006; Jikeli e al., 2015), which is de ined by he o hono mal basis e1(s, ),e2(s, )
and e3(s, ):∂ (s, )/∂s =e3(s, ),∂e3(s, )/∂s =κ(s, )e1(s, ),∂e1(s, )/∂s =
−κ(s, )e3(s, )+τ(s, )e2(s, ),∂e2(s, )/∂s =−τ(s, )e1(s, ). No e e3 ep esen s
he cen eline along he lagellum, (0, ) = {−a, 0,0},e1(0, ) = {0,−1,0},
e2(0, ) = {0,0,1}and e3(0, ) = {−1,0,0}. A a gi en ime , he cen oid o
he head has ex insic posi ion Rh( )and o ien a ion ma ix Θ( ), which o a es
om ex insic o in insic coo dina es. We can de ine he o ien a ion ma ix as
combina ion o o a ions on he X,Yand Zaxes – Eule angles. In ou speci ic case
Θ=RXRYRZ, whe e:
RX=


1 0 0
0 cos(νX) sin(νX)
0−sin(νX) cos(νX)


;RY=


cos(νY) 0 −sin(νY)
0 1 0
sin(νY) 0 cos(νY)


;RZ=


cos(νZ) sin(νZ) 0
−sin(νZ) cos(νZ) 0
0 0 1



a e o a ion ma ices on he X,Yand Zaxes, espec i ely. No ice we d opped he
ime om he p e ious equa ions o simplici y. Thus, he o ien a ion o a cell a a
gi en ime is speci ied by he pa ame e s νX( ),νY( )and νZ( ). In he con ined
swimming case we assume νX( )=0and νY( ) = 0.
Mechanics
As he cell mo es wi hin a iscous luid, he la e exe s d ag o ces which, unde some
condi ions, p opel he cell in he media. The physics behind i ha e been shown o be
app oxima ed by Resis i e Fo ce Theo y (RFT) (G ay and Hancock, 1955; F ied ich
e al., 2010; Jikeli e al., 2015), which we will use he e. B ie ly, he d ag o ce densi y
ha luid exe s on he lagella piece is (s, ) = ξk∂˙ ,k(s, )/∂ +ξ⊥∂˙ ,⊥(s, )/∂ ,
whe e ˙ ,kand ˙ ,⊥a e he angen and no mal componen s, espec i ely, o he mean
lagella piece eloci y ˙ (s, ) = (Ψ0(δθ( ))· (s, +δ)+δ ( )− (s, +δ))/δ du ing
he ime in e al δ. No e and θa e he in insic head ansla ional and o a ional
eloci ies, espec i ely, Ψ0(ν)is he an app oxima ion o Rod igues o a ion o mula
(Ψ0(ν)) o small angles (sin β≃βand cos β≃1) and ξkand ξ⊥a e he angen and
no mal d ag coe icien s o he o ce he luid exe s on he lagellum, espec i ely.
45
CHAPTER 2. MORPHODYNAMICAL IMAGE ANALYSIS OF SPERMATOZOA
SWIMMING IN THE PLANE
We can ob ain he in insic head eloci ies by sol ing he sys em o o ce and o que
equilib ia de ined by ξT ( )RL
0 (s, )∂s = 0∧ξRθ( )−RL
0 (s, )× (s, )∂s = 0,
whe e ξTand ξRa e he ansla ional and o a ional d ag coe icien s o he head
and he symbol ×deno es he c ossp oduc ope a o . The in insic head eloci ies a e
ans o med in o ex insic eloci ies by mul iplica ion o he ansposed o ien a ion
ma ix and applied o he cell o calcula e he new ex insic posi ion Rh( +δ) =
Rh( ) + δΘ( )T· ( )and o ien a ion Θ( +δ) = Ψ(δΘ( )T·θ( )) ·Θ( ). No e
we used he app oxima ed Rod igues o a ion ma ix o calcula e he local lagella
eloci y in o de o make he sys em linea bu his app oxima ion is no needed when
we calcula e he ex insic eloci ies om he in insic ones.
Assuming a dynamic iscosi y ηwe can calcula e he ansla ional and o a ional
d ags o he head based on i s size using Pe in’s o mulas (Pe in, 1936). As we
used Eule in eg a ion me hod, we will upda e he sys em e e y δ ime s eps and
hen we sol e i o he in insic ansla ional and o a ional eloci ies assuming hey
a e cons an wi hin each ime pe iod δ. Fo spa ial in eg a ion o he lagellum, we
disc e ized i in Nsegmen s and calcula ed he in insic midpoin posi ion κ[i, ]
whe e i=L/N We de e mined N= 149 and δ= 50 µs o p oduce e o s below
1% when calcula ing he in insic ansla ional and o a ional eloci ies, compa ing
o N= 499 and δ= 10 µs.
The spe ma ozoon model S(Σ,Φ)is hus comple ely de ined by i s shape
pa ame e s Σ= (a, b, c, L, N, Rh,Θ, K0, A0, A1, φ0, φ1, ωκ, λκ, T0)and i s physical
o mechanical pa ame e s Φ= (ξT,ξR, ξk, ξ⊥). Al hough no explici he e, some o
hese pa ame e s a e dependen on ime, as shown abo e.
To measu e he di e ence be ween wo models ins ances, say Saand Sb, we
de ined a con enien lis o 14 pa ame e s, composed o some o basic pa ame e s
and hei a ios. The dis ance be ween he pa ame e lis s, Paand Pb, o he wo
models is compu ed as:
χ2=X
j
(pa
j−pb
j)2
(pb
j),(2.1)
whe e j∈ {1, ..., 14}is he index o he pa ame e o a io in he lis
{a, a/b, L, K0, A0, A1, ωκ, λκ, φ0, φ1, ξTx, ξTy/ξTx, ξk, ξ⊥/ξk}. No e we did no
include in his measu e some pa ame e s as hey will no a ec he cell’s mo ili y i we
assume plana lagella bea ing, which we did in his chap e . Also no e only he a ios
46
2.2. MATERIALS AND METHODS
be ween some o pa ame e s a ec he swimming pa h o he cell, he eason why we
compa e he dis ance o hose a ios and no o he pa ame e s hemsel es. We will
usually e e o he dis ance o he g ound u h model o o he ue pa ame e s,
meaning his se o pa ame e s is conside ed as he model Sbin he o mula abo e.
2.2.2 Compa ing and i ing he model o imaging da a
We gene a ed a model ins ance a ime iwi h a gi en se o pa ame e s Sand
ende ed i in a new image (IS; = i) wi h weigh s (i.e. adimensional pixel in ensi y)
whand w o he head and lagellum, espec i ely. Then, we con ol ed he IS; = i
wi h a Gaussian il e G(0,(σ/6)2)(ke nel o size σ, down-scaled by a ea in e pola ion
( e e o OpenCV unc ion esize) o ma ch he spa ial and empo al esolu ion o
he expe imen al imaging da a (I = i)). The co ela ion sco e o a pa ame e se
o he model is he sum o he co ela ion coe icien s be ween he imaging da a
(I = i) and IS; = i, only o he pixels ha a e wi hin he ρ adius a ound he objec :
Sco e(S, i, j, ρ , σ, wh, w ) = Pj
k=iCo ela ion(I = k,IS; = k|ρ , σ, wh, w ), o
i≤j, i≤ j,∀i, j ∈N0, whe e Nis se o he na u al numbe s. No e ha we will
also e e o he a e age co ela ion coe icien as Sco e(S, i, , ρ , σ, wh, w )/(j−
i+ 1).
The likelihood is he p obabili y o a pa ame e se gi en he da a. Thus, we
sea ched he pa ame e space o he se which bes desc ibes he da a by maximizing
his p obabili y. Ou model has many pa ame e s, is non-linea and he solu ion space
is no con ex, which makes i di icul o de ine he maximum likelihood algeb aically
and also o ind i wi h op imiza ion algo i hms. To o e come his di icul y we
used he esul o (Zucke , 2003) who ha e shown ha unde some condi ions, by
maximizing he c oss-co ela ion be ween a model and imaging da a one is e ec i ely
maximizing he likelihood. We implemen ed a simple Mon e Ca lo (MC) e olu iona y
algo i hm ( ig. 2.2 B), whe e he i ness unc ion is gi en by he co ela ion sco e.
To unde s and he p ocedu e, le us in oduce he ollowing symbol: Sg
kis he spe m
model indexed ka i e a ion go he e olu iona y algo i hm. A i e a ion ze o (g= 0)
we inpu ou ini ial pa ame e se S0and eplica ed i Nc= 1000 imes while adding
Gaussian noise o he pa ame e s, excep o one se which is an exac eplica e.
We hen s a ed he i e a ion 1 a ime iand compu ed he indi idual sco e o
each pa ame e se (Sco e(S1
k), no ice he abuse o no a ion by disca ding all he
emainde pa ame e s). Then we selec ed he Nb= 10 i es pa ame e se s (i.e.
47

CHAPTER 2. MORPHODYNAMICAL IMAGE ANALYSIS OF SPERMATOZOA
SWIMMING IN THE PLANE
he op highes Sco e(S1
k)) and each one o hese gene a ed 100 new child en (S2
k)
o be e alua ed and a nex i e a ion. We epea ed his o Ni= 20 i e a ions.
Unless s a ed o he wise, we i s pe o med p elimina y op imiza ion o only he
shape pa ame e s (Σ0) a he ini ial ime ( i= ) and subsequen ly pe o med he
op imiza ion o he all he mechanical and empo al pa ame e s (see sec ion 2.3.1) on
he whole imaging da a se ( i6= ). The Gaussian pe u ba ions o he pa ame e s
we e pe o med using a ze o-cen e ed Gaussian wi h s anda d de ia ion 5% o he
pa ame e alue, excep o angles which we e 5% o π ad. A each passing i e a ion
g∈N0, he pe cen age d opped as 5%/(1+g). Some logical cons ains we e applied,
e.g. 1< ξ⊥/ξk<2and a>b=c > 0. To inc ease he speed o he algo i hm
while i ing he mechanical and empo al pa ame e s, we abo ed he compu a ion
o he co ela ion sco e o child en wi h a e age co ela ion ha was below 0.5 a
ime ∅, whe e i≤ ∅< , and he co ela ion sco e o ha model was assumed
o be Sco e(Sg
k, i, ∅, ρ , σ, wh, w ). Simila ly, he compu a ion o he sco e was
abo ed o candida e solu ions wi h an a e age sco e ha is lesse o equal o hal
he a e age sco e o he bes solu ion ob ained un il ha s age in he execu ion o
MC e olu iona y algo i hm.
2.2.3 Implemen a ion de ails
Unless s a ed o he wise, all he algo i hms o image manipula ion, model de ini ion
and op imiza ion we e de eloped and encoded in C/C++ using he ee compu e
ision lib a y OpenCV 3 (In el, San a Cla a, Uni ed S a es o Ame ica). Simula ions
we e un in a In el®i7-6700HQ CPU @ 2.60GHz×8 p ocesso (In el, San a Cla a,
Uni ed S a es o Ame ica) in Ubun u 16.04 (Canonical, London, Uni ed Kingdom)
using he pa allel (Tange, 2011) o un se e al simula ions a he same ime. Plo s,
s a is ics and igu es we e pe o med using R .3.0.3 (R Founda ion o S a is ical
Compu ing, Vienna, Aus ia) and L
A
T
EXusing he ikZ package, espec i ely.
2.2.4 Imaging da a
L. pic us and S. pu pu a us
All in i o imaging da a was oba ined as desc ibed in Gue e o e al., (2010), wi h a
ew al e a ions. A b ie desc ip ion ollows.
48
2.2. MATERIALS AND METHODS
Ma e ials. L. pic us and S. pu pu a us spe ma ozoa (Ma inus Inc., Long Beach,
CA, USA; Pamanes S. A. de C.V., Ensenada, Mexico) we e ex ac ed undilu ed
by in acoelomic injec ion o 0.5 M KCl, s o ed on ice and used wi hin 24 hou s.
a i icial sea wa e (ASW) was p epa ed wi h 486 mM NaCl, 10 mM KCl, 10 mM
CaCl2, 26 mM MgCl2, 30 mM MgSO4, 2.5 mM NaHCO3, 10 mM HEPES and 1
mM EDTA, up o 950-1000 mOsm. Final pH was 8.0 and 7.4 o S. pu pu a us and
L. pic us, espec i ely. Low Ca2+ ASW was p epa ed simila ly o ASW bu using
1 mM CaCl2and se ing he pH o 7.0. Fluo-4-AM and plu onic F-127 we e om
Molecula P obes, Inc. (Eugene, OR, USA). All o he eagen s we e om Sigma-
Ald ich (Toluca, Edo de Mexico, Mexico), unless s a ed o he wise.
Labeling spe ma ozoa and loading o incuba o chambe . Ten olumes o
low Ca2+ ASW con aining 0.2% w / ol plu onic F-127 and 20 µM o Fluo-4 AM
we e used o suspend undilu ed spe ma ozoa o S. pu pu a us. A e incuba ion o
wo hou s a 14 °C, spe ma ozoa we e s o ed in he da k and on ice. To p e en cells
o adhe e o he glass, all co e slips we e coa ed in PolyHEME (poly(2-hyd oxye hyl
me hac yla e)). Spe m om ei he species we e dilu ed in ASW in o a eusable
chambe and main ained a 15 °C h oughou he expe imen .
Fluo escence imaging o S. pu pu a us spe ma ozoa. Images we e acqui ed
wi h Nikon Plan Fluo 40×1.3 NA objec i e using a Ch oma il e se (ex, HQ470/
40×; DC, 505DCXRU; em, HQ510LP) and eco ded on a EMCCD Ando came a
(DV887, Ando iXon). S oboscopic ligh ing was used such ha 2 ms o lash was
synch onized wi h he came a exposu e (also 2 ms). Images we e collec ed wi h Ando
iQ 1.8 so wa e (Ando Bioimaging, NC) wi h ame a e 200 Hz in c opped-chip mode
(window =60×60 µm). Pixel esolu ion is 1.56 µm.pixel-1.
A ligh spli e was used o p oduce side-by-side image ames in which he
whole spe ma ozoa o only he spe m heads we e isible. We no iced ha he
ligh spli e p oduced misaligned images. The image side whe e spe m head whe e
labelled and isible was misaligned (−3.15,2.36) and -0.0605 ad ela i e o he side
o he image whe e he whole-cell label was isible. The misalignmen was co ec ed
by es ima ing o he a ine ans o ma ion ma ix ha maximizes he Enhanced
Co ela ion Coe icien (E angelidis and Psa akis, 2008) o he p ojec ions o he
wo imaging da a se s. The p ojec ion was pe o med using he bi wise OR ope a o
o he i s i een images a en by en in e als o each da a se (i.e. ames numbe
49
CHAPTER 2. MORPHODYNAMICAL IMAGE ANALYSIS OF SPERMATOZOA
SWIMMING IN THE PLANE
{0,10, ..., 140}). The algo i hm was implemen ed using he OpenCV and NumPy
lib a ies in Py hon 2.7.12. The a ine ans o ma ion was applied o he ini ial posi ion
and o ien a ion o he model i ed o he image wi h in isible lagella o S. pu pu a us
o p ojec he coo dina es on o he image whe e he lagella a e isible.
Imaging o L. pic us spe ma ozoa L. pic us images we e acqui ed wi h
Op onics CR500X2 came a a a ame a e o 500 Hz in ull chip in a b igh - ield
Olympus in e ed mic oscope (IX71) wi h a 60 ×1.6 0.7 NA long wo king dis ance
objec i e. Pixel esolu ion is 0.33 µm.pixel-1. La e hese we e p ocessed in ImageJ
as ollows: (1) Image ype 16 bi s, (2) Smoo h, (3) Sub ac backg ound (Rolling
ba adius = 13px, Ligh backg ound), (4) C ea e an a e age ime p ojec ion (1000
ames) (5) Sub ac he esul an image o each ame o he s ack (32 bi esul ),
(6) Enhance con as (No malize, Use s ack his og am), (7) In e and (8) Image
ype 16 bi s. T ajec o ies and lagella we e acked using BohBoh so wa e 3.29
(BohBohSo , Tokyo, Japan).
In silico
The model coo dina es we e ans o med o image coo dina es aking in o accoun
he image o igin, spa ial esolu ion and ime. Wi h he pa ame e s used, he e can
be an e o up o 50 µs be ween image and model, due o hei empo al esolu ions.
The head was ende ed as an ellipse wi h he app op ia e pa ame e s (see sec ion
Compa ing imaging da a o model) wi h basal in ensi y mul iplied by whand he
disc e e lagellum was ende ed as linea segmen s wi h wid h wand basal in ensi y
mul iplied by w (w = 1 o d aw he lagellum and w = 0 o no d aw i ). The
poin -sp ead unc ion (PSF) unc ion was app oxima ed by a Gaussian il e wi h σ/6
s anda d de ia ion (i.e. app oxima ed as an odd in ege in image disc e e dimensions)
ha was con ol ed o he ende ed model image. The mask o be used o compu e
he co ela ion (see sec ion 2.2.2) was p oduced by he same p ocedu e ha ende s
he model spe m image using a adius ha was added o bo h head hal axis and
lagella wid h. Pa ame e s used: a= 2.50 µm, b=c= 1.25 µm, L= 45 µm,
N= 150,Rh(0) = {45,45,0}µm, Θ(0) = Iden i y Ma ix, ξT={3.06,3.51,3.51}
pN.s.µm-1,ξR={8.55,15.96,15.96}pN.s.µm, ξk= 0.300 pN.s.µm-2,ξ⊥= 0.525
pN.s.µm-2,K0= 0.035 ad.µm-1,A0= 0.144 ad.µm-1,A1= 0.100,ωκ= 182.485
ad.s-1,λκ= 0.184 ad.µm-1,φ1= 1.571 ad and he emainde a e ze o.
50
2.3. RESULTS
2.3 Resul s
2.3.1 Co ela ion sco es sensi i i y o RFT pa ame e s
We buil a mo phodynamical model o a sea u chin spe ma ozoon whe e he shape
o he head is gi en by an ellipse, he lagellum is pa ame e ized by he cu a u e
along i s leng h and he mechanics o he in e ac ions wi h he luid is desc ibed by
Resis i e Fo ce Theo y (RFT). Al hough an app oxima ion o he physical eali y
(mo e ealis ically desc ibed by Slende Body Theo y (SBT) (Johnson and B okaw,
1979)), his amewo k has been shown o model he swimming beha io o spe m
cells qui e accu a ely (G ay and Hancock, 1955; F ied ich e al., 2010; Jikeli e al.,
2015). The s a e o he model a any gi en ime is a speci ica ion o he posi ion,
o ien a ion and o m o he spe ma ozoon. To compa e he model o imaging da a,
he s a e was used o ende a syn he ic image, his image was con ol ed wi h PSF
expec ed o he mic oscopy se up used and hen co ela ed i wi h he co esponding
ame o he expe imen al imaging da a. The maximiza ion o co ela ion be ween he
image ende ed om he model and he expe imen al image allows o maximize he
likelihood o he model pa ame e s gi en he la e image. To accele a e compu a ion
and educe e ec s o spu ious noise he co ela ion is only pe o med up o a maximum
adius a ound he modeled cell ( ig. 2.1 A).
To assess whe he his me hod allows o make p ecise and accu a e es ima es o
he pa ame e s o spe ma ozoon model, we gene a ed a syn he ic imaging da a se (in
which he eal pa ame e se by which i was gene a ed a e known by de ini ion, he
g ound u h) and we explo ed how changing he di e en pa ame e s independen ly
o in g oups a ec ed he co ela ion. The i s ques ion was how does he p ocees
o ende he model image a ec s he co ela ion coe icien ? To add ess his, we
ook he ini ial ame and co ela ed i wi h he g ound u h model bu changing he
wid h o he lagellum (w), he s anda d de ia ion (σ/6) o he Gaussian unc ion
ha app oxima es he PSF and he adius (ρ ) a ound he cell ha de ines he a ea
used o co ela ion ( ig. 2.1B). We con i med ha he co ela ion is maximal o
he expec ed s anda d de ia ion and lagella wid h (σ= 5 µm, w= 0.5µm). As
we inc ease he a ea o co ela ion by inc easing ρ , he peak o e e y lagella wid h
used con e ges o he expec ed s anda d de ia ion. Fo his eason we decided o
use highe adius alues bu no so high i would be a ec ed by o he spe m cells o
deb is on he expe imen al da a so we se = 15 µm o all subsequen analysis.
51
CHAPTER 2. MORPHODYNAMICAL IMAGE ANALYSIS OF SPERMATOZOA
SWIMMING IN THE PLANE
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−0.5
0.0
0.5
1.0
1.5
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
ac o (i )
log10(RMSE. ab.calc)
log(MSE) (µm2)
I e a ion
= 0.000 s = 0.200 s = 0.400 s = 0.600 s = 0.798 s
Ini (i . 0)
= 0.000 s = 0.200 s = 0.400 s = 0.600 s = 0.798 s
Final (i . 40)
Figu e 2.4: Fi ing a model o L.pic us da a. Please e e o igu e 2.3 legend o
de ails. Scale ba is 10 µm.
58

2.4. DISCUSSION
on he whole cell and o he ha ma ks only he head. Using a ligh spli e , he
pho ons wi h wa eleng hs o he wo ma ke s we e sepa a ed and cap u ed by he
same cha ged coupling de ice (CCD) came a, appea ing side by side on each ame.
One side ha can be used o i he model when he in o ma ion on lagella posi ion
and o m is missing and he o he can be used o con i m he quali y o he p edic ions
o he i ing.
This s a egy was used on a p epa a ion o S. pu pu a us swimming con ined o
a plane in he absence o any s imulus. We i ed he model o he side o he image
da a se con aining in o ma ion on he whole S. pu pu a us cell ollowing he app oach
desc ibed abo e o L. pic us spe ma ozoon images. This i ed spe ma ozoon
became he pseudo-g ound u h o his expe imen ( ig. 2.5). We can see easonable
ag eemen be ween imaging da a and i ed model in all imaging da a se , wi h he
excep ion o he ea lie ime-poin s, p esumably because he a ge ed cell is no uly
s a iona y. We hen p oceeded o i he model o he side o he images whe e he
in o ma ion on he lagella is missing. To se unbiased ini ial alues o he pa ame e s
we did spa ial and empo al escaling o he pa ame e s o he e e ence L. pic us,
acco ding he lagella size and bea ing equency desc ibed o he S. pu pu a us.
No e his escaling was enough o ob ain a small pa ame e dis ance om he s a
(χ2∼0.37). Because he lagellum is no isible in he images, we skipped he
ini ial s ep o i ing lagella con o ma ions o he ini ial ames and p oceeded o
i all pa ame e s o he whole da a. The inal i ing o he head ma ke da a was
e y simila o he one ob ained wi h he whole cell ma ke , bo h in e ms o model
pa ame e s and cellula posi ion and con o ma ions ( ig. 2.5). The pa ame e s se s
esul ing om i ing he whole cell o he head ha e a χ2∼0.66. Al hough he
pa ame e dis ance o he inal i ing was highe han he one he ini al guess, i is
clea ly isible he o me p o ides be e i ing.
2.4 Discussion
In his a icle we explo ed he possibili y ha a mechanis ic model o a spe ma ozoon
can be success ully deployed o quan i a i e image analyses. We ha e shown ha one
can use a biologically meaning ul model o ack cells and o es ima e he di e en
cha ac e is ics o he cells as de ailed in he model. Gi en a pa ame e se we ende ed
syn he ic images which can be di ec ly compa ed o he expe imen al imaging da a.
59
CHAPTER 2. MORPHODYNAMICAL IMAGE ANALYSIS OF SPERMATOZOA
SWIMMING IN THE PLANE
= 0.00 s = 0.25 s = 0.50 s = 0.75 s = 1.00 s
Cell (Final)
= 0.00 s = 0.25 s = 0.50 s = 0.75 s = 1.00 s
Head (Ini ial)
= 0.00 s = 0.25 s = 0.50 s = 0.75 s = 1.00 s
Head (Final)
Figu e 2.5: Fi ing a model o noisy, low- esolu ion S. pu pu a us da a by i ing
da a whe e he whole spe ma ozoon is p esen ( op) o by i ing o da a whe e only
he head is isible (middle and bo om). The middle sequence is he ini ial condi ion
upon escaling o he L. pic us mechanis ic model and he op and bo om ep esen
he bes i ed models. No e only he imaging da a wi h he lagellum isible is shown
o be e compa ison o he expe imen al and model lagella o m and de o ma ion.
Please e e o igu e 2.3 legend o de ails. The g een channel in ensi y was escaled
o be e depic ion o he cell. Scale ba is 20 µm.
60
2.4. DISCUSSION
Be e ye , we we e able o in e in isible s uc u es and hei dynamics by making
good p edic ions o he lagella posi ions and o ms on se ies o images whe e only
he head was isible. Al hough i ing pa ame ic models o s a is ical da a o o
a se o poin s ex ac ed om imaging da a has been done o a long ime, o ou
knowledge, his is he i s ime a mechanis ic model, able o ealis ically ep oduce
he beha io o cells, was i ed di ec ly o imaging da a in o de o ack and measu e
such cells. The powe o his app oach is e idenced by he measu emen o cellula
componen s which a e no p esen no can be di ec ly measu ed in he imaging da a.
Visual examina ion o he obse ed and i ed spe m con o ma ions indica es ha
he e a e small de ia ions. One can in e p e his pessimis ically as indica ing ha
he image analysis based on model- i ing is no pe o ming well enough. Models
ha e a pu pose and i ou pu pose was o es ima e a e age pa ame e s o he
swimming ajec o y o o he lagella bending wa es he quali y o he i ing
would be su icien . In con as , he i ing would no be good enough i one
would be in e es ing in using he model p edic ions as a ine esolu ion mask o
make u he measu emen s on he image. This leads o he o he , pe haps mo e
in e es ing in e p e a ion o he small disc epancies be ween modeled and obse ed
con o ma ions: he image analysis me hod p oposed he e allows o in e ha model
is o e simpli ying he mechanics o deg ees o eedom o he lagella bending wa es.
As a model in e ence ool hese esul s a e a he p omising.
The e is no gene al and de ini i e conclusion on wha is he bes sco ing c i e ia
o compa e wo images, om which co ela ion-based and lpno m-based a e he mos
commonly used (E angelidis and Psa akis, 2008). While using he sum o pe pixel
in ensi y dis ance o he wo images can be used o calcula e he likelihood o he
pa ame e s gi en he da a di ec ly, his measu emen is mo e sensi i e o di e ences
in con as and b igh ness le els. On he o he hand, maximizing he c oss-co ela ion
is equi alen o maximizing he likelihood (Zucke , 2003) and co ela ion no malizes
he in ensi y dis ances by he mean o each image, which inc eases he obus ness o
he me hod o hose pho ome ic e ec s.
We decided o use he sum o co ela ion as he c i e ia o op imize in he ime-
lapse imaging da a. Fo compu a ional e iciency we abo ed he calcula ion o he
co ela ion sco e o pa ame e s se s in which he a e age co ela ion d opped below
0.5 o below hal he maximal a e age co ela ion coe icien ound by he op imiza ion
p ocedu e un il ha i e a ion. This implies ha a leas one co ela ion coe icien
61
CHAPTER 2. MORPHODYNAMICAL IMAGE ANALYSIS OF SPERMATOZOA
SWIMMING IN THE PLANE
a a gi en ame was below his h eshold. By doing his we migh ha e missed
pa ame e se s which ha e highe co ela ion sum o e he se ies o images bu poo
co ela ion in a subse o he images. As he inal solu ions ob ained do no ha e
any ime-poin wi h co ela ion coe icien below his h eshold (da a no shown) we
a e con iden ha he solu ion ound is be e han all he candida es solu ions ha
we e disca ded.
In he op imiza ion p ocedu es we used ixed lagella wid h, PSF s anda d
de ia ion and adius o co ela ion a ea. We could ha e ed he op imize wi h hese
pa ame e s and i hem along he o he s. Doing so migh imp o e he co ela ion
coe icien s ob ained when compa ing ende ed images and expe imen al images.
This should ha e highe impac when i ing a model o imaging da a wi hou p e-
p ocessing. No e howe e ha co ela ion assumes he cons an a e age backg ound
so phenomena such as une en ligh ing should be add essed be o e i ing he model.
Ano he possible app oach would be o include such e ec s in he model i sel a he
expense o inc easing he numbe o pa ame e s o i and o es ima e.
To pe o m an unbiased es when in e ing he lagellum by i ing he model o
ime-lapse images whe e only he head is isible, we used he spa ial and empo al
escaling o L. pic us high esolu ion model as ini ial guess o he pa ame e s o
model S. pu pu a us cell. We emained wi hin he spi i o ou p emiss which is o
use a p io i in o ma ion o i a model o da a. To use he mean lagella leng h o
a measu e o he so and he lagella bea ing equency, which can be es ima ed
by he Fou ie analysis o he head o ien a ion, is hus accep able. The poin is ha
we we e able o ob ain simila pa ame e s, shapes, posi ions and o ien a ions when
acking he whole cell in images whe e on he heads o isible and in images whe e
he lagellum was no isible.
We did no explo e he e ec o noise on he abili y o ou me hod o es ima e he
co ec pa ame e s. To es bo h he obus ness and limi o his amewo k we can
gene a e in silico imaging da a wi h di e en signal- o-noise a ios (SNRs). Howe e ,
we do no expec his o be di e en han om a o dina y s a is ics co ela ion whe e
a lowe bound o SNR exis s om which he me hod canno di e en ia e he cell
om he backg ound.
62
2.4. DISCUSSION
2.4.1 Expanding ou knowledge-based model
The objec i e o his wo k was o me ely do a p oo -o -concep and no o de elop,
es o compa e di e en algo i hms o pa ame e ini ializa ion and op imiza ion.
Fo his eason we implemen ed a simple MC e olu iona y algo i hm and some ad
hoc guess ima es we e used as ini ial pa ame e s. I is possible o u he au oma e
he image analysis p ocessing wi h ou amewo k and o inc ease bo h he speed
o con e gence and he goodness-o - i o he inal model by aking ad an age o a
g ea body o wo k ega ding he es ima ion o pa ame e s om imaging da a (Coo es
e al., 1998; Bake and Ma hews, 2004). These me hods include bo h he issue o he
ini ial es ima ion (Wu e al., 2013), on he sea ch me hod and some o hem also deal
wi h e ec s such as une en illumina ion o backg ound. Also, pa alleliza ion using
g aphics p ocessing uni s o p oducing he model image, o co ela e hem wi h he
imaging da a and e en o he pa ame e op imiza ion a e expec ed o inc ease he
compu a ional speed and will help o deploy ou me hod in acking mul iple cells in
a use ul ime ame.
Ou model assumes ha a cell has cons an beha io , meaning i will swim
in pe ec ci cles wi h he same pe iodic oscilla ions in ansla ion and o a ion.
Howe e , he ajec o y o a cell can be al e ed i i is pe u bed by local changes in
physicochemical p ope ies o he en i onmen (media o su ace), i i bumps in o
o he cells, o e en by endogenous changes o he cellula s a e . Also, ci cula d i ing
o ajec o ies ha e been epo ed unde chemoa ac an e ec and ou model does
no accoun o i (B¨
ohme e al., 2005; F ied ich and J¨
uliche , 2007; Gue e o e al.,
2010). Fo all hese easons, he bes possible i o expe imen al da a wi h ou
cu en model will e en ually accumula e e o s due o ajec o y pe u ba ions. A
p io i, he e is no hing p e en ing us o ex end he mechanis ic model o he cell o
include ime-dependen a iables ha adap he swimming beha io . The addi ional
complexi y would ende he da a i ing mo e challenging. Ano he al e na i e is
o i he same model in a piecewise manne , and choose he mos pa simonious
combina ion o ime-dependen pa ame e s se s ha bes i s he da a. Fo example,
a cell which changed i s bea ing equency om a cons an alue o ano he can be
desc ibed by no less and no mo e han wo single model pa ame e se s. This global
model can be s a ed as a dynamical p og amming (DP) p oblem and i s pa simony
could be achie ed using Akaike’s in o ma ion c i e ia (AIC) o Bayesian in o ma ion
c i e ia (BIC) as sco ing c i e ia (see Chap e 3 o an combined implemen a ion o
63

CHAPTER 2. MORPHODYNAMICAL IMAGE ANALYSIS OF SPERMATOZOA
SWIMMING IN THE PLANE
hese amewo ks).
Ano he challenge o u u e esea ch is o i he cellula model o a se o images
con aining se e al spe ma ozoa. This was no implemen ed in his wo k because ou
objec i e was o es a new amewo k whe e a pa ame ic model o a cell gene a es
he obse ed da a. While acking mul iple spe ma ozoa was no essen ial o his
seminal p oo -o -concep i is impo an o add ess i he compa ison c i e ia chosen
(i.e. co ela ion coe icien ) is sui able o ack di e en spe ma ozoa in mo e complex
scena ios. A highly co ela ed model image sugges s mos pixels a e explained by
ha numbe o cells a hose posi ions, o ien a ions, shapes and wi h hose physical
p ope ies. Using ou cu en pa ame e op imiza ion we en isage ha choosing he
co ec igh numbe o cells can be a challenge. This is a p oblem simila o he
inding he app op ia e numbe o clus e s in clus e analysis, and s a egies de ised
o unsupe ised clus e analysis may se e as inspi a ion. This no wi hs anding,
he e a e some speci ics o he p oblem a hand ha a e wo h men ioning. While
he need o one mo e cell can be assessed by di e en pa simony c i e ia (e.g. AIC
o BIC), including one mo e cell wi h he w ong pa ame e s can dec ease ei he he
co ela ion coe icien o he pa simony c i e ia. To a oid his, he new added cell
should be allowed o con e ge be o e calcula ing i s con ibu ion o he global pic u e.
Simila ly o he case o a cell which changes i s beha io we can use DP o choose
be ween models wi h di e en numbe o cells a di e en ime-poin s.
The e is no limi o he biological de ail ha can be in oduced in a
mo phodynamical model o a cell. We can apply i o o he s cells by changing he
shape module acco dingly and we can u he expand i o e.g. g ow acco ding
o cell cycle ules o o use a di e en mechanics module o e.g. locomo ion
by ilopodia o e en he signaling module o e.g. egula ion o immune cells.
In ac he ma hema ical and compu a ional biology ield is plen y o models o
he cell capable o gene a ing meaning ul beha io s. In he case o spe m cells,
he mechanics module can be implemen ed as SBT, which models he long ange
hyd odynamic o ces and how hese a ec he lagella shape, o o ini e elemen s
(Yang e al., 2008), which allow o model con inemen and di ec in e ac ion wi h
o he spe m. The shape unc ion can also be modi ied. One can also inc ease he
complexi y o he shape module by modeling he axoneme di ec ly (Riedel-K use e al.,
2007). Finally, he signaling cascades om ecep o binding o in e nal calcium (II)
concen a ion ([Ca2+]i) ha e been modeled (Espinal e al., 2011) and we know ha
64
2.5. CONCLUSION
he lagella cu a u e can be es ima ed by he de i a i e o [Ca2+]i(Al a ez e al.,
2012). This aspec can be o u mos alue because spe ma ozoa agglome a e a he
cen e o he chemoa ac an g adien and i becomes ex emely di icul o ei he
compu e o human o ack cells in hese condi ions. Using an adequa e model o
chemo axis should allow ou me hod o ack cells e en in hese condi ions and o
dis inguish how di e en ea men s o condi ions a e a ec ing he cells. No e ha
ex ending he cell model can inc ease i s numbe o pa ame e s and hei es ima ion
om he imaging da a migh be mo e di icul . The di icul ies can be mi iga ed by
no icing ha once a mechanis ic model is success ully i ed o ime-lapse imaging we
ha e mo e han jus acked he cells, we ha e in e ed all he biological meaning ul
pa ame e s con ained in he model.
2.5 Conclusion
In his chap e , we we e able o p o e a new concep : we gene a ed a
mo phodynamical model, based on biological and ma hema ical p inciples, and i ed
i di ec ly o imelapsed wo dimensional imaging da a in o de o de ec and ack
spe ma ozoa. Using he a p io i knowledge o a spe m mo phology, lagella bending
wa es and he physics o hei mo ili y, we could expand he capabili ies o he
mic oscope by being able o es ima e he posi ion o he lagellum in imaging da a
whe e his s uc u e was no labeled and he e o e no isible. This ea u e o ou
app oach can be gene ally applied o o he sys ems, p o ided he app op ia e model
is de eloped, and will allow o es ima e wha was is cu en ly di icul o be measu ed
di ec ly. O e all, his echnological achie emen opens he doo o new a enues o
imaging analysis in gene al.
Re e ences
Al a ez, L., Dai, L., F ied ich, B. M., Kashika , N. D., G ego , I., Pascal, R., and Kaupp, U. B.
(2012).“The a e o change in Ca 2+ concen a ion con ols spe m chemo axis”. Jou nal
o Cell Biology 196.(5), pp. 653–663.
Baba, S. A. and Mogami, Y. (1985).“An app oach o digi al image analysis o bending shapes
o euka yo ic lagella and cilia”. Cell Mo ili y 5.(6), pp. 475–489.
Bake , S. and Ma hews, I. (2004). “Lucas-Kanade 20 Yea s On: A Uni ying F amewo k”.
In e na ional Jou nal o Compu e Vision 56.(3), pp. 221–255.
65
CHAPTER 2. MORPHODYNAMICAL IMAGE ANALYSIS OF SPERMATOZOA
SWIMMING IN THE PLANE
B¨
ohme , M. e al. (2005). “Ca2+ spikes in he lagellum con ol chemo ac ic beha io o
spe m.”The EMBO jou nal 24.(15), pp. 2741–2752.
B okaw, C. J. (1984). “Au oma ed me hods o es ima ion o spe m lagella bending
pa ame e s.”Cell mo ili y 4.(6), pp. 417–430.
Cao, D., Liu, D., and Wang, C. H.-T. (2006). “Th ee-dimensional nonlinea dynamics o
slende s uc u es: Cosse a od elemen app oach”. In e na ional Jou nal o Solids and
S uc u es 43.(3-4), pp. 760–783.
Coo es, T. F., Edwa ds, G. J., and Taylo , C. J. (1998). “Ac i e Appea ance Models”.
P oceedings o Eu opean Con e ence on Compu e Visison 2, pp. 484–498.
Espinal, J., Aldana, M., Gue e o, A., Wood, C. D., Da szon, A., and Ma ´ınez-Mekle , G.
(2011). “Disc e e dynamics model o he spe ac -ac i a ed Ca2+ signaling ne wo k
ele an o spe m mo ili y.”PloS one 6.(8), e22619.
E angelidis, G. D. and Psa akis, E. Z. (2008).“Pa ame ic Image Alignmen Using Enhanced
Co ela ion Coe icien Maximiza ion”. IEEE T ansac ions on Pa e n Analysis and
Machine In elligence 30.(10), pp. 1–8.
F ied ich, B. M. and J¨
uliche , F. (2007). “Chemo axis o spe m cells.” P oceedings o he
Na ional Academy o Sciences o he Uni ed S a es o Ame ica 104.(33), pp. 13256–61.
F ied ich, B. M., Riedel-K use, I. H., Howa d, J., and J¨
uliche , F. (2010). “High-p ecision
acking o spe m swimming ine s uc u e p o ides s ong es o esis i e o ce heo y.”
The Jou nal o expe imen al biology 213.(P 8), pp. 1226–34.
G ay, J. and Hancock, G. (1955). “The p opulsion o sea-u chin spe ma ozoa”. Jou nal o
Expe imen al Biology 32.(4), pp. 802–814.
Gue e o, A., Nishigaki, T., Ca nei o, J., Yoshi o Ta su, Wood, C. D., and Da szon, A. (2010).
“Tuning spe m chemo axis by calcium bu s iming.” De elopmen al Biology 344.(1),
pp. 52–65.
Jikeli, J. F., Al a ez, L., F ied ich, B. M., Wilson, L. G., Pascal, R., Colin, R., Pichlo, M.,
Rennhack, A., B enke , C., and Kaupp, U. B. (2015). “Spe m na iga ion along helical
pa hs in 3D chemoa ac an landscapes.”Na u e Communica ions 6, pp. 1–10.
Johnson, R. E. and B okaw, C. J. (1979). “Flagella hyd odynamics. A compa ison be ween
esis i e- o ce heo y and slende -body heo y”. Biophysical Jou nal 25.(1), pp. 113–127.
Ke ann, C., So zano, C., Ac on, S. T., Oli o-Ma in, J.-C., and Unse , M. (2016).“A guided
ou o selec ed image p ocessing and analysis me hods o luo escence and elec on
mic oscopy”. IEEE Jou nal o Selec ed Topics in Signal P ocessing 10.(1), pp. 6–30.
Pe in, F. (1936).“Mou emen B ownien d’un ellipsoide (II). Ro a ion lib e e d´epola isa ion
des luo escences. T ansla ion e di usion de mol´ecules ellipsoidales”. Le Jou nal de
Physique e le Radium S´e ie VII.(1), pp. 1–11.
Riedel-K use, I. H., Hil inge , A., Howa d, J., and J¨
uliche , F. (2007).“How molecula mo o s
shape he lagella bea .”HFSP jou nal 1.(3), pp. 192–208.
66
REFERENCES
Su, T.-W., Choi, I., Feng, J., Huang, K., McLeod, E., and Ozcan, A. (2013). “Spe m
ajec o ies o m chi al ibbons.”Scien i ic epo s 3.(1664).
Su, T.-W., Xue, L., and Ozcan, A. (2012).“High- h oughpu lens ee 3D acking o human
spe ms e eals a e s a is ics o helical ajec o ies”. P oceedings o he Na ional Academy
o Sciences 109.(40), pp. 3–7.
Tange, O. (2011).“GNU Pa allel: The Command-Line Powe Tool”. ;login: 36.(1), pp. 42–47.
Vulo i´c, M., Ra elli, R. B. G., Vlie , L. J. an, Kos e , A. J., Lazi´c, I., L¨
ucken, U.,
Rullg˚a d, H., ¨
Ok em, O., and Riege , B. (2013). “Image o ma ion modeling in c yo-
elec on mic oscopy”. Jou nal o S uc u al Biology 183.(1), pp. 19–32.
Wu, Y., Lim, J., and Yang, M.-h. (2013). “Online Objec T acking: A Benchma k”.
P oceedings IEEE Con e ence on Compu e Vision and Pa e n Recogni ion,
pp. 2411–2418.
Yang, Y., Elge i, J., and Gomppe , G. (2008). “Coope a ion o spe m in wo dimensions:
Synch oniza ion, a ac ion, and agg ega ion h ough hyd odynamic in e ac ions”.
Physical Re iew E 78.(6), pp. 1–9.
Zucke , S. (2003). “C oss-co ela ion and maximum-likelihood analysis: A new app oach
o combining c oss-co ela ion unc ions”. Mon hly No ices o he Royal As onomical
Socie y 342.(4), pp. 1291–1298.
67
CHAPTER 3. COMPARATIVE STUDY OF SEA URCHIN SPERM MOTILITY –
CONFINED AND FREE SWIMMING
hyd oxye hyl me hac yla e)) and o he eagen s, unless indica ed, we e om Sigma-
Ald ich (Toluca, Edo de Mexico, Mexico). A dilu ion o cells was p epa ed wi h ASW
s o ed on ice and 2µl we e added o 200 µl o ASW a 14◦C in a imaging came a
p e iously coa ed wi h PolyHEME.
Acquisi ion o 2D+Z( )imaging da a o bo h sea u chin species we e ob ained
using he mic oscopy se up de eloped by Co kidi e al., (2008), using he se ings
de ined in Pimen el e al., (2012). B ie ly, a piezoelec ic de ice coupled o he
objec i e (Olympus 40X/0.60 N.A., Olympus Ame ica Inc. U.S.A.) ins alled in
an in e ed op ical mic oscope Olympus IX71, oscilla ing a 30 Hz wi h ∼250 µm
ampli ude was used. F ame a e acquisi ion was a 2000 Hz wi h a high-speed came a
(Op onics CR5000x2, Op onics GmbH, Ge many). Images we e p e-p ocessed by
sub ac ing he backg ound (a e age in ensi y o he i s 100 ames), ans o ming
o 8-bi and ho izon al line il e ing by applying a ho izon al low-pass il e o 95%.
3.2.2 3D ajec o ies o ee swimming spe ma ozoa
To cha ac e ize he ajec o ies o ee swimming spe ma ozoa we p ocessed he 2D+
imaging da a as illus a ed on igu e 3.2. B ie ly, he da a a e a se ies o ime-s amped
bidimensional image ames I i(wi h i∈ {1, ..., }) cap u ed by a high speed came a
while he piezoelec ic oscilla ions scan a olume in an independen and concu en
manne . Since he mic oscope sys em does no e u n he Zposi ion a which each
imaging ame was acqui ed his needs o be in e ed om he image da a i sel .
Once he unc ion Z( i)is econs uc ed one has o iden i y and measu e he spa ial
and empo al coo dina es o he cen oid o he head o each spe m cells index s.
This implies es ima ing he posi ions o he cen oids {X, Y, Z}s[ i]o each cell s
a each ime iand hen acking he cell du ing he pe iod o he analysis. The
me hods used in each o hese s eps a e desc ibed in he ollowing sec ions. They
we e inspi ed on he p e iously p oposed me hods Pimen el e al., (2012) bu ha e
been ully e ised a each s ep.
In e ence o he dep h unc ion Z( )
To accu a ely es ima e he Z( i)o each ame i, we buil on he seminal idea by
Pimen el e al., (2012). Spe m cells ha e an a e age pa h eloci y o 250 µm/s and
he equency o he piezoelec ic and he high acquisi ion ame a e ensu e spe m
cells a e no signi ican ly displaced be ween wo consecu i e ames a he same Z
74

3.2. MATERIALS AND METHODS
posi ion, i.e., wi hin a piezoelec ic pe iod deb is a e i ually s a ic and cells do no
displace mo e han wo imes he a e age head size. Unde hese condi ions, ames
a he same Zposi ion should be highly co ela ed whe eas ames wi h di e en
Zshould no , and he e o e he Zposi ions can be in e ed by co ela ion. This
no wi hs anding, he e is a h ee old challenge: i s , nei he he posi ion Z( i)no
he eloci y a e p oduced by he mic oscopy se up; second, he piezoelec ic mo es
non-linea ly due o he mass o he objec i e such ha consecu i e images a e no a
he same dis ance in he Z-dimension; and hi d, he ib a ions o he piezoelec ic
p oduces ho izon al wobbling o he objec i e ansla ing he e e en ial o he (X, Y )
coo dina es ha can spoil he expec ed image co ela ions.
The new algo i hm o econs i u e he unc ion Z( i)implemen s a s a egy
o o e come hese challenges. Fi s , we cons uc ed a co elog am in which he
co ela ion coe icien ρi,i+kis he maximum o he c oss-co ela ion be ween he
no malized i h image ame and a no malized inse o he (i+k) h image, compu ed
in Fou ie space. Using he inse o 7.8µm (o 10 pixels) allowed o accommoda e
and co ec o he ansla ions in he ho izon al plane in oduced by he small
bu no negligible ib a ions o he piezoelec ic. The bidimensional co elog am
ob ained shows a conspicuous pe iodic s uc u e ( ig. 3.2B). The nex s ep in ol es
ealizing ha i a some ins an he ocal plane is a he minimum o maximum Z
posi ion hen he images acqui ed be o e and a e a ixed ime lapse a e a he same
dep h and he e o e should be highly co ela ed (i he unc ion Z( )is app oxima ely
symme ic). Based on his ealiza ion we de ined a new image index m=i+k/2
such ha he local maxima o he unc ion ρm=A e age(ρi,i+k)co espond o he
ex emes o he e ical posi ions o he ocal plane. The do s ollowing slan ed lines
on op o he co elog am igu e 3.2B, co espond o he indica ed m alues. As m
inc eases he slan ed line slides o he igh and he a e age co ela ion unde he line
is maximal a he ex emes. The igu e also explains he choice o k∈ {3, ..., 64}as
he expec ed pe iod o 30 Hz co esponds o abou 66.7 ames. The a e age ρmis
plo ed in igu e 3.2C as a unc ion o m. Since by cons uc ion he i s ex eme is a
minimum hen he se ies o minima and maxima is ob ained in s aigh o wa d way.
Finally, o econs uc he unc ion Z( )a linea escaling o each hal o cha ac e is ic
piezoelec ic pe iodic unc ion (Pimen el and Co kidi, 2009) was applied in ime o
i he se ies o ex emes in a piecewise manne . An example o he econs uc ed
unc ion Z( )is depic ed in igu e 3.2D.
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CHAPTER 3. COMPARATIVE STUDY OF SEA URCHIN SPERM MOTILITY –
CONFINED AND FREE SWIMMING
Cell de ec ion and acking
The p e iously p oposed algo i hm (Pimen el e al., 2012) segmen ed egions
con aining concen ic ings o da k and ligh in ensi y, cha ac e is ic o he ou -
o - ocus pa e ns o he spe m cells, selec ed hose in which he pa e n appea ed
in e ed in consecu i e neighbo ing image ames and a e aged he Zposi ion. I
u ned ou ha his me hod yields Zes ima es o he spe m head cen oids wi h
signi ican unce ain y. To imp o e he pe o mance o he me hod we explo ed
quan i a i ely he ela ionship be ween he di ac ion pa e ns cha ac e is ic o he
ou -o - ocus cells and he o se be ween he plane o he spe m cell and he ocal
plane ( ig. 3.2E). No e ha he asymme y o he pa e ns when he ocal plane o
he objec i e is imaging abo e and below he ocal plane o he cell (a ea u e ha was
inco ec ly misin e p e ed by Pimen el e al., (2012)). A e aging o e hund eds o
images spe m heads, we cons uc ed an o de ed empla e se , (Pk)wi h k= 1, ..., 9,
whe e pa e ns a e o de ed acco ding o he o se om he e ical posi ion o he
ocus plane o he cell o he ocal plane o he objec i e ( ig. 3.2E). I is impo an
o emphasize ha hese pa e ns a e o de ed bu he exac o se alues a e unknown.
Each pa e n kin he empla e se is used o pe o m c oss-co ela ion wi h e e y
image ame iin he downwa d mo emen o he piezoelec ic ( o educe he amoun
o da a o be p ocessed), and he de ec ion e en s a e collec ed e e y ime he
coe icien is highe han 0.7. In his way, one ob ains a collec ion o de ec ed objec s
{X, Y, Z( i), k, }, whe e Xand Ya e he coo dina es o he cen e pa e n on he
image ame, Z( i)is he e ical coo dina e o dep h o he ame, kis he index o
he pa e n in he o de ed empla e se and is he co ela ion coe icien be ween
he empla e and he ame. The objec s a e clus e ed in space (X,Yand Z) and
ime ( ) by a iends o iends algo i hm using a link- h eshold o 5µm and 1.5 ms.
The numbe o objec s in each clus e is educed by e aining he objec wi h highes
co ela ion coe icien o each ( i, k)pai and elimina ing he emaining ones.
Associa ed o each spe m head we ob ain a se o up o 9 o de ed objec s indexed
kco esponding o he highes co ela ion o each empla e (cells mo ing close o
he limi s o he imaged olume, inside o ou side, will be ep esen ed by less han 9
objec s). We call his he di ac ion se o a cell ( ig. 3.2E). A e his selec ion, we
eg essed he Zcoo dina e o e he empla e index kwi h weigh s w= ( −0.7)/0.3
wi hin each di ac ion se o a cell. The eg ession slope is a e aged o e all he
di ac ion se s ob aining an a e age slope β. We ecalcula e he eg ession o Z
76
3.2. MATERIALS AND METHODS
o e kimposing he a e age slope βac oss he di ac ion se and he Z-coo dina e
co esponding o Z(k= 5) is ob ained and aken as he e ical coo dina e o he
cell. The bidimensional coo dina e (X, Y )is ob ained by weigh ed a e age wi h he
weigh s w, and is ob ained by linea in e pola ion. I is impo an o emphasize
ha he coo dina es o he cell {X, Y, Z, }will ha e highe p ecision han nominal
p ecision associa ed wi h image esolu ion and empo al acquisi ion a e.
T acking o he indi idual spe m heads in he whole da a se ( ig. 3.2H) was
pe o med using wo clus e ing s eps. In he i s s ep, we clus e ed he poin s by
no malized {X, Y, Z, }Euclidean dis ances wi h he iends-o - iends agglome a ion
me hod, using a no malized maximal dis ance o 0.238. To esol e dis inc cells ha
happen o c oss nea each o he , we applied space-wise iends-o - iends clus e ing
(i.e. agglome a ing poin s by inc easing spa ial dis ance) ensu ing a maximal spa ial
(X,Yand Z) euclidean dis ance o 25 µm be ween all clus e membe s i hei
empo al dis ance was equal o o below 0.1 s. This me hod b eaks he ajec o ies
o wo c ossing cells in o exac ly i e di e en clus e s. The egion whe e wo cells
mee in space and ime we e disca ded and he clus e s belonging o he same cell
we e de e mined as such by clus e ing in no malized ajec o ies pa ame e space (see
below). We now ha e collec ions o dis inc se s o poin s, each co esponding o
ime se ies o he cen oid o he head o dis inc spe ma ozoa.
Piecewise helix i ing
The inal s ep in acking he spe ma ozoa in he imaging da a se was o es ima e
he indi idual ajec o ies and hei pa ame e s. To his end, we pe o med piecewise
helical segmen i ing o he se s o poin s ob ained in he p e ious s ep using
Dynamical P og amming and Bayesian in o ma ion c i e ia (BIC) ( ig. 3.2I). This
me hod chooses he mos pa simonious model ha bes i s he da a, meaning he
minimal numbe o helical pa hs necessa y o app oxima e he poin s in a ajec o y.
Fo single helix i ing, he coo dina e-based Poin Dis ance Me hod (Liu and Wang,
2008) was applied o measu e he dis ance be ween model and da a poin s. No e
he se o helices which desc ibe a ajec o y a e non-con inuous in space. Also, ou
me hod allows o di e en weigh s o di e en dimensions. In his case, we used
weigh 1 o ei he Xand Yand 0.2 o Zaxis, o accoun o he highe unce ain y
in he Zes ima es.
Conside a pa ame ic cu e (p, )de ined by i s pa ame e s p={p1, ..., pm},
77
CHAPTER 3. COMPARATIVE STUDY OF SEA URCHIN SPERM MOTILITY –
CONFINED AND FREE SWIMMING
m∈Nand p, ∈R. We will conside a cu e in Euclidean R3space. In his s udy,
e e y cu e is an helix desc ibed as ollows:
(p, ) = 


X( )
X( )
Z( )


=RXRYRZ



cos(2π /T)
sin(2π /T)
kak


+c1(3.1)
whe e he pa ame e ec o is p={kak, , T, νX, νY, νZ, cX, cY, cZ}T,Ria e
e e ence ame o a ion ma ices on iaxis, is he helix adius, Tis he helix
e olu ion pe iod and kakis he p og essi e speed (i.e. helix pi ch o e e olu ion
pe iod). The o a ion ma ices a e also a unc ion o a subse he pa ame e ec o
p:
RX=


1 0 0
0 cos(νX) sin(νX)
0−sin(νX) cos(νX)


;RY=


cos(νY) 0 −sin(νY)
0 1 0
sin(νY) 0 cos(νY)


;RZ=


cos(νZ) sin(νZ) 0
−sin(νZ) cos(νZ) 0
0 0 1



and he ansla ion ec o is c1={cX, cY, cZ}T.
Conside he expe imen al da a column ec o ˙eT={ei}n
i=1, whe e eiis he
posi ion de ined expe imen ally a ime i( ∈ { 1, . . . , n}), n∈N. The sum o
squa ed e o s dis ance (coo dina e-based (Ahn, 2004)) be ween expe imen al and
modeled s a e is gi en by he sum o squa ed e o s
SSE = ( ˙m−˙e)TWTW(˙m−˙e)(3.2)
whe e ˙mT={ ( i)}n
i=1 and W=diag({w}n
i=1)is a weigh (non-singula ) ma ix
de ined by w={w1, . . . , wm}. As his e o measu e depends on he model and
hese depend on he pa ame e s used, he e o also depends on he model pa ame e s
(SSE(p( ))).
Suppose you ha e expe imen al da a ˙eTas de ined abo e and ha p( )is cons an
(p). I is possible o calcula e which pa ame e s o you model bes i he da a. Using
equa ion 3.2 as ou dis ance measu e be ween model and expe imen al da a, we wan
o ind which pa ame e s minimize his dis ance:
a gmin
p
SSE(p)(3.3)
By using ini ial condi ions o pclose o he solu ion we can (hope ully) ind he global
78
3.2. MATERIALS AND METHODS
minima using hill-climbing algo i hms like he quasi-new on me hods. In pa icula we
applied he op im unc ion wi h a gumen me hod="BFGS" in R s a is ical so wa e.
The ini ial condi ions we e se as ollows, assuming he da a de ines a helix wi h
se e al e olu ions and adius smalle han he helix heigh . We s a ed by es ima ing
he imescale o he line ˆa=k n− 1k/∆ 1,n. P incipal Componen Analysis gi es
us he p incipal axis ec o o he helix (z0) and he cen e o mass o he poin
cloud (cc). Then, c1is es ima ed by he displacemen be ween he closes poin o
he cen e o mass ( c) along he p incipal axis: ˆ
c1=cc− cˆaz0. The e olu ion
pe iod Tcan be es ima ed as he low equency (1/ˆ
Tless han 10% o he calcula ed
Nyquis equency) wi h highe magni ude o he angles be ween each sampled poin
and PC2as a unc ion o ime (∠ ( −cc, PC2)). PC2gi es us a adial ec o
(e.g. passes h ough he cen e o he ci cle) and he adius can be es ima ed using
h ee poin s as in Coeu jolly and S ensson, (2003): le di,j =k j− ikand s=
(di−k,idi−k,i+kdi,i+k)/2 hen ˆ =s/(2A ea(4 i−k, i, i+k)). In his wo k, iwas
chosen o be he closes poin o cc( c) and kis a andom poin unde he es ic ion
| i±k− i| ≤ ˆ
T/4. I we sol e z0=RX(νX)RY(νY)RZ(0){0,0,1}Twe ob ain
νY= a csin(−z0
X)and νX= a c an{(−z0
Y/cos(νY))/(z0
Z/cos(νY))}. No e we
should calcula e νX aking he quad an s in o accoun , so we mus use he unc ion
a an2. Also no e ha Eule angles sys ems (i.e. he o a ion sys em used he e) ha e
wo solu ions and he e we conside he one whe e −π/2≤νY≤π/2. Finally, we
es ima e νZas he angle be ween ( i,ˆ
p)and i.
Conside he possible ime uni o m kno se Ω = {˙
1,...,˙
o},˙
i= 1+ ( n−
1)(i−1)/(o−1),i≤o,o > 1and i, o ∈N, we will de e mine he subse ˙
Ω⊆Ω,
˙
1,˙
o∈˙
Ω ha minimizes he BIC. This is needed o make a balance be ween numbe
o pa ame e s (i.e. numbe o helices i ed) and he e o o ha i . In de ail, we
wan o ind ˙
Ω ha minimizes he cos unc ion d(˙
o) = d(˙
1,˙
o). As we a e no
assuming C0no G0con inui ies, he gene al p oblem ha e independen smalle sub-
p oblems p oblems ha a e pa o he gene al p oblem solu ion. Thus, we can use
sol e i using Dynamical P og amming. Fo mally, he gene al p oblem d(˙
o)has
sub-p oblems
d(˙
j) = min
i,j∈N(ni+ni,j) ln (SSEi+SSEi,j
ni+ni,j
)+(Ki+Ki,j + 1) ln (ni+ni,j)(3.4)
whe e 1≤i < j ≤o,ni,j is he numbe o sample poin s whe e ˙
i≤ k≤˙
j,
79

CHAPTER 3. COMPARATIVE STUDY OF SEA URCHIN SPERM MOTILITY –
CONFINED AND FREE SWIMMING
SSEi,j =SSE(pi,j )and Ki,j is he numbe o pa ame e s o he i ing be ween
kno s ˙
iand ˙
j, espec i ely. Also, ni=n1,i and his is simila in SSEiand Ki.
Finally, d(˙
1)=0. No e he BIC o mula in he equa ion abo e.
The solu ion o a gmin
˙
Ω
d(˙
o)is ob ained i e a i ely by o de ly sol ing sub-
p oblems om d(˙
1) o d(˙
o). We hen ex ac he pa ame e s ec o s pi,j whe e
˙
i,˙
j∈Ω0 o a ain ou piecewise helix i ing.
A e ob aining he i s i ings wi h his me hod, we i ed all ajec o ies wi h
h ee ini ial condi ions: he one es ima ed and he median pa ame e s o ei he sea
u chin species. Finally, cells we e uni o mly esampled using hei espec i e piecewise
helix model a he piezoelec ic equency.
3.2.3 2D imaging da a o spe ma ozoa and ajec o ies
L. pic us and S. pu pu a us imaging da a was kindly p o ided by Gue e o e al.,
(2010). T ajec o ies we e manually ob ained using he M akJ plugin (Meije ing
e al., 2012) o ImageJ 1.4. As caged Spe ac was p esen , we only analyzed da a
in he 0-3 s in e al, be o e UV i adia ion in o de o s udy spe m mo ili y in non-
chemo ac ic condi ions.
T ajec o y pa ame e s we e es ima ed as in he 3D case, bu cons aining
p og essi e speed o ze o, so we i ed ci cle a cs ins ead o helices. No e ha his
me hod calcula es he eloci y, cu a u e and o sion along he a e age pa h.
T ajec o y pa ame e s we e es ima ed as in he 3D case, bu cons aining
p og essi e eloci y o ze o, so we i ed ci cle a cs ins ead o helices. No e ha
his me hod calcula es he eloci y, cu a u e and o sion along he a e age pa h.
3.2.4 Da a
All piecewise i ed ajec o ies included in his s udy we e manually inspec ed
and disca ded i app op ia e, keeping ajec o ies wi h one and only one cell wi h
easonable speed (50 ≤ k k ≤ 300 µm.s-1), cu a u e (0≤ |κ| ≤ 1 ad.µm-1) and
o sion (0≤ |τ| ≤ 1 ad.µm-1). Fu he mo e, only ajec o ies spanning mo e han
one second we e conside ed.
Da a was analyzed using R s a is ical p og am .3.0.3 (R Founda ion o
S a is ical Compu ing, Vienna, Aus ia). Compa ison o empi ical cumula i e
dis ibu ions was pe o med by wo-sample Kolmogo o -Smi no while he median
80
3.2. MATERIALS AND METHODS
was compa ed using he Mann-Whi ney es . Fo all es s we assumed a ype I e o
o α= 0.05. When s a ed, Bon e oni co ec ion was applied wi h a α/α0 ac o ,
whe e is α0 he ype I e o conside ed o ha speci ic s a is ical es .
3.2.5 Mo phodynamical model
Conside a ma hema ical model o a cell in which he mo phology changes a e de ined
wi hin he model i sel – his is a mo phodynamical model. In ou pa icula case,
he spe ma ozoon cell model (S) is de ined by he shape (Σ) and mechanics (Φ)
modules.
Shape
We can de ine he mo phology o a spe ma ozoon by de ining wo egions, he
head and he lagellum. We assume he head is a e olu ion ellipsoid wi h hal
axes aand b=c. The lagellum is composed o N ods wi h o al leng h
L. The con o ma ion o he lagellum a a pa icula ime ( ) is gi en by i s
lagella cu a u e (κ) along lagella a cleng h (s), a i s o de a eling wa e:
κ(s, ) = K0+A0cos (ωκ −λκs+φ), which is de ined by he mean lagella
cu a u e (K0), basal cu a u e ampli ude (A0), angula eloci y o bea ing (ωκ),
cu a u e wa eleng h (λκ) and phase (φ). We assume he lagella o sion (τ(s, ) =
T0) along a cleng h o be cons an . The in insic lagella posi ion ( (s, )) is
calcula ed by a cleng h in eg a ion o he Cosse a ame (Cao e al., 2006; Jikeli
e al., 2015), which is de ined by he o hono mal basis e1(s, ),e2(s, )and
e3(s, ):∂ (s, )/∂s =e3(s, ),∂e3(s, )/∂s =κ(s, )e1(s, ),∂e1(s, )/∂s =
−κ(s, )e3(s, )+τ(s, )e2(s, ),∂e2(s, )/∂s =−τ(s, )e1(s, ). No e e3 ep esen s
he cen eline along he lagellum, (0, ) = {−a, 0,0},e1(0, ) = {0,−1,0},
e2(0, ) = {0,0,1}and e3(0, ) = {−1,0,0}. A a gi en ime , he cen oid o
he head has ex insic posi ion Rh( )and o ien a ion ma ix Θ( ), which o a es
om ex insic o in insic coo dina es.
Mechanics
As he cell mo es wi hin a iscous luid, he la e exe s d ag o ces which, unde some
condi ions, p opel he cell in he media. The physics behind i ha e been shown o be
app oxima ed by Resis i e Fo ce Theo y (RFT) (G ay and Hancock, 1955; F ied ich
81
CHAPTER 3. COMPARATIVE STUDY OF SEA URCHIN SPERM MOTILITY –
CONFINED AND FREE SWIMMING
e al., 2010; Jikeli e al., 2015), which we will use he e. B ie ly, he d ag o ce densi y
ha luid exe s on he lagella piece is (s, ) = ξk∂˙ ,k(s, )/∂ +ξ⊥∂˙ ,⊥(s, )/∂ ,
whe e ˙ ,kand ˙ ,⊥a e he angen and no mal componen s, espec i ely, o he
mean lagella piece eloci y ˙ (s, ) = (Ψ0(δθ( ))· (s, +δ)+δ ( )− (s, +δ))/δ
du ing he ime in e al δ. No e and θa e he in insic head ansla ional and
o a ional eloci ies, espec i ely, Ψ0(ν)is he an app oxima ion o Rod igues o a ion
o mula (Ψ0(ν)) o small angles (sin β≃βand cos β= 1) and ξkand ξ⊥a e he
angen and no mal d ags exe ed on he lagellum, espec i ely. We can ob ain he
in insic head eloci ies by sol ing he sys em o o ce and o que equilib ia de ined
by ξT ( )RL
0 (s, )∂s = 0∧ξRθ( )−RL
0 (s, )× (s, )∂s = 0, whe e ξTand ξR
a e he ansla ional and o a ional d ag coe icien s o he head and he symbol ×
deno es he c ossp oduc ope a o . The in insic head eloci ies a e ans o med in o
ex insic eloci ies by mul iplica ion o he ansposed o ien a ion ma ix and applied
o he cell o calcula e he new ex insic posi ion Rh( +δ) = Rh( ) + δΘ( )T· ( )
and o ien a ion Θ( +δ) = Ψ(δΘ( )T·θ( )) ·Θ( ). No e we used he app oxima ed
Rod igues o a ion ma ix o calcula e he local lagella eloci y in o de o make he
sys em linea bu his app oxima ion is no needed when we calcula e he ex insic
eloci ies om he in insic ones.
Assuming a dynamic iscosi y ηwe can calcula e he ansla ional and o a ional
d ags o he head based on i s size using Pe in’s o mulas (Pe in, 1936). As we
used Eule in eg a ion me hod, we will upda e he sys em e e y δ ime s eps and
hen we sol e i o he in insic ansla ional and o a ional eloci ies assuming hey
a e cons an wi hin each ime pe iod δ.Fo spa ial in eg a ion o he lagellum, we
disc e ized i in Nsegmen s and calcula ed he in insic midpoin posi ion κ[i, ],
whe e i=L/N. We de e mined N= 49 and δ= 50 µs o p oduce e o s
(P(obs −exp)/exp ×100) below 5% when calcula ing he in insic ansla ional and
o a ional eloci ies, compa ing o N= 499 and δ= 10 µs. No e he simula ions
we e pe o med wi h N=49 in o de o he explo a ion o he pa ame e space o be
accomplished in easible ime.
The spe ma ozoon model S(Σ,Φ)is hus comple ely de ined by i s shape
pa ame e s Σ= (a, b, c, L, N, Rh,Θ, K0, A0, φ, ωκ, λκ, T0)and i s physical o
mechanical pa ame e s Φ= (ξT,ξR, ξk, ξ⊥). Al hough no explici he e, some o
hese pa ame e s a e dependen on ime, as shown abo e.
82
3.2. MATERIALS AND METHODS
Table 3.1: Pa ame e s used in he RFT model.
Pa ame e Uni s Lp Sp Ap*
a, b µm 3.25,1.625 2.88,1.44 2.50,1.25
L µm 44 38 41
N– 49 49 49
ηPa.s 0.00900 0.01115 0.01080
ξkpN.s.µm-2 0.01427867 0.01041734 0.010692
ξ⊥pN.s.µm-2 0.02855734 0.02073051 0.01935252
ξTpN.s.µm-1 (0.332,0.380,0.380) (0.364,0.417,0.417) (0.306,0.351,0.351)
ξRpN.s.µm (1.567,2.924,2.924) (1.353,2.522,2.522) (0.855,1.596,1.596)
s µm 0.898 0.776 0.837
K0 ad.µm-1 0.03483757 0.04110697 0.0351
A0 ad.µm-1 0.17597653 0.20163048 0.160
A1 ad.µm-1 000
λκ ad.µm-1 0.1927098 0.22425032 0.2122698
ωκ ad.s-1 180.72 195.42 273.32
T0 ad.µm-1 0 0 0.00477
Pa ame e desc ip ion and i ed alues o a con ined spe ma ozoon o L. pic us (Lp) and S.
pu pu a us (Sp). (*) F ee swimming pa ame e s o A. punc ula a (Ap) as epo ed on o he wo k
(Jikeli e al., 2015).
One o he p oposed mechanisms o he con ined swimming mode nea he wa e -
glass in e ace is ha he bounda y and he hyd odynamic o ces he ein cons ain he
(quasi-)plana bea ing wa es o he lagellum o be pa allel o ha su ace (Cosson
e al., 2008). This sugges ed he ollowing simpli ica ion o he modeling o he
con ined swimming. We p ojec ed he 3D ee swimming ansla ional eloci y on he
plane de ined by he i s wo p incipal componen s o all lagella posi ions wi hin a
bea ing pe iod. The angula eloci y was p ojec ed in o he hi d p incipal componen .
An Eule in eg a ion wi h he same imes ep as in he ee swimming mode was used
o calcula e he con ined posi ion and o ien a ion o he cell along ime using he
p ojec ed eloci y ec o s.
3.2.6 Compa ing and i ing he model o ajec o ies
To i he model o expe imen al ajec o ies we i s es ima ed he lagella shape
pa ame e s by andom walk and hen by g adien descen op imiza ion using high
empo al and spa ial esolu ion imaging da a o each species spe m in con ined
swimming (Table 3.1). The cos unc ion o minimize was he mean lagella dis ance
be ween da a and model as e u ned by an alignmen by he Kabsch me hod (see R
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CHAPTER 3. COMPARATIVE STUDY OF SEA URCHIN SPERM MOTILITY –
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Table 3.2: T ajec o y pa ame e s in he li e a u e o he species s udied he e.
Species Swim Pa ame e (mean±sd)1(mean±se)2(mean±sd)*
| |(µm) n.a. 24.9±1.0 25.8±7.7
|a|(µm/s) – – –
Con ined 1/|T|(s−1) n.a. 1.67±n.a. 0.9±0.3
(µm/s) n.a. ( 261.3) 143.6±30.4
1/κ (µm) n.a. 24.9±1.0 25.8±7.7
L. pic us τ(µm−1) – – –
(µm) n.a. n.a. 13±3.7
a(µm/s) n.a. n.a. 89.4±25.5
F ee 1/T (s−1) n.a. n.a. 1.5±0.5
(µm/s) n.a. n.a. 149.4±24.4
1/κ (µm) n.a. n.a. 24.6±12.3
τ(µm−1) n.a. n.a. 0.04±0.01
(µm) 16.3±0.3 17.8±1.0 16.1±3.8
a(µm/s) – – –
Con ined 1/T (s−1) 1.3±0 1.45±n.a. 1.3±0.2
(µm/s) 134.7±9.3 ( 162.2) 128.6±17.2
1/κ (µm) 16.3±0.3 17.8±1.0 16.1±3.8
S. pu pu a us τ(µm−1) – – –
(µm) 6.8±0.6 n.a. 6.7±0.9
a(µm/s) ( 55.5) n.a. 74.6±15
F ee 1/T (s−1) 4±0.5 n.a. 3.7±0.7
(µm/s) 179.7±11.4 n.a. 175.5±30.9
1/κ (µm) ( 7.5) n.a. 8.3±1.1
τ(µm−1) ( 0.04) n.a. 0.06±0.01
Mean spe ma ozoon adius ( ), p og essi e eloci y (a, pi ch/pe iod), equency
(1/T), eloci y ( ), adius o cu a u e (1/κ) and o sion (τ). Values in pa en hesis
we e no a ailable in he s udy bu we e es ima ed using he expe imen al da a ha
was a ailable o ha expe imen . Fo his we assumed helical ajec o ies. (1)
Co kidi e al., (2008); (2) Gue e o e al., (2010); (*) P esen s udy; (n.a.) no
a ailable; (–) no applicable.
90

3.3. RESULTS
A B
Figu e 3.4: Fi ing he 3DRFT model o con ined swimming expe imen al da a.
Model i ing (whi e) o a spe ma ozoon o L. pic us (A, b own) and S. pu pu a us
(B, pu ple) in con ined swimming. Scale ba is 5 µm.
by andom walk ( ig. 3.4). The pa ame e iza ion o A. punc ula a was p e iously
desc ibed by o he s (Jikeli e al., 2015).
A p e ious s udy analyzed he ee swimming mo ili y o spe ma ozoa by assuming
cons an o sion along he lagellum (Jikeli e al., 2015). Ano he one discussed and
sugges ed di e en models o 3D quasi-plana lagella shape (Cosson e al., 2003).
He e, we inc eased lagella asymme y in Zby assuming cons an o sion along he
lagellum τ(s, ) = T0. To model he bounda y in e ac ion, we assumed he o ces
a play a e such ha he plane de ined by he wo i s p incipal componen s o he
lagella posi ions wi hin a bea ing pe iod a e always pa allel o he bounda y su ace
(see sec ion 3.2.5 o de ails). The d ag coe icien s a e known o inc ease as an
objec app oaches he bounda y, ela i e o i s size (Ramia e al., 1993). Because
spe m cells ha e a quasi-plana bea ing pa e n when con ined and he disc e iza ion
o he lagellum used makes a lagella piece e y small ela i e o he head adius,
we assume he d ag a io be ween head and lagellum can inc ease when he cell
is con ined. We model his by changing he e ec i e dynamic iscosi y used o
calcula e he d ag coe icien s o he head. Also, i has been desc ibed ha bo h
lagella mean cu a u e and lagella cu a u e ampli ude dec ease wi h inc eased
iscosi y (B okaw, 1966; F ied ich e al., 2010; Chen e al., 2015). Finally, we also
assume he mechanism by which inc eased iscosi y educes he lagella cu a u e
can a ec he lagella o sion in a simila manne .
We ook he i ed models o con ined L. pic us,S. pu pu a us and A. punc ula a
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CHAPTER 3. COMPARATIVE STUDY OF SEA URCHIN SPERM MOTILITY –
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Table 3.3: Quan iles o pa ame e a ios be ween ee and con ined swimming.
Ra ios L. pic us A. punc ula a*S. pu pu a us
η0.21 1.00 4.95 -0.10 0.79 3.7 0.22 0.70 1.40
K00.68 1.01 1.59 0.21 1.00 1.70 0.98 1.56 2.05
A00.68 0.99 1.45 0.58 1.22 2.26 0.61 1.03 1.62
T00.11 0.98 657.74 0.10 0.53 535.52 0.31 4.62 2592.69
Minimum, median and maximum quan iles o he e ec i e iscosi y (η), mean
lagella cu a u e (K0), lagella cu a u e mean ampli ude (A0) and mean lagella
o sion (T0) a ios ob ained by he model. (*) Pa ame e ized as epo ed on o he
wo k (Jikeli e al., 2015).
and sea ched he pa ame e space (i.e. e ec i e iscosi y, mean lagella cu a u e,
lagella cu a u e ampli ude and lagella o sion) ha gene a ed ajec o ies ha lied
wi hin he i s and hi d qua iles o he expe imen ally obse ed speed, cu a u e
and o sion o he ajec o ies ( ig. 3.5A). We hen compu ed he a io be ween ee
and con ined pa ame e s o each species ( ig. 3.6 and able 3.3), which we expec
o be one o he pa ame e s ha do no change signi ican ly be ween hese wo
swimming modes and depa om he uni o hose ha change. The pa ame e
a ios ob ained when L. pic us spe m a e con ined do no change signi ican ly as
compa ed o when hey swim eely (p obabili y o pa ame e a ios being equal o
below 1 is 0.50, 0.47, 0.53 and 0.52 o d ag, mean cu a u e, cu a u e ampli ude and
o sion, espec i ely). Fo A. punc ula a spe m, bo h mean cu a u e and cu a u e
ampli ude a ios ha e lowe p obabili y bu hese a e no s a is ically signi ican
(p obabili ies 0.71, 0.51, 0.15 and 0.79). Bo h d ag and cu a u e ampli ude a ios
a e simila be ween he wo swimming modes o S. pu pu a us (p obabili ies o 0.98
and 0.42, espec i ely). Al hough no signi ican , he lagella o sion is highe o
ee swimming spe m his species, ela i e o con ined (p obabili y o o sion a io is
0.066). Also, he mean cu a u e is signi ican ly highe when S. pu pu a us spe m is
ee swimming (p obabili y o <0.001). These esul s sugges he lagella bending o
S. pu pu a us spe m is mo e a ec ed when hey con ine, when compa ed o bo h L.
pic us and A. punc ula a spe m. Using he median pa ame e a ios ( able 3.3) we
we e able o eco e bo h ee and con ined median ajec o ies o all h ee species
( ig. 3.5B).
92
3.4. DISCUSSION
|κ|(µm−1)
k k(µm.s−1)
|τ|(µm−1)
AB
20 µm
Figu e 3.5: 3DRFT model eco e s he expe imen al median ajec o ies o L. pic us
(b own), S. pu pu a us (pu ple) and A. punc ula a (pink). (A) Pa ame e s ha
gene a e ajec o ies which p ope ies a e wi hin he i s and hi d qua iles o he
expe imen al alues obse ed. (B) Using he median pa ame e iza ion a ios we
gene a ed ajec o ies (black) ha i he median ajec o y (colo ).
3.4 Discussion
Spe ma ozoa mus deli e hei gene ic ma e ial o he emale game e. In mos
species hese cells ha e e ol ed elabo a e mechanisms o egula e hei lagella
bea ing and swimming beha io o ind he oocy e and achie e e iliza ion. The
en i onmen whe e hese cells swim is a 3D one bu his o icaly hey ha e been s udied
mos ly in wo dimensions (2D), when he cell is con ined o he wa e -glass in e ace.
The ques ion whe he hese s udies we e ep esen a i e o he ee swimming mode
was aised, as small ansien changes in he pa h cu a u e esul in small changes
in he o e all con ined ajec o y bu signi ican changes in 3D helical ajec o ies
(Gue e o e al., 2011).
He e we p oposed a new me hod o add ess de ec ion and acking o spe ma ozoa
imaged in 2D a oscilla ing Zdep hs which was used o measu e he di e en
con ining beha io o L. pic us and S. pu pu a us spe m. Then, o add ess he
mechanism by which his di e ence a ises, we used a ma hema ical app oach based
on Resis i e Fo ce Theo y o model con ined and ee swimming spe m o hese wo
species and also o A. punc ula a. Ou model sugges s mean lagella cu a u e is
signi ican ly diminished when S. pu pu a us spe m con ines, while lagella o sion is
93
CHAPTER 3. COMPARATIVE STUDY OF SEA URCHIN SPERM MOTILITY –
CONFINED AND FREE SWIMMING
012345
0.0 0.2 0.4 0.6 0.8 1.0
D ag a io ( ee/con ined)
P obabili y
P obabili y
η( ee)/η(con ined)
0.5 1.0 1.5 2.0
0.0 0.2 0.4 0.6 0.8 1.0
K0 a io ( ee/con ined)
P obabili y
P obabili y
K0( ee)/K0(con ined)
0.5 1.0 1.5 2.0 2.5
0.0 0.2 0.4 0.6 0.8 1.0
A1 a io ( ee/con ined)
P obabili y
P obabili y
A0( ee)/A0(con ined)
−1 0 1 2 3
0.0 0.2 0.4 0.6 0.8 1.0
log(T0 a io) ( ee/con ined)
P obabili y
P obabili y
ln(T0( ee)/T0(con ined))
Figu e 3.6: Cumula i e his og ams o pa ame e a ios be ween ee and con ined
swimming ob ained o L. pic us (b own), S. pu pu a us (pu ple) and A. punc ula a
(pink). The pa ame e s ep esen ed a e he e ec i e iscosi y (η), he mean lagella
cu a u e (K0), he lagella cu a u e mean ampli ude (A0) and he mean lagella
o sion (T0) a ios ob ained by he model.
94
3.4. DISCUSSION
no .
3.4.1 F om 2D+Z( ) o 3D ajec o ies
The use o p io s a e known o imp o e s a is ical in e ence. I is a classical issue how
much we belie e ou p io s and how much we belie e he da a. In he p esen case
we made use o he knowledge ha spe ma ozoa display helical ajec o ies when
swimming eely in h ee dimensions. Piecewise i ing o helical pa h segmen s o
se s o poin s (X, Y, Z, )allowed us o pa ame e ize and esol e he pa hs o wo
species o spe ma ozoa. I is in e es ing o no e ha a emp s o use mo ing a e aged
da a and/o splines i ing did gene a e ajec o ies and pa h pa ame e s wi h local
noise ha ailed o esol e he wo species (da a no shown; implici in igu e 3.1).
The piecewise i ing o helical pa h segmen s allows o each he mos pa simonious
accoun o a ull ajec o y, iden i ying in an objec i e way when and whe e spe m
changed hei beha io , as he disc e e ansi ions be ween wo segmen s. These
disc e e ansi ions may e lec esponses o he cell o ex e nal cues on in insic
dynamics o he cell. The heu is ic po en ial o hese new me hods in scena ios o
chemo axis emains o be explo ed. A c i icism ha can be made o he me hod
is ha i will end o p esen as disc e e e en s smoo h con inuous changes in he
pa ame e s o a pa h. Such scena ios should lea e a signa u e in he esiduals o
he i ing. De ining whe he swimming cells unde go con inuous changes o mo e
disc e e ansi ions in beha io emains o be cla i ied.
He e we p oposed a new me hod o add ess de ec ion and acking o spe ma ozoa
imaged in 2D a oscilla ing Zposi ions. Ou me hod equi ed li le human
in e en ion, p ima ily o echecking acking e iciency. Many ajec o ies we e
disca ded due o poo i ing, mos wi h a small helix adius. This was because unde
his si ua ion he noise becomes signi ican and he i ing ended up app oxima ing he
helix axis ins ead o he helix i sel . This was pa icula ly ele an o S. pu pu a us,
as i s helical adius is subs an ially smalle . Fo his eason, we migh ha e a biased
es ima ion o he ajec o y pa ame e s. Al hough we canno exclude he possibili y
ha o he non-helical swimming pa e ns do exis o hese species and ha simple
models (i.e. wi h less pa ame e s) would cha ac e ize he ajec o ies adequa ely, he
piecewise helix i ing me hod should able o desc ibe hose ajec o ies by segmen s
o helical a cs.
O he wo k ocused on es ima ing swimming pa ame e s by helical i ing o by
95

CHAPTER 3. COMPARATIVE STUDY OF SEA URCHIN SPERM MOTILITY –
CONFINED AND FREE SWIMMING
ob aining helical pa ame e s om 2D in o ma ion only (C enshaw e al., 2000; Gu a ie
e al., 2011; Che in e al., 2014). No wi hs anding, hese me hods canno be applied
o ou s udy as some o hei assump ions do no apply o ou da a. A numbe o
me hods a e unable o esol e helices whe e he helical axis is pa allel o Z, some
need mo e da a poin s pe ajec o y han wha we ha e and o he s assume equal
noise in all dimensions. The e a e o he al e na i es me hods o s udy changes in
animal beha io (i.e. mo emen ) (Gu a ie e al., 2016). These, howe e , need many
pa ame e s o assume da a independece, which does no occu o he same cell.
Al hough ha ing he disad an age o only i ing helices, he only pa ame e o he
piecewise helix i ing me hod de eloped he e is he desi ed ime esolu ion, making
i easie and s aigh o wa d o be applied o helicoidal da a han he o he me hods.
The s a egy we ha e used is discon inuous on ansi ion poin s, o example
ime-poin s whe e a helix ends and ano he begins in he same ajec o y; cau ion
mus be ad ised when in e p e ing swimming beha io nea hese poin s. This should
no a ec he conclusions a which his wo k a i es, as we a e s udying he main
swimming beha io s ha hese species display in wo di e en bounda y condi ions.
In he s udy o axis, ou me hod should be able o de ec changes in speed, cu a u e,
o sion o di ec ion. Howe e , his me hod has limi a ions when s udying ine e en s
such as acu e u ning, as hese occu nea he ansi ion poin s.
We can apply his me hod since ou s udy deals wi h he a e age pa h (i.e. he
maximum empo al esolu ion we ha e o his olume in his mic oscopy sys em). I
we ake he lagella bea ing in o accoun , he ajec o y should be a chi al ibbon as
desc ibed o some human and ho se spe ma ozoa (Su e al., 2012; Su e al., 2013).
This can be implemen ed by changing he helix unc ion o be i ed bu one mus
be awa e ha mo e in o ma ion migh be needed o i he ex a pa ame e s o mo e
complex unc ions.
Finally, i is wo h no ing ha new mic oscopy sys ems based on holog aphy a e
eme ging, being able o ack as cells in 3D wi h submic on esolu ion (Su e al.,
2012; Jikeli e al., 2015). Al hough ou sys em has less esolu ion, i has he po en ial
o be coupled o luo escen measu emen s o he lagellum. Also, i is possible o
inc ease he spa ial and empo al esolu ion o ou sys em by scanning a smalle
olume (i.e. lowe Zampli ude), which allowed us o image he lagellum in 3D
(Sil a-Villalobos e al., 2014). Toge he wi h he imp o emen s made in his wo k,
he 2D+Z( )mic oscopy sys em emains an up- o-da e, powe ul ool o s udy sea
96
3.4. DISCUSSION
u chin chemo axis.
3.4.2 F ee swimming ajec o ies
A ew ee swimming ajec o ies o sea u chin spe m ha e been cha ac e ized, namely
o A. punc ula a (C enshaw e al., 2000; Jikeli e al., 2015) and S. pu pu a us
(Co kidi e al., 2008) species. Al hough he i s ocuses on he speed, cu a u e
and o sion o a ajec o y, he second ocuses on he helical pa h pa ame e s like
adius and e olu ion speed, bu does no analyze p og essi e speed. In spi e o
he ac ha hese wo de ini ions o helical ajec o ies a e edundan , as one can
be calcula ed om he o he , he six pa ame e s he e desc ibed a e impo an o
es ablish di e ences and simila i ies be ween cellula ajec o ies.
3.4.3 Con ined s ee swimming
A disc epancy be ween ee swimming helix adius and con ined oscula ing ci cle
adius was been p e iously desc ibed o S. pu pu a us spe m. In he p esen
wo k we con i med ha his di e ence is also obse ed in L. pic us spe ma ozoa.
Howe e , he ee swimming oscula ing ci cle adius had no been desc ibed o
hese species un il now. In e es ingly, he ajec o y cu a u e was compa able in
ee and con ined swimming o L. pic us bu ma kedly di e en o S. pu pu a us
spe ma ozoa. Acco ding o he pa ame e s o he ajec o ies o A. punc ula a spe m
epo ed in he li e a u e hese spe m seem o be an in e pola ion be ween he wo.
We hypo hesized and hen p o ed ha highe asymme y in he zcomponen
o he lagella bea ing (i.e. in his case p o ided by cons an lagella o sion on an
inc easingly asymme ical cu ed lagellum) is su icien o inc ease he ee swimming
ajec o y o sion, bu no o inc ease he ajec o y’s eloci y and cu a u e. As
p e iously epo ed, we canno disca d he possibili y ha hyd odynamic in e ac ions
nea he bounda y a ec he swimming beha io (Fauci and Dillon, 2006; Smi h
e al., 2009) bu he e we simpli ied such in e ac ions by o cing he plane o lagella
bea ing o be pa allel o he bounding su ace. Using his a ionale we we e able
o quan i a i ely explain he expe imen al da a o all h ee spe m species, assuming
ha a e age lagella cu a u e and a e age o sion a e lowe when he S. pu pu a us
spe m a e con ined. No e ha by cons uc ion, he amewo k used in his s udy
assumes he bounda y exe s o ces on he lagellum bu hese do no de o m i i he
lagella shape pa ame e a e he same (e.g. lagella pa ame e a io be ween ee
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CHAPTER 3. COMPARATIVE STUDY OF SEA URCHIN SPERM MOTILITY –
CONFINED AND FREE SWIMMING
and con ined a e equal o one). The e o e, one does no need o e oke any change in
he spe ma ozoon in e nal machine y o explain con inemen and plana swimming o
L. pic us and A. punc ula a spe m bu ha such al e a ions a e equi ed o explain
he obse a ions on hose o S. pu pu a us. Ins ead, di e en lagella s i ness migh
accoun o he di e ences obse ed. Ano he in e p e a ion would be ha highe
lagella o sion o S. pu pu a us spe m p o okes highe o -plane bea ing asymme y
which p opels he cell close o he bounda y. As he e ec i e d ag exe ed on
he cell inc eases exponen ially as he cell ge s close o he su ace, he lagella
bea ing quickly becomes plana i he in e nal axonemal o ces emain he same o
bo h ee and con ined swimming. Nos a i e al., (2015) s udied human and bull
spe m swimming close o su aces using o al in e nal e lec ion luo escence (TIRF)
mic oscopy and concluded ha bull spe ma ozoa swim close o he bounda y and
ha e mo e ma ked changes in hei ajec o y. Also, hese spe m expe ience mo e
la ening o he lagella bea ing wa es. Likewise, we specula e ha S. pu pu a us
spe m swim so close o he bounda y ha i s lagella bea ing becomes mo e plana
han ha o L. pic us, e en hough he o me ha e a highe lagella o sion in ee
swimming.
The compa ison o he pa ame e s o he swimming ajec o ies and he modeling
iden i ied a end in he way spe ma ozoa al e hei swimming pa h when in e ac ing
wi h a su ace. The speed and he cu a u e o he swimming pa h ends o dec ease
when he spe m ge con ined, bu in he case o he S. pu pu a us his change is
e y ma ked. Ma hema ical modeling o he mo phodynamics o hese cells led us
o hypo hesize ha he con inemen b ings he cells o a mo e iscous en i onmen
a he liquid-solid in e ace ha changes bo h he d ag a ios and he o sion and
cu a u e o he lagella bending wa es. The smalle S. pu pu a us spe ma ozoa
would be mo e sensi i e o hese changes in iscosi y and/o may swim close o he
solid su ace being subjec ed o a highe iscosi y han L. pic us and A. punc ula a
spe ma ozoa.
We ep oduced he obse ed ee swimming ajec o ies using a model
pa ame e ized based on he da a om con ined spe ma ozoa and se ing lagella
o sion o a posi i e alues. This sugges s ha is possible o in e A. punc ula a o
L. pic us ee swimming ajec o ies om he s udies on con ined se ings, o a leas
de ine a sho e amily o possibili ies. Howe e , he p ope ies o plana mo ili y
canno be ex apola ed o 3D in he case o S. pu pu a us spe m. This aises a
98
REFERENCES
cau iona y no e when ying o ex apola e swimming beha io om expe imen s
whe e na u al ee-swimme s a e s udied unde plana con inemen , a leas o some
species. We canno disca d he possibili y ha in p e ious chemo axis s udies using
S. pu pu a us (Gue e o e al., 2010) he cells esponded in such a way ha would
b ing hem close o he g adien cen e i hey we e swimming a om he su ace
bu he p oximi y o he bounda y impai ed such esponse by cancelling ei he o bo h
lagella cu a u e and o sion.
3.4.4 Conclusion and u u e wo k
He e we p esen he i s cha ac e iza ion o L. pic us spe m ee swimming
ajec o ies in he absence o any nominal s imulus and complemen ed his wi h
a me a-analysis o he a ailable da a on S. pu pu a us and A. punc ula a. Fo he
i s ime, sea u chin species we e compa ed ega ding hei swimming beha io in
uncons ained and cons ained en i onmen s. This compa a i e s udy allowed us o
o be e unde s and how spe ma ozoa swim and is an example o he impo ance
o in e species s udies. Ou esul s indica e ha he conclusions on he mo ili y
and chemo axis s udies pe o med on spe m con ined o plana mo emen migh
be con enien bu no di ec ly ex apola ed o he beha io in na u al condi ions.
This emphasizes he need o cha ac e ize he s a egies used by spe m o s ee he
swimming pa h in 3D.
Re e ences
Ahn, S. J. (2004).“Leas squa es o hogonal dis ance i ing o cu es and su aces in space”.
PhD hesis. Be lin Heidelbe g: Uni e si y o S u ga , Ge many.
An, Y., Shao, C., Wang, X., and Li, Z. (2011).“Geome ic p ope ies es ima ion om disc e e
cu es using disc e e de i a i es”. Compu e s & G aphics 35.(4), pp. 916–930.
B okaw, C. J. (1966).“E ec s o inc eased iscosi y on he mo emen s o some in e eb a e
spe ma ozoa”. Con ol 45, pp. 113–139.
Cao, D., Liu, D., and Wang, C. H.-T. (2006). “Th ee-dimensional nonlinea dynamics o
slende s uc u es: Cosse a od elemen app oach”. In e na ional Jou nal o Solids and
S uc u es 43.(3-4), pp. 760–783.
Chen, D. T. N., Heymann, M., F aden, S., Nicas o, D., and Dogic, Z. (2015). “ATP
consump ion o euka yo ic lagella measu ed a a single-cell le el.” Biophysical jou nal
109.(12), pp. 2562–73.
99
CHAPTER 4. ANALYSIS OF SPERM CHEMOTAXIS IN THREE DIMENSIONS
which did no hold (A. Pimen el, pe sonal communica ion).
The objec i e o his chap e is o eassess he 3D chemo ac ic beha io o bo h
L. pic us and S. pu pu a us spe m. We will s ill use he same da a analysed be o e
p oduced by he 2D+Z( )mic oscope se up. We ook ad an age o he piecewise
helix i ing me hod de eloped in Chap e 3 o es ima e he ajec o y pa ame e s and
used linea mixed models o add ess he dependence o he da a poin s. We applied
his me hod o da a on ee (3D) swimming spe m in he p esence and absence o
a Spe ac g adien and we could no de ec chemo axis o ei he species o he
condi ions es ed. We show ha he chemoa ac an g adien does no a ec he
a e age ajec o y speed, cu a u e, o sion and speed o g adien cen e .
4.2 Me hods
4.2.1 Spe m imaging da a
Spe m o ei he species we e collec ed a e in acoelomic injec ion o 0.5 M o KCl
s o ed on ice and used wi hin 24 hou s. Immedia ely be o e imaging, cells we e
ans e ed o wo di e en solu ions o a i icial sea wa e , one wi h and ano he
wi hou 0.1 µM o caged spe ac (see Gue e o e al., (2010) o u he de ails).
The 2D+Z( )imaging sys em was p e iously desc ibed in Co kidi e al., (2008) and
Pimen el e al., (2012). B ie ly, a piezoelec ic de ice mo es he objec i e up and
down allowing o ake di e en Zsec ions a a as a e. This is needed o scan
a conside able olume whe e cells swim e y as (spe ma ozoa swim a 200-300
µm.s-1). A subse o he imaged ield-o - iew was i adia ed wi h ul a iole (UV)
ligh wi hin he ime in e al 2-4 s. Please e e o sec ion 3.2.1 o u he de ails
on spe m p epa a ion, mic oscopy se ings and image acquisi ion.
4.2.2 Imaging da a analysis
The 2D+Z( )imaging da a was analyzed acco ding o sec ion 3.2.2. B ie ly, we
in e ed he dep h o each ame using a co elog am o he imaging da a, de ec ed
he posi ion o cells using a 3D cell empla e o he di ac ion pa e ns, acked
he cells by clus e ing and es ima ed he pa ame e s o he ajec o ies by piecewise
helix i ing. He e we also de ine he eloci y o he g adien cen e as ollows.
Conside he angen ec o o he helical ajec o y ( )and he uni ec o u( ) =
106

4.2. METHODS
(g− ( ))/kg− ( )kde ined by cu en cell posi ion ( ( )) and he ixed g adien
cen e g. Then, he eloci y componen ela i e o he g adien cen e is he do
p oduc o he wo: g( ) = ( )·u( ).
4.2.3 S a is ical analysis
All piecewise i ed ajec o ies included in his s udy we e manually inspec ed and
disca ded i inadequa e, keeping ajec o ies wi h one and only one cell wi h ealis ic
speed (50 ≤ k k ≤ 300 µm.s-1), cu a u e (0≤ |κ| ≤ 1 ad.µm-1) and o sions
(0≤ |τ| ≤ 1 ad.µm-1). Fu he mo e, only ajec o ies spanning mo e han one
second we e conside ed.
Da a was analyzed using R s a is ical p og am .3.0.3 (R Founda ion o
S a is ical Compu ing, Vienna, Aus ia). The da a can be ca ego ized in o he
ollowing ac o s: Species (L.pic us o S.pu pu a us), T ea men (wi h (CS) o
wi hou (None) caged Spe ac ), UV i adia ion (acco ding o ime in e als o UV
i adia ion: 0-2 s (Be o e), 2-4 s (Du ing), 4-6 s (A e I) and 6-8 s (A e II))
and Spe m (unique cell iden i ica ion numbe ). Linea mixed models we e i ed
using he R unc ion lme 4::lme assuming he andom e ec o Spe m ac o
wi hin he UV I adia ion ac o (UV |Spe m), aking in o accoun he dependence
o he obse a ions on each indi idual Spe m wi hin he ime in e als ela i e o UV
uncaging. Models spanning all possible combina ions o Species, T ea men and UV
I adia ion as main e ec s and hei in e ac ions we e i ed o he da a. The model
ha bes i s he da a was selec ed by log-likelihood a io.
No e he ollowing no a ion used in he de ini ion o he se e al linea mixed
models: symbol (+) means an added e ec (e.g.,p∼A+B,pis modeled by he
main e ec s o A and B), symbol (:) ep esen s and in e ac ion (co ela ion) be ween
ac o s (e.g.,p∼A+B+C+A:B,pis modeled by he main e ec s o A, B
and C and also he in e ac ion o A and B) and he symbol (|) ep esen s a andom
e ec (e.g.,p∼A+(B|C),pis modeled by he main e ec s o A, gi en he andom
e ec s o C wi hin B).
107
CHAPTER 4. ANALYSIS OF SPERM CHEMOTAXIS IN THREE DIMENSIONS
4.3 Resul s
4.3.1 3D expe imen s wi h caged Spe ac
To in es iga e chemo axis in 3D, spe m o S. pu pu a us and L. pic us whe e imaged
using he 2D+Z( )mic oscope se up (Co kidi e al., 2008). UV i adia ion o caged
Spe ac was used o p oduce a g adien o his SAP unde condi ions p e iously shown
o elici esponses by spe m o hese wo species (Gue e o e al., 2010; Gue e o
e al., 2011). Spe ac is a small pep ide ex ac ed om he eggs o S. pu pu a us
and i s caged o m con ains a 2-ni obenzyl g oup a a backbone amide, which lowe s
i s a ini y o i s ecep o s by se e al o de s o magni ude. This caging-g oup can
be eleased unde UV i adia ion making he Spe ac ac i e and de ec able by he
cell. The spe m o he wo species we e imaged unde wo di e en condi ions, one
whe e caged Spe ac is p esen (CS) and o he whe e i is no (None). Wi hin he
ime in e al 2-4 s, all samples we e i adia ed wi h UV, hus eleasing Spe ac and
c ea ing a g adien o his SAP in he CS ea men bu no in he None ea men .
Cells ha de ec and eac o he SAP should al e he a e age ajec o y pa ame e s
such as eloci y, cu a u e o o sion. I , addi ionally, he cells display chemo ac ic
beha io , he magni ude o he eloci y componen ela i e o he ec o be ween he
posi ion o he cell and he cen e o he g adien should inc ease and become mo e
posi i e (Pimen el, 2013) (see sec ion 4.2.2).
We used he me hodology de eloped in Chap e 3 o i piecewise helical segmen s
o he da a. We agg ega ed he pa ame e es ima es o hese ajec o ies by ime
in e als ela i e o UV i adia ion (UV ): 0-2s (Be o e), 2-4 s (Du ing), 4-6 s
(A e I) and 6-8 s (A e II). No e we only conside ed cells swimming a speeds
abo e 50 µm.s-1, neglec ing all slowe objec s. As he same cell is being measu ed
se e al imes wi hin and ac oss UV i adia ion pe iods, he obse a ions a e no
independen . To deal wi h such cases we can use linea mixed models (Hende son,
1982). The in e es ing ea u e o hese models is ha hey accoun o dependence in
he esiduals o epea ed measu emen s (Ba es e al., 2014). In his case, we expec o
ha e a andom e ec pe cell wi hin each ime in e al o UV i adia ion (UV |Spe m).
To de ec wha a e he main ac o s ha ha e an e ec on he ajec o y pa ame e s
we es ed di e en models by log-likelihood a io. These di e en models we e a
combina ion o he possible main e ec s (T ea men ∈ {None, CS};Species ∈
{L.pic us, S.pu pu a us}and UV ∈ {Be o e, Du ing, A e I, A e II}) and
108
4.4. DISCUSSION
he in e ac ions be ween hem, assuming he andom e ec o cell wi hin each UV
ime pe iod (UV |Spe m). Choosing one model o e he o he by log-likelihood a io
means he da a suppo s ha model mo e han i suppo s he compe ing model. In
he case o a chemo ac ic esponse, we expec ha bo h he main ac o s T ea men
( ela i e o he p esence o absence o caged Spe ac ) and UV ( ela i e o he ime
in e al o Spe ac uncaging and i s subsequen di usion: Du ing,A e I and
A e II) o be signi ican , as well as hei in e ac ion (T ea men :UV ). In ac ,
his in e ac ion is essen ial o a chemo ac ic model as i sugges s he combina ion o
he p esence o caged Spe ac and o he ime o UV i adia ion ha e an e ec , which
could be due o Spe ac being uncaged and di used. The null hypo hesis would be
ha he e is no e ec o any o he ac o s o hei in e ac ions.
We can see ha he only s a is ically signi ican e ec on he a e age speed,
cu a u e and o sion o he ajec o ies is he ac o Species ( ig. 4.1A, B and C).
This ag ees wi h he esul s o he p e ious chap e ha hese pa ame e s a e di e en
o spe m o S. pu pu a us and L. pic us. Mo e impo an ly, i indica es ha he e is
no s a is ically signi ican e ec o he combina ion o caged Spe ac , CS, wi h any
o he in e als Du ing and A e UV i adia ion on he a e aged ajec o y speed,
cu a u e o o sion. In o he wo ds, no chemo ac ic esponses we e de ec able unde
he condi ions o he expe imen . As a cu iosi y, he e is a s a is ical e ec o he
in e ac ion be ween species and o some condi ions o he eloci y o g adien cen e
g( ig. 4.1D). Howe e , a close examina ion a he da a indica es ha he e ec is
obse ed in L. pic us spe m in he condi ion Du ing UV i adia ion, when no caged
Spe ac is p esen , meaning his di e ence canno be accoun ed o by a chemo ac ic
esponse, and mos likely ep esen s a ype I e o .
4.4 Discussion
To assess i he mo ili y o spe ma ozoa swimming eely is a ec ed by he p esence
o Spe ac , we used piecewise helix i ing o ob ain he a e age ajec o y speed,
cu a u e, o sion and eloci y componen ela i e o he cen e o he g adien .
Using linea mixed models we ound ha none o hese spe m ajec o y pa ame e s
we e signi ican ly a ec ed by he p esence o he chemoa ac an in any o he
species s udied. These esul s do no con i m he p e ious epo on L. pic us spe m
chemo axis unde he same condi ions (Pimen el, 2013). The p e ious conclusion
109
CHAPTER 4. ANALYSIS OF SPERM CHEMOTAXIS IN THREE DIMENSIONS
L.pic us
S.pu pu a us
0
100
200
300
0
100
200
300
None
CS
Be o e
Du ing
A e I
A e II
Be o e
Du ing
A e I
A e II
Uncaging
|| || (µm.s−1)
L.pic us
S.pu pu a us
0.0
0.1
0.2
0.0
0.1
0.2
None
CS
Be o e
Du ing
A e I
A e II
Be o e
Du ing
A e I
A e II
Uncaging
κ (µm−1)
L.pic us
S.pu pu a us
−0.1
0.0
0.1
0.2
−0.1
0.0
0.1
0.2
None
CS
Be o e
Du ing
A e I
A e II
Be o e
Du ing
A e I
A e II
Uncaging
τ (µm−1)
L.pic us
S.pu pu a us
−3
−2
−1
0
1
2
−3
−2
−1
0
1
2
None
CS
Be o e
Du ing
A e I
A e II
Be o e
Du ing
A e I
A e II
Uncaging
Veloci y o cen e o Spe ac g adien (µm.s−1)
k k ∼ Species + (UV |Spe m)
k k(µm.s−1)
T ea men
UV
A
Species
L.pic us
S.pu pu a us
0
100
200
300
0
100
200
300
None
CS
Be o e
Du ing
A e I
A e II
Be o e
Du ing
A e I
A e II
Uncaging
|| || (µm.s−1)
L.pic us
S.pu pu a us
0.0
0.1
0.2
0.0
0.1
0.2
None
CS
Be o e
Du ing
A e I
A e II
Be o e
Du ing
A e I
A e II
Uncaging
κ (µm−1)
L.pic us
S.pu pu a us
−0.1
0.0
0.1
0.2
−0.1
0.0
0.1
0.2
None
CS
Be o e
Du ing
A e I
A e II
Be o e
Du ing
A e I
A e II
Uncaging
τ (µm−1)
L.pic us
S.pu pu a us
−3
−2
−1
0
1
2
−3
−2
−1
0
1
2
None
CS
Be o e
Du ing
A e I
A e II
Be o e
Du ing
A e I
A e II
Uncaging
Veloci y o cen e o Spe ac g adien (µm.s−1)
|κ| ∼ Species + (UV |Spe m)
|κ|( ad.s−1)
T ea men
UV
B
Species
L.pic us
S.pu pu a us
0
100
200
300
0
100
200
300
None
CS
Be o e
Du ing
A e I
A e II
Be o e
Du ing
A e I
A e II
Uncaging
|| || (µm.s−1)
L.pic us
S.pu pu a us
0.0
0.1
0.2
0.0
0.1
0.2
None
CS
Be o e
Du ing
A e I
A e II
Be o e
Du ing
A e I
A e II
Uncaging
κ (µm−1)
L.pic us
S.pu pu a us
−0.1
0.0
0.1
0.2
−0.1
0.0
0.1
0.2
None
CS
Be o e
Du ing
A e I
A e II
Be o e
Du ing
A e I
A e II
Uncaging
τ (µm−1)
L.pic us
S.pu pu a us
−3
−2
−1
0
1
2
−3
−2
−1
0
1
2
None
CS
Be o e
Du ing
A e I
A e II
Be o e
Du ing
A e I
A e II
Uncaging
Veloci y o cen e o Spe ac g adien (µm.s−1)
|τ| ∼ Species + (UV |Spe m)
|τ|( ad.s−1)
T ea men
UV
C
Species
L.pic us
S.pu pu a us
0
100
200
300
0
100
200
300
None
CS
Be o e
Du ing
A e I
A e II
Be o e
Du ing
A e I
A e II
Uncaging
|| || (µm.s−1)
L.pic us
S.pu pu a us
0.0
0.1
0.2
0.0
0.1
0.2
None
CS
Be o e
Du ing
A e I
A e II
Be o e
Du ing
A e I
A e II
Uncaging
κ (µm−1)
L.pic us
S.pu pu a us
−0.1
0.0
0.1
0.2
−0.1
0.0
0.1
0.2
None
CS
Be o e
Du ing
A e I
A e II
Be o e
Du ing
A e I
A e II
Uncaging
τ (µm−1)
L.pic us
S.pu pu a us
−3
−2
−1
0
1
2
−3
−2
−1
0
1
2
None
CS
Be o e
Du ing
A e I
A e II
Be o e
Du ing
A e I
A e II
Uncaging
Veloci y o cen e o Spe ac g adien (µm.s−1)
k gk ∼ Species +T ea men +UV
T ea men :Unc + (Unc|Spe m)
Speed o g adien cen e (µm.s−1)
T ea men
UV
D
Species
Figu e 4.1: Linea mixed models o chemo axis o ei he a e age pa h speed (A),
cu a u e (B), o sion (C) and speed o cen e (D) o di e en T ea men s – wi h
(CS) and wi hou (None) caged Spe ac – a di e en imes a e UV i adia ion
(UV ): 0-2s (Be o e), 2-4 s (Du ing), 4-6 s (A e I) and 6-8 s (A e II). The
da a is ep esen ed as box-and-whiske s subplo s wi h he mean alue ( ed do ). On
op o each plo is he model chosen by log-likelihood a io.
110
4.4. DISCUSSION
migh ha e been mislead by ype one e o s a ising om he iola ion o he
assump ion o independence o da a poin s and he con ounding e ec s gene a ed
by cells en e ing and lea ing he olume. We canno disca d he unlikely hypo hesis
ha he independence o da a poin s be ween he ime in e als ( ac o UV ) could
ha e also a ec ed ou esul s. This could be add essed by assuming an addi ional
andom e ec only dependen on cell (1|Spe m, acco ding o he no a ion o he
R s a is ical so wa e) bu he a ailable sample size was oo small o i models
based on his assump ion. The ac ha indi idual cells a ely swam in he imaged
olume du ing he eigh seconds o he expe imen also impai ed he possibili y o
his assump ion (i.e. unbalanced da a).
O e all hese esul s sugges ha spe m o L. pic us and S. pu pu a us do
no exhibi chemo axis in unde he condi ions o he expe imen . As L. pic us
spe m we e p e iously shown o be chemo ac ic in con ining s udies (Gue e o e al.,
2010), we sugges he explo a ion o di e en chemoa ac an g adien s (i.e. using
di e en caged Spe ac concen a ions o UV i adia ion imes and condi ions). Jikeli
e al., (2015) showed 3D chemo axis o A. punc ula a spe ma ozoa by uncaging he
chemoa ac an h oughou he expe imen . This should also be es ed wi h he
wo species used in his s udy. Also no e ha he app oach aken he e only allowed
o es he signi icance o a beha io simila o he ‘o - esponse’ chemo ac ic mode
sugges ed o ee-swimming spe m o A. punc ula a. In he p esen s udy, he UV
i adia ion was pe o med using he objec i e, while i was mo ing up and down d i en
by he piezoelec ic de ice. Unde hese condi ions, he uncaging o ms a hou -glass
shaped g adien in h ee dimensions (Pimen el, 2013). I is an in iguing possibili y
ha such a i icial hou -glass g adien may no be decodable by he senso imo o
sys em o he spe ma ozoa ha has been e ol ed o loca e an app oxima e adial
g adien a ound he egg.
In his wo k, we es ed whe he he mo ili y pa ame e s o spe m di e ed
be ween di e en condi ions in he expe imen , including hose p esumably con aining
meaning ul concen a ions o Spe ac . The e a e di e en s a is ical amewo ks ha
es he agg ega ion o clus e ing o cells which ha e been p e iously de eloped (i.e.
posi ional in o ma ion). Fo example, Ripley’s K- unc ion measu es he expec ed
numbe o neighbo s wi hin a ce ain adius o a cell and compa es o he null
hypo hesis ha his numbe is gi en by a Poisson p ocess (i.e. he poin s a e uni o mly
dis ibu ed) (Ripley, 1977). The es ima e o Kcan summa ize he aspec s like in e -
111

CHAPTER 4. ANALYSIS OF SPERM CHEMOTAXIS IN THREE DIMENSIONS
poin dependence and clus e ing. This kink o summa y s a is ics a e po en ially good
candida es o assess chemo axis as cells end o accumula e in he g adien cen e ,
hus inc easing he expec ed occu ence o cells o smalle adius nea he g adien
cen e . No wi hs anding, hose me hods do no epo he dis ance o g adien cen e
whe e cells a e agg ega ing and he dis ances om whe e cells a e mo ing om,
no allow o quan i y he chemoa ac an powe unde di e en concen a ions o
g adien s. A s a is ical amewo k which conside s hese aspec s is impo an .
Chemo axis models such as Kelle and Segel, (1970) o Jikeli e al., (2015) could
be used o add ess hese issues. The i s is an o dina y di e en ial equa ion (ODE)
sys em ha models bac e ial popula ional chemo axis assuming biased andom walk.
By i ing such model o whole da a se (including ime), we could es ima e he
chemo ac ic pa ame e s and compa e hem be ween ea men s, using he simpli ied
pa ame e iza ion o non-chemo axis as null model. The models would hen be
compa ed by log-likelihood a io. Howe e , his app oach equi es ini ial pa ame e
es ima ion ha should no be easy o ge o a gi en condi ion. Also he assump ion
o andom walk o he Kelle -Segel model migh be oo s ingen o cells ha swim
in ci cles on in helices in non-chemo ac ic condi ions.
The o he me hod is also a sys em o ODEs based on Resis i e Fo ce Theo y
(G ay and Hancock, 1955) and a simple chemo ac ic sys em. Again, we i s need o
es ima e some pa ame e s o ed o he algo i hm, e.g. he g adien being gene a ed in
o de o apply his amewo k. These me hods can be imp ac ical in many si ua ions
as hey may also equi e indi idual cell acking and e en need inc eased sample size.
No e bo h app oaches a e easible and he e we only men ion he expec ed di icul ies,
should one wish o implemen ei he app oach.
4.5 Conclusion
In his chap e , we applied linea mixed models o assess he chemo ac ic beha io
ee-swimming spe m o L. pic us and S. pu pu a us. We did no ind e idence
ha he mo ili y pa ame e s we e a ec ed in he p esence o he chemoa ac an .
Should we ha e ound such an e idence, i would be in e es ing o add a signaling
module o he spe m model p esen ed in he p e ious chap e s. This model would
encompass om Spe ac binding o he ecep o , memb ana channels hype - and
depola iza ion up o changes in in e nal calcium (II) concen a ion ([Ca2+]i) and in
112
REFERENCES
lagella con o ma ion. Recen ly, 3D chemo axis was e i ied in A. punc ula a and
i s da a can be used in conjunc ion wi h he expanded mo phodynamical models o
p o ide u he knowledge on spe ma ozoan mo ili y.
Re e ences
Ba es, D., M¨
achle , M., Bolke , B., and Walke , S. (2014).“Fi ing linea mixed-e ec s models
using lme4”. Jou nal o S a is ical So wa e 67.(1).
Co kidi, G., Taboada, B., Wood, C. D., Gue e o, A., and Da szon, A. (2008).“T acking spe m
in h ee-dimensions.” Biochemical and Biophysical Resea ch Communica ions 373.(1),
pp. 125–129.
G ay, J. and Hancock, G. (1955). “The p opulsion o sea-u chin spe ma ozoa”. Jou nal o
Expe imen al Biology 32.(4), pp. 802–814.
Gue e o, A., Ca nei o, J., Pimen el, J. A., Wood, C. D., Co kidi, G., and Da szon, A. (2011).
“S a egies o loca ing he emale game e: he impo ance o measu ing spe m ajec o ies
in h ee spa ial dimensions”. Molecula Human Rep oduc ion 17.(8), pp. 511–523.
Gue e o, A., Nishigaki, T., Ca nei o, J., Yoshi o Ta su, Wood, C. D., and Da szon, A. (2010).
“Tuning spe m chemo axis by calcium bu s iming.” De elopmen al Biology 344.(1),
pp. 52–65.
Hende son, C. R. J. (1982). “Analysis o co a iance in he mixed model: highe -le el,
nonhomogeneous, and andom eg essions”. Biome ics 38.(3), pp. 623–640.
Jikeli, J. F., Al a ez, L., F ied ich, B. M., Wilson, L. G., Pascal, R., Colin, R., Pichlo, M.,
Rennhack, A., B enke , C., and Kaupp, U. B. (2015). “Spe m na iga ion along helical
pa hs in 3D chemoa ac an landscapes.”Na u e Communica ions 6, pp. 1–10.
Kelle , E. F. and Segel, L. A. (1970). “Ini ia ion o slime mold agg ega ion iewed as an
ins abili y”. Jou nal o Theo e ical Biology 26.(3), pp. 399–415.
Lillie, F. R. (1912). “The p oduc ion o spe m iso-agglu inins by o a”. Science 36.(929),
pp. 527–530.
Pimen el, J. A. (2013).“Sis ema de as eo idimensional de mic o-pa ´ıculas: aplicaci´on en
el as eo de espe ma ozoides al nado lib e”. PhD hesis. Uni e sidad Nacional Au ´onoma
de M´exico, pp. 1–120.
Pimen el, J. A., Ca nei o, J., Da szon, A., and Co kidi, G. (2012).“A segmen a ion algo i hm
o au oma ed acking o as swimming unlabelled cells in h ee dimensions.”Jou nal o
Mic oscopy 245.(P 1), pp. 72–81.
Ripley, B. D. (1977). “Modelling spa ial pa e ns”. Jou nal o he Royal S a is ical Socie y.
Se ies B (Me hodological) 39.(2), pp. 172–212.
113
CHAPTER 4. ANALYSIS OF SPERM CHEMOTAXIS IN THREE DIMENSIONS
Wa d, G. E., B okaw, C. J., Ga be s, D. L., and Vacquie , V. D. (1985). “Chemo axis o
A bacia punc ula a spe ma ozoa o esac , a pep ide om he egg jelly laye ”. Jou nal o
Cell Biology 101.(6), pp. 2324–2329.
114
Chap e 5
Gene al Discussion
In his hesis, we se ou o de elop au oma ic imaging analysis me hods en iched
wi h p io s based on biological models o cope wi h he la ge imaging da a se s.
Fo his we used a mechanis ic model o a spe ma ozoon in o de o desc ibe hese
cells in he syn he ic and expe imen al imaging da a o sea u chin spe m. We i ed
he model pa ame e s by maximizing hei likelihood and showed we we e able o
ob ain es ima es indis inguishable om he ue pa ame e s used o gene a e he
syn he ic imaging da a. Using he same p ocedu e on expe imen al da a led o model
pa ame e s ha gene a ed cellula posi ions and lagella con o ma ions as good as
hose gene a ed by human-assis ed so wa e. Fu he mo e, he mo phological and
physical cons ains imposed by he model allowed us o ack he in isible lagellum
using only he in o ma ion o he head, showing we can es ima e ea u es absen om
he imaging da a using models wi h he su icien de ail. Taking his al oge he , we
alida ed ou hypo hesis ha we can use a mo phodynamical model o a cell and
i i di ec ly o he imaging da a by maximum likelihood es ima ion (MLE). The
op imiza ion p ocess, i.e. o ind he likelihood maximum, is he majo bo leneck
in ou app oach and p obably he g ea es challenge o i s deploymen o ou ine
da a analysis. Reducing he ime he model-based image analysis me hod akes o
p oduce esul s o alues close o he ime i akes o acqui e he da a is a desi able ye
eachable objec i e. The op imiza ion p ocess in ol ed in pa ame e es ima ion can
be imp o ed in se e al ways. These can include he usage o mo e e icien algo i hms
o unc ion maximiza ion and he usage o g aphic p ocessing uni s (GPUs) o
accele a e bo h Resis i e Fo ce Theo y (RFT) calculus and compa ison o model
and biological imaging da a. In chap e 3, we i ed he mechanis ic model o
115
CHAPTER 5. GENERAL DISCUSSION
es s and s anda ds o imaging da a analysis pipelines. Howe e , hese a e no usually
in eg a ed oge he , only he mo e common (and usually ou da ed) p ocedu es a e.
Fo a pa icula da a sou ce, a plugin. Also, many in e na ional s anda ds and
algo i hms a e no open-sou ced o need a paid licence o be used, he eason why
hei in eg a ion should be comme cially non iable. Thus, a p ojec o his magni ude
can only be possible by public unding. Taking in o accoun how i s use will p omo e
good p ac ices in da a analysis, we expec i o boos bo h scien i ic ou pu speed
and quali y. O e all, we al eady ha e he componen s (algo i hms), a mo i e and he
u gency o do i . I hink we can achie e i in he ollowing decade.
5.3 B inging i oge he
The powe o mechanis ic models. Using a mechanis ic model ha desc ibes
s uc u es ha a e missing in da a allowed us o expand he in o ma ion gi en by ha
same da a. In Chap e 2 we we e able o inc ease he spa ial and empo al esolu ions
o sea u chin spe ma ozoa labelled wi h luo escen dyes by i ing a mo phodynamical
model wi h a bi a y empo al and spacial esolu ions. Mo e no ably, we we e able
o in e he lagella posi ions and con o ma ions by i ing o imaging da a whe e
only he head was labelled and isible. In o he wo ds, he in o ma ion in he
ime-lapse images o he head was su icien o allow he in e en ial s uc u e o
he model o decode he missing in o ma ion on he lagellum. This esul can
be ins umen al o measu e in e nal calcium (II) concen a ion ([Ca2+]i) along he
lagellum. Conside he ollowing example. In o de o s udy how [Ca2+]iis coupled
o cu a u e gene a ion on he lagellum, Gue e o e al., (2013) used a ligh spli e
in conjunc ion wi h whole cell and calcium ma ke s (i.e. one o he assays used in
Chap e 2) o image spe ma ozoa unde di e en d ugs (e.g. con ol and ni lumic
acid). This s a egy allowed o ack he lagellum in he se ies o images and hen
o e lay i wi h he signal o he Ca2+ epo e o quan i y in ensi y o he in lux
o his ca ion. Since he spe m heads a e cons i u i ely labeled by he calcium
indica o (Gue e o e al., 2010; Gue e o e al., 2013) i would be possible o he
mo phodynamical model (o any o he sugges ed ex ensions) o in e he lagella
posi ions and con o ma ions di ec ly om his da a.
App oxima ely one o e e y six couples in sub e ile, making human e ili y a
subjec o g owing medical and economic impo ance (Ga ney e al., 2011). Male
122

5.3. BRINGING IT TOGETHER
ac o s accoun o a ound hal o he cases and spe m mo ili y is a majo playe
(Ba a e al., 2009). A signi ican ac ion o he in e ile cases ha e unknown
cause. I is an in iguing possibili y ha hese can be explained by abe a ions a
highe -le el con ol o spe m mo ili y and o ien a ion by en i onmen al cues, such
as chemo axis, which canno be diagnosed by cu en Compu e Assis ed Spe m
Analysis (CASA) sys ems ha measu e low le el p ope ies such as p og essi e a e
and bea ing equencies. In ac , one o he mos ema kable aspec s o mammalian
e iliza ion p ocess is he jou ney o he spe m o he egg. Spe m cells mus swim a
pa h housands o imes hei own body leng h h ough a complex in e io geome y,
o en illed wi h highly iscous and po en ially hos ile immune cells. Ini ially o e
hund eds o millions, he o e whelming majo i y do no e en each he Fallopian
ubes, le alone he si e o e iliza ion (Ga ney e al., 2011). Any abe a ion in
he spa ial- empo al coo dina ion o lagella bea ing dynamics wi h spa ial cues will
p e en he spe m o ind i s way and he e o e dec ease he e ili y a e. The
ex ension o he me hods de eloped in his hesis o model he human spe m is
a he s aigh o wa d. Se e al adap a ions can be o eseen. The shape module
should accoun o he di e en igidi y, leng h and wid h o he lagella midpiece.
As hese cells swim wi hin an in ica e landscape, he physical module should model
he bounda y explici ly as should he e ec o iscosi y and luid in e ac ion wi h
he lagella shape. Finally, a module desc ibing he signalling ansduc ion o he
chemoa ac an signals down o changes in [Ca2+]ishould also be added i one wishes
o accoun o chemo axis. By applying such humanized model as a p io i knowledge
and pe o ming he app op ia e assays, one could e en ually measu e ea u es ha
ha e been omi ed in spe m mo ili y and chemo axis analysis. E en ually, hese would
p o ide some insigh s on some o he unknown causes o e ili y and allow o u u e
ea men s o be de eloped.
2D+Z( )sys em o chemo axis analysis. Spe m chemo axis is an essen ial
p ocess o he li e cycle o many species. As pe de ini ion, chemo axis depends on
how cells eo ien hemsel es and spe ma ozoa do i by modula ing he asymme y
in he lagella bending wa e cu a u e (and pe haps o sion), which is co ela ed
wi h he de i a i e o whole cell [Ca2+]i(Wood e al., 2003; Al a ez e al., 2012).
In Chap e 1 we e e ed pa o an ex ensi e li e a u e on how and why di e en
memb ana channels (i.e. ei he a he plasma o a mi ochond ial memb anes)
123
CHAPTER 5. GENERAL DISCUSSION
ope a e in o de o modula e [Ca2+]ia e chemo ac ic s imulus. I is clea om
hese s udies ha i is c ucial o unde s and calcium dynamics in spe ma ozoa in
o de o unde s and hei mo ili y and, hus, chemo axis. A mic oscopy sys em which
is able o image calcium ma ke -loaded cells in 3D becomes he ob ious choice o
s udy chemo axis and he 2D+Z( )sys em (Co kidi e al., 2008) has po en ially
such capabili ies. The majo challenge ha mus be o e come o use his sys em
o ha pu pose is o imp o e he luo escence de ec ion as he acquisi ion exposu e
ime mus be e y sho , due o bea ing equency o he lagella ha equi es a
apid mo emen o he piezoelec ic de ice. Using small ampli udes o he de ice
o cap u e a single indi idual will g ea ly dec ease he di icul y. The compu a ional
analysis me hods de eloped he e demand mino adjus men o be able o deal wi h
4D+ imaging da a ha is expec ed o be gene a ed om such sys em in he coming
yea s. In pa icula , simila o he analysis o he plana mo ili y in Chap e 2, acking
o he cell in 3D using ou mechanis ic model would allow o measu e he calcium
wi hou ex a in o ma ion o he lagella posi ion. This would bene i , ob iously, i
he model is calib a ed and alida ed wi h measu emen o he 3D lagella bea ing
con o ma ions be o e deploying i o he ask o 4D+ image analysis.
Holog aphic mic oscopy sys ems based on cohe en ligh (Su e al., 2012) a e
no able o measu e luo escence, which is an incohe en ligh sou ce. Incohe en
holog aphy has been de eloped since he la e 00’s and has been used o acqui e
biological da a (Rosen and B ooke , 2008). Howe e , because o he sca e ing o
pho ons emi ed om he luo escence sou ce poin , he signal- o-noise a io (SNR) o
he econs uc ed image is low and he e o e his echnique equi es immobile o ixed
ma e ial. The e is ac i e de elopmen o econs uc ion algo i hms ha imp o e his
SNR bu he bes hey achie e is a SNR∼2 o a o al exposu e ime o 1.5 ms (Jang
e al., 2016). As luo escen imaging o lagella [Ca2+]iand in acellula pH signals
equi es pho omul iplie s (Gue e o e al., 2010; Gonz´alez-Co a e al., 2015; Jansen
e al., 2015), he SNR is e ec i ely lowe , implying ha hese signals canno be
esol ed. This means holog aphic luo escen mic oscopy will ha e o ma u e be o e
i can be applied o he 3D s udy o spe m mo ili y and chemo axis. Un il hese
limi a ions o incohe en holog aphy a e o e come, he Co kidi’s piezoelec ic-d i en
mic oscopy sys em o simila ins umen a ion seems o be sa es be on he nea
u u e. This 3D mic oscopy sys em is also able o image spe ma ozoa a high spa ial
esolu ion, inclusi e o ob ain he posi ion o he lagellum (Sil a-Villalobos e al.,
124
REFERENCES
2014). Being able o use calcium indica o s wi h high spa ial and empo al esolu ion
o he lagellum is an impo an s ep owa ds comp ehension o chemo axis. Howe e ,
we’ll loose spa ial esolu ion i we ake he app oach o using he ligh spli e o ob ain
he whole cell and he calcium ma ke s in he same de ec o (i.e. wo ligh channels
a e scaled down so hey can be eco ded simul aneously by he same ha dwa e piece).
We can use he al e na i e ha we al eady men ioned o using a mechanis ic model
o i he 3D mo phodynamical model o he head da a and disca d he need o he
whole cell ma ke , allowing us o use he ull chip o he calcium indica o hus
inc easing he esolu ion.
The p esen hesis, by o e coming he ea lie sho comings o he 2D+Z( )da a
analysis me hods and by making he p oo -o -p inciple ha mechanis ic models o
he spe ma ozoon can be deployed o mo e be e image analysis, se he g ound o
enable his new expe imen al a enue in o spe m chemo axis.
Re e ences
Al a ez, L., Dai, L., F ied ich, B. M., Kashika , N. D., G ego , I., Pascal, R., and Kaupp, U. B.
(2012).“The a e o change in Ca 2+ concen a ion con ols spe m chemo axis”. Jou nal
o Cell Biology 196.(5), pp. 653–663.
Ba a , C. L. R., Kay, V., and Oxenham, S. K. (2009). “The human spe ma ozoon - a
s ipped down bu e ined machine.”Jou nal o biology 8.(7), p. 63.
Co kidi, G., Taboada, B., Wood, C. D., Gue e o, A., and Da szon, A. (2008).“T acking spe m
in h ee-dimensions.” Biochemical and Biophysical Resea ch Communica ions 373.(1),
pp. 125–129.
Elge i, J., Winkle , R. G., and Gomppe , G. (2015).“Physics o mic oswimme s - single pa icle
mo ion and collec i e beha io : a e iew”. Repo s on P og ess in Physics 78.(056601),
pp. 1–50.
F ied ich, B. M., Riedel-K use, I. H., Howa d, J., and J¨
uliche , F. (2010). “High-p ecision
acking o spe m swimming ine s uc u e p o ides s ong es o esis i e o ce heo y.”
The Jou nal o expe imen al biology 213.(P 8), pp. 1226–34.
Ga ney, E. A., Gadˆelha, H., Smi h, D. J., Blake, J. R., and Ki kman-B own, J. C. (2011).
“Mammalian spe m mo ili y: Obse a ion and Theo y”. Annual Re iew o Fluid Mechanics
43.(1), pp. 501–528.
Gonz´alez-Co a, A. L., Sil a, P. ˆ
A., Ca nei o, J., and Da szon, A. (2015).“Single cell imaging
e eals ha he mo ili y egula o spe ac induces a lagella alkaliniza ion ha p ecedes
and is independen o Ca 2+ in lux in sea u chin spe ma ozoa”. FEBS le e s 589,
pp. 2146–2154.
125
G ay, J. and Hancock, G. (1955). “The p opulsion o sea-u chin spe ma ozoa”. Jou nal o
Expe imen al Biology 32.(4), pp. 802–814.
Gue e o, A., Espinal, J., Wood, C. D., Rend´on, J. M., Ca nei o, J., Ma ´ınez-Mekle , G., and
Da szon, A. (2013). “Ni lumic acid dis up s ma ine spe ma ozoan chemo axis wi hou
impai ing he spa io empo al de ec ion o chemoa ac an g adien s.” Jou nal o cell
science 126.(P 6), pp. 1477–87.
Gue e o, A., Nishigaki, T., Ca nei o, J., Yoshi o Ta su, Wood, C. D., and Da szon, A. (2010).
“Tuning spe m chemo axis by calcium bu s iming.” De elopmen al Biology 344.(1),
pp. 52–65.
Jang, C., Cla k, D. C., Kim, J., Lee, B., and Kim, M. K. (2016).“Signal enhanced holog aphic
luo escence mic oscopy wi h guide-s a econs uc ion”. Biomedical Op ics Exp ess 7.(4),
pp. 1271–1283.
Jansen, V., Al a ez, L., Balbach, M., S ¨
unke , T., Hegemann, P., Kaupp, U. B., and
Wach en, D. (2015). “Con olling e iliza ion and cAMP signaling in spe m by
op ogene ics”. eLi e 2015.(4), pp. 1–15.
Nos a i, R., D iouchi, A., Yip, C. M., and Sin on, D. (2015). “Two-dimensional sli he
swimming o spe m wi hin a mic ome e o a su ace.”Na u e communica ions 6, p. 8703.
Rosen, J. and B ooke , G. (2008). “Non-scanning mo ionless luo escence h ee-dimensional
holog aphic mic oscopy”. Na u e Pho onics 2, pp. 190–195.
Rosen, R. (1991).“Li e i sel : a comp ehensi e inqui y in o he na u e, o igin, and ab ica ion
o li e”. Columbia Uni e si y P ess.
Sil a-Villalobos, F., Pimen el, J. A., Da szon, A., and Co kidi, G. (2014).“Imaging o he 3D
dynamics o lagella bea ing in human spe m.”Con e ence P oceedings IEEE Enginee ing
in Medicine and Biology Socie y 2014, pp. 190–193.
Su, T.-W., Xue, L., and Ozcan, A. (2012).“High- h oughpu lens ee 3D acking o human
spe ms e eals a e s a is ics o helical ajec o ies”. P oceedings o he Na ional Academy
o Sciences 109.(40), pp. 3–7.
Tu k, M. and Pen land, A. (1991). “Eigen aces o Recogni ion”. Jou nal o Cogni i e
Neu oscience 3.(1), pp. 71–86.
Wood, C. D., Da szon, A., and Whi ake , M. (2003).“Spe ac induces calcium oscilla ions in
he spe m ail”. The Jou nal o Cell Biology 161.(1), pp. 89–101.
Yang, Y., Elge i, J., and Gomppe , G. (2008). “Coope a ion o spe m in wo dimensions:
Synch oniza ion, a ac ion, and agg ega ion h ough hyd odynamic in e ac ions”.
Physical Re iew E 78.(6), pp. 1–9.
Apoio inancei o da FCT e do FSE no ˆambi o do Quad o Comuni ´a io de Apoio,
Bolsa noSFRH/BD/79261/2011.