RESEARCH ARTICLE
Insigh s in o he dynamic ajec o ies o
p o ein ilamen di ision e ealed by
nume ical in es iga ion in o he ma hema ical
model o pu e agmen a ion
Magali Tou nusID
1
*, Miguel EscobedoID
2
, Wei-Feng Xue
3,4☯
, Ma ie DoumicID
4☯
1Cen ale Ma seille, I2M, UMR 7373, CNRS, Aix-Ma seille uni e si e
´, Ma seille, F ance, 2Uni e sidad del
Paı
´s Vasco, Depa amen o de Ma ema
´icas, Bilbao, Spain, 3School o Biosciences, Uni e si y o Ken ,
Can e bu y, Uni ed Kingdom, 4INRIA Rocquencou , e
´quipe-p oje BANG, domaine de Voluceau,
Rocquencou , F ance
☯These au ho s con ibu ed equally o his wo k.
*[email p o ec ed].
Abs ac
The dynamics by which polyme ic p o ein ilamen s di ide in he p esence o negligible
g ow h, o example due o he deple ion o ee monome ic p ecu so s, can be desc ibed by
he uni e sal ma hema ical equa ions o ‘pu e agmen a ion’. The a es o agmen a ion
eac ions e lec he s abili y o he p o ein ilamen s owa ds b eakage, which is o impo -
ance in biology and biomedicine o ins ance in go e ning he c ea ion o amyloid seeds
and he p opaga ion o p ions. He e, we de ised om ma hema ical heo y in e sion
o mulae o eco e he di ision a es and di ision ke nel in o ma ion om ime-dependen
expe imen al measu emen s o ilamen size dis ibu ion. The nume ical app oach o sys-
ema ically analyze he beha iou o pu e agmen a ion ajec o ies was also de eloped.
We illus a e how hese o mulae can be used, p o ide some insigh s on hei obus ness,
and show how hey in o m he design o expe imen s o measu e ib il agmen a ion dynam-
ics. These ad ances a e made possible by ou cen al heo e ical esul on how he leng h
dis ibu ion p o ile o he solu ion o he pu e agmen a ion equa ion aligns wi h a s eady dis-
ibu ion p o ile o la ge imes.
Au ho summa y
Amyloid ib ils a e ib illa p o ein s uc u es in ol ed in many neu odegene a i e ill-
nesses, such as Pa kinson’s disease o Alzheime ’s disease. To p opaga e in disease, hese
mis olded p o ein agg ega es mus g ow and di ide o p oli e a e. The e o e, he in insic
cha ac e is ics o hei di ision, including he di ision a e and he pa e n o di ision in
e ms o whe he he ib ils a e likely o b eak in he middle o a he edges, impac he dis-
ease ae iology. He e, we disco e ed ma hema ical o mulae ha can be used o di ec ly
ex ac he ib il di ision cha ac e is ics om ecen expe imen s da a ob ained om
ime-dependen ib il leng h dis ibu ion measu emen s. We explain how hese o mulae
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OPEN ACCESS
Ci a ion: Tou nus M, Escobedo M, Xue W-F,
Doumic M (2021) Insigh s in o he dynamic
ajec o ies o p o ein ilamen di ision e ealed by
nume ical in es iga ion in o he ma hema ical
model o pu e agmen a ion. PLoS Compu Biol
17(9): e1008964. h ps://doi.o g/10.1371/jou nal.
pcbi.1008964
Edi o : Philip K. Maini, Ox o d, UNITED KINGDOM
Recei ed: Janua y 6, 2021
Accep ed: Ap il 13, 2021
Published: Sep embe 3, 2021
Pee Re iew His o y: PLOS ecognizes he
bene i s o anspa ency in he pee e iew
p ocess; he e o e, we enable he publica ion o
all o he con en o pee e iew and au ho
esponses alongside inal, published a icles. The
edi o ial his o y o his a icle is a ailable he e:
h ps://doi.o g/10.1371/jou nal.pcbi.1008964
Copy igh : ©2021 Tou nus e al. This is an open
access a icle dis ibu ed unde he e ms o he
C ea i e Commons A ibu ion License, which
pe mi s un es ic ed use, dis ibu ion, and
ep oduc ion in any medium, p o ided he o iginal
au ho and sou ce a e c edi ed.
Da a A ailabili y S a emen : All ele an da a a e
wi hin he manusc ip and i s Suppo ing
in o ma ion iles. The code is a ailable on Gi Hub,
can be used, and p o e he obus ness o he di ision a e o mula whe e small e o s in
he measu emen leads o small e o s in he di ision a e. We also demons a e ha he
ma hema ical o mula is no obus enough o p ecisely deciphe he pa e n o di ision in
he da a, and sugges ins ead new u u e expe imen al design wi h sho ime measu e-
men s in expe imen s s a ing wi h ib il suspensions whe e all ib ils ha e simila size,
which would be sui able o p o ide imp o ed es ima es.
In oduc ion
How can we ex ac in o ma ion on he s abili y and dynamics o p o eins nano- ilamen s
om popula ion dis ibu ion da a? This gene al ques ion is o opical in e es due o he e e -
inc easing e idence o sugges ha he agmen a ion o amyloid and p ion p o ein ib ils [1]
a e associa ed wi h hei biological esponse anging om being ine , unc ional o oxic,
in ec ious and pa hological [2]. The expe imen al me hods o cha ac e ize he dynamics o
amyloid ib il agmen a ion has been e ol ing om indi ec bulk kine ics measu emen s [3]
o di ec obse a ions in popula ion le el ime-dependen nano-imaging expe imen s ([4,5]).
To analyze he di ision o p o ein ilamen s when he expe imen al in o ma ion we ha e is a
he le el o he popula ion dis ibu ion, o ins ance when he ype o da a we cu en ly can
acqui e a e ime-poin samples o ib il leng h dis ibu ions and indi idual di iding pa icles
canno ye be isola ed and acked, he pu e agmen a ion equa ion e eals o be a powe ul
ma hema ical ool. The pu e agmen a ion equa ion desc ibes he ime e olu ion o a popula-
ion o ib il pa icles s uc u ed by hei size x ha di ide in o smalle pa icles. The unde ly-
ing assump ion is ha he dimensions o each pa icle go e n i s di ision dynamics: each
pa icle o leng h xis assumed o di ide wi h a a e B(x), and when a pa icle o size ydi ides,
i gi es ise o a pa icle o size xwi h a p obabili y encoded in he agmen a ion ke nel κ(x,y).
Though he agmen a ion equa ion desc ibes he dynamics a he le el o he whole popula-
ion, he p ope ies Band κha e a na u al in e p e a ion in e ms o he mic oscopic s abili y
o he polyme s. In his epo , we add ess he ques ion o de e mining he pa ame e s Band κ
om he size dis ibu ion o he p o ein ilamen suspension a di e en imes.
The applica ion o he pu e agmen a ion equa ion can be aced back o almos 100 yea s.
In he seminal pape by Kolmogo o [6], a agmen a ion model o g inding pa icles was
p oposed. The model is disc e e wi h espec o ime bu con inuous in he s uc u ing a iable
co esponding o he size o he pa icle. This allowed Kolmogo o o wo k wi h explici o -
mulae. The unknown p ope y in he Kolmogo o model is he cumula i e dis ibu ion unc-
ion o he pa icle sizes, and he assumed a cons an agmen a ion a e and a gene ic ke nel
p e en ing he c ea ion o oo many small pa icles. Unde hese assump ions, he ob ains ha
he cumula i e dis ibu ion ollows, asymp o ically in ime, a log-no mal dis ibu ion. A he
e y end o he pape , Kolmogo o sugges s ha his s udy should be ex ended o gene ic ag-
men a ion a es, and especially he ones wi h a powe law dependence on pa icle size, i.e.
BðxÞ ¼ axg:ð1Þ
In pa allel, Mon oll and Simha [7] de eloped a disc e e model o pu e agmen a ion o
long-chain molecules such as s a ch wi h he es ic ions ha he ke nel ollows a uni o m dis-
ibu ion (each bond has he same p obabili y o ission), and only ission in o wo pa s is
allowed (compa ed o Kolmogo o ’s model ha allows he ission in o npa icles). In he la e
70’s [8], he p oblem was again conside ed o he pu pose o s udying he deg ada ion o long
chains unde high shea mechanical ac ion. This was encou aged by new echniques o ob ain
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Nume ical in es iga ion in o he ma hema ical model o pu e agmen a ion
PLOS Compu a ional Biology | h ps://doi.o g/10.1371/jou nal.pcbi.1008964 Sep embe 3, 2021 2 / 21
ia he ollowing link h ps://gi hub.com/m ou nus/
F agmen a ion.
Funding: M.E. is suppo ed by DGES G an
MTM2014-52347-C2-1-R and Basque Go e nmen
G an IT641-13. M. D. and M.T. we e suppo ed by
he ERC S a ing G an SKIPPERAD (numbe
306321). W.-F.X. was suppo ed by unding om
In ia, and Bio echnology and Biological Sciences
Resea ch Council (BBSRC) UK g an s BB/J008001/
1 and BB/S003312/1. The unde s had no ole in
s udy design, da a collec ion and analysis, decision
o publish, o p epa a ion o he manusc ip .
Compe ing in e es s: The au ho s ha e decla ed
ha no compe ing in e es s exis .
measu emen s o he so-called “molecula weigh dis ibu ion” in a closed sys em wi h con-
s an o al mass. This was he i s ime a dependence o he agmen a ion a e on he size o
pa icles is s udied. The au ho s sugges ed ha o he molecula sys em hey s udied, i he
powe γ= 1, hen mechanis ically, he agmen a ion ke nel should be uni o m. Fo γ<1,
which is he alue hey ob ained (γ= 2/3), hey sugges ed ha he bonds on he edges o he
molecules a e mo e eac i e han hose in he cen e. In his case, he agmen a ion equa ions
we e sol ed nume ically, and he agmen a ion a e was de e mined om he a e age leng h
o molecules, based on an app oxima ion alid o a monodispe se suspension (we de ail his
app oxima ion in S1 Appendix). The o he pa ame e s (αand κ) we e ob ained by i ing hei
model o he e olu ion o he o al numbe o molecules. To es ima e he agmen a ion ke nel,
hey conside ed h ee di e en ypes o ke nels and sugges ed ha he bes - i ke nel was one
desc ibed by a pa abolic unc ion, al hough he selec ion c i e ia we e no de ailed. In a heo-
e ical pape by Ballau and Wol [9], he same disc e e model was s udied and h ee agmen-
a ion ke nels we e conside ed: a uni o m ke nel, a Gaussian ke nel, and a Di ac ke nel whe e
pa icles can only spli exac ly a hei cen e. An example o he ime-dependen solu ion is
plo ed in each case, howe e , again, he o e all c i e ion o ke nel selec ion is h ough simula-
ions, wi h no p ecise objec i e p o ocol sugges ed. A se ies o heo e ical wo ks by McG ady
and Zi ollowed in he 1980’s, ocusing exclusi ely on analy ical o mulae o he con inuous
model. In [10], hey p o ided undamen al solu ions ha in ol e hype geome ic unc ions
o a uni o m ke nel, and o a monomial agmen a ion a e wi h γ= 2/mwi h m2Z. In
[11], hey p o ided explici o mulae o he undamen al solu ion o he pu e agmen a ion
equa ion wi h a uni o m ke nel and monomial agmen a ion a e o any γ, a uni o m ke nel
in he case whe e pa icles b eak in o 3 pieces ins ead o 2, as well as o γ= 3 combined wi h a
pa abolic ke nel cen e ed a he pa icle cen e, jus i ied by he pa abolici y o he Poiseuille
low. Typically, hei solu ion is made o a sum o wo e ms, one e m whe e he ini ial condi-
ion anishes exponen ially, and he o he e m whe e he p o ile o a s a iona y s a e a ises.
Using hese explici solu ions, hey no iced, jus like Kolmogo o did, ha a s a iona y dis i-
bu ion shape p o ile a ises asymp o ically a e escaling. F om he 1970’s onwa d, size s uc-
u ed popula ion models we e ex ensi ely de eloped by ma hema icians o biological
applica ions, (see [12]). The pa icles unde conside a ion we e bac e ial and non-bac e ial
cells, mic o ubules, e c. Fo hese sys ems, he ‘pa icles’ unde go di ision as well as g ow h,
which led o he de elopmen and applica ion o g ow h- agmen a ion equa ions. F om he
1990’s, a la ge se o ma hema ical s udies we e ocused on he di ision equa ions and ela ed
models [13], in pa icula on he long- ime beha iou s [14–16]. To deal wi h he majo issue o
model calib a ion, ma hema icians also de eloped heo ies o eco e some pa ame e s, o
ins ance [17,18] whe e he au ho s de e mined a obus es ima e o he di ision a e o bac e-
ial cells om noisy measu emen s o he size dis ibu ion p o iles o he cells a he end o he
expe imen s, and he ime e olu ion o he o al numbe o cells, see also [19] and he e e -
ences he ein. Mo e ecen ly, a heo y was de eloped [20] o es ima e bo h he di ision a e
and he di ision ke nel om he measu emen o he pa icle dis ibu ion p o ile a he end o
he expe imen , unde assump ions on he di ision a e being gi en by he simple powe law
αx
γ
. Ano he app oach eme ging o es ima e he di ision ke nel is he use o s ochas ic indi id-
ual based models by s udying he unde lying s ochas ic b anching p ocesses [21].
While he uni e sali y o he agmen a ion equa ion is demons a ed in i s applicabili y
anging om physical p ocesses such as he g inding o ocks, o chemical p ocesses such as
he deg ada ion o long chain s a ch molecules and biological p ocesses such as cell di ision,
he applica ion we exempli y he e is he mechanis ic laws go e ning he di ision and p opaga-
ion o ilamen ous amyloid s uc u es. These p o einaceous ib ils can be associa ed wi h
human diseases such as Alzheime ’s disease, Pa kinson’s disease [22], ype 2 diabe es, p ion
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Nume ical in es iga ion in o he ma hema ical model o pu e agmen a ion
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diseases and sys emic amyloidosis. The agmen a ion o amyloid ib ils has been shown o
enhance hei cy o oxic po en ial by gene a ing la ge numbe s o small ac i e pa icles [23].
Likewise, he agmen a ion o p ion pa icles ha a e ansmissible amyloid esul s in an
inc ease in hei in ec i e po en ial [24]. Recen ly, as a p oo o concep , we epo ed a new
expe imen al app oach [5] whe e he s abili y owa ds b eakage unde mechanical pe u ba-
ion o di e en ypes o amyloid ib ils we e analyzed and quan i a i ely compa ed. We
de e mined he di ision a es and he ype o agmen a ion ke nels associa ed o each ype o
amyloid ib ils. These da a sugges ed ha he p o eins ha a e in ol ed in diseases may be
o e all less s able owa d b eakage and gene a e la ge numbe s o small ac i e pa icles han
hei non-disease associa ed coun e pa s. In he con ex o he expe imen al da a p esen ed in
[5], and as poin ed ou in [9], he expe imen al con ex may ha e a conside able impac on he
loci a which he ib il is mo e likely o b eak up. The e o e, i is impo an o de elop a gene al
me hod based on a common ma hema ical pla o m, which can be applied o he analysis and
compa ison o expe imen al da a om a wide ange o amyloid sys ems and condi ions.
In his epo we p o ide a de ailed explana ion o he ma hema ical me hod based on he
analysis o he pu e agmen a ion equa ion used in [5], oge he wi h a ho ough nume ical
in es iga ion o he in luence o he h ee key pa ame e s o he model, and he nume ical algo-
i hm used o es ima e he agmen a ion a e and ke nel om expe imen al measu emen s.
We ocus on he case o ‘pu e agmen a ion’ o amyloid p o ein ib ils, i.e. on expe imen s
whe e o he g ow h eac ions such as nuclea ion, polyme iza ion and/o coagula ion could be
neglec ed. We also do no conside nonlinea agmen a ion eac ions, which may be induced
by collisions o in e ac ions be ween ib il pa icles, since in ou con ex he ib il pa icles can
be conside ed dilu e so ha his e ec may be neglec ed. We p o ide in e sion o mulae o
eco e he h ee pa ame e s γ,αand κ om expe imen al measu emen s o he pa icle leng h
dis ibu ion a di e en imes whe e samples o ib il leng hs a e aken bu no in o ma ion on
he o al numbe o pa icles o he o al mass o he suspension is di ec ly a ailable. In pa icu-
la , ou me hod does no ely on inding he bes - i o model dis ibu ions o he da a o on
he goodness-o - i compa ison be ween models. Ins ead we demons a e obus analy ical
in e sion o mulae ha exp ess he pa ame e s as unc ions ha can be di ec ly compu ed
om he solu ion o he equa ion. The me hod and he analysis p esen ed he e a e gene al
and can be use ul in o he con ex s. Bu impo an ly, he ma hema ical esul s will in o m he
design o expe imen s ailo ed o e alua e and compa e he dynamical s abili ies o p o ein
ilamen s.
Theo y
In his sec ion, we summa ize he ma hema ical esul s ha a e he heo e ical ounda ion o
ou me hod.
The pu e agmen a ion model
We conside a popula ion o amyloid p o ein ib ils, which a e ilamen ous and pseudo linea
pa icles, unde going a p ocess o ‘pu e agmen a ion’, whe e he only phenomenon aken
in o accoun is he di ision o any pa en pa icle in o wo daugh e pa icles. In his case, he
a es o g ow h p ocesses such as nuclea ion, polyme iza ion, coagula ion e c. a e conside ed
o be negligible in he expe imen s, o example due o he lack o monome ic p ecu so s. The
modelling assump ions we make on he agmen a ion p ocess a e as ollows.
Assump ion 1: he agmen a ion a e depends only upon he size o he pa en pa icle
unde going di ision, and ollows a powe law, namely, he i s o de a e cons an o pa icles
o size xb eaking in o wo pieces is B(x) = αx
γ
o some α>0. We also impose γ>0, which
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Nume ical in es iga ion in o he ma hema ical model o pu e agmen a ion
PLOS Compu a ional Biology | h ps://doi.o g/10.1371/jou nal.pcbi.1008964 Sep embe 3, 2021 4 / 21
means ha la ge pa icles a e mo e likely o b eak up han small pa icles. This assump ion is
necessa y o he asymp o ic beha iou (4) o happen. Fo γ= 0, no sel -simila beha iou
occu s [16], whe eas o γ<0, sha e ing (some imes e e ed o as ‘dus o ma ion’) occu s in
ini e ime [25].
Assump ion 2: he agmen a ion eac ion is sel -simila , meaning ha he si es o agmen-
a ion on pa icles a e in a ian wi h size escaling, ha is he si e o agmen a ion on a pa i-
cle can be desc ibed as a a io be ween he posi ion and he o al leng h o he pa icle.
The agmen a ion ke nel κis a p ope y ha desc ibes he p obabili y dis ibu ion o he
leng h o he daugh e pa icles o med in each agmen a ion e en , assuming ha such a ag-
men a ion e en akes place. Assump ion 1 is jus i ied since he agmen a ion eac ion consid-
e ed in he expe imen s is p omo ed due o a single ype o pe u ba ion, in he case o [5]
mechanical in na u e. The pa icles in he sample suspension a e also homogeneous in e ms
o being o med by he same monome p ecu so s and only di e by hei size. In pa icula ,
he agmen a ion a e is conside ed o be independen o he his o y o each pa icle, and on
he a e o o he pa icles. Assump ion 2 is jus i ied by he ac ha he agmen a ion beha -
iou o ods ollow he scaling pa e n as discussed in [26].
As amyloid p o ein ib ils a e sup amolecula polyme s uc u es, he ib il pa icles consid-
e ed he e a e made o monome ic uni s. The e a e wo main app oaches o desc ibe he e olu-
ion o he ib il popula ion. The popula ion o pa icles can be desc ibed by he numbe o
pa icles u
ℓ
( ) composed wi h ℓmonome s a ime ,
u0
‘ð Þ ¼ að‘ Þgu‘ð ÞþaX
þ1
j¼kþ1
1
j k‘
j
��ðj Þgu‘ð Þ; >0; ‘ ¼0 . . . N;
u‘ð0Þ ¼ u0
‘; ‘ ¼0 . . . N;
8
>
<
>
:ð2Þ
whe e is he a e age leng h o one monome . We e e o Eq (2) as he disc e e model. Al e -
na i ely, when he numbe o monome s composing each pa icle is assumed o be su icien ly
la ge, we can w i e a con inuous e sion o he model. The unknown is he densi y u( ,x) o
pa icles o leng h xa ime and he model is w i en as ollows:
@u
@ ð ;xÞ ¼ axguð ;xÞþaZ1
x
1
ykx
y
��yguð ;yÞdy; >0;x�0;
uð0;xÞ ¼ u0ðxÞ;x>0:
8
>
<
>
:ð3Þ
The ad an age o he disc e e amewo k is i s alidi y e en when he numbe o monome s in
he pa icles is small, which could be he case a e y long ime scales o agmen a ion expe i-
men s. As o he con inuous amewo k, he main ad an age i ha i is ma hema ically con-
enien since some explici o mulae exis o some speci ic pa ame e s, and i enables pa ial
di e en ial analysis esul s o be used o unde s and he quali a i e beha iou o he sys em.
The beha iou o hese wo models should no di e in he ime o he expe imen s we analyse.
The e o e o ou analysis, we ocus on he con inuous amewo k. Fo he solu ions o (3),
mass conse a ion (i.e. Rxu( ,x)dx does no depend on ime) is gua an eed by he condi ion
Rzκ(z)dz = 1. We also assume ha he ib ils can only di ide in o wo, i.e. we impose Rκ(z)
dz = 2 (no e na y b eak-up).
An in e sion o mula o γ: Dynamics o he momen s. The long ime beha iou o he
solu ions o (3) is now well-known by ma hema icians [14]: he solu ion con e ges a e escal-
ing o some s eady p o ile gin he sense ha
2=guð ; 1=gxÞ !
!1 gðxÞ;ð4Þ
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Nume ical in es iga ion in o he ma hema ical model o pu e agmen a ion
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and whe e gis he solu ion o
yg0ðyÞþð2þagygÞgðyÞ ¼ agZ1
y
1
ky
�� ggð Þd ;Z1
0
ygðyÞdy ¼ ;ð5Þ
whe e ρonly depends on he ini ial condi ion u
0
h ough ρ=Rxu
0
(x)dx. In imaging expe i-
men s, we sample leng hs o pa icles p esen in he popula ion a each ime poin . The e o e,
we in oduce he measu ed quan i y:
ð ;xÞ ¼ uð ;xÞ
R1
0uð ;xÞdx :ð6Þ
We de ine he momen o o de qo he dis ibu ion as
Mqð Þ ¼ Z1
0
xq ð ;xÞdx:ð7Þ
We deduce di ec ly om (4) ha
1=g ð ; 1=gxÞ !
!1
gðxÞ
R1
0gðyÞdy :ð8Þ
A space in eg a ion o he abo e o mula gi es us
log Mqð Þ ¼ q
glog ð ÞþCðqÞ; la ge;ð9Þ
o he cons an CðqÞ ¼ log R1
0gðyÞyqdy
R1
0gðyÞdy
� �. In pa icula , he i s momen (q = 1), being he
a e age leng h, can be e alua ed di ec ly om he leng h measu emen s. This p o ides us wi h
a me hod o ex ac γ om he da a because he log-log dynamics o he mass ends o a
s aigh line whose slope is equal o −1/γ, p o ided ha he egime wi h s eady dis ibu ion
shape p o ile has been eached. No ice ha C(0) = 1 and Cð1Þ ¼
R1
0gðyÞdy. Impo an ly, he
asymp o ic s aigh line depends on he pa ame e s o he model (e.g. i s slope depends on γ,
and i s posi ion depends on γ,α,κ h ough g) bu no on he ini ial leng h dis ibu ion.
Eq (9) shows elegan ly ha , when applied wi h q= 1, he numbe a e age molecula weigh
(p opo ional o he a e age leng h o ib ils) M
1
( ) decays linea ly o la ge imes indepen-
den ly o he ini ial leng h dis ibu ion when plo ed on a loglog scale. We e e o his cha ac-
e is ic line as he asymp o ic line. A sho e ime scales, M
1
is also decaying.
We no e ha ou me hod o eco e γwo ks e en i he pa icles can b eak up in o mo e han
wo pa icles, indeed Eq (9) does no use he in o ma ion o he numbe o pa icles p oduced by
each b eakage. The au ho s o [8] also use he dynamics o he momen s o es ima e γ, in he case
o b eakage o dex an molecules h ough acid hyd olysis. Howe e , he app oach in [8] is a spe-
cial case wi h an assump ion o monodispe si y, and a model selec ion app oach compa ing di -
e en solu ions wi h di e en γ alues was used (see a ull compa ison de ailed in S1 Appendix).
The Mellin ans o m
The in e sion o mula o αand κs ongly elies on he Mellin ans o m, which appea s o be
an in insic ea u e o he pu e agmen a ion equa ion. Fo any unc ion (o gene alized unc-
ion) o e Rþ, we ecall ha he Mellin ans o m M½m�o μis de ined h ough he in eg al
M½m�ðsÞ ¼ Zþ1
0
xs1dmðxÞ;ð10Þ
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o hose alues o sin he complex plane o which he in eg al exis s. We de ine o <e(s)>1
GðsÞ≔M½g�ðsÞand KðsÞ≔M½k�ðsÞ. The Mellin ans o m u ns he di e en ial Eq (5) in o
he ollowing non-local unc ional equa ion:
ð2sÞGðsÞ ¼ agðKðsÞ 1ÞGðsþgÞ 8s2C;<eðsÞ>1:ð11Þ
An in e sion o mulae o αand κ.Since he ission is only bina y, K(1) = 2. Thus, using
he Mellin ans o m, we ob ain (see [20] o he ma hema ical jus i ica ion)
a¼Gð1Þ
gGð1þgÞ¼
Gð1þgÞ:ð12Þ
We emphasize ha , con a ily o γ, he es ima e on αmainly elies on he bina y ission
assump ion.
Es ima ing he di ision ke nel κ e eals a much ha de and mo e ill-posed p oblem com-
pa ed o ha o γand α. Once αand γa e known, we may o mally di ide Eq (11) by G(s+γ)
and ob ain
KðsÞ ¼ 1þð2sÞGðsÞ
agGðsþgÞ:ð13Þ
The p ope ies o he ke nel κa e such ha he in e se Mellin ans o m o Kis well de ined
and equal o κ(see o ins ance [27](Theo em 11.10.1). The e o e he agmen a ion ke nel κ
is gi en by he in e se Mellin ans o m o 1þð2sÞGðsÞ
agGðsþgÞ, p o ided ha he (complex alued)
denomina o does no anish. In ac , i is ma hema ically p o ed in [20] ha he e exis s
s
0
>2 such ha he denomina o G(s+γ) does no anish. Then, o his speci ic s
0
we ha e
kðzÞ ¼ 1
2ipZ
<eðsÞ¼s0
zs1þð2sÞGðsÞ
agGðsþgÞ
� �ds:ð14Þ
The de ailed ma hema ical jus i ica ions and p oo s o he o mulae gi en he e can be ound
in [20]. The main idea unde lying he me hod is he cen al ollowing heo e ical esul : he
leng h dis ibu ion p o ile o he solu ion o he pu e agmen a ion equa ion aligns wi h a
s eady shape o la ge imes, and all he momen s o he p o ile decay p edica i ely on an
asymp o ic line in log-log space. See Box 1 o a summa y o he heo y.
Box 1: Summa y o he heo y
• In e sion o mula o γ:γis ob ained using Eq (9) as g¼ 1
S, whe e Sis he slope o
he s aigh line ep esen ing he i s momen (e.g. a e age leng h) as a unc ion o
ime, in log-log scale. The cu e unde ques ion is a s aigh line o la ge ime poin s.
• In e sion o mula o α:αis ob ained using Eq (12), whe e Gis he Mellin ans o m
o he s eady shape o he leng h dis ibu ion o la ge imes.
• In e sion o mula o κ:κis ob ained using o mula (14) oge he wi h γand α. Again,
Gis he Mellin ans o m o he s eady shape o he leng h dis ibu ion o la ge imes.
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Resul s and discussion
Explo a ion o ajec o ies
In his sec ion, we gi e an o e iew o he in luence o he pa ame e s on he s a iona y p o ile
o he sel -simila leng h dis ibu ion and on i s ansien beha iou .
In luence o γ. I is p o en in he heo e ical pape [28] ha he pa ame e γimpac s he s a-
iona y p o ile o la ge x, and mo e speci ically ha g(x) beha es like Cexp
−x
γ
/γ
as x!1 o
some C>0. This p ope y canno be used o ex ac he pa ame e om he s a iona y p o ile
g, since i would equi e o ha e p ecise expe imen al in o ma ion o la ge sizes. This p ope y
is illus a ed in S1(A) Fig, whe e he s a iona y p o ile co esponding o di e en alues o γ
and o a gaussian ke nel is plo ed. Fo la ge alues o γ, since decay a la ge pa icle sizes is
as e , he s a iona y p o ile is mo e concen a ed a ound x= 0 ( he in eg al o xg(x) is equal o
1) compa ed o smalle γ alues. The ole o γon he o e all shape o gis highly non-linea ,
and o all o he pa ame e s ixed, he o e all shape can a y wi h γ. This is illus a ed on Fig
1A o α= 1 and he speci ic ke nel κdisplayed in he inse , he s a iona y p o ile has di e en
quali a i e beha iou s o γ= 0.8, 1, 1.5 and 2. The in luence o γon he ime e olu ion o he
leng h dis ibu ion is desc ibed by Fo mula (9). The momen s o o de zo he p o ile ( o
example i s momen o o de 1: he a e age size o ib ils) dec ease linea ly wi h ime a log-log
scale. Depending on he ini ial momen s, he e olu ion o he momen s can ha e wo di e en
shapes as illus a ed on Fig 2. Fo example, he a e age leng h M
1
( ) can s ay comple ely below
he asymp o ic line as illus a ed by he g een line, o abo e he asymp o ic line as illus a ed
by he blue line. See Fig 1B, o an illus a ion o he ajec o ies wi h simula ed da a s a ing
om di e en ini ial a e age leng hs.
In luence o α. I o he ini ial da a u
0
(x), he solu ion o he agmen a ion equa ion o α= 1
is u( ,x), hen he solu ion o he agmen a ion equa ion o he same ini ial da a, he same
alues o γand κ, and α>0 is uað ;xÞ ¼ 1
auða ;xÞ. Fu he i he s a iona y s a e o α= 1 is g,
hen, o α>0, i is g
α
(y) = α
2/γ−1
g(α
1/γ
y). Indeed hen,
2=guð ; 1=gxÞ ! gðxÞ;ð15Þ
Fig 1. In luence o γon he s a iona y leng h dis ibu ion p o ile, and ansien dynamics o he a e age leng h. A: S a iona y p o ile o di e en alues o γand
α= 1,
= 200. B: Time e olu ion o he mass M
1
( ) in a log-log scale. The ini ial condi ions a e sp ead gaussian wi h di e en masses 1, 10, 50 o 100 and γ= 1.
h ps://doi.o g/10.1371/jou nal.pcbi.1008964.g001
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hen, se ing τ=α
2=guað ; 1=gxÞ ¼ 2=g
auða ; 1=gxÞ ¼ a2=g1 2=guð ; 1=ga1=gxÞ ! gaðxÞ:ð16Þ
This is illus a ed on Fig 3A. We conclude ha he pa ame e αac s as a ime scaling e m. This
p ope y canno be used o eco e α, since om expe imen we only know gup o a mul iplica-
i e ac o .
In luence o he ke nel κ. We i s explo ed he in luence o κon he s a iona y p o ile g. In
i s app oxima ion, smoo h ke nels can be classi ied in o wo classes: Wi hin class A, he
ke nels a e such ha κ(0) = κ(1) = 0, and wi hin class B, ke nels a e such ha κ(0) >0 and
κ(1) >0. On Fig 3B he s a iona y p o iles o a selec ion o six di e en ke nels a e displayed.
As seen, on Fig 3, igh , whe he he ke nel belongs o class A o B can be ead di ec ly on he
shape o he s a iona y p o ile. Fo ke nels o class A, he s a iona y p o ile is ze o a x= 0 and
is unimodal (one peak), and o ke nels o class B, he s a iona y p o ile is non-ze o a x= 0
and dec easing in a neighbo hood o 0. This is consis en wi h he heo e ical esul s o [28]
which s a e ha i k(z)�C
κ
z
�
o C
κ
>0 and � > −1 a ound z= 0, hen g(x)�C
g
x
�
o some
cons an C
g
>0. Howe e , wi hin a gi en class, i is di icul o ex ac he shape o he ke nel
om he me e in o ma ion o he s a iona y p o ile. In pa icula , wi hin class A, using only
he s a iona y in o ma ion, i is no possible o dis inguish one peak ke nels om wo peaked
ke nels, no dis inguish be ween Gaussian wi h small o la ge sp ead (see Fig 3B and S1(C)
Fig). S1 Fig shows he s a iona y p o iles o ke nels ha ha e ea u es esembling bo h classes.
Fo ke nels o class A, we only obse e s a iona y p o iles wi h one peak. On S1(B) Fig, we
show s a iona y p o iles co esponding o a ke nel composed o he sum o wo Gaussian
Fig 2. Illus a ion o wo possible scena ii o >1 depending on he ini ial momen o he sys em. The i s
momen M
1
( ) ( he dis ibu ion mean) can s ay below (g een) o abo e he asymp o ic line (blue). Bo h beha iou s
ha e been obse ed nume ically. In all cases, he momen M
1
( ) is dec easing wi h ime and aligns o he asymp o ic
line (s aigh line in ed) o la ge ime.
h ps://doi.o g/10.1371/jou nal.pcbi.1008964.g002
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Nume ical scheme
We de ail he e he nume ical scheme we use o sol e (3) o gene a e simula ed dis ibu ions
and ajec o ies. Ou me hod is based on [30]. We se w= log(x) and ins ead o di ec ly w i ing
a scheme on , we simula e he e olu ion o he quan i y n( ,w)≔e
2w
u( ,e
w
) which sa is ies
o >0 and w2R
@n
@ ð ;wÞ ¼ aegwnð ;wÞþZ1
0
kðeyÞegye2ynð ;yþwÞdy
� �;nð0;wÞ ¼ e2wuð0;ewÞ:ð18Þ
The ad an age o using a scheme on he a iable n( ,w) ins ead o u( ,x) is ha he quan i y n
sa is ies he conse a ion p ope y
d
d ZRþ
nð ;wÞdz ¼0:ð19Þ
We disc e ize he ime axis wi h a uni o m ime s ep Δ . Fo he w a iable, we conside a
uni o m g id [w
1
,. . .,w
p
,. . .,w
I
] o s ep Δw(which co esponds o an exponen ial g id o x),
wi h w
p
= 0. We deno e by nk
i he app oxima ed alues o he a iable na ime kΔ and a
w
i
= (i−p)Δw. Le us obse e ha w
i
+w
j
= (i+j−2p)Δw=w
i+j−p
. We se he ini ial da a
n0
i¼e2wiuð0;zwiÞ o i2 ½1;I�;ð20Þ
and he i e a ion p ocess o i2[1, I], k�0,
nkþ1
i¼nk
iaD egwinkþ1
iþaD egwiX
min Ii;Ipg
j¼0
e2wpþjkðewpþjÞegwpþjnk
iþj;ð21Þ
which is o i2[1, I], k�0
nkþ1
i¼1
1þaD egwink
iþaD egwiX
min Ii;Ipg
j¼0
e2wpþjkðewpþjÞegwpþjnk
iþj
!:ð22Þ
Rema k 1 We use an implici scheme ins ead o he explici scheme
nkþ1
i¼nk
iaD egwink
iþaD egwiX
min Ii;Ipg
j¼0
e2wpþjkðewpþjÞegwpþjn;
iþjð23Þ
since o he explici o mula ion, he CFL s abili y condi ion ([31]) ha gua an ees posi i i y o
he solu ion imposes he ollowing uppe bound on Δ
D �1
aexpðgwIÞ:ð24Þ
In some cases, o ins ance o eal da a, he CFL s abili y condi ion leads o impose Δ �0.01
whe eas he inal ime is 1million.On he con a y, he implici e sion (22) o he scheme is s a-
ble wi h no s abili y condi ion on Δ and allows us o ake la ge alues o Δ .
An al e na i e nume ical scheme ha uses he disc e e modeling app oach (2) based on [4]
is also used o compa ison and o alida ing he abo e nume ical scheme. Explici solu ions
o (3) a e summa ized in Table 1. We use hese explici solu ions o alida e ou nume ical
scheme.
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S a is ical es s
We de ail he e he s a is ical es desc ibed and used in he Resul s and discussion sec ion.
A each ime poin o he expe imen , we es he null hypo hesis H
0
“H
0
: S a ing wi h a
ixed ini ial dis ibu ion, he samples
a
and
b
espec i ely ob ained wi h γ=α= 1 and he wo
di e en ke nel κ
a
and κ
b
, ha e he same dis ibu ion”.
Gi en wo samples
a
and
b
o espec i e size N
a
and N
b
, we de ine he dis ance
dab ¼sup
xjFaðxÞ FbðxÞj;ð25Þ
whe e F
a
and F
b
a e he empi ical cumula i e dis ibu ion unc ions associa ed wi h he sam-
ples u
a
and u
b
. The Kolmogo o -Smi no es wo ks as ollows: he H
0
null hypo hesis is said
o be ejec ed a he signi icance le el ℓi
d2
ab >1
2ln ð‘ÞNaþNb
NaNb
:ð26Þ
No e ha in he li e a u e, he le el o signi icance is deno ed by αins ead o ℓ. The symbol α
being al eady used o he agmen a ion a e, we decided o deno e he signi icance le el by ℓ.
I he abo e condi ion is sa is ied, he Kolmogo o -Smi no es ecommends no o ejec he
H
0
hypo hesis. We ecall ha no conclusion can be d awn i he e e se inequali y is sa is ied
(in pa icula , we can ne e say ha H
0
can be s a is ically ejec ed, see [33] o a comple e he-
o y on s a is ical es s).
The p- alue associa ed wi h a s a is ical es is he le el ℓ
lim
om which we conside ha we
canno s a is ically ejec he null hypo hesis. The p- alue is hen a non-linea unc ion o his
dis ance d
ab
exp essed as
p alue ¼exp 2NaþNb
NaNb
d2
ab
� �:ð27Þ
Wha is done in gene al is building an es ima e o he cumula i e unc ions F
a
and F
b
using
an in e pola ion o wo samples o size N
a
and N
b
. (e.g. S7 Fig). In ou case, we use he exac N
a
and N
b
o compu e d
ab
. Le us also men ion ha in ou con ex , in he case whe e he hypo he-
ses H
0
canno be ejec ed, i means one ke nel canno be dis inguished om he o he using
only a measu emen o size Na he ime .
Conclusions
In his s udy, we p esen ed he ma hema ical analysis o he pu e agmen a ion equa ion.
Based on he heo e ical analysis, in e sion o mulae o di ec ly eco e in o ma ion ega ding
di ision a es αand γpa ame e s, and di ision ke nel κ om ime dependen expe imen al
measu emen s o ilamen size dis ibu ion a e de i ed. These in e sion o mulae allow analy-
sis o he dynamical ajec o ies o ib il agmen a ion wi hou goodness o i analysis o mod-
els. This is he basis o an analy ical me hod ha enables he sys ema ic compa ison o he
s abili y owa ds di ision o amyloid ilamen o di e en ypes. We belie e ex ac ing and
compa ing he a es and he ke nel desc ibing agmen a ion eac ions e lec he s abili y o
he p o ein ilamen s owa ds b eakage, which is o impo ance in amyloid seed p oduc ion
and he p opaga ion o he amyloid s a e in unc ional and disease-associa ed amyloid.
He e, ou conclusions a e ha he s a iona y leng h dis ibu ion p o ile depends non-line-
a ly on γand κ. The pa ame e γcan be es ima ed using he measu emen o wo o mo e la e-
ime leng h dis ibu ion p o iles. The pa ame e αis a scaling pa ame e ha can be es ima ed
om one la e- ime leng h dis ibu ion p o ile combined wi h he es ima ed alue o γ. Ou
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in e sion o mulae o he pa ame e s γand αa e p o ed o be obus wi h espec o sampling
noise. We also p o ide an algo i hm (code w i en in Ma lab) ha ake as an inpu he mea-
su ed leng h dis ibu ion p o iles a di e en imes and gi e o he use , as an ou pu , he es i-
ma ed alues o γand αco esponding o he measu ed dynamics.
As o smoo h agmen a ion ke nels κ, we show ha hey can be sepa a ed in o wo g oups:
he ke nels such as κ(0) = κ(1) = 0, (e.g. a Gaussian unc ion), ha lead o a unimodal s a ion-
a y leng h dis ibu ion p o ile, and he ke nels such ha κ(0) and κ(1) a e la ge enough, ha
lead o a dec easing s a iona y leng h dis ibu ion p o ile a ound 0. Howe e , non- i ial com-
bina ions be ween hese wo ough ypes o ke nels may lead o highly non- i ial s a iona y
dis ibu ion p o iles. Despi e hese wo ough classes o ke nels, ou wo k demons a es ha
he knowledge o la e- ime leng h dis ibu ion p o iles is no enough o iden i y he p ecise
agmen a ion ke nel. In pa icula , i he ke nel is a Gaussian unc ion, i s sp ead canno be
deduced om la e- ime measu emen s. Ins ead, ea ly leng h dis ibu ions con ain mo e
de ailed in o ma ion on κ. This sugges s ha he expe imen s ha can p o ide he bes da a o
es ima e γand αa e long- ime expe imen s s a ing wi h any ini ial dis ibu ion. In his case,
leng h dis ibu ions a se e al ime poin s a e needed a e he asymp o ic egime is eached o
ensu e good es ima es o γand α. On he con a y, o es ima e he agmen a ion ke nel κ, he
expe imen should a he s a wi h a highly peaked dis ibu ion, ib ils o simila leng h, and
he e olu ion o he sample leng h dis ibu ions should be measu ed a sho ime poin s be o e
asymp o ic egime is eached. Such an ini ial dis ibu ion is e y complica ed o ob ain expe i-
men ally, and we explo e in a u u e wo k how he sp ead o he ini ial dis ibu ion a ec s he
es ima e o κ. Such expe imen s a e challenging o pe o m, and u u e wo k e ealing how
he sp ead o he ini ial dis ibu ion a ec s he es ima e o κis also needed. A p ac ical consid-
e a ion o he expe imen alis is o de e mine whe he o when he asymp o ic egime has
been eached. This a heo e ically challenging ques ion, bu a p ac ical p o ocol (as ollows)
can be used o in o m he design o expe imen s. Fi s ly, un a simula ion o he agmen a ion
expe imen (Ma lab code is made a ailable, see Me hods) using he ini ial dis ibu ion ha can
be expe imen ally de e mined, and a i s guess o he agmen a ion a e and ke nel pa ame-
e s γ,αand κ. Secondly, es ima e he ime T
e
a e which he cu e (log( ), log(M
1
( ))) has
become a line. Thi dly: pe o m he simula ion un il ime 5T
e
.
Finally, we emphasize ha wha is assumed in he p esen pape is ha he pa ame e s γ,α
and κa e in insic and independen cha ac e is ics o each and e e y indi idual ypes o amy-
loid ib ils. Then, an app op ia e expe imen o es ima e γ,αand κis one ha obse es he
popula ion o ib ils o one gi en ype in he absence o g ow h, o example using dilu e sam-
ples wi h deple ed ee monome s. I may be o in e es o also es ima e he in insic g ow h
a e o he ib ils. The p o ocol we sugges is o sepa a e he g ow h expe imen om he ag-
men a ion expe imen , and o i s es ima e he agmen a ion cha ac e is ics as p esen ed in
his pape , and hen ocus on es ima ing he g ow h a e sepa a ely. F agmen a ion equa ions
a e used in many di e en applica ions, o which ou me hod can apply o. In pa icula , wi h
a mode n expe imen al app oach ha would p o ide ime-dependen size dis ibu ion p o-
iles, he agmen a ion a e and ke nel can be ob ained o polyme s o any ype [8].
Suppo ing in o ma ion
S1 Appendix. Reco e ing γ om he da a: Compa ison wi h [8].
(PDF)
S1 Fig. In luence o he pa ame e s γand κon he s a iona y leng h dis ibu ion p o ile.
(PDF)
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S2 Fig. S a iona y p o ile o di e en alues o γand α= 1.
(PDF)
S3 Fig. Addi ional igu es.
(PDF)
S4 Fig. Plo o he ime e olu ion o he p- alue co esponding o he null compa ison
hypo hesis H
0
, o 3 di e en ini ial condi ions.
(PDF)
S5 Fig. Illus a ion o he p o ocol—Example 2.
(PDF)
S6 Fig. Con e gence o he nume ical scheme.
(PDF)
S7 Fig. Plo o he ime e olu ion o he p- alue co esponding o he Kolmogo o -Smi -
no es o he H
0
hypo hesis.
(PDF)
Au ho Con ibu ions
Concep ualiza ion: Magali Tou nus, Miguel Escobedo, Wei-Feng Xue, Ma ie Doumic.
Fo mal analysis: Magali Tou nus.
In es iga ion: Magali Tou nus, Miguel Escobedo, Wei-Feng Xue, Ma ie Doumic.
Supe ision: Wei-Feng Xue, Ma ie Doumic.
Valida ion: Wei-Feng Xue, Ma ie Doumic.
W i ing – o iginal d a : Magali Tou nus.
W i ing – e iew & edi ing: Magali Tou nus, Miguel Escobedo, Wei-Feng Xue, Ma ie
Doumic.
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