biomolecules
A icle
P obing he Occu ence o Soluble Oligome s
h ough Amyloid Agg ega ion Scaling Laws
Alexand a Sil a 1,2,† , Zsuzsa Sá kány 1,2,†,‡ , Joana S. F aga 1,2,3 , Pablo Taboada 4,5,
Sand a Macedo-Ribei o 1,2 and Ped o M. Ma ins 1,2,3,*
1IBMC—Ins i u o de Biologia Molecula e Celula , Uni e sidade do Po o, 4200-135 Po o, Po ugal;
[email p o ec ed] (A.S.); [email p o ec ed] (Z.S.); [email p o ec ed] (J.S.F.);
[email p o ec ed] (S.M.-R.)
2Ins i u o de In es igação e Ino ação em Saúde, Uni e sidade do Po o, 4200-135 Po o, Po ugal
3ICBAS—Ins i u o de Ciências Biomédicas Abel Salaza , Uni e sidade do Po o, 4050-313 Po o, Po ugal
4Á ea de Física de la Ma e ia Condensada, Facul ad de Física, Uni e sidad de San iago de Compos ela,
15782 San iago de Compos ela, Spain; [email p o ec ed]
5Ins i u o de In es igación Sani a ia (IDIS), 15706 de San iago de Compos ela, Spain
*Co espondence: [email p o ec ed]; Tel.: +351-220-408-800
† These au ho s con ibu ed equally o his wo k.
‡ P esen Add ess: LEPABE—Depa amen o de Engenha ia Química, Faculdade de Engenha ia da
Uni e sidade do Po o.
Recei ed: 4 Sep embe 2018; Accep ed: 1 Oc obe 2018; Published: 4 Oc obe 2018


Abs ac :
D ug disco e y equen ly elies on he kine ic analysis o physicochemical eac ions ha
a e a he o igin o he disease s a e. Amyloid ib il o ma ion has been ex ensi ely in es iga ed in
ela ion o p e alen and a e neu odegene a i e diseases, bu hus a no he apeu ic solu ion has
di ec ly a isen om his knowledge. O he agg ega ion pa hways p oducing smalle , ha d- o-de ec
soluble oligome s a e inc easingly appoin ed as he main eason o cell oxici y and cell- o-cell
ansmissibili y. He e we show ha amyloid ib illa ion kine ics can be used o un eil he p o ein
oligome iza ion s a e. This is illus a ed o human insulin and a axin-3, wo model p o eins o which
he amyloidogenic and oligome ic pa hways a e well cha ac e ized. Agg ega ion cu es measu ed
by he s anda d hio la in-T (ThT) luo escence assay a e shown o e lec he ela i e composi ion
o p o ein monome s and soluble oligome s measu ed by nuclea magne ic esonance (NMR) o
human insulin, and by dynamic ligh sca e ing (DLS) o a axin-3. Uncon en ional scaling laws o
kine ic measu ables we e explained using a single se o model pa ame e s consis ing o wo a e
cons an s, and in he case o a axin-3, an addi ional o de -o - eac ion. The same i ed pa ame e s
we e used in a disc e ized popula ion balance ha adequa ely desc ibes ime-cou se measu emen s
o ib il size dis ibu ions. Ou esul s p o ide he oppo uni y o s udy oligome ic a ge s using
simple, high- h oughpu compa ible, biophysical assays.
Keywo ds: p o ein agg ega ion; amyloid; soluble oligome s; kine ic analysis; nuclea ion
1. In oduc ion
The deposi ion o amyloid ib ils in he b ain is a pa hological hallma k o se e al di e en
neu odegene a i e diso de s, ye he pa hogenic ole o hese insoluble agg ega es is no ully
unde s ood [
1
]. On he o he hand, he e is now subs an ial
in i o
e idence o amyloidogenic
p o eins also o ming small soluble oligome s ha sp ead o neighbo ing cells and induce downs eam
p ocesses associa ed wi h neu odegene a ion [
1
,
2
]. Chemical kine ics, a classical co ne s one o d ug
disco e y [
3
], is ha dly applicable o he s udy o his new and p e-eminen a ge [
4
–
6
], in pa due
o he lack o s aigh o wa d me hods o moni o he o ma ion o a highly he e ogeneous g oup o
Biomolecules 2018,8, 108; doi:10.3390/biom8040108 www.mdpi.com/jou nal/biomolecules
Biomolecules 2018,8, 108 2 o 20
species anging om p o ein dime s o complex n-me s [
7
,
8
]. In con as , ex ensi e esea ch has been
de o ed o p o ein agg ega ion kine ics based on he cha ac e is ic inc o ial p ope ies o amyloid
ib ils [9].
An impo an s ep owa ds he kine ic quan i ica ion o o -pa hway agg ega ion was aken a e
he obse a ion o p o ein p ecipi a ion occu ing in pa allel wi h he o ma ion o amyloid ib ils o
lysozyme [
10
]. Be o e, kine ic analysis o amyloid agg ega ion o he isle amyloid polypep ide (IAPP)
sugges ed he o ma ion o in e media e on- and o -pa hway phases du ing IAPP ib illogenesis [
11
].
The p esence o non-amyloidogenic species p oduces pe cep ible de ia ions om he ime e olu ion
o he amyloid signal expec ed o he gene ic nuclea ion and g ow h p ocesses o he phase ansi ion:
α=1−1
kb[exp(ka )−1]+1(1)
whe e
α
is he no malized amyloid con e sion, and
ka
and
kb
a e combina ions o elemen a y a e
cons an s [
12
]. One o he kine ic signa u es ound o be associa ed wi h o -pa hway agg ega ion
was he unusually weak dependence o he lag phase du a ion on he ini ial concen a ion o
lysozyme [
10
]. Simila beha io s obse ed wi h o he p o ein models ha e p o ided he basis
o a ied in e p e a ions o he amyloid agg ega ion mechanism encompassing, o example,
Michaelis-Men en-like sa u a ion o he elonga ion s ep [
13
], complex sub-s eps o nuclea ion
and g ow h [
14
], s ochas ic luc ua ions in he nuclea ion ime [
15
], and he supp ession o
ib il agmen a ion a high ib il concen a ions [
16
]. Unlike hese possible explana ions o he
unde pe o ming scaling laws, o -pa hway agg ega ion can be di ec ly in es iga ed by analy ical and
mic oscopic echniques such as hose used o iden i y insoluble agg ega es o lysozyme [
10
], and la e
on, soluble oligome s o a axin-3 [17], and me as able oligome s o Aβ40 and Aβ42 pep ides [18,19].
Because he o ma ion o soluble and insoluble assemblies is ed by a common pool o p o ein
monome s, we p opose ha amyloid ib illa ion kine ics can be used o e eal he p esence o he
pa allel oligome ic pa hway. To es his hypo hesis, we chose wo sys ems, human insulin and
a axin-3, o which he ib illa ion kine ics ha e been measu ed unde condi ions o known oligome ic
composi ion. Insulin is a p o ein ho mone exis ing in solu ion in a he modynamic equilib ium o
monome s, dime s, e ame s, hexame s and highe -o de oligome s [
20
,
21
]. Changes in he p o ein
molecula s uc u e induced by low pH, high empe a u e o he p esence o o ganic sol en s lead
o he o ma ion o amyloid ib ils h ough he di ec associa ion o insulin monome s [
20
], o by
he assembly o in e media e on-pa hway oligome s [
22
]. A axin-3 is a mul i-domain p o ein wi h a
globula Josephin domain and a C- e minal lexible ail con aining a polyglu amine (polyQ) epea
whose expansion ul ima ely causes Machado–Joseph disease. A axin-3 agg ega ion in ol es an ini ial
s ep media ed by he Josephin domain, and a second s ep dependen on he expanded polyQ ac
ha accele a es p o ein agg ega ion and p omo es he o ma ion o ma u e amyloid ibe s [
23
,
24
].
The analysis o he hio la in-T (ThT) binding assay un a di e en concen a ions o human insulin
and a axin-3 unco e s mechanis ic aspec s o he oligome ic and ib illa pa hways. The dis inc
agg ega ion mechanisms p edic ed o each p o ein a e expe imen ally alida ed by ime-cou se
dynamic ligh sca e ing (DLS) measu emen s.
2. Ma e ials and Me hods
2.1. P o ein P epa a ion
Human insulin pu chased om Sigma-Ald ich (Sain Louis, MO, USA) (I2643) was dissol ed
wi hou u he pu i ica ion in 20% ace ic acid 0.5 M NaCl (pH 1.8) o a inal concen a ion o 5 mg/mL.
Be o e incuba ion, samples we e il e ed wi h 0.22
µ
m sy inge il e uni s (Millex-GV, Millipo e, Co k,
I eland). Non-expanded a axin-3 was exp essed and pu i ied as p e iously desc ibed [17].
Biomolecules 2018,8, 108 3 o 20
2.2. T ansmission Elec on Mic oscopy
T ansmission elec on mic oscopy (TEM) isualiza ion o insulin and a axin-3 ib ils was
pe o med using a TEM JEM-1400 (JEOL, Tokyo, Japan) a an accele a ing ol age o 80 kV. 100
µ
L
samples o 5 mg/mL insulin we e incuba ed o 6 h a 45
◦
C wi hou mechanical shaking in 1.5 mL
eppendo ubes (DNA LoBind, Eppendo AG, Hambu g, Ge many). 700
µ
L samples o 5
µ
M
(0.218 mg/mL) a axin-3 in 20 mM sodium phospha e pH 7.5, 150 mM NaCl, 1 mM di hio h ei ol (DTT)
we e incuba ed in 1.5 mL eppendo ubes (DNA LoBind, Eppendo AG, Hambu g, Ge many) o
65 h a 37
◦
C wi hou mechanical shaking. P o ein samples we e dilu ed in wa e (1:20 o insulin and
1:10 o a axin-3), adso bed o ca bon-coa ed 200 mesh nickel g ids (FCF300-NI, Elec on Mic oscopy
Sciences, Ha ield, PA, USA), nega i ely s ained wi h 2% (w/ ) u anyl ace a e, d ied and obse ed a a
magni ica ion o 80,000–100,000×.
2.3. Dynamic Ligh Sca e ing
Dynamic ligh sca e ing measu emen s we e pe o med using an ALV/DLS/ SLS-5000F, SP-86
goniome e sys em (ALV-GmbH, Langen, Ge many) equipped wi h a CW diode-pumped Nd:YAG
solid-s a e Compass-DPSS lase wi h a symme ize (Cohe en Inc., San a Cla a, CA, USA). The lase
ope a es a 488 nm wi h an ou pu powe o 400 mW. The in ensi y scale was calib a ed agains
sca e ing om oluene. 700
µ
L samples o 5 mg/mL insulin we e incuba ed in glass cu e es a
45
◦
C wi hou mechanical shaking and pe iodically analyzed a a sca e ing angle 90
◦
o he inciden
beam. Hyd odynamic adii o he pa icles in solu ion we e es ima ed om he di usion coe icien (s)
deli e ed om CONTIN analysis [
25
]. Discon inuous au o-co ela ion unc ions we e no conside ed
o CONTIN analysis.
3. Resul s and Discussion
Fo he expe imen al condi ions adop ed in each model p o ein, human insulin and a axin-3
p oduce ib illa species wi h dis inc mo phologies (Figu e 1A,B) and a ma kedly di e en
agg ega ion a es (Figu e 1C). Long, s aigh ilamen s o human insulin a e o med much as e
han he small, wo m-like ib ils o a axin-3, he eby sugges ing ha phase ansi ion mechanisms a e
di e en ly a ec ed by he ib il elonga ion s ep. Chemical kine ic analysis pinpoin s hese di e ences,
and e eals how he p esence o soluble oligome s in luences each ype o p o ein agg ega ion cu es.
The quan i a i e me hods p oposed he e a e expec ed o con ibu e o he iden i ica ion o mechanis ic
changes p o oked, e.g., by he p esence o agg ega ion modula o s o by di e en condi ions o
empe a u e, pH, ionic s eng h, e c.
Figu e 1.
Case s udy examples o human insulin and a axin-3 agg ega ion. T ansmission elec on
mic oscopy (TEM) mic og aphs o nega i ely s ained ib ils o (
A
) 5 mg/mL human insulin and
(
B
) 5
µ
M (0.218 mg/mL) a axin-3 cap u ed a e 6 h and 65 h incuba ion, espec i ely (scale ba s,
100 nm). (
C
) Schema ic amyloid ib illa ion cu es ep esen ing he p og ess o no malized hio la in-T
(ThT) luo escence (
F/FF
) du ing he agg ega ion o human insulin and a axin-3 in he ange o p o ein
concen a ions s udied by Fode àe al. [
26
] and Sil a e al. [
17
], espec i ely. The hal -li e coo dina es
50 and 50 a e indica ed by he a ows and by he slopes o dashed lines, espec i ely.
Biomolecules 2018,8, 108 4 o 20
3.1. Mechanis ic Analysis o Insulin Agg ega ion
In he simpli ied mechanism ep esen ed in Figu e 2A, he elemen a y in e media e s eps
pa icipa ing in he p ima y nuclea ion, seconda y nuclea ion and elonga ion o insulin ib ils a e
summed up in o he o e all a e cons an s
kn
,
k2
, and
k+
, espec i ely [
17
]. The sigmoidal ( a he han
hype bolic) p og ess cu e o insulin ib illa ion (Figu e 1C) poin s o low alues o he pa ame e
kb=kn/ka
, which gi es he ela i e weigh o p ima y nuclea ion o e he au oca aly ic s eps o
seconda y nuclea ion and elonga ion (
ka=k2+k+
) [
10
]. The as elonga ion a es sugges ed by he
mo phology o insulin ib ils (Figu e 1A) a e con i med by he high alue o
ka
associa ed o he
s eep bu s phase (and high
50
alue) in Figu e 1C. The speci ic weigh o seconda y nuclea ion and
elonga ion in de e mining he alue o
ka
canno be dis inguished om single p og ess cu e analysis
because hese s eps ollow simila a e laws [
17
]. Mo eo e , since ib il b eakage ( a e cons an
k−
)
does no change he o al mass o ThT-posi i e ilamen s bu only hei numbe [
17
], complemen a y
measu emen s o ib il size dis ibu ions a e equi ed o di ec ly assess he ole o he b eakage s ep.
P io knowledge o he p o ein oligome iza ion s a e is equi ed be o e we can mo e in o he
deepe le els o he di e en agg ega ion pa hways [
27
]. The oligome iza ion equilib ium o human
insulin (Figu e 2B) has been cha ac e ized by Bocian e al. [
21
] using 2D and pulsed ield g adien spin
echo (PFGSE) nuclea magne ic esonance (NMR). I is, he e o e, possible o es ima e he a ailabili y
o insulin monome s unde condi ions o o al p o ein concen a ion, p esence o zinc, and acidic pH
ha a e simila o hose adop ed by Fode àe al. [
26
] while measu ing amyloid ib illa ion kine ics.
Based on he knowledge o he alues o he monome concen a ion
C1−me
(Figu e 2C), peculia
scaling laws o equilib ium (Figu e 2D) and kine ic (Figu e 2E,F) pa ame e s can be explained using
a numbe o i ed pa ame e s commensu a e wi h he numbe o independen obse a ions. As an
indica o o he amoun o amyloid ib ils p oduced, he inal ThT luo escence in ensi y (
FF
) (pink line
in Figu e 2D) is no di ec ly de e mined by he o al p o ein a ailable (closed ci cles in Figu e 2D) o
e en by he monome concen a ion alone. Since p o ein agg ega ion akes place un il he monome
concen a ion
C1−me
equals he he modynamic solubili y
C∗
, he alue o
FF
e lec s he di e ence
(C1−me −C∗)
o he wise known as supe sa u a ion (
∆C
) [
12
]. This is illus a ed in Figu e 2D (blue
line) wi h no o he i ing pa ame e s han he luo escence p opo ionali y cons an (in a bi a y uni s)
and he insulin solubili y, which is a measu able quan i y. Besides con i ming amyloid ib illa ion as a
phase ansi ion p ocess d i en by supe sa u a ion, he
FF
scaling law is consis en wi h a mechanism
o monome addi ion admi ing no supplemen a y con ibu ion om p e-exis ing soluble oligome s o
he inal ThT luo escence signal. Consequen ly, he amyloid pa hway (Figu e 2A) and he oligome ic
equilib ium (Figu e 2B) a e ound o ake place o e dis inc imescales, wi h insulin monome s being
consumed by he i s p ocess a much as e a es han hey a e p oduced by he second.
Biomolecules 2018,8, 108 5 o 20
Figu e 2.
Agg ega ion pa hways o human insulin in es iga ed h ough amyloid ib illa ion kine ics.
(
A
) Reac ion s eps and co esponding a e cons an s pa icipa ing in he amyloid pa hway. G een glows
ep esen an inc ease in he mass o ib ils. This a ia ion is de ec ed by amyloid binding assays and can
be used o es ima e wo pa ame e s,
ka
and
kb
, consis ing o combina ions o he o he a e cons an s
(see ex o de ails). (
B
) Oligome ic equilib ium o insulin as de e mined by Bocian e al. [
21
] using
2D and pulsed ield g adien spin echo (PFGSE) nuclea magne ic esonance (NMR) (
K12 =
4.9
×
10
5
,
K24 =
5.0
×
10
4
,
K46 =
2.7
×
10
3
and
Kiso =
1.35
×
10
4
). (
C
) Concen a ion o insulin monome s
(
C1−me
) p edic ed by he oligome ic equilib ium (
B
) o he alues o o al p o ein concen a ion (
CT
)
used in (
D
–
F
) (symbols). Pink line: polynomial i o he da a. (
D
–
F
) Reac ion scaling laws measu ed
by Fode àe al. [
26
] (symbols) and p edic ed by he model equa ions shown in blue o he monome
concen a ions es ima ed in (
C
) (solid blue lines). (
D
) The inal ThT luo escence (
FF
) is a di ec
p opo ion o supe sa u a ion
∆C=C1−me −C∗
(p opo ionali y cons an
cns =
1.95
×
10
5
) o an
in e ed solubili y alue o
C∗=
0.029 mg/mL. Pink line: The polynomial i in (
C
) is used o es ima e
FF
wi hou he solubili y co ec ion (
cns =
1.15
×
10
5
and
C∗=
0
)
. (
E
) Double-loga i hmic plo o
hal -li e coo dina e
50
as a unc ion o
CT
. Red lines: Limi scaling exponen s
|γ|
o 1 (dashed line) and
0.5 (solid line) a e s ill oo high o ep esen he measu ed end. (
E
,
F
) Bo h
ka
and
kb
a e conside ed
i s -o de dependen on
∆C
( i ed alues:
ka=
1.34
×
10
2∆C
h
−1
and
kb=
2.41
×
10
−7∆C
). Measu ed
da a we e adap ed wi h pe mission om Fode àe al. [
26
]. Copy igh 2017 Ame ican Chemical Socie y.
The sepa a ion o imescales simpli ies he applica ion o analy ic model equa ions ha we e
o iginally de i ed by assuming he soluble p o ein ully dissocia ed [
12
]. Theo e ical cu es o
50
and
50
s. p o ein concen a ion can be compu ed using he equa ions in Figu e 2E,F (see Appendix A o
de ails), a e exp essing
ka
and
kb
as a unc ion o
∆C
(and o
C1−me
). I , as i seems o be he case o
insulin, ib il elonga ion p edomina es o e seconda y nuclea ion (i.e.,
ka≈k+
and
kb≈kn/k+
), hen
bo h kaand kba e p opo ional o he ini ial supe sa u a ion (∝ ∆C) conside ing ha [10,12]:
(k+∝ ∆C
kn∝ ∆C2(2)

Biomolecules 2018,8, 108 6 o 20
These simple p emises and wo model pa ame e s a e su icien o elucida e he uncon en ionally
weak
CT
-dependence o
50
(Figu e 2E) as being he esul o he lowe mola ac ions o insulin
monome obse ed o highe p o ein concen a ions (Figu e 2C). I he associa ed s a es o soluble
insulin we e igno ed, he lowe limi s usually admi ed o he absolu e scaling ac o
|γ|
would be oo
high o ep oduce he measu ed end in Figu e 2E ( ed lines). Rema kably, he se o pa ame e s,
ka
and
kb
i ed o he lag- ime scaling da a in Figu e 2E a e he same as hose ha desc ibe he agg ega ion
a e da a in Figu e 2F (Appendix B). In bo h cases, he used alue o
C∗
is he one esul ing om
he in e p e a ion o Figu e 2C. Fa om being edundan , he con i ma ion o kine ic p edic ions by
di e en and independen measu emen s p o ides unequi ocal e idence ha he p esen heo e ical
amewo k, wi h only wo model pa ame e s, is indeed alid.
3.2. Mechanis ic Analysis o A axin-3 Agg ega ion
The s udy o a axin-3 agg ega ion ollows he same unde lying p inciple ha was adop ed
o human insulin, and has a simila pu pose: o show how adi ional kine ics can be ma kedly
dis o ed by he p esence o soluble oligome s. As in he case o o he polyQ- epea p o eins [
28
],
he o ma ion o a axin-3 ib ils and he dissocia ion o a axin-3 oligome s occu simul aneously
(Figu e 3A), and hus, imescale sepa a ion canno be assumed as a simpli ying hypo hesis. Suppo ed
by DLS, size-exclusion ch oma og aphy and TEM da a, a de ailed accoun o he di e en s eps shown
in Figu e 3A was ecen ly p o ided [
17
], including quan i a i e es ima ions o he a e cons an s
κ1+
,
κ1−
,
κn+
,
κn−
cha ac e izing he elemen a y s eps o a axin-3 oligome iza ion. The wo m-like
ib ils shown in Figu e 1B a e p edominan ly o med by seconda y nuclea ion (
ka≈k2
) and p ima y
nuclea ion
(kb≈kn/k2)
, wi h mino con ibu ions om he ib il elonga ion (
k+≈
0) and ib il
b eakage
(k−≈0)
s eps [
17
]. The measu ed e ec o p o ein concen a ion on he ThT luo escence
p og ess cu es (Figu e 3B, symbols) is no ully assessed i he oligome ic pa hway is no aken
in o accoun ; on he whole, he black lines in Figu e 3B a e indica i e o good nume ical i s, ye
hey a e based on Equa ion (1), which igno es he occu ence o he pa allel eac ions o soluble
oligome o ma ion/dissocia ion. Rega dless o how elabo a ed he heo e ical model can be, he i ed
pa ame e s a e, in his limi ed scena io, compa able o semi-empi ical coe icien s showing no e iden
undamen al meaning. In he illus a i e case o Figu e 3B, he empi ically de e mined alues o
ka
and
kb
would ollow a p opo ional ela ionship wi h p o ein concen a ion, which is no econcilable
wi h es ablished heo ies (Figu e S1).
Biomolecules 2018,8, 108 7 o 20
Figu e 3.
Agg ega ion pa hways o a axin-3 in es iga ed h ough amyloid ib illa ion kine ics.
(
A
) The oligome ic and amyloid pa hways ake place simul aneously. The a e cons an s o oligome
o ma ion/dissocia ion we e p e iously de e mined (
κ1+=
7.99
×
10
−4µ
M
−1
h
−1
,
κ1−=
9.73 h
−1
,
κn+=
0.167
µ
M
−1
h
−1
, and
κn−=
0.775 h
−1
) [
17
]. The s eps o amyloid ib il o ma ion a e he same as
in Figu e 2A. The mass o amyloid ib ils is a unc ion o only
ka
and
kb
, whe eas he numbe o ilamen s
is also in luenced by ib il b eakage and by he c i ical size o ib ils o med by p ima y and seconda y
nuclea ion (
R∗
and
R∗
2
, espec i ely). (
B
) Symbols: ThT luo escence inc ease measu ed o a axin-3
concen a ions o ( om op o bo om)
CT=
10
µ
M, 7
µ
M, 5
µ
M, 4
µ
M and 2
µ
M [
17
]. Lines: indi idual
(black) and global (blue) i ings o he expe imen al da a by Equa ions (1) and (S7), espec i ely.
Fi ing s a is ics gi en in Figu e S1A. Global i ing:
ka=
0.364
Cn2
T
h
−1
,
kb=
2.91
×
10
−10C2−n2
T
and
n2=
0.160). (
C
–
E
) Reac ion scaling laws co esponding o he kine ic measu emen s (symbols)
and global i ing (blue lines) shown in (
B
). (
C
) Double-loga i hmic plo . Red lines: Limi scaling
exponen s
|γ|
o 1 (dashed line) and 0.5 (solid line) a e s ill oo high o ep esen he measu ed end.
(
D
) Red-shadowed a ea: ypically,
50
is posi i ely co ela ed wi h
CT
(and wi h
−1
50
) [
29
]. Measu ed
da a we e adap ed wi h pe mission om Sil a e al. [17]. Copy igh 2018 John Wiley and Sons.
Ins ead o using he amyloid ib illa ion model in i s closed o m solu ion, he o iginal di e en ial
equa ion, was sol ed simul aneously wi h he oligome iza ion a e equilib ium, Equa ions (A6)
and (A7) (Appendix A), and hen i ed o he ThT luo escence p og ess cu es (Figu e 3B, blue
lines). Al hough compu a ionally mo e demanding han he app oach ollowed wi h human insulin,
he numbe o deg ees o eedom emains unusually low as ega ds o complex biophysical p oblems:
h ee independen scaling laws o
50
(Figu e 3C),
50
(Figu e 3D) and
FF
(Figu e 3E) a e used o
es ima e no o he unknowns bu he scaling cons an s associa ed o
ka
and
kb
. Unlike he case o
insulin, he alue o a axin-3 solubili y is known be o ehand o be e y low (
C∗≈
0) as e idenced by
alues o monome concen a ion lowe han he de ec ion limi s unde equilib ium condi ions [
17
].
In con as , since he au oca aly ic a e cons an o a axin-3 is de e mined by he seconda y nuclea ion
Biomolecules 2018,8, 108 8 o 20
s ep (
ka≈k2
), a scaling exponen
n2
is now in oduced o accoun o he poo ly unde s ood
k2
s.
∆C ela ionship:
(k2∝ ∆Cn2
kn∝ ∆C2(3)
In p ac ice, di e en i ed pa ame e s a e p o ided by he indi idualized analysis o each
p og ess cu e in Figu e 3B (black lines), whe eas he global i (blue lines) equi es a single se
o a e cons an s
ka
and
kb
o model bo h he agg ega ion assay and i s scaling laws (Figu e 3C–E).
The be e goodness-o - i s a is ics o he o me p ocedu e (Figu e S1) is no su p ising since, as in
he case o o e pa ame e ized p oblems, he indi idual nume ical analysis is no c oss- alida ed
and ends o o e i he expe imen al e o , he e o e, comp omising he model’s p edic i e
powe
[17,30–32]
. The global i ing con i ms ha p e-de e mined oligome iza ion cons an s can
be in eg a ed in agg ega ion eac ion ne wo ks o explain highly peculia kine ics, such as he e y
weak
CT
-dependence o
50
(Figu e 3C), and no ably, he nega i e
CT
-dependence o
50
(Figu e 3D).
Al hough a mo e con en ional esul in he absence o quenching phenomena [
33
,
34
], he linea scaling
law o he end-poin ThT luo escence (Figu e 3E) is explained by he dissocia ion o a axin-3 oligome s
occu ing in he same ime scale as amyloid ib illa ion. The obse ed s aigh line c ossing he o igin
also indica es ha he soluble p o ein was con e ed in o amyloid-like ib ils wi hou he occu ence o
signi ican monome deg ada ion du ing incuba ion [
17
]. The complex, ye sel -consis en beha io s
o hal -li e and end-poin eadings c oss- alida e he molecula -le el implica ions a ising om he
de ini ion o he seconda y nuclea ion a e cons an (
k2
), and pa icula ly, om he ob ained alue o
he scaling exponen
n2
close o 0. A di ec compa ison wi h he ib il elonga ion s ep would sugges
a i s -o de dependence o
k2
on he ini ial supe sa u a ion
∆C
since bo h a es linea ly inc ease
wi h he ins an aneous alues o supe sa u a ion and ib il mass [
10
,
17
]. Howe e , mo e han jus a
collisional a e coe icien ,
k2
is an o e all a e cons an accoun ing o he a e-limi ing s eps leading
o he o ma ion o seconda y nuclei [
35
]. Acco ding o classical nuclea ion heo y [
36
], he nuclea ion
p omo ing e ec elici ed by highe supe sa u a ion le els (and lowe ene ge ic ba ie s o phase
ansi ion) can be, in pa , coun e ac ed by he concomi an dec ease in he c i ical sizes o he p ima y
(
n∗
1
) and seconda y (
n∗
2
) nucleus. This ex a con ibu ion, which is no e iden o p ima y nuclea ion
o amyloid ib ils [
12
], seems ele an o he seconda y nuclea ion o a axin-3. Somewha unde alued
in ega d o induc ion ime measu emen s, hal -li e agg ega ion a es
50
(o , equi alen ly, maximum
agg ega ion a es) o e he oppo uni y o iden i y he p edominan au oca aly ic p ocess. Whils
he scaling o
50
is g ea ly in luenced by p ima y nuclea ion, he scaling o
50
is de e mined by he
balance be ween elonga ion and seconda y nuclea ion a es, wi h he e ec o
CT
ge ing weake
as seconda y nuclea ion becomes mo e impo an . The e o e, and simila ly o wha was concluded
o insulin, he measu ed scaling laws o a axin-3 agg ega ion a e de e mined by he ib illa ion
mechanism i sel and by he p esence o he modynamically s able, soluble agg ega es ha u he
deple e he concen a ion o ee monome in solu ion.
3.3. Model P edic ions A e Fu he Con i med by Size Dis ibu ion Analysis o Insulin Agg ega ion
The p e ious models p esen a de ailed pic u e o he di e en s eps a ec ing he o ma ion o he
insoluble ilamen s ha can be u he es ed using DLS measu emen s o pa icle size dis ibu ions
(PSDs). Con a y o wha was obse ed o a axin-3 [
17
], he size o insulin ib ils ends o inc ease
o e ime un il eaching hyd odynamic adii (
Rh
) abo e he mic ome e scale—Figu e 4A–D (insulin)
and Figu e 4E (a axin-3). This is no su p ising aking in o accoun he TEM images ob ained a he
end o each agg ega ion assay (Figu e 1A,B), and he negligible ole o he elonga ion s ep du ing
a axin-3 ib illa ion. Ano he ob ious di e ence o a axin-3 is he pe sis ing dominance o he le -side
peak (
Rh<
10 nm) up o he end o he agg ega ion eac ion (Figu e 4A–C). To a ce ain ex en ,
his is explained by he alue o p o ein solubili y (
C∗
), which as al eady discussed, is much highe
in he case o human insulin. While he inal concen a ion o soluble a axin-3 was oo low o be
Biomolecules 2018,8, 108 9 o 20
de ec ed by DLS [
17
], he
C∗
alue o insulin is esponsible o he popula ion o soluble p o ein o
con inue p edomina ing, e en a e la ge insoluble agg ega es a e o med (Figu e 4C). Ano he eason
explaining he modes inc ease in he in ensi y o sca e ed ligh o la ge pa icles is associa ed wi h
he dispe sion o sizes and consequen ial b oadening o PSDs p o oked by he con inuous elonga ion
o old and newly- o med insulin ib ils, as opposed o he o ma ion o a axin-3 ilamen s wi h he
cons an dimension cha ac e is ic o he a axin-3 seconda y nucleus. In common wi h a axin-3, ib il
b eakage has a mino ole in de e mining he ime a ia ion o he PSD in quiescen insulin solu ions:
in he case o a axin-3, he shape o hese dis ibu ions did no change signi ican ly du ing he bu s
and pla eau phases o agg ega ion despi e he inc eased ela i e impo ance o he popula ion o
a axin-3 ib ils [
17
]. In he case o insulin, he elonga ion-domina ed mechanism can be disce ned
om he expec ed ib il size inc ease du ing he bu s phase (Figu e 4A,B,D), whe eas, a e
∼
4.5 h
incuba ion, he mean agg ega e size s abilizes a a cons an alue o
Rh≈
1100 nm wi hou any isible
signs o ib il agmen a ion (Figu e 4C,D).
Figu e 4.
Time-cou se DLS analysis o human insulin agg ega ion—di e ences and common aspec s
wi h a axin-3. (
A
–
C
) Symbols connec ed by lines: in ensi y-based size dis ibu ions measu ed a di e en
ime poin s as indica ed by he colo ba in (
A
). La ge symbols: alues o he hyd odynamic adius (
Rh
)
used as es ima es o he mean size (
Rh
) o insulin ib ils. Ve ical dashed lines: isual e e ence o he
i s Rh alue o each panel. (D) Measu ed (symbols) and simula ed (lines) ime e olu ion o Rh.
Biomolecules 2018,8, 108 16 o 20
n∞=
9
×
10
6
monome s was adop ed as he maximum dimension o soluble a axin-3 oligome s,
and he condi ion
dCn/d =
0 was imposed o
n≥n∞
. The ini ial condi ions we e se assuming no
ib illa agg ega es p esen in solu ion (
M(0)=
0) and ha he ac ional composi ions o monome s
and
n
-me s co espond o hose ex ac ed om DLS measu emen s [
17
]. Since he p edominan
au oca aly ic s ep du ing a axin-3 ib illa ion is seconda y nuclea ion,
ka≈k2
and
kb≈kn/k2
. Owing
o he low solubili y o a axin-3 (
C∗≈
0), he ini ial supe sa u a ion is a di ec p opo ion o
CT
and he
CLM pa ame e s a e gi en as
ka=k0
aCn2
T
and
kb=k0
bC2−n2
T
. The p opo ionali y cons an s
k0
2
and
k0
n
,
and he o de -o - eac ion
n2
we e es ima ed by minimizing he absolu e e o be ween p edic ed and
measu ed ThT luo escence (
F
) p og ess cu es using he expe imen ally de e mined calib a ion cu e
F(a.u.)=
0.70
×M(µM)
and he known oligome iza ion a e cons an s
κ1+=
7.99
×
10
−4µ
M
−1
h
−1
,
κ1−=9.73 h−1,κn+=0.167 µM−1h−1, and κn−=0.775 h−1[17].
To simula e he
50
and
50
scaling laws in Figu e 6, Equa ions (A6) and (A7) we e nume ically
sol ed as desc ibed o a axin-3 using illus a i e alues o
C∗=
0,
ka=
0.4
Cn2
T
h
−1
and
kb=
5
×
10
−4(CT/5)2−n2
. Two limi si ua ions co esponding o p edominan elonga ion s ep
(
n2=
1) o p edominan seconda y nuclea ion (
n2=
0) p ocesses we e conside ed. Soluble p o ein
was admi ed o occu ei he as a monome o as a dime (
κn+=
0 and
κn−=
0) wi h he dis ibu ion
o ini ial species gi en by he exempla unc ion
C1=
10
×1−e−CT/10
o
CT
alues comp ised
be ween 0 and 20 in a bi a y concen a ion uni s. The ela i e weigh o he oligome dissocia ion a e
was in es iga ed by changing he alue o
κ1−
be ween 0 and 5 h
−1
while keeping a ixed alue o
κ1+=
0. The no malized amyloid signal (
M/MF
) was compu ed o e ime and he co esponding
hal -li e coo dina es, 50 and 50 we e ep esen ed as a unc ion o CT(Figu e 6).
Appendix B.2. Disc e ized Popula ion Balance
The ime e olu ion o he size dis ibu ion o insulin ib ils was simula ed using he disc e e
popula ion balance de i ed o gene al phase ansi ion p ocesses comp ising he s eps o p ima y
nuclea ion, seconda y nuclea ion, g ow h/elonga ion and b eakage [
12
,
17
]. As p e iously desc ibed
o a axin-3, he concen a ion o ilamen s composed by jmonome s ( j) is gi en as [17]:
d j
d =1
jhkn(1−α)2M∞δj,n∗+k2(1−α)αM∞δj,n∗
2i+
+k+(1−α)(j−1) j−1−k+(1−α)j j+
+1
j"−k− j(j−n∗
2)H(j−2n∗−1)+
∞
∑
i=j+1
k− iH(i−2n∗−1)#(A8)
The K onecke del a unc ions in Equa ion (A8) se he sizes o he p ima y and seconda y nuclei
o ixed alues o
n∗
and
n∗
2
, espec i ely, wi h he la e being adop ed as he smalles possible
ilamen size (
j≥n∗
2
). The Hea iside unc ion es ablishes a minimum ib il size o 2
n∗+
1 molecules
abo e which agmen a ion s a s o occu [
46
]. In he case o insulin, his equa ion is simpli ied
since seconda y nuclea ion and ib il b eakage ake place o a negligible ex en (
k2≈
0 and
k−≈
0).
The p e ious Equa ion (A2) can be ob ained om Equa ion (A8) by ex ending he sum o
j×d j/d
o
all ilamen s [17]:
dM
d =
∞
∑
j=n∗
2
jd j
d (A9)
A popula ion o p e-exis ing clus e s wi h hyd odynamic adii
Rh>
100 nm was iden i ied in
he analyzed insulin solu ions (Figu e S2). The DLS in ensi y-peak co esponding o his popula ion
g adually anished om he measu ed size dis ibu ions as ib il elonga ion ook place (Figu e 4A–D).
To simula e his beha io , a simple pa icle adhesion mechanism is p oposed in which p e-exis ing
clus e s composed by
k
monome s a e conside ed o join o he a ailable
j
-me ib ils and o m
he e ogeneous agglome a es composed by
l=k+j
monome s. Du ing he ini ial phases o he
eac ion, a numbe o p e-exis ing clus e s emains isola ed because he e a e ewe ib ils han clus e s.

Biomolecules 2018,8, 108 17 o 20
This scena io is hen e e sed as newly o med ib ils ou numbe he ini ial clus e s. The e o e, he less
abundan species a e he ones dic a ing he numbe o agg ega es ha will pa icipa e in he adhesion
p ocess (Pa):
Pa=min P=
∞
∑
j
j,Pc(0)=
∞
∑
k
k(0)!(A10)
whe e
P
is he concen a ion o insulin ib ils,
Pc(0)≈
5.5
×
10
−16
M is he o al concen a ion o
clus e s and
k(0)
is he concen a ion o
k
-me clus e s—
Pc(0)
and
k(0)
a e es ima ed om he ini ial
DLS measu emen s. A a gi en ins an , he numbe o
k
-me clus e s joining o
j
-me ib ils is a
unc ion o he ela i e amoun s o each ype o agg ega es,
k→j=Pa× j
P× k(0)
Pc(0), (A11)
meaning ha he concen a ions o isola ed ib ils ( 0
j) and isola ed clus e s ( 0
k) a e
0
j= j−
∞
∑
k
k→j, (A12)
0
k= k(0)−
∞
∑
j
k→j, (A13)
and ha he concen a ion o he e ogeneous agglome a es composed by a o al lmonome s ( 1
l) is
1
l=
∞
∑
k
∞
∑
j
k→jδk+j,l. (A14)
He e, he K onecke del a unc ion is used o limi he sum e ms o he possible adhesion con ac s
p oducing l-me agglome a es. Finally, he concen a ion o all l-me agg ega es ( l) is gi en as:
l= 0
j=l+ 0
k=l+ 1
l, (A15)
which can be used o compu e he heo e ical size dis ibu ions. I ollows om he p oposed one- o-one
adhesion mechanism ha
Pa=
∞
∑
l
1
l, (A16)
and ha he o al numbe o sca e ing pa icles co esponds o he concen a ion o he mos
abundan species: ∞
∑
l
l=max(P,Pc(0)). (A17)
The ollowing addi ional app oxima ions we e adop ed du ing he de i a ion o his model:
(i) ib il elonga ion is una ec ed by he p esence o joined clus e s, (ii) complex adhesion pa hways
a e no conside ed and (iii) he p esence o isola ed ib ils and isola ed clus e s is a necessa y and
su icien condi ion o p oduc i e adhesion. O he mechanisms explaining he obse ed decline o
he clus e popula ion a e admissible, e.g., by admi ing p og essi e clus e dissocia ion. The goal
o desc ibing he expe imen al esul s in Figu e 4A–D could, howe e , be achie ed using p e iously
de e mined alues o
ka
and
kb
(Figu e S3A–C) and adop ing he physical limi o pa icle de ec ion
(
l>
1.6
×
10
−23
M) as he only adjus able pa ame e se o minimize he di e ences be ween he
measu ed DLS da a and he heo e ical size dis ibu ions. Ma hwo ks
®
MATLAB R2013b was used
o nume ically sol ed Equa ion (A8) and Equa ions (A10)–(A15) o a cu -o size o isola ed insulin
ib ils o
n=
1
×
10
13
monome s. The ob ained
l( )
p o iles we e con e ed in o in ensi y-based size
Biomolecules 2018,8, 108 18 o 20
dis ibu ions ollowing he Rayleigh law o ligh sca e ing [
17
]. Impo an ly, he inal size dis ibu ions
we e no signi ican ly a ec ed by he p esence o ini ial clus e s as indica ed by he simula ed esul s
o Pc(0)=0 (Figu e S3D–F).
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