Bioin o ma ics, YYYY, 0–0
doi: 10.1093/bioin o ma ics/xxxxx
Ad ance Access Publica ion Da e: DD Mon h YYYY
Manusc ip Ca ego y
S uc u al Bioin o ma ics
Sphe eCon - A me hod o p ecise es ima ion
o esidue ela i e sol en accessible a ea
om limi ed s uc u al in o ma ion.
Alexande G ess1,2,* and Olga V. Kalinina1,3
1Helmhol z Ins i u e o Pha maceu ical Resea ch Saa land (HIPS), Helmhol z Cen e o In ec ion
Resea ch (HZI), Campus E8.1, 66123 Saa b ücken, Ge many
2G adua e School o Compu e Science, Saa land Uni e si y, 66123 Saa b ücken, Ge many.
3Medical Facul y, Saa land Uni e si y, 66421 Hombu g, Ge many.
*To whom co espondence should be add essed.
Associa e Edi o : XXXXXXX
Recei ed on XXXXX; e ised on XXXXX; accep ed on XXXXX
Abs ac
Mo i a ion: In p o eins, sol en accessibili y o indi idual esidues is a ac o con ibu ing o hei
impo ance o p o ein unc ion and s abili y. Hence one migh wish o calcula e sol en accessibili y
in o de o p edic he impac o mu a ions, hei pa hogenici y, and o o he biomedical applica ions.
A di ec compu a ion o sol en accessibili y is only possible i all a oms o a p o ein h ee-
dimensional s uc u e a e eliably esol ed.
Resul s: We p esen Sphe eCon, a new p ecise measu e ha can es ima e esidue ela i e sol en
accessibili y (RSA) om limi ed da a. The measu e is based on calcula ing he olume o in e sec ion
o a sphe e wi h a cone cu ou in he di ec ion opposi e o he esidue wi h su ounding a oms. We
p opose a me hod o es ima ing he posi ion and olume o esidue a oms in cases when hey a e
no known om he s uc u e, o when he s uc u al da a a e un eliable o missing. We show ha in
cases o eliable inpu s uc u es, Sphe eCon co ela ed almos pe ec ly wi h he di ec ly compu ed
RSA, and ou pe o ms o he p e iously sugges ed indi ec me hod. Mo eo e , Sphe eCon is he only
measu e ha yield accu a e esul s when he iden i ies o amino acids a e unknown. A signi ican
no el ea u e o Sphe eCon is ha i can es ima e RSA om in e - esidue dis ance and con ac
ma ices, wi hou any in o ma ion abou he ac ual a om coo dina es.
A ailabili y: h ps://gi hub.com/kalininalab/sphe econ
Con ac : [email p o ec ed]
Supplemen a y in o ma ion: Supplemen a y da a a e a ailable a Bioin o ma ics online.
1 In oduc ion
P o ein unc ion is in ima ely ela ed o p o ein h ee-dimensional (3D)
s uc u e, which, in u n, is de e mined by p o eins amino acid sequence.
P o ein 3D s uc u es a e mo e conse ed han hei sequences, meaning
ha dis an ly ela ed p o eins, o which homology is di icul o de ec
ia sequence compa ison, s ill can old in o e y simila 3D s uc u es
(Ille gå d e al., 2009). Homology modelling app oaches allow o
compu a ionally p edic 3D s uc u es o p o eins wi h no
expe imen ally esol ed s uc u es a ailable using empla e 3D s uc u es
o p o eins wi h as low as 30-35% sequence iden i y (Ros , 1999).
Howe e , in hese models, he coo dina es o amino acid esidues can be
econs uc ed imp ecisely, pa icula ly o he esidues a he p o ein
su ace (Eisenmenge e al., 1993).
Amino acid esidues loca ed on p o ein su ace and wi hin i s co e play
di e en unc ional oles (Aloy e al., 2001), and a e unde di e en
e olu iona y cons ain s (Bo do and A gos, 1990; Sasidha an and
Cho hia, 2007; Bloom e al., 2006). Fo example, mu a ions o esidues
A.G ess e al.
in p o ein co e can in luence p o ein o e all s abili y (Jackson e al.,
1993; B ockwell e al., 2002; Lim e al., 1992; Bullock e al., 1997) and
in ex eme cases lead o p o ein mis olding and agg ega ion (Xu e al.,
2011; an de Kamp and Dagge , 2010), whe eas mu a ions o esidues
on he p o ein su ace may in luence he s abili y and speci ici y o
p o ein in e ac ions (Yi e al., 2017). I he mu a ed esidue is no a pa
o an in e ac ion in e ace o binding pocke , and he mu a ion does no
in oduce an un a ou able hyd ophobic esidue a he su ace, i is likely
o be o no consequence. In line wi h his, i has been epo ed ha
pa hogenic mu a ions can ha e a des abilizing e ec on p o ein s uc u e,
a ec con o ma ional dynamics o hyd ogen bond ne wo ks (Kucukkal
e al., 2015; S e l e al., 2013) and a e en iched in p o ein co e and
deple ed om hei su ace (Wang and Moul , 2001; G ess e al., 2017).
Thus he ela i e posi ion o esidues wi h espec o p o ein co e a e
p edic i e o he unc ional impo ance o hese esidues and
pa hogenici y o mu a ing hem.
The sol en accessible a ea (SA) is a measu e o he ex en o which a
esidue lies on p o ein su ace o in i s co e. Fo an expe imen ally
esol ed p o ein 3D s uc u e, i can be compu ed by olling a p obe
sphe e o e he an de Waals su ace o he p o ein (Sanne e al.,
1996). The su ace aced by he cen e o he p obe sphe e is hen called
he p o ein accessible su ace. The whole accessible su ace o a p o ein
can be di ided wi h espec o he con ibu ions o each esidue, and
hese con ibu ions a e called accessible su ace o indi idual esidues.
To compu e he a ea o a esidue accessible su ace (SA) di ec ly, he
coo dina es o all hea y a oms mus be known.
SA alone gi es limi ed in o ma ion, since esidues o di e en ypes a y
in size conside ably. The ela i e sol en accessible a ea (RSA) is he
esidue SA di ided by a esidue maximal SA, which is cons an o all
amino acid esidues o he same ype. RSA p o ides a easonable way o
compa ing di e en esidues wi h espec o hei posi ion ela i e o he
p o ein su ace. Gi en a ull-a om s uc u e o a p o ein, i can be
de e minis ically compu ed by di e en ools (Kabsch and Sande ,
1983). Howe e , he compu a ion ails i coo dina es o some a oms a e
missing, e.g. in cases when he posi ion o some esidues has no been
expe imen ally esol ed. RSA can also be compu ed o homology
models using he same ools, bu his in oduces ano he sou ce o e o :
i coo dina es o some esidues ha e been econs uc ed inco ec ly,
which can happen in pa icula o p o ein su ace esidues, he di ec
calcula ion o RSA can be inco ec .
Fo such cases, o he measu es o es ima ing SA and RSA om limi ed
in o ma ion ha e been de eloped. When only he coo dina es o he Cα
a oms o he p o ein a e known, which is o en he case o s uc u es o
low esolu ion, and is also in an in e media e s ep o p o ein s uc u e
modeling pipelines (Fasnach e al., 2007; Gadka i e al., 2009),
specialized su ace measu es can be used. One o he i s such measu es
was he coo dina ion numbe (CN) (Simons e al., 1997) ha is based on
coun ing he numbe o Cα a oms in a sphe e a ound he Cα a om o he
esidue in ques ion. Fo esidues on he p o ein su ace, one expec s a
hal o such a sphe e o be less popula ed compa ed o esidues lying in
he p o ein co e, hus CN nega i ely co ela es wi h RSA. While
measu ing CN can sa is ac o ily di e en ia e be ween esidues
p edominan ly exposed o he sol en and esidues comple ely bu ied in
he p o ein co e, i lacks he abili y o ecognize ine di e ences o he
deg ee o sol en exposu e among pa ially exposed esidues.
To mi iga e his p oblem, he hal -sphe e exposu e (HSE) me hod
(Hamel yck, 2005) was de eloped. In his app oach, he sphe e used o
de e mining he CN is cu in o wo hal sphe es by a plane o hogonal o
he Cα-Cβ ec o o he esidue in ques ion. This esul s in wo new
measu es: he alues o CN o he lowe and uppe hal -sphe es, whe e
uppe hal -sphe e co esponds o he di ec ion owa ds he Cβ a om.
Compa ing he alues o CN o he uppe and bo om hal -sphe es o
s uc u es, o which RSA can be compu ed di ec ly om he all-a om
coo dina es, p o es ha RSA can be in e ed om he CN alues
co esponding o he uppe hal -sphe e, whe eas he CN alues o he
lowe hal -sphe e a e no co ela ed o i . Thus he CN alue o he uppe
hal -sphe e, which is called he hal -sphe e exposu e (HSE), can be used
as a p oxy o RSA. Howe e , o compu e i , one needs o know no only
he coo dina es o he Cα a oms, bu also he coo dina es o he Cβ
a oms. These la e coo dina es can be es ima ed using he posi ions o
he Cα a oms o he neighbo ing esidues. The co ela ion be ween HSE
and he di ec ly compu ed RSA om all-a om s uc u es eaches 82%,
which is enough o di e en ia e be ween bu ied esidues and
p edominan ly exposed esidues. Fo pa ially exposed esidues, HSE
pe o ms be e han CN, bu s ill is no pe ec .
In his s udy, we p esen a no el app oach o assess he RSA om
limi ed da a ha ou pe o ms all o he ools o he case when only he
coo dina es o Cα a oms a e a ailable. The ypical use case o ou
measu e is when he p o ein sequence is known, howe e , we also
p esen a a ian , whe e his in o ma ion is also missing. Simila ly o CN
and HSE, we assume ha he mo e space a ound a esidue in ques ion is
occupied by o he esidues and he mo e e enly i is occupied by hem,
he less i is exposed o he sol en . The main no el y o ou me hod is
no o ep esen esidues as poin s o sphe es in he 3D space
co esponding o he coo dina es o hei Cα a oms, bu as space- illing
bodies, which di e o di e en amino acid ypes. Each esidue is
ep esen ed by a sphe e, he adius o which is amino acid ype-speci ic.
A sea ch space is de ined as a sphe e wi h a cu -ou cone placed a ound
he esidue, and he in e sec ions be ween he sea ch space and he
sphe es co esponding o he neighbo ing esidues a e calcula ed. In his
way, neighbo ing esidues in luence he es ima ed RSA mo e han he
dis an ones. Addi ionally, we in oduce di e en ia ion be ween amino
acid esidues o di e en size. We demons a e ha hese ea u es make
Sphe eCon mo e accu a e han ea lie app oaches. In e es ingly, e en
when he esidue-speci ic adii a e igno ed, and all esidues a e
ep esen ed by sphe es o he same size and only he coo dina es o Cα
a om a e used (which co esponds o he case when p o ein sequence is
unknown), Sphe eCon s ill pe o ms be e han o he ools, all o which
use addi ional in o ma ion.
Pe haps he mos in e es ing pa o Sphe eCon is a no el simple me hod
o es ima ing RSA om da a on esidue- esidue con ac s, wi hou
knowing hei 3D coo dina es. This makes ou me hod po en ially use ul
in no el de no o modelling schemes, whe e in e - esidue con ac s a e
p edic ed om mul iple sequence alignmen s a is ics (Wang e al., 2017;
Seemaye e al., 2014; Zhang e al., 2016; Adhika i e al., 2018;
O chinniko e al.; Jones e al., 2015; Tegge e al., 2009; Eickhol and
Cheng, 2012). Ano he ool in he ield ha can p edic RSA om
p edic ed esidue con ac s, SPOT-1D (Hanson e al., 2019), employs
deep lea ning. We show ha e en wi h a simple geome ic me hod using
e en his limi ed in o ma ion, we a e able o econs uc he RSA alues
wi h a posi i e co ela ion. We apply ou me hods o he mos ecen
CASP a ge s and show ha e en o his mos challenging da ase we
achie e easonable esul s.
2 Me hods
We de eloped ou al e na i e measu es ha use he same p inciple, bu
a e designed o di e en ypes o inpu in o ma ion. The key s eps o
he algo i hm a e he ollowing: de ine a sea ch space a ound a gi en
esidue using a sphe e o a la ge diame e , possibly wi h a cu -ou cone;
Sphe eCon
de ine in e sec ing olumes o each su ounding esidue colliding wi h
he sea ch space; and calcula e he unoccupied ac ion o he sea ch
space, esul ing in a alue be ween 0 and 1.
The sea ch space (see Fig.1) is de ined as a sphe e o adius (pa ame e
o he op imiza ion p oblem, see below) and a cone cu ou om he
sphe e. The sea ch sphe e is cen e ed a he cen oid o all he side-chain
a oms o a gi en esidue. The apex o he cone coincides wi h he cen e
o he sphe e and he cone axis is collinea o he cen oid- o-Cα ec o
C. In ou main applica ion scena io, we assume ha he coo dina es o
he side-chain a oms a e unknown, so we also de eloped a me hod o
es ima ing he cen oids coo dina es (see below). The apex angle a
egula es he olume cu ou om he sea ch sphe e. The case when he
apex angle is ze o co esponds o he ull sphe e as he sea ch space, he
apex angle equal o 180° amoun s o he same sea ch space as in he HSE
me hod.
Figu e 1: Schema ic ep esen a ion o he design o he sea ch space.
Thus he sea ch space is de ined by h ee pa ame e s: he posi ion o he
sphe e cen e ; he adius o he sea ch sphe e; and he apex angle o he
cone. We desc ibe h ee scena ios based on di e en amoun o
s uc u al in o ma ion a ailable. In he di e en scena ios we ha e o
employ di e en echniques o es ima e he pa ame e s necessa y o
de ine he sea ch space.
3.1 Scena io 1 (SC-S1): All a om coo dina es a ailable
The i s scena io co esponds o he case, when he coo dina es o all
a oms in a 3D s uc u e o he p o ein a e a ailable, and hence he RSA
can be calcula ed di ec ly. The cen e o he sea ch sphe e is placed a he
cen oid o he coo dina es o he side chain a oms o he esidue in
ques ion. The sea ch sphe e adius and he cosine o he apex angle o
he cu -ou cone a e esidue-speci ic cons an s ha yield he bes
co ela ion wi h RSA o a es da ase , de e mined by a g id sea ch. The
hea y a oms o all o he esidues a e ep esen ed by hei an de Waals
sphe es, and he in e sec ion wi h he sea ch space is calcula ed o hem.
The sum o all in e sec ing olumes is called o al occupied olume.
Sphe eCon is de ined as he a io o he olume o he sea ch space and
he o al occupied olume. Since o his scena io he RSA can be
di ec ly calcula ed, Sphe eCon has he e no p ac ical use case, bu he
esul s o Sphe eCon in his scena io a e impo o e alua ing he
me hod.
3.2 Scena io 2 (SC-S2): Coo dina es o Cα a oms a e a ail-
able and p o ein sequence is known
In he second scena io o each esidue only he coo dina es o he Cα
a om and he amino acid ype a e known. Since he coo dina es o he
side chain a oms a e no a ailable now, he cen oid has o be p edic ed.
As in SC-S1, he adius o he sea ch sphe e and he cone apex angle a e
de e mined based on a g id sea ch. The a om-speci ic sphe es
co esponding o he a om an de Waals adii om SC-S1 a e eplaced
by la ge esidue speci ic sphe es cen e ed a hei p edic ed cen oids.
The adius is chosen, such ha he olume o he sphe e is equal o he
sum o he an de Waals sphe e olumes o he hea y a oms o he
speci ic amino acid. The adii o he in e sec ing esidue sphe es ange
om 2.57 Å o glycine o 4.02 Å o yp ophan (Supplemen a y Table
S1).
3.3 Scena io 3 (SC-S3): Only coo dina es o Cα a oms a e
a ailable
The leas amoun o in o ma ion, o which he RSA es ima ion by
Sphe eCon is possible, a e he 3D coo dina es o he p o ein Cα a oms
wi hou he in o ma ion on wha amino acid esidues hey co espond o.
Simila ly o SC-S2, we ha e o p edic he cen oids o all esidues. The
sea ch sphe e adii canno be esidue speci ic anymo e, since he ype o
amino acid a each posi ion is no known. Again, a g id sea ch is used o
ind he sea ch sphe e adius and he cosine o he apex angle o he
cu ing cone wi h he bes pe o mance in a esidue-obli ious manne ,
i.e. he same alue se o all posi ions. The in e sec ing olumes a e
sphe es cen e ed a ound all o he p edic ed cen oids wi h adius 3.23 Å
(a adius ha co esponds o he a e age olume o he wen y na u al
amino acids weigh ed by hei equency in ou aining se , see below).
3.4 Scena io 4 (SC-S4): Only p edic ion o a dis ance ma ix
o con ac ma ix is a ailable
In his scena io we depa comple ely om 3D coo dina es o any a oms
and use he ou pu o sequence-based esidue- esidue con ac p edic ion
me hods as inpu . In pa icula , we use pai wise con ac s o dis ances
p edic ed by Rap o X (Wang e al., 2016) o a simila ool. The
in o ma ion on he iden i y o amino acids in p o ein sequence is
e ained, and hus he di e en adii o he sea ch sphe es can be
op imized by he g id sea ch p ocedu e. Since he coo dina es o he
backbone a e no known, p edic ion o he cen oid coo dina es
(o ien a ion o he side chain) is no possible and he di ec ion o he
cu ing cone canno be es ima ed. Ins ead we use sphe es in his scena io.
Since he dis ances o o he esidues as well as iden i y o hose esidues
a e assumed o be known in his scena io, we can ell, which esidues a e
wi hin he sea ch sphe e o in e sec wi h i , and calcula e he in e sec ion
o he sea ch sphe e wi h esidue-speci ic sphe es co esponding o hese
esidues. A e hese in e sec ion olumes a e calcula ed, Sphe eCon can
be calcula ed as in o he scena ios. Fo con ac ma ices, we jus
conside all esidues p edic ed o be in con ac o lie a a dis ance o 8 Å,
and all o he esidues so a away ha hey do no in e sec wi h he
sea ch sphe e. As, he p edic ed dis ance ma ices used as inpu may be
spa se and con ain e o s, we calcula e he spa si y o he inpu ma ix
and use a se o pa ame e s (sphe e adii) op imized o his pa icula
spa si y.
3.5 T aining da ase s
We c ea ed a gold s anda d se o p o ein 3D s uc u es by choosing a
ep esen a i e s uc u e om each SCOP (Mu zin e al., 1995) amily
(only SCOP classes a, b, c and d) wi h all he hea y a oms esol ed
(Supplemen a y Table S2). Fo es ing, we pe o med a ou - old c oss
alida ion, whe e we selec ed one SCOP class as he es se and used he
A.G ess e al.
o he h ee o aining (Supplemen a y Table S2). This c oss- alida ion
scena io is he mos s ingen in he sense ha aining and es ing a e
pe o med o p o eins wi h d ama ically di e en opological p ope ies.
Fo op imizing pa ame e s in SC-S4, we used dis ance ma ices
calcula ed om he 3D s uc u es and emo ed all dis ances be ween he
esidues ha a e sepa a ed by less han h ee esidues in he sequence, as
hese pai s a e ypically no conside ed by con ac p edic ion ools. Then
we inc easingly emo ed andom dis ances om he ma ix in 10% bins.
Beside he desc ibed abo e c oss- alida ion, o es ou me hod in SC-
S4, we used dis ance ma ices p edic ed o he sequences o 13 a ge s
om CASP13 (K ysh a o ych e al., 2019), o which a 3D s uc u e was
a ailable in he PDB o he ime o his w i ing, using he Rap o X
(Wang e al., 2016) webse e .
3.6 Cen oid p edic ion
Fo all s uc u es in he gold s anda d da ase , we compu e he Cα-
cen oid ec o o each esidue. Then o each amino acid ype we
calcula e he a e age o ien a ion o hese ec o s o he plane spanned by
he C-alpha coo dina es o he esidue and i s wo neighbo esidues.
These amino acid ype speci ic a e age Cα-cen oid ec o s can now be
used o p edic he cen oid o esidues wi h missing coo dina es by
placing hem on he espec i e planes. Fo scena io 3, one a e age ec o
o all amino acid ypes in he gold s anda d da ase was used.
3.7 Op imiza ion
The goal o he op imiza ion s ep is o de ine he op imal pa ame e s in
o de o maximize he Pea son's co ela ion be ween Sphe eCon and
RSA. In o de o ind he op imal sea ch space pa ame e s, i.e. he adius
o he sea ch sphe e and he apex angle o he cone, we desc ibe he
ollowing op imiza ion p oblem:
❑❑❑❑❑❑
(
(
()
)
)
,
whe e is sea ch sphe e adius, a is he cosine o he apex angle, co is
he Pea son’s co ela ion. Then *, a* a e he op imal sea ch space
pa ame e s. We pe o med a wo-dimensional g id sea ch in he space o
he adii alues in he in e al [4 Å, 20 Å] wi h he s eps o 0.5 Å and he
cosine o he apex angle in he in e al [0.5, 1] wi h he s eps o 0.05. Fo
each esidue o all s uc u es in he aining da ase , he RSA and he
Sphe eCon o all combina ions o he wo pa ame e s we e compu ed,
sepa a ely o each scena io.
3 Resul s
3.1 Op imiza ion
The op imal pa ame e s o all scena ios we e ound in he g id sea ch
(Supplemen a y Table S3). The adii o he sea ch sphe es co ela e wi h
he size o he amino acid, anging om 6.5 Å o 8.0 Å o SC-S1 and
om 6.75 Å o 9.0 Å o SC-S2. In SC-S1 sha p cu ing cone angles
we e p e e ed o mos o he esidues, while o SC-S2 hal o he
esidues he p e e ence is using no cu ing cone (cos(a*) = 1.0). The e is
a end ha la ge esidues a o wide angles, in pa icula yp ophan
co esponds o an apex angle wi h a cosine o -0.6. In SC-S4 only one
pa ame e is subjec o he op imiza ion, yielding * = 7.5 Å. A po ion
o he co esponding hea map ha ep esen s he co ela ion o all
s uc u es in he es se can be seen in Supplemen a y Figu e S1. The
maximum co ela ion ha could be achie ed is 0.893. The change o he
co ela ion alue is small be ween he neighbo ing pa ame e alues, and
hus ha ou op imiza ion landscape is smoo h and well-de ined.
3.2 C oss- alida ion
When compa ing he op imal sea ch pa ame e s lea ned om he c oss
alida ion se up o he op imal pa ame e s lea ned on he whole da ase
(Supplemen a y Table S3), we no ice only sligh di e ences, which
e idences he s abili y o he lea ned pa ame e s. Pe o mance o
Sphe eCon is e y li le in luenced by he exac aining and es ing se up
(Table 1). When compa ed o he o e all pe o mance o Sphe eCon
lis ed in Table 2, one no es ha hey a e iden ical and hence he e a e no
signs o any o e aining e ec s.
Table 1: Each cell con ains he Pea son’s co ela ion be ween
Sphe eCon and RSA o he di e en scena ios and di e en es se
se ups.
SCOP
class A
SCOP
class B
SCOP
class C
SCOP
class D
SC-S1 0.95 0.94 0.95 0.95
SC-S2 0.91 0.92 0.91 0.91
SC-S3 0.89 0.89 0.89 0.89
SC-S4 0.87 0.87 0.87 0.88
3.3 Benchma k
We benchma ked Sphe eCon agains CN (Simons e al., 1997) and HSE
(Hamel yck, 2005), which we e-implemen ed wi h he o iginal
pa ame e s (Table 2). We we e able o ep oduce he same alues o
Pea son's co ela ion be ween he CN, HSE and RSA measu es as
epo ed in (Hamel yck, 2005) in spi e o he ac ha we use a da ase
di e en om he one in he o iginal publica ion. This indica es ha
hese measu es a e obus wi h espec o he es da ase . Sphe eCon
co ela es well wi h bo h CN and HSE in all h ee scena ios. As
expec ed, he mo e inpu in o ma ion is used, he be e is he co ela ion
wi h he RSA. E en in he wo s -case scena io, when he sequence o
esidues in he backbone is unknown, we achie e a clea imp o emen
o e HSE, which uses in o ma ion no only on he posi ion o Cα bu
also on he posi ion o he esidue’s Cβ a oms.
Table 2: Pea son's co ela ion coe icien s be ween all es ed measu es
o he ull da ase .
RSA CN HSE SC-S4 SC-S3 SC-S2 SC-S1
RSA n/a -0.770 -0.823 0.878 0.890 0.916 0.945
CN n/a 0.817 -0.795 -0.866 -0.830 -0.778
HSE n/a -0.819 -0.882 -0.860 -0.835
SC-S4 n/a 0.899 0.949 0.903
Sphe eCon
SC-S3 n/a 0.949 0.907
SC-S2 n/a 0.944
SC-S1 n/a
Ano he quali y c i e ion o a measu e o es ima ing he RSA is how
well can i be used in o de o dis inguish be ween su ace and co e
esidues. We classi ied each esidue in he da ase as su ace o co e
based on an RSA h eshold o 16% (Ros , 1997), using CN, HSE and
Sphe eCon as p edic o s (Figu e 2). Wi h an AUC o 0.943, HSE was
al eady an excellen su ace/co e p edic ion measu e, bu Sphe eCon
pe o med e en be e , achie ing he AUC o 0.963 e en in he mos
di icul scena io SC-S3, in which much less in o ma ion is used as in
HSE. The balance poin yields he h eshold o 0.27, which esul s in
only 23041 mis-classi ica ions o e he o al o 211539 esidues (Figu e
3).
Figu e 2: ROC cu es o all measu es based on he bina y class
assignmen (su ace/co e) done by RSA.
Figu e 3: Sca e plo s o di e en measu es agains RSA, ed lines
indica e he classi ica ion h eshold and he balance poin o di e en
me hods.
Fu he , we analyzed how s able Sphe eCon is in ega ds o di e en
seconda y s uc u e elemen s (assigned wi h DSSP (Kabsch and Sande ,
1983), Supplemen a y Table 4). In e es ingly, in all scena ios he
pe o mance was be e o α-helices and coiled egions compa ed o β-
shee s.
3.4 Spa se dis ance and con ac ma ices in SC-S4
Since he use case o SC-S4 is undamen ally di e en om o he
scena ios, , we modi ied he c oss- alida ion se up. Fo he es se
s uc u es, we calcula ed he dis ance ma ices and andomly emo ed
en ies o non-neighbo ing esidues c ea ing a spa se dis ance ma ix. In
hese spa se ma ices, be ween 50% and 100% o all con ac s we e
e ained. In his way, we simula e he po en ial missing alues in he
con ac o dis ance ma ices de i ed om co ela ions in he mul iple
sequence alignmen s. As expec ed, he pe o mance d ops wi h he
inc ease o spa si y (Table 3), bu e en wi h a dis ance ma ix con aining
only 70% o all ue en ies, Sphe eCon pe o ms as well as HSE wi h
ull in o ma ion.
Table 3: Pea son’s co ela ion o Sphe eCon and RSA o SC-S4.
Spa si y (% o dis-
ances emo ed)
Tes se ( aining se consis s o he o he
h ee SCOP classes)
SCOP
class A
SCOP
class B
SCOP
class C
SCOP
class D
10% 0.83 0.83 0.83 0.83
20% 0.82 0.82 0.82 0.83
30% 0.80 0.80 0.80 0.81
40% 0.75 0.76 0.74 0.76
50% 0.77 0.77 0.76 0.78
60% 0.73 0.74 0.72 0.74
70% 0.75 0.75 0.73 0.75
80% 0.72 0.73 0.71 0.73
90% 0.71 0.71 0.70 0.72
HSE ( ull coo dina e
in o ma ion) -0.82 -0.81 -0.82 -0.81
Al hough he communi y now is expe iencing an ad en o me hods ha
a e capable o p edic ing in e - esidue con ac s om sequence da a (Xu
and Wang, 2019), me hods ha only p edic con ac s, de ined as wo
esidues sepa a ed by a dis ance below a ce ain h eshold ( ypically 8 Å)
a e s ill mo e common (Ma ks e al., 2011; Jones e al., 2015; Seemaye
e al., 2014). Hence we modi ied SC-S4 o use his kind o da a. In his,
we con e ed he dis ance ma ices in con ac ma ices by d opping all
dis ance la ge han 8 Å and se ing he es o 5 Å. Then we modi ied he
esul ing con ac ma ices by andom emo ing o adding up o 50% o
spu ious con ac s o mimic he e o s ha he p edic ions migh en ail.

A.G ess e al.
The co ela ion wi h he ue compu ed RSA is s ill ema kably high
(Supplemen a y Table S4), and d ops slowly as mo e andom con ac s
a e added o emo ed. Fo 10% andom con ac s, he quali y o ou
p edic ion is s ill be e han ha o CN, which has he ull a omic
coo dina es.
In a simila app oach SPOT-1D (Hanson e al., 2019) deep lea ning is
used o p edic RSA om p edic ed con ac maps, ou pe o ming
Sphe eCon ( he epo ed co ela ion o SPOT-1D o ue RSA alues is
be ween 0.79 and 0.82 depending on he es se ). Compa ing Tables 3
and 4, one can see ha he pe o mance o Sphe eCon can be much
imp o ed by using speci ic dis ance alues ins ead o con ac s, which
may be also he case o SPOT-1D.
3.5 Tes ing Sphe eCon on p edic ed dis ance ma ices
Finally, we ha e applied Sphe eCon o dis ance ma ices p edic ed by
Rap o X (Wang e al., 2016) o he a ge sequences om he las CASP
ound, which ep esen p obably a da ase mos challenging one can
imagine. The spa si y o he p edic ed ma ices a ies conside ably,
because o a p o ein o leng h L he dis ance ma ix has a size L×L, and
only L, L/5, o L/10 op-quali y dis ances a e usually epo ed. So o
longe sequences he dis ance ma ices a e spa se . We compa ed he
p edic ed Sphe eCon alues o he ac ual RSA om he expe imen ally
esol ed s uc u es (Table 4). The co ela ion be ween Sphe eCon and
RSA a ies be ween 0.404 and 0.771, whe eas he spa si y o he
dis ance ma ices anges be ween 60% and o e 90%. Thus, he
co ela ion is ypically lowe han in ou benchma k and, unlike he
benchma k, he e is no appa en co ela ion be ween he ma ix spa si y
and he Sphe eCon p edic ion quali y.
Table 4: Pea son’s co ela ion o Sphe eCon and RSA o CASP a ge s.
Ta ge PDB
ID
Sequence
leng h
Spa -
si y
Pea son’s
co ela ion o RSA
6EK4 chain A 342 92.3% 0.600
6F45 chain A 68 70.5% 0.484
5W9F chain A 72 58.8% 0.460
6CP8 chain A 157 78.8% 0.740
6BTC chain A 84 67.1% 0.771
6CP9 chain A 116 73.2% 0.742
6CP9 chain B 114 75.1% 0.706
6CCI chain A 354 93.2% 0.631
6G57 chain A 97 66.0% 0.404
6GNX chain A 98 72.3% 0.729
6D7Y chain A 89 69.2% 0.620
6Q64 chain A 319 90.6% 0.552
6MSP chain A 80 59.7% 0.719
Thus wi h Sphe eCon we can o e an al e na i e ool o p edic ing RSA
om p edic ed con ac /dis ance ma ices, ha can be used, o example,
in he pipelines o p o ein 3D s uc u e p edic ion ha a e based on
e olu iona y coupling da a.
4 Discussion
In his s udy, we p esen Sphe eCon, a new and mo e p ecise me hod o
es ima ing esidue ela i e sol en accessibili y. We o e di e en usage
scena ios, based on how much in o ma ion abou he s uc u e o he
p o ein o in e es is a ailable, om known coo dina es o all a oms (SC-
S1) o only p edic ed con ac s and no coo dina e in o ma ion (SC-S4).
SC-S1 has no much p ac ical use, since RSA can be compu ed di ec ly
om h a om coo dina es, bu i se es o compa ison and shows ha
Sphe eCon co ela es wi h he ue RSA alues be e han o he ools in
he ield (co ela ion coe icien 0.945). The balance poin (0.15) o he
bina y classi ica ion in SC-S1 is also nea ly iden ical o he h eshold
(0.16) commonly used o dis inguish be ween su ace and bu ied
esidues (Ros , 1996). Addi ionally, Sphe eCon is much less
compu a ionally expensi e, since he e is no need o un cos ly p ocedu e
o p obe olling o e he whole p o ein su ace, and can be ecommended
when only RSA o one o ew esidues is o in e es . This migh no be
o impo ance when a ew p o eins a e analyzed, bu may become an
issue o la ge-scale s udies.
The amoun o in o ma ion a ailable in SC-S2 and SC-S3 is compa able
o HSE, and hus po en ially hey sha e he use cases in p ac ice wi h his
measu e. HSE es ima es a e mos ly used as ea u e in a ious p edic ions
me hods (Jam oz e al., 2012; Wang e al., 2012; Zheng e al., 2012;
Magnan and Baldi, 2014; Liu e al., 2018; Sanchez-Ga cia e al., 2019;
Sha ma e al., 2019). He e, subs i u ing HSE wi h Sphe eCon could
esul in imp o emed pe o mance, when Sphe eCon es ima es a e close
o he ue RSA alues. Fo example, in he 3D s uc u e o he
phospho ibosyl-AMP cyclohyd olase (PDB id 1ZPS), Gly105 is
comple ely bu ied (RSA = 0), and s ill HSE is equal 5 (Figu e 4). On he
o he hand, he Sphe eCon es ima e o i is below 0. In his case, HSE
migh ha e go diso ien ed by he posi ion o he amino acid in a bulge
on he p o ein su ace, close o i , bu no exac ly exposed o he sol en .
Gene ally, Sphe eCon is likely o be e ea he complex case o
glycine, which lacks he Cβ esidue, since HSE elies on i s posi ion o
compu a ion.
Sphe eCon
Figu e 4: Glycine 105 (magen a) o Chain B (g een) in he s uc u e
wi h he PDB-Id 1ZPS; he su ace accessible a ea in anspa en g ey.
Using an app oxima e measu e can be a be e choice han di ec
compu a ion o RSA also on he case when he coo dina es o side chain
a oms a e poo ly esol ed o modeled wi h a isk o in oducing e o s.
In SC-S3, no in o ma ion on he ype o amino acid a each posi ion is
assumed. This can be use ul i expe imen ally a ailable s uc u es ha e
e y poo esolu ion, o when he s uc u e was modeled using empla es
wi h low sequence iden i y o de no o. HSE as well o e s a possibili y
o compu e he Cα-Cβ ec o in a esidue-obli ious manne , bu he
epo ed co ela ion o i is lowe han o he implemen a ion ha is
awa e o amino acid ypes (Hamel yck, 2005). Sphe eCon in SC-S3, on
he o he hand, pe o ms be e han amino acid ype-awa e HSE
implemen a ion.
A no el ea u e o Sphe eCon is he abili y o p edic RSA om esidue
dis ance o con ac maps, ha is, wi hou any coo dina e in o ma ion a
all. Such maps a e ou inely gene a ed hese days in he pipelines ha
a emp o p edic p o ein 3D s uc u e de no o using da a on
e olu iona y couplings (Wang e al., 2017; O chinniko e al., 2015;
Ma ks e al., 2011). Ano he ing edien o hese pipelines is p edic ion o
RSA, o en using deep o shallow neu al ne wo ks (Magnan and Baldi,
2014; Fa aggi e al., 2012; Ros , 1996; Wang e al., 2016). The accu acy
o con ac /dis ance p edic ion in hese app oaches is limi ed (Wang e al.,
2016). Using Sphe eCon allows o econs uc RSA wi h high accu acy,
compa able o hose o s uc u e-based ools, e en o noisy da a whe e
up o hal o ue con ac s a e missing o up o hi d o p edic ed con ac s
a e spu ious. Thus Sphe eCon can na u ally i in he pipelines p edic ing
h ee-dimensional s uc u e om sequence and po en ially p o ide ye
ano he bi o pe o mance imp o emen o hem.
Con lic o In e es : none decla ed.
Acknowledgmen s
We wan o o e ou hanks o ui ul discussions and c i ical eading
o he manusc ip o Sebas ian Kelle , Sanjay kuma S ikakulam and
Fawaz Dabbaghie.
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