Con en s lis s a ailable a ScienceDi ec
Compu e s in Biology and Medicine
jou nal homepage: www.else ie .com/loca e/compbiomed
Classical Simula ions on Quan um Compu e s: In e ace-D i en Pep ide
Folding on Simula ed Memb ane Su aces
Daniel Conde-To esa,b, Ma iamo Mussa-Juanec, Daniel Faíldec, And és Gómezc,
Rebeca Ga cía-Fandiñob,∗, Ángel Piñei oa,∗∗
aDepa amen o de Física Aplicada, Facul ade de Física, Uni e sidade de San iago de Compos ela, Campus Vida, San iago de Compos ela, E-15782, A
Co uña, Spain
bO ganic Chemis y Depa men , Cen o Singula de In es igación en Química Biolóxica e Ma e iais Molecula es (CiQUS), Uni e sidade de San iago de
Compos ela, Campus Vida, San iago de Compos ela, E-15782, A Co uña, Spain
cGalicia Supe compu ing Cen e (CESGA), A enida de Vigo, s/n, San iago de Compos ela, E-15782, A Co uña, Spain
ARTICLE INFO
Keywo ds:
An imic obial pep ides
Quan um compu ing
Lipid memb anes
In e ace
Pep ide olding
ABSTRACT
Backg ound: An imic obial pep ides (AMPs) a e c ucial in he igh agains in ec ions and play signi ican
oles in a ious heal h con ex s, including cance , au oimmune diseases, and aging. A key aspec o AMP
unc ionali y is hei selec i e in e ac ion wi h pa hogen memb anes, which o en exhibi al e ed lipid
composi ions. These in e ac ions a e hough o induce a con o ma ional shi in AMPs om andom coil
o alpha-helical s uc u es, essen ial o hei ly ic ac i i y. T adi ional compu a ional app oaches ha e aced
challenges in accu a ely modeling hese s uc u al changes, especially in memb ane en i onmen s, he eby
opening and oppo uni y o mo e ad anced app oaches.
Me hod: This s udy ex ends an exis ing quan um compu ing algo i hm, ini ially designed o pep ide olding
simula ions in homogeneous en i onmen s, o add ess he complexi ies o AMP in e ac ions a in e aces. Ou
app oach enables he p edic ion o he op imal con o ma ion o pep ides loca ed in he ansi ion egion
be ween hyd ophilic and hyd ophobic phases, esembling lipid memb anes. The new me hod was es ed on
h ee 10-amino-acid-long pep ides, each cha ac e ized by dis inc hyd ophobic, hyd ophilic, o amphipa hic
p ope ies, ac oss di e en media and a in e aces be ween sol en s o di e en pola i y.
Resul s: The de eloped me hod success ully modeled he s uc u e o he pep ides wi hou inc easing he num-
be o qubi s equi ed compa ed o simula ions in homogeneous media, making i mo e easible wi h cu en
quan um compu ing esou ces. Despi e he cu en limi a ions in compu a ional powe and qubi a ailabili y,
he indings demons a e he signi ican po en ial o quan um compu ing in accu a ely cha ac e izing complex
biomolecula p ocesses, pa icula ly AMP olding a memb ane models.
Conclusions: This esea ch highligh s he p omising applica ions o quan um compu ing in biomolecula
simula ions, pa ing he way o u u e ad ancemen s in he de elopmen o no el he apeu ic agen s. We
aim o o e a new pe spec i e on enhancing he accu acy and applicabili y o biomolecula simula ions in he
con ex o AMP in e ac ions wi h memb ane models.
1. In oduc ion
An imic obial pep ides (AMPs) a e c i ical componen s o he inna e
immune sys em p esen in all li ing o ganisms [1]. These pep ides
ha e been p ima ily associa ed wi h a de ensi e ole agains exoge-
nous in ec ions caused by bac e ia, i uses, and ungi, and hey a e
conside ed powe ul and e sa ile endogenous an ibio ics, capable o
esis ing bac e ial adap a ion o millions o yea s. Howe e , ecen e-
sea ch ad ances ha e poin ed o he link be ween AMPs and a b oade
∗Co esponding au ho .
∗∗ Co esponding au ho .
E-mail add esses: [email p o ec ed] (R. Ga cía-Fandiño), [email p o ec ed] (Á. Piñei o).
spec um o diseases, such as cance and a ious human in lamma o y
and au oimmune diseases, including aging [2,3]. Al hough mo e han
3000 AMPs ha e been iden i ied so a in dis inc cells and issues
o animals, insec s, plan s, and bac e ia, only a ew ha e eached he
pha maceu ical ma ke [4]. Challenges o he clinical applica ion o
AMPs include cy o oxic e ec s, p oduc ion cos s, and p oblems ela ed
o sus ained, a ge ed, and e ec i e deli e y [5–7]. The ques o dis-
co e and e ine new an imic obial pep ides (AMPs), including bo h
h ps://doi.o g/10.1016/j.compbiomed.2024.109157
Recei ed 5 Ma ch 2024; Recei ed in e ised o m 14 Augus 2024; Accep ed 12 Sep embe 2024
Compu e s in Biology and Medicine 182 (2024) 109157
A ailable online 24 Sep embe 2024
0010-4825/© 2024 The Au ho s. Published by Else ie L d. This is an open access a icle unde he CC BY license ( h p://c ea i ecommons.o g/licenses/by/4.0/ ).
D. Conde-To es e al.
Fig. 1. Two scena ios o an imic obial pep ide (AMP) in e ac ions wi h cell memb anes. Panel A shows di e en AMP uni s in he p esence o heal hy mammalian cell memb anes
( he lipid head g oups ep esen ed by blue sphe es), whe e he pep ides main ain a andom coil con o ma ion. Panel B depic s he in e ac ion o AMPs wi h a pa hogenic o
pa hological memb ane model ypical o bac e ial memb anes, some i uses, cance cells, o senescen cells. He e, he AMPs adop a helical con o ma ion upon in e ac ing wi h
he memb ane. The o ange sphe es ep esen he head g oups o se e al anionic lipids commonly ound in hese al e ed memb anes, highligh ing he s uc u al adap a ions o
AMPs in di e en cellula en i onmen s.
na u al and enginee ed a ian s, ep esen s a dynamic and p omising
ield o esea ch. The goal is o o e come hese challenges, op imizing
hese pep ides o medical use and le e aging hei ull po en ial as
he apeu ic agen s. Un a eling his ma e is c ucial when an ibio ic
esis ance is a g owing global h ea bu also in he igh agains cance
and aging, a eas whe e AMPs can s ill p o ide signi ican solu ions.
Despi e a ying leng h, sequence, and con o ma ion, mos AMPs
sha e c ucial s uc u al and physicochemical p ope ies: hey a e ypi-
cally sho , ca ionic, and amphipa hic pep ides. This unique combina-
ion o cha ac e is ics enables hem o selec i ely a ge and in e ac
wi h pa hogenic o pa hological memb anes, such as hose ound in
cance , bac e ia, and senescen cells. This selec i e a ge ing s ems
om a common ea u e in hese memb anes: a high p opo ion o
nega i ely cha ged lipids, in con as o wha happens in heal hy
mammalian cells whose elec os a ic cha ge densi y is no mally negli-
gible. AMPs a e known o unde go con o ma ional shi s, ansi ioning
om andom s uc u es in solu ion o helical s uc u es upon encoun-
e ing a memb ane, a change d i en by hei inhe en amphiphilic
na u e (Fig. 1). This ans o ma ion enhances he alignmen o hei
hyd ophobic dipole momen s ac oss he memb ane, acili a ing op imal
in e ac ion wi h he lipid bilaye . The spa ial a angemen o he amino
acid esidues in AMPs is indispensable o hei biological unc ion.
Following memb ane binding, AMPs exe hei e ec h ough a i-
ous mechanisms, including he ba el s a e, ca pe , and o oidal po e
models, among o he con o ma ions [8,9]. A deepe unde s anding o
hese ac ion mechanisms is essen ial o imp o e AMP design, mo -
ing om cu en ial-and-e o me hods owa ds mo e p ecise and
e ec i e s a egies. Fo example, uning he modeling is especially
ele an when he e is an al e a ion in he lipid composi ion. While
he e a e models desc ibing AMP in e ac ions wi h cell memb anes,
comp ehensi e a omic-le el de ails a e sca ce, indica ing a need o
mo e in-dep h esea ch in his a ea.
The ans o ma ion o a polypep ide chain in o i s unc ional h ee-
dimensional s uc u e ep esen s a cen al challenge in molecula biol-
ogy, especially a in e aces such as he su ace o a cell memb ane o
upon he in luence o some he e ogeneous en i onmen . Despi e hei
undamen al ole in nume ous biological mechanisms, he speed and
dynamics o hese olding p ocesses emain puzzling. The Le in hal
pa adox illus a es his complexi y by highligh ing he seeming im-
possibili y o amino acid chains in inding hei na i e, unc ional
con o ma ion in a biologically ele an imescale i hey we e o explo e
all possible con o ma ions [10]. To add ess his challenge, a a ie y
o compu a ional and expe imen al app oaches ha e been employed.
AlphaFold ini ia i e [11] is a signi ican ad ance capable o p edic ing
he h ee-dimensional s uc u e o p o eins wi h unp eceden ed accu-
acy. Ne e heless, his imp essi e echnology is s ill unable o eliably
p edic ing he in e ac ion be ween he 3D s uc u e o sho pep ides
and he memb ane models ha accoun o speci ic lipid composi ions.
In pa allel, molecula dynamics (MD) simula ions ha e eme ged as a
powe ul ool o in es iga ing pep ide and p o ein olding dynamics o
a leas s uc u al s abili y unde di e en condi ions [12,13]. S anda d
MD simula ions explo e he ene gy landscape o he polypep ide chain,
p o iding insigh s in o he olding o s uc u al e olu ion pa hway.
Howe e , limi a ions in compu a ional powe o en es ic he simu-
la ion imescales, hinde ing he obse a ion o comple e olding e en s
o ansi ions be ween di e en s a es sepa a ed by signi ican ene gy
ba ie s, e en o ela i ely sho sequences. Biased MD echniques
o e come hese limi a ions by nudging he simula ion o sample di e se
s a es [14,15]. These me hods can signi ican ly accele a e he olding
p ocess, allowing o s udy i wi h g ea e de ail, including he p esence
o speci ic he e ogeneous en i onmen s. Despi e hese ad ancemen s,
p edic ing pep ide and p o ein olding emains a complex ask, pa -
icula ly in he p esence o memb ane models due o hei in ica e
in e ac ions be ween he mac omolecule and he lipid bilaye . This
coupling unde lines a c i ical need o enhanced me hodologies ha
Compu e s in Biology and Medicine 182 (2024) 109157
2
D. Conde-To es e al.
Fig. 2. Te ahed al la ice in which a ep esen a i e pep ide will mo e, wi h subla ices A ( ed) and B (g een) indica ed.
can accu a ely p edic pep ide s uc u es as a unc ion o hei speci ic
en i onmen , which would allow signi ican ad ances in he cha ac e -
iza ion o known s uc u es and u he he de elopmen o new AMP
candida es.
Thus, s udying p o ein and pep ide olding is an in insically e y
complex p oblem whose p ac ical solu ion is beyond he each o clas-
sical algo i hms [11,16]. In his scena io, quan um compu e s eme ge
as a p omising ool despi e he noisy in e media e scale quan um
(NISQ) e a. Recen wo k has a emp ed o sol e his p oblem o
ela i ely sho amino acid sequences wi hin homogeneous media [17–
19]. These s udies adop se e al simpli ying app oaches ha neglec
speci ic chemical de ails, such as mapping amino acids on o single
sphe es and modeling hei in e ac ion ene gy using a simpli ied pai -
wise po en ial. Addi ionally, o a ions o hese sphe es a e limi ed o
disc e e angles ela i e o hei nea es neighbo s, u he educing
compu a ional complexi y. Mo eo e , unde es ima ing explici in e ac-
ions wi h sol en molecules is ano he e en ual sou ce o imp ecision.
While hese simpli ica ions signi ican ly imp o e compu a ional e i-
ciency and educe he equi ed numbe o qubi s (𝑁𝑞), hey come a
he cos o educed accu acy. Ne e heless, hese app oaches o e a
aluable ool o gaining ini ial insigh s in o pep ide s uc u e, p ecisely
in he quan um compu ing con ex , whe e compu a ional esou ces
a e limi ed. While hese me hods emains e ineable, he e is oom
o make hem mo e e sa ile. In pa icula , ou ocus on AMPs and
hei in e ac ion wi h memb ane models necessi a es ex ending hese
app oaches o inco po a e a smoo h in e ace be ween wo media o
di e ing pola i ies, ying o mimic he in e ace be ween a lipid bilaye
and he aqueous phase in con ac wi h i . In his scena io, he olding
p ocess becomes signi ican ly mo e complex, as he inhomogeneous
and aniso opic en i onmen subs an ially in luences he s uc u e and
unc ion o AMPs. This a emp highligh s he c i ical demand o in en-
si ied e o s in de eloping quan um compu ing echniques, po en ially
leading o b eak h oughs in s udying and designing no el AMPs.
Ou wo k ex ends a quan um-compu ing ou ine o pep ide olding
in homogeneous media o p edic he op imal s uc u e o amino acid
sequences a he ansi ion egion be ween hyd ophilic and hyd opho-
bic en i onmen s, used as memb ane models. The o iginal p oposal o
Robe e al. [18] demons a ed he e ec i e use o quan um algo i hms
in op imizing he con o ma ion o small pep ides, employing a Hamil-
onian () model o olding polyme chains on a la ice. This app oach
b idged he gap be ween simpli ied models and mo e de ailed pep ide
ep esen a ions.
Th ee amino acid sequences, chosen o hei dis inc cha ac e -
is ics: pola , non-pola , and ha ing a high ans e sal hyd ophobic
dipola momen when o ming an alpha helix, we e employed o es
he new me hod in a ious homogeneous and non-homogeneous en-
i onmen s. Ou p oposal in oduces a aluable new dimension o
exis ing compu a ional models wi hou adding subs an ial compu a-
ional esou ce demands o unnecessa y complexi y. This ep esen s a
signi ican s ep owa ds e ining mo e sophis ica ed and p ecise pep ide
modeling echniques, enhances ou unde s anding o p o ein chemis y
in complex en i onmen s and lays he ounda ion o u u e ad ance-
men s in he ield. We ha e con idence ha his wo k will inspi e
u he esea ch, ul ima ely leading o he c ea ion o obus pep ide
s uc u es ha e ec i ely conside di e en en i onmen al condi ions.
This expansion o scien i ic knowledge holds p omise o he apeu ic
applica ions, ha nessing he unique capabili ies o quan um compu ing
o explo e he in ica e de ails o p o ein s uc u es.
2. Ma e ials and me hods
2.1. In e ace implemen a ion
2.1.1. Backg ound
The p edic ion o pep ide s uc u e in homogeneous media wi hin
he p o ein_ olding module o he qiski _ esea ch [20]
lib a y u ilizes a quan um compu a ional app oach [18] ha employs
a model Hamil onian and a a ia ional quan um algo i hm o old a
polyme chain on a e ahed al la ice. The Hamil onian is based on
he pai wise Miyazawa–Je nigan (MJ) po en ial [21,22], whe e each
amino acid is ep esen ed by a single sphe e. The MJ coa se-g ained
ep esen a ion igno es chemical de ails bu i is expec ed o desc ibe
easonably well he in amolecula in e ac ions be ween he amino
acid esidues. The la ice model simpli ies he ep esen a ion o he
pep ide o make i compu a ionally easible o quan um simula ions.
Speci ically, wo se s o non-equi alen la ice poin s (Aand B) a e
de ined as subla ices. A si es A, he polyme can only g ow in he
di ec ions 𝑡𝑖∈ {0,1,2,3} while a si e B, he possible di ec ions a e 𝑡𝑖∈
{
0,
1,
2,
3} (Fig. 2). Th oughou he sequence, he Aand Bsi es al e na e,
allowing us o adop he con en ion ha Aand Bsi es co espond
o e en and odd alues o i, espec i ely. Wi hou loss o gene ali y,
he i s wo u ns can be se o 𝑡1=
1and 𝑡2= 0 due o symme ic
degene acy. The u ns a e encoded by assigning a combina ion o wo
qubi s pe axis. Each pai o qubi s can be in one o ou possible s a es:
00, 01, 10 and 11, hus allowing o a p ecise and e icien encoding o
u ns.
The e o e, a bi s ing ep esen s he h ee-dimensional s uc u e o
he pep ide, which codi ies he sequen ial u ns o he coa se-g ain
beads. A ela i ely low numbe o con o ma ion and in e ac ion qubi s
is equi ed unde his app oach, including penal y e ms o p e en
Compu e s in Biology and Medicine 182 (2024) 109157
3
D. Conde-To es e al.
Fig. 3. Schema ic ep esen a ion o how he VQE algo i hm wo ks.
Fig. 4. Compa ison be ween he sign unc ion (black line) and i s 7 h-deg ee polynomial app oxima ion (blue line). The e ical g een line indica es he loca ion o a i ual plane
sepa a ing bo h sol en s and he dashed ed lines indica e he dis ances beyond which he di e ence be ween he sign unc ion and i s polynomial app oxima ion di e ges.
meaningless con o ma ions o he pep ide, such as esidue o e laps and
chi al iola ions. The numbe o qubi s equi ed o his model scales
quad a ically wi h he numbe o amino acid esidues in he pep ide
sequence (𝑁)while he numbe o e ms in he Hamil onian scales in
𝑂(𝑁4). Adding sidechains and inco po a ing s a e-o - he-a classical
o ce ields based on Lenna d-Jones and Coulomb in e ac ions is also
possible by keeping he s uc u e o he employed Hamil onian, albei
his would equi e a highe numbe o pa icles and so a highe numbe
o qubi s.
Since he p o ein_ olding module akes ad an age o a Va i-
a ional Quan um Eigensol e (VQE) [23] (Fig. 3), he Hamil onian is
minimized o each i e a ion o he pa ame ized quan um ci cui . This
means ha he Hamil onian should be sel -consis en o be execu ed
in he quan um p ocesso uni wi hou depending on he s a e o he
qubi s.
2.1.2. Amino acids loca ion conce ning he in e ace
The posi ional displacemen o each bead along a speci ic e ahe-
d al axis 𝑎is quan i ied as:
𝛥𝑛𝑎(𝑗) =
𝑗−1
∑
𝑘=1
(−1)𝑘𝑓𝑎(𝑘) + 𝛥𝑎(1)
whe e he sum is pe o med om he i s o he cu en bead (𝑗), 𝛥𝑎
ep esen s he dis ance along axis 𝑎 om he i s bead o he pep ide
(which is always ixed in ou app oach) o he phase-sepa a ing plane.
The unc ion 𝑓𝑎(𝑘) e u ns 1i he e is a displacemen along axis 𝑎 o
he u n o amino acid 𝑘, and 0o he wise. The e m (−1)𝑘indica es he
di ec ionali y o he u n ela i e o subla ices Ao B, e ec i ely show-
ing whe he he mo emen b ings he bead close o o u he om he
phase-sepa a ing plane. 𝛥𝑛𝑎(𝑗)inhe en ly de e mines he phase loca ion
o amino acid 𝑗, as well as he dis ance o he phase-sepa a ing plane.
I is impo an o no e ha he con ibu ion o he in e ac ion be ween
he amino acids and i s co esponding phase canno be p opo ional o
𝛥𝑛𝑎(𝑗)since, in ha case, such in e ac ion would linea ly inc ease he
a ini y o epulsion (depending on he sign o he in e ac ion) o each
amino acid o each phase as a unc ion o he dis ance o i . On he o he
hand, ex ac ing di ec ly he sign o his unc ion is no a i ial ask
wi hou eading he s a e o he qubi s. While auxilia y qubi s could
acili a e his, hey would also inc ease he compu a ional demands,
which is incon enien . Addi ionally, di ec ly using a s ep unc ion o
iden i y he loca ion o he bead a each medium would be an unsui -
able app oach since ac ual in e aces, such as ha be ween an aqueous
media and a lipid memb ane, a e smoo h. The oughness o such
in e aces is compa able o he diame e o a wa e molecule (3–6 Å), as
Compu e s in Biology and Medicine 182 (2024) 109157
4
D. Conde-To es e al.
es ima ed om neu on e lec ome y analysis [24], so a g adual an-
si ion be ween bo h phases is o eseeable. To add ess all hese issues,
we decided o use a polynomial app oxima ion o he sign unc ion
(see Fig. 4) as a scaling ac o o he Hamil onian con ibu ion o he
in e ac ion be ween each amino acid and he co esponding medium:
𝑓(𝑥)=0.48175𝑥− 0.0182𝑥3+ (2.95 ⋅10−4)𝑥5− (1.56 ⋅10−6)𝑥7(2)
This app oach p o ides a smoo h ansi ion be ween he wo phases,
wi h an in e ace hickness o app oxima ely 5 a bi a y uni s, co e-
sponding o he dis ance be ween wo beads in he pep ide’s la ice.
F om his egion and a dis ances lowe han 9 uni s, he polynomial
unc ion exhibi s ela i ely small oscilla ions. A longe dis ances his
polynomial di e ges om he sign unc ion. Since he s udied pep ides
a e qui e sho (maximum 10 amino acids) and unlikely o ex end u -
he om he in e ace cen e , his app oxima ion conside ably ailo s
ou pu poses.
2.1.3. Tuning he in e ac ion be ween amino acids as a unc ion o he
media
The app oach al eady implemen ed in he qiski _ esea ch
lib a y is well-designed o modeling pep ide olding in homogeneous
media. De ails o he implemen a ion a e well documen ed in p e-
ious publica ions [18] as well as in he o icial eposi o y o he
p o ein_ olding module [20]. Howe e , i s applicabili y o unc-
ional an imic obial pep ides is limi ed, as hese pep ides exe hei
biological unc ion by in e ac ing wi h he su ace o pa hological mem-
b anes, which could be oughly modeled as a hyd ophilic/hyd ophobic
in e ace. Thus, se e al modi ica ions we e in oduced in o he o igi-
nal model. Fi s , he Miyazawa–Je nigan (MJ) pa ame e s we e mod-
i ied ollowing he wo k o Leonha d e al. [25,26] o accoun o
in e ac ions be ween esidue beads in di e en phases:
𝑒𝐿𝑒𝑜𝑛ℎ𝑎𝑟𝑑
𝑖,𝑗 =𝑒𝑀𝐽
𝑖,𝑗 −𝑒𝑖,𝑝ℎ𝑎𝑠𝑒 −𝑒𝑗,𝑝ℎ𝑎𝑠𝑒 (3)
whe e 𝑒𝑀𝐽
𝑖,𝑗 is he o iginal alue o he MJ in e ac ion e ms be ween
amino acids 𝑖and 𝑗and 𝑒𝑘,𝑝ℎ𝑎𝑠𝑒 (wi h 𝑘=𝑖o 𝑗) ep esen s he
in e ac ion o amino acid 𝑘wi h he phase i esides in. The alue
o 𝑒𝑘,𝑝ℎ𝑎𝑠𝑒 o a homogeneous phase is calcula ed using he ollowing
equa ion:
𝑒𝑘,𝑝ℎ𝑎𝑠𝑒 =1
2(1 − 𝐶𝑠)𝑒𝑀𝐽
𝑘𝑘 +𝜔 +𝐶𝑠
2𝑛
20
∑
𝑖=1
𝑒𝑀𝐽
𝑖𝑖
=1
2(1 − 𝐶𝑠)𝑒𝑀𝐽
𝑘𝑘 +𝜔′
(4)
whe e 𝑖i e a es o e he o al numbe o amino acid ypes, and 𝐶𝑠
de e mines he con as be ween phases. A posi i e 𝐶𝑠 a o s con ac
be ween sol en and hyd ophilic esidues, while a nega i e 𝐶𝑠 a o s
con ac be ween sol en and hyd ophobic esidues. 𝜔 de e mines he
a e age in e ac ion be ween amino acids and he sol en . Nega i e
𝜔 indica es a ac ion, while posi i e 𝜔 indica es epulsion. A he
ansi ion egion be ween wo phases o di e en pola i y 𝑒𝑘,𝑝ℎ𝑎𝑠𝑒 will
be eplaced by 𝑒𝑘,𝑝ℎ𝑎𝑠𝑒′:
𝑒𝑘,𝑝ℎ𝑎𝑠𝑒′=1
2[(1 − 
𝑆)⋅𝑒𝑘,𝑝ℎ𝑎𝑠𝑒1+ (1 + 
𝑆)⋅𝑒𝑘,𝑝ℎ𝑎𝑠𝑒2](5)
whe e 
𝑆can ake alues be ween 1and −1, depending on whe he he
amino acid is in he pola o nonpola phase. In ou case, 
𝑆will be
eplaced by he unc ion p o ided by Eq. (2). Depending on he alue
o 
𝑆,𝑒𝑘,𝑝ℎ𝑎𝑠𝑒′can be close o he alue o 𝑒𝑘,𝑝ℎ𝑎𝑠𝑒1o 𝑒𝑘,𝑝ℎ𝑎𝑠𝑒2. This e m
can be swi ched o in he Hamil onian, in case he s udy is pe o med
in an homogeneous media and so he o iginal MJ po en ial is employed,
as i is a boolean pa ame e .
Table 1
Fauche e and Pliska [27] hyd ophobici y scale.
Amino acid 𝛾Residue ype
ASP −0.77 Cha ged (−)
GLU −0.64 Cha ged (−)
LYS −0.99 Cha ged (+)
ARG −1.01 Cha ged (+)
HIS 0.13 Cha ged (+)
GLY 0.00 Nonpola
ALA 0.31 Nonpola
VAL 1.22 Nonpola
LEU 1.70 Nonpola
ILE 1.80 Nonpola
PRO 0.72 Nonpola
MET 1.23 Nonpola
PHE 1.79 A oma ic
TRP 2.25 A oma ic
TYR 0.96 A oma ic
THR −0.04 Pola
SER 0.26 Pola
CYS 1.54 Pola
ASN −0.60 Pola
GLN −0.22 Pola
2.1.4. In e acial con ibu ion o he Hamil onian
The p e ious modi ica ions o he MJ po en ial accoun o he
di e en occu ing in e ac ions be ween amino acids based on hei
loca ion wi hin he aqueous o memb ane phases. Besides in e ac ing
wi h each o he , amino acids also di ec ly in e ac wi h he sol en in
bo h media. Thus, a new con ibu ion, 𝑠𝑜𝑙(𝑞𝑐𝑓 ), has been added o he
o al Hamil onian:
(𝑞) = 𝑔𝑐 (𝑞𝑐𝑓 ) + 𝑐ℎ(𝑞𝑐𝑓 ) + 𝑖𝑛(𝑞) + 𝑠𝑜𝑙(𝑞𝑐𝑓 )(6)
whe e 𝑞=𝑞𝑐𝑓 , 𝑞𝑖𝑛 ep esen s he comple e se o qubi s used in he
model, including bo h he con o ma ion qubi s (𝑞𝑐𝑓 ) and he in e ac ion
qubi s (𝑞𝑖𝑛). The i s h ee- e ms desc ip ion is a ailable in [17,18].
B ie ly:
•𝑔𝑐 (𝑞𝑐𝑓 )accoun s o he geome ical cons ain s imposed by he
e ahed al la ice s uc u e o he amino acids.
•𝑐ℎ(𝑞𝑐𝑓 )en o ces he co ec s e eochemis y o he sidechains
(when p esen ), ensu ing he accu acy o he amino-acid-chi ali y
ep esen a ion.
•𝑖𝑛(𝑞)accoun s o he in e ac ions be ween neighbo ing beads
using he Miyazawa–Je nigan (MJ) po en ial.
The new e m 𝑠𝑜𝑙(𝑞𝑐𝑓 )accoun s o he in e ac ion be ween he
amino acids and each sol en . This e m has been de ined he e as:
𝑠𝑜𝑙(𝑞𝑐𝑓 ) = ∑
𝑖
𝛥𝑃 ⋅𝛾𝑖⋅
𝑆(7)
𝛥𝑃 ep esen s he pola i y di e ence be ween he wo media and
𝛾𝑖 ep esen s a quan i a i e measu emen o he hyd ophobici y, o
a ini y o each esidue o a hyd ophobic media. In he p esen wo k,
he pa ame e s used we e p oposed by Fauche e and Pliska [27] (see
Table 1), al hough he e a e di e en p oposals o his pa ame e in
he li e a u e, ob ained om a a ie y o me hods [28–31].
No e ha cha ged esidues, wi h he s onges a ac ion o pola
sol en s, ha e he lowes (mos nega i e) 𝛾 alues; pola esidues
exhibi mode a e alues depending on hei speci ic side chains, ang-
ing om sligh ly nega i e o sligh ly posi i e; nonpola esidues ha e
consis en ly posi i e alues; while a oma ic esidues, wi h hei la ge
hyd ophobic ings, possess he highes posi i e alues o 𝛾. The i-
nal exp ession o 𝑠𝑜𝑙(𝑞𝑐𝑓 )p o ides a nega i e con ibu ion o he
Hamil onian, a o s he in e ac ion o amino acids wi h 𝛾 < 1(mainly
nonpola and a oma ic) a he posi i e side o he in e ace (phase I)
and o amino acids wi h 𝛾 > 1(mainly cha ged) a he nega i e side
o he in e ace (phase II) i 𝛥𝑃 > 1. The highe he alue o 𝛥𝑃 he
s onge his con ibu ion.
Compu e s in Biology and Medicine 182 (2024) 109157
5

D. Conde-To es e al.
Fig. 5. Wheel ep esen a ion o P1,P2, and P3 wi h sequences WLWLWLWWLW,
DRDRDRDRDR and WRDWGSGWDR, espec i ely. W, L, R, D, G and S deno e
T yp ophan, Leucine, A ginine, Glu amic Acid, Glycine and Se ine, espec i ely. P1
and P2 a e expec ed o exhibi a high a ini y o a nonpola and o a pola media,
espec i ely, while P3 is expec ed o acqui e a helical con o ma ion a he in e ace
be ween wo media o di e en pola i y, wi h a high ans e sal componen o he
hyd ophobic dipola momen . Posi i ely cha ged amino acids (A ginine) a e in blue,
nega i ely cha ged amino acids (Glu amic Acid) a e in ed, pola amino acids (Se ine)
a e in o ange, and neu al-nonpola (Glycine and Leucine) and a oma ic (T yp ophan)
amino acids a e in g ay.
Bo h he pai wise MJ po en ial and he hyd ophobici y 𝛾ha e
a bi a y uni s and bo h a e o he same o de , so hey compe e wi h
each o he o modula e he op imal s uc u e o he pep ide a he in-
e ace. Impo an ly, his Hamil onian implemen a ion does no equi e
addi ional qubi s, and he numbe o ex a ope a ions is modes . In
pa icula , he calcula ion o a sequence o 10 amino acids he numbe
o equi ed qubi s is 𝑁𝑞= 22. Hence, including he ex a dimension
o he in e ace, he inal compu a ional cos is no ema kably highe
compa ed o he o iginal model o homogeneous media.
All he desc ibed modi ica ions o he model we e implemen ed in
he p o ein_ olding module o he qiski _ esea ch lib a y.
The whole code is w i en in Py hon [32], making special use o he
Qiski [33], Numpy [34], Ma plo lib [35] and Maya i [36] lib a ies
and i is publicly a ailable a h ps://gi hub.com/TeamMduse.
2.2. S udied sys ems and pa ame e s
We ha e employed h ee amino acid sequences deno ed P1,P2, and
P3 o alida e ou app oach. These sequences we e chosen o exhibi
dis inc a ini ies o media o di e en pola i y, based on he 𝛾 alues
p esen ed in Table 1.
1. P1 (WLWLWLWWLW) comp ises exclusi ely hyd ophobic amino
acids (Leucine and T yp ophan), maximizing i s a ini y o non-
pola en i onmen s (See Fig. 5).
2. P2 (DRDRDRDRDR) consis s solely o cha ged amino acids (Glu-
amic Acid and A ginine), p omo ing i s in e ac ion wi h pola
media (See Fig. 5).
3. P3 (WRDWGSGWDR) ep esen s a mo e in ica e sequence, con-
aining cha ged amino acids o opposing cha ges (Glu amic Acid
and A ginine), highly pola and neu al esidues (Se ine), highly
pola (T yp ophan) and neu al-nonpola (Glycine) amino acids,
dis ibu ed such ha gene a es a signi ican ans e sal compo-
nen o he hyd ophobic dipole momen when adop ing a helical
con o ma ion (See Fig. 5).
This selec ion o sequences allows o a comp ehensi e e alua ion
o he abili y o ou app oach o accu a ely cap u e and p edic he
beha io o pep ides wi h a ange o physicochemical p ope ies in
en i onmen s o di e ing pola i y. By analyzing hei pa i ioning be-
ween phases o opposi e pola i y unde he in luence o ou model, we
can assess i s e icacy in e lec ing he unde lying p inciples o amino
acid–sol en in e ac ions.
I is wo h eminding ha he model ou pu is a bi s ing wi h he
sequen ial u ns o he amino acids ela i e o hei p e ious closes
neighbo s. Subs an ially, he i s wo beads, ep esen ing he i s wo
amino acids, ha e ixed posi ions. The loca ion o hese beads de ines
hei dis ance and o ien a ion conce ning he plane sepa a ing he wo
phases. Upon hese es ain s and hose p o ided by he model (chem-
ical consis ency and e ahed al la ice), he u ns o he emaining
beads es ablish he s uc u e o he pep ide. Axis 1 o he e ahed al
la ice (see Fig. 2) was chosen o de ine he pola i y g adien . The i s
bead o he pep ides, ep esen ing he i s amino acid, was placed a
di e en posi ions along he same axis (−1,−0.5, 0, 0.5, and 1). This se
o con igu a ions led o a ious dis ances be ween such a bead and he
phase-sepa a ing plane owa ds bo h sol en s. In all cases, he second
bead was aligned along he same axis in he di ec ion o he mo e
pola sol en . Addi ionally, di e en alues o 𝛥𝑃 (0.1, 1 and 10) we e
essayed in o de o balance he compe i ion be ween he weigh o he
in e ac ion be ween amino acids and he weigh o hei in e ac ion
wi h he sol en . Finally, he weigh s o he exis ing penal y e ms in
he o iginal app oach we e inc eased om 10 o 1000. This was done o
p e en hem om being o e shadowed by he new con ibu ion o he
Hamil onian. The esul s ob ained om his combina ion o pa ame e s
o he h ee s udied sequences a e p esen ed in he nex sec ion.
3. Resul s and discussion
The op imal con o ma ions o he pep ide sequences desc ibed in
he me hods sec ion we e ob ained h oughou he minimiza ion o he
Hamil onian, using he VQE algo i hm, unde di e en condi ions: in
pola and nonpola homogeneous phases as well as a pola /nonpola
in e aces. The loca ion and o ien a ion o he i s wo amino acids
o each sequence conce ning he phase-sepa a ing plane (when wo
di e en media a e conside ed) we e es ained.
3.1. Homogeneous media
The con o ma ion o each pep ide is highly sensi i e o he pola i y
o he en i onmen in homogeneous media (Fig. 6). The pep ide con-
sis ing jus o hyd ophobic amino acids (P1) and ha o med jus by
cha ged amino acids (P2) exhibi an opposi e beha io , as expec ed. P1
is olded in pola en i onmen s and ully ex ended in nonpola media,
while P2 is ully ex ended in pola en i onmen s and olded in nonpola
media. We obse e ha he beha io o P3 is simila o ha o P1. The
h ee pep ides acqui e di e en olded con o ma ions when using he
o iginal MJ po en ial.
The ex ended con o ma ions o P1 and P2 in nonpola and pola
media, espec i ely, a e due o highly a o able in e ac ions be ween
he amino acids consis ing o hose pep ides and he model sol en in
hose scena ios. In hese cases he in e ac ion wi h he media la gely
domina es he Hamil onian while he in amolecula in e ac ions a e
less impo an . Con e sely, he olded con o ma ions o he same pep-
ides in he opposi e media (P1 in pola sol en and P2 in nonpola
sol en ) a ise om he a o able in e ac ions be ween he amino acids
consis ing o hose pep ides combined wi h un a o able in e ac ions
wi h he media. No clea seconda y s uc u e pa e ns a e obse ed in
any o he olded con o ma ions.
3.2. Pola /nonpola in e aces
Acco ding o he p esen ed calcula ions wi hin wo media, pep ide
con o ma ion elies hea ily on se e al compe ing ene gy ac o s. These
ac o s p ima ily include he in e ac ions among he amino acids, as
well as hei in e ac ions wi h he wo model sol en s used in he s udy.
Fo ins ance, i is possible o iden i y scena ios whe e wo di e en
amino acids ha e a s ong mu ual a ac ion bu an e en s onge
a ini y o opposi e phases. This disbalance can de ini ely in luence he
pep ide con o ma ion, po en ially esul ing in he spa ial sepa a ion o
hese amino acids despi e hei in insic a ac ion. The eme gence o
dispa a e pep ide con igu a ions in he e ogeneous en i onmen s high-
ligh s he complex in e play be ween in a-pep ide and pep ide-sol en
in e ac ions and he ele ance o implemen ing an in e ace model.
Compu e s in Biology and Medicine 182 (2024) 109157
6
D. Conde-To es e al.
Fig. 6. Con o ma ions o pep ides P1,P2, and P3 in homogeneous media o di e en pola i y and also igno ing he pola i y o he media (i.e. using he o iginal MJ po en ial).
Di e en colo s a e employed o each esidue: W in b own, L in g een, R in blue, D in ed, G in cyan, and S in yellow.
The esul s obse ed o sequences P1,P2, and P3 clea ly show ha
hyd ophobic esidues a e mo e s able in he nonpola en i onmen ,
e en i he pep ide sequence needs a u n o eo ien a e he co e-
sponding coa se-g ained beads. No e ha he loca ion o he i s wo
amino acids is ixed in ou app oach, so he pep ide canno a el as
a whole om one media o he o he , and he o ien a ion o he i s
wo amino acids conce ning he phase-sepa a ing plane is no op imized
by minimizing he Hamil onian. The beha io o cha ged esidues (D
and R) is opposi e o ha o hyd ophobic esidues (W and L). Thus,
he con o ma ion o pep ides P1 and P2 a he in e acial model could
be easily p edic ed (Fig. 7). Besides, P3 was designed o ideally old
in o a helical s uc u e in his he e ogeneous en i onmen . This pep ide
does no exhibi a clea end o s ay in one o o he phase, bu he
amino acids a e dis ibu ed be ween he wo media, as expec ed. The
ob ained con o ma ion is no an ideal helix. Mo eo e , some amino
acids a e loca ed in he w ong phase, p obably due o he limi a ions o
he employed e ahed al model. While he possibili y o con e gence
o local minima in he VQE algo i hm canno be comple ely uled ou ,
we ook ho ough measu es o mi iga e his issue. The calcula ions we e
epea ed mul iple imes o he mos con o e sial cases, employing
a conse a i ely high numbe o i e a ions and a ying he seeds, ye
hese adjus men s did no al e he inal s uc u e ob ained.
Addi ional calcula ions using di e en ini ial coo dina es o he
wo i s amino acids o he employed sequences p o ided di e en
con o ma ions o he pep ides (as expec ed) bu hey ollowed he same
quali a i e beha io as he esul s shown in Fig. 7.
4. Conclusions
This s udy aims o con ibu e signi ican ly o he ield o pep ide
olding simula ions using quan um compu ing by in oducing a new
dimension o a p eexis ing model [18]. Ou esea ch ocuses on he
olding o pep ides in di e en en i onmen s, pa icula ly a he in-
e ace be ween hyd ophobic and hyd ophilic phases, which is c i ical
o unde s anding he unc ion o an imic obial pep ides (AMPs) in
biological sys ems. Based on a modi ied e sion o he Miyazawa–
Je nigan po en ial, ou app oach employs a e ahed al la ice model
o ep esen pep ide s uc u es, combined wi h he in oduc ion o a
Hamil onian con ibu ion accoun ing o he in e ac ion be ween he
amino acids and he sol en in each phase. The ansi ion egion om
one o ano he media is modeled as an smoo h unc ion, ying o mimic
he ac ual in e ace a he icini y o a cell memb ane. Fu he mo e,
ou implemen a ion is compu a ionally e icien and does no equi e
addi ional qubi s compa ed o he o iginal model ha only conside s
an homogeneous phase. Ou indings demons a e ha pep ides exhibi
dis inc olding pa e ns in esponse o he pola i y o hei su ounding
en i onmen . Resul s poin ou he po en ial o quan um compu ing
o simula e complex biological p ocesses, which classical compu ing
app oaches s uggle o accomplish due o compu a ional limi a ions.
While in eg a ing a pola /nonpola in e ace in pep ide olding
ep esen s a signi ican achie emen , he ex ended model leans on
app oxima ions o iginally p oposed o calcula ions in homogeneous
media. In pa icula , he conside a ion o a e ahed al la ice ha
es ains he u ns o he amino acids combined wi h he minimalis
MJ pai wise po en ial in e ac ion seems o be inaccu a e in success ully
p edic ing pep ide seconda y s uc u e. The limi ed numbe o a ailable
qubi s cu en ly makes i un easible o add mo e deg ees o eedom
and a mo e eliable po en ial o amino acid in e ac ion.
Al hough ou s udy was limi ed o pep ides wi h 10 amino acids,
longe sequences can be s udied ollowing he same me hodology.
Howe e , such calcula ions would equi e a la ge numbe o qubi s,
inc easing he compu a ional cos and po en ially educing accu acy
due o he app oxima ions inhe en in he model. Fu u e wo k migh
explo e his ex ension as quan um compu ing esou ces and echniques
e ol e, bu he cu en wo k in oduces a iable app oach, showing ha
pep ide olding a pola -nonpola in e aces can be simula ed wi h a
easonable use o compu a ional esou ces.
The speci ic aim o his s udy is o in oduce, o he i s ime,
an e icien me hod o le e age quan um compu ing o p edic ing
easonable pep ide s uc u es a he in e ace be ween media o di e -
en pola i y. This s a ing poin opens new a enues o unde s anding
pep ide in e ac ions a he molecula le el, which could lead o sig-
ni ican ad ances in de eloping new he apeu ic agen s, pa icula ly
in he ealm o an imic obial pep ides. Fu u e esea ch should aim o
e ine he quan um compu a ional app oach o enhance i s accu acy
and applicabili y o a b oade ange o biomolecules. Fu he mo e,
in eg a ing mo e de ailed chemical p ope ies and in e ac ions in o he
model could yield e en mo e nuanced insigh s in o pep ide olding
dynamics. The gene al goal is o de elop a quan um compu a ional
amewo k capable o simula ing a ious biological p ocesses. Ad anc-
ing ou unde s anding and capabili ies in molecula biology unde sco e
he c i ical impo ance o ongoing esea ch and de elopmen in he
ield o quan um compu ing, pa icula ly in i s applica ion o complex
biological sys ems.
Compu e s in Biology and Medicine 182 (2024) 109157
7
D. Conde-To es e al.
Fig. 7. Con o ma ions o pep ides P1,P2, and P3 a he in e ace be ween wo media o di e en pola i y. Fo each pep ide a side iew (le ), a iew om he nonpola phase
(middle) and ano he iew om he pola phase ( igh ), a e shown. In he side iew he in e ace is shown as an ho izon al line. Di e en colo s a e employed o each esidue:
W in b own, L in g een, R in blue, D in ed, G in cyan, and S in yellow.
CRediT au ho ship con ibu ion s a emen
Daniel Conde-To es: W i ing – e iew & edi ing, W i ing – o igi-
nal d a , Visualiza ion, Valida ion, So wa e, Me hodology, In es iga-
ion, Fo mal analysis, Da a cu a ion. Ma iamo Mussa-Juane: W i ing
– e iew & edi ing, Supe ision, Me hodology, In es iga ion, Concep-
ualiza ion. Daniel Faílde: W i ing – e iew & edi ing, Supe ision,
Me hodology, In es iga ion, Concep ualiza ion. And és Gómez: W i -
ing – e iew & edi ing, Supe ision, P ojec adminis a ion, Funding
acquisi ion, Concep ualiza ion. Rebeca Ga cía-Fandiño: W i ing – e-
iew & edi ing, W i ing – o iginal d a , Supe ision, Resou ces, P ojec
adminis a ion, Me hodology, In es iga ion, Funding acquisi ion, Con-
cep ualiza ion. Ángel Piñei o: W i ing – e iew & edi ing, W i ing –
o iginal d a , Supe ision, So wa e, Resou ces, P ojec adminis a ion,
Me hodology, In es iga ion, Funding acquisi ion, Concep ualiza ion.
Decla a ion o compe ing in e es
The au ho s decla e ha hey ha e no known compe ing inan-
cial in e es s o pe sonal ela ionships ha could ha e appea ed o
in luence he wo k epo ed in his pape .
Decla a ion o Gene a i e AI and AI-assis ed echnologies in he
w i ing p ocess
Du ing he p epa a ion o his wo k he au ho s used cha GPT4o
om OpenAI and Claude 3.5 Sonne om An h opic in o de o imp o e
language and eadabili y. A e using hese ools, he au ho s e iewed
and edi ed he con en as needed and ake ull esponsibili y o he
con en o he publica ion.
Acknowledgmen s
D.C.T hanks o he Minis e io de Uni e sidades o his p edoc o al
con ac (FPU22/00636). This wo k was suppo ed by he In e eg
Sudoe and he ERDF (S1/1.1/P0033), by he Spanish Agencia Es a al de
In es igación (AEI) and he ERDF (PID2022-141534OB-I00, PDC2022-
133402-I00, CNS2023-144353 and PID2019111327GBI00), by MICINN
h ough he Eu opean Union Nex Gene a ionEU eco e y plan (PRTR-
C17.I1), by Xun a de Galicia h ough he “Planes Complemen a ios de
I+D+I con las Comunidades Au onomas” in Quan um Communica ion,
by Xun a de Galicia, Spain and he ERDF (ED431C 2021/21, ED431B
2022/36) and Cen o singula de in es igación de Galicia acc edi a ion
2016–2019, ED431G/09 and Axencia Galega de Inno ación, Spain
h ough he G an Ag eemen ‘‘Desp egamen o dunha in aes u u a
baseada en ecnoloxías cuán icas da in o mación que pe mi a impulsa
a I+D+I en Galicia’’ wi hin he p og am FEDER Galicia 2014–2020.
Simula ions on his wo k we e pe o med using he Finis e ae III
Supe compu e , unded by he p ojec CESGA-01 FINISTERRAE III.
Re e ences
[1] S. Nayab, M.A. Aslam, S.u. Rahman, Z.u.D. Sindhu, S. Sajid, N. Za a , M. Razaq,
R. Kanwa , Amanullah, A e iew o an imic obial pep ides: i s unc ion, mode
o ac ion and he apeu ic po en ial, In . J. Pep . Res. The . 28 (1) (2022) 46,
h p://dx.doi.o g/10.1007/s10989-021-10325-6.
[2] A. Ja a i, A. Babajani, R. Sa ami Fo ooshani, M. Yazdani, M. Rezaei-Ta i ani,
Clinical applica ions and an icance e ec s o an imic obial pep ides: om bench
o bedside, F on . Oncol. 12 (2022) 819563, h p://dx.doi.o g/10.3389/ onc.
2022.819563.
[3] B.A. S ua , A.L. F ani za, e al., Regula o y oles o an imic obial pep ides in
he ne ous sys em: Implica ions o neu onal aging, F on . Cell. Neu osci. 16
(2022) 843790.
[4] G.S. Dijks eel, M.M. Ul ich, E. Middelkoop, B.K. Boekema, Lessons lea ned om
clinical ials using an imic obial pep ides (AMPs), F on . Mic obiol. 12 (2021)
616979.
[5] K. Bo elho Sampaio de Oli ei a, M. Lopes Lei e, V. Albuque que Cunha, N. B i o
da Cunha, O. Luiz F anco, Challenges and ad ances in an imic obial pep ide
de elopmen , D ug Disco . Today 28 (8) (2023) 103629.
[6] X. Luo, H. Chen, Y. Song, Z. Qin, L. Xu, N. He, Y. Tan, W. Dessie, Ad ance-
men s, challenges and u u e pe spec i es on pep ide-based d ugs: Focus on
an imic obial pep ides, Eu . J. Pha maceu . Sci. 181 (2023) 106363.
[7] C.B. Ma gi Mahlapuu, J. Ekblom, An imic obial pep ides as he apeu ic agen s:
oppo uni ies and challenges, C i . Re . Bio echnol. 40 (7) (2020) 978–992.
[8] W.C. Wimley, Desc ibing he mechanism o an imic obial pep ide ac ion wi h
he in e acial ac i i y model, ACS Chem. Biol. 5 (10) (2010) 905–917.
Compu e s in Biology and Medicine 182 (2024) 109157
8
D. Conde-To es e al.
[9] J. Li, J.-J. Koh, S. Liu, R. Lakshmina ayanan, C.S. Ve ma, R.W. Beue man,
Memb ane ac i e an imic obial pep ides: ansla ing mechanis ic insigh s o
design, F on . Neu osci. 11 (2017) 73.
[10] R. Zwanzig, A. Szabo, B. Bagchi, Le in hal’s pa adox, P oc. Na l. Acad. Sci. 89
(1) (1992) 20–22.
[11] J. Jumpe , R. E ans, A. P i zel, T. G een, M. Figu no , O. Ronnebe ge , K.
Tunyasu unakool, R. Ba es, A. Žídek, A. Po apenko, e al., Highly accu a e
p o ein s uc u e p edic ion wi h AlphaFold, Na u e 596 (7873) (2021) 583–589.
[12] N. Palme , J.R. Maasch, M.D. To es, C. de la Fuen e-Nunez, Molecula dynamics
o an imic obial pep ide disco e y, In ec . Immun. 89 (4) (2021) 10–1128.
[13] F. Sua ez-Les on, M. Cal elo, G.F. Tolu ashe, A. Muñoz, U. Velei o, C. Po o,
M. Bas os, Á. Piñei o, R. Ga cia-Fandino, SuPepMem: A da abase o inna e
immune sys em pep ides and hei cell memb ane in e ac ions, Compu . S uc .
Bio echnol. J. 20 (2022) 874–881.
[14] P.W. Simcock, M. Bubli z, F. Cipcigan, M.G. Ryadno , J. C ain, P.J. S ans eld,
M.S. Sansom, Memb ane binding o an imic obial pep ides is modula ed by lipid
cha ge modi ica ion, J. Chem. Theo y Compu . 17 (2) (2021) 1218–1228.
[15] I. Kabelka, R. B ozek, R. Vácha, Selec ing collec i e a iables and ee-ene gy
me hods o pep ide ansloca ion ac oss memb anes, J. Chem. In . Model. 61
(2) (2021) 819–830.
[16] B. Be ge , T. Leigh on, P o ein olding in he hyd ophobic-hyd ophilic (HP) is
NP-comple e, in: P oceedings o he Second Annual In e na ional Con e ence on
Compu a ional Molecula Biology, 1998, pp. 30–39.
[17] P. Chanda ana, P.S. Viei es, N.N. Hegade, E. Solano, Y. Ban, X. Chen, Me a-
lea ning digi ized-coun e diaba ic quan um op imiza ion, Q. Sci. Technol. 8 (4)
(2023) 045007.
[18] A. Robe , P.K. Ba kou sos, S. Woe ne , I. Ta e nelli, Resou ce-e icien quan um
algo i hm o p o ein olding, Npj Q. In . 7 (1) (2021) 38.
[19] S. Bopa dika , K. Dasgup a, S. Senapa i, e al., An app oach o sol e he coa se-
g ained p o ein olding p oblem in a quan um compu e , 2023, a Xi p ep in
a Xi :2311.14141.
[20] T.Q.R. de elope s, con ibu o s, Qiski esea ch, 2023, h p://dx.doi.o g/10.
5281/zenodo.7776174.
[21] S. Miyazawa, R.L. Je nigan, Es ima ion o e ec i e in e esidue con ac ene gies
om p o ein c ys al s uc u es: quasi-chemical app oxima ion, Mac omolecules
18 (3) (1985) 534–552.
[22] S. Miyazawa, R.L. Je nigan, Residue– esidue po en ials wi h a a o able con ac
pai e m and an un a o able high packing densi y e m, o simula ion and
h eading, J. Mol. Biol. 256 (3) (1996) 623–644.
[23] J. Tilly, H. Chen, S. Cao, D. Picozzi, K. Se ia, Y. Li, E. G an , L. Wossnig, I.
Rungge , G.H. Boo h, e al., The a ia ional quan um eigensol e : a e iew o
me hods and bes p ac ices, Phys. Rep. 986 (2022) 1–128.
[24] L. Lee, D. Lange in, B. Fa noux, Neu on e lec i i y o an oil-wa e in e ace,
Phys. Re . Le . 67 (19) (1991) 2678.
[25] K. Leonha d, J.M. P ausni z, C.J. Radke, Sol en –amino acid in e ac ion ene gies
in h ee-dimensional-la ice Mon e Ca lo simula ions o a model 27-me p o ein:
Folding he modynamics and kine ics, P o . Sci. 13 (2) (2004) 358–369.
[26] K. Leonha d, J.M. P ausni z, C.J. Radke, Th ee-dimensional la ice Mon e Ca lo
simula ions o model p o eins. IV. P o eins a an oil- wa e in e ace, Langmui
22 (7) (2006) 3265–3272.
[27] J.-L. Fauche e, V. Pliska, Hyd ophobic pa ame e s pi o amino-acid side chains
om he pa i ioning o N-ace yl-amino amides, Eu . J. Med. Chem. (1983).
[28] J. Ky e, R.F. Dooli le, A simple me hod o displaying he hyd opa hic cha ac e
o a p o ein, J. Mol. Biol. 157 (1) (1982) 105–132.
[29] W.C. Wimley, S.H. Whi e, Expe imen ally de e mined hyd ophobici y scale o
p o eins a memb ane in e aces, Na u e S uc . Biol. 3 (10) (1996) 842–848.
[30] T. Hessa, H. Kim, K. Bihlmaie , C. Lundin, J. Boekel, H. Ande sson, I. Nilsson,
S.H. Whi e, G. on Heijne, Recogni ion o ansmemb ane helices by he
endoplasmic e iculum anslocon, Na u e 433 (7024) (2005) 377–381.
[31] C.P. Moon, K.G. Fleming, Side-chain hyd ophobici y scale de i ed om ans-
memb ane p o ein olding in o lipid bilaye s, P oc. Na l. Acad. Sci. 108 (25)
(2011) 10174–10177.
[32] G. Van Rossum, F.L. D ake, e al., Py hon Re e ence Manual, ol. 111, Cen um
oo Wiskunde en In o ma ica Ams e dam, 1995.
[33] H. Ab aham, I.Y. Akhalwaya, G. Aleksand owicz, T. Alexande , G. Alexand owics,
E. A bel, A. As aw, C. Azaus e, P. Ba kou sos, G. Ba on, e al., Qiski : An open-
sou ce amewo k o quan um compu ing, 2019, URL h ps://doi.o g/10.5281/
zenodo2562111, 2019.
[34] C.R. Ha is, K.J. Millman, S.J. Van De Wal , R. Gomme s, P. Vi anen, D.
Cou napeau, E. Wiese , J. Taylo , S. Be g, N.J. Smi h, e al., A ay p og amming
wi h numpy, Na u e 585 (7825) (2020) 357–362.
[35] J.D. Hun e , Ma plo lib: A 2D g aphics en i onmen , Compu . Sci. Eng. 9 (03)
(2007) 90–95.
[36] P. Ramachand an, G. Va oquaux, Maya i: 3D isualiza ion o scien i ic da a,
Compu . Sci. Eng. 13 (2) (2011) 40–51.
Compu e s in Biology and Medicine 182 (2024) 109157
9