This is a sel -a chi ed e sion o an o iginal a icle. This e sion
may di e om he o iginal in pagina ion and ypog aphic de ails.
Au ho (s):
Ti le:
Yea :
Ve sion:
Copy igh :
Righ s:
Righ s u l:
Please ci e he o iginal e sion:
CC BY 4.0
h ps://c ea i ecommons.o g/licenses/by/4.0/
Bes P ac ices in Cons an pH MD Simula ions : Accu acy and Sampling
© 2022 The Au ho s. Published by Ame ican Chemical Socie y
Published e sion
Buslae , Pa el; Aho, Noo a; Jansen, An on; Baue , Paul; Hess, Be k; G oenho ,
Ge i
Buslae , P., Aho, N., Jansen, A., Baue , P., Hess, B., & G oenho , G. (2022). Bes P ac ices in
Cons an pH MD Simula ions : Accu acy and Sampling. Jou nal o Chemical Theo y and
Compu a ion, 18(10), 6134-6147. h ps://doi.o g/10.1021/acs.jc c.2c00517
2022
Bes P ac ices in Cons an pH MD Simula ions: Accu acy and
Sampling
Pa el Buslae ,*
,#
Noo a Aho,
#
An on Jansen, Paul Baue , Be k Hess,*and Ge i G oenho *
J. Chem. Theo y Compu . 2022,10.1021/acs.jc c.2c00516
Ci e This: h ps://doi.o g/10.1021/acs.jc c.2c00517
Read Online
ACCESS Me ics & Mo e A icle Recommenda ions *
sı Suppo ing In o ma ion
ABSTRACT: Va ious app oaches ha e been p oposed o include he e ec
o pH in molecula dynamics (MD) simula ions. Among hese, he
λ-dynamics app oach p oposed by B ooks and co-wo ke s [Kong, X.;
B ooks III, C. L. J. Chem. Phys. 1996,105, 2414−2423] can be pe o med
wi h li le compu a ional o e head and h o each ypeence be used o
ou inely pe o m MD simula ions a mic osecond ime scales, as shown in
he accompanying pape [Aho, N. e al. J. Chem. Theo y Compu . 2022, DOI:
10.1021/acs.jc c.2c00516]. A such ime scales, howe e , he accu acy o he
molecula mechanics o ce ield and he pa ame iza ion becomes c i ical.
He e, we add ess hese issues and p o ide he communi y wi h guidelines on
how o se up and pe o m long ime scale cons an pH MD simula ions. We
ound ha ba ie s associa ed wi h he o sions o side chains in he
CHARMM36m o ce ield a e oo high o eaching con e gence in cons an
pH MD simula ions on mic osecond ime scales. To a oid he high compu a ional cos o ex ending he sampling, we p opose small
modi ica ions o he o ce ield o selec i ely educe he o sional ba ie s. We demons a e ha wi h such modi ica ions we ob ain
con e ged dis ibu ions o bo h p o ona ion and o sional deg ees o eedom and hence consis en pKaes ima es, while he
sampling o he o e all con igu a ional space accessible o p o eins is una ec ed as compa ed o no mal MD simula ions. We also
show ha he esul s o cons an pH MD depend on he accu acy o he co ec ion po en ials. While hese po en ials a e ypically
ob ained by i ing a low-o de polynomial o calcula ed ee ene gy p o iles, we ind ha highe o de i s a e essen ial o p o ide
accu a e and consis en esul s. By esol ing p oblems in accu acy and sampling, he wo k desc ibed in his and he accompanying
pape pa es he way o he widesp ead applica ion o cons an pH MD beyond pKap edic ion.
■INTRODUCTION
Thanks o imp o emen s in algo i hms, o ce ields, and
compu e ha dwa e, molecula dynamics (MD) simula ions
ha e become a e sa ile ool o in es iga ing he con o ma-
ional landscape o complex biomolecula sys ems a he
a omic le el.
1−5
An impo an algo i hmic imp o emen has
been he explici inclusion o pH in MD simula ions,
6−12
as pH
is an impo an expe imen al pa ame e ha a ec s he
s uc u e and dynamics o biomolecules. To p o ide he
use s o he GROMACS MD package
13
wi h access o
simula ions a cons an pH, we ha e implemen ed he
λ-dynamics-based cons an pH app oach by B ooks and co-
wo ke s.
9,14
In con as o he p e ious implemen a ion in a
o k o GROMACS 3.3,
10
he new implemen a ion, which is
desc ibed in an accompanying pape ,
15
is e icien and cons an
pH MD simula ions can be pe o med wi h abou 25% o
CPU, 30−40% o CPU + GPU compu a ional o e head
compa ed o no mal MD simula ions i espec i e o he
numbe o i a able si es in he sys em. In he accompanying
me hodological pape ,
15
we also demons a e ha he me hod
can be success ully applied o calcula e pKa alues o i a able
si es in a p o ein. The pu pose o his pape is o p o ide use s
wi h guidelines and ecommenda ions on how o se up and
pe o m cons an pH MD simula ions, including he necessa y
pa ame iza ion s eps.
In cons an pH MD simula ions, i a able g oups can
dynamically change hei p o ona ion s a e. These changes a e
d i en by in e ac ions be ween he g oup and he chemical
en i onmen (modeled wi h a o ce ield) and he aqueous
p o on concen a ion (modeled wi h a pH po en ial). Because
a he o ce ield le el a numbe o con ibu ions o he ee
ene gy o (de)p o ona ion a e no included explici ly, i.e.,
quan um mechanical in e ac ions associa ed wi h bond b eak-
age and o ma ion as well as he ac ual p o on pa icle,
co ec ions o he o ce ield a e needed in λ-dynamics-based
Recei ed: May 18, 2022
A iclepubs.acs.o g/JCTC
© XXXX The Au ho s. Published by
Ame ican Chemical Socie y A
h ps://doi.o g/10.1021/acs.jc c.2c00517
J. Chem. Theo y Compu . XXXX, XXX, XXX−XXX
Downloaded ia UNIV OF JYVASKYLA on Sep embe 19, 2022 a 12:16:48 (UTC).
See h ps://pubs.acs.o g/sha ingguidelines o op ions on how o legi ima ely sha e published a icles.
cons an pH MD. In GROMACS, hese co ec ions a e
implemen ed as analy ical unc ions, VMM(λj), i ed o he
ee ene gy p o ile associa ed wi h he dep o ona ion o a
i a able esidue ja he o ce ield le el.
15
The accu acy o such ee ene gy p o iles depends no only
on how closely he o ce ield model ep esen s he ue
po en ial ene gy su ace bu also on he con e gence o
sampling o all o he deg ees o eedom in he sys em.
The e o e, whe eas in no mal MD he accu acy o he
dynamics depends solely on he quali y o he o ce ield, he
accu acy o λ-dynamics-based cons an pH MD depends
addi ionally on whe he all ele an deg ees o eedom a e
sampled su icien ly in he simula ions equi ed o pa a-
me izing he co ec ion po en ials.
We ound ha insu icien sampling o he dihed al deg ees
o eedom in he amino acid side chains can lead o poo
con e gence in he dep o ona ion ee ene gy p o iles, as also
obse ed by Klimo ich and Mobley in simula ions wi hou
cons an pH.
16
We aced he lack o he dihed al sampling o
he ba ie s ha sepa a e he minima in he o sion po en ials.
These ba ie s a e oo high o each a con e ged sampling o
he dihed al ee ene gy landscape on he ime scales o ypical
cons an pH MD simula ions. Because he in e ac ion be ween
he i a able g oup and he en i onmen depends c i ically on
he dihed al angles o he side chain, a lack o con e gence in
hese dihed al angles also a ec s he sampling o he
p o ona ion s a es.
Ra he han inc easing he ime scale o he MD simula ions
o ob ain con e ged dihed al and p o ona ion s a e dis ibu-
ions o in oducing enhanced sampling echniques,
12,17−21
we
p opose educing he ba ie s o dihed al o a ions in a
sys ema ic way. We will demons a e ha such op imized
dihed al o ce ield pa ame e s imp o e pKaes ima es o amino
acids wi hou comp omising he o e all con o ma ional
sampling o he p o ein.
Wi h a highe accu acy o he unde lying dep o ona ion ee
ene gy p o iles, we ound ha he co ec sampling o
p o ona ion s a es also depends c i ically on he o de o he
polynomial i used o ob ain an analy ical o m o he
co ec ion po en ial. We show ha he commonly accep ed
i s -o de i ,
12
al hough i mly based on linea esponse
heo y,
22
is no su icien ly accu a e and can lead o e oneous
p o ona ion dynamics in cons an pH MD simula ions.
Because he dominan ene ge ic con ibu ion o p o on
a ini y comes om elec os a ic in e ac ions,
22
i is o u mos
impo ance o use an accu a e desc ip ion o such in e ac ions.
Cons an pH MD simula ions ha e been pe o med wi h
a ious elec os a ic models, including gene alized Bo n,
9
shi ed cu o ,
23
and pa icle mesh Ewald (PME).
10,12
O
hese me hods, he Ewald summa ion-based PME me hod
24,25
is gene ally conside ed o p o ide he mos accu a e
desc ip ion o he elec os a ic in e ac ions in pe iodic
biomolecula sys ems.
26
Because Ewald summa ion can only
p o ide accu a e esul s i he simula ion box emains neu al,
27
he cha ge luc ua ions associa ed wi h he dynamic p o o-
na ion and dep o ona ion in cons an pH MD simula ions
need o be compensa ed o p e en a i ac s. Ti a able si es
can be di ec ly coupled o special pa icles, modeled as ions o
wa e molecules,
21,28
such ha cha ge is ans e ed di ec ly
be ween he i a able si e and ha pa icle. Al e na i ely, all
si es can be coupled collec i ely o a su icien ly la ge numbe
o bu e pa icles.
29
The la e app oach has he ad an age ha
spon aneous luc ua ions in he in e ac ion o he bu e
pa icles wi h hei en i onmen a ec all i a able si es o he
same ex en . The disad an age is ha he se up and
pa ame iza ion o he bu e app oach a e mo e in ol ed, as
hese equi e selec ing he numbe o bu e s and pa ame izing
hei in e ac ion wi h he es o he sys em. To acili a e he
use o bu e s in cons an pH MD, we p o ide a pa a-
me iza ion s a egy aimed a p e en ing bu e clus e ing,
bu e binding o i a able si es, and bu e pe mea ion in o
hyd ophobic egions. We demons a e ha bu e s pa a-
me ized wi h his s a egy also a oid ini e-size e ec s
associa ed wi h he pe iodici y o small simula ion boxes.
30,31
■METHODS
He e, we go h ough all o he impo an aspec s o he
simula ion se ups.
Simula ed Sys ems. We pe o med s anda d and cons an
pH MD simula ions o he sys ems lis ed in Table 1. The
o iginal and modi ied (desc ibed in de ail below)
CHARMM36m
32,33
o ce ields we e used in all simula ions.
Table 1. Table o Simula ed Sys ems
a
sys em box size (nm3) no. o wa e s no. o ions o ce ield
BUF15×5×5∼4000 11 Na, 11 Cl, 2 Bu CHARMM36m
BUF23×3×3∼3000 1 o 2 Bu CHARMM36m
ADA 5 ×5×5∼4000 11 Na, 11 Cl, 1 o 10 Bu CHARMM36m, CHARMM36m-cph
ADA33×3×3∼850 2 Na, 2 Cl, 10 Bu CHARMM36m-cph
ADA77×7×7∼11100 31 Na, 31 Cl, 10 Bu CHARMM36m-cph
ADAlow sal 5×5×5∼4000 4 Na, 4 Cl, 10 Bu CHARMM36m-cph
ADAhigh sal 5×5×5∼4000 38 Na, 38 Cl, 10 Bu CHARMM36m-cph
AEA 5 ×5×5∼4000 11 Na, 11 Cl, 1 o 10 Bu CHARMM36m, CHARMM36m-cph
AKA 5 ×5×5∼4000 11 Na, 12 Cl, 1 o 10 Bu CHARMM36m, CHARMM36m-cph
AHA 5 ×5×5∼4000 11 Na, 12 Cl, 1 o 10 Bu CHARMM36m, CHARMM36m-cph
AAA15×5×5∼4000 11 Na, 12 Cl, 1 o 10 Bu CHARMM36m, CHARMM36m-cph
AAA25×5×5∼4000 11 Na, 11 Cl, 1 o 10 Bu CHARMM36m, CHARMM36m-cph
1CVO16.8 ×7.8 ×7.1 ∼12400 35 Na, 48 Cl, 20 Bu CHARMM36m, CHARMM36m-cph
1CVO27.1 ×7.1 ×7.1 ∼11400 13 Cl, 150 Bu CHARMM36m
MEMB15.9 ×5.9 ×8.1 ∼4200 8 Na, 8 Cl, 50 Bu CHARMM36m
MEMB25.9 ×5.9 ×8.1 ∼4200 8 Na, 8 Cl, 1 Bu CHARMM36m
SOL 4.3 ×4.3 ×4.3 ∼2600 50 Bu CHARMM36m
a
We deno e he modi ied CHARMM36m o ce ield as CHARMM36m-cph.
Jou nal o Chemical Theo y and Compu a ion pubs.acs.o g/JCTC A icle
h ps://doi.o g/10.1021/acs.jc c.2c00517
J. Chem. Theo y Compu . XXXX, XXX, XXX−XXX
B
The able also p esen s he box size, he numbe o
CHARMM36 TIP3P wa e molecules,
34−36
he numbe o
ions, and bu e pa icles included in each sys em. The i ing
coe icien s o he VMM co ec ion po en ial o he bu e
pa icles we e ob ained wi h sys em BUF1. To ind he op imal
cha ge ange and Lenna d−Jones pa ame e s o he bu e
pa icles, enhanced sampling simula ions wi h he accele a ed
weigh his og am (AWH) me hod we e pe o med on sys em
BUF2. Sys ems ADA, AEA, AKA, and AHA a e alanine
ipep ides wi h capped e mini and as he cen al esidue
aspa ic, glu amic, lysine, and his idine amino acids,
espec i ely. AAA1and AAA2sys ems a e alanine ipep ides
wi h p o ona ed e mini. C- and N- e mini we e made
i a able in AAA1and AAA2sys ems, espec i ely. Two se s
o simula ions o he ca dio oxin V p o ein we e pe o med.
Sys em 1CVO1was used o calcula e he pKa alues o
i a able esidues, while he la ge sys em 1CVO2was used o
compu e he adial dis ibu ion unc ion o he bu e pa icles
a ound he p o ein. The memb ane sys ems MEMB1and
MEMB2con ained 106 1-palmi oyl-2-oleoyl-glyce o-3-phos-
phocholine (POPC) lipids. The s a ing coo dina es and
opologies o all sys ems a e p o ided as Suppo ing
In o ma ion.
Simula ion Se up. Pe iodic bounda y condi ions we e
applied in all sys ems. Elec os a ic in e ac ions we e modeled
wi h he pa icle mesh Ewald me hod,
24,25
while an de Waals
in e ac ions we e modeled wi h Lenna d−Jones po en ials
which we e smoo hly swi ched o ze o in he ange om 1.0 o
1.2 nm. Simula ions we e pe o med a a cons an empe a u e
o 300 K, main ained by he - escale he mos a ,
37
wi h a ime
cons an o 0.5 ps−1and a a cons an p essu e o 1 ba ,
main ained by he Pa inello−Rahman ba os a ,
38
wi h a
pe iod o 2.0 ps. The leap og in eg a o wi h an in eg a ion
s ep o 2 s was used. Bond leng hs o hyd ogens in he solu e
we e cons ained wi h he LINCS algo i hm,
39
while he
in e nal deg ees o he CHARMM TIP3P wa e molecules
35,36
we e cons ained wi h he SETTLE algo i hm.
40
P io o he
cons an pH MD simula ions, he ene gy o all sys ems was
minimized using he s eepes descen me hod, ollowed by a
1 ns equilib a ion.
Cons an pH MD Simula ion Se up. In he cons an pH
MD simula ions, he mass o λ-pa icles was se o 5 a omic
uni s and he empe a u e was kep cons an a 300 K wi h a
sepa a e - escale he mos a o he λdeg ees o eedom
37
wi h a ime cons an o 2.0 ps−1. The single-si e ep esen a ion,
de ined and desc ibed in he accompanying pape ,
15
was used
o Asp, Glu, Lys, C- e , and N- e , whe eas he mul isi e
ep esen a ion, also desc ibed in ha pape , was used o
His.
15,41,42
The mul isi e ep esen a ion models each physical
s a e o g oups wi h chemically coupled i a able si es wi h an
independen λ-coo dina e and akes chemical coupling in o
accoun by applying he linea cons ain on hese
λ-coo dina es equi ing
=1
ii
g oup
g oup
.
15
The same pH
and biasing po en ials we e used as in Aho e al.
15
In he
sampling simula ions o single i a able esidues, he pH was
se equal o he pKaand he ba ie heigh o he biasing
po en ial was se o ze o. The i a ion o he ca dio oxin V
(PDB ID 1CVO
43
) p o ein was pe o med by unning 10
independen eplicas o 100 ns each o 15 equidis an ly spaced
pH alues in he ange om 1.0 o 8.0 using bo h he o iginal
and a modi ied CHARMM36m o ce ield. In he i a ion
simula ions, he ba ie heigh o he biasing po en ial was se
o 7.5 kJ mol−1 o g oups modeled wi h a single-si e
ep esen a ion and o 5 kJ mol−1 o g oups modeled wi h a
mul isi e ep esen a ion.
Re e ence Simula ions. The cons an pH simula ions
equi e a co ec ion po en ial VMM(λj) o each i a able
esidue j. These co ec ion po en ials a e he in eg als o
polynomial i s o he expec a ion alue o ⟨∂V/∂λ⟩λin
e e ence s a e simula ions a ixed λ- alues.
15
Thus, a e
in eg a ion, an n h-o de polynomial i o ⟨∂V/∂λ⟩λyields an
(n+ 1) h-o de polynomial unc ion ha ep esen s VMM(λ).
Howe e , in ou implemen a ion, he i o ⟨∂V/∂λ⟩λwas used,
a he han VMM(λ). We hus e e o he i ing o de as he
o de o he polynomial i o ⟨∂V/∂λ⟩λ.
We pe o med he e e ence simula ions as ollows: The
pa ial cha ges in he ipep ide sys ems we e linea ly
in e pola ed be ween λ=−0.1 and λ= 1.1 wi h a s ep o
0.05. Fo His, all h ee λcoo dina es we e changed unde he
cons ain λ1+λ2+λ3= 1. Fo each se o λ- alues, called a
g id poin , we pe o med an 11 ns MD simula ion in which he
∂V/∂λjwe e sa ed e e y picosecond and accumula ed. The
o al cha ge o he sys em was kep neu al by simul aneously
changing he cha ge o a single bu e pa icle. The i ing
p ocedu e is desc ibed in ull de ail in he accompanying
pape .
15
Dihed al F ee Ene gy P o iles. Because he sampling o
p o ona ion s a es is igh ly coupled o he sampling o side
chain dihed al deg ees o eedom, we compu ed he ee
ene gy p o iles associa ed wi h he o a ion o he dihed als in
he side chain o he cen al amino acid in he capped
ipep ide sys ems (Table 1) by means o umb ella
sampling.
44,45
As he i s s ep, we pe o med 20 ns MD
simula ions wi h a ime-dependen po en ial on he dihed al
angle wi h a o ce cons an o 418.4 kJ mol−1 ad−2. The cen e
o his po en ial was mo ed om 0° o 360°wi h a a e o 18°
ns−1. F om hese simula ions, ames wi h dihed al angles
closes o 0°, 10°, 20°, e c., we e selec ed as e e ences o he
umb ella eplicas. The di e ence be ween he dihed al angle in
he selec ed ames and he a ge angle was always below 0.1°.
Then, we pe o med 36 umb ella sampling simula ions o 11 ns
wi h a ha monic es aining po en ial cen e ed a he e e ence
dihed al angle and a o ce cons an o 418.4 kJ mol−1 ad−2. We
used he WHAM p ocedu e,
46
implemen ed in GROMACS,
47
o unbias hese umb ellas and ob ain ee ene gy p o iles
associa ed wi h he ull o a ion o he dihed al angle.
Dihed al Po en ial Ene gy P o iles a he QM and MM
Le els. To check he alidi y o he p oposed o ce ield
modi ica ions, we compu ed he po en ial ene gy p o iles o
he N−Cα−Cβ−Cγdihed al o aspa ic acid wi h capped
esidues. The p o iles we e compu ed a bo h quan um
mechanical (QM) and molecula mechanical (MM) le els.
The QM p o iles we e compu ed a he MP2/6-31+G*le el o
heo y using he Fi e ly QC package,
48
which is pa ially based
on he GAMESS (US)
49
sou ce code. The MM p o iles we e
compu ed o bo h he o iginal and he modi ied
CHARMM36m o ce ields. The po en ial ene gy was
compu ed o he N−Cα−Cβ−Cγdihed al angle wi h 10°
inc emen s. Fo each dihed al alue, he s uc u es we e ene gy
minimized p io o po en ial ene gy calcula ion.
Accele a ed Weigh His og am Alchemical Simula-
ions. Bu e pa icles a e used in cons an pH MD o main ain
he neu ali y o he simula ed sys em. Ideally, bu e s should
no in oduce any a i ac s due o binding o i a able g oups,
binding o each o he , o pene a ing in o hyd ophobic egions.
Jou nal o Chemical Theo y and Compu a ion pubs.acs.o g/JCTC A icle
h ps://doi.o g/10.1021/acs.jc c.2c00517
J. Chem. Theo y Compu . XXXX, XXX, XXX−XXX
C
To p e en such beha io , we op imized he cha ge ange and
Lenna d−Jones pa ame e s o he bu e s. To his end, we
pe o med a se ies o enhanced sampling simula ions wi h he
accele a ed weigh his og am me hod (AWH).
50,51
In one se
o simula ions wi h wo bu e s in he simula ion box (BUF2,
Table 1), we quan i ied he sampling e iciency om he
ic ion me ic
50,51
as a unc ion o he absolu e cha ge pe
bu e pa icle. In hese simula ions, he cha ge o one bu e
was changed om 0 o +0.8 while simul aneously he cha ge o
he o he bu e was changed om 0 o −0.8 in o de o
main ain neu ali y. The CHARMM36m Lenna d−Jones
pa ame e s o sodium we e used o he bu e s in hese
simula ions. In he o he se o simula ions, we compu ed he
ee ene gy di e ence be ween in oducing a neu al bu e in
wa e (BUF2) and inside he hyd ophobic egion o a POPC
bilaye sys em (MEMB2,Table 1) o a ious alues o he
Lenna d−Jones pa ame e s o he bu e . In hese simula ions,
he Lenna d−Jones in e ac ions be ween he bu e and he
es o he sys em we e inc eased om nonin e ac ing a λ= 0
o ully in e ac ing (λ= 1) in 10 disc e e s eps. Fo all sys ems,
we simula ed 10 eplicas o 10 ns, om which he ee ene gy
di e ences and ic ion coe icien s we e ob ained as a e ages
o e he eplicas.
Dihed al Analysis. Dis ibu ions o side chain dihed al
angles in p o eins we e de i ed om publicly sha ed MD
ajec o ies o p o eins wi h PDB IDs 1U19,
52
2RH1,
53
2Y02,
54
and 5UEN
55
ob ained om he GPCRMD
56
and
SARS-CoV-2 da abases (h ps://co id.molssi.o g/).
57
Fo
each ajec o y, he dis ibu ions o he ollowing dihed al
angles we e calcula ed:
(1) N−Cα−Cβ−Cγin aspa ic acid
(2) N−Cα−Cβ−Cγin glu amic acid
(3) Cα−Cβ−Cγ−Cδin glu amic acid
(4) N−Cα−Cβ−Cγin his idine
(5) H−Oϵ2−C−Oϵ1in aspa ic and glu amic acids.
In his wo k, we also compu ed he dis ibu ions o hese
dihed als om s anda d MD ajec o ies o ca dio oxin V
(PDB ID 1CVO
43
).
Compa isons o Dis ibu ions. To compa e wo
dis ibu ions Pi(x) and Pj(x), wi h he co esponding
cumula i e dis ibu ion unc ions Fi(x) and Fj(x), we used
Kolmogo o −Smi no s a is ics (KSS)
58
= [ ]F F F x F xKSS( , ) sup ( ) ( )
i j
x
i j
(1)
The la ge he KSS, he less simila he dis ibu ions
a e. The dis ibu ions we e conside ed consis en when
KSS(Fi,Fj) < 0.03. The KSS was compu ed using he sc ip
om he PCAlipids package.
59,60
Ti a ion. To es ima e he pKa alues o i a able g oups
om mul iple simula ions a a ious pH alues, we compu ed
he a e age ac ion o dep o ona ed ames (Sdep o ) o e all
eplicas a each pH alue. Fo a g oup wi h a single i a able
si e, his a e age was ob ained as
=
+
SN
N N
(pH)
dep o dep o
p o dep o
(2)
whe e Np o and Ndep o a e he o al numbe o ames in which
he si e is p o ona ed and dep o ona ed, espec i ely. Fo
i a able si es modeled wi h he single-si e ep esen a ion, we
Figu e 1. Dis ibu ions o he λ-coo dina e (A and E) and dihed als (B−D and F−H) in cons an pH MD simula ion o Asp wi h he o iginal
(A−D) and modi ied CHARMM36m (E−H) o ce ield. While he simula ions we e pe o med o an ADA ipep ide, only he cen al aspa ic
acid is shown o cla i y in he inse o A. (A and D) Dis ibu ions o hi d-o de i s o ⟨∂V/∂λ⟩λob ained wi h he o iginal o ce ield. Di e en
colo s co espond o independen eplicas. Dis ibu ions o he λ-coo dina e (A) as well as dis ibu ions o N−Cα−Cβ−Cγ(B) and
O
1
-Cγ-
O
2
-
H
2
(D) dihed als a e no iden ical. Righ column (E−H) shows he dis ibu ions o he modi ied CHARMM36m o ce ield wi h hi d-o de i
o ⟨∂V/∂λ⟩λ. Dis ibu ions a e iden ical.
Jou nal o Chemical Theo y and Compu a ion pubs.acs.o g/JCTC A icle
h ps://doi.o g/10.1021/acs.jc c.2c00517
J. Chem. Theo y Compu . XXXX, XXX, XXX−XXX
D

conside ed he si e p o ona ed i λis below 0.2 and
dep o ona ed i λis abo e 0.8. Fo si es ha a e desc ibed
wi h he mul isi e desc ip ion, we conside ed a s a e
p o ona ed i he λassocia ed wi h he p o ona ed o m o
he esidue is abo e 0.8 and dep o ona ed i he λassocia ed
wi h he dep o ona ed o m o he esidue is abo e 0.8.
To es ima e he mac oscopic pKa alue o his idine, which
con ains wo i a able si es Nϵand Nδ, we calcula ed o each
pH alue he a e age ac ion o ames in which he esidue is
dep o ona ed a ei he o he wo si es
=
+ +
SN
NNN
(pH) 1
mac o
dep o p
p
(3)
whe e
Np
,
N
, and
N
a e he numbe o ames in which
λp> 0.8, λϵ> 0.8, and λδ> 0.8.
The a e aged ac ions a each pH alue we e i ed o he
Hende son−Hasselbalch equa ion
=
+
S1
10 1
K
dep o
(p pH)
a
(4)
which yielded he pKa alues as i ing pa ame e s.
■RESULTS AND DISCUSSION
Fi s , we demons a e ha a lack o sampling o he ele an
dihed al deg ees o eedom in amino acid side chains wi h he
CHARMM36m o ce ield educes he accu acy o he
co ec ion po en ials o λ-dynamics. To o e come hese
con e gence p oblems, we modi y he o ce ield by educing
he ba ie s in he o sion po en ial and show ha his
signi ican ly imp o es he accu acy o he co ec ion po en ials
and hence he esul s o cons an pH simula ions, including
pKaes ima es, wi hou a ec ing he p o ein con o ma ional
dynamics. A e he alida ion o he modi ied o ce ield
pa ame e s, we show how he bu e pa icles ha main ain he
neu ali y o he simula ion box ha e o be pa ame ized o
p e en ini e-size e ec s on p o on a ini ies due o
pe iodici y.
30,31
Sampling. Klimo ic and Mobley ha e shown ha
calcula ed hyd a ion ee ene gies o single amino acids depend
on he s a ing con o ma ion.
16
Because a ew picoseconds
ypically su ice o sample bond and angle deg ees o eedom
in he amino acid as well as he o a ional deg ees o eedom
o he wa e molecules, we specula e ha hei obse a ion
implies a lack o sampling in he dihed al deg ees o eedom in
he amino acid side chain. The e o e, we sys ema ically
analyzed he con e gence o bo h λ-coo dina es and dihed al
angles du ing cons an pH simula ions o single amino acids in
wa e .
We pe o med 100 ns cons an pH MD simula ions a
pH = pKao sys ems ADA, AEA, AKA, AHA, AAA1, and AAA2
(Table 1). To enhance he sampling o he λ-coo dina e in
hese sys ems, we an he simula ions wi hou a ba ie in he
biasing po en ial (Vbias(λ), eq 5 in Aho e al.
15
). The co ec ion
po en ial (VMM(λ), eq 5 in Aho e al.
15
) was ob ained by i ing
a hi d-o de polynomial unc ion o he ⟨∂V/∂λ⟩λ alues o he
e e ence ajec o ies. We will show la e ha o accu a e and
ep oducible cons an pH MD esul s, a highe o de i is
equi ed. Ne e heless, in spi e o i s limi ed accu acy, using
he same hi d-o de i o all sys em su ices o sys ema ically
compa e he dis ibu ions o he ele an deg ees o eedom
and assess hei con e gence.
Figu e 2. Dis ibu ions o dihed al angles o which he o sion po en ials we e co ec ed om s anda d MD simula ions. (A) Dihed al dis ibu ions
om publicly a ailable ajec o ies
56,57
( o al simula ion ime ≈10 μs). P obabili y densi y is signi ican only a ound local minima. (B) Dihed al
dis ibu ions ob ained om simula ions o ca dio oxin V wi h bo h he o iginal and he modi ied ba ie s o di e en p o ona ion s a es o he
i a able esidues. Local minima a e p ese ed, and no addi ional con igu a ions a e obse ed.
Jou nal o Chemical Theo y and Compu a ion pubs.acs.o g/JCTC A icle
h ps://doi.o g/10.1021/acs.jc c.2c00517
J. Chem. Theo y Compu . XXXX, XXX, XXX−XXX
E
In Figu e 1A, we show he dis ibu ions o he λ-coo dina e
in i e cons an pH MD eplicas o he ADA sys em wi h he
o iginal CHARMM36m o ce ield pa ame e s. Dis ibu ions o
he λ-coo dina es in he o he sys ems (AEA, AKA AHA, and
AAA1and AAA2) a e shown in he Suppo ing In o ma ion
(SI, Figu es S1−S5). The dissimila i y be ween he
λ-dis ibu ions in he eplicas (maximum KSS be ween eplicas
o 0.29, 0.11, 0.04, and 0.095 o ADA, AEA, AHA, and AAA1,
espec i ely) indica es a lack o con e gence. In addi ion, he
dis ibu ions o he dihed al angles, shown in Figu e 1B−D, a e
also no iden ical o all eplicas. Because he e is no ba ie
om he biasing po en ial o he λ-coo dina e, we conclude
ha he lack o con e gence in λis due o insu icien sampling
o he dihed al deg ees o eedom.
Fo ce Field Modi ica ions. To o e come he lack o
sampling, one can inc ease he simula ion ime o enhance
sampling by means o special algo i hms such as eplica
exchange MD.
61,62
Replica exchange has been applied
equen ly in he con ex o cons an pH MD wi h exchanges
be ween eplicas a di e en empe a u e (T-REMD), pH
(pH-REMD), o bo h.
20,63−66
Howe e , because he p o o-
na ion s a e has li le e ec on he o sion ba ie heigh
(Figu e 3), changing he pH ac oss he pH ladde would do
li le o enhance he sampling o he dihed al deg ees o
eedom. Fu he mo e, REMD me hods a e compu a ionally
mo e demanding han pe o ming a single MD simula ion and
also p e en access o he dynamical p ope ies o he sys em
due o he jumps be ween eplicas. As o some applica ions
he dynamical p ope ies may be highly needed, we conside i
impo an ha all ele an p o ona ion s a es can be sampled
co ec ly in a single cons an pH MD ajec o y. Because he
sampling o λ-coo dina es is igh ly coupled o he sampling o
he dihed al angles, con e gence wi hin a single ajec o y can
in p inciple also be eached by lowe ing he ba ie s o he
o sion po en ials.
P e iously, modi ica ions o he o ce ield ha e been
in oduced o ca boxyl g oups. To imp o e he sampling o
he syn and an i con o ma ions o he ca boxyl p o on in Glu,
Asp, and he C- e minus, B ooks and co-wo ke s educed he
ba ie o his o a ion by a ac o o 8 and also scaled he
ca boxyl oxygen adii by 0.95.
9
In con as , G ubmulle and co-
wo ke s modi ied his o sion po en ial o p e en sampling he
an i-con o ma ion al oge he .
67
Howe e , acco ding o ou
analysis (Figu e 1), he e is a lack o con e gence no only in
he ca boxyl dihed al angle bu also in he o he side chain
dihed al angles.
Reducing he o sional ba ie s wi hou a ec ing he o e all
sampling o he con o ma ional space is possible only i he
egions nea such ba ie s a e spa sely sampled. We, he e o e,
analyzed he dis ibu ions o he dihed al angles in
he side chains o i a able amino acids in he publicly
a ailable ajec o ies o G-p o ein-coupled ecep o s and o
SARS-CoV-2 p o eins.
56,57
The dis ibu ions o hese dihed al
angles, plo ed in Figu e 2A, e lec he shape o he unde lying
o sion po en ials wi h maxima coinciding wi h local minima o
he po en ial p o iles. The low densi y nea ba ie s sugges s
ha hese ba ie s a e a he high and migh be educed
wi hou a ec ing he dihed al dis ibu ions.
To achie e con e gence o bo h dihed al angles and
λ-coo dina es on a 100 ns ime scale, we hus al e ed he
maxima o he o sion po en ials by adding a small dihed al
Figu e 3. Modi ica ion o he o sional ba ie s in he Asp side chain. (A) Aspa ic acid and i s a omic nomencla u e. (B and C) Co ec ions added
o dihed al o sions o he o iginal o ce ield. Co ec ions wi h wo (B) and h ee (C) local minima we e used o o sions wi h wo and h ee local
minima, espec i ely. Heigh s o he co ec ions we e selec ed h ough he i e a i e p ocess, aimed a achie ing consis en λ-dis ibu ions wi hou
in oducing addi ional local minima in he ee ene gy p o iles. (D−F) O iginal and modi ied o sional ba ie s o Asp o bo h p o ona ed (H+)
and dep o ona ed (H−) s a es.
Jou nal o Chemical Theo y and Compu a ion pubs.acs.o g/JCTC A icle
h ps://doi.o g/10.1021/acs.jc c.2c00517
J. Chem. Theo y Compu . XXXX, XXX, XXX−XXX
F
angle (ϕ)-dependen co ec ion o he CHARMM36m o ce
ield
=V n n( ) cos( ) ( 1) 1
4cos(2 )
i i i
n
i
i
Ä
Ç
Å
Å
Å
Å
Å
Å
Å
Å
É
Ö
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
(5)
whe e niis he mul iplici y o he o sion angle i(i.e., he
numbe o minima) wi h ni= 2 o conjuga ed bonds and ni= 3
o alipha ic bonds. The pa ame e ϵiis an empi ical coe icien
ha is op imized such ha he ba ie s a e low enough o
con e ge he dis ibu ion o ϕiwi hou in oducing addi ional
minima on he po en ial ene gy su ace.
Fo each side chain dihed al angle o he i a able amino
acids, he coe icien ϵiwas op imized in an i e a i e ashion:
A e an ini ial guess, we compu ed he ee ene gy p o iles
associa ed wi h o a ion o he dihed al as well as i e unbiased
100 ns ajec o ies a pH = pKawi h di e en s a ing
condi ions and a biasing po en ial (Vbias(λ), eq 5 in Aho e
al.
15
) wi hou ba ie . P io o hese cons an pH MD
simula ions, we ecompu ed he co ec ion po en ial,
VMM(λ), by i ing a hi d-o de polynomial o he ⟨∂V/∂λ⟩λ
alues ob ained om he modynamic in eg a ion simula ions
pe o med wi h he cu en alue o ϵi. F ee ene gy p o iles
we e inspec ed isually o a i icial minima, while dis ibu ions
o bo h dihed al angles and λ-coo dina es we e compa ed
be ween he i e unbiased eplica uns based on hei simila i y.
The coe icien ϵiwas g adually inc eased un il he
dis ibu ions in he di e en eplicas we e su icien ly simila
(KSS < 0.03), while a he same ime no addi ional minima
appea ed in he ee ene gy p o iles.
In Figu e 3 we show he op imized o sion po en ials as well
as hei e ec on he ee ene gy p o iles o he dihed al angles
in Asp. The co ec ions and ee ene gy p o iles o Glu, His,
and he C- e minus a e shown in Figu es S6−S8 o he SI.
Wi h he excep ion o he Cβ−Cγ−Oδ−H dihed al, he
co ec ions in oduce no addi ional minima on he ee ene gy
p o ile o hese o sions. Fu he mo e, as shown on he igh
panels o Figu e 1, he dis ibu ions o he dihed als and
λ-coo dina es a e nea ly indis inguishable o all eplicas a e
co ec ion. Because wi h he co ec ed po en ials he
Kolmogo o −Smi no s a is ics o Asp, Glu, His, and he
C- e minus a e 0.028, 0.015, 0.027, and 0.022, we conclude
ha he co ec ions imp o e he con e gence o bo h he
λ-coo dina es and he dihed al deg ees o eedom in cons an
pH simula ions. No e ha al hough he dis ibu ions o he
λ-coo dina e a e su icien ly simila , he sampling o hese
coo dina es is no ye uni o m (Figu e 1E). We will show
below ha his disc epancy is due o he low o de o he
polynomial i used o ob aining he co ec ion po en ial
VMM(λ).
Wi h he excep ion o he
O
1
−Cγ−
O
2
−H dihed al, he
co ec ions we p opose he e lead o changes in he o sion
ba ie o a mos 16 kJ mol−1(i.e., o he Glu N−Cα−Cβ−Cγ
o sion). Fo many biomolecula o ce ields, he pa ame e s o
he o sion po en ials a e ob ained by i ing sui able pe iodic
unc ions o ene gies e alua ed a he MP2 le el o heo y.
68−71
The pa ame e s o each ype o o sional po en ial a e
simul aneously i ed o mul iple amino acids. The e o e, he
a e age oo mean squa ed (RMS) di e ence be ween he
o sional ene gy a he CHARMM36m le el and ha a he
MP2 le el o heo y is on he o de o 10 kJ mol−1. The RMS
de ia ion be ween he modi ied and he o iginal o sion
po en ials is a mos 8 kJ mol−1, and he RMS de ia ion
be ween he ab ini io po en ial a he MP2/6-31+G*le el and
he N−Cα−Cβ−Cγ o sion po en ial in ASP is educed om 4
kJ mol−1 o he o iginal CHARMM36m o ce ield o
3.5 kJ mol−1 o he modi ied CHARMM36m o ce ield
(Figu e S9). The e o e, we conclude ha wi h he co ec ions
o he o sion po en ials, he modi ied o ce ield p o ides an
equally good i o QM po en ial p o iles as he o iginal
CHARMM36m o ce ield.
33,71
We also pe o med s anda d MD simula ions and simula ed
i e eplicas o 100 ns o he wo p o ona ion s a es o he
Asp ipep ide in wa e using bo h he o iginal and he
modi ied CHARMM36m o ce ield pa ame e s. Wi hou he
modi ica ions, he local minima a e no consis en ly sampled in
all eplicas (Figu e S10). In con as , wi h he co ec ions,
iden ical dis ibu ions o he dihed al angles a e ob ained also
in s anda d MD simula ions. In addi ion, he modi ica ions a e
essen ial o sample bo h syn and an i con o ma ions o he
ca boxyl p o on, in line expe imen .
72
We no e, howe e , ha
he co ec ion equi ed o sampling bo h o hese con-
o ma ions signi ican ly al e s he shape o he ba ie (Figu e
3F). Ne e heless, because o hei low mass, p o on can
unnel h ough such ba ie s, and he e o e, we conside he
shape and heigh o he o sional ba ie less ele an o his
speci ic dihed al han o he o he dihed als.
Finally, we demons a e ha he modi ica ions do no al e
he dis ibu ions o he dihed al angles in p o ein simula ions.
We pe o med MD simula ions o he 1CVO1sys em bo h
wi h and wi hou he modi ica ions o he o sion po en ials o
i a able amino acids wi h ei he (i) all o hese esidues
p o ona ed, (ii) all dep o ona ed, o (iii) all Asp esidues
dep o ona ed and all o he esidues p o ona ed. In Figu e 2B,
we plo he dis ibu ions o he dihed al angles o which
co ec ions we e in oduced. The high simila i y be ween he
dis ibu ions sugges s ha he co ec ions do no lead o he
sampling o di e en dihed al dis ibu ions, e en i he ela i e
weigh s o he minima a e sligh ly al e ed, in pa icula , o he
H−O−C−O dihed al. We conclude, he e o e, ha he
co ec ions in oduced o acili a e sampling o he dihed al
and λ-coo dina es do no signi ican ly al e he p o ein
con o ma ional landscape and can hence be used o pe o m
bo h no mal and cons an pH MD simula ions.
Quali y o he Co ec ion Po en ial VMM.Re e ence
Po en ial. To e i y ha he modi ied o sion po en ials
o e come he con e gence p oblems, we pe o med cons an
pH simula ions a pH = pKaand wi hou a ba ie in he
biasing po en ial. Because wi h such a se up he po en ial
ene gy p o ile o he λ-pa icle should be la , we expec ed a
uni o m λ-dis ibu ion, p o ided ha he dihed al deg ees o
eedom a e su icien ly sampled. Howe e , as shown in Figu e
1E, he dis ibu ions a e iden ical be ween eplicas bu no
uni o m despi e he co ec ions o he o sion po en ials.
Because bo h he pH-dependen po en ial VpH(λ) and he
biasing po en ial a e la by cons uc ion a pH = pKa, he
de ia ions mus o igina e om disc epancies be ween he
co ec ion po en ial VMM(λ) and he unde lying ee ene gy
p o ile associa ed wi h dep o ona ion. The co ec ion po en ial
is ob ained as a polynomial i o he ⟨∂V/∂λ⟩λ alues om
he modynamic in eg a ion simula ions. Because linea e-
sponse (LR) heo y p edic s a linea dependence be ween he
hyd a ion ee ene gy and he magni ude o a (poin ) cha ge, a
i s -o de i has o en been used o ob ain he co ec ion
po en ial o cons an pH MD.
9,12
Howe e , e en i he change
in he cha ge domina es he ee ene gy o changing he
Jou nal o Chemical Theo y and Compu a ion pubs.acs.o g/JCTC A icle
h ps://doi.o g/10.1021/acs.jc c.2c00517
J. Chem. Theo y Compu . XXXX, XXX, XXX−XXX
G
p o ona ion s a e, hyd ogen-bond ea angemen s can con ib-
u e as well. Because he e ec s due o such s uc u al
ea angemen s a e neglec ed in LR heo ies, we hypo hesized
ha highe o de i s may be necessa y o ob aining
su icien ly accu a e co ec ion po en ials.
To es ou hypo hesis, we in es iga ed he accu acy o he
polynomial i o he co ec ion po en ial. In Figu e 4A, we
show he mean e o o he co ec ion po en ial wi h espec o
he compu ed ee ene gy di e ence associa ed wi h dep o o-
na ion as a unc ion o he i ing o de . Fo he LR
app oxima ion he i ing e o s a e highe han 10 kJ/mol.
In he wo s -case scena io, such e o s could lead o de ia ions
in p edic ed pKa alues o mo e han one pKauni . Wi h an
e o o 4 kJ mol−1, he hi d-o de i , used abo e o add ess
he con e gence issues, does no yield a su icien ly accu a e
ep esen a ion o he unde lying ee ene gy p o ile. Inc easing
he o de o he polynomial i educes his e o , and as shown
in Figu e 4B, a leas a se en h-o de i is equi ed o p o ide a
uni o m dis ibu ion o he λcoo dina e o he Asp ipep ide
in cons an pH MD simula ions a pH = pKa.
Also, o ca boxyl g oups in he side chains o Glu and in he
C- e minus, a polynomial i o ⟨∂V/∂λ⟩λo a leas se en h-
o de is needed o p o ide a su icien ly accu a e co ec ion
po en ial (Figu e 4 and Figu es S11 and S14 in SI). Fo he
imidazole ing o His wi h h ee coupled i a able si es, a
se en h-o de i su ices as well (Figu e S13), while o he
amino bases in he side chain o Lys and he N- e minus, a
leas an eigh h-o de i is equi ed (Figu es S12 and S15 in
SI). We specula e ha he highe o de i is needed o he
la e si es due o he la ge change in he cha ge on he cen al
ni ogen a om om −0.3 e o −0.96 eupon dep o ona ion.
The change in he cha ge o he ca boxylic oxygen om 0.55 e
o −0.76 eis smalle as a e he changes on he ni ogen a oms
o he imidazole ing o His ( om −0.36 e o −0.7 e).
Pa ame e iza ion o Bu e Pa icles. A change in he
p o ona ion s a e a ec s he o al cha ge o he simula ed
sys em, which can lead o a i ac s when Ewald summa ion is
used o ea he elec os a ic in e ac ions.
27,31
In ou
implemen a ion o cons an pH MD, we a oid his p oblem
by in oducing i a able bu e s in o he simula ion box ha
compensa e o he cha ge luc ua ions o he i a able
esidues.
29
In he o iginal implemen a ion o cons an pH MD in
GROMACS,
10
he bu e s we e hyd onium molecules ha
compensa ed o he o e all cha ge luc ua ions by changing
hei cha ge be ween 0 and +1 e. To p e en sampling cha ges
beyond his in e al, a biasing po en ial wi h s eep edges a
λ= 0 and 1 was in oduced o es ic he ange o λ- alues.
Howe e , his po en ial s ill in oduces addi ional o ces a he
edges o he λ-in e al. To a oid he e ec s o such o ces, we
use a comple ely la biasing po en ial o he bu e s, also
ou side o he cha ge in e al.
Because changing he cha ge o a bu e pa icle in solu ion
induces local ea angemen s o he hyd ogen-bonding ne -
wo k ha in u n could a ec he p o on a ini y o a nea by
i a able g oup, we wan o minimize he impac o cha ging
he bu e pa icles. To de e mine he cha ge ange in which
he bu e s do no cause signi ican hyd ogen-bond ne wo k
ea angemen , we an AWH simula ions wi h wo ions, he
cha ges o which a e changed simul aneously in opposi e
di ec ions (BUF2sys em). F om he ic ion me ic a ailable in
he AWH me hod,
50
we es ima ed he local di usion
coe icien , which is ela ed o he e iciency o sampling:
The highe he ic ion, he slowe he dynamics and he mo e
sampling is equi ed o each con e gence. We calcula ed he
ic ion coe icien (Figu e 5A) o he coo dina e associa ed
wi h changing he cha ge on he bu e . Fo cha ges highe
han 0.5 e, he ic ion was mo e han 50% highe han ha o
ze o cha ges, e lec ing longe co ela ion imes and hence
slowe dynamics. We, he e o e, conclude ha he op imal
ange o he bu e cha ge is be ween −0.5 eand 0.5 e.
The collec i e λ-coo dina e o he bu e pa icles is no
es ic ed o a ixed in e al by a wall-like po en ial. To a oid
he bu e cha ge exceeding he op imal ange, mul iple bu e
pa icles a e needed in he simula ion box. The op imal
numbe o bu e s can be calcula ed based on he analysis o
cha ge luc ua ions pe o med by Donnini e al.
29
Wi h a small cha ge, a bu e pa icle is apola . To p e en
clus e ing o such apola pa icles in wa e , pe mea ion in o
hyd ophobic a eas, such as memb ane in e io s, o in e ac ions
wi h he p o ein, he Lenna d−Jones pa ame e s (σand ϵ) o
he bu e s we e chosen such ha he bu e s ha e only
epulsi e in e ac ions wi h all o he a oms, excep wa e . A e
expe imen ing wi h he pa ame e s o he bu e pa icles, we
se led on a σo 0.25 nm and an ϵo 4 kJ mol−1. This choice
leads o dec eased clus e ing o bu e s, low bu e concen-
a ions in he p oximi y o i a able si es, and educed
pene a ion in o hyd ophobic egions (Figu e 5). The esul ing
ee ene gies o neu al bu e inse ion in o wa e and he
hyd ophobic egion o he memb ane a e −2.09 ±0.07 and
1.2 ±0.6 kJ mol−1compa ed o 8.45 ±0.05 and 7.8 ±0.3
Figu e 4. Quali y o he VMM(λ) co ec ion po en ial as a unc ion o
he o de o he polynomial i o ⟨∂V/∂λ⟩λ o Asp. (A) Fi ing e o
as a unc ion o i ing o de (black line). G ay dashed line shows he
a e age e o in he calcula ed ⟨∂VMM/∂λ⟩. (B) Dis ibu ions o
λ-coo dina es o he hi d- and se en h-o de polynomial i s o
⟨∂V/∂λ⟩λ. Whe eas wi h he lowe o de i he dis ibu ion is
signi ican ly ugged, he dis ibu ion becomes nea ly la and uni o m
on he [0, 1] in e al o he λ-coo dina e i a se en h-o de i is used.
Jou nal o Chemical Theo y and Compu a ion pubs.acs.o g/JCTC A icle
h ps://doi.o g/10.1021/acs.jc c.2c00517
J. Chem. Theo y Compu . XXXX, XXX, XXX−XXX
H