# Free-energy landscapes and insertion pathways for peptides in membrane
environment

Ganga P. Sharma  Department of Physics, University of Rhode Island, Kingston
RI 02881, USA  Aaron C. Meyer  Department of Physics, University of Rhode
Island, Kingston RI 02881, USA  Suhail Habeeb  Department of Physics,
University of Rhode Island, Kingston RI 02881, USA  Michael Karbach
Fachgruppe Physik, Bergische Universität Wuppertal, D-42097 Wuppertal, Germany
Gerhard Müller  Department of Physics, University of Rhode Island, Kingston RI
02881, USA

###### Abstract

Free-energy landscapes for short peptides – specifically for variants of the
pH Low Insertion Peptide (pHLIP) – in the heterogeneous environment of a lipid
bilayer or cell membrane are constructed, taking into account a set of
dominant interactions and the conformational preferences of the peptide
backbone. Our methodology interprets broken internal H-bonds along the
backbone of a polypeptide as statistically interacting quasiparticles,
activated from the helix reference state. The favored conformation depends on
the local environment (ranging from polar to non-polar), specifically on the
availability of external H-bonds (with $\mathrm{H_{2}O}$ molecules or lipid
headgroups) to replace internal H-bonds. The dominant side-chain contribution
is accounted for by residue-specific transfer free energies between polar and
non-polar environments. The free-energy landscape is sensitive to the level of
pH in the aqueous environment surrounding the membrane. For high pH, we
identify pathways of descending free energy that suggest a coexistence of
membrane-adsorbed peptides with peptides in solution. A drop in pH raises the
degree of protonation of negatively charged residues and thus increases the
hydrophobicity of peptide segments near the C terminus. For low pH, we
identify insertion pathways between the membrane-adsorbed state and a stable
trans-membrane state with the C terminus having crossed the membrane.

##  I Introduction

This work reports the second stage of a three-stage project aiming at a more
complete theoretical understanding of membrane-associated protein or peptide
folding Jaeh83 ; BBH96 ; BBNH96 ; KB02 ; RGHM04 ; Bowi05 ; HKB+05 ; OSMG15 ;
VCC+04 ; ORV05 ; WV04 ; SSDI10 ; BYMB06 ; Rent10 . The first stage, reported
in Ref. cohetra , involved the design and solution of a microscopic model for
the coil-helix transition of a long polypeptide adsorbed to a planar water-
lipid interface. The methodology interprets the polypeptide and its
homogeneous, effectively two-dimensional environment as a system of
statistically interacting quasiparticles activated from the (ordered) helix
state. The particles represent broken internal H-bonds along the backbone,
producing segments of a (disordered) coil conformation, sprawled across the
interface in the shape of a self-avoiding random walk.

In one experimentally realized scenario RSA+07 ; ADS+07 ; RAS+08 ; AKW+10 ;
KWW+12 ; WMT+13 ; WAR16 ; VSS+18 ; JW89 ; ES81 ; PE90 ; ECC+03 ; RMAE20 , the
coil-helix conversion is triggered by a drop in pH. The ensuing protonation of
negatively charged side chains enhances the hydrophobicity of the polypeptide
and pushes its backbone deeper into the (non-polar) membrane. This
environmental change favors the formation of internal hydrogen bonds, which
stabilize the $\alpha$-helix conformation. Depending on the parameter
settings, the model predicts the conformation to change as a crossover, a
first-order transition, or a second-order transition cohetra .

Here, in the continuation of this project, we begin by considering long
polypeptides which are no longer confined to a flat water-lipid interface, but
are positioned and oriented along some line in the heterogeneous environment
comprising the lipid bilayer of a liposome or a biological cell and the
surrounding water. The model parameter identified in Ref. cohetra  to drive
the conformational change then turns into a (scalar) field with values
reflecting the nature of the local medium, specifically its degree of
polarity. Such circumstances pose a challenge to any method of analysis. The
methodology used here has already proven to be adaptable to heterogeneous
environments in different applications NA14 ; janac2 ; inharo .

From the analysis thus carried out, profiles emerge for the densities of free
energy, enthalpy, entropy, and helicity pertaining to segments of long
polypeptides in the heterogeneous membrane environment. The profiles reflect
enthalpic and entropic consequences of the interactions between the backbone
of the polypeptide and the lipid membrane or the surrounding hydrogen-bonded
network of water molecules. These profiles are then taken to represent
propensities for the statistical mechanical behavior of short peptides in the
same environment. In this step, enthalpic and entropic effects involving the
side chains and the semi-fluid bilayer of lipid amphiphiles can be built into
the model. The result are landscapes of free energy, enthalpy, entropy, and
helicity for short peptides of given composition.

The free-energy landscapes incorporate road signs for pathways that guide the
peptide from solution to adsorption under one set of environmental
circumstances and from adsorption to insertion under a different set. The
enthalpy and entropy landscapes offer insights into the dominant forces that
shape the pathways of descending free energy. This analysis, which is grounded
in equilibrium statistical mechanics, will be a key ingredient for the third
stage of this project kinpep : a kinetics study of trans-membrane insertion
and exit of peptides.

The theoretical study reported here is custom-made for variants of the _pH Low
Insertion Peptide_ (pHLIP) family RAS+08 ; AKW+10 ; KWW+12 ; WAR16 ; NWK+16 ;
HSQ+16 , but not to the exclusion of other peptides with similar attributes.
The sequences of three pHLIP variants are shown in Fig. 1 with some relevant
features highlighted. The positioning in the sequence of charged residues and
polar residues is instrumental for the solubility of pHLIP in water. The
hydrophobic residues provide the affinity for adsorption of pHLIP to the
water-lipid interface. The protonatable negative charges at and near the
(inserting) C terminus make pHLIP sensitive to the experimentally controllable
and reversible change in pH.

The Trp residues are markers for fluorescent spectroscopy, by which the
insertion and exit processes are monitored KWW+12 ; TYM+09 ; RKB01 . The three
pHLIP variants with Trp at different positions in the sequence of residues are
designed to yield clues about the modes of insertion and exit. The
conformational changes between coil and helix segments that accompany both
insertion and exit can be monitored by circular dichroism (CD) spectroscopy
KWW+12 ; VSS+18 . The helix-inhibiting Pro residue near the center of the
sequence may be instrumental for the mechanics of the insertion process,
allowing a kink between two fully formed helical segments note2 ; Heij91 ;
CBS02 ; WD91 .

Figure 1: Sequence of the $N_{\mathrm{R}}=32$ residues that make up variants
W6, W17, W30 (top to bottom) of pHLIP. Highlighted are the residues Arg, Asp,
which have electrically chargeable side chains, and the Trp residue used as a
marker in fluorescent spectroscopy.

The following sections are about fields, profiles, landscapes, and pathways:
fields of environmental parameters (Sec. II), profiles for local properties of
long polypeptides (Sec. III), landscapes for global properties of short
peptides (Sec. IV), and pathways associated with descending free energy of
short peptides (Sec. V). Finally, we briefly discuss effects not yet accounted
for and outline the challenges facing kinetic study (Sec. VI).

##  II Membrane environment

All heterogeneity in a lipid bilayer considered here is associated with the
normal spatial coordinate $x$. We set $x=0$ at the center of the bilayer. The
outside of the cell or liposome at positive $x$. Any effects of curvature are
set aside as higher-order corrections to results presented here. The membrane
environment is characterized by several parameters. We take the dominant
parameter field to be the concentration of $\mathrm{H_{2}O}$ molecules.
Hydrophobic interactions are prevalent TG12 . Subdominant parameter fields
involve electrostatic interactions including trans-membrane, surface, and
dipole potentials Wang12 . Further parameters are related to properties of
lipids, notably the profile of lateral pressure and the entropy reduction
along the contact line with the peptide BenS95 ; Seel04 ; KIM08 ; LVA12 ;
NWK+16 . We examine the effects of the dominant environmental parameter in
some detail and discuss those of subdominant parameters summarily at the end.

###  II.1 Density field of water

The dominant environmental parameter, the density field $\rho_{\mathrm{w}}(x)$
of $\mathrm{H_{2}O}$ molecules is symmetric under reflection about $x=0$. It
is a dimensionless quantity varying between $\rho_{\mathrm{w}}(x)=1$
sufficiently far from the lipids and $\rho_{\mathrm{w}}(x)\ll 1$ near the
center of the bilayer. We use a smoothed-ramp density field as a model
representation in our statistical mechanical analysis:

$\displaystyle\rho_{\mathrm{w}}(x)=1-\frac{x_{s}}{x_{\mathrm{a}}-x_{\mathrm{b}}}\ln\\!\left(\frac{\cosh\left(\frac{x}{x_{\mathrm{s}}}\right)+\cosh\left(\frac{x_{\mathrm{a}}}{x_{\mathrm{s}}}\right)}{\cosh\left(\frac{x}{x_{\mathrm{s}}}\right)+\cosh\left(\frac{x_{\mathrm{b}}}{x_{\mathrm{s}}}\right)}\right)\\!.$
(1)

It has two control parameters, $x_{\mathrm{b}}/x_{\mathrm{a}}>1$ and
${x_{\mathrm{s}}/x_{\mathrm{a}}>0}$. A density field of such shape (Fig. 2) is
well-established from experiment ALE+03  and computer simulations MBT08 .

Figure 2: Smoothed-ramp density field of $\mathrm{H_{2}O}$ molecules. The
parameters $x_{\mathrm{a}}$ and $x_{\mathrm{b}}$ in (1) locate the bottom and
the top of the ramp symmetrically, and $x_{\mathrm{s}}$ controls the softening
of the edges.

Throughout Secs. II-IV we use the following specifications:
$x_{\mathrm{a}}=15$Å represents the distance from the center of the bilayer to
a point just inside the lipid headgroups, and $x_{\mathrm{b}}=25$Å the
distance from the center to a point outside, where $\mathrm{H_{2}O}$ molecules
are in contact with the polar ends of the lipid headgroups; the value
$x_{\mathrm{s}}\simeq 3$Å ensures smoothness over an atomic length scale note3
.

A peptide of 32 residues in helix conformation would have a length of roughly
$50$Å, if we assume that each helical link note5  adds an element
$l_{\mathrm{h}}\simeq 1.5$Å of length in the direction of the axis FP02 . The
length (end-to-end distance) of the peptide in the coil conformation is a
fluctuating quantity. With a size $l_{\mathrm{a}}\simeq 4$Å for each link, the
contour length of the peptide becomes more than double its length in helix
conformation. The coil conformation, modeled as a random walk, has an average
end-to-end distance of $\sim 24$Å if unrestricted. Geometrical and dynamical
constraints make the average end-to-end distance considerably longer GK94 .

###  II.2 Free-energy

The term _free-energy landscape_ of the title requires some explanation. The
system under consideration includes a peptide in an environment consisting of
a lipid bilayer surrounded by water. For a homogeneous system, the Gibbs free
energy can be expressed in the form,

$G(T,P)=U+PV-TS=H-TS,$ (2)

where $U$ is the internal energy, $H$ the enthalpy, $S$ the entropy, and $V$
the volume. The pressure $P$ and the temperature $T$ are control variables.

Our system is not homogeneous in all respects. We consider situations at
uniform $T$, typically room temperature ($T_{\mathrm{rm}}=293$K). The pressure
is uniform in the aqueous environment and the normal pressure,
$P_{\mathrm{N}}$, also across the bilayer. However, the lateral pressure,
$P_{\mathrm{L}}$, has a characteristic profile that averages out to the value
of the normal pressure Mars96 ; Cant97 ; Cant99 ; GS04 ; GBM06 ; Mars07 ;
BMT10 ; ZL13 ; DFST16 ; DM16 . For the purpose of this study, we only consider
quasistatic processes of a restricted type in which both $T$, and
$P_{\mathrm{N}}$ remain constant and uniform. One natural energy scale uses
$k_{\mathrm{B}}T_{\mathrm{rm}}\simeq 4.0\times 10^{-21}\mathrm{J}\simeq
0.58\mathrm{kcal/mol}$ as its unit. The processes involve the translocation of
the peptide, accompanied by changes in conformation of the peptide and in its
interactions with water and lipids.

Each quasistatic process can be described as a path in a space of independent
variables. The variation of $G$ ascends or descends in a landscape of sorts,
just as a path on a topographic map does. Path segments with $\Delta G<0$ are
favorable and path segments with $\Delta G>0$ unfavorable regarding
spontaeneous occurrence. Paths that are all downhill are likely to have fast
realizations in experiments. By contrast, paths that have significant barriers
between initial and final points are much slower if realized at all.

All changes of $G$ are a combination of enthalpic and entropic contributions.
For the processes under consideration here we can write,

$\Delta G=\Delta H-T\Delta S.$ (3)

We refer to changes $\Delta H<0$ as associated with an _enthalpic gain_ and
changes $\Delta S>0$ as associated with an _entropic gain_ in the sense that a
gain lowers $G$ while a loss does the opposite.

Entropic losses, $\Delta S<0$, that are relevant for this study include the
following: (i) a complete or partial conformational change of the peptide from
(disordered) coil to (ordered) helix; (ii) the immobilization of
$\mathrm{H_{2}O}$ molecules via the formation of H-bonds with polar contacts
on the backbone or the side chains of the peptide or with polar contacts on
lipid head-groups; (iii) the formation of an ordered contact line between
lipid head-groups and the peptide in adsorbed or trans-membrane states.

Enthalpic losses related to a change $\Delta U>0$ are all associated with
molecular interactions. The two main sources of this type in the context of
this study involve (i) the breaking of H-bonds and (ii) the translocation of
charges or polar contacts from a polar environment (water) into a non-polar
environment (membrane). The H-bonds in question include internal ones between
backbone contacts, and external H-bonds between the peptide (backbone or side
chain), $\mathrm{H_{2}O}$ molecules, and lipid head-groups, all of which have
polar contacts. A different type of enthalpic loss,
$(P\mathrm{{}_{L}^{h}}-P\mathrm{{}_{L}^{l}})\Delta V>0$, comes into play when
a peptide segment (e.g. a residue) of volume $\Delta V$ translocates from a
position of low lateral pressure, $P\mathrm{{}_{L}^{l}}$, to a position of
high lateral pressure, $P\mathrm{{}_{L}^{h}}$.

It is quite challenging, in general, to estimate all these contributions with
some accuracy. Existing estimates found in the literature vary widely, in part
due to differences in underlying assumptions. In what follows, a case will be
made for identifying and quantifying dominant contributions.

###  II.3 Enthalpic cost of H-bonds

In the $\alpha$-helix conformation the backbone of each residue participates
in two H-bonds. The CO group of residue $n$ is acceptor to the NH group of
residue $n+4$ acting as donor. The helix conformation thus involves one
internal H-bond per residue. The conversion of a helix segment into a coil
segment breaks internal H-bonds, for which there is an enthalpic cost.

Deep inside the lipids the full price is due, up to 5kcal/mol per H-bond,
which is considerable in units of ambient thermal fluctuations. When the
peptide is positioned in the polar environment of the lipid headgroups and the
adjacent water, there are opportunities for internal H-bonds to be replaced by
external ones. The replacement bonds reduce the (enthalpic) cost of breaking
internal H-bonds. At the same time, it increases the maximum number of H-bonds
per residue from one to two. The replacement bonds (with reduced
directionality) are likely to be weaker.

Whether the enthalpic bottom line in this case is a gain or loss depends on
how the exposed backbone of the polypeptide affects the internal H-bonds of
liquid water. In ice there are two intact H-bonds per $\mathrm{H_{2}O}$
molecule. Each molecule shares four bonds, two in the role of donor and two as
acceptor. In liquid water about 12% of H-bonds between $\mathrm{H_{2}O}$
molecules are broken FP02 ; note6 . The intact H-bonds form a dynamic network
with $\mathrm{H_{2}O}$ molecules, sharing less than four bonds on average.

If the fraction of unsatisfied H-bonds between $\mathrm{H_{2}O}$ molecules is
higher at the lipid-water interface than inside the bulk, then the exposed CO
groups and NH groups along a coil segment of the backbone are more likely to
encounter partners for external H-bonds in adjacent $\mathrm{H_{2}O}$
molecules. This reduces the need for breaking H-bonds between
$\mathrm{H_{2}O}$ molecules when external H-bonds are formed along the
backbone of the peptide. The formation of external H-bonds in water is then
more likely to result in an enthalpic gain.

The methodology developed in Ref. cohetra  encodes the enthalpic contribution
to the conformational affinity in the activation energy $\epsilon$ of a coil
link from the helix reference state. The heterogeneous membrane environment is
accounted for by turning this parameter into a field. We use the ansatz,

$\epsilon(x)=\epsilon_{\mathrm{Hb}}\big{[}1-\alpha_{\mathrm{H}}\rho_{\mathrm{w}}(x)\big{]},$
(4)

for the dependence on position, the premise being that the density field
$\rho_{\mathrm{w}}(x)$ of water represents the dominant environmental
influence. Near the center of the lipid bilayer we have
$\rho_{\mathrm{w}}(x)\ll 1$, which maximizes $\epsilon(x)$ to roughly the
strength of an internal H-bond. At positions closer to the lipid-water
interface, $\epsilon(x)$ decreases as $\rho_{\mathrm{w}}(x)$ increases. This
change is due to the growing probability that internal H-bonds are being
replaced by external ones.

The parameter $\alpha_{\mathrm{H}}$ determines whether in the aqueous
environment, with $\rho_{\mathrm{w}}(x)\simeq 1$, we have an enthalpic gain
$(\alpha_{\mathrm{H}}>1)$ or an enthalpic loss $(\alpha_{\mathrm{H}}<1)$. Here
we use, $\epsilon_{\mathrm{Hb}}/k_{\mathrm{B}}T\simeq 9$, i.e.
$\epsilon_{\mathrm{Hb}}\simeq 5$kcal/mol. The strength of H-bonds varies and
depends on geometrical constraints and the charge distribution on the polar
contacts. H-bonds between $\mathrm{H_{2}O}$ molecules in liquid water are
highly fluctuating in strength, and could be as low as 2kcal/mol on average
SYSS03 ; FR05 ; LH06 ; JMF+08 ; GBPK09 ; CHSB17 .

###  II.4 Entropic cost of H-bonds

The enthalpic cost reduction associated with external H-bonds in the polar
environment of liquid water comes with an entropic price that has yet to be
included in the accounting. Every H-bond formed between an exposed backbone CO
or NH group with an $\mathrm{H_{2}O}$ molecule reduces the mobility of that
water molecule to some extent and thus lowers its contribution to the entropy.

It is hard to estimate the magnitude
$|\Delta\bar{S}_{\mathrm{H}}|/k_{\mathrm{B}}$ of this entropy reduction from
first principles. At a glance, we might expect it to be comparable in
magnitude to the entropic gain per residue produced when a segment of
(ordered) helix transforms into a segment of (disordered) coil. In Ref.
cohetra  we have calculated that entropy gain per residue to range between
$\ln 2\simeq 0.7$ and $\ln 3\simeq 1.1$ in units of $k_{\mathrm{B}}$. Backbone
segments in coil conformation are less mobile than $\mathrm{H_{2}O}$ molecules
in the dynamic network of H-bonds that make up liquid water. The unstructured
nature of the coil reduces the entropy loss of $\mathrm{H_{2}O}$ molecules
forming H-bonds with it. All this makes it reasonable to operate with an
estimated upper limit,

$\frac{|\Delta\bar{S}_{\mathrm{H}}|}{k_{\mathrm{B}}}\lesssim 1.$ (5)

The actual value of $|\Delta\bar{S}_{\mathrm{H}}|$ could be lower on account
of the role played by the lipid headgroups. The zwitterionic headgroups of
POPC, for example, offer alternative contacts for external H-bonds to backbone
segments in coil conformation. They come at lower entropic cost.

The overall message for what follows in Sec. IV is that the entropic
contribution to the free energy of the peptide backbone in coil conformation
is significantly reduced from what the disorder of the coil conformation would
suggest when the peptide is in contact with water.

###  II.5 Model for peptide conformation

In Ref. cohetra , we exactly solved a microscopic statistical mechanical model
for the conformational transformation between coil and helix of a long
polypeptide positioned in a plane parallel to the interface of a polar and a
non-polar medium such as realized by water and a lipid bilayer. Depending on
the settings and variations of its parameters, the model predicts a
conformational crossover or a transition of first or second order between coil
and helix.

All microstates of the peptide are characterized, in this model, by
combinations of $2\mu$ species of statistically interacting nested particles:
hosts $(m=1)$ nucleate coil segments, whereas hybrids $(m=2,\ldots,\mu)$ and
tags $(m={\mu+1},\ldots,2\mu)$ grow such segments in two perpendicular
directions of a self-avoiding random walk as illustrated in Fig. 3.

Figure 3: Coil conformation of polypeptide modeled as a self-avoiding random
walk generated by statistically interacting nested particles activated from
the helix reference state via _nucleation_ (control parameter $\tau$) and
_growth_ (control parameter $t$) with _range_ limited by control parameter
$\mu$.

The activation energies $\epsilon_{m}$, $m=1,\ldots,2\mu$, can be tailored to
meet the physical requirements at hand. In the plane of the water-lipid
interface all activation energies are spatially uniform. Nucleating a coil
segment requires that several internal H-bonds along the backbone of the
$\alpha$-helix are broken simultaneously, whereas the growth of a coil segment
already nucleated proceeds by the sequential breaking of single H-bonds.

The model, therefore, assigns different activation energies for the control of
nucleation and for the control of growth, namely
$\epsilon_{1}\doteq\epsilon_{\mathrm{n}}$ for hosts,
$\epsilon_{2}=\cdots=\epsilon_{\mu}\doteq 2\epsilon_{\mathrm{g}}$ for hybrids,
and $\epsilon_{\mu+1}=\cdots=\epsilon_{2\mu}\doteq\epsilon_{\mathrm{g}}$ for
tags. These activation energies in units of the thermal energy
$k_{\mathrm{B}}T$ are usefully expressed by the nucleation parameter $\tau$
(also named cooperativity) and growth parameter $t$:

$\displaystyle\tau=e^{\beta(\epsilon_{\mathrm{g}}-\epsilon_{\mathrm{n}})}\quad:~{}0<\tau\leq
1,$ (6a) $\displaystyle t=e^{\beta\epsilon_{\mathrm{g}}}\quad\hskip
21.33955pt:~{}0<t<\infty.$ (6b)

In addition to these two continuous control parameters, the discrete model
parameter $\mu$ controls the range of the random walk away from the axis of
the local helix segments. All model features from Ref. cohetra  used in the
following are reviewed in Appendix A.

###  II.6 Model parameter field

Of the three model parameters, the growth parameter $t$ is the one most
sensitive to the environment by far. We begin the model adaptation to the
membrane environment by keeping the cooperativity parameter $\tau$ and the
range parameter $\mu$ uniform, while we turn $t$ into a field. For this
purpose, we use the ansatz (4) as we construct two fields of scaled activation
energy, $\epsilon_{\mathrm{n}}(x)$ for hosts, $\epsilon_{\mathrm{g}}(x)$ for
hybrids and tags note8 . We leave the former $(m=1)$ undetermined for now and
link the latter $(m=2,3,\ldots,2\mu)$ to the density field of water:

$\displaystyle
K_{\mathrm{n}}(x)\doteq\frac{\epsilon_{\mathrm{n}}(x)}{k_{\mathrm{B}}T},$ (7)
$\displaystyle
K_{\mathrm{t}}(x)\doteq\frac{\epsilon_{\mathrm{g}}(x)}{k_{\mathrm{B}}T}=\frac{\epsilon_{\mathrm{Hb}}}{k_{\mathrm{B}}T}\big{[}1-\alpha_{\mathrm{H}}\rho_{\mathrm{w}}(x)\big{]},$
(8)

where $\epsilon_{\mathrm{Hb}}/k_{\mathrm{B}}T$ represents the scaled energy of
an H-bond and $\alpha_{\mathrm{H}}\simeq 1$ is the enthalpy parameter
introduced previously (Sec. II.3), assumed to be equal for hybrids and tags.
The growth parameter field,

$t(x)=e^{K_{\mathrm{t}}(x)}\quad:~{}0<t(x)<\infty,$ (9)

is environmentally sensitive via the shape of $\rho_{\mathrm{w}}(x)$ and the
value of $\alpha_{\mathrm{H}}$. The cooperativity,

$\tau=e^{K_{\mathrm{t}}(x)-K_{\mathrm{n}}(x)}=t(x)e^{-K_{\mathrm{n}}(x)}\quad:~{}0<\tau\leq
1,$ (10)

is kept as a position-dependent parameter. The function $K_{\mathrm{n}}(x)$ is
determined from (10).

This choice of modeling is informed by the following facts. Cooperativity,
which controls the nucleation of coil segments, is a process initiated by
thermal fluctuations within the backbone of an intact segment of
$\alpha$-helix. Multiple H-bonds must be broken simultaneously. They are all
protected from direct environmental contact. Nucleation is only affected
indirectly by an environmental change from non-polar to polar. In the non-
polar environment, the nucleation energy barrier is followed by a high plateau
and in the polar environment by a low plateau. The former favors a reversal of
nucleation events, whereas the latter favors the growth of nucleated segments.

Figure 4: (a) Model density field of water $\rho_{\mathrm{w}}(x)$ and (b)
growth parameter field $t(x)$, with specifications as indicated. The dot-
dashed lines represent the endpoints of the ramp and the dashed lines the
range $1\leq t\leq 3$ in which a coil-helix phase transition may occur.

Figure 4(a) shows the model density field of water $\rho_{\mathrm{w}}(x)$ used
henceforth. It has the smoothed-ramp profile (1). The growth parameter field
$t(x)$ inferred from the predominant environmental field
$\rho_{\mathrm{w}}(x)$ via (8) and (9) is shown in Fig. 4(b). When the
polypeptide is in coil conformation while adsorbed to the water-lipid
interface, its position is near the outer dot-dashed line. For the coil
conformation to be stable, the growth parameter $t(x)$ must be below the upper
horizontal dashed line (at $t=3$). That is barely the case if
$\alpha_{\mathrm{H}}<1$. A robust coil conformation requires that
$\alpha_{\mathrm{H}}\gtrsim 1$, implying that breaking internal H-bonds along
the backbone of the polypeptide and replacing them by external H-bonds with
available $\mathrm{H_{2}O}$ molecules is an enthalpic gain.

The extended model system is now ready for applications to the heterogeneous
membrane environment. We have converted $t$ into the field (9) and kept the
control parameters $\tau$ and $\mu$ uniform. We have already stated reasons
for not turning the nucleation parameter $\tau$ into a field. Regarding the
discrete parameter $\mu$ note9 , we will consider the cases $\mu=2$ and
$\mu=\infty$, for which analytic solutions are available in Ref. cohetra . The
two values span a range that is correlated with a range of entropy generated
inside coil segments of given length.

##  III Profiles

The analysis reported in the following yields profiles for specific position-
dependent attributes of a polypeptide backbone in the heterogeneous membrane
environment. The relevant quantities are the helicity $\bar{N}_{\mathrm{hl}}$
as well as densities of free energy $\bar{G}$, enthalpy $\bar{H}$, and entropy
$\bar{S}$. What is taken into account at this stage are the internal H-bonds
along the backbone of a long, generic polypeptide and external H-bonds with
water or lipid headgroups depending on their local availability. Also taken
into account is the entropy of the backbone in coil conformation and an
entropic contribution associated with external H-bonds (Sec. II.4).

###  III.1 Internal H-bonds

We construct profiles from the expressions for helicity
$\bar{N}_{\mathrm{hl}}(t,\tau)$, entropy density $\bar{S}(t,\tau)$, free-
energy density $\bar{G}(t,\tau)$, and enthalpy density $\bar{H}(t,\tau)$
stated in Appendix A, in combination with the field $t=t(x)$ for the growth
parameter (9), and a value of choice for the nucleation parameter $\tau$. The
growth parameter field $t(x)$ determines all profiles via local relations.
Profiles for $\mu=2$ (narrow range of conformational disorder) are shown Fig.
5 and profiles for $\mu=\infty$ (broad range) in Fig. 6.

Figure 5: Profiles of (a) helicity, (b) entropy density, (c) free-energy
density, and (d) enthalpy density for a long polypeptide. The model parameter
values are $\mu=2$ and $\tau=0.5$. The growth parameter field $t(x)$ uses
(8)-(9) with $\rho_{\mathrm{w}}(x)$ from Fig. 4. The solid, dashed, and dotted
curves pertain to $\alpha_{\mathrm{H}}=1$, $\alpha_{\mathrm{H}}=1.05,1.1$, and
$\alpha_{\mathrm{H}}=0.95,0.9$, respectively.

Figure 6: Profiles of helicity, entropy density, free-energy density, and
enthalpy density for a long polypeptide. The model parameter values are
$\mu=\infty$ and $\tau=0.5$. The growth parameter field $t(x)$ uses (8)-(9)
with $\rho_{\mathrm{w}}(x)$ from Fig. 4. The solid, dashed, and dotted curves
pertain to $\alpha_{\mathrm{H}}=1$, $\alpha_{\mathrm{H}}=1.05,1.1$, and
$\alpha_{\mathrm{H}}=0.95,0.9$, respectively.

Well inside the lipid bilayer, at $x\simeq 0$, the helix conformation is
firmly established. All internal H-bonds are intact. There is no
configurational disorder. Therefore, the order parameter (helicity) is close
to saturation whereas the densities of enthalpy note10  and entropy are near
zero. In consequence, the free-energy density of the polypeptide rises only
imperceptibly above its (zero) reference value as well. At positions away from
the center of the bilayer, the helicity decreases and the entropy density
increases, the former reflecting a drop in (helical) order and the latter a
rise in (coil-like) disorder, both associated with the same conformational
change.

In Ref. cohetra  we identified one source of order and two sources of disorder
involving the secondary structure of the polypeptide backbone alone. Order
increases with the growth of segments of helix conformation. Disorder is
contained (i) in the spatial distribution of boundaries between segments of
coil and helix and (ii) inside each (unstructured) coil segment.

The enthalpy density produces an energy barrier at locations near the lipid-
water interface [Figs. 5(d) and 6(d)]. Near the peak position the thermal
fluctuations are just strong enough to break internal H-bonds, but the
environment is not yet sufficiently polar to produce an adequate supply of
external H-bonds as replacements. The enthalpy density decreases (for
different reasons) on each side of the peak. On one side, broken bonds are
rare, on the other side, they are energetically inexpensive. Outside the
interface, the enthalpy profile levels off in a high or low plateau depending
on the value of the physical parameter $\alpha_{\mathrm{H}}$. That parameter
also affects the drop in helicity and the rise in entropy. Enthalpic loss
$(\alpha_{\mathrm{H}}<1)$ favors order and suppresses disorder (for the
peptide) in the aqueous environment.

There are some qualitiative and some quantitative differences between the
curves for $\mu=2$ and for $\mu=\infty$. The case $\mu=\infty$ produces pure
coil with maximum entropy in the aqueous environment. Pure coil conformation
means zero helicity. Coil segments generate significantly more entropy for
$\mu=\infty$ than for $\mu=2$. The enthalpic barriers near the interface are
more pronounced in the case $\mu=\infty$. This difference is attributable to
an entropic effect. The breaking of an H-bond at significant enthalpic cost is
more likely to happen if the entropy produced is large $(\mu=\infty)$ than if
it is small $(\mu=2)$.

The free-energy profiles in Figs. 5 and 6 tell us that the incentives for the
insertion of peptides must come from a source other than what has already been
taken into account. The free-energy density $\bar{G}(x)$ is significantly
higher in the membrane environment than in the surrounding water. The
dependences of $\bar{H}(x)$ and $\bar{S}(x)$ on $\alpha_{\mathrm{H}}$ are
strong, but not decisively so. The entropic contributions accounted for thus
far are dominating the free energy. Coil is favored over helix. Water beats
lipids as the favored environment for the peptide.

###  III.2 External H-bonds

At this point in the analysis, we add one other contribution to the free-
energy density profile. Further contributions, associated with side chains and
their interaction with lipids, will be introduced in Sec. IV.3. The
contribution discussed here is entropic in nature and favors insertion. The
replacement of backbone internal H-bonds with external H-bonds that immobilize
$\mathrm{H_{2}O}$ molecules from the aqueous environment, while providing an
enthalpic discount as already accounted for, comes at an entropic cost DMA+10
. This effect is taken into account via an amended free-energy density
constructed as follows:

$\frac{\bar{G}_{\mathrm{H}}(x)}{k_{\mathrm{B}}T}=\frac{\bar{G}(x)}{k_{\mathrm{B}}T}+2\frac{|\Delta\bar{S}_{\mathrm{H}}|}{k_{\mathrm{B}}}\Big{[}1-\bar{N}_{\mathrm{hl}}(x)\Big{]},$
(11)

with $|\Delta S_{\mathrm{H}}|$ as discussed and estimated in Sec. II.4. The
factor in square brackets represents the fraction of backbone segments in coil
conformation with each segment offering docks for two $\mathrm{H_{2}O}$
molecules. Profiles for the amended free-energy densities are shown in Fig. 7.

Figure 7: Profiles of free-energy density (11) which includes an entropic
contribution from external H-bonds. The two panels pertain to different values
of the range parameter $\mu$. The curves from bottom to top are for increasing
values of $|\Delta\bar{S}_{\mathrm{H}}|/k_{\mathrm{B}}$. Additional model
parameter values are $\tau=0.5$ and $\alpha_{\mathrm{H}}=1$. Note the
different vertical scales left and right.

This amendment does indeed contribute an incentive for insertion. Its impact
depends on the size of $|\Delta\bar{S}_{\mathrm{H}}|/k_{\mathrm{B}}$, which
controls the entropy reduction due to external H-bonds, and on the range
parameter $\mu$, which controls entropy production of coil segments. An
increase of $|\Delta\bar{S}_{\mathrm{H}}|/k_{\mathrm{B}}$ switches the global
minimum in the free-energy density from the exterior region to the interior
region. The switch happens at
$|\Delta\bar{S}_{\mathrm{H}}|/k_{\mathrm{B}}\simeq 0.5$ for $\mu=2$. A
somewhat larger value, $|\Delta\bar{S}_{\mathrm{H}}|/k_{\mathrm{B}}\simeq
0.55$ is needed to cause the switch for $\mu=\infty$. These values are well
below the estimated upper bound (5). Unsurprisingly, the larger range
parameter, which produces more backbone entropy, requires more entropy
reduction in compensation before it yields an insertion incentive.

Interestingly, an energy barrier between the exterior and interior levels of
free-energy-density builds up as $|\Delta\bar{S}_{\mathrm{H}}|/k_{\mathrm{B}}$
increases in size. This barrier, which is more pronounced in panel (b), helps
stabilize the (experimentally confirmed) coexistence of short peptides in
states of solution and adsorption. Energy barriers such as emerge here quite
naturally, play an important role in kinetic studies of the insertion process.

##  IV Landscapes

In the continuation of the analysis, we interpret the profiles calculated in
Sec. III as propensities of backbone segments of short peptides such as pHLIP.
The helicity profile and the density profiles for enthalpy, entropy, and free
energy are employed here as one factor affecting the behavior of residues of
short peptides in the same environment, specifically their conformational
preference (coil or helix). Other factors depend on attributes of the specific
side chains and on further (enthalpic and entropic) effects of peptide-lipid
interactions.

Landscapes of interest include those of free energy, enthalpy, entropy, and
helicity. It would take a molecular dynamics simulation to explore entire
landscapes, i.e. the full range of configurations that a peptide can assume in
the membrane environment. The approach taken here, which is more limited in
scope (see Sec. V), is more selective with configurations. This selectivity is
justifiable, at least in part, by the fact that the range of configurations is
naturally and severely restricted by conformational constraints, specifically
by the rigidity of the helix conformation.

###  IV.1 Three-variable landscapes

In the following, we investigate three-variable free-energy landscapes for
short peptides in varying positions and orientations. The model peptide has
$N_{\mathrm{R}}$ residues and is assumed to consist of two straight segments
as schematically represented in Fig. 8. In applications to pHLIP, a likely
candidate for the kink position is the helix inhibiting Pro residue Heij91 ;
CBS02 ; WD91 . One simulation study VSS+18  places a kink at the position of
the Asp residue which is somewhat closer to the N terminus than the Pro
residue.

Figure 8: Variables $x_{n_{\mathrm{K}}}$ (depth of kink) and
$\theta_{\mathrm{N}}$, $\theta_{\mathrm{C}}$ (angles between normal and
segments). The counting of residues begins at the N terminus as is custom.

We use the position $x_{n_{\mathrm{K}}}$ of the kink on the normal to the
bilayer and the angles $\theta_{\mathrm{N}}$, $\theta_{\mathrm{C}}$ of the
segments ending in the N, C terminus, respectively, as the variables that
specify the position and orientation of the peptide in the membrane
environment. The lengths of both segments depend on the local conformation,
which, in turn, depend on the location of a given segment in the membrane
environment as specified below.

We intend to calculate the free energy of peptides along specific pathways in
this three-variable landscape. Our focus will be on the pHLIP variants with
the sequences as stated in Fig. 1. We consider three principal states I, II,
and III of pHLIP in the membrane environment and specific pathways between
them RSA+07 ; RMAE20 :

I: pHLIP is in aqueous solution and in coil conformation. We shall start
adsorption pathways at coordinates $x_{n_{\mathrm{K}}}=27$Å,
$\theta_{\mathrm{N}}=\theta_{\mathrm{C}}=90^{\circ}$.

II: pHLIP is adsorbed to the outside interface of a cell membrane with
interstitial fluid or of a liposome with water. At high pH the adsorption is
rather loose and the conformation is coil. At low pH the adsorption is deeper
and the conformation is largely $\alpha$-helix. We shall see that the adsorbed
state has the kink of the peptide positioned deeper in the membrane and both
termini sticking out into the water: $x_{n_{\mathrm{K}}}\simeq 17$Å,
$\theta_{\mathrm{N}}\simeq\theta_{\mathrm{C}}\simeq 30^{\circ}$.

III: pHLIP is in a trans-membrane state with a helical central part and short
coil-like flanking parts. The coordinates roughly are
$x_{n_{\mathrm{K}}}\simeq 0$, $\theta_{\mathrm{N}}\simeq 0$,
${\theta_{\mathrm{C}}\simeq 180^{\circ}}$.

In order to calculate the peptide free energy in any of the three principal
states and in states that connect them we need to know the position $x_{n}$ of
every residue in the membrane environment. We determine these positions
recursively from distances between residues beginning at the kink. Writing,

$\displaystyle
x_{n-1}=x_{n}+l(x_{n})\cos\theta_{\mathrm{N}}\quad:~{}n=n_{\mathrm{K}},\ldots,2,$
(12) $\displaystyle
x_{n+1}=x_{n}+l(x_{n})\cos\theta_{\mathrm{C}}\quad:~{}n=n_{\mathrm{K}},\ldots,N_{\mathrm{R}}-1,$

takes into account that the distance between adjacent residues depends on the
local conformation of the residue position in the membrane environment. That
conformation is either coil or helix with probabilities for which we use the
propensity profile as calculated in Sec. III:

$\displaystyle x_{n+1}-x_{n}\doteq
l(x_{n})=l_{\mathrm{c}}-(l_{\mathrm{c}}-l_{\mathrm{h}})\bar{N}_{\mathrm{hl}}(x_{n}).$
(13)

Whereas the length of a helical segment is given $l_{\mathrm{h}}=1.5$Å, the
(averaged) length $l_{\mathrm{c}}$ of coil segments becomes a physical model
parameter. Its maximum value is the contour length per residue of the
backbone: $l_{\mathrm{c}}\lesssim 4$Å.

###  IV.2 Backbone contributions

It is useful to look at some free-energy and helicity landscapes that
represent the effects of the backbone alone. This is facilitated in Appendix
B. The main message from the results presented there to what follows is
twofold: (i) The backbone-lipid interaction favors coil conformation near the
water-lipid interface and helix conformation inside the membrane. (ii)
Insertion lowers the backbone contribution to the free energy.

Here we merely state the expression for the backbone contribution to the free-
energy landscape and the expression for the helicity (fraction of peptide in
helix conformation) to be used in the following:

$\displaystyle
G_{\mathrm{BB}}(x_{n_{\mathrm{K}}},\theta_{\mathrm{N}},\theta_{\mathrm{C}})=\sum_{n=1}^{N_{\mathrm{R}}}\bar{G}_{\mathrm{H}}(x_{n}),$
(14) $\displaystyle
N_{\mathrm{hl}}^{(\mathrm{K})}(x_{n_{\mathrm{K}}},\theta_{\mathrm{N}},\theta_{\mathrm{C}})=\sum_{n=1}^{N_{\mathrm{R}}}\bar{N}_{\mathrm{hl}}(x_{n}),$
(15)

where the function $\bar{G}_{\mathrm{H}}(x)$ is taken from (11) and the
functions $\bar{G}(x)$, $\bar{N}_{\mathrm{hl}}(x)$ from Appendix A. The
dependence of $x_{n}$ on
$x_{n_{\mathrm{K}}},\theta_{\mathrm{N}},\theta_{\mathrm{C}}$ is given in (12).

###  IV.3 Side-chain contributions

The contributions to the free-energy landscape originating from the side
chains of a peptide with a given sequence of residues are manifold NLS66 .
Their interactions with lipid molecules include aspects of hydrophobicity,
pressure differentials, entropy reductions, and electrostatics. Ranking the
relative importance has been challenging and not without controversy. For the
adsorption and insertion pathways presented in Sec. V we use well-established
transfer-free-energy data as the dominant side-chain contribution. The
potential impact of other contributions will be discussed in Sec VI as an
outline of future work.

The side chains of pHLIP residues range from strongly hydrophobic to strongly
hydrophilic. Their transfer between polar and non-polar environments
contributes significantly to the free-energy landscape. Some side chains carry
(positive or negative) electric charges. The protonation at low pH of the
negatively charged Asp residues and C terminus (see Fig. 1) changes the
overall hydrophobicity critically as we shall see, destabilizing the adsorbed
state II in favor of the trans-membrane state III. An estimate of the side-
chain transfer-free energy between states I, II, and III based on the Wimley-
White interface and octanol scales is given in Appendix C. That scheme takes
into account free-energy differences between three levels, w, i, o,
representing (polar) water, (mixed) interface, and (non-polar) octanol or
lipid-hydrocarbon-tail environments, respectively. Our modeling with these
data strongly indicates that the aforementioned instability is real and in
agreement with experiments AKW+10 ; WAR16 ; HSQ+16 .

For the investigation of insertion pathways we replace the three environmental
levels w, i, o by an environmental field in the manner discussed in Sec. II.
We use the density field of water (1) to specifically convert the steps
$\Delta G_{\mathsf{wo}}^{(n)}$ from Wimley-White data WW96 ; WCW96 ; WW98 ;
WW99 ; Guy85 ; HW11 ; MF11  for the residue at sequential position $n$ into
fields of gradual change:

$G_{\mathrm{res}}^{(n)}(x)=\Delta
G_{\mathsf{wo}}^{(n)}\big{[}1-\rho_{\mathrm{w}}(x)\big{]}.$ (16)

The side-chain contribution to the free-energy of the peptide then becomes,

$G_{\mathrm{SC}}(x_{n_{\mathrm{K}}},\theta_{\mathrm{N}},\theta_{\mathrm{C}})=\sum_{n=1}^{N_{\mathrm{R}}}G_{\mathrm{res}}^{(n)}(x_{n}),$
(17)

with the dependence of $x_{n}$ on
$x_{n_{\mathrm{K}}},\theta_{\mathrm{N}},\theta_{\mathrm{C}}$ from (12). From
here on we use the specifications $x_{\mathrm{a}}=17.8$Å,
$x_{\mathrm{b}}=26.3$Å, $x_{\mathrm{s}}=2.0$Å for the profile (1).

The transfer free energies relative to an exterior position for the
hydrophobic Val residue and the mildly hydrophilic Asp residue (when
protonated) are shown in Fig. 9 (solid curves) as an illustration of this
scheme. The deprotonated Asp residue is much more strongly hydrophilic, which
costs significant extra free energy, as indicated by the dotted line. The
dashed line represents a modification of the Asp profile that takes into
account the complication that during the insertion process, the pH is
different in the exterior and interior regions. This last point requires some
explanation.

Figure 9: Transfer-free-energy profile of one hydrophobic residue, Val, and
one hydrophilic residue Asp, when protonated (solid line) or deprotonated
(dashed line). The dashed line represents a hybrid profile for the same Asp
residue, which accounts for a kinetic effect as described in the text.

When the pH is lowered in the exterior region, all Asp residues quickly become
protonated and, in consequence, less hydrophilic. Their transfer-free-energy
profile changes from the dotted line to the solid line in Fig. 9. It is well
established in experiments with liposomes that the insertion process is faster
than the equilibration of the pH across the membrane. Hence the three Asp
residues closest to the C terminus will only remain protonated, during the
insertion process, until they come into contact with the aqueous environment
in the interior region, which is still at higher pH. The consequence in the
framework of quasistatic translocations considered here is that the transfer-
free energies of these Asp residues drop to a level below that of the exterior
region, as is reflected in the dashed profile of Fig. 9. As the pH slowly
drops inside the liposome, the modified (dashed) profile gradually turns back
into the profile characterizing the protonated Asp (solid line).

In the projected kinetic study of peptide insertion kinpep , this complication
will, of course, be accounted for in a very different way. The goal here is
more modest. We aim to identify the circumstances under which insertion
pathways for pHLIP in the free-energy landscape exist. There are no intrinsic
time scales associated with these pathways. However, it appears to be a
necessary condition for insertion, as will be explained in Sec. V, that a
differential in pH between exterior and interior regions is maintained. In
liposome experiments, this pH differential only exists for a limited time,
during which insertion must be completed.

##  V Pathways

Here we pick up the thread from Sec. IV.1, where we designed a template for
exploring adsorption and insertion pathways of a peptide consisting of two
straight segments with a kink at the Pro position (our choice). Any pathway
taken by the peptide must be downhill in the free-energy landscape, assembled
from the backbone part (14) and the side-chain part (17),

$\displaystyle G(x_{n_{\mathrm{K}}},\theta_{\mathrm{N}},\theta_{\mathrm{C}})$
$\displaystyle=G_{\mathrm{BB}}(x_{n_{\mathrm{K}}},\theta_{\mathrm{N}},\theta_{\mathrm{C}})+G_{\mathrm{SC}}(x_{n_{\mathrm{K}}},\theta_{\mathrm{N}},\theta_{\mathrm{C}})$
$\displaystyle=\sum_{n=1}^{N_{\mathrm{R}}}\left[\bar{G}_{\mathrm{H}}(x_{n})+G_{\mathrm{res}}^{(n)}(x_{n})\right],$
(18)

with $x_{n}$ from (12).

We have adopted a very simple search procedure for descending pathways. For
each step we consider positive and negative infinitesimal increments for the
three variable $x_{n_{\mathrm{K}}},\theta_{\mathrm{N}},\theta_{\mathrm{C}}$
(see Fig. 8) and execute the step that produces the most negative $\Delta G$.
The pathway stops when none of the variable changes generates a $\Delta G<0$.
Guided by the circumstances associated with experimental investigations of
pHLIP insertion into liposomes, we divide the pathways toward lower free
energy into three consecutive phases. Different environmental conditions
pertain to each phase.

(A) During the _adsorption_ phase, the pH is high and all Asp residues are
deprotonated. The peptide starts out in solution close to the membrane.
Pathways with $\Delta G<0$ end with the peptide adsorbed at the interface of
the membrane with the exterior aqueous region.

(B) The _insertion_ phase starts with the peptide in that adsorbed state but
with the environment changed. The pH is low in the exterior region and all Asp
residues have become protonated. Pathways with $\Delta G<0$ end with the
peptide in a trans-membrane state. The pH remains high in the interior region.

(C) The _stabilization_ phase comes into play once the pH has dropped in the
interior region (of a liposome), which typically happens more slowly than the
insertion. It affects the stability of the trans-membrane state. However,
pathways with $\Delta G<0$ keep the peptide inserted.

###  V.1 Adsorption

The gray background of Fig. 10 is a representation of the water-density
profile [Fig. 4(a)] across the heterogeneous membrane environment. The
exterior region is at the top. At the starting point of the pathway explored
here, the adsorption phase is already on its way. The peptide is sprawled wide
near the interface.

Figure 10: Variant W6 of pHLIP modeled as two straight segments with a kink at
the Pro position during the adsorption phase of the pathway.

The adsorption pathway from here on consists of two legs. First the peptide
moves toward the membrane without changing its orientation. Along this
stretch, it undergoes a conformational change from coil to mostly helix, which
shortens its length. The initial and final configurations of this first leg
are shown in Fig. 10(a). Along the second leg of the adsorption pathway, the
two segments on either side of the Pro kink change their orientations and the
kink position continues to move toward the center of the membrane. This brings
the hydrophobic residues deeper into the non-polar environment, yet keeps the
hydrophilic residues closer to the polar environment. The final, adsorbed
equilibrium state is shown in Fig. 10(b).

Additional information of a more quantitative nature about the adsorption
pathway is compiled in Fig. 11. The two legs are clearly discernible in the
variations of the kink position [panel (a)] and the angles of orientation of
the two segments [panel (b)]. The first leg ends after step $i=8$. Here the
angles begin to vary and the kink moves at a slower rate, meaning down a more
shallow slope. The pathway ends after step $i=100$. The two legs are more
clearly recognizable in some data than in others. The zigzag lines visualize
the search protocol described earlier.

Figure 11: (a) Kink position, (b) angles (away from vertical orientation) of
the two segments, (c) helicity, (d) change in free energy (backbone, side-
chain contributions and total) versus step number $i$ along the adsorption
pathway.

The conformational change from coil to helix happens almost entirely during
the first leg. Here the decrease in free energy is driven by the backbone
contribution. The conformational change, which involves the backbone, takes
place during the first leg. An environmental differentiation restricted by
conformational constraints takes place during the second leg. Hydrophobic
residues (near the center of the peptide) move further into membrane while
hydrophobic residues stay closer to water. We have chosen the starting point
of the adsorption phase such that the overall change of free energy
($\sim-6$kcal/mol) is consistent with caloric experiments RAS+08 ; WMT+13 .

###  V.2 Insertion

Figure 12: Variant W6 of pHLIP modeled as two straight segments with a kink at
the Pro18 position during the insertion phase of the pathway.

When the pH is lowered in the exterior region, the final state of the
adsorption pathway becomes unstable. It becomes the initial state of the next
phase of the pathway [Fig. 12(a)]. All Asp residues are now protonated ADS+07
; MKA+10 . The destabilizing agent is the switch of their transfer-free-energy
profile from the deprotonated to the hybrid version. This modification opens
up pathways of descending free energy, which we explore using the previously
described protocol. What emerges turns out to be an insertion pathway. It
consists of three legs, as visualized in Fig. 12.

Along the first leg [panels (a)-(b)], the peptide sinks somewhat deeper into
the membrane with no significant change in the orientation of the two segments
on either side of the kink at the Pro position. The protonation has made the
Asp residues significantly less hydrophilic. This has weakened their
resistance against the pull of the hydrophobic residues into the non-polar
environment. Hence the downward translocation of the entire peptide.

Along the second leg [panels (b)-(d)], the segment on the left with the N
terminus at its end straightens up while the other segment does not rotate
significantly. The Pro kink position moves further toward the center of the
membrane. Protonation has taken place primarily near the C terminus. The
positive charges at the N terminus and at the Arg residue on the same segment
are still present. Reorienting that segment into a trans-membrane direction
keeps those charges in or near the polar environment and allows the
hydrophobic residues to move deeper into non-polar environment.

The third leg of the insertion pathway [panels (d)-(f)] is mainly driven by
the forces acting on the segment with the C terminus at its end. There are
competing forces in action. The hydrophobic forces lead the initial descent in
free energy toward a horizontal orientation of that segment. The subsequent
descent in free energy toward the trans-membrane orientation is guided by the
forces acting on the protonatable Asp residues and the C terminus. Recall that
the interior region is still at high pH, which significantly enhances the pull
of the protonatable contacts toward the non-polar environment, where
deprotonation takes place.

Figure 13: (a) Kink position, (b) angles (away from vertical orientation) of
the two segments, (c) helicity, (d) change in free energy (backbone, side-
chain contributions and total) along the insertion pathway.

Figure 13 provides additional, more quantitative information about the
insertion pathway. The three legs are most clearly discernible in the kink
position and the two angles of orientation. The reorientation of the two
segments during the second and third legs is almost completely sequential:
first the N terminus backs out into the exterior region while the kink
position moves toward the center; then the C terminus moves across into the
interior region while the kink position remains stationary.

The helicity (already strong) reaches near saturation during the first leg,
but then decreases somewhat as the N terminus reestablishes closer contact
with the exterior aqueous region. The change in free energy during insertion
originates predominantly from the side chains. The drop of backbone free
energy is less than 0.3 kcal/mol and happens quickly while the helicity
increases. The drop in side-chain free energy is, for the most part, spread
across the first two legs of the insertion pathway. Its amount of $\sim
4$kcal/mol is a bit higher than what caloric experiments predict, but not by
much RAS+08 ; WMT+13 .

###  V.3 Stability

In an experiment that uses liposomes instead of biological cells, the low
level of pH imposed on the exterior region will slowly leak into the interior
region. This happens on a significantly slower time scale than the insertion
process. There is clear experimental evidence that pHLIP stays inserted as the
pH equilibrates at a low level. Hence the third phase of the pathway must
ensure that when we change the Asp transfer-free-energy profiles from the
hybrid version to the protonated version the trans-membrane state remains
stable RSA+07 ; KWW+12 .

This turns out to be the case indeed as illustrated in Fig. 14. Panel (a)
shows the final state of the second phase and panel (b) the final state of the
third phase. The low pH in the interior region reprotonates the C terminus and
the Asp residues near it. Hence their counteraction against the hydrophobic
forces on the Leu residues (also near the C terminus) weakens. This has the
effect that the segment with the C terminus sticking into the interior region
changes its orientation somewhat to find the new local free-energy minimum.

Figure 14: Variant W6 of pHLIP modeled as two straight segments with a kink at
the Pro18 position during the third phase of the pathway.

The kink position and the orientation of the segment with the N terminus at
its end do not undergo significant changes. The descent in free energy is
small in comparison to the previous two stages. It amounts to less than
10cal/mol. The important message is that the peptide stays in a trans-membrane
state. Note that insertion (second stage) only happens, according to the
protocol of our pathway exploration, if the pH remains high in the interior
region until the trans-membrane state has been realized. However, once the
trans-membrane state is realized it remains stable even after the pH has
dropped in the interior region.

Let us emphasize that Figs. 11 and 13 are not meant to suggest any times
scales for the adsorption and insertion processes, respectively. Our protocol
for the exploration of free-energy landscapes merely searches for downhill
directions and then takes steps accordingly with no attempts undertaken to
optimize directions for steepest descent. A kinetic study of the pHLIP
adsorption and insertion processes is an entirely different project kinpep .

##  VI Conclusion and outlook

Processes of peptide insertion into a membrane are highly complex in their
kinetics and can be very diverse in their energetics. In this work an effort
has been described that focuses on the energetics of a particular scenario.
The specific goal has been to demonstrate plausible insertion pathways in a
free-energy landscape assembled from contributions identified as dominant. We
built that landscape from enthalpic and entropic contributions associated with
internal and external H-bonds along the peptide backbone and from transfer
free energies of side chains between hydrophilic and hydrophobic regions of
the membrane environment.

The advantage of the approach centered on energetics is that the problem
associated with the inevitably broad range of time scales has come into play
only marginally. The investigation of free-energy landscapes and the search
for insertion pathways is not meant to be a substitute for a molecular
dynamics simulation or other form of kinetic study DQL13 ; DFST16 ; GV09 ;
VSS+18 ; MBT08 ; MS92 ; MS93 ; IB05 ; NWG05 ; UU08 ; USU10 ; UDK+10 . It is an
exploration of the circumstances under which kinetic processes associated with
peptide insertion are likely to take place.

Insertion pathways as explored in this work are characteristic of quasistatic
processes. The free-energy landscapes provide road signs, but no time tables.
The road signs thus established are not infallible because the free-energy
landscapes in use are manifestly incomplete. A strong case can be made that
what has been left out is more productively taken into account in a kinetic
study or a molecular dynamics simulation. The four most relevant missing
pieces represent different aspects of lipid-peptide interactions. The first
two are enthalpic and the last two describe closely related entropic effects.

(i) The lateral pressure profile of the membrane, which provides mechanical
stability to the bilayer structure against perturbations of various kinds, is
known to have a characteristic shape with regions of positive deviations from
ambient pressure sandwiching a narrow band of negative deviation just inside
the lipid headgroups. These empirically established pressure variations are
significant Mars96 ; Cant97 ; Cant99 ; GS04 ; GBM06 ; Mars07 ; BMT10 ; ZL13 ;
DFST16 ; DM16 .

(ii) In the (trans-membrane) state III the predominently hydrophobic and
$\alpha$-helical center of pHLIP is flanked by coil segments that are more
hydrophilic. In a positive (negative) mismatch, the hydrophobic center is too
long (short) to be comfortably accommodated in the membrane. Each type of
mismatch elicits a distinct response from both the peptide and the membrane.
In a positive mismatch, the membrane tends to increase its width locally.
Sheltering additional hydrophobic residues from water costs elastic energy.
That cost can be lowered if the helical axis tilts its trans-membrane
orientation away from the bilayer normal. To a negative mismatch the membrane
responds with a local thinning, which puts the helix under tension. Some of
that tension may be released by a partial change from $\alpha$-helix to the
more tightly wound $3_{10}$-helix BK92 ; Mill95 ; Kill98 ; PK03 ; SEUS12 ;
DA13 ; note15 .

(iii) The lipid molecules in the bilayer, typically bent into a liposome of
spherical shape, are in a macrostate which represents a two-dimensional fluid.
The positional ordering of the headgroups is then of short range over some
coherence length $\xi$. The undisturbed lipid bilayer is characterized by a
uniform entropy density with two contributions, one representing positional
disorder of the headgroups and the other orientational and conformational
disorder of the hydrocarbon tails BBH96 ; CKT05 . Headgroups and tails perform
a free-energy balancing act of sorts. An increase in bilayer width produces
more space for the tails to explore and requires work by the headgroups
against the pressure of the surrounding water, thus implying $T\Delta S>0$ and
$\Delta H>0$, respectively. A decrease in the average number of headgroups per
unit area increases their positional freedom, implying $T\Delta S>0$, but
compresses the tails, implying $\Delta H>0$. Balance is reached when $\Delta
G=\Delta H-T\Delta S=0$.

(iv) The presence of pHLIP produces a contact line with lipid molecules. Along
that contact line, the aforementioned thermal equilibrium of headgroups and
tails is being disturbed LHH95 ; WHLH95 . The dominant changes involve an
entropy reduction in both headgroups and tails. The former tend to line up
tightly against the foreign object and form an ordered layer of width roughly
equal to the aforementioned coherence length $\xi$. pHLIP going into the
(adsorbed) state II throws the normal pressure out of balance. Fewer tails
must exert the same force per unit area. They can do that only under higher
compression, facilitated by membrane thinning as noted before. This, in turn,
leads to an entropy reduction. When pHLIP inserts the contact line becomes
much shorter. The associated decrease in $\Delta G$ dominated by a positive
$T\Delta S$ is a factor favoring insertion thermodynamically. Jähnig Jaeh83
investigated this effect under the name _lipohobic effect_ and used a
coherence length of $\xi\simeq 15$Å.

Taking into account effects (i)-(iv) calls for a more detailed model of the
peptide than is being used in this study. The stochastic modeling intended to
be used for pHLIP insertion kinetics kinpep  is a Markov chain model, which
promises interesting points of comparison with molecular dynamics simulations.
One strong point of Markov chain modeling is that it can deal with a range of
different time scales in a most transparent and efficient way. The price to be
paid for that advantage is that interactions on a microscopic scale are
accounted for more summarily than a simulation does at the molecular dynamics
level.

###### Acknowledgements.

We are grateful to Yana K. Reshetnyak and Oleg A. Andreev for suggesting this
study and for generously offering much help along the way.

##  Appendix A Statistically interacting polymer links

The origin in quantum many-body theory Hald91a ; Wu94 ; Isak94 ; Anghel ;
PMK07  of the methodology used here and its adaptation to problems of current
interest in classical statistical mechanics LVP+08 ; copic ; picnnn ; pichs ;
GKLM13 ; sivp ; janac2  has already been presented from several different
angles. Among these works are papers dedicated to polypeptides cohetra  and to
double-stranded DNA mct1 .

Here we summarize those results from Ref. cohetra  which are being used (first
in Sec. III) as the main building blocks for extensions reported in this work.
The microscopic model for the coil-helix transition of a long polypeptide at a
water-lipid interface solved in cohetra  has three parameters: the growth
parameter $t$, the nucleation parameter $\tau$, both continuous, and the
(discrete) range parameter $\mu$, which numbers the coil states available to
each residue (see Fig. 3). Here we consider the cases $\mu=2$ and $\mu=\infty$
at $\tau>0$ and use selected results. We quote the relevant expressions in
ways easy to trace back to their derivations.

The scaled Gibbs free energy in closed form reads

$\frac{\bar{G}(t,\tau)}{k_{\mathrm{B}}T}=-\ln\left(\frac{1+w(t,\tau)}{t}\right),$
(19)

where, for $\mu=2$,

$\displaystyle
w(t,\tau)=\frac{1}{3}\left[x+2\sqrt{x^{2}+3y}\cos\frac{\varphi}{3}\right],$
(20)
$\displaystyle\tan\varphi=\frac{\sqrt{27(4y^{3}+y^{2}x^{2}+18yx^{2}+4x^{4}-27x^{2})}}{x(2x^{2}+9y-27)},$

with $x\doteq t-1$, $y\doteq 1+t\tau$ and $0\leq\varphi<\pi$, and, for
$\mu=\infty$,

$w(t,\tau)=\left\\{\begin{array}[]{ll}2&:0\leq t\leq t_{\mathrm{c}},\\\
t-1+{\displaystyle\frac{t\tau}{\lambda}}&:t>t_{\mathrm{c}},\end{array}\right.$
(21)

with

$\lambda(t,\tau)\doteq\frac{1}{2}\left[t-1+\sqrt{(t+1)(t-3)+4t\tau}\,\right]$
(22)

and

$t_{\mathrm{c}}\doteq\frac{3}{1+\tau}.$ (23)

The expressions for helicity (order parameter) and entropy, inferred from
first derivatives of $\bar{G}$ become

$\bar{N}_{\mathrm{hl}}(t,\tau)=\frac{t}{1+w}\frac{\partial w}{\partial t},$
(24) $\displaystyle\frac{\bar{S}(t,\tau)}{k_{\mathrm{B}}}$
$\displaystyle=\ln(1+w)-\frac{1}{1+w}\left[t\ln t\frac{\partial w}{\partial
t}+\tau\ln\tau\frac{\partial w}{\partial\tau}\right],$ (25)

respectively. They are too unwieldy for $\mu=2$ to be reproduced here but
fairly concise for $\mu=\infty$:

$N_{\mathrm{hl}}(t,\tau)=\begin{cases}\displaystyle 0&:0\leq t<t_{c},\\\
\displaystyle 1-\frac{t\tau}{\lambda^{2}-1+t\tau}&:t\geq t_{c},\end{cases}$
(26) $\displaystyle\frac{\bar{S}(t,\tau)}{k_{\mathrm{B}}}$
$\displaystyle=\ln\left(1+\frac{\tau}{\lambda}\right)$ $\displaystyle\hskip
14.22636pt+\frac{t\tau}{\lambda^{2}-1+t\tau}\left(\ln
t-\frac{\lambda^{2}-1}{t(\lambda+\tau)}\ln\tau\right).$ (27)

Finally the scaled enthalpy,

$\displaystyle\frac{\bar{H}(t,\tau)}{k_{\mathrm{B}}T}$
$\displaystyle=\frac{\bar{G}(t,\tau)}{k_{\mathrm{B}}T}+\frac{\bar{S}(t,\tau)}{k_{\mathrm{B}}}$
$\displaystyle=\frac{t\tau}{\lambda^{2}-1+t\tau}\left(\ln
t-\frac{\lambda^{2}-1}{t(\lambda+\tau)}\ln\tau\right),$ (28)

is of importance in this work.

##  Appendix B Backbone effects

It is instructive to take a look at the backbone contributions to the
landscapes of helicity and free energy for the case of fixed angles
$\theta_{\mathrm{N}}=0$, $\theta_{\mathrm{C}}=\pi$, notwithstanding its
limitations. The simplest case replaces the second Eq. (13) by

$l(x)=l_{\mathrm{h}}=1.5\mathrm{\AA},$ (29)

implying that the position coordinate of successive residues progresses
uniformly and independently of conformation. The value chosen in (29) is
accurate for a helical segment but shorter than most coil segments.

Free-energy and helicity landscapes thus predicted are shown in Fig. 15 for
$\mu=2,\infty$ and $N_{\mathrm{R}}=35,47,23$. The first row of landscapes
$(N_{\mathrm{R}}=35)$ is tailored to represent some variant of pHLIP, the
second row a significantly longer longer peptide $(N_{\mathrm{R}}=47)$, and
the third row a significantly shorter peptide $(N_{\mathrm{R}}=23)$. We only
consider enthalpy parameters that are neutral $\alpha_{\mathrm{H}}=1$ or
represent a gain $\alpha_{\mathrm{H}}=1.05,1.1$.

Figure 15: Helicity (peaked at center) and scaled free-energy of peptide with
$N_{\mathrm{R}}$ residues oriented and positioned as described in the text
versus the coordinate $x_{0}$ of the central residue for $\mu=2$ (left) and
$\mu=\infty$ (right) and three sizes. The solid and dashed curves pertain to
the values $\alpha_{\mathrm{H}}=1$ and $\alpha_{\mathrm{H}}=1.05,1.1$ of the
enthalpy parameter, respectively.

The case $\mu=\infty$ assigns more entropy to coil segments than the case
$\mu=2$. The effect on the results is significant, but produces no qualitative
changes. The helicity landscapes are almost independent of the enthalpy
parameter. That parameter affects the free-energy primarily in the aqueous
environment as expected.

Insertion into the membrane is clearly favored in all three cases and for both
variants of the model. The plots also tell us that insertion is accompanied by
a conformational change from coil to helix. For the longest peptide the
minimum in free energy is not as deep and the maximum in helicity is not as
high as is the case for the two shorter ones. The obvious reason is that the
former has significant flanking ends that remain in water.

Of particular interest is the free energy barrier that separates states with
the center of the peptide in aqueous or membrane environments, the former
mostly in coil conformation and the latter in helix conformation. This free-
energy barrier is very shallow for $\mu=2$ and only exists if
$\alpha_{\mathrm{H}}>1$. For $\mu=\infty$, on the other hand, it is more
conspicuous and present even for $\alpha=1$. This difference is related to the
higher entropy that coil segments must shed if $\mu=\infty$ when they order
into helix segments before they can cash in the enthalpic benefit of the lipid
environment.

One message we take from this simplest case is that insertion is not
automatic. An environmental change may be needed to push the peptide over the
barrier. An increase in acidity is known to do the trick. It neutralizes
negatively charged residues (e.g. Asp and Glu) via protonation. The
consequences for pHLIP are well documented by experiments RSA+07 ; RAS+08 ;
WMT+13 ; ADS+07 . Let us recall that the extent of insertion as predicted by
free-energy landscapes and the extent of ordering as predicted by helicity
landscapes can be directly monitored experimentally, namely by Trp
fluorescence and by circular dichroism experiments, respectively KWW+12 ;
TYM+09 ; KWW+12 ; VSS+18 .

An improved level of modeling takes into account that the distance between
adjacent residues is different in the coil and helix conformations as
reflected in Eqs. (13). The distance between successive residues now depends
on the local conformation of the backbone at their position in the membrane
environment. The main features in the results of the improved model (not
shown) remain qualitatively the same.

##  Appendix C Environmental levels

In a first round of estimates we may consider each residue placed in one of
three distinct environments: polar (w), interface (i), or non-polar (o). For
the transfer free energies between any two environments we use the Wimley-
White interface and octanol scales from WW96 ; WCW96 ; WW98 ; WW99 . State I
(see Sec. IV.1) has all residues in solution. For state II we assume that all
residues are at the interface. With that assumption we use $\Delta
G_{\mathrm{wi}}$ for all residues to calculate the transfer free energy of the
entire peptide. In this way we get one peptide transfer free energy,

$\Delta G_{\mathrm{I-II}}^{\mathrm{hpH}}=+0.18\mathrm{kcal/mol},$ (30)

if the water is at high pH and another peptide transfer free energy,

$\Delta G_{\mathrm{I-II}}^{\mathrm{lpH}}=-7.62\mathrm{kcal/mol},$ (31)

if the water is at low pH. In the former we assume that the negatively charged
residues are deprotonated and in the latter we assume that they are
protonated.

The more important result is (30). It is well-established that at high pH
pHLIP coexists in states I and II. Our rough estimate, which favors state I
but only slightly, by a small fraction of $k_{\mathrm{B}}T$, is consistent
with empirical evidence. A refined model will take into account that adsorbed
pHLIP is largely in coil conformation. The mechanical flexibility of this
conformation allows the hydrophobic residues to be closer to the interface and
some hydrophilic ones to stick out into water. This will lower the free-energy
prediction for state II relative to state I sufficiently to make $\Delta
G_{\mathrm{I-II}}^{\mathrm{hpH}}$ negative but not nearly as much as (31). A
slightly negative value of $\Delta G_{\mathrm{I-II}}^{\mathrm{hpH}}$ is most
consistent with experimental evidence.

Next we investigate, again by rough estimate, the peptide transfer free
energies $\Delta G_{\mathrm{II-III}}$ associated with insertion. However,
during the insertion process, from state II to state III, not all residues
switch environment. We assume that in state III the trans-membrane part of the
peptide is in $\alpha$-helix conformation. The relevant bilayer width, $\sim
35$Å, then accommodates 23 residues in an $\alpha$-helix conformation. These
residues experience the transfer io. The five residues closest to the N
terminus and the four residues closest to the C terminus are assumed to remain
in the interface region. We thus obtain the following results for the variants
W6 and W17 (see Fig. 1):

$\displaystyle\Delta G_{\mathrm{II-
III}}^{\mathrm{hpH}}=-0.5\mathrm{kcal/mol}\quad:~{}\mathrm{W6},$ (32)
$\displaystyle\Delta G_{\mathrm{II-
III}}^{\mathrm{lpH}}=-4.32\mathrm{kcal/mol}\quad:~{}\mathrm{W17},$ (33)

at high and low pH, respectively. The numbers are slightly lower for variant
W30. The more important result is (33). At low pH, the experimental evidence
shows that insertion is clearly favored over the adsorbed state. We also see
that the transfer free energy of insertion due to hydrophobic forces
associated with side chains is considerably higher than the corresponding
backbone transfer free energy investigated in Sec. IV. The result (32) is
significant for the pHLIP exit process (to be discussed elsewhere).

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