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# Characterizing the Interplay between Polymer Solvation and Conformation

Debdas Dhabal  Department of Chemical and Biomolecular Engineering, University
of Pennsylvania, Philadelphia, Pennsylvania 19104, United States  Zhitong
Jiang  Department of Chemical and Biomolecular Engineering, University of
Pennsylvania, Philadelphia, Pennsylvania 19104, United States  Amish J. Patel
<EMAIL_ADDRESS>Department of Chemical and Biomolecular
Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104,
United States

###### Abstract

Conformational transitions of flexible molecules, especially those driven by
hydrophobic effects, tend to be hindered by desolvation barriers. For such
transitions, it is thus important to characterize and understand the interplay
between solvation and conformation. Using specialized molecular simulations,
here we perform such a characterization for a hydrophobic polymer solvated in
water. We find that an external potential, which unfavorably perturbs the
polymer hydration waters, can trigger a coil-to-globule or collapse
transition, and that the relative stabilities of the collapsed and extended
states can be quantified by the strength of the requisite potential. Our
results also provide mechanistic insights into the collapse transition,
highlighting that polymer collapse proceeds through the formation of a
sufficiently large non-polar cluster, and that collective water density
fluctuations play an important role in stabilizing such a cluster. We also
study the collapse of the hydrophobic polymer in octane, a non-polar solvent,
and interestingly, we find that the mechanistic details of the transition are
qualitatively similar to that in water.

###### keywords:

Flexible, Hydrophobicity, Enhanced Sampling, Free Energy Landscape

##  1 Introduction

Aqueous or organic solutions of conformationally flexible solutes, such as
polymers or peptides, play an important role in diverse materials 1, 2, 3, 4
and biomolecular contexts 5, 6, 7, 8. Although solvated flexible molecules can
exist in a multitude of conformations, they often display two or more distinct
basins that are separated by free energy barriers, e.g., polymers can be in
extended or collapsed states, whereas proteins can be in their native folded
structures, become denatured, or even adopt well-defined misfolded
configurations 9, 10. Molecular simulations have been used extensively, often
with the aid of enhanced sampling methods, to study a variety of flexible
molecules, and characterize their conformational free energy landscapes,
typically as a function of one or more solute co-ordinates, such as radius of
gyration or dihedral angles 7, 11.

In characterizing such landscapes, solvent degrees of freedom are usually
integrated out. Such integration does not lead to any loss of mechanistic
information into conformational transitions when solvent degrees of freedom
equilibrate rapidly with respect to the conformational co-ordinates 12.
However, certain transitions, such as those driven by hydrophobic effects,
tend to feature slow solvent degrees of freedom, which relax on timescales
that are comparable to, or even longer than, the relaxation times of the
configurational co-ordinates 13, 14, 15, 16, 17, 18. In particular, both
theoretical and simulation studies have highlighted the importance of slow
solvent degrees of freedom in the coil-to-globule transition of non-polar
polymers in water 19, 13, 14, 20, 21.

For transitions characterized by slow solvent degrees of freedom, attempts to
integrate out the slow solvent co-ordinates can lead to hysteresis in the
sampling of the conformational co-ordinates, and make it challenging to
accurately estimate the conformational free energy landscape 22. Importantly,
the resulting loss of mechanistic information can also obfuscate how the
solute conformational landscape might respond to changes in the solvent, e.g.,
due to the introduction of co-solutes 23, 24, 25, 26 or proximity to
interfaces 27, 28, 29, 30. Thus, characterizing the interplay between
conformation and solvation, e.g., through a free energy landscape that is a
function of both conformational and solvent co-ordinates, can be valuable,
particularly for systems that are expected to feature slow solvent degrees of
freedom 13, 14, 31. However, the enhanced sampling of solvent co-ordinates,
which must be performed to obtain such a landscape, can be challenging because
the solvation shell of a flexible solute is inherently dynamic, and changes
along with the conformation of the solute.

To address this challenge, we recently introduced a method for sampling the
number of solvent molecules, $\tilde{N}_{v}$, in a dynamical volume, $v$,
which evolves with, and continuously conforms to, the shape of a flexible
solute of interest 31. Here, we use this method to study the coil-to-globule
or collapse transition of a flexible hydrophobic polymer solvated in water. We
find that polymer collapse can be triggered by an unfavorable potential,
$\phi\tilde{N}_{v}$, that perturbs the polymer hydration waters 32, 33, and
that the potential strength, $\phi$, needed to trigger the transition can
serve as a measure of the relative stabilities of the collapsed and extended
states. We also characterize the free energy landscape as a function of both
the radius of gyration of the polymer (conformational co-ordinate), and the
number of waters, $\tilde{N}_{v}$, in its hydration shell (solvent co-
ordinate), and use this characterization to uncover mechanistic insights into
the collapse transition. We find that polymer collapse proceeds through the
formation of a sufficiently large non-polar cluster, and that collective water
density fluctuations play an important role in nucleating such a cluster 13,
14. We also study the collapse of the hydrophobic polymer in the non-polar
solvent, octane, and find that it is remarkably similar to the collapse of the
polymer in water.

##  2 Methods

To interrogate the interplay between the solvation and conformation of
flexible solutes, here we study a linear alkane chain consisting of 45 beads
that is designated as C45. The solvation shell, $v$, of the C45 chain, is
defined as the union of 45 spherical sub-volumes with radius, $r_{v}$, which
are pegged to the C45 monomers. Thus, the solvation shell, $v$, of the
flexible C45 molecule is not static, but changes its shape and/or size
dynamically in response to the conformational fluctuations of the polymer.
Using the recently developed dynamic INDUS method 31, here we bias the coarse-
grained number, $\tilde{N}_{v}$, of solvent heavy atoms in $v$;
$\tilde{N}_{v}$ is closely related to the actual number of solvent atoms in
$v$, but is chosen to be a continuous function of particle positions to permit
biasing $\tilde{N}_{v}$ without resulting in impulsive forces 34, 35. By using
harmonic biasing potentials to sample $\tilde{N}_{v}$ over its range of
interest, and using the Weighted Histogram Analysis Method (WHAM) to combine
the biased probability distributions 36, 37, 38, 39, we are able to
characterize the free energetics, $\beta G_{v}(\tilde{N})\equiv-\ln
P_{v}(\tilde{N})$, where $\beta^{-1}\equiv k_{\rm{B}}T$ is the thermal energy,
and $P_{v}(\tilde{N})$ is the probability of observing $\tilde{N}$ solvent
atoms in $v$. The probability,
$P_{v}(\tilde{N})\equiv\langle\delta(\tilde{N}-\tilde{N}_{v})\rangle_{0}$, is
the ensemble average of the Dirac delta function obtained using the unbiased
Hamiltonian, $\mathcal{H}_{0}$. Once $G_{v}(\tilde{N})$ is estimated, we use
standard reweighting techniques 36 to interrogate how biasing potentials, such
as $U_{\phi}\equiv\phi\tilde{N}_{v}$, modulate polymer solvation and determine
the corresponding averages, $\langle\dots\rangle_{\phi}$, obtained using the
biased Hamiltonians, $\mathcal{H}_{\phi}=\mathcal{H}_{0}+U_{\phi}$. For
example, the free energetics in a biased ensemble, $\beta
G_{v}^{\phi}(\tilde{N})\equiv-\ln P_{v}^{\phi}(\tilde{N})$, can be obtained
using
$P_{v}^{\phi}(\tilde{N})=\langle\delta(\tilde{N}-\tilde{N}_{v})\rangle_{\phi}$.
Finally, to characterize the two-dimensional free energetics, $\beta
G_{v}(\tilde{N},R_{\rm g})\equiv-\ln P_{v}(\tilde{N},R_{\rm g})$, where
$P_{v}(\tilde{N},R_{\rm g})$ is the joint probability of observing $\tilde{N}$
solvent atoms in $v$, and the polymer with a radius of gyration, $R_{\rm g}$,
we combine the joint probability distributions, observed in our biased
simulations, using the WHAM weights determined above 35, 40.

###  2.1 Simulation Details

All simulations were performed using GROMACS package 41 (version 4.5.3),
suitably modified to perform indirect umbrella sampling (INDUS) using dynamic
probe volumes 35, 31. The equations of motion were integrated using the leap-
frog algorithm with a time step of 2 fs, and periodic boundary conditions were
employed in all three dimensions. The system temperature, $T$, was maintained
at 298 K using the canonical velocity rescaling thermostat 42 with a time
constant of 0.5 ps, and the system pressure, $P$, was maintained at 1 atm
using the Parrinello-Rahman barostat 43 with a time constant of 1 ps. The
n-alkane chain containing 45 carbons (designated as C45) was modeled using the
TraPPE-UA (Transferable Potentials for Phase Equilibria – United Atom)
forcefield developed by Siepmann et. al. 44. The TraPPE-UA forcefield models
individual -CH2\- or -CH3 units as pseudo-atoms, so that our solute is
comprised of 45 united-atom beads. The C45 chain was solvated using either
water or n-octane, and Lorentz-Berthelot combination rules were used to
determine the Lennard Jones parameters for cross-interactions. In all cases,
the simulation box was first energy minimized using the steepest descent
algorithm, followed by a 2 ns NVT simulation, and then a 4 ns NPT simulation
to equilibrate the system. The biased simulations were run for 10 ns, with
either the first 2 ns (for C45/water) or 1 ns (for C45/octane) being discarded
for equilibration; the coarse-grained number of solvent atoms,
$\tilde{N}_{v}$, in the polymer hydration shell, $v$, and the polymer radius
of gyration, $R_{\rm g}$, were estimated every 0.1 ps. Additional details of
the C45/water and C45/octane simulations are included below.

C45/Water:  Water was modeled using the SPC/E (extended simple point charge)
model 45 with bonds involving hydrogen atoms being constrained using the
SETTLE algorithm 46. The particle Mesh Ewald (PME) method 47 was used to
compute long-range electrostatic interactions; short-range electrostatic and
Lennard Jones interactions were truncated at 1 nm. The C45 chain was hydrated
using 11,614 water molecules giving rise to a roughly $7$ nm cubic simulation
box. Before performing biased simulations, a 10 ns long unbiased simulation
was performed to estimate the average number of solvent atoms in $v$, and
ascertain the $\tilde{N}_{v}$-range that must be sampled; the radius of the
spherical sub-volumes used to define $v$ was chosen to be $r_{v}=0.6$ nm. The
biased simulations used harmonic potentials with a spring constant of 0.0243
kJ/mol.

C45/Octane: Octane was modeled using the TraPPE-UA forcefield 44 with Lennard
Jones interactions being truncated at 1.4 nm. The C45 chain was solvated using
1,000 octane molecules. The sub-volume radius used to define $v$ was chosen to
be $r_{v}=0.7$ nm, and a spring constant of 0.033 kJ/mol was used for the
harmonic biasing potentials.

##  3 Results and Discussion

###  3.1 How Polymer Hydration Waters Respond to Perturbations

To characterize how the hydration of the flexible C45 chain influences its
conformation, we focus on water molecules in its first hydration shell, $v$
(Figure 1a, cyan/transparent). We account for the dynamic nature of the
polymer hydration shell, i.e., changes in its size and shape in response to
polymer conformational fluctuations, by defining $v$ to be the union of 45
spherical sub-volumes, where every sub-volume is pegged to a polymer bead, and
has a radius, $r_{v}=0.6$ nm. We then interrogate how the polymer hydration
waters respond to perturbations by employing a biasing potential,
$U_{\phi}=\phi\tilde{N}_{v}$, where $\tilde{N}_{v}$ is the (coarse-grained)
number of waters in $v$, and $\phi$ is the potential strength. Such a biasing
potential provides a convenient way to modulate $\tilde{N}_{v}$, and thereby
polymer hydration. In particular, hydration is disfavored for $\phi>0$ with
the perturbation becoming more pronounced as $\phi$ is increased; conversely,
for $\phi<0$, polymer hydration is favored. From a physical standpoint, a
biasing potential of strength, $\phi$, effectively decreases the pressure in
the polymer hydration shell by roughly $\phi\rho_{\rm w}$, where $\rho_{\rm
w}$ is the density of bulk water 48.

Figure 1:  (a) Simulation snapshots of the C45 polymer (cyan, space-fill)
solvated in water (red/white) are shown. Water molecules in the polymer
hydration shell, $v$ (cyan, transparent), are highlighted. As the polymer
undergoes conformational fluctuations, both the shape and size of its
hydration shell change; to account for such changes, $v$ is defined to be the
union of 45 spherical sub-volumes that are centered on the polymer beads and
have radii, $r_{v}=0.6$ nm. (b) As the strength, $\phi$, of a linear biasing
potential, $U_{\phi}=\phi\tilde{N}_{v}$, is increased, the average number of
waters, $\langle\tilde{N_{v}}\rangle_{\phi}$, in the hydration shell
decreases; interestingly, two sharp drops are observed in
$\langle\tilde{N_{v}}\rangle_{\phi}$. (c) Correspondingly, the susceptibility,
$-\partial\langle\tilde{N_{v}}\rangle_{\phi}/\partial(\beta\phi)=\langle\delta\tilde{N}_{v}^{2}\rangle_{\phi}$,
displays two peaks at $\phi$-values denoted by $\phi_{1}^{*}$ and
$\phi_{2}^{*}$. (d) Simulation snapshots corresponding to three select
$\tilde{N}_{v}$-values are shown. As shown in snapshot \raisebox{-.9pt}
{\smallI}⃝ , the polymer adopts an extended configuration for
$\tilde{N}_{v}>\langle\tilde{N_{v}}\rangle_{\phi_{1}^{*}}$, whereas snapshot
\raisebox{-.9pt} {\smallII}⃝ highlights that the polymer is in a collapsed
configuration for
$\langle\tilde{N_{v}}\rangle_{\phi_{2}^{*}}<\tilde{N}_{v}<\langle\tilde{N_{v}}\rangle_{\phi_{1}^{*}}$.
As illustrated in snapshot \raisebox{-.9pt} {\smallIII}⃝ , the collapsed
polymer is dewetted for
$\tilde{N}_{v}<\langle\tilde{N_{v}}\rangle_{\phi_{2}^{*}}$.

The response of the C45 hydration waters to the biasing potential is shown in
Figure 1. As expected, the average number of hydration waters,
$\langle\tilde{N_{v}}\rangle_{\phi}$, decreases as $\phi$ is increased (Figure
1b). However, the decrease in $\langle\tilde{N_{v}}\rangle_{\phi}$ is not
gradual, but is punctuated by two sharp drops. The corresponding
susceptibility,
$\partial\langle\tilde{N_{v}}\rangle_{\phi}/\partial(-\beta\phi)=\langle\delta\tilde{N}_{v}^{2}\rangle_{\phi}$,
shown in Figure 1c, displays marked peaks at $\phi$-values, which are denoted
by $\phi_{1}^{*}$ and $\phi_{2}^{*}$ . The sharp drops in
$\langle\tilde{N_{v}}\rangle_{\phi}$, and the corresponding peaks in
susceptibility, suggest that the polymer undergoes collective transitions at
potential strengths, $\phi_{1}^{*}$ and $\phi_{2}^{*}$. To shed light on the
nature of these transitions, we include representative snapshots of the system
for select $\tilde{N}_{v}$-values in Figure 1d. Snapshot \raisebox{-.9pt}
{\smallI}⃝ suggests that the polymer is in an extended configuration for
$\phi<\phi_{1}^{*}$; snapshot \raisebox{-.9pt} {\smallII}⃝ shows the polymer
in a collapsed, but hydrated configuration for
$\phi_{1}^{*}<\phi<\phi_{2}^{*}$; and snapshot \raisebox{-.9pt} {\smallIII}⃝
highlights that the collapsed polymer is dewetted for $\phi>\phi_{2}^{*}$.
These snapshots suggest that at potential strength, $\phi_{1}^{*}$, the
polymer undergoes a collapse (or folding) transition, whereas at potential
strength, $\phi_{2}^{*}$, the collapsed polymer dewets.

Figure 2:  (a) The free energetics of water number fluctuations,
$G_{v}(\tilde{N})$, in the dynamical hydration shell of the C45 polymer are
shown. The basins corresponding to the collapsed (C) and extended (E) states
of the polymer are denoted with arrows. (b) To illustrate the interplay
between the hydration of the hydrophobic polymer in water and its
conformation, we plot the two-dimensional free energy landscape,
$G_{v}(\tilde{N},R_{\rm g})$, where $\tilde{N}$ is the number of waters in
$v$, and $R_{\rm g}$ is the radius of gyration of the C45 polymer. The
relatively sharp increase in $G_{v}(\tilde{N},R_{\rm g})$ at low $\tilde{N}$
and $R_{\rm g}$ corresponds to the free energetic cost of dewetting the
collapsed polymer globule; whereas the increase in $G_{v}(\tilde{N},R_{\rm
g})$ at high $\tilde{N}$ and $R_{\rm g}$ corresponds to the unfavorable
wetting of the extended polymer.

###  3.2 Interplay Between Polymer Hydration and Conformation

To characterize the free energy landscape underpinning the collapse and
dewetting transitions, we estimate the free energetics, $G_{v}(\tilde{N})$, of
water number fluctuations in the hydration shell of the C45 polymer (Figure
2a), as well as the corresponding two-dimensional free energy landscape,
$G_{v}(\tilde{N},R_{\rm g})$ (Figure 2b). The conformational free energy
landscape, $G_{v}(R_{\rm g})$, can be readily obtained from
$G_{v}(\tilde{N},R_{\rm g})$ by integrating out $\tilde{N}$, and is included
in Figure S1 of the Supplementary Material. Figure 2 confirms the presence of
two nearly degenerate basins, observed at low $\tilde{N}\approx 130$ and
$R_{\rm g}\approx 0.5$ nm, and at higher $\tilde{N}\approx 200$ and $R_{\rm
g}\approx 1.2$ nm, corresponding to the collapsed (C) and extended (E) states,
respectively. Figure 2 also suggests that the solvation and conformational co-
ordinates are coupled to one another across the collapse transition, and that
the transition incurs a free energetic cost of less than $8~{}k_{\rm{B}}T$. In
contrast, decreasing $\tilde{N}$ below 100 results in a sharp increase in
$G_{v}(\tilde{N},R_{\rm g})$; such a decrease in $\tilde{N}$ is not
accompanied by a further decrease in $R_{\rm g}$, suggesting that it
corresponds to the dewetting of the collapsed polymer. Similarly, increasing
$\tilde{N}$ above 220 also incurs a large free energetic penalty, and is not
accompanied by an increase in $R_{\rm g}$, suggesting that it represents the
unfavorable compression of waters in the hydration shell of the extended
polymer.

Although the free energy difference between the collapsed and extended states
or the folding free energy, $\Delta G_{\rm fold}$ is relatively small, Figure
2a suggests that for the hydrophobic $C_{45}$ chain in water, the collapsed
state is nevertheless stable relative to the extended state, with $\beta\Delta
G_{\rm fold}=-3.3$. The relative stability of the collapsed state is also
evident from the fact that the potential strength, $\phi^{*}_{1}$,
corresponding to the collapse transition is negative (Figure 1c). In fact, as
shown below, the two quantities are proportional to one another, with $\Delta
G_{\rm fold}\approx\phi_{1}^{*}(\tilde{N}_{\rm E}-\tilde{N}_{\rm C})$, where
$\tilde{N}_{\rm E}$ and $\tilde{N}_{\rm C}$ are $\tilde{N}$-values
corresponding to the extended and collapsed basins, respectively. The
collapsed and extended states are expected to be in coexistence with one
another in the $\phi^{*}_{1}$-ensemble, i.e.,
$G_{v}^{\phi_{1}^{*}}(\tilde{N}_{\rm C})=G_{v}^{\phi_{1}^{*}}(\tilde{N}_{\rm
E})$, and because $G_{v}^{\phi}(\tilde{N})$ equals
$G_{v}(\tilde{N})+\phi\tilde{N}$ within a constant 40: $G_{v}(\tilde{N}_{\rm
C})+\phi_{1}^{*}\tilde{N}_{\rm C}=G_{v}(\tilde{N}_{\rm
E})+\phi_{1}^{*}\tilde{N}_{\rm E}$; thus, the polymer folding free energy,
$\Delta G_{\rm fold}\approx G_{v}(\tilde{N}_{\rm C})-G_{v}(\tilde{N}_{\rm
E})=\phi_{1}^{*}(\tilde{N}_{\rm E}-\tilde{N}_{\rm C})$. For the C45 polymer in
water, we estimate $\phi_{1}^{*}(\tilde{N}_{\rm E}-\tilde{N}_{\rm
C})\approx-3.47~{}k_{\rm{B}}T$, which in good agreement with $\Delta G_{\rm
fold}$ obtained directly from $G_{v}(\tilde{N})$. Thus, $\phi^{*}_{1}$ can
serve as a quantitative measure of $\Delta G_{\rm fold}$.

###  3.3 Free Energy Landscape at the Collapse Transition

To better understand the transition of the C45 polymer between its extended
and collapsed states, we now focus on its behavior in the presence of a
biasing potential of strength, $\phi_{1}^{*}$, wherein the two states are
expected to be in coexistence with one another. The corresponding free energy
landscapes, $G_{v}^{\phi^{*}_{1}}(\tilde{N})$ and
$G_{v}^{\phi^{*}_{1}}(\tilde{N},R_{\rm g})$ are shown in Figure 3. As seen in
Figure 3a, $G_{v}^{\phi^{*}_{1}}(\tilde{N})$ features two basins: the
low-$\tilde{N}$, collapsed (C) and the high-$\tilde{N}$, extended (E) basins,
which are in coexistence with one another, and are separated by a barrier of
roughly $2~{}k_{\rm{B}}T$. Figure 3a also highlights the locations of the
basins (C, E) and the maximum in $G_{v}^{\phi^{*}_{1}}(\tilde{N})$ at
intermediate $\tilde{N}$ (I).

Figure 3:  To better understand the coil-to-globule collapse transition of the
C45 polymer, here we study its behavior in the $\phi_{1}^{*}$-ensemble; that
is, in the presence of the biasing potential, $\phi_{1}^{*}\tilde{N}_{v}$,
where $\beta\phi_{1}^{*}=-0.05$. In such a biased ensemble, the coil and
globule states are expected to be in coexistence with one another. (a) The
free energetics, $G_{v}^{\phi_{1}^{*}}(\tilde{N})$, of water number
fluctuations in the $\phi_{1}^{*}$-ensemble, display two distinct basins that
are separated by a barrier of roughly $2~{}k_{\rm{B}}T$. The basins
corresponding to the collapsed (low $\tilde{N}$) and extended (high
$\tilde{N}$) states of the polymer are labelled “C” and “E”, respectively,
whereas the location of the maximum in $G_{v}^{\phi_{1}^{*}}(\tilde{N})$ at
intermediate $\tilde{N}$ is denoted by “I”. (b) The two-dimensional free
energy landscape in the $\phi^{*}_{1}$-ensemble,
$G_{v}^{\phi_{1}^{*}}(\tilde{N},R_{\rm g})$, is shown here, and sheds light on
the minimum free energy path that the polymer is likely to adopt as it
undergoes collapse. Representative configurations of the polymer in the “C”,
“I”, and “E” states are also shown.

The two-dimensional free energy landscape,
$G_{v}^{\phi^{*}_{1}}(\tilde{N},R_{\rm g})$, shown in Figure 3b, sheds light
on the mechanistic pathways that the polymer follows as it undergoes the
collapse transition, i.e., it elucidates the minimum free energy path in the
($\tilde{N}$, $R_{\rm g}$) space, with $\tilde{N}$ and $R_{\rm g}$
representing the polymer hydration and conformational degrees of freedom. The
$G_{v}^{\phi^{*}_{1}}(\tilde{N},R_{\rm g})$ landscape highlights that in its
extended state, the polymer can undergo substantial fluctuations in $R_{\rm
g}$, but only small fluctuations in $\tilde{N}$. To approach the saddle point
in $G_{v}^{\phi^{*}_{1}}(\tilde{N},R_{\rm g})$, the polymer must undergo a
marked low-$R_{\rm g}$ fluctuation, which is followed by a substantive
decrease in its hydration waters, $\tilde{N}$. Upon traversing the saddle
point, polymer collapse proceeds through a co-ordinated decrease in both
$R_{\rm g}$ and $\tilde{N}$, which is downhill in free energy.

Representative configurations of the polymer in the extended (E), saddle point
(I) and collapsed (C) states are also shown Figure 3b, and shed further light
on the pathways involved in polymer collapse. In particular, at the saddle
point, the polymer adopts a partially collapsed conformation with some, but
not all, of its monomers being clustered together. Thus, the formation of a
sufficiently large hydrophobic cluster, which not only lowers the $R_{\rm g}$
of the polymer, but also leads to a decrease in its hydration waters,
represents the barrier to polymer collapse. Importantly, as the polymer
crosses this barrier, the decrease in its $R_{\rm g}$ is small relative to the
corresponding decrease in its hydration waters, $\tilde{N}$. These findings
highlight the importance of collective water density fluctuations in
facilitating polymer collapse.

Our results also lend support to the observations of ten Wolde and Chandler
13, who studied the collapse of a purely repulsive, idealized hydrophobic
polymer using a coarse-grained model of water, and those of Miller et al.14,
who studied the same polymer using atomistic simulations; both studies found
that the formation of a critical non-polar cluster and the associated
dewetting represent the barrier to polymer collapse. Moreover, the agreement
between these studies, performed on ideal hydrophobic polymers, and our
results, obtained using realistic polymers, which have favorable dispersion
interactions with water, highlight that weak polymer-water attractions do not
qualitatively alter the polymer hydration landscape 49.

###  3.4 Polymer Collapse in Octane

We now study the solvation of the hydrophobic C45 polymer in octane, a non-
polar solvent, and draw comparisons to the solvation of the polymer in water.
In Figure 4a, we plot the response of the average number of solvent heavy
atoms, $\langle\tilde{N_{v}}\rangle_{\phi}$, in the polymer solvation shell,
$v$, to the strength, $\phi$ of the linear biasing potential,
$U_{\phi}=\phi\tilde{N}_{v}$, and in Figure 4b, we plot the corresponding
susceptibility,
$-\partial\langle\tilde{N_{v}}\rangle_{\phi}/\partial(\beta\phi)=\langle\delta\tilde{N}_{v}^{2}\rangle_{\phi}$.
Once again, two sharp drops are observed in
$\langle\tilde{N_{v}}\rangle_{\phi}$ with increasing $\phi$, and
correspondingly, two peaks are seen in
$\langle\delta\tilde{N}_{v}^{2}\rangle_{\phi}$ at $\phi$-values denoted by
$\phi_{1}^{*}$ and $\phi_{2}^{*}$. The configurations of C45 in octane, shown
for three select $\tilde{N}_{v}$-values in Figure S2 of the Supplementary
Material, suggest that the transitions at $\phi_{1}^{*}$ and $\phi_{2}^{*}$,
once again, correspond to polymer collapse and the dewetting of the collapsed
polymer. Vertical lines in Figures 4a,b (blue) correspond to $\phi=0$, and
highlight that in contrast with C45 in water, $\phi_{1}^{*}$ is positive for
C45 in octane. Thus, the folding (or collapse) free energy, $\Delta G_{\rm
fold}\approx\phi^{*}_{1}(\tilde{N}_{\rm E}-\tilde{N}_{\rm C})$, must also be
positive, suggesting that in octane, the extended state of C45 is stable
relative to its collapsed state. Figure 4b also shows that the dewetting of
the collapsed polymer occurs at a lower $\phi^{*}_{2}$-value in octane than in
water; this observation can be readily rationalized in terms of the smaller
liquid-vapor surface tension of octane, which makes it easier to form cavities
50.

Figure 4:  The C45 polymer solvated in octane. (a) The average number of
solvent heavy atoms, $\langle\tilde{N_{v}}\rangle_{\phi}$, in the solvation
shell, $v$, of the C45 polymer are shown as a function of the strength,
$\phi$, of the linear biasing potential, $U_{\phi}=\phi\tilde{N}_{v}$. For the
C45 polymer solvated in both octane (red, solid) and in water (green, dashed
lines), two sharp drops in $\langle\tilde{N_{v}}\rangle_{\phi}$ are observed
as $\phi$ is increased; the vertical line (blue) depicts $\phi=0$. (b)
Correspondingly, the susceptibility,
$-\partial\langle\tilde{N_{v}}\rangle_{\phi}/\partial(\beta\phi)=\langle\delta\tilde{N}_{v}^{2}\rangle_{\phi}$,
displays two peaks in both cases; the locations of the peaks, $\phi_{1}^{*}$
and $\phi_{2}^{*}$, are included for octane. In contrast with C45 in water,
$\phi_{1}^{*}>0$ for C45 solvated in octane, suggesting that polymer is in its
extended state at equilibrium ($\phi=0$). (c) The free energetics of solvent
number fluctuations, $G_{v}(\tilde{N})$, in the hydration shell of the C45
polymer are shown, and the locations of the collapsed (C) and extended (E)
states of the polymer are denoted with arrows. In octane, the extended state
(high $\tilde{N}$) is stable, whereas the collapsed state (low $\tilde{N}$) is
unstable. (d) The two-dimensional free energy landscape,
$G_{v}(\tilde{N},R_{\rm g})$, for the C45 polymer solvated in octane is shown.
(e) The free energetics, $G_{v}^{\phi_{1}^{*}}(\tilde{N})$, of solvent number
fluctuations in the $\phi_{1}^{*}$-ensemble, display two distinct basins that
are separated by a barrier. (f) The two-dimensional free energy landscape,
$G_{v}^{\phi_{1}^{*}}(\tilde{N},R_{\rm g})$, shown here for the C45 polymer
solvated in octane, as well as the representative configurations of the
polymer, are qualitatively similar to those for C45 in water (Figure 3b).

The unbiased free energy landscapes, $G_{v}(\tilde{N})$ and
$G_{v}(\tilde{N},R_{\rm g})$, for C45 in octane, are shown in Figures 4c,d,
and the conformational free energy landscape, $G_{v}(R_{\rm g})$, is included
as Figure S3 of the Supplementary Material. Apart from the fact that the
collapsed state is metastable with respect to the extended state, with
$\beta\Delta G_{\rm fold}=4.5$, the landscapes are qualitatively similar to
those for C45 in water (Figure 2). The corresponding free energy landscapes in
the $\phi_{1}^{*}$-ensemble are shown in Figures 4e,f. As with C45 in water,
the collapsed (C) and extended (E) states are degenerate in the
$\phi_{1}^{*}$-ensemble, and a barrier is observed at intermediate $\tilde{N}$
and $R_{\rm g}$ for C45 in octane. Although the barrier height is somewhat
lower for C45 in octane (relative to water), both the
$G_{v}^{\phi_{1}^{*}}(\tilde{N},R_{\rm g})$ landscape and representative
configurations of the polymer (Figure 4f), suggest that polymer collapse in
octane proceeds through mechanistic pathways, which are remarkably similar to
those in water (Figure 3b). In particular, as in water, the collapse of the
C45 polymer in octane also proceeds through the formation of a sufficiently
large cluster, which is both partially collapsed and partially desolvated.

To understand these similarities, we first recognize that polymer collapse
involves the solvent-mediated assembly of well-solvated sub-units, which are
smaller than 1 nm, into a cluster that is larger than 1 nm in size 51, 52.
According to the Lum-Chandler-Weeks theory of hydrophobicity 19, the solvation
free energy of the former scales as their excluded volume, whereas the
solvation free energy of the latter scales as its surface area 53, 54, 55, 56,
57; this interplay between solvation at small and large length scales is
responsible for the emergence of a critical cluster, which must be nucleated
for the collapse transition to proceed 13, 14. Interestingly, a solute size-
dependent crossover in solvation free energies has also been observed in non-
polar solvents, such as octane 58, 50. Although the corresponding solvophobic
effect, experienced by solutes in non-polar solvents, is expected to be weaker
than the hydrophobic effect 54, 59, our results suggest that it may
nevertheless be responsible for the qualitative similarities between the
polymer collapse pathways in water and in octane.

##  4 Conclusions and Outlook

Using the recently developed dynamic Indirect Umbrella Sampling (INDUS) method
for sampling the number of solvent molecules, $\tilde{N}_{v}$, in the
dynamical solvation shell of a flexible solute, here we study the coil-to-
globule or collapse transition of a hydrophobic polymer solvated in water and
in the non-polar solvent, octane. We find that an unfavorable potential, which
perturbs $\tilde{N}_{v}$, can trigger the collapse transition, and that the
requisite potential strength quantifies the relative stabilities of the
collapsed and extended states. To understand the interplay between polymer
hydration and conformation, we characterize the free energy landscape of the
system as a function of both the number of solvent molecules, $\tilde{N}_{v}$,
in the polymer hydration shell, and its radius of gyration, $R_{\rm g}$. We
find that polymer collapse proceeds through the formation of a critical non-
polar cluster, which represents the bottleneck to the coil-to-globule
transition. In particular, assisted by thermal fluctuations, an extended
polymer must first adopt a relatively compact configuration, which then
undergoes collective dewetting; if the resulting non-polar cluster is
sufficiently large, the rest of the polymer then collapses spontaneously.

In agreement with previous studies on ideal hydrophobic polymers 13, 14, our
findings thus confirm the presence of slow solvent co-ordinates in the coil-
to-globule transition, and highlight the importance of collective solvent
density fluctuations in overcoming the corresponding barriers. In contrast
with water, which is a poor solvent for the hydrophobic polymer, the non-polar
solvent, octane, is expected to be a good solvent for the polymer.
Interestingly, we find that the mechanistic details of the collapse transition
in octane are nevertheless remarkably similar to that in water. Our results
suggest that the size-dependent crossover in solvation free energies, which
has been observed not just in water 54, but also in non-polar solvents, such
as octane 58, 50, may underpin the similar polymer collapse pathways in these
two solvents.

A characterization of the free energetics of water density fluctuations in
static volumes, both in bulk water 60, 53 and in the vicinity of surfaces 61,
62, 63, 64, has provided numerous insights into the collective, solvent-
mediated hydrophobic effects, ranging from micelle formation and interfacial
assembly, to superhydrophobicity and protein interactions 65, 66, 67, 68, 69.
By characterizing the free energetics of solvent density fluctuations in the
dynamic solvation shells of conformationally flexible molecules, we hope that
our work will similarly pave the way for a richer understanding of the
conformational transitions 70, 71 and the phase behavior 72, 73 of a large
classes of interesting solutes, ranging from hydrocarbons and polymers, to
peptides and nucleic acids 30, 74, 31. Our approach is also likely to shed
light into how co-solutes (e.g., osmolytes, salts, etc.), which have a
propensity to be included or excluded from the vicinity of a polymer, might
modulate the solvation of a polymer as well as its conformational landscape
75, 76, 77, 78, 79, 80.

A.J.P. gratefully acknowledges financial support from the National Science
Foundation (NSF grants CBET-1652646, CHE-1665339 and DMR-1720530), and awards
from the Alfred P. Sloan Research Foundation (FG-2017-9406) and the Camille
and Henry Dreyfus Foundation (TC-19-033). D.D. was supported by NSF grant
CHE-1665339. The authors thank Akash Pallath for helpful discussions.

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