Evolution of the Free Energy Landscapes of n-Alkane
Guests Bound within Supramolecular Complexes
Confinement within nanoscale spaces can dramatically alter the ensemble of conformations flexible species explore. For example, chaperone complexes take advantage of confinement to fold misfolded proteins, while viral capsids transport genomic materials in tight packings. Here we examine the free energy landscapes of n-alkanes confined within supramolecular dimeric complexes of deep-cavity cavitand octa-acid, which have been experimentally demonstrated to force these chains with increasing length to adopt extended, helical, hairpin, and spinning top conformational motifs, using molecular simulations. Alkanes up to n-docosane in both vacuum and water predominantly exhibit a free energy minimum for elongated conformations with a majority of trans dihedrals. Within harmonically sealed cavitand dimers, however, the free energy landscapes as a function of the end-to-end distance between their terminal methyl units exhibit minima that evolve with the length of the alkane. Distinct free energy basins are observed between the helical and hairpin motifs and between the hairpin and chicane motifs whose relative stability changes with the number of carbons in the bound guest. These changes are reminiscent of two state-like protein folding, although the observed alkane conformations confined are more insensitive to temperature perturbation than proteins are. While the chicane motif within the harmonically sealed dimers has not been observed experimentally, this conformation relaxes to the observed spinning top motif once the harmonic restraints are released for the complexes in aqueous solution, indicating that these motifs are related to one another. We do not observe distinct minima between the confined extended and helical motifs, suggesting these conformers are part of a larger linear motif family whose population of gauche dihedral angles grows in proportion to the number of carbons in the chain to ultimately form a helix that fits the alkane within the complex.
Special Issue<!>Introduction<!><!>Introduction<!>Simulation
Methods<!>Results and Discussion<!><!>Results and Discussion<!><!>Results and Discussion<!><!>Results and Discussion<!><!>Results and Discussion<!><!>Results and Discussion<!><!>Results and Discussion<!>Conclusions<!>
<p>Published as part
of The Journal of Physical Chemistry virtual special
issue "Dor Ben-Amotz Festschrift".</p><!><p>The spontaneous organization
of molecular host and guest species
in solution into well-defined supramolecular assemblies offers an
attractive route toward building synthetic complexes that potentially
possess biomimetic functions. Indeed, organisms utilize nanoscale
compartmentalization to enable processes like catalysis, transport,
storage, and protection. The GroEL/GroES1,2 and CCT/TRiC3−5 complexes, barrel-like assemblies of chaperonin proteins, internalize
misfolded proteins and facilitate their refolding to their native
structures to prevent the buildup of toxic protein aggregates. Virus
capsids, on the other hand, are large supramolecular protein complexes
that serve as transport containers to traffic viral RNA/DNA under
pressure between organisms.6−8 In 2004, Gibb and co-workers synthesized
deep-cavity cavitand octa-acid (OA), a host molecule possessing three
rows of aromatic rings that compose a hydrophobic pocket approximately
8 Å in diameter and 8 Å deep (Figure 1a).9,10 The rim and foot of
this pocket are functionalized with eight carboxylic acid coating
groups that engender aqueous solubility. The nonpolar pockets of OA
readily bind to hydrophobic guest species in water to form 1:1, 2:1,
and 2:2 supramolecular complexes (denoted X:Y, where X and Y are the
number of host/cavitand and guest species in a complex) depending
on the guest's size and shape.11−13 Guests confined within
these supramolecular complexes can adopt conformations rarely observed
in bulk solution, which in turn can enable the cavitand to mimic an
enzymatic pocket.14,15 Encapsulation of α-alkyl
dibenzyl ketones within 2:1 complexes with OA, for example, was found
to yield distinct photoreaction products depending on the packing
of constraints imposed by packing of the alkyl side chain within the
nonpolar interior.16 Cyclization of α,ω-thio-alkane
halides within cavitand complexes, on the other hand, has been shown
to be promoted by both templation by the host for the guest to adopt
a J-like/hairpin conformation and
the charge (anionic or cationic) of the host complex to lower the
barrier for creating the negatively charged transition species.17,18 Experimental analysis of the packing of n-alkanes
within 2:1 complexes with OA using one- and two-dimensional NMR found
these guests exhibited a succession of conformational motifs with
increasing chain length (Figure 1b). Beginning with n-nonane, the shortest
guest to stabilize 2:1 complexes, the guest adopted an extended motif enriched in trans dihedrals down its backbone.
The inherent spatial limitations inside the complex compressed alkanes
of increasing length, forcing the population of gauche dihedrals to increase so that alkanes like n-tetradecane
were found to adopt a helical motif. Around n-heptadecane the strain imposed by the growing gauche population forced the guest to adopt a hairpin motif, in which the two methyl termini are directed toward a single
OA host pocket while the hairpin turn is directed toward the opposing
pocket in the dimer. Ultimately, guests longer than tricosane were
found to adopt a spinning top motif where the terminal
methyl units are redirected toward the opposing host pockets while
the hairpin turn is extruded between the two hosts and partially exposed
to the surrounding aqueous environment. While the interiors of the
cavitand dimers are dry, the capacity for water to enter into such
confined spaces is anticipated to play a role in determining the conformation
of encapsulated alkanes as well.19</p><!><p>(a) Chemical structure of deep-cavity cavitand octa-acid.
At pH 7 the carboxylic acids ringing the rim of the hydrophobic pocket
at the top of the cavitand are deprotonated and carry a negative charge
(−), while only two of the carboxylic acids of the four feet
at the bottom of the cavitand are deprotonated. (b) Simulation
snapshots of alkanes contained within a cavitand dimer complex exhibiting
an extended (C11), helical (C14), hairpin (C18), or spinning top (C22) conformational motif. The
alkanes are illustrated as a teal (carbon) and white (hydrogen) van
der Waals surface, while the two cavitands are illustrated as the
red or orange solvent accessible surfaces. Part of the cavitand dimer
structure has been peeled away to show the alkane buried within each
complex.</p><!><p>Over the past 10 years Ashbaugh
and Gibb have examined the stability
and function of cavitand complexes in water in a joint experimental
and simulation collaboration.20 In one
of our first studies, we demonstrated that molecular simulations accurately
reproduce the succession of alkane packing motifs inferred from NMR
as a function of the guest length.21 The
bound alkane motifs were identified by visual inspection of the most
probable conformation over a simulation trajectory for each guest
obtained by dihedral principal component analysis.22−26 Gauge invariant atomic orbital27 calculations performed on the most probable bound alkane
conformation from simulation yielded semiquantitively accurate predictions
of the experimental shifts in the one-dimensional NMR chemical shifts
of the bound guest protons providing confidence in the accuracy of
our simulation results. Free energies associated with growing the
alkanes within OA dimers by addition of −CH2–
groups found breaks in the incremental free energy change at the chain
lengths at which the dominant motif changed. While this suggested
that the conformational motifs observed were associated with transitions
between distinct basins in the underlying free energy landscape, we
could not directly address the thermodynamic differences between conformers.
Characterization of the free energy landscapes of free and bound guests
within supramolecular complexes would provide valuable insight into
the impact of confinement on orchestration of the guest conformation.</p><p>Here we revisit our earlier simulation study of n-alkane conformations within cavitand host dimers to examine the
evolution of guest free energy landscapes under confinement. Molecular
dynamics (MD) simulations are performed to analyze the potentials-of-mean
force (PMFs) between the terminal methyl units of the alkanes n-undecane to n-docosane in vacuum, in
water, and confined within harmonically sealed cavitand complexes.
The changes in the PMFs with increasing guest length subsequently
enable characterization of the impact of confinement on the conformational
landscape of the chains, determination of the equilibrium conformational
motifs exhibited by the confined chains, and examination of the changes
in the PMFs with increasing guest length to determine the thermodynamic
driving forces determining the dominant conformational motif. Finally,
we release the harmonic restraint between the cavitands to unseal
the dimer complexes in aqueous solution. This allows the dimer capsule
some latitude to "breathe" and hence reveal the equilibrium
guest conformation that would be experimentally observable.</p><!><p>MD simulations of n-alkanes and
OA dimeric capsules
were performed using the GROMACS 2016.328 simulation package. Four sets of simulations were carried out: n-alkanes in a vacuum; alkanes in bulk water; alkanes confined
within a harmonically sealed OA dimer in vacuum; and alkanes within
an unsealed OA dimer in water. The alkanes (denoted Cn, were n is the number of carbons
in the alkane chain) considered ranged in length from undecane (C11) to docosane (C22). The alkanes were modeled
using the L-OPLS all-atom force field.29 The OA cavitands were modeled using the generalized Amber force
field (GAFF)30 with partial charges obtained
from AM1-BCC calculations.31 The net charge
of each cavitand was set to −6e to match the expected protonation
state at pH 7.32 To achieve this charge
state, the four acidic coating groups ringing the hydrophobic pocket
at the top of OA and two acids diagonal to one another at the foot
of OA were deprotonated (Figure 1a). Anionic charges on OA were neutralized by 12 sodium
cations (6 per OA) modeled using GAFF. Water was modeled using the
TIP4P-Ew potential.33 Bonds involving hydrogens
for the hosts and guests were constrained using the LINCS algorithm,34 while water was held rigid using SETTLE.35 The equations of motion were integrated using
a leapfrog algorithm with a time step of 2 fs.</p><p>In the first
set of simulations, individual alkanes were placed
in vacuum to determine the potential of mean force (PMF) as a function
of the end-to-end distance (ree) between
the terminal methyl end groups of the chain. The alkanes C12, C13, C15, C17, C18,
C19, C21, and C22 were considered
here, while determination of the PMFs of the remaining chains in vacuum
is described immediately below. Simulations were conducted at 25 °C
in the canonical ensemble. The temperature was controlled using the
Nosé–Hoover thermostat.36,37 No potential
truncation or periodic boundary conditions were applied in the vacuum.
The end-to-end PMFs were determined using umbrella sampling38 over a series of overlapping windows from the
terminal methyl units in contact to the fully elongated chain. Overlapping
windows (30–55) were used depending on the alkane length. The
harmonic umbrella potential utilized a force constant of 50 kJ/(mol
Å2) with successive minima evenly spaced in 0.5 Å
increments. Each window was simulated for 10 ns for sampling following
2 ns of equilibration. The PMFs were reconstructed using the weighted
histogram analysis method (WHAM).39</p><p>In the case of C11, C14, C16,
and C20, we performed replica exchange molecular dynamics
(REMD) simulations to determine their vacuum PMFs as a function of
temperature so that the enthalpic and entropic contributions to the
free energy could be determined. Simulations were conducted for a
single chain in vacuum in the canonical ensemble with the temperature
maintained using the Parrinello velocity rescaling algorithm.40 Ten replicas were considered at temperatures
ranging from 25 to 226.85 °C (298.15, 317.79, 338.46, 360.20,
383.13, 407.22, 432.55, 459.21, 487.29, and 500 K). These temperatures
were assigned using the Patriksson and van der Spoel41 algorithm to ensure maximal exchanges between neighboring
temperatures. An exchange rate of 55% was obtained. Following equilibration,
the end-to-end PMF between the alkane methyl groups was determined
using umbrella sampling in combination with REMD. Here, 50–80
overlapping windows determined by the alkane length were simulated
with a harmonic umbrella potential utilizing a force constant of 50
kJ/(mol Å2) with successive minima evenly spaced in
0.32 Å increments. Equilibration runs were conducted for 4 ns,
after which each window was simulated for 20 ns to gather statistics.</p><p>In the second set of simulations, an individual alkane (C11–C22) was placed in a bath of 5000 water molecules
to determine the end-to-end PMF of the alkane. Simulations were conducted
in the isothermal–isobaric ensemble at 25 °C and 1 bar.
In addition to using the Nosé–Hoover thermostat, the
pressure was maintained using the Parrinello–Rahman barostat.42 Nonbonded Lennard-Jones interactions were truncated
beyond a separation of 9 Å with a mean-field dispersion correction
for longer-range contributions to the energy and pressure. Electrostatic
interactions were evaluated using particle mesh Ewald summation with
a real space cutoff of 9 Å.43 The
PMF was evaluated using the same protocol and number of windows as
our first set of simulations of alkanes in vacuum.</p><p>In the third
set of simulations, we examined the end-to-end PMFs
of alkanes confined within a cavitand dimer in vacuum. The presumption
is that the chain conformation under confinement in vacuum is comparable
to that in solution, that is the solvent plays only a secondary role
in the chain conformation. To ensure the 2:1 supramolecular complexes
did not disassemble in vacuum, harmonic restraints were applied between
eight pairs of carbons ringing the mouths of the cavitands to seal
the dimer.21 The harmonic bond length and
spring constant used to seal the dimer were 4 Å and 25 kJ/(mol
Å2), respectively. These simulations were conducted
in the canonical ensemble with the temperature maintained using the
Parrinello velocity rescaling algorithm.40 Given that the chains are conformationally constrained within the
dimer, initial equilibration of the alkanes to relax the chain conformation
was conducted using REMD.44 Pre-equilibration
was conducted for 100 ns over 10 temperatures from 25 to 226.85 °C
(298.15, 317.79, 338.46, 360.20, 383.13, 407.22, 432.55, 459.21, 487.29,
and 500 K). These temperatures correspond to the same temperatures
used in the vacuum REMD simulations above. An exchange rate of 40%
was obtained. Following equilibration, the end-to-end PMF between
alkane methyl groups was determined using umbrella sampling in combination
with REMD. Fifty overlapping windows were simulated with a harmonic
umbrella potential utilizing a force constant of 50 kJ/(mol Å2) with successive minima evenly spaced in 0.32 Å increments.
The final configurations of the pre-equilibration runs were used as
the initial configuration of the secondary equilibration run with
the umbrella potential applied. The secondary equilibration runs were
conducted for 4 ns. Following secondary equilibration, each window
was simulated for 20 ns to gather statistics for the PMF. The PMFs
were reconstructed using WHAM. Here 10 000 configurations of
each window were saved for dihedral principal component analysis (vide infra) to determine the dominant chain conformation
within the complex for specified windows.</p><p>In the final set of
simulations, we considered the unsealed host–guest
complex in aqueous solution. The initial guest configuration for these
simulations were taken from the final configuration of the dimers
in vacuum above whose umbrella potential minimum was closest to the
minimum in the end-to-end PMF of the alkane. The harmonic restraints
sealing the dimer were released and the complex was placed in a bath
of 2500 waters. These simulations were conducted at 25 °C and
1 bar pressure using the Nosé–Hoover thermostat and
Parrinello–Rahman barostat, respectively. Following equilibration
of 4 ns, these simulations were conducted for 100 ns. Also, 50 000
configurations were saved for dihedral principal component analysis
(vide infra) to determine the dominant chain conformation
within the complex.</p><p>The REMD simulations described above provided
PMFs over a wide
temperature range from which the enthalpies and entropies of interaction
can be evaluated. Specifically, we fit the simulation PMFs to the
function1where G(T, ree) is the temperature dependent PMF
as a function of ree, A(ree), B(ree), and C(ree) are temperature independent functions of ree, and T0 is a reference temperature
assumed here to be 298.15 K. This function effectively assumes the
interaction heat capacity is independent of temperature. While higher
order terms could be included in this expression to incorporate a
temperature dependence for the heat capacity, the fits obtained were
not statistically improved. The temperature independent coefficients
were obtained by least-squares fitting of eq 1 to the simulation PMFs. The enthalpy and
entropy are obtained from the temperature derivatives of eq 12aand2bAt T0 the free
energy, enthalpy, and entropy are G(T0,ree) = A(ree), H(T0, ree) = A(ree) – [B(ree) + C(ree)]T0, and −T0S(T0, ree) = [B(ree) + C(ree)] T0, respectively.</p><p>The dominant
conformation of the encapsulated n-alkanes was obtained
from dihedral principal component analysis
(DPCA)22,23,25,26 which has previously been applied to examine the
structures of proteins in bulk solution and n-alkanes
in water45 and confined within cavitand
dimers.21 To determine the dominant alkane
conformation from a given set of simulation configurations, DPCA was
used to establish the two dominant eigenvectors of the dihedral covariance
matrix exhibiting the greatest variance during the simulations. A
probability distribution of chain configurations mapped to the eigenvector
coordinate system was subsequently obtained. From this probability
distribution, the simulation alkane configuration that fell closest
to the maximum in probability was subsequently judged to be the dominant
chain configuration. For the simulations of the dimers in vacuum this
analysis was conducted over the specified window containing the local
free energy minima in the end-to-end PMFs of the alkanes, while for
the simulations of the dimers in water the analysis was conducted
over the entire simulation trajectory.</p><!><p>The end-to-end PMFs between the terminal methyl units of the alkanes
C11–C22 in vacuum and water at 25 °C
are reported in Figure 2. In vacuum (Figure 2a), the PMF for C11 exhibits a dominant free energy minimum
at an end-to-end separation (ree) of 11.3
Å. This separation is slightly less than expected if the chain
was in an all-trans conformation. We observe a secondary
PMF minimum at a higher free energy at ree = 12.6 Å corresponding to the all-trans conformation.
As discussed below, the primary minimum occurring at 11.3 Å is
a result of the entropic need of the alkane to incorporate a minority
population of gauche dihedrals down the chain's
backbone despite the fact the all-trans conformation
has the lowest intramolecular energy. Unsurprisingly, increasing the
alkane length shifts the primary minimum in the end-to-end PMF in
vacuum to more distant separations. Concurrently, the secondary, all-trans minimum becomes even less favorable (i.e., higher
free energy) and less prominent. In addition, the free energy at a
ring closing end-to-end separation of ∼5 Å becomes lower
and more favorable with increasing size, such that for C22 a shallow secondary minimum is even observed. The free energy of
the extended chain is still lower than that of the ring closing state,
so that C22 is expected to be elongated in vacuum.</p><!><p>Potentials-of-mean
force between the terminal methyl units of the
alkanes (C11–C22) as a function of their
end-to-end separation (ree) at 25 °C
in vacuum (a) and water (b). The figure
lines for each alkane are defined in the legend of a.
Error bars are neglected for clarity.</p><!><p>The vacuum PMFs of C11, C14, C16, and C20 are broken down into their enthalpic and entropic
contributions at 25 °C in Figure 3. In difference to the PMF, the minimum in the end-to-end
enthalpy for C11 occurs for the all-trans chain (Figure 3a).
This energetic minimum of −10 kJ/mol is significantly deeper
that the free energy minimum (normalized to 0 kJ/mol), amounting to
∼4RT. This enthalpy minimum is opposed, however,
by an even more unfavorable entropy for the chain to adopt the all-trans conformation. This repulsive entropic contribution
almost certainly arises from the fact that there is only one way for
the chain to adopt an all-trans conformation while
there are considerably more ways to adopt less extended conformations.
Indeed, the minimum in the end-to-end entropy of C11 occurs
at ree = 10.5 Å, shorter than the
free energy minimum at 11.3 Å. Below the end-to-end separation
of 10.5 Å, both the enthalpy and the entropy of C11 become increasingly unfavorable. The enthalpies and entropies of
C14, C16, and C20 are qualitatively
similar to that of C11 (Figures 3b–d). The all-trans enthalpy minimum of the alkanes gets deeper with increasing length,
however, the increasing number of configurations available for intermediate
separations gives rise to a concomitant rise in the entropic penalty
for adopting the all-trans conformation. Interestingly,
a secondary minimum in the enthalpy at methyl unit contact arises
with increasing chain length. This secondary minimum is not as deep
as the all-trans primary enthalpy minimum. Like the
all-trans conformation, the secondary minimum is
entropically unfavorable. Given that when the terminal methyl units
are in contact the chain can adopt more conformations than those available
in the all-trans conformation, this "ring
closed" conformation is not as entropically unfavorable as
the all-trans state. We may expect that for alkanes
longer than C20 the chain may transition from an elongated
conformation to a ring closed conformation. That chain length, however,
is longer than the alkanes considered here. Nevertheless, the enthalpy
reduction for ring closure with increasing chain length arising from
attractive intramolecular van der Waals interactions underlies the
stabilization of this state observed in vacuum (Figure 2a).</p><!><p>Decomposition of the alkane terminal methyl
unit potentials of
mean force into their enthalpic and entropic components for alkanes
in a vacuum at 25 °C. Results as reported for (a) C11, (b) C14, (c) C16, and (d) C20. The lines
for the enthalpic (H) and entropic (−TS) contributions to the free energy/PMF (G) are defined in the legend of a.</p><!><p>The end-to-end PMFs for the alkanes in water are, to a first approximation,
nearly the same as those in vacuum (Figure 2b). That is water does not appear to impact
the overall conformation of the alkanes in difference to what would
be expected if they underwent a hydrophobic collapse. Our results
are in qualitative agreement with previous findings that alkanes on
the order of 20 carbons in length and shorter do not collapse,45 although longer chains adopt globular conformations.46 The most significant difference between the
alkane PMFs in water and vacuum is that the free energies of the ring
closing state in water are more unfavorable by ∼RT. So at least up to C22 water appears to disfavor the
ring closed state. Resultantly, the dominant conformations for the
alkanes in vacuum and water are nearly indistinguishable. This can
be more easily observed by considering the root-mean-square end-to-end
separation, ⟨ree2⟩1/2, of the alkanes
(Figure 4). Notably,
⟨ree2⟩1/2 for the alkanes in vacuum
and water are effectively the same, so that we may conclude that the
alkane conformations in water are largely dictated by intramolecular
rather than solvent-mediated forces. Comparing ⟨ree2⟩1/2 for the alkanes evaluated from simulation against that
expected if the chains adopted an all-trans conformation,
we find that the chains in vacuum and water are shorter than the all-trans chains. Moreover, the difference between the end-to-end
length of the all-trans alkanes and ⟨ree2⟩1/2 increases with increasing chain length. This
is consistent with an increasing minority population of gauche dihedrals with increasing alkane length.</p><!><p>Root mean square end-to-end
separations (⟨ree2⟩1/2) of the alkanes
(C11–C22)
as a function of the number of carbons in the chain (n) at 25 °C. Results are reported for alkanes in vacuum, water,
confined within the sealed cavitand dimer in vacuum, and within the
unsealed dimer in water. In addition, we also include the end-to-end
length of the alkanes if they adopted an all-trans conformation. The figure symbols are defined in the legend. Errors
are comparable in size to the figure symbols.</p><!><p>The conformational free energy landscapes of the alkanes confined
within harmonically sealed OA dimer complexes in vacuum are distinct
from those of the free chains in vacuum and water (Figure 5). Beginning with C11 (Figure 5a), the
end-to-end PMF between its methyl units is qualitatively similar to
that of the free chain in that it has a dominant primary minimum at
∼11.9 Å slightly shorter than an all-trans chain. The secondary minimum at 12.6 Å, however, has a free
energy ∼RT lower than that of the free chain
in vacuum. In addition, the shorter end-to-end separations to the
left of the primary minimum are significantly more unfavorable under
confinement than for the free chain. These observations indicate that
under confinement, C11 favors extended conformations even
more than the free chain does. Visual inspection of the dominant conformation
of the chain at the primary minimum is consistent with an extended motif observed experimentally, although a slight
bend near the middle of the chain is observed suggesting this conformation
may not be readily distinguished from the helical motif. The interior of the dimer is akin to a prolate ellipsoid,
wider in the middle and narrower at either end in the opposing cavitands.
The bend likely arises from the chain attempting to maximize favorable
van der Waals interactions by associating with the inner wall of the
complex. Considering ⟨ree2⟩1/2 for C11 under confinement (Figure 4), the alkane adopts a conformation longer than those
of the free chains just shy of the all-trans chain.</p><!><p>Potentials-of-mean
force between the terminal methyl unit of the
alkanes contained within sealed cavitand dimers in vacuum as a function
of the end-to-end separation (ree) between
the terminal methyl units at 25 °C. The figures correspond to
results for (a) C11, (b) C12, (c) C13, (d) C14, (e) C15, (f) C16, (g) C17, (h) C18, (i) C19, (j) C20, (k) C21, and (l) C22. The dominant chain conformations at the primary minimum
(identified by the blue arrow) as determined following DPCA analysis
are illustrated as the teal and white licorice structures for all
alkanes. The dominant chain conformations at other prominent minima
(identified by the red arrow) are also shown for alkanes C13–C22.</p><!><p>The PMF profiles of C12 (Figure 5b) and C13 (Figure 5c) are similar to that of C11,
exhibiting a primary minimum for an extended chain in the neighborhood
of ∼12.7 Å. In difference to C11, however,
we do not observe secondary minima for C12 or C13 corresponding to an all-trans chain. Given that
the depth of an individual OA pocket is comparable to the length of
a C6 chain, it stands to reason that the linear extent
of the OA dimer's interior volume is comparable to C12. As a result, it may not be possible for C12, let alone
C13, to even adopt an all-trans conformation
within the dimer. Visualizing the dominant conformational motifs of
these chains at their primary minima, we categorize them as extended, although like C11, gauche dihedrals are observed. The root-mean-square end-to-end lengths
of C12 and C13 (Figure 4) in this case are less than that of an all-trans chain and are comparable to that of C11 suggesting that the internal volume of the dimer is at most 13 Å
in length. While the length of confined C12 is slightly
greater than the free chain, the length of confined C13 is approximately equal to that of the free chain. Examining the
PMF of C13 (Figure 5c) we also observe the potential initiation of a secondary
minimum at end-to-end separations just to the left of the primary
minimum, suggesting the potential for a second stable chain conformation
for the confined guest with increasing length. Indeed, at the emerging
secondary minimum of C13 the chain appears to adopt a hairpin motif.</p><p>For C14 (Figure 5d) the secondary minimum inferred
from the PMFs of the shorter
chains is more apparent. Specifically, we observe a dominant primary
minimum at an end-to-end methyl separation of 12.6 Å and a secondary
minimum near ∼7 Å. Visual inspection of the dominant conformation
at the primary free energy minimum finds that instead of being extended the chain incorporates a conformational twist consistent
with assignment of a helical motif. This free energy
minimum, however, does not appear to be distinct from that associated
with the extended motif assigned for the shorter
alkanes. Rather, the chain adopts a helical twist so that the alkane
fits within the dimeric complex. At the secondary minimum, the dominant
conformation corresponds to a hairpin motif with
the two methyl units directed toward one cavitand in the dimer and
a turn directed toward the opposing cavitand. The free energy of the
secondary minimum, however, is ∼10 kJ/mol greater than the
primary minimum. As a result, the chain is expected to adopt a predominantly helical motif with brief excursions to a hairpin motif. Based on ⟨ree2⟩1/2 the length of
C14 is practically the same as that for C11–C13 and shorter than that of the free chain (Figure 4), reflecting an increase in
the population of gauche dihedrals to form the helical
twist to fit the guest within the capsule.</p><p>The end-to-end PMF
profile of C15 (Figure 5e) is qualitatively similar
to that of C14, although the free energy difference between
the helical motif at 12.6 Å and hairpin motif at 6.6 Å drops from 10 to 5 kJ/mol. The hairpin becomes the dominant motif for C16 when we observe that
the primary minimum in the PMF switches from 12.6 to 6.3 Å (Figure 5f). The helical conformer in this case is 10 kJ/mol less stable than the hairpin. The transition between the helical and hairpin free energy basins is accompanied by
a drop in ⟨ree2⟩1/2 from 12.2 to 6.7 Å
(Figure 4). This transition
from one conformational motif to another under confinement is reminiscent
of a two-state transition in protein folding. Different from protein
folding, however, the relative change in stability of the two free
energy minima is not driven by changes in temperature but through
changes in the length of the confined guest. Depending on the relative
contributions of the enthalpy and entropy (vide infra), the observed motifs may be temperature dependent or independent.</p><p>The hairpin motif continues to dominate the free
energy landscapes of C17 and C18 (Figures 5g and h). At the same time,
the extended chain basin becomes increasingly unstable rising from
10 kJ/mol higher in free energy than the hairpin for C16 (Figure 5f) to 130
kJ/mol for C18 (Figure 5h). Interestingly, visualizing C17 and C18 at their secondary minimum near 12.6 Å, we find the gauche dihedrals of these alkane appear to migrate to the
middle of the chains. This suggests the helical motif
may be evolving into a fourth motif. Given that only ree is being used to quantify the conformational transition,
however, we cannot assess whether this emerging motif is thermodynamically
distinct from the helical motif. Increasing the alkane
length to C19, we find that the free energy basin at 12.6
Å increases in stability with its free energy dropping from 130
kJ/mol for C18 to 85 kJ/mol for C19 relative
to the hairpin (Figure 5i), suggesting the potential for a fourth
conformational motif to emerge. The end-to-end lengths of C16–C19 are comparable to one another consistent with
all of these chains adopting a hairpin motif (Figure 4).</p><p>Increasing
the alkane length to C20, the dominant motif
transitions once again (Figure 5j), with the free energy minimum shifting back to 12.5 Å.
This conformational motif, however, appears to be distinct from the helical motif the alkanes previously adopted at this end-to-end
length. Specifically, the twist for C20 now appears wider
than the helices observed for C14 and C15. In
difference to the spinning top motif (Figure 1b), however, the harmonically
sealed dimer does not permit the turns to herniate between the two
cavitands and partially expose the chain to the bulk. C20 subsequently appears to adopt two hairpin turns along its backbone
that we refer to as the chicane motif. Below we examine
the impact of releasing the dimer restraints for the complex in aqueous
solution. In difference to the PMF landscapes of the shorter chains,
we observe three additional progressively less stable minima in the
end-to-end PMF of C20 at 10.5, 8.6, and 4.2 Å (Figure 5j). While the chain
in the 4 Å basin appears to adopt a hairpin-like
conformation, the alkane actually includes two turns such than the
chain looks more like the letter "C". These observations
suggest the transition between the hairpin and chicane motifs is distinct and more complex than that between
the helical and hairpin motifs discussed
above. The transition between C19 and C20 is
accompanied by a discrete change in ⟨ree2⟩1/2, jumping from 4.7 to12.6 Å (Figure 4). The interior volume of the harmonically
sealed cavitand dimer is 740 Å3 while the van der
Waals volume of C20 is 350 Å3. The resultant
packing fraction of the alkane within the dimer is 0.47, which is
comparable to the packing fraction at which hard spheres freeze (0.4947,48). We surmise then that C20 and longer chains will experience
significant packing frustration within the sealed dimers. The chicane motif remains the dominant conformation of C21 and C22 (Figure 5k and l). In qualitative agreement with C20, three progressively less stable minima in the end-to-end PMF are
observed at shorter end-to-end separations than the primary minimum
for C21 and C22. The value of ⟨ree2⟩1/2 for C21 and C22 is nearly
constant and ∼12.6 Å (Figure 4), equal to that of C20. This
points to the fact that sealing the complexes in vacuum limits the
ability of the alkanes to crack open the dimer to relieve packing
frustration despite the fact the guest packing fractions of C21 and C22 within the complexes are 0.49 and 0.51,
respectively.</p><p>Representative end-to-end PMFs of C11, C14, C16, and C20 within the sealed
dimers are
broken down into their enthalpic and entropic contributions at 25
°C in Figure 6. These alkanes were chosen to be representative of the extended, helical, hairpin, or chicane conformational motifs. In the case of C11 (Figure 6a), the
PMF is dictated almost entirely by the end-to-end enthalpy. While
the enthalpy favors the all-trans conformation, this
conformation is disfavored by the entropy similar to that observed
for the free chains in vacuum (Figure 3a). At shorter end-to-end separations, however, the
entropy of the confined chain is nearly zero so that the enthalpy
dominates. The enthalpy of the confined chain rises faster than that
of the free chain with decreasing end-to-end length. We conclude then
that confinement significantly limits the conformations available
to C11 compared to the free chains. Moreover, the dimer
presents an energetic confinement as suggested by the dominance of
the enthalpy. Given the negligible role of the entropy in C11's PMF, we expect over a reasonable temperature range (i.e.,
temperatures less than 500 K) that the extended motif
is stable, consistent with the results of our REMD simulations at
elevated temperatures.</p><!><p>Decomposition of the alkane terminal methyl unit potentials
of
mean force into their enthalpic and entropic components for alkanes
sealed in a dimer in vacuum at 25 °C. Results as reported for
(a) C11, (b) C14,
(c) C16, and (d) C20. These four chains are representative of the succession of conformational
motifs from extended, to helical, to hairpin, and to chicane. The
lines for the enthalpic (H) and entropic (−TS) contributions to the free energy/PMF (G) are defined in the legend of a.</p><!><p>The end-to-end enthalpy of C14 within the sealed dimer
exhibits two minima associated with the helical (ree = 12.6 Å) and hairpin (ree = 5.3 Å) motifs (Figure 6b). The helical basin is largely controlled by the enthalpy while the end-to-end
entropy near this free energy minimum is negligible and close to zero.
The entropy of the hairpin basin is comparatively
unfavorable and plays a minor role in shifting the free energy minimum
from the local minimum in the enthalpy to slightly longer chain lengths.
Nevertheless, extrapolating the temperature to absolute zero, while
the hairpin motif will become more stable it will
not overcome the stability of the helical motif.
As with C11 then, we do not anticipate significant changes
in the dominant conformation of C14 over a reasonable temperature
range. The observation of a secondary minimum in the enthalpy at shorter
end-to-end separations is qualitatively similar to what was observed
for the free chain in vacuum (Figure 3b). The secondary enthalpy minimum under confinement,
however, is enthalpically more favorable compared to the primary minimum
than what was observed in vacuum, again pointing to a significant
role for confinement to alter the enthalpic landscape of the confined
chain.</p><p>The end-to-end enthalpy of C16 dominates its
free energy
landscape, controlling the shape and relative stability of the hairpin and helical basins at 6.2 and 12.6
Å (Figure 6c),
respectively. The entropy plays a negligible role in determining the
dominant chain conformation. As such, while the hairpin motif is expected to be dominant, the helical basin
will be visited more frequently with increasing temperature. While
the end-to-end enthalpy of the free chain does exhibit a minimum at
shorter terminal methyl separations (Figure 3c), it is not the primary minimum in the
enthalpy. Moreover, the entropy plays a more significant role in determining
the free energy profile of the free chain than for the confined chain,
for which the entropy plays almost no role.</p><p>In difference to
the other guests considered, the end-to-end enthalpy
and entropy of C20 confined within the sealed dimer plays
a significant role in determining the observed guest conformation
(Figure 6d). Notably,
while the free energy favors the chicane motif (ree = 12.5 Å) the enthalpy exhibits a broad
minimum from 3 to 7 Å associated with collapsed, ring closing
conformations. These conformations, however, are more strongly opposed
by an unfavorable entropy. As such, it might be expected that with
diminishing temperature the collapsed states could dominate the observed
conformer population. Given the magnitude of the product of the temperature
and entropy, however, the potential for stabilizing these conformers
lie below the freezing point of water and would subsequently not be
observed experimentally. The disparity between the enthalpy and entropy
of C20 and C11, C14, and C16 (Figures 6a–c)
strongly suggests the transition to the chicane motif
is distinct from that between the other motifs. Notably the large
entropic penalty for C20 to adopt shorter conformations
(ree < 7 Å) suggests that the
frustration associated with trying to insert the two methyl units
into a single cavitand results from the chain being only able to adopt
limited conformations. This packing frustration is alleviated by redirecting
the two terminal methyl units toward the two opposing host pockets
so that the two turns settle near the wider middle of the dimer's
internal volume (Figure 5j) to enjoy more configurational freedom.</p><p>The chicane motif of C20–C22 found within the sealed
dimers is not observed experimentally.
While we believe the free energy landscapes of the guests within the
sealed dimers in vacuum mirrors what would be observed for guests
in an unsealed dimer in water, the imposed restraints limit breathing
of the dimer complex that allow for separation of the two hosts and
partial herniation of the guests. To assess the impact of unsealing
the dimer on the bound alkane conformation, we harvested simulation
configurations from the primary minima of each of the guests, placed
the complex in aqueous solution, released the harmonic restraints,
and simulated the guests within the unsealed complex for sufficient
time to allow the chains to relax to their final equilibrium conformation.
Snapshots of the dominant alkane guest conformations as determined
from DPCA over these simulations are plotted in Figure 7. The equilibrated conformations of C11–C19 that adopt either extended/helical or hairpin motifs in the
sealed dimers (Figures 5a–i) remain in that motif in the unsealed dimers. The chains
exhibiting the chicane motif within the sealed dimers
(C20–C22), however, relax to a spinning top motif within the unsealed dimers (Figure 7). In this case the
turns migrate to the middle of the complex, merge, and are extruded
between the partially separated hosts (Figure 1b). The greatest difference between ⟨ree2⟩1/2 in the sealed and unsealed dimers occurs for
C20–C22, for which the end-to-end separation
grows by ∼1 Å as a result of the extrusion of alkane turn
between the two cavitands in the unsealed dimer (Figure 4). The coincidence between
the spinning top and chicane suggests
that their stability has a common thermodynamic origin. The ability
for the chains to relax from the chicane to spinning top morphology, however, indicates the free energy
of the spinning top may be even lower than inferred
from the sealed dimer calculations as a result of the chain lowering
its packing fraction by partially opening the capsule.</p><!><p>Relaxed alkane (C11–C22) guest conformations
in unsealed cavitand dimers in water. The dominant alkane conformation
was determined following DPCA analysis. The alkanes are illustrated
in teal (carbon) and white (hydrogen) licorice format. In these snapshots
the distinction between the extended and helical motifs (C11–C15) is
subjective, although the degree of helicity appears to increase with
the chain length. This highlights the conclusion that the extended and helical motifs do not appear
to be thermodynamically distinct (Figure 5) but are members of a larger "linear" motif family. The alkanes are subsequently
identified as belonging to either a linear, hairpin, or spinning top motif below each
snapshot.</p><!><p>Finally, we suggested in our initial
simulation study of alkane
conformations in cavitand dimers that the evolution from the extended and helical motifs does not correspond
to a thermodynamic transition.21 The main
difference between the sequence from C11 to C15 in the unsealed dimers in solution appears to be only a question
of the extent of gauche dihedrals down the backbone
for the chain to adopt a helical twist (Figure 7). As found in Figure 5, it is impossible for the longer alkanes
to adopt an all-trans conformation and the population
of gauche dihedrals must increase with increasing
chain length so that the guest will even fit within the complex. The
assignment of a helical motif is subsequently only
a result of visual inspection of the dominant chain conformation.
Indeed, comparing the dominant conformations of C12 and
C13 in Figures 5 and 7 suggests an increase in the
helical nature of the chains upon releasing the harmonic constraints.
In addition, no clear differences in the primary minima of C11–C15 is found (Figure 5b–e). These observations provide deeper
support for our previous proposition that extended and helical motifs are members of a larger linear motif family. The equilibrium between distinct free
energy basins for the linear (extended/helical) to hairpin transition
and hairpin to chicane/spinning
top transition, however, indicates that these motifs are
thermodynamically distinct. We subsequently identify the dominant
guest conformers within the unsealed host dimers in water as belonging
to the linear, hairpin, and spinning top motifs in Figure 7.</p><!><p>While molecular simulation
of free alkanes up to C22 in vacuum and aqueous solution
do not exhibit conformational transitions,
when these guests are confined within dimeric supramolecular complexes
composed of cavitand hosts, a succession of conformational motifs
from extended, to helical, to hairpin, to chicane/spinning top are observed. The extended and helical motifs do not appear to be thermodynamically distinct but fall within
a larger linear motif family, visually distinguished
only by the fraction of gauche dihedrals down the
guest backbone forming a helical twist. The transition between the linear, hairpin, and chicane/spinning top motifs, however, appears to be thermodynamically
distinct, with the transitions between the linear and hairpin motifs and hairpin and chicane motifs associated with jumps between
free energy basins in the PMF profiles of the confined guests. Different
from typical thermodynamic transitions driven by temperature changes,
the conformational transitions under confinement are triggered by
changes in the guest length. Specifically, examination of the PMF
landscapes of the guests confined within harmonically sealed dimers
found that the relative depths of primary and secondary minima in
the free energy change systematically with increasing alkane length.
For example, beginning with C11, the hairpin's free energy basin, occurring at a shorter end-to-end distance
than the linear basin, became increasingly stable
with increasing chain length until it became the lowest free energy
state beginning with C16. Similarly, the chicane free energy basin did not become dominant until C20,
leading to a transition from the hairpin to chicane motifs. Examination of the thermodynamic components
of the free energy found that confinement induces significant changes
in the enthalpic and entropic landscape. Notably, the end-to-end entropy
of free alkanes in vacuum was found to favor alkanes adopting conformations
intermediate between ring closing and a fully extended chain. Confinement
was found to significantly limit the conformations available, on the
other hand, reducing the entropic penalties associated with adopting
either ring closed or extended conformations for alkanes adopting
the linear or hairpin motifs. As
such, the end-to-end enthalpy plays a dominant role in determining
the equilibrium conformation observed. In the case of the chicane motif, however, the entropy was found to play a
dominating role in pushing the chain out of the hairpin basin and into the chicane basin despite the fact
the enthalpy favors hairpin-like conformations. While
the chicane motif is stable for chains C20 and longer within a harmonically sealed cavitand dimer, it is not
observed experimentally. We attribute this to the fact that sealing
the dimers does not permit the guests to potentially crack the complex
open and partially expose themselves to the surrounding media. When
we removed the harmonic restraints and equilibrated the dimer complexes
in aqueous solution, we found the chains adopting the chicane relaxed to a spinning top motif with the alkanes
partially herniating between the two hosts. The guests shorter than
C20, however, retained the conformations observed in the
sealed dimers in vacuum in the unsealed dimers in solution. This is
attributable to the fact that alkanes C20 and longer exhibit
packing fractions comparable to that of hard spheres near their freezing
point, suggesting considerable packing frustration within the capsules
that can be alleviated by pushing the hosts apart. Our simulations
illustrate that confinement of alkanes significantly alters the free
energy landscapes of guests bound within supramolecular complexes
that can force the chains to adopt conformations that are not readily
observed for unconfined, free chains. This, in turn, is reminiscent
of the role of chaperones like the GroEL/GroES complex in refolding
misfolded proteins by encapsulation within their internal volumes.</p><!><p>The authors declare no
competing financial interest.</p>