Article pubs.acs.org/Macromolecules
Polymer Folding in Slitlike Nanoconfinement Mark P. Taylor* Department of Physics, Hiram College, Hiram, Ohio 44234, United States S Supporting Information *
ABSTRACT: A flexible homopolymer chain with sufficiently shortrange interactions undergoes a discontinuous transition from an expanded coil to a compact crystallite analogous to the all-or-none folding transition exhibited by fast-folding proteins. In this work we investigate the effects of both tethering and confinement on this type of folding transition. In particular, we study a flexible square-well Nmer chain (monomer diameter σ) located between two hard walls forming a slitlike pore (width W) with the chain end tethered to one wall. We carry out Monte Carlo simulations with Wang−Landau sampling to construct the single-chain density of states and use both microcanonical and canonical analyses to characterize phase transitions. We find that in a very wide pore (W > Nσ) the all-or-none folding transition is only slightly affected by the tethering. As the pore width is reduced to near the size of the folded polymer chain (W ≈ N1/3σ) there is a modest entropy reduction for the unfolded states leading to a stabilization of the folded state. Below a critical slit width the chain is unable to fold into the native ground state. However, discontinuous all-or-none folding still occurs to higher energy ground-state structures commensurate with the slit width. All-or-none folding persists even to the limit of a very narrow pore (W ≈ σ) where the groundstate structure is a quasi-two-dimensional crystal. Both structural and thermodynamic effects of confinement are studied including quantitative analyses of entropy reduction and changes to the free energy barrier to folding. confinement12 on polymer and, in particular, protein folding. If one assumes that the crowding or confinement greatly reduces the number of accessible conformations of the unfolded chain while having little effect on the folded chain, then the transition should shift in favor of the folded state. Thus, the entropy reduction in the ensemble of unfolded states results in an entropic stabilization of the folded state. The experimental observation of confinement induced stabilization in a number of proteins13−18 appears to validate the entropic stabilization idea. However, in these experimental systems there are also energetic interactions with the confining surfaces, and thus, both entropic and enthalpic contributions need to be considered. In the case of macromolecular crowding in protein systems, a stabilization effect is often found, but cases of destabilization are also reported.18−22 Computer simulation studies allow for direct investigation of the entropic aspects of confinement by using inert hard confining surfaces. A very simple test of the anticipated entropic stabilization effect is to study chains tethered to a hard flat wall. The hard wall should prohibit a large fraction of unfolded chain conformations, thereby stabilizing the compact state. However, simulation studies do not universally find this result. In fact, for a very simple homopolymer model the folded state is slightly destabilized23 while studies of Go̅-type protein models find cases of both stabilization and destabilization, depending on the
1. INTRODUCTION Polymer molecules can undergo conformational transitions that are analogous to the phase transitions exhibited by bulk materials. Thus, the polymer collapse or coil−globule transition is analogous to gas−liquid condensation while the crystallization of a single long polymer chain into a single crystal is a small system analogue to the freezing of a bulk liquid. Protein folding shares some similarities with a freezing transition as proteins typically fold via a first-order-like process with a free energy barrier separating the unfolded and folded states.1 In contrast, the coil−globule transition for a flexible polymer is a continuous or second-order-like process.2 One might anticipate that confinement and other geometric constraints, with length scales on the order of the polymer size, may have large effects on these transitions. In particular, for a transition between a compact and expanded state it is the expanded state that is expected to be most affected by confinement. For an expanded polymer chain confined between two parallel hard walls, the resulting perturbation to conformational and thermodynamic properties of both flexible and semiflexible polymers is well studied.3−10 Scaling laws for polymer size, anisotropy, and free energy as a function of confinement length have all been obtained for good solvent conditions. With worsening solvent conditions the confined polymer will collapse or fold from this ensemble of perturbed states, and thus details of the transition should differ from the free chain case, being dependent on the type and degree of confinement. A simple entropic argument is commonly used to understand the effects of both macromolecular crowding11 and geometric © XXXX American Chemical Society
Received: April 4, 2017 Revised: June 27, 2017
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DOI: 10.1021/acs.macromol.7b00709 Macromolecules XXXX, XXX, XXX−XXX
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repeat units, the monomer size σ is on the order of nanometers, and thus, W ≈ (a few σ) corresponds to nanoscale confinement. We use the Wang−Landau (WL) simulation algorithm46,47 to construct the density of states g(E) for this model. We have previously described in some detail our implementation of this algorithm for a free chain.49 For the end-tethered chain a Monte Carlo cycle consists of on average N single bead axial rotation moves, N end-bridging moves, 1 regular pivot, 1 pivot about a local z-axis, and 1 pivot about the tethering site. Any move attempt leading to a sphere−sphere or sphere−wall hardcore overlap is immediately rejected. For most of our results we use a flatness criterion of 20% for both static and dynamic histogram flatness checks, and in all cases we carry out 30 levels of WL iteration. As described previously, before carrying out a full WL simulation, we do a ground state search to estimate Egs of the system. On the basis of the visitation histogram accumulated in the ground state search, we choose a minimum energy, which is typically a few percent above Egs, that will allow the simulation to complete in a workable amount of time. In all cases, we run between 3 and 6 independent simulations in order to estimate statistical error. The ultimate precision of the WL algorithm is limited by a number of implementation specific factors.50 In the Supporting Information we provide a detailed analysis of the convergence and sampling properties of our implementation of the algorithm and provide exact results51 for the N = 3 tethered chain which we have used in our code validation. The thermodynamic behavior of the polymer chain is encoded in the density of states function. In a microcanonical analysis one defines the entropy function
specific protein and on the location of the tethering site.24−27 Similarly, a variety of behaviors are found for the collapse and folding of both simple homopolymers and protein models in more confining geometries including slits, pores, and spheres.28−39 Results from these studies include the observation of a stabilization effect in a wide slit crossing over to destabilization in a narrow slit,39 destabilization vs stabilization for flexible vs semiflexible chains in spherical confinement,29,31 the stabilization of a non-native ground state in a narrow pore,38 and the suppression of folding in extreme confinement.35 One potential weakness of some of the confined protein simulation studies33,35−37 is the use of Go̅-type models which encode the native state of the free protein, and thus confinement-induced perturbations to the native state may not be properly explored.34 In this work we investigate the entropic effects of confinement on a polymer folding (i.e., direct coil → crystal) transition.40−43 This transition is analogous to the “all-or-none” or two-state folding transition displayed by many fast-folding proteins,1 and thus, we hope to elucidate some of the underlying physics of protein folding in confinement. Our results should also provide some insight into the effects of confinement on polymer crystallization.44,45 We consider both a polymer tethered to a hard surface and in slitlike confinement between parallel hard walls. We carry out computer simulations using the Wang−Landau algorithm46,47 which is a flat histogram technique48 that allows us to directly compute entropy (up to an additive constant). Thus, we are able to quantify the entropy changes associated with confinement.
2. MODEL AND METHODS In this work we study a flexible interaction-site polymer chain composed of N square-well (SW) spheres connected by universal joints of fixed bond length L. Monomers are numbered sequentially 1 through N, and the center of monomer i is located by the vector ri⃗ = {xi, yi, zi}. Nonbonded sites i and j (|i − j| > 1) interact via the potential ⎧∞ r < σ ⎪ u(r ) = ⎨−ϵ σ < r < λσ ⎪ ⎩ 0 r > λσ
S(E) = kB ln g (E)
whose curvature properties give information about phase transitions.40,41,52 In particular, a so-called convex intruder in this function is the signature of a first-order-like transition where one can define distinct coexisting states. This behavior results in a “loop” in the microcanonical temperature
T (E) = [dS(E)/dE]−1
(4)
where multiple energy states E correspond to a single T(E). Continuous or second-order-like transitions are signaled by an inflection point in T(E). In the WL simulations we actually construct g(E)/g(Eref), where Eref is a reference state, typically chosen as the energy state with the maximum g(E). Thus, for each W, S(E) is given only up to an undetermined additive constant a(W) = −kB ln[g(Eref)]. However, the curvature properties of S(E) are given absolutely. The more standard canonical thermodynamic analysis starts from the canonical partition function
(1)
where r = |ri⃗ − rj⃗ |, σ and λσ are the hard-sphere and square-well diameters, respectively, and ϵ is the square-well depth. We set L = σ (tangent hard spheres) and define a reduced temperature T* = kBT/ϵ. We study the free chain, the chain end tethered to a hard flat wall, and the chain end tethered to one of two parallel hard flat walls forming a slitlike pore. For the tethered chains, the tethering site defines the origin of the coordinate system (i.e., r1⃗ = 0), and the tethering plane is parallel to the xyplane located at z = −σ/2. For the slit case the upper wall is located at z = W − σ/2 forming a slit of width W. The chain− wall interaction is simply ⎧∞ z < 0 or z > W − σ w(z) = ⎨ ⎩ 0 otherwise
(3)
Z (T ) =
∑ g(E)e−E /k T B
(5)
E
and the associated probability function (2)
p(E , T ) = g (E )
This model has a discrete energy spectrum given by E = −nϵ, where n is the number of square-well overlaps in a chain conformation, with a ground state energy Egs = −nmaxϵ that depends on both N and λ and, for a sufficiently narrow slit, will also depend on W. Since the monomers in this course-grained polymer model are meant to represent one or more chemical
e−E / kBT Z (T )
(6)
In the canonical approach, phase transitions are typically associated with peaks in the single-chain specific heat ⎛ dE ̅ ⎞ 1 C W (T ) = ⎜ ⎟ = ( E2 − E ̅ 2 ) ⎝ dT ⎠W kBT 2 B
(7)
DOI: 10.1021/acs.macromol.7b00709 Macromolecules XXXX, XXX, XXX−XXX
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determine the entropic effects of geometric confinement on this folding transition. 3.1. Folding of a Free vs Tethered Chain. We begin by comparing the folding transition of a free chain and a chain end tethered to a hard flat wall. In Figure 1 we show the single chain
where the overbar notation indicates an average over all states: q(T) = ∑Eq(E)p(E,T). For a first-order-like transition p(E,T) ̅ will be bimodal in the neighborhood of the transition with equal areas in the two peaks at the transition. The canonical entropy, which we determine up to the additive constant a(W), is defined as S(T ) = E ̅ /T + kB ln[Z(T )]
(8)
One can also define a configurational free energy G(E , T ) = E − TS(E) = −kBT ln[p(E , T )Z(T )]
(9)
which we determine up to the additive function of temperature −Ta(W). G(E,T) is in fact a projection of a multidimensional (coarse-grained) free energy functional onto the single variable E, which we take as a reaction coordinate for the folding transition.43 In terms of mechanical properties, the average force exerted by the chain on the walls of the slit is given by a derivative of the Helmholtz free energy (E − TS(T)) with respect to W as follows: ⎛ dE ̅ ⎞ ⎛ dS ⎞ ⎟ ⎟ + T⎜ Q (W , T ) = − ⎜ ⎝ dW ⎠T ⎝ dW ⎠T
Figure 1. Logarithm of the density of states ln[g(E)] (i.e., microcanonical entropy S(E)/kB), relative to the value at E = Emin, vs energy E/ϵ for a SW-chain with N = 64 and λ = 1.04 that is free (dashed line) or end-tethered to a hard wall (solid line). Inset: inverse microcanonical temperature 1/T*(E), given by the derivative of the entropy function S(E). The horizontal dashed lines illustrate the equal area construction for the transition temperature.
(10)
Since the entropy contribution a(W) is undetermined, we are unable to compute the second term in this expression; however, in the following we will examine the first term, defining σ ⎛ dE ̅ ⎞ ⎟ fT (W ) = − ⎜ ϵ ⎝ dW ⎠T
(11)
microcanonical entropy S(E)/kB, relative to the value at E = Emin, for a free and end-tethered SW chain with N = 64, λ = 1.04, and Emin = −196ϵ (where, for both the free and tethered chain, Egs = −200ϵ). These two entropy curves are seen to be essentially identical. Since the tethering plane excludes many chain conformations, the absolute entropy of the tethered chain is in fact smaller than for the free chain; however, we are unable to determine this entropy reduction since our simulation approach only determines S(E) up to an arbitrary additive constant. In a microcanonical thermodynamic analysis, the phase behavior of the single chain is determined directly from the curvature properties of S(E). In the Figure 1 inset we show the derivative of S(E)/kB with respect to E/ϵ, which defines the inverse microcanonical temperature 1/T*(E). The van der Waals type loop in the dS/dE curve indicates the presence of a convex intruder in S(E) (which is visually apparent in Figure 1) and is the signature of a first-order-like transition in a finite size system. The transition temperature, obtained from an equal areas construction on the loop in the 1/T*(E) curve (which is equivalent to a double tangent construction on S(E)), is found to be T* = 0.384(1) for the both the free and tethered chain. The tie-line shown in the Figure 1 inset connects states E = −56ϵ (coil) and −180ϵ (folded). A more conventional canonical thermodynamic analysis of the free vs tethered chain folding transition, using the specific heat CW(T)/NkB, is presented in Figure 2. Both chains display a strong specific heat peak near T* = 0.384, in agreement with the microcanonical analysis, but the free chain peak is shifted slightly to the right and the transition temperatures are found to be T* = 0.3841(2) and 0.3836(2) for the free and tethered chains, respectively. Thus, contrary to a general expectation of confinement-induced stabilization of compact structures,12 here tethering slightly destabilizes the folded state. Regarding the
as a mechanical response function analogous to the thermal response given by CW(T). To obtain structural information, one can use the Wang− Landau derived g(E) to carry out subsequent multicanonical simulations which uniformly sample configuration space.53 For example, for each energy state one can determine the average site−site distances ⟨rij2⟩E which allows one to compute the mean-square radii of gyration ⟨Rg2⟩E = (1/N2)∑i − 50ϵ) are affected by the confinement with a maximum reduction of about 2.5kB for the E = 0 state. For the narrower slits (W = 6σ and 5σ) both the folded (low energy) and coil (high energy) states have reduced entropy with the effect being largest for the most expanded (i.e., highest energy) states. For example, in the W = 5σ slit there is an entropy decrease of about 7kB in the E = 0 state corresponding to over a 1000-fold reduction in the “number” of accessible chain conformations. The wiggles evident in Figure 4 ΔS curves are due to a slight shift in the location of the convex intruder in the S(E) functions with changing W. In the inset to Figure 4 we show the canonical single-chain entropy S(T), relative to the value at T = 0, for chains tethered in a slit and for the simple tethered chain (W →
( R g 2)1/2 ≈ 3σ with zm̅ ax ≈ 7σ. Thus, we might expect a slit to begin perturbing the folding transition for this chain when W ≈ zcoil m ̅ ax + σ ≈ 8σ. In the folded state the average chain size is ( R g 2)1/2 ≈ 1.7σ with zm̅ ax ≈ 4σ. Thus, for W > zfolded m ̅ ax + σ ≈ 5σ we expect little effect of the slit on the entropy of the folded ensemble, and this should allow for a direct determination of the absolute entropy reduction due to confinement (compared with the simple tethered chain). We present such entropy reduction results in Figure 4 where we show the difference in microcanonical entropy S(E) between the simple tethered chain (i.e., W = ∞) and the tethered chain in a slit of width W/σ = 8, 6, and 5. For W ≥ 5σ D
DOI: 10.1021/acs.macromol.7b00709 Macromolecules XXXX, XXX, XXX−XXX
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Macromolecules ∞). These curves are systematically shifted down with decreasing W, indicating the confinement-induced entropy reduction. The folding transition is signaled by a very steep drop in S(T) (by ∼300kB) near T* ≈ 0.385, and the entropy reduction is seen to be larger on the high temperature (unfolded) side of the transition, approaching the microcanonical ΔS(E = 0) value in the high temperature limit. This entropy reduction results in a shift of the folding transition to slightly higher temperature. The abrupt change in S(T) at the folding transition produces a large peak in the single chain specific heat (which can be written as CW = T(dS/dT)W). As seen in Figure 5, the peak in
3.55σ since for smaller slit widths the chain is unable to access the W → ∞ ground state ensemble. The effects of the slit confinement are apparent for 3.6 < W/σ < 8, producing an increase in transition temperature due to the entropic stabilization of the folded state (arising from the entropy reduction for the coil states). The maximum effect is near W = 5σ while for W > 8σ the confinement has little effect. For W < 5σ the slit perturbs the ensemble of folded states, and thus both the coil and folded states will experience an entropy reduction leading to a smaller overall confinement effect. More details on the thermodynamics of confinement are provided by the configurational free energy GW(E,T), which we show in Figure 7 for a series of slit widths. These free energy
Figure 5. Single chain specific heat CW(T)/NkB vs model temperature T* (upper scale) or physical temperature in °C (lower scale) for a SWchain with N = 64 and λ = 1.04 end-tethered in a slit of width W, as indicated.
Figure 7. Configurational free energy GW(E,T) (in units of kBT) vs energy E/ϵ at the W → ∞ folding temperature for a SW-chain with N = 64 and λ = 1.04 end-tethered in a slit of width W/σ = ∞, 8, 6, 5 (solid lines from bottom to top), 4 (long dash), and 3.6 (short dash). The lowest energy state is taken as the reference for GW(E,T), and the W/σ = 4 and 3.6 curves are shifted up by kBT and 2kBT, respectively, for clarity.
CW(T) systematically shifts to higher temperature with decreasing W. In terms of physical temperature, again defining 60 °C as the W → ∞ transition, there is a 3 °C increase in the folding temperature for W = 5σ. In Figure 6 we show a configurational phase diagram for the N = 64 and λ = 1.04 SWchain in a slit. This phase diagram only extends down to W =
functions are all computed at the folding temperature of the simple tethered chain (W → ∞) and are plotted in units of kBT. As reference, we use the E = Emin state which we assume is unperturbed by the slit for W ≥ 5σ (and thus we can make absolute comparisons between free energies for these W values). These free energy functions show the ensembles of coexisting coil and folded states at the transition (located in the neighborhood of E = −60ϵ and −180ϵ, respectively), and the barrier that must be crossed to convert between these states (barrier height ≈9kBT). The confinement slightly shifts the basins of coil and folded states to lower energies. Destabilization of the coil states with decreasing W is evident from the increase in free energy for these states. For the case of W = 8σ only coil states are affected by the confinement, but for W = 6σ the transition (top of the barrier) states are perturbed by the slit and for W = 5σ the folded states are also perturbed (although we assume the Emin state is not perturbed at W = 5σ). With decreasing W the barrier to folding decreases (for W ≥ 5σ), suggesting a confinement induced speed-up in the folding kinetics. In the case of a simple self-avoiding walk, Daoud and de Gennes3 predict a free energy change that should scale as (GW − G∞)/kBT ∼ (W/σ)−γ where γ = 5/3 for confinement in a slit of width W. Our GW results for the E = 0 state (i.e., the state closest to a simple self-avoiding walk) exhibit such power law scaling; however, we find using 5 ≤ W/σ ≤ 10 the value γ = 2.2(1).
Figure 6. Phase diagram for a SW-chain with N = 64 and λ = 1.04 endtethered in a slit of width W. Symbols locate the folding transition temperature while the solid line is a guide for the eye marking the phase boundary. The horizontal dashed line indicates the transition temperature in the W → ∞ limit, and the shift of the phase boundary above this line is the result of a confinement-induced entropic stabilization of the folded state. The snapshots show representative folded and unfolded conformations for W = 5σ and 8σ. For W < 3.55σ folding to the ground state ensemble is not possible. E
DOI: 10.1021/acs.macromol.7b00709 Macromolecules XXXX, XXX, XXX−XXX
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Macromolecules 3.3. Folding in a Narrow Slit. As slit width is decreased below a critical value, the chain is prevented from folding into its “native” ground state ensemble due to steric effects (i.e., none of the structures comprising this ensemble will fit within the slit). For the case of N = 64 and λ = 1.04 this critical slit width is W ≈ 3.55σ. Decreasing the slit width below this critical value will, depending on the temperature, either melt (unfold) the chain or cause a structural solid−solid type of transition to a new ground state structure commensurate with the narrower slit width. To further investigate such confinement-induced transitions, we have computed the chain ground state energy as a function of W under these nanoconfinement conditions (W < N1/3σ). These ground state energy results are shown in Figure 8a where we observe a steplike switching to higher energy
This should lead to a more dramatic entropic stabilization effect than observed for the wide slit case. These entropic stabilization effects lead to a rich phase diagram for the single-chain system under nanoconfinement as shown in Figure 9 for the N = 64 and λ = 1.04 SW-chain. The
Figure 9. Phase diagram for a SW-chain with N = 64 and λ = 1.04 endtethered in a slit of width W. Symbols locate the folding transition temperature while the solid line is a guide for the eye marking the phase boundary. The horizontal dashed line indicates the transition temperature in the W → ∞ limit, and the vertical dashed lines approximately locate solid−solid transitions between different groundstate structures and identify a crossover to pseudo-2D behavior for the coil phase.
Figure 9 phase diagram includes the wide-slit regime, detailed in Figure 6, and three additional narrow slit regimes each associated with a different ground-state structure. In each case the folding transition is located by a single sharp peak in the single-chain fixed-W specific heat CW(T), and in all cases these transitions are found to be first-order-like with an equal area bimodal probability distribution p(E,T) at the transition. The behavior of the phase boundary in the narrow slit regions mirrors the behavior seen for the wide slit. Thus, for each ground-state region the transition temperature increases with decreasing slit width due to an entropic reduction for the unfolded states and thus an entropic stabilization of the folded states. For the wide slit this increase in transition temperature is quite modest while for the narrow slits the stabilization effect is amplified and the temperature increase is significant. In terms of the physical temperature scale introduced for the tethered chain, we see temperature shifts in the transition of up to 4, 30, and 55 °C for the folded III, II, and I ground-state regimes, respectively. As the system crosses over from the IV → III, III → II, and II → I regions the transition temperature displays a sudden decrease of approximately the same magnitude as the increase realized across the W-range of the newly entered region. The rather interesting structure of the Figure 9 phase digram indicates that the confined chain system studied here can undergo a sequence of transitions upon isothermal compression of the slit. In Figure 10 we show both thermal (CW(T)) and mechanical (f T(W)) response functions for such isothermal compressions. In Figure 10a these functions are shown across the unfolded region of the phase diagram. With decreasing W the specific heat for these coil states decreases as the structural and energy fluctuations are suppressed by the geometric confinement. The mechanical response shows a peak as the system crosses from a strongly perturbed three-dimensional coil to a pseudo-two-dimensional coil regime (W < 1.87σ, where the chain is unable to cross over itself). The vertical line in the
Figure 8. (a) Ground state energy estimates (filled symbols, solid line) Egs vs slit width W for a SW-chain with N = 64 and λ = 1.04 endtethered in a slit. The open symbols (dashed line) show the minimum energy actually used in the Wang−Landau simulations. Distinct ground state morphologies are indicated by labels and vertical lines. (b) Microcanonical entropy difference between energy states E = 0 (coil) and Emin (folded) vs slit width W. Here the solid line is a guide for the eye, and the horizontal dashed line locates the W → ∞ result.
values associated with sequential reductions of the slit width by approximately one monomer diameter σ. Each distinct ground state energy Egs corresponds to an ensemble of crystallite structures with a number of crystal layers commensurate with the slit width W. Thus, the system switches from an ensemble of n → n − 1 layers for n = 4, 3, and 2 at the slit widths W/σ ≈ 3.5, 2.7, and 1.8, respectively. Illustrations of these four different grounds structures are included in Figure 11. In Figure 8b, we show how the entropy range (i.e., the entropy difference between the highest and lowest energy states) accessible to the chain varies as a function of slit width. With decreasing W this entropy range decreases (due to the decrease in volume accessible to the chain) with steplike behavior when the system switches between ground states. Within each ground-state region one might expect the decreasing slit width to primarily affect the coil states, and thus the smooth decrease in entropy range for each groundstate represents a confinement induced entropy reduction. For the wide slit (folded IV ground state) the entropy reduction is quite modest, however, as the system moves to more extreme confinement the entropy reduction becomes more pronounced. F
DOI: 10.1021/acs.macromol.7b00709 Macromolecules XXXX, XXX, XXX−XXX
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Figure 11. Configurational snapshots for a SW-chain with N = 64 and λ = 1.04 at T* = 0.360, end-tethered in a slit of width W/σ = 4.0, 3.0, 2.4, 2.0, 1.6, and 1.3. This figure illustrates the isothermal sequence of transitions driven by changing slit width W. The W values at each transition (located by the peaks in Figure 10b) are given in the figure. For the pseudo-2D cases (W < 1.8σ) we include a top view of the chain structure. The tethered and free end sites are colored yellow and dark blue, respectively.
leading to the observed weak destabilization effect. As shown in the Supporting Information, this is an exact result for the N = 3 SW-chain. In a hard-wall slit we do find entropic stabilization once the slit width is smaller than the average size of the unfolded chain, and we find a maximum in the stabilization when the slit width is approximately equal to the average size of the folded chain (W ≈ N1/3σ). Further reduction of the slit width lessens the stabilization due to the reduction in number of folded conformations that can fit into the slit. At a critical width the chain is sterically prohibited from folding into any conformation of the native ground state ensemble. For slit widths smaller than this critical value, the confined chain still undergoes a first-order-like folding transition but now into a different ensemble of low-energy structures. We observe a set of such ensembles, each composed of low-energy structures with a fixed number of layers, down to the extreme limit of a pseudo2D system with a monolayer ground state. Across the range of slit widths for each ground-state ensemble we again observe an entropic stabilization effect (increasing transition temperature) with decreasing W. The resulting configurational phase diagram allows for sequences of isothermal folding and unfolding transitions through these various low-energy structures. Although we have focused our analysis on the flexible SWchain model with N = 64 and λ = 1.04, we have also studied the N = 32, λ = 1.02 and N = 128, λ = 1.05 versions of this model, both of which also display a direct first-order-like coil → folded transition. We have carried out a full analysis of the N = 32 model and have studied the ground-state structure and wide-slit behavior of the N = 128 model (see Supporting Information). Both of these models display entropic stabilization in a wide slit with the maximum stabilization occurring near W ≈ N1/3σ. For smaller slit widths both models display a series of ground-state structures with the N = 32 chain going from a “native” foldedIII state to a folded-II and then the monolayer folded-I state. The overall topology of the N = 32 configurational phase diagram is very similar to Figure 9 with one less ground-state region. The N = 128 chain goes through the ground-state sequence folded-V → folded-I, although the energy step between the “native” folded-V and the folded-IV structure is quite small. Presumably the discrete step structure in the ground-state energy becomes increasingly washed out for the many-layer crystal structures of longer chains. We can compare our findings for both the simple tethered chain and the chain in slit confinement with other simulation
Figure 10. Thermal and mechanical response functions vs slit width W/σ for a SW-chain with N = 64 and λ = 1.04 end-tethered in a slit at reduced temperature T* = (a) 0.410, (b) 0.360, and (c) 0.310. Shown are the specific heat CW(T)/NkB (left scale, solid line with open symbols) and the energy contribution to the force f T(W)/N (right scale, dashed line with filled symbols). Peaks in panels (b) and (c) correspond to the structural transitions seen in the Figure 8 phase diagram. The inset in (b) gives an expanded view of the III → IV transition.
unfolded region of the Figure 9 phase diagram marks the location of this peak in f T(W). In Figure 10c, the temperature is sufficiently low that all transitions are between folded structures. In this case, both the specific heat and mechanical response exhibit peaks coincident with the crossover in the ground-state location with the peak height increasing as the slit is narrowed. Finally, in Figure 10b we show the interesting case of crossing the Figure 9 phase diagram at a temperature of T* = 0.360 (T ≈ 40 °C). In this case the chain undergoes the following sequence of transitions with decreasing slit width: folded IV → folded III → 3D-coil → folded II → 2D-coil → folded I. Each transition is marked by nearly coincident peaks in the specific heat and mechanical response function. The latter function shows negative peaks for the coil → folded transitions, indicating a decrease in average energy through these isothermal transitions. Figure 11 provides an illustration of the Figure 10b transition sequence, showing configurational snapshots for each region.
4. DISCUSSION AND CONCLUSIONS In this work we have investigated the entropic stabilization effects of both tethering and slitlike confinement on the all-ornone folding transition of a square-well-sphere polymer chain. Contrary to expectations from simple entropic stabilization arguments, we find the folded state of the chain end-tethered to a hard wall to be slightly destabilized with respect to the free chain. The original stabilization argument due to Zhou and Dill12 assumes the tethering plane has a negligible effect on the entropy of the folded state. However, the wall reduces the entropy of both the folded and coil states, and what matters is not the total entropy reduction but rather the fractional loss of entropy. For the N = 64 and λ = 1.04 SW-chain the introduction of a tethering plane produces a slightly larger fractional loss in entropy for the folded state than the coil state, G
DOI: 10.1021/acs.macromol.7b00709 Macromolecules XXXX, XXX, XXX−XXX
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currently studying the SW-chain model in both two- and threedimensional confinement and hope to clarify (at least for a simple model) these issues regarding scaling and universality at the folding transition. As the slit width W is reduced to less than the optimal value (approximately 4σ, 5σ, and 6.5σ for our N = 32, 64, and 128 SW-chain models, respectively), we find a reduction in transition temperature and an increase in the barrier to folding. Continued reduction in W leads to either chain unfolding or to an abrupt transition to a different folded structure. Both a reduction in folding temperature and a slowdown in folding kinetics are found in simulations of model proteins in slitconfinement when W is reduced below the optimal value.37−39 Additionally, Javidpour and Sahimi find the stabilization of a non-native folded structure in a sufficiently narrow slit.38 In addition to biopolymer folding, another area where confinement effects have been of much interest is polymer crystallization.44,45 Studies of polymer crystallization in moderate confinement find that the arrangement of the crystallized polymer can be modified by the confinement as to be commensurate with the confinement length. For example, the crystal structure of poly(ethylene oxide) (PEO), confined between polystyrene layers, evolves from 3D spherulites → 2D spherulites/disks → oriented layered lamellae → highly anisotropic, oriented single crystals, as the layer spacing is reduced from the micrometer to the 10 nm length scale.44 Extreme confinement is found to suppress crystallization. In the case of PEO, studies of confinement in nanometer wide silicate layers55 or subnanometer wide graphite oxide layers56 show a complete suppression of crystallization and a dense disordered polymer structure. Although the SW-chain model studied here does not crystallize into a lamellar structure, the transitions between different layered structures and the isothermal melting into a low temperature pseudo-2D coil phase, seen in the Figure 9 phase diagram, are analogous to the observed PEO behavior. Thus, in comparison with both experimental and computer simulation results, we find that the simple model analyzed here describes much of the underlying physics regarding the entropic effects of slitlike confinement on a first-order-like folding transition of a flexible polymer chain. One of the more intriguing aspects of the present study is the possibility of sequential isothermal melting and freezing transitions through different low energy folded structures. It would be desirable to describe this behavior in terms of applied pressure rather than slit width (i.e., work in an NPT thermodynamic ensemble rather than the NWT ensemble used here). To reformulate our results in this way requires knowledge of the undetermined additive entropy term we have denoted a(W). Determination of this term is possible through an umbrella sampling approach to link the thermodynamics for different values of W, or we could carry out a full 2D Wang−Landau simulation to construct the 2D density of states g(E,W). Either approach will require significant computation and thus is left for future study.
results as well as with experimental results. In the case of simple hard-wall tethering, our result of a slight destabilization effect is in agreement with the work of Luettmer-Strathmann et al., who report a similar result for a bond-fluctuation model lattice chain.23 However, there are experimental results showing a stabilization effect due to hard-wall tethering. In particular, tethering a self-complementary DNA strand, that folds into a stem-loop structure, to a gold surface (at high ionic strength where entropic effects should dominate) is found to stabilize the folded state.54 (In these experiments both the free and tethered DNA are subject to the same chemical modification required for tethering.) It seems likely that this qualitative difference in behavior between the simple chain models and DNA is related to the fact that DNA is a semiflexible polymer. Differences in chain stiffness can produce different behaviors under confinement.29,31 Simulation studies of Go̅-model proteins tethered to a surface find both stabilization and destabilization effects depending on the protein and choice of tethering site.24−27 For a chain in a slit, our findings of stabilization in a wide slit and an optimal slit width for maximum stabilization are in agreement with previous simulation studies of slitlike confinement for a flexible (and semiflexible) homopolymer32 and Go̅type and other model proteins.37−39 For a sufficiently wide slit the confinement will primarily affect the unfolded states, increasing their free energy and thus decreasing the free energy barrier to folding ΔG‡f as seen in Figure 7. In this case it might be expected that both the change in folding temperature ΔTf = Tf(W) − Tf(∞) and free energy barrier to folding ΔΔG‡f = ΔG‡f (∞) − ΔG‡f (W) will follow the same scaling behavior as the free energy of a confined self-avoiding walk,33 i.e., ΔTf ∼ ΔΔG‡f ∼ (W/σ)−γ, where γ = 5/3 for slitlike confinement.3 The reduction in barrier height implies a speed up in the folding rate which, assuming simple Arrhenius kinetics, is given by ln[kf(W)/kf(∞)] ∼ ΔΔG‡f /kBT. Simulation studies of model proteins confined in a slit report values of γ (obtained from both ΔTf and Δkf) in the range 1.4−2.3.37,39 For the confined SW-chain we also find power-law behavior for ΔTf and ΔΔG‡f ; however, our results for chains with N = 32, 64, and 128 all give exponents consistent with γ = 3.2(2) (see Supporting Information). While Rathore et al.36 have noted the apparent nonuniversality of the exponent γ as applied to Go̅-model proteins in spherical confinement, we do expect universality for a simple homopolymer model (and the consistency of our γ result across different versions of the SW-chain model supports this). A likely reason for our seemingly large exponent is that for the range of slit widths we are considering, the confinement perturbs not only the coil states but also the transition and folded states (see Figure 7). A more direct estimate of γ, that is independent of the folding transition, is provided by the scaling of the free energy increase for the E = 0 state with decreasing W. Using this approach, we find γ = 2.2(1) for the SW-chain models (N = 32, 64, and 128) considered here. This is closer to but still not in agreement with the expected 5/3 value. This discrepancy is presumably due to our use of short chains, as the original Daoud−deGennes scaling theory is rigorously valid only in the limit Rg ≫ W ≫ σ, and our chains are not long enough to satisfy the Rg ≫ W part of this limit. (It is somewhat surprising that the protein models noted above, which also use short chains, come so close to the asymptotic exponent.) Additionally, there is an ambiguity regarding the definition of slit width used in the scaling analysis, as discussed in refs 5 and 6, which may be contributing to the discrepancy. We are
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.7b00709. Figures S1−S6 (PDF) H
DOI: 10.1021/acs.macromol.7b00709 Macromolecules XXXX, XXX, XXX−XXX
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Mark P. Taylor: 0000-0002-3281-432X Notes
The author declares no competing financial interest.
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ACKNOWLEDGMENTS The author thanks Jutta Luettmer-Strathmann for many helpful discussions. Financial support from Hiram College and the National Science Foundation (DMR-1607143) is gratefully acknowledged.
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