On the inverse temperature transition and development of an entropic elastomeric force of the elastin mimetic peptide [LGGVG]_{3, 7}

Elastin, the principal protein component of the elastin fiber, is an extracellular insoluble protein responsible for the remarkable elasticity of many vertebrate tissues. The concept that the elasticity of elastin is entropic in origin was introduced early on by Hoeve and Flory.¹ The source of entropy based elasticity was made evident by observations of the dampening of internal chain dynamics derived from force-extension data of elastin model molecules.² These results led to the development of the librational entropy model, wherein the development of the entropic elastic force results from a decrease in the elastomer entropy.

A key feature that is related to the thermal properties of elastin is the observation of an inverse temperature transition; a process in which elastin and short elastin mimetic peptides become more ordered by expelling water and reducing volume when the temperature is raised from ∼5 °C to 40 °C.³ Various experimental studies, including transmission electron microscopy (TEM),³ light scattering, circular dichroism (CD),⁴ composition,⁵ and nuclear magnetic resonance (NMR) (Refs. 4 and 6) have provided evidence of the inverse temperature transition in elastin and in the short mimetic peptides consisting of repeats of [VPGVG]_n.³ The observation of an inverse temperature transition in short peptides, such as [VPGVG]_n, has led many researchers to study their dynamical and structural properties via experiment and simulation.^7–9 Some of the commonly studied synthetic elastin-like peptides include residues of glycine (G), valine (V), proline (P), and leucine (L) in repeating motifs of VPGVG and LGGVG.¹² It was demonstrated, via short 9 ns simulations, that the signatures of an inverse temperature transition in repeats of [VPGVG]_n (n = 18) include a decrease in solvent-accessible surface area (SASA), radius of gyration, number of peptide-water hydrogen bonds, and number of proximal water molecules concomitant with an increase in number of peptide-peptide hydrogen bonds and side chain contacts as the temperature is raised from 7 °C to 45 °C.⁷ In these studies, a type II β-turn was observed in the Ramachandran maps of glycine, valine, and proline segments at 10 °C and 42 °C.⁷ The type II β-turn and β-spiral are structural models proposed by Urry and co-workers^4,13,14 whose formation at high temperature is correlated to the inverse temperature transition.³ Type II β-turn structures were also observed in experimental studies of [(VPGVG)₄(VPGKG)]₃₉ and [VPGVG]₃ peptides using solid-state NMR.^15,16 More recently it was demonstrated that the oligopeptide GVG[VPGVG] behaves as a two-state system that undergoes an inverse temperature transition and that a re-entrant unfolding occurs near the boiling point of water.⁹ In addition, the role of the proline residue on the inverse temperature transition and dynamics of the model elastin peptide GVG(VPGVG) was recently investigated via molecular dynamics simulation and experiment.¹⁰ As further evidence of the observed inverse temperature transition, when a sample of a [VPGVG] polypentapeptide was stretched to 60% of its original length and subsequently heated, the elastomer shortened with largest changes observed in the 20 °C–40 °C temperature range.³ The inverse temperature transition of short elastin mimetic peptides is now well known and applied in novel anticancer therapeutics.¹¹ 

In addition to the experimental observations relating to the inverse temperature transition of elastin, several measurements have also confirmed that the remarkable properties of the elastomer hinge on a complex interaction with the solvent. Solid state NMR studies showed that elastin exhibits more overall mobility when hydrated,¹⁷ and it has been demonstrated that the motion of a short mimetic sequence, poly[VPGVG], is also facilitated by the presence of water.¹⁸ The protein's Young modulus depends inherently on the polarity of the solvent, as demonstrated in numerous experimental studies.¹⁹ 

In experiment, it has been observed that the polypeptide [LGGVG] also exhibits type II β-turns that are found in [VPGVG]_n, as well as type I β-turns. The structure of [LGGVG]_n (n = 2–15) has been shown to be an ensemble of type I, type II β-turns as well as unfolded regions via CD, NMR, and TEM studies.²⁰ Short 150 ps molecular dynamics simulation studies of LGGVG have given some insight into the distribution of γ and type-II β-turns and also point to a conformational floppiness due to low motional barriers.²¹ Recent NMR studies of the [LGGVG]₆ motif agree with these studies; similar structural properties found in VPGVG sequences are present in this model system, however, a mixture of conformations are also observed.²² In this work we report on a simulation based study of [LGGVG]_n (n = 3, 7) and compare the derived dynamical, thermal, and mechanical characteristics to [VPGVG]_n (n = 3, 7). Using a quasi-harmonic approach, it is shown that both model peptides exhibit a reduction in the population of low-frequency modes and an increase in population of high-frequency modes upon elongation. The shift in population of frequency modes causes the peptide entropy to decrease upon elongation and is responsible for the development of an entropic force that gives rise to elasticity. These observations are in disagreement with a previously published notion that model elastin peptides, such as [VPGVG]₁₈, increase in entropy upon elongation.²³ 

Molecular dynamics simulations were performed in GROMACS (Ref. 24) using the OPLS-AA/L force field model²⁵ and a Berendsen thermostat²⁶ at temperatures ranging from 5 °C to 60 °C. The starting structures for all simulations were linear-chains of [VPGVG]₃, [VPGVG]₇, [LGGVG]₃, or [LGGVG]₇. In the simulations, the peptides were placed in a cubic box solvated with water using the SPC216 water model.²⁷ Our recent experimental studies of the dynamics of an elastin mimetic peptide [VPGVG]₃ correlated well with 4 ns simulations using the same water model and thus we implement the same water model here.³² The peptides were terminated by the amine (–NH₂) group and the carboxyl (–COOH) group at the N- and C-termini, respectively. The dimensions of the box and number of water molecules varied for each simulation. Short 4 ns simulations of the [VPGVG]₃ and [LGGVG]₃ peptides were performed in 113 nm³ and 27 nm³ cubic boxes containing 3656 and 828 water molecules, respectively. Longer 20 ns simulations of the [VPGVG]₃ and [LGGVG]₃ peptides were performed in a 125 nm³ cubic box and the two peptides were solvated with 4072 and 8382 water molecules, respectively. Short 4 ns simulations of the [VPGVG]₇ and [LGGVG]₇ peptides were performed in 991 nm³ and 2346 nm³ cubic boxes with 32 704 and 73 200 water molecules, respectively. In all simulations an energy minimization was performed using the method of steepest descent with the maximum step size set to 1 and the force tolerance set to 2000 kJ/mol nm to remove overlapping atoms.²⁸ We performed a short simulation restraining the peptide position and equilibrating the pressure for 10 ps before executing the full simulations.

For the simulations of the mechanically strained [VPGVG]₃ and [LGGVG]₃ peptides, the equilibrated structures obtained at 25 °C were used as the initial structures. The atoms of Val4 were fixed while a pull force was applied on the Gly13 atoms in the ± z direction with a 50 000 kJ/mol nm² force constant at a pulling rate of 0.005 nm/ps; a similar pull force was implemented on a recent simulation of [VPGVG]₁₈.²³ Short 1 ns molecular dynamics simulations were performed with the same OPLS-AA/L force field and preparatory steps indicated above in a 125 nm³ cubic box solvated with 4080 water molecules. The resulting radius of gyration of the C_α of [LGGVG]₃ as a function of time is shown in Figure 1(a), and indicates that the peptide was indeed stretched. Following this step we performed a 2 ns simulation on the stretched peptide with an applied 20 000 kJ/mol nm constant force using the structure from the 774 ps time frame in Figure 1(a) as the initial structure (the time frame choice was randomly chosen). The resulting radius of gyration of [LGGVG]₃ is shown in Figure 1(b), and demonstrates that the peptide remained stretched for the duration of the simulation. Using the same applied forces, similar features of Figure 1 were also observed for the strained simulation of [VPGVG]₃.

To estimate the entropy of the peptides of interest we implemented the quasi-harmonic approach.²⁹ In this formalism, the peptide entropy is computed by assuming that any motion may be modeled as a series of harmonic oscillators with individual frequencies ω_i. The total entropy is computed by performing a sum over all frequencies and is given by

In the above expression k_B is Boltzmann's constant, T is the temperature of the reservoir and the sum is performed over the number of available modes, 3n-6.²⁹ In this approach the frequencies ω_i are given by

where λ_i is the ith eigenvalue of the mass-weighted covariance matrix, σ^′. The eigenvalues are determined by diagonalizing the expression,

where M is the mass matrix. The elements of the covariance matrix, σ, are determined by the usual method,

where r_{i, j} are the Cartesian coordinates.

Figure 2(a) highlights examples of the resulting averaged C_α root mean square displacement (RMSD) as a function of time at various temperatures of [VPGVG]₃. The figure shows that the RMSD of the C_α of [VPGVG]₃ appears to reach an equilibrium value after 2 to 3 ns. Similar results for the RMSD of the C_α were obtained for all temperatures studied indicating conformational equilibrium. While it has been observed that simulations on the order of a few nanoseconds might overlook certain slow dynamical processes associated with long correlation times,³⁰ the time duration and findings of the simulations presented in this work are analogous to those on a much larger [VPGVG]₁₈ peptide simulated for only 6–9 ns.⁷ We recently experimentally investigated the dynamics of a short elastin mimetic VPGVG peptide via NMR;³² our experimental findings correlated well with short 4 ns simulations of [VPGVG]₃ under mechanical strain. In addition, we have also reported on an experimental study of the thermal behavior of [VPGVG]₃ via deuterium NMR and found that a thermal hysteresis was observed in the NMR T₁ and T₂ relaxation times of Gly-α and Val-α and -β deuterons.³² The observed thermal behavior was correlated to the inverse temperature transition of the peptide which was again observed in short 4 ns simulations of [VPGVG]₃.

Table I summarizes the observed variations in the physical characteristics of [VPGVG]₃ for 4 ns as well as 20 ns simulations. The caption in Table I includes the metrics that were used (similar metrics were implemented in Ref. 7). The thermal dependence reported in our simulation data indicate that the peptide decreases in size, expels water molecules and develops greater intramolecular interactions with the temperature increasing from 10 °C to 42 °C; these are signatures of the well-known inverse temperature transition in elastin and agree very well with those of Ref. 7. Table I highlights a decrease in the RMSF of the C_α, the radius of gyration, SASA, non-polar SASA, number of water molecules, as well as the number of hydrogen bonds between the peptide and the proximal waters of hydration with increasing temperature. In addition, an increase in the number of side-chain contacts and the number of hydrogen bonds within the peptide is observed with increasing temperature. The fluctuations of the data in Table I observed between 20 °C and 30 °C for the 4 ns simulation are all within the statistical error of our analysis and are also evident in the work of Ref. 7 in the same temperature range. We also performed the same analysis on 4 ns simulations of [VPGVG]₇. The averaged C_α RMSD are highlighted in Figure 2(b). Similar to the results of the 15-residue peptide, shown in Figure 2(a), these larger peptides also seem to reach an equilibrated structure after 2 to 3 ns of simulation time. The data in Figure 2(b) also suggest that the RMSD fluctuates more so at 10 °C than at 42 °C, which may be related to the observation that the peptide is almost double in size at the lower temperature (as measured by the Rg). The thermal properties of the 35-residue peptide at 10 °C and 42 °C are reported in Table I and exhibit similar trends as the shorter 15-residue peptide. Finally, Table I summarizes results from longer 20 ns simulations of [VPGVG]₃; the tabulated results from these simulations appear to be within our statistical uncertainty of the 4 ns simulations.

We performed a similar analysis on the [LGGVG]_n (n = 3, 7) peptides. To check equilibration, we investigated the C_α RMSD at various temperatures and the results are highlighted in Figure 3(a) for n = 3 and Figure 3(b) for n = 7; the figures demonstrate that both peptides reach conformational equilibrium after 2 to 3 ns for all temperatures. Table II summarizes the thermal properties of the [LGGVG]_n (n = 3, 7) peptides for 4 ns and 20 ns simulations. The similarities in the trends of the temperature dependence are also observed among the three simulations, suggesting that the short time 4 ns simulations of [LGGVG]₃ are similar to the 4ns [LGGVG]₇ simulation and the 20 ns [LGGVG]₃ simulation. With the temperature raised from 10 °C to 42 °C, a reduction is observed in the SASA, non-polar SASA, number of water molecules, and the number of peptide-water hydrogen bonds. In addition, an increase in the number of side-chain contacts and the number of hydrogen bonds within the peptide are also observed over this temperature range. These observations appear to follow the trends observed in the [VPGVG]_n (n = 3, 7) peptides shown in Table I and indicate that the [LGGVG]_n (n = 3, 7) peptides also undergo an inverse temperature transition in the same temperature range as that observed in the [VPGVG]₃ peptide.

In addition to the similarities observed in the thermal behavior of the mimetic peptides [VPGVG]₃ and [LGGVG]₃, there are similarities and differences observed in the dynamical properties and in the secondary structures. Referring to Table II, the RMSF of the [LGGVG]₃ peptide is observed to not continually decrease with increasing temperature, but fluctuates within our statistical uncertainty and is at its minimum at 42 °C. The simulation results of the radius of gyration between 5 °C and 35 °C seem to suggest a decreasing trend, notwithstanding the fact that the values between 20 °C and 30 °C are within the respective error bars of the numbers shown. Similar findings are observed on the [VPGVG]₃ peptide. Second, the comparison of the Ramachandran maps of the valine residues of both peptides at 10 °C and 42 °C, shown in Figure 4, reveal some key differences at different temperatures. Urry's studies indicate that the β-II turns are an important structural element of the elastomeric force of VPGVG.³ This β-II turn configuration is observed in our [VPGVG]₃ peptide but we also note the presence of an α-helix as well – the α-helix configuration is more prominent in the peptide at higher temperatures. However, for the [LGGVG]₃ peptide the α-helix configuration is present at 10 °C but not at 42 °C. Instead, the [LGGVG]₃ peptide exhibits only a β-II turn configuration at 42 °C, which has been observed in experiments as one of the major populations of the valines in [LGGVG]₆.²² 

We also studied the entropy of the [LGGVG]₃ and [VPGVG]₃ as a function of temperature. Figure 5 highlights the results of our computations and indicate that the two peptides have relatively similar entropies at all temperatures studied. The data also reveal that the entropy of both peptides decreases with increasing temperature between 25 °C and 35 °C; we attribute this reduction to the inverse temperature transition which results in an ordering of the peptide in this temperature range. It should be pointed out, however, that while the peptide undergoes a phase change and becomes more ordered, the diffusion coefficient of water increases with increasing temperature in elastin, as confirmed by experiment.³³ The increase in diffusion of water with increasing temperature will affect the tumbling motion of the peptide and counteract local ordering, thereby causing an increase in entropy when the temperature is raised above 35 °C.

We analyzed the strained simulations of [LGGVG]₃ and [VPGVG]₃ at 25 °C and the results are tabulated in Table III. All the values shown in Table III were computed between the atoms of Val4 and Gly13 during the last 1 ns for both the relaxed and strained simulations. Upon extension the RMSF, number of side-chain contacts and number of peptide-peptide hydrogen bonds decreases for both peptides. With the peptides strained, we also observe an increase in the radius of gyration, number of proximal water molecules, and the number of peptide-water hydrogen bonds. Qualitatively the SASA and non-polar SASA for both peptides exhibit little change upon extension. While the trends in the data for both peptides appear to be similar there are some differences in the relative values shown. The simulation data show that the amount of water within 0.35 nm of both peptides in their relaxed states are different; we measured 224 ± 20 for [LGGVG]₃ and only 47 ± 14 for [VPGVG]₃. The differences may be expected given that the proline residues of [VPGVG]₃ may impose some degree of steric hindrance and restrict the range of torsional motion that may allow water to come in contact with the peptide surface. This may also explain the difference in peptide-water hydrogen bonds observed in [VPGVG]₃ and [LGGVG]₃ in their relaxed states; referring to Table III, we observed 21.7 ± 2.4 peptide-water hydrogen bonds for [LGGVG]₃ and only 3.8 ± 1.9 for [VPGVG]₃ in their relaxed states. It should be noted that the number of peptide-peptide hydrogen bonds is between 0 and 1 in the relaxed state, but no peptide-peptide hydrogen bonds are observed upon extension.

We determined the entropy as a function of the time sampling window, Δt, for both peptides under extension and highlight the results from this computation in Figure 6. The time sampling window refers to the time frame over which the entropy was computed. The purpose of this figure is to indicate that the entropy calculation can be underestimated if the time sampling time is too small; the computation appears to have converged after 2 ns. A similar analysis of the convergence of the entropy determination via the quasi-harmonic approach has been presented in on two DNA sequences d(GC)₃₀ and d(AT)₃₀.³⁴ The figure shows that there is an overall reduction in the entropy for both peptides when taken out of mechanical equilibrium, as expected. This behavior is analogous to that of natural rubber – its entropy is known to decrease upon extension.³⁵ This observation is in disagreement with a statement that an extended peptide chain has greater entropy because it is more dynamic, as made in Ref. 23. Another interesting characteristic of the two peptides that is revealed by the simulation data, shown in Figure 6, is that the total entropies appear to be relatively similar. Taking the time sampling window of 1ns as an example, the relaxed [VPGVG]₃ peptide has a total entropy of 2.2 kJ/mol K which is similar to that of the relaxed [LGGVG]₃ peptide that has a total entropy of 2.4 kJ/mol K. The total entropy of the strained [VPGVG]₃ peptide is 1.1 kJ/mol K, and a similar value for the strained [LGGVG]₃ peptide of 1.3 kJ/mol K was computed. The values determined via the quasi-harmonic approach appear to be similar to the entropy computed by a previous molecular dynamics study of [VPGVG]₁₈.³⁶ In the study of Ref. 36 the entropy of a strained peptide was ∼76 J/mol K per residue at 27 °C. Using this value, the average entropy in their computation for a 10-residue peptide is ∼0.8 kJ/mol K in good agreement to what was obtained in our study.

Figure 7 highlights histograms of the frequency distributions for [LGGVG]₃ and [VPGVG]₃ obtained via the quasi-harmonic approach described above. In both peptides, the data demonstrate an increase in the number of high-frequency modes (greater than 2.5 THz for [LGGVG]₃ and greater than 1.5 THz for [VPGVG]₃) that are present upon extension. In addition, the figures show a reduction in low-frequency modes that dominate when the peptides are in their relaxed states. One is reminded from the field of solid state physics, that an applied strain may induce a shift in the frequency of vibrational modes that may be observed by Raman spectroscopy.^37–39 Based on the second law of thermodynamics, the entropy of the system should decrease under mechanical strain; our findings appear to indicate that the peptide entropy is reduced when mechanically strained. Equation (1) is a positive definite function; the entropy reduces with increasing frequency at any given temperature – a well-known result of statistical thermodynamics for any object undergoing oscillatory motion. With the assumption that the motion of the atoms of the peptide can be modeled as a series of harmonic oscillators, Eq. (1) predicts that the entropy should reduce in a situation of high-frequency oscillation. In agreement with this model, Figure 7 shows that both peptides occupy higher frequency modes of oscillation in situations of extension resulting in a reduction in entropy when taken out of equilibrium.

It should also be noted that the entropy obtained by the quasi-harmonic approach may be overestimated under certain conditions.⁴⁰ As discussed in Ref. 29, Eq. (1) would approximate the entropy when fluctuations derived from a classical simulation are used, rather than fluctuations determined by a quantum mechanical simulation. If classical fluctuations are implemented, Eq. (1) is justified in the limit when ω ≪ k_BT/ℏ. In our work k_BT/ℏ is ∼6.2 THz at 25 °C and a large fraction of the frequency components shown in Figure 7 satisfy the above approximation. However, although Eq. (1) breaks down in the limit for high-frequency quantum oscillations, they contribute less to the entropy compared to low frequency components⁴⁰ and, therefore, their contribution to the total entropy is negligible. Finally, nanosecond time scale simulations such as that presented in this work cannot capture low-frequency oscillatory motions in the kHz to MHz range that have been observed in elastin and model elastic protein-based polymers,² as the Nyquist-Shannon sampling theorem cannot be satisfied.

Experimental studies of a poly[VPGVG] peptide indicate that large-amplitude low-frequency librations can occur between β-turns.² When distressed, the peptide studied in Ref. 2 exhibited a reduction in the librational amplitude. Our simulation findings also point to a reduction in the amplitude of the backbone motion–referring to Table III, when the peptides are stretched there is a significant decrease in the RMSF averaged over all C_α. It is therefore clear that the reduction in amplitude also contributes to the reduction in the peptide entropy, shown in Figure 6, as the amplitude and the frequency of the peptide motion are correlated.

To further probe the peptide water interface we examined the lifetime of hydrogen bonds in situations of mechanical strain. As discussed above, when a mechanical deformation is applied there is an increase in the frequency of the peptide's oscillatory motion. This would result in a decrease in the lifetime of the peptide-peptide and peptide-water hydrogen bonds, and is highlighted in Table III. We computed a lifetime of 0.81 ps for the peptide-peptide hydrogen bonds in [LGGVG]₃ and 0.40 ps in [VPGVG]₃ when relaxed, and no peptide-peptide hydrogen bonds when strained. For the peptide-water hydrogen bonds, we computed a lifetime of 0.97 ± 0.03 ps for [LGGVG]₃ and 0.83 ± 0.03 ps for [VPGVG]₃ when relaxed and 0.35 ± 0.01 ps for [LGGVG]₃ and 0.38 ± 0.01 ps for [LGGVG]₃ when stretched.

Using the resulting changes in the entropies determined by the quasi-harmonic approach, we provide an estimate of the internal tension of the two peptides. From statistical thermodynamics, the internal tension Y of a one-dimensional linear polymer chain at constant temperature may be determined by

where δL is the change in length, δS is the change in entropy, and T is the temperature of the system. From our simulation data we have determined the resulting change in length due to the applied force; δL = 0.33 nm for [LGGVG]₃ and δL = 0.42 nm for [VPGVG]₃. These values were determined by considering the change in the radius of gyration of the two peptides computed by the average C_α positions from Val4 to Gly13. From the quasi-harmonic approach, we found δS = –1.1 kJ/mol K for [LGGVG]₃ and δS = –1.1 kJ/mol K for [VPGVG]₃. Thus, at 25 °C we obtain Y = 993kJ/mol nm for [LGGVG]₃ and Y = 780 kJ/mol nm for [VPGVG]₃ due to the respective changes in entropy and length of the peptides. It should be pointed out that the internal tension determined via this analysis relies on a simplified treatment of the change in length δL of the peptides, and depends on the applied force on the peptide. Second, the computation of the entropy determined by nanosecond time scale simulations preclude the ability to sample low-frequency oscillations that would contribute significantly to the entropy and have been observed in experiment.² Together, these approximations prevent us from determining a spring constant that can be compared with atomic force microscopy² or single molecule force spectroscopy⁴¹ of short elastin polymers. While the side chains contribute to the change in entropy, this one-dimensional treatment considers the change in length due to displacements of the backbone only and appears to indicate that the mechanical properties of the [VPGVG]₃ and [LGGVG]₃ peptides are similar within the bandwidth of modes that one may sample with short time simulation.

In this work we have reported on the thermal and mechanical properties of the elastin mimetic peptide [LGGVG]₃ by simulation. The temperature-dependence of the simulation findings indicate that the [LGGVG]₃ peptide undergoes an inverse temperature transition, resulting in a local ordering of the peptide with the temperature raised from 10 to 42 °C. The [LGGVG]₃ peptide has similar characteristics to the well-studied [VPGVG]₃ peptide both in relaxed and mechanically strained states. Using the quasi-harmonic approach we estimated the entropy of both peptides and found that the values are very similar at all temperatures. For both peptides, the entropy decreases upon deformation and the associated changes to the entropy of both peptides when stretched also appear to be similar. We also provide an estimate of the internal tension of the peptides upon strain. The findings of the simulation results also indicate that the lifetimes of the peptide-peptide and peptide-solvent hydrogen bonds both decrease upon extension.

G.S. Boutis acknowledges support from award No. SC1GM086268-04 from the National Institute of General Medical Sciences. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institute of General Medical Sciences or the National Institutes of Health (NIH).