3.1. Canonical Analysis
In the past, the common approach to the characterization of phase transitions has been canonical statistical analysis. For infinitely large systems, one or more derivatives of an appropriate thermodynamic potential such as the free enthalpy or the free energy would exhibit nonanalytic behavior, if one of the potential’s natural variables was altered at the transition point. Because it is easy to control in experiment, the analysis has typically been done in the space of the canonical (or heat bath) temperature . Discontinuities in the canonical entropy and specific heat help identify and characterize discontinuous and continuous phase transitions, respectively, in the thermodynamic limit. Later, this approach was simply extended to systems of finite size that do not allow for the hypothetical extrapolation toward the thermodynamic limit (for example, heterogeneous polymers such as proteins).
However, transition features in finite systems cannot be linked to nonanalyticities. Hence, peaks and shoulders in response quantities like heat capacity or order parameter fluctuations have usually been considered signals of pseudotransitions in finite systems. The problem with this approach is that extremal points in fluctuating quantities are not safe indicators of phase transitions. The probably most prominent example is the specific-heat curve for the one-dimensional Ising model, which exhibits a pronounced maximum at a certain temperature, but there is no obvious feature that links this extremal thermal activity to a phase transition. In the thermodynamic limit, this peak does not develop into a divergence.
Interestingly, if one plots the heat capacity for the 1D Ising system as a function of the inverse temperature
, though, the peak vanishes and there is no pseudotransition signal. Note that in the microcanonical analysis the inverse temperature
is the more obvious thermodynamic variable than the temperature. Consequently, microcanonical inflection-point analysis does not show any transition signal for the 1D Ising system in equilibrium [
19].
In fact, historically the (canonical) temperature was introduced when the need arose to assign a reliable scale to the thermoscope to quantify the environmental heat content, which ultimately lead Galilei to the introduction of the thermometer at the end of the 16th century. It is ironic that the nowadays most commonly used centigrade scale, the Celsius scale, was introduced in 1742 by Celsius in inverted form, i.e., Celsius chose 0
as the boiling point of water under norm conditions and 100
as the reference point for the melting of ice. At about the same time, French scientist Christin independently used a centigrade scale that essentially inverted the original scale by Celsius, which is what we nowadays refer to as the
Celsius Scale [
23].
This brief historic recollection shows that the temperature and its scale were introduced in an era, when it was still believed to be a quantitative measure for heat, and the scale was ultimately linked to the instrument and substances used for its measurement. Thus, there is no physical reason that favors the historic quantity temperature over the inverse temperature, which is the more natural variable in the microcanonical analysis. To distinguish the microcanonical temperature from the heat bath temperature, we denote the latter by in the following canonical analysis of structural transitions.
Consequently, we plot the canonical mean energy
as a function of
in
Figure 1 for the three models studied:
(flexible chain) and
(semiflexible variants). All curves show a sharp drop at
. Around
, the flexible chain (
) experiences another significant drop in energy, which is less pronounced for the semiflexible polymer with bending stiffness
. No obvious signal is visible on this level for the stiffer chain with
.
The plots of fluctuating quantities, such as the heat capacity
and the fluctuations of the mean square radius of gyration,
, allow for a more detailed analysis. Both quantities are shown as functions of
in
Figure 2a,b, respectively. Whereas we certainly see changes in the curvature of the
curves at
, only the plots of the structural fluctuations show sharp peaks. These indicate the well-known
collapse transition between extended, random-coil structures and compact globular conformations. The transition signal is more pronounced for the stiffer chains. Since this transition is more entropy than energy driven, it is not surprising that it shows up more prominently in the structural rather than the energetic fluctuations. This is different for the freezing transition at about
, which is strongest for the flexible chain (
). Large error bars at very small temperatures (
) prevent a conclusion about another separate transition for the semiflexible polymer with
, which, if it exists, marks the transition into the global energy minimum basin.
3.2. Microcanonical Analysis of Phase Transitions
In this section, we perform a full-scale microcanonical inflection-point analysis of the three models that aims at the identification of all structural transitions in these systems up to third order.
Figure 3,
Figure 4 and
Figure 5 show the microcanonical entropy and its derivatives up to third order as functions of the reduced energy
, i.e., we subtracted from the energy the respective putative global energy minima
obtained for each system in the parallel tempering simulations and verified by simulated annealing. This shift allows for an easier comparison of the results.
Figure 3.
(a) Microcanonical entropy S and its derivatives (b) , (c) , (d) for the different models with plotted as functions of the energy difference from the -dependent global energy minimum estimate. In this figure, we focus on the high-energy (or high-temperature) regime. Least-sensitive inflection points are marked by a dot.
Figure 3.
(a) Microcanonical entropy S and its derivatives (b) , (c) , (d) for the different models with plotted as functions of the energy difference from the -dependent global energy minimum estimate. In this figure, we focus on the high-energy (or high-temperature) regime. Least-sensitive inflection points are marked by a dot.
As in the canonical analysis, we first discuss the low-
(or high-temperature) regime in the relevant energy space. The entropies plotted in
Figure 3a for all cases do not possess least-sensitive inflection points and thus there is no first-order transition, as expected. However, least-sensitive inflection points in the first derivative (
) signal second-order phase transitions in all three systems, which reflect the mostly entropic
collapse from random-coil to globular polymer structures. The corresponding peak locations in the second entropy derivative (
), shown in
Figure 3c, allow for a unique determination of the transition points in energy space and thus also in
, thereby rendering the ambiguous canonical analysis of response functions in the previous section obsolete. Since there are no least-sensitive inflection points in the
plots, none of the systems undergoes a third-order transition in this energy region.
Figure 4 shows the same microcanonical quantities as plotted in
Figure 3, but for an intermediate energy range that covers the inverse temperatures in the interval
. As expected for flexible polymers, the entropy curve for
does exhibit a least-sensitive inflection point, which corresponds to the minimum in the backbending region found in the
plot (
Figure 4b) at about
. The inverse temperature associated with it is approximately
, which confirms earlier results for flexible polymers [
6]. The polymer undergoes a freezing transition from the liquid-globular states into the solid phase, which for this model is dominated by icosahedral structures. Remarkably, for the semiflexible polymers with
, the situation changes. For
, there is still a weak inflection point in the entropy, but the
curve has already almost plateaued at about the same energy value, where we found the first-order transition for the flexible polymer. Consequently, the peak value of the next derivative (
) is virtually zero at this energy (
Figure 4c). Therefore, the freezing transition seems to turn from first order for the flexible polymer (
) to second order for the semiflexible case with
. However, even more surprisingly, the transition behavior changes again for the stiffer chain with
: In fact, the transition signal has completely vanished. The subsequent structural analysis of ground-state properties will lend deeper insight into the reasons for these changes.
The second derivative
in
Figure 4c shows signs of independent third-order transitions for
and
in the ordered phase, but not for
. These results suggest that freezing into a unique and characteristic global energy minimum state is not possible if bending effects overwhelm non-bonded interactions that are commonly responsible for tertiary structure formation in polymeric systems. Local bending effects turn into restraints that inhibit or at least suppress the formation of symmetries. Previous studies have shown that bending restraints play an important role in the formation of stable conformations in tertiary assemblies of helical segments [
24].
Eventually, the sequence of figures shown in
Figure 5 covers the lowest-energy regime for all models compared in this study. In the flexible case, no additional transitions are expected and, correspondingly, we do not see signals in any of the derivatives up to third order that would indicate a transition. The situation is potentially different for the systems with bending restraint. Whereas we do not find indications of transition signals in the entropy and inverse temperature curves for
and
in
Figure 5a,b, respectively, the
results shown in
Figure 5c might tell a different story. The error bars are too large for an ultimate conclusion, but it looks like a third-order transition develops close to the ground state for the semiflexible polymer models.
The quantitative results of the microcanonical inflection-point analysis obtained for the polymer systems studied here are listed in
Table 1.
Figure 4.
Same quantities as in
Figure 3, but plotted for an intermediate energy region.
Figure 4.
Same quantities as in
Figure 3, but plotted for an intermediate energy region.
Figure 5.
Plots of the same microcanonical quantities as in
Figure 3 and
Figure 4 in the lowest-energy regions of the three models.
Figure 5.
Plots of the same microcanonical quantities as in
Figure 3 and
Figure 4 in the lowest-energy regions of the three models.
3.3. Structural Analysis of Lowest-Energy Conformations
The lowest energies found in the simulations, which we consider best estimates of the ground state energies, are for the flexible polymer (), for the semiflexible chain with , and if . All values are given in units of the energy scale of the Lennard-Jones potential, . Since the bonded interactions (bond vibrations) are negligible near the ground state, the differences in the ground-state energy estimates for the different models must be attributed to the penalty paid for chain bending. The ground-state conformation for the flexible polymer that does not experience these restraints is formed by optimizing the non-bonded interaction and results, for this model, in an ideally icosahedral structure. Turning on the bending restraint and setting , the competition between nonlocal attraction and local repulsion increases the tension in the icosahedral structure, although it still stays in shape. For , however, it cannot maintain the optimal icosahedral arrangement of monomers and rather forms an entangled structure of longer, less bent, segments. As we have seen in the thermodynamic analysis of the transitions, the phase behavior changes significantly with increasing bending stiffness.
Representative lowest-energy conformations are shown in
Figure 6. For a more quantitative analysis let us take a look at the pair distribution functions and the contact maps. We introduce the pair distribution function as
where
As a robust threshold for the necessary binning of the
r space, we chose
. The histograms for the lowest-energy conformations shown in
Figure 6 are plotted in
Figure 7. The perfect icosahedral structure is only found for
, whereas its decay is already visible for the weaker semiflexible polymer (
). The maximum number of nearest-neighbor contacts (
) found for
is not reached in the semiflexible cases. The broadening of the peaks and additional spikes not present for
are clear indicators that the ground-state structure for the semiflexible polymer with
is not icosahedral. Bending restraints prevent the formation of perfect symmetries and thus a characterization of the
ground-state conformation as a distinct crystalline or quasicrystalline structure is not possible. It rather resembles tertiary folds of protein conformations, where effective bending restraints and the local stable secondary segments purposefully prevent symmetric arrangements of monomers. This enables different heteropolymers of similar size to form distinct and functional individual conformations.
Figure 6.
Representatives of lowest-energy conformations for . Labels help to follow the chain from the first monomer (1, blue) to the last (55, red).
Figure 6.
Representatives of lowest-energy conformations for . Labels help to follow the chain from the first monomer (1, blue) to the last (55, red).
This is obvious from the plots of the contact maps for the lowest-energy states shown in
Figure 8. In the grid spanned by the monomer labels, we mark pairs of monomers with distances
. Bonded monomer pairs are not included. For the flexible polymer, the contact map does not exhibit particularly remarkable structural features. Since there are no bond angle restraints, each monomer tries to maximize its number of nearest neighbors for the energetic benefit. Icosahedral conformations with one of the end monomers in the center are optimal. For the semiflexible polymer with
, we already see that changes occur. Monomers do not only try to maximize the number of contacts, but three monomers connected by two adjacent bonds now have to cooperate to minimize bending. Close to the diagonal, we find that short anti-diagonal streaks form. These are clear indicators of turns with two linear strands in contact with each other (hairpins). In protein folds, these would be referred to as building blocks for
-sheets. For stronger bending rigidity (
), there are fewer, but longer such segments. We also observe the formation of streaks parallel to the diagonal, which can be associated with helical alignments. Both types of these secondary structure elements can be seen in the geometric representation in
Figure 6. The hairpin-like sections are located in the interior and the helical segments wrap around it. Of course, under these conditions, energetically favored structures are not icosahedral. In fact, no global symmetries can be identified in this tertiary fold. Although it is tempting to think of these structures in analogy to protein folds, the lack of dihedral (torsion) constraints, which separates secondary structure elements from each other in the tertiary fold, prevents a more direct comparison.