Thermodynamics of tubelike flexible polymers

Thermodynamics of tubelike flexible polymers


In this work we present the general phase behavior of short tubelike flexible polymers. The geometric thickness constraint is implemented through the concept of the global radius of curvature. We use sophisticated Monte Carlo sampling methods to simulate small bead–stick polymer models with Lennard-Jones interaction among non-bonded monomers. We analyze energetic fluctuations and structural quantities to classify conformational pseudophases. We find that the tube thickness influences the thermodynamic behavior of simple tubelike polymers significantly, i.e., for given temperature, the formation of secondary structures strongly depends on the tube thickness.

I Introduction

To resolve the conformational mechanics of secondary-structure formation is one of the major tasks in polymer science. While in the “real world” experiments are restricted to specific molecules under specific conditions, in the “virtual world” of computer simulations there are no such limitations. Using reasonably simplified models, systematic studies of classes of polymers in different environments are possible Binder (1995).

A common, effective and widely-used coarse-grained model for polymers is the bead–stick model Flory (1953); de Gennes (1979). Here the polymer is modeled as a linear chain of pointlike monomers, which correspond to molecular units, e.g., amino acid residues in the case of proteins. The monomers are connected by stiff bonds and interact via simple effective pair potentials. This class of models enables computer simulations of very large polymer systems and is, for example, quite useful for studying global structure formation or structural transitions Grassberger (1997); Vogel et al. (2007); Schnabel et al. (2009). On the other hand, due to the simple pairwise interactions, it is hardly possible to investigate the formation of secondary structures in a systematic way, which is due to missing specific extensions like hydrogen bonds, anisotropy, explicit stiffness, etc. Noguchi and Yoshikawa (1998); Kemp and Chen (1998); Rapaport (2002); Sabeur et al. (2008).

In this work, we therefore follow the approach introduced by Banavar and Maritan et al. Banavar et al. (2003a, b); Banavar et al. (2004), where a tubelike model is considered instead of linelike chains. The virtual thickness of the tube caused by the bulky shape of the monomers (e.g., because of side chains connected to the backbone) is introduced via a three-body interaction. The general tertiary phase behavior of tubelike polymers with 40 and more monomers has already been investigated using a square-well model Banavar et al. (2003a), identifying the folding and collapse transitions in a structural phase diagram parameterized by thickness and temperature. In our study, we investigate in detail the thickness and temperature dependence of secondary-structure formation of tube polymers, employing a continuum model with inter-monomeric Lennard-Jones potential. For this reason, we consciously investigate rather small chains (with up to 13 monomers). For longer chains, tertiary folding effects become apparently important and symmetry, anisotropy, and marginal compactness of globular protein structures are then doubtlessly of interest (14); (15). However, the globular arrangement of secondary segments in tertiary folds is not in the focus of this study and it is also hardly feasible to perform a similarly precise analysis of the present work for longer chains.

The present work extends our recent study of ground-state properties of tubelike polymers as a function of their thickness Vogel et al. (2009, 2008). Thus, the conformations identified in these former studies represent the dominant structures in the fluctuation-free, i.e., lowest-temperature region () of the entire conformational phase diagram that we will discuss in detail in the following. Thus, the goal of this study is to identify independently of the chain length the relevant pseudophases in the thermodynamic phase diagram, based on the shape of ground-state structures. The notion “phase” shall be handled with some care; conformational phase transitions of small systems are not thermodynamic phase transitions in a strict sense. Nonetheless, there is a strong similarity of these structural transitions and thermodynamic phase transitions, as both are typically governed by the competition of energy and entropy. However, to make clear that there can also be significant differences (no collapse of fluctuating quantities, i.e., there are transition regions rather than transition points), we call conformational phases of short chains “pseudophases” in the following (18).

Recent related studies also apply other tube models for homopolymers to investigate secondary-structure formation (see, e.g., Refs. Hoang et al. (2004); Snir and Kamien (2005, 2007); Auer et al. (2007); Wolff et al. (2008)). These are, however, based on different approaches to influence or potentiate structure formation. See, e.g., the discussion in Ref. Vogel et al. (2009).

The structure of the rest of the paper is as follows: In Sec. II, we describe the model and specify the simulation methods we employed. In Sec. III, we present the complete thermodynamic phase diagrams for various chain lengths of homopolymers and analyze and classify the different pseudophases therein. In Sec. IV, we introduce a hydrophobic-polar heteropolymer tube model and analyze the ensuing pseudophase behavior for an exemplified sequence of monomers. Finally, our main findings are summarized in Sec. V.

Ii Model and Methods

As outlined above, we employ in this study a linear, flexible polymer model with thickness, i.e., we consider tubelike chains instead of linelike objects. The bond length in this model is kept fixed, i.e., , where denotes the distance between two monomers. The monomers interact via a standard Lennard-Jones (12,6)-potential resulting from pairwise attractive van-der-Waals and short-range repulsion forces:


In the following, we set , such that vanishes for and is minimal at with . The total energy of a conformation is then calculated as the sum of all LJ contributions, .

To define the thickness of a conformation , we apply the concept of the global radius of curvature Gonzalez and Maddocks (1999). Accordingly, we measure all (see technical remark below) radii of curvature , i.e., the radii of the circles defined by the monomer positions , , and . The minimal radius of curvature is called the global radius of curvature:


The thickness of the polymer tube is simply twice the global radius of curvature, . For illustration, Fig. 1 shows two radii of curvature of a conformation with monomers. As a technical remark: The explicit calculation of all radii of curvature is obviously needless and would be very expensive in terms of computing time as the number of radii grows with the third power of the monomer number (). By excluding a huge number of a priori too large radii with much less effort, the calculation can be done nearly in steps (possibly plus some marginal higher-order terms) Neuhaus et al. (2007).

Figure 1: Two examples of circumcircles of three monomers and the corresponding radii of curvature. The small circle corresponds to the radius of curvature of three consecutive monomers, i.e., to the local radius of curvature of the monomers . The bigger circle corresponds to the radius of curvature of the monomers .

In order to simulate the model, we restrict the conformational space by a thickness constraint , such that conformations with are forbidden, i.e., the Heaviside function is included in the partition function,


where is the functional integral measure and is the inverse temperature (with in natural units). For a more detailed description and discussion of the concept and its applicability to polymer models, see, for example, Refs. Banavar et al. (2003b); Banavar et al. (2004); Vogel et al. (2009, 2008).

In the Monte Carlo simulations, we use multicanonical flat histogram sampling Berg and Neuhaus (1991, 1992) to estimate the density of states. To determine the weight factors, we employ the recursive method of Wang and Landau Wang and Landau (2001), with the control parameter initialized and subsequently decreased to as described in Ref. Zhou and Bhatt (2005). We remark that for any finite value the Markov chain of configurations, as generated with the Wang–Landau algorithm does not possess a proper Gibbs measure. Rather, the density of states, entering here the Metropolis criterion, is constantly updated and hence varies as the Markov chain proceeds. Thus detailed balance is violated in particular in the initial simulation phase. We therefore decided to freeze the weights at some point of the Wang–Landau iteration and to perform a multicanonical production run with a Gibbs measure as determined by the multicanonical weight factor. Furthermore, we also checked our results for reliability against data obtained by parallel tempering simulations Hukushima and Nemoto (1996); Geyer (1991); Neuhaus et al. (2007); Bittner et al. (2008), which generate simultaneous ensembles of polymers at a multitude of temperature values. The checks are done for selected parameter sets, as well as against data from the study presented in Ref. Bachmann et al. (2005). The simulations of different polymer lengths and thickness constraint values were carried out separately to avoid correlations and statistical imbalances.

Iii Conformational phase diagrams of tubelike homopolymers

iii.1 General

Figure 2: (color online) Phase diagrams of the homopolymers with (left) and (right). The labels , , , and indicate the different pseudophases at finite temperature. Figures a) and b) show the perspective view of the specific-heat landscape, and in c) and d), the top-views are plotted with marked peak positions for various parameters . The specific-heat values are encoded in gray scale. The pictures in the insets in c) and d) correspond to the ground states presented in Ref. Vogel et al. (2008), the pictures in the regions show relevant conformations there.
Figure 3: Energy histograms for various thickness constraints at . Histograms corresponding to specific-heat maxima are marked with arrows. a) polymer. Histograms correspond to the following thickness parameters: (solid line), , , , (solid line, ), , , , (solid line, ), , , , (solid line, ), . b) polymer. Histograms correspond to (solid line), , , (solid line, ), , , , (solid line, ), , , (solid line, ), . The histograms were obtained by reweighting the density of states and are consistent with histograms obtained from independent canonical simulations at this temperature. These histograms contain about entries. Statistical errors are less than 1% and, almost everywhere, smaller than the line width.

In this work, we study homopolymers consisting of , , , and monomers. After having considered the low-temperature regime, i.e., ground states, in a recent paper Vogel et al. (2008), we here concentrate on the conformational phase behavior at finite temperatures. As common, we calculate the specific heat and consider the peak regions of this observable as indicators of relevant thermodynamical activity. Figure 2 shows these specific-heat landscapes for the and polymer. The points (+) plotted in the top-view representation of Fig. 2 c) resp. Fig. 2 d) indicate the positions of the crest lines in this landscape, i.e., the lines signaling structural changes. We notice four major pseudophases, which we denote by , , , and . In Fig. 3, we show the corresponding canonical energy histograms at temperature for different thickness constraints , with the histograms at the transition values of marked by arrows. Both plots, for and , do not differ qualitatively, i.e., have all interesting features in common. The phase structure will be discussed in the subsequent detailed analysis of the pseudophase diagrams.

In the insets of Figs. 2 c) and 2 d), ground-state conformations, according to their thickness, are shown. They provide a first indication for the population of the respective pseudophase at finite temperatures. Deeper analyses will strengthen the expectation that the ground-state conformations are the relevant conformations in the corresponding pseudophases at finite temperatures as well. This includes, for example, the analyses of distributions of structural observables like end-to-end distance, radius of gyration, radial distribution of monomers, bond angles and torsion angles, as well as comparisons with reference structures and “counting” structural components, e.g., using pattern recognition Tenenbaum et al. (2000), during additional canonical simulations at fixed temperatures. Let us note, that we neglect data for , which corresponds to the pure Lennard-Jones volume exclusion, as the thickness constraint does not influence the system at all below this “natural thickness” Vogel et al. (2008).

iii.2 Analysis of structural phases

Figure 4: Measured histograms from simulations at fixed temperatures for the polymer a) at the transition from the bended-ring phase to the sprawled-coil phase and b) deep inside these two phases. a) End-to-end distance (solid line) and radius of gyration (dashed line) at the transition ( and ). b) Radial distribution () in the bended-ring phase (solid line, ) and in the sprawled-coil phase (dashed line, ). The histograms are differently scaled for better visibility, each contains more than entries. Statistical errors are less than 1% and smaller than the line width.

We begin the detailed discussion of the different structural phases with the high-thickness region, i.e., with the phase and the transition to . Based on the knowledge of the ground states and some general structural properties of polymers, we assume in a population of bended rings, which undergo a structural change to sprawled random coils in , which become more and more rodlike with increasing thickness. This assumption can be illustrated and strengthened by an example in little more detail. For monomers, let us consider the geometrical objects “octagon” and “straight line” as limiting prototypes of these regions. Calculating the properties of these prototypes, one expects for the end-to-end distance distributions a sharp peak at the position of the LJ potential minimum, i.e., at , and a diffuse peak at , for the radius of gyration distribution a sharp peak at and a diffuse peak at , and for the radial distribution function sharp peaks at , , , and and smooth peaks below integer values for the respective conformations. In Fig. 4, these distributions are shown, measured in canonical simulations at the transition temperature and within both phases. In Fig. 4 a), the end-to-end distance and radius of gyration histogram are plotted, and Fig. 4 b) shows the radial distribution function. These quantities exhibit exactly the assumed behavior, i.e., the peaks of the measured distributions appear exactly at the calculated values for the anticipated “prototypes”. Additionally, the bimodal shapes of the distributions in Fig. 3 at the transition are an indication for the first-order-like character of the transition with coexisting conformational phases. The energy histograms near the transition point exhibit two distinct peaks separated by broad energy gaps. During simulations at the transition line, both structures appear equally, as can be seen for example in Fig. 4 a).

Figure 5: Measured histograms in phase for and for the polymer. a) The end-to-end distance histogram exhibiting three separate peaks indicating three different major contributing groups of conformations. b) The energy histogram and c) the histogram of torsional angles. Error bars were obtained from independent simulations and are shown exemplarily. In b) and c), the histograms for each group of conformations, distinguished by its end-to-end distance, are shown in addition. Each histogram contains at least entries. d) Representatives of each group of this energetic pseudophase and their corresponding properties.

Reducing the thickness parameter , we reach the phase , which we call the sheet phase. Figure 5 shows the results of simulations at and for the polymer, which belongs to the region called in Fig. 3 a). There are mainly three structures dominating the phase , amongst them the two ground-state conformations in the range (cp. Ref. Vogel et al. (2008) and Fig. 2). As shown in Fig. 5 a), they can be distinguished with the help of the end-to-end distance, where three distinct peaks in the distribution appear, whereas they cannot be resolved in terms of the specific heat. The plot in Fig. 5 b) shows the overall energy distribution as well as the contributions from the three regions corresponding to the peaks in the end-to-end distribution. As illustrated in Fig. 5 d), the peak in the energy distribution is associated with ring-like conformations and their excitations, whereas the shoulder is caused by hairpin-like conformations. In Fig. 5 c), we plot the distribution of torsion angles. The contributions of the different structural classes can be distinguished very well again. One notes for example an accumulation of torsion angles around in the contribution of the hairpin-like conformations, an indication for the planar structure of the conformation. At , the conformations extend into the third dimension, i.e., bonds within the conformations begin to overlap. An analogous behavior is found for , see Fig. 3 b).

Figure 6: Torsion angle distributions at for the a) and b) polymers in phase at and , respectively (cuboid or sc-helical region). Each histogram contains about entries. For visualizations of corresponding conformations see, e.g., Fig. 2 c).
Figure 7: (color online) Phase diagrams of the (left) and (right) polymers analogously to Fig. 2.

The region of lowest thickness, , is the helical phase. This phase can be further separated into subphases, where in one of them the exact -helix resides as a ground state for and  Vogel et al. (2008). In a further region, simple-cubic helical structures Chan and Dill (1990), or cuboids for , corresponding to the ground-state conformations in the range , respectively, dominate foo (a). These regions are separated by noticeable, but in the context of the whole phase diagram less important, transition lines. For illustration, we show in Fig. 6 the distribution of torsional angles in the cuboid region for , and , at temperature . For the polymer, it can clearly be seen that only conformations with torsional angles of and , i.e., cuboids, occur. For the polymer, these angles are still dominant, although not occurring exclusively. In any case, the existence of that region is insofar worth mentioning as the corresponding conformations do not appear as ground states for this length and as it shows that it is a characteristic feature and not only a length-dependent artefact.

Figure 7 shows the phase diagrams for the longer tubes consisting of and monomers analogously to Fig. 2. In general, beside the short-length artefacts near , the phase diagrams at different lengths do not differ qualitatively much from each other. The general thermodynamic behavior is quite similar for all system sizes, especially we find again the four major phases discussed above. Also, the characteristics of the sprawled-coil and bended-ring regions do not depend, beside an obvious shift of the thickness parameter, on the polymer length. We note, however, the onset of the formation of tertiary structures, as also discussed in Ref. Vogel et al. (2008), especially the helical phase becomes internally more complex. Furthermore, the relevant thermodynamical activity shifts to lower temperatures.

The ground-state conformations for these systems, plotted again in the insets of Fig. 7 c) and d), support our interpretation of the phases given above. Especially the motivation for denoting the sheet phase becomes clearer, as we found almost planar, “two-dimensional” ground states seeming to crystallize on a honeycomb lattice. These conformations are the dominant conformations in at finite temperatures as well and form, in the case of the polymer, three LJ contacts, in the sense of a contact map Vogel et al. (2008). We find a further interesting detail here, which occurs only for these longer chains. The 13mer is long enough, that an intermediate phase emerges between and . This phase is populated, as indicated by the ground-state conformation shown in Fig. 7 d), by two small bended circles such that two LJ contacts are formed.

Since we focus in our study on the very precise investigation of short chains only in order to elaborate the thickness and temperature dependence of secondary-structure formation, noticeable tertiary effects, such as the globular arrangement of secondary-structure segments, are not yet relevant. For longer chains, a classification of structural phases is only possible by accounting for the globular tertiary folding behavior as it was shown in Ref. (14), where protein-like structures were identified as marginally compact, thus representing a particular globular conformational phase.

Iv Secondary-structure pseudophases of a hydrophobic-polar tube model

As the central result of this work, we have shown above how the sole introduction of a thickness constraint enhances the formation of different secondary structures, including helix and sheet formation, for classes of homopolymers. Here, we modify the homopolymer tube model by introducing two species of monomers: hydrophobic (A) and hydrophilic or polar (B) ones. The nonbonded Lennard-Jones interaction between pairs of monomers now depends on their types:



Besides the strong attraction of A-type monomers we thus have a weak attraction between B-type monomers and a weak repulsion between monomers of different type, favoring “hydrophobic” core formation of A monomers. To enable a direct comparison with the literature on the standard linelike AB model Stillinger et al. (1993, 1995); Hsu et al. (2003); Bachmann et al. (2005), we introduce here in addition a bending term and take the total energy as


where the ’s are the bending angles of adjacent bond vectors.

Figure 8: (color online) Pseudophase diagram of the Fibonacci AB heteropolymer. a) The plot shows the top-view with marked peak positions of the specific heat for various parameters , b) the qualitative view of the specific-heat landscape. Gray scales encode the value of the specific heat. The pictures in c) illustrate selected ground-state conformations. Conformations are shown from different viewpoints, A monomers are marked by red color (dark gray), B monomers are white.

Just to acquire a taste for the effects of these changes, we show as an example results for the 13mer Fibonacci sequence ABABABABAB, which has been studied in the linelike AB model, i.e., with , some time ago Stillinger et al. (1995); Hsu et al. (2003); Bachmann et al. (2005). Figure 8 shows the phase diagram analogously to Figs. 2 and 7, as well as selected ground-state conformations. The general structure including several separated structural subphases is similar to that for the presented homopolymers. The most prominent finding is definitely the very stable -sheet region in the interval , as . The conformations there are neither of - nor -type, i.e., they have neither constant bond nor torsion angles Vogel et al. (2008), but they are indeed “planar” (data not shown, see Fig. 8 for visualization). These qualitative properties do not change over the entire region. A quantitatively remarkable fact is the variation of the intra-monomer distances. We note, that the interaction length between the opposite hydrophobic A monomers (, see Fig. 8 c) for monomer numbering) and () in the sheet conformation does not change in the whole thickness region at all. On the other hand, the distances between the B monomers and increase () and decrease between the A monomers and (, differences respecting the conformations at and ). The van-der-Waals attraction between the A monomers is thus the dominant factor that stabilizes the -sheet. Remarkably, as becomes clear by the listed geometrical quantities above, the bending energy is even increasing with increasing thickness in this region, contrarily to the general overall trend, that the bending energy decreases with increasing thickness. We discuss the influence of the bending term further below. Remember that there are planar six-ring conformations at comparable thicknesses for the , , and homopolymer ground states Vogel et al. (2008). These structures are now stabilized by the specific monomer sequence. We emphasize that the tube thickness keeps playing an important role. Just simulating the given sequence in a two-dimensional space without thickness leads to completely different conformations, consisting of a hydrophobic core and a polar shell Stillinger et al. (1995).

At lower thickness parameters we find structures with helical properties, which, however, depend on the monomer sequence. We note here a very pronounced conformational transition from random coils to native conformations at , which is in detail discussed for the linelike limit “” in Ref. Bachmann et al. (2005). With increasing thickness the ground-state conformation becomes a ring and finally switches to a stretched rod, which, contrarily to the homopolymers discussed above, appears as ground-state conformation. This is a qualitative difference to the results in Sec. III.

Finally two remarks are in order. Firstly, using the described model, we make two independent changes compared to the homopolymer model used before. We introduce different kinds of monomers with different interactions and in addition a bending stiffness. To evaluate the influence of each of the two changes, we simulated the 13mer with a homopolymer sequence consisting of just hydrophobic A monomers (A), which is equal to the homopolymer studied without bending stiffness in Sect. III. We made sure, that the influence of the bending stiffness is marginal for both, ground-state structures and thermodynamical behavior in the relevant structural regions. The ground-state energies change by 1% to 5% in the and region, the structures themselves remain qualitatively the same. The effect on the thermodynamical behavior is marginal, in particular peak positions in the specific heat are not influenced. We conclude, therefore, that the described behavior is predominantly based on the influence of different monomer types. Remember also the example discussed above on this observation. Note that choosing a B homopolymer (B) would correspond to and in Eq. (1), with and . Absorbing the energy scale in the definition of temperature (i.e., ), we would work with .

Secondly, as a methodological remark, knowing that ground states of one-dimensional linelike models do intrinsically have some measurable “natural thickness” in the meaning of the interpretation of the global radius of curvature, see Eq. (2), it may be favorable to search for ground states by simulating the polymer with a thickness constraint slightly below this value. One restricts the conformational space significantly and may travel much faster through the remaining phase space. That way, we could confirm for the 13mer Fibonacci sequence and other widely-used AB polymers with monomers the ground-state energies and conformations presented over the past years Bachmann et al. (2005); Kim et al. (2005); Elser and Rankenburg (2006); Huang et al. (2006).

V Summary

We present in this article results of a computer simulation study of the thermodynamical behavior of a tube model for simple homopolymers as well as for an exemplified hydrophobic-polar heteropolymer. The thickness of the tube in our simulations is controlled by a single parameter, the global radius of curvature, which depends on three-body interactions foo (b).

After focusing on ground states of homopolymers and their properties in a previous work Vogel et al. (2008), we identified dominant structural pseudophases at finite temperatures, i.e., specific-heat landscapes depending on the thickness parameter and temperature, representing the conformational phase diagram. Independently of the polymer length, we find four major structural phases. These include helices, sheetlike planar structures, bended rings and sprawled random coils. These different secondary structure phases can be assigned to different ranges of the tube thickness. The thickness parameter is therefore suitable for a classification of the secondary structures of polymers. Concentrating on the analysis of the secondary-structure formation of short chains, tertiary effects could widely be excluded. Symmetries and anisotropy in the arrangement of secondary-structure segments in globular domains (14), which are particularly interesting for proteins, are necessarily of importance in the discussion of the folding behavior of longer chains. A precise investigation of the thickness-dependent influence of thermal fluctuations on the phase structure is future work.

In an extension of the tube polymer concept, we also introduced the AB tube model for hydrophobic-polar heteropolymers and discussed results for a given sequence of monomers, which has extensively been studied before without thickness. We showed that a sequence of hydrophobic and polar monomers can stabilize the general secondary structures. In particular we find a very pronounced and stable region of a -sheet structure.

Our results are qualitative in a sense that they represent the general frame of possible conformational phases of secondary structures for thick polymers and proteins. This is the basis of the further analysis of pseudophases of models designed for specific polymers or proteins.

To conclude, the tube picture is well suited to mimic the volume extension of polymers, for example due to side chains of amino acids in biopolymers. It may be employed in other contexts as well, for example, for simulations of a tube model for entangled networks of polymers, where the hypothetical tube around a polymer models the suppression of transverse undulation by the network de Gennes (1979); Hinsch et al. (2007). Finally also the diffusion of knots in knotted DNA can proceed via the solitonic diffusion of compact knot shapes Grosberg and Rabin (2007). The tube picture also may be applicable here.

This work is partially supported by the DFG (German Science Foundation) under Grant Nos. JA 483/24-1/2/3 and the Leipzig Graduate School of Excellence “BuildMoNa”. Support by the supercomputer time grant of the John von Neumann Institute for Computing (NIC), Forschungszentrum Jülich, is acknowledged. We thank Sebastian Schöbl for interesting discussions.


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