Introduction

For a system under thermal conditions in a heat bath with temperature T, the dynamics of each of the system particles is influenced by interactions with the heat-bath particles. If quantum effects are negligible (what we will assume in the following), the classical motion of any system particle looks erratic; the particle follows a stochastic path. The system can “gain” energy from the heat bath by these collisions (which are typically more generally called “thermal fluctuations”) or “lose” energy by friction effects (dissipation). The total energy of the coupled system of heat bath and particles is a conserved quantity, i.e., fluctuation and dissipation refer to the energetic exchange between heat bath and system particles only. Consequently, the coupled system is represented by a microcanonical ensemble, whereas the particle system is in this case represented by a canonical ensemble: The energy of the particle system is not a constant of motion. Provided heat bath and system are in thermal equilibrium, i.e., heat-bath and system temperature are identical, fluctuations and dissipation balance each other. This is the essence of the celebrated fluctuation-dissipation theorem [74]. In equilibrium, only the statistical mean of the particle system energy is constant in time.

This canonical behavior of the system particles is not accounted for by standard Newtonian dynamics (where the system energy is considered to be a constant of motion). In order to perform molecular dynamics (MD) simulations of the system under the influence of thermal fluctuations, the coupling of the system to the heat bath is required. This is provided by a thermostat, i.e., by extending the equations of motion by additional heatbath coupling degrees of freedom [75].

The intrinsic nature of secondary structures

Resolving structural properties of single molecules is a fundamental issue as molecular functionality strongly depends on the capability of the molecules to form stable conformations. Experimentally, the identification of substructures is typically performed, for example, by means of single-molecule microscopy, X-ray analyses of polymer crystals, or NMR for polymers in solution. With these methods, structural details of specific molecules are identified, but frequently these can not be generalized systematically with respect to characteristic features being equally relevant for different polymers. Therefore, the identification of generic conformational properties of polymer classes is highly desirable. To date the most promising approach to attack this problem is to analyze polymer conformations by means of comparative computer simulations of polymer models on mesoscopic scales, i.e., by introducing relevant cooperative degrees of freedom and additional constraints. In these modeling approaches – we have already made use of it in the previous chapters – the linear polymer is considered as a chain of beads and springs. Monomeric properties are accumulated in an effective, specifically parametrized single interaction point of dimension zero (“united atom approach”). Noncovalent van der Waals interactions between pairs of such interaction points are typically modeled by Lennard-Jones (LJ) potentials. In such models, only the repulsive short-range part of the LJ potentials keeps the monomers pairwisely apart. Although such models have proven to be quite useful in identifying universal aspects of global structure formation processes, these approaches are less useful in this form to describe local symmetric arrangements of segments of the chain.

Aggregation of semiflexible polymers

In the following, we will discuss the aggregation of interacting semiflexible polymers by analyzing the order and hierarchy of subphase transitions that accompany the aggregation transition.

Cluster formation and crystallization of polymers are processes that are interesting for technological applications, e.g., for the design of new materials with certain mechanical properties or nanoelectronic organic devices and polymeric solar cells. From a biophysical point of view, the understanding of oligomerization, but also the (de)fragmentation in semiflexible biopolymer systems like actin networks is of substantial relevance. This requires a systematic analysis of the basic properties of the polymeric cluster formation processes, in particular, for small polymer complexes on the nanoscale, where surface effects are competing noticeably with structure-formation processes in the interior of the aggregate.

A further motivation for investigating the aggregation transition of semiflexible homopolymer chains derives from the intriguing results of the similar aggregation process for peptides [254, 255] discussed in Chapter 11, which were modeled as heteropolymers with a sequence of two types of monomers, hydrophobic (A) and hydrophilic (B). By specializing the previously employed heteropolymer model to the apparently simpler homopolymer case, we now, by comparison, aim at isolating those properties that were driven mainly by the sequence properties of heteropolymers. In fact, while in both cases the aggregation transition is a first-order-like phase-coexistence process, it will turn out that for the homopolymer model considered in the following, aggregation and crystallization (if any) are separate conformational transitions, if the bending rigidity of the interacting homopolymers is sufficiently small. This was different in the example of heteropolymer aggregates that we discussed in the previous chapter, where these transitions were found to coincide [254, 255].

Structure formation at hybrid interfaces of soft and solid matter

The requirement of higher integration scales in electronic circuits, the design of nanosensory applications in biomedicine, and also the fascinating capabilities of modern experimental setup with its enormous potential in polymer and surface research, have produced an increased interest in macromolecular structural behavior at hybrid interfaces of organic and inorganic matter [273–278]. Relevant processes include, e.g., wetting [279–281], pattern recognition [282–284], protein–ligand binding and docking [285–287], effects specific to polyelectrolytes at charged substrates [288] as well as electrophoretic polymer deposition and growth at surfaces [289].

The recent developments of experimental single-molecule techniques at the nanometer scale, e.g., by means of atomic force microscopy (AFM) [290, 291] and optical tweezers [292, 293], allow for a more detailed exploration of structural properties of polymers in the vicinity of adsorbing substrates [278]. The possibility to perform such studies is of essential biological and technological significance. From the biological point of view, the understanding of the binding and docking mechanisms of proteins at cell membranes is important for the reconstruction of biological cell processes. Similarly, specificity of peptides and binding affinity to selected substrates are relevant features that need to be taken into account for potential applications in nanoelectronics and pattern recognition nanosensory devices on a molecular basis [282].

In this chapter, theoretical approaches to the modeling of hybrid systems of soft and solid matter will be discussed. A comprehensive analysis of conformational transitions experienced by polymers in the binding process, of the phase-diagram structure, and of dominant polymer conformations in these phases can only be performed efficiently by means of computer simulations of hybrid physisorption models on mesoscopic scales.

Folding of linear chains of amino acids, i.e., bioproteins and synthetic peptides, is, for single-domain macromolecules, accompanied by the formation of secondary structures (helices, sheets, turns) and the tertiary hydrophobic-core collapse. While secondary structures are typically localized and thus limited to segments of the peptide, the effective hydrophobic interaction between nonbonded, nonpolar amino acid side chains results in a global, cooperative arrangement favoring folds with compact hydrophobic core and a surrounding polar shell that screens the core from the polar solvent. Systematic analyses for unraveling general folding principles are extremely difficult in microscopic all-atom approaches, since the folding process is strongly dependent on the “disordered” sequence of amino acids and the native-fold formation is inevitably connected with, at least, significant parts of the sequence. Moreover, for most proteins, the folding process is relatively slow (microseconds to seconds), which is due to a complex, rugged shape of the freeenergy landscape [190–192] with “hidden” barriers, depending on sequence properties. Although there is no obvious system parameter that allows for a general description of the accompanying conformational transitions in folding processes (as, for example, the reaction coordinate in chemical reactions), it is known that there are only a few characteristic folding behaviors known, such as two-state folding, folding through intermediates, and glass-like folding into metastable conformations [193–199].

Thus, if a classification of folding characteristics is useful at all, strongly simplified coarse-grained models should allow to describe generic statistical [46,47] and kinetic [200] pseudo-universal properties.