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.

Biomolecular research has so many facets that it is impossible to cover all important aspects of this highly interdisciplinary field in a single book. However, by definition, generic physics-based approaches have the potential to introduce concepts and tools that enable systematic and consistent investigations of complex systems, even in cases where these systems (cell systems, individual cells, molecular composites, biomolecules, solvent molecules, etc.) do not seem to possess any similarities. The physical concepts are based on quantum and classical theories, intertwined by the basic theory of complexity under the influence of thermodynamic effects: statistical mechanics. All biological, biochemical, and biophysical processes are caused by the interaction of basic units such as atoms, chemical groups, molecules. None of those processes can be thought of as being disconnected from cooperative ordering (or disordering) effects, and for our understanding of these effects on nanoscopic to mesoscopic scales only the basic theory of statistical physics is available to unravel the macroscopic consequences of these processes. The macroscopic description is what we call thermodynamics.

The currently most successful theoretical tool to investigate and to analyze thermal fluctuations statistically is the computer simulation. The computer has not replaced the human brain, but it has changed the way we deal with complex problems.

Steps toward bionanotechnology

The advancing progress in manipulating soft and solid materials at the nanometer scale opens up new vistas for potential bionanotechnological applications of hybrid organicinorganic interfaces [278, 326]. This includes, e.g., nanosensors being sensitive to specific biomolecules (“nanoarrays”), as well as organic electronic devices on polymer basis which have, for example, already been realized in organic light-emitting diodes [327]. An important development in this direction is the identification of proteins that can bind to specific compounds. Over the last decade, genetic engineering techniques have been successfully employed to find peptides with affinity for, e.g., metals [273, 328], semiconductors [274–276], and carbon nanotubes [329]. However, the mechanisms by which peptides bind to these materials are not completely understood; it is, for example, unclear what role conformational changes play in the binding process.

In these mainly experimental studies, it has also been shown that the binding of peptides on metal and semiconductor surfaces depends on the types of amino acids [330] and on the sequences of the residues in the peptide chain [273–276]. These experiments reveal many different interesting and important problems, which are related to general aspects of the question why and how proteins fold. For example, this pertains to the character of the adsorption process, i.e., whether the peptides simply dock to the substrate without noticeable structural changes or whether they perform conformational transitions while binding. Another point is how secondary structures of peptide folds in the bulk influence the binding behavior to substrates. In helical structures, for example, side chains are radially directed and – due to the helical symmetry – residues with a certain distance in the sequence arrange linearly.

Conformational transitions of flexible homopolymers

In this chapter, we will discuss general similarities and differences in crystallization and collapse transitions under the influence of finite-size effects for flexible polymer chains. The analysis of conformational transitions a single polymer in solvent can experience is surprisingly difficult. In good solvent (or high temperatures), solvent molecules occupy binding sites of the polymer and, therefore, the probability of noncovalent bonds between attractive segments of the polymer is small. The dominating structures in this phase are dissolved or random coils. Approaching the critical point at the Θ temperature, the polymer collapses and in a cooperative arrangement of the monomers, globular conformations are favorably formed. At the Θ point, which has already been studied over many decades, the infinitely long polymer behaves like a Gaussian chain, i.e., the effective repulsion due to the volume exclusion constraint is exactly balanced by the attractive monomer–monomer interaction. Below the Θ temperature, the polymer enters the globular phase, where the influence of the solvent is small. Globules are very compact conformations, but there is little internal structure, i.e., the globular phase is still entropy-dominated. For this reason, a further transition toward low-degenerate energetic states is expected to happen: the freezing or crystallization of the polymer. Since this transition can be considered as a liquid-solid phase separation process, it is expected to be of first order, in contrast to the Θ transition, which exhibits characteristics of a second-order phase transition [117, 118].

Beside receptor-ligand binding mechanisms, folding and aggregation of proteins belong to the biologically most relevant molecular structure formation processes. While the specific binding between receptors and ligands is not necessarily accompanied by global structural changes, protein folding and oligomerization of peptides are typically cooperative conformational transitions [246]. Proteins and their aggregates are comparatively small systems. A typical protein consists of a sequence of some hundred amino acids and aggregates are often formed by only a few peptides. A very prominent example is the extracellular aggregation of the Aβ peptide, which is associated with Alzheimer's disease. Following the amyloid hypothesis, it is believed that these aggregates (which can also take fibrillar forms [247]) are neurotoxic, i.e., they are able to fuse into cell membranes of neurons and create pores that are penetrable to calcium ions. It is known that extracellular Ca2+ ions intruding into a neuron can promote its degeneration [248–250].

In this chapter, we will investigate thermodynamic properties of aggregation transitions of polymers and peptides from different perspectives of statistical analysis.

Pseudophase separation in nucleation processes of polymers

We have already discussed why conformational transitions polymers experience in structure formation processes are not phase transitions in the strict thermodynamic sense. This will be similar for the aggregation of a finite number of finitely long polymers, because surface effects are also not negligible in these cases.

In our discussion of flexible polymers of finite length, we have seen that finite-size effects essentially influence the formation of “crystalline” structural phases. We will now investigate how the arrangement of monomers in this crystalline regime changes, if the flexibility is successively reduced by bond–bond correlations. The class of linear macromolecules that effectively fills the gap between purely flexible and stiff chains is called semiflexible polymers. The most prominent representatives in nature are DNA and RNA, but also many proteins possess an effective stiffness. Examples are myosin fibers and actin filaments, which are sufficiently stiff to prevent self-interaction. In these cases, the continuum wormlike-chain model (1.22) is reasonable to investigate thermal fluctuations. However, if the persistence length is small enough, self-interaction occurs, which can cause conformational transitions, depending on environmental conditions. For the basic introduction to semiflexible polymers and their modeling, the reader may consult Section 1.6.

Meanwhile, many examples of single- and double-stranded nucleic acids are known, where the structural behavior deviates from the one predicted by the wormlike-chain model. In these cases, the persistence length is not constant, but depends on chain length and external conditions such as temperature and salt concentrations [171–173]. Consequently, the persistence length is no longer an appropriate length scale for the description of structural behavior.