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.

Relevance of biomolecular research

More than 30 million people, infected by the human immunodeficiency virus (HIV), suffer from the acquired immune deficiency syndrome (AIDS); 2 billion humans carry the hepatitis B virus (HBV) within themselves, and in more than 350 million cases the liver disease caused by the HBV is chronic and, therefore, currently incurable. These are only two examples of worldwide epidemics due to virus infections. Viruses typically consist of a compactly folded nucleic acid (single- or double-stranded RNA or DNA) encapsulated by a protein hull. Proteins in the hull are responsible for the fusion of the virus with a host cell. Virus replication, by DNA and RNA polymerase in the cell nucleus and protein synthesis in the ribosome, is only possible in a host cell. Since regular cell processes are disturbed by the virus infection, serious damage or even the destruction of the fine-tuned functional network within a biological organism can be the consequence.

Another class of diseases is due to structural changes of proteins mediated by other molecules, so-called prions. As there is a strong causal connection between the three-dimensional structure of a protein and its biological function, refolding can cause the loss of functionality. A possible consequence is the death of cells. Examples for prion diseases in the brain are bovine spongiform encephalopathy (BSE) and its human form Creutzfeldt–Jakob disease (CJD).

The idea to write this book unfolded when I more and more realized how equally frustrating and fascinating it can be to design research projects in molecular biophysics and chemical physics – frustrating for the sheer amount of inconclusive and contradicting literature, but fascinating for the mechanical precision of the complex interplay of competing interactions on various length scales and constraints in conformational transition processes of biomolecules that lead to functional geometric structures. Proteins as the “workhorses” in any biological system are the most prominent examples of such biomolecules.

The ability of a “large” molecule consisting of hundreds to tens of thousands of atoms to form stable structures spontaneously is typically called “cooperativity.” This term is not well defined and could easily be replaced by “emergence” or “synergetics” – notions that have been coined in other research fields for the same mysterious feature of macroscopic ordering effects. There is no doubt that the origin of these net effects is of “microscopic” (or better nanoscopic) quantum nature. By noting this, however, we already encounter the first major problem and the reason why heterogeneous polymers such as proteins have been almost ignored by theoretical scientists for a long time. From a theoretical physicist's point of view, proteins are virtually “no-no's.” Composed of tens to thousands of amino acids (already inherently complex chemical groups) linearly lined up, proteins reside in a complex, aqueous environment under thermal conditions. They are too large for a quantum-chemical treatment, but too small and too specific for a classical, macroscopic description. They do not at all fulfill the prerequisites of the thermodynamic limit and do not scale.

A central result of the discussion in the last chapter was the strong influence of finite-size effects on the freezing behavior of flexible polymers constrained to regular lattices. Thus, (unphysical) lattice effects interfere with (physical) finite-size effects and the question remains what polymer crystals of small size could look like. Since all effects in the freezing regime are sensitive to system or model details, this question cannot be answered in general. Nonetheless, it is obvious that the surface exposed to a different environment, e.g., a solvent, is relevant for the formation of the whole crystalline or amorphous structure. This is true for any physical system. If a system tries to avoid contact with the environment (a polymer in bad solvent or a set of mutually attracting particles in vacuum), it will form a shape with a minimal surface. A system that can be considered as a continuum object in an isotropic environment, like a water droplet in the air, will preferably form a spherical shape.

But what if the system is “small” and discrete? Small crystals consisting of a few hundred cold atoms, e.g., argon [154], but also as different systems as spherical virus hulls enclosing the coaxially wound genetic material [155, 156] exhibit an icosahedral or icosahedral-like shape. But why is just the icosahedral assembly naturally favored?

The capsid of spherical viruses is formed by protein assemblies, the protomers, and the highly symmetric morphological arrangement of the protomers in icosahedral capsids reduces the number of genes that are necessary to encode the capsid proteins. Furthermore, the formation of crystalline facets decreases the surface energy, which is particularly relevant for small atomic clusters.

Evolutionary aspects

The number of different functional proteins encoded in human DNA is of the order of about 100 000, which is an extremely small number compared to the total number of possibilities: Recalling that 20 types of amino acids occur in natural proteins and typical proteins consist of N ∼ O(102−103) amino acid residues, the number of possible primary structures, 20N, lies somewhere far, far above 20100 ∼ 10130. Assuming all proteins were of size N = 100 and a single folding event would take 1 ms, a sequential enumeration process would need about 10119 years to generate structures of all sequences, irrespective of the decision about their “fitness,” i.e., the functionality and ability to efficiently cooperate with other proteins in a biological system. Of course, one might argue that the evolution is a highly parallelized process that drastically increases the generation rate. So, we can ask the question, how many processes can maximally run in parallel. The visible universe contains of the order of 1080 protons. Assuming that an average amino acid consists of at least 50 protons, a chain with N = 100 amino acids has of the order O(103) protons, i.e., 1077 sequences could be generated in each millisecond (forgetting for the moment that some proton-containing machinery is necessary for the generation process and only a small fraction of protons is assembled in Earth-bound organic matter).

From the exact solution of the Ising model by Onsager in 1944 up to that of the hard hexagon model by Baxter in 1980, the statistical mechanics of two-dimensional systems has been enriched by a number of exact results. One speaks (in quick manner) of exact models once a convenient mathematical expression has been obtained for a physical quantity such as the free energy, an order parameter or some correlation, or at the very least once their evaluation is reduced to a problem of classical analysis. Such solutions, often considered as singular curiosities upon their appearance, often have the interest of illustrating the principles and general theorems rigorously established in the framework of definitive theories, and also enabling the control of approximate or perturbative methods applicable to more realistic and complex models. In the theory of phase transitions, the Ising model and the results of Onsager and Yang have eminently played such a reference role. With the various vertex models, the methods of Lieb and Baxter have extended this role and the collection of critical exponents, providing new useful elements of comparison with extrapolation methods, and forcing a refinement of the notion of universality. Intimately linked to two-dimensional classical models (but of less interest for critical phenomena), one-dimensional quantum models such as the linear magnetic chain and Bethe's famous solution have certainly contributed to the understanding of fundamental excitations in many-body systems. One could also mention the physics of one-dimensional conductors.