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.

The ice model

Introduction

The six-vertex model, which is the object of this chapter, is a special case of the eight-vertex model on a two-dimensional square lattice introduced by Fan and Wu (1970) in order to summarize a class of exactly solvable models in classical statistical mechanics. The thermodynamics of the six-vertex model is by now known in its full generality (Yang and Yang, 1966a–d; Lieb, 1967a) which is not the case for the eight-vertex model, but only the self-conjugate one (Baxter, 1971a).

The general model can be considered as a two-dimensional idealization of a crystalline system in which pairs of adjacent atoms or radicals on the network are linked by ‘hydrogen bonds’. Of ionic type, this link between two neighbouring electronegative atoms is realized by a proton H+ which is located closer to one of the atoms than to the other. On each link of the network there thus exist two equilibrium positions for H+ (Pauling, 1960). If the coordination number at each site equals 4, as is the case for the oxygen atom in hexagonal ice, we indeed have eight possible proton configurations on a given site, which give rise to the eight vertices.

In contrast, for the physical systems for which this model could be viewed as a valid idealization – ice H2O, the ferroelectric PO4H2K, the antiferroelectric PO4H2NH4 – there exist at most two H+ next to each site.