All proteins are linear polymers of amino acids, large sequences of which constitute a peptide chain. Our focus is on globular proteins, whose peptide chain has a folded structure. In general, they are soluble in water and in other polar solvents. Although their structure is complex, we will see in this chapter that they often assume similar forms and that their shapes, sequence, and conformation can be understood by considering some fundamental aspects of their structure.

Amino acids and primary structure

The fundamental unit (monomer) of the protein molecule is the α amino acid. It consists of an acidic carboxyl group and an amino group attached to a single carbon atom, referred to as the α-carbon, and a hydrogen atom. This is illustrated in Fig. 2.1. A side-chain of molecules, designated as “R,” is also attached to the amino acid. This side-chain is specific to each amino acid and is what differentiates them from each other. The side-chains can vary in complexity; examples are simple hydrogen atoms, an extra amino group, an extra carboxylic group, a sulphydryl group, a hydroxyl group, or a simple hydro-chain or hydro-carbon ring. There are 20 biologically important amino acids listed in Table 2.1. The table also lists some proteins and their amino acid composition.

In this chapter we provide a brief summary of some of the methods used to study the physical properties of proteins in solution. This is not meant as a thorough discussion of experimental techniques, but rather as an introduction with sufficient references to provide the interested reader with a guide to the literature.

Methods to determine three-dimensional protein structure

As noted earlier, knowing the structure of a protein molecule is crucial to understanding its biological function. Over the past century, efforts have been focused on methods which allow protein structure to be determined. X-ray crystallography and NMR studies are two popular techniques, the former being predominant.

X-ray crystallography

X-ray crystallography [13, 14] of protein crystals is a well known technique, which we briefly summarize here. Its use for protein crystals, of course, depends upon the regular arrangement of protein molecules on a crystal lattice. When electromagnetic radiation of a given wavelength, such as X-rays, is incident upon a grating of appropriately sized spacings, the waves interfere constructively and destructively in a process known as diffraction. The diffraction pattern is a three-dimensional image of the crystal in reciprocal space and can be used ultimately to determine the real structure of the lattice; it represents the scattering of the X-rays by the electrons of the atoms in the crystal.

Classical nucleation theory

In this section we give a brief review of nucleation theories sufficient for understanding the following chapters. Very good reviews of nucleation theories can be found in refs. [208]–[212].

Nucleation is a stochastic process by which droplets of the new (stable) phase are formed in the background of the metastable phase. The stochastic nature of fluctuation means that the supersaturated system does not switch to the new stable state immediately; before this, a sufficiently large fluctuation with properties of the new phase should occur. This fluctuation is called a critical nucleus or critical droplet, and the formation of the droplet is called a nucleation event. Each nucleation event occurs randomly. The average rate of the nucleation events per unit of time per volume is called the nucleation rate. Homogeneous nucleation is the occurrence of nucleation events in the bulk, with no surfaces or impurities involved. Heterogeneous nucleation is nucleation that occurs on surfaces or impurities. Homogeneous nucleation is an intrinsic property of a system, whereas heterogeneous nucleation is not.

In the case of liquid condensation from vapor or crystallization from dilute solution, it is straightforward to identify the clusters of new phase, as they differ in density. However, this is not always the case.

Introduction

A major stumbling block in developing a first principles understanding of crystal nucleation is the lack of a detailed understanding of the forces that exist between protein molecules in solution. Proteins are complex molecules whose surfaces consist of amino acid residues that possess charged and uncharged regions and in general have complicated interactions with the surrounding electrolyte. The forces between protein molecules in solution include, for example, screened repulsive Coulomb interactions and attractive van der Waals interactions. In addition, protein–ion dispersion forces and hydrophobic and hydration forces play an important role. To further complicate matters, subtle mutations on the surface of these molecules (e.g. sickle hemoglobin, gamma crystallin) can have profound effects on their intermolecular interactions. One can imagine, therefore, how difficult it is, in principle, to determine a realistic model of protein–protein interactions. Progress to date has been based on rather simple models which are spatially isotropic, with short range attractive interactions. Clearly, more realistic models are needed to obtain a more quantitative understanding of the protein phase diagram, nucleation rate, and subsequent growth. One important aspect of the interactions between globular protein molecules, which is only recently being considered by theorists, is that many of the interactions are highly directional.

In this chapter we review some of the existing theoretical studies of models for the phase diagram of globular proteins. Almost all studies to date treat the multicomponent solution as a quasi-one-component solution, with a potential energy of interaction (e.g. potential of mean force) between the protein molecules that is usually assumed to be spatially isotropic. Typically, the interactions consist of a repulsive hard core and a short range attractive interaction. The majority of these studies involve simulation techniques, particularly the Gibbs ensemble Monte Carlo method, although studies based on thermodynamic perturbation theory, finite size scaling theory, and various theories for the radial distribution function also exist. These models have in common the feature that, for sufficiently short range attractive interactions, the fluid–fluid coexistence curve is metastable. Their phase diagrams look qualitatively similar to experimental diagrams, such as that shown in Fig. 4.4. In addition to these studies, we discuss a theoretical estimate for the rather narrow window of values for the range R of the attractive interaction within which the boundary between stable and metastable fluid–fluid transitions is located. We also discuss recent theoretical studies of spatially non-isotropic models that are more realistic descriptions of protein–protein interactions.

The idea that the range of the attractive interaction between protein molecules should be small as compared with the molecular size has its roots in the seminal observation by George and Wilson [25] that for several globular proteins the optimal window of crystallization corresponds to small negative values of B2.

Introduction

Another globular protein, amyloid β protein (Aβ), plays an important role in Alzheimer's disease (AD). This is a devastating disease that, if left unchecked and untreated, threatens to affect over 13 million Americans by the year 2050 [520]. About one in ten people over the age of 65 and as many as five out of ten people over the age of 85 are affected by AD. In the twentieth century, it was found by looking at deceased patients suffering from Alzheimer's that the brains of such individuals contained plaques and neurofibrillary tangles [521–523]. Over the past 20 years, we have gained a more molecular understanding of the causes of AD. Indeed, the primary protein component of these plaques was sequenced [524, 525]; in addition, the protein that is the building block of the neurofibrillary tangles has now been sequenced. Such knowledge has obviously led to a focused concentration on these phenomena. Indeed, at first it was believed that these plaques and tangles were the prime causes of the disease. Recent studies, however, point to another possible cause. In this chapter we summarize our progress in understanding AD. Although our understanding of the disease is still developing, the mechanisms causing AD are better understood.

Introduction

Membrane proteins play a significant role in biological processes, including biological energy conversion and photosynthesis. They also can act as signal receptors and transmitters of hormones, light, or chemicals. However, there is far less structural information about these proteins than about globular proteins. A major factor is their insolubility in aqueous solutions. As a consequence, far fewer membrane protein structures are known than for globular proteins; the number of successfully crystallized membrane proteins is well below 100.

Membrane proteins are different from globular proteins in one main respect: the former are usually “anchored” to a membrane lipid layer. Two predominant types of interactions are responsible for the overall structure observed in a membrane: hydrophobic and hydrophilic interactions. Other non-covalent interactions are present as well, such as hydrogen bonding and electrostatic interactions. However, hydrophobic and hydrophilic interactions are the main forces responsible for the overall structure of a membrane and its attached proteins [350]. Hydrophobic interactions are responsible for keeping a non-polar object away from polar groups such as water. The well known example of the immiscibility of hydrocarbons (i.e. oil) and water applies. An object is termed hydrophobic if it prefers to be oriented away from water. More specifically, it would require a significant expenditure of free energy to transfer a hydrophobic particle from a non-polar environment to a polar one.

The fact that interactions and bonds play an essential role in physical phenomena can hardly be disputed. Their role in the social world is no less important. The parallel between these two classes of phenomena can be illustrated by two key episodes: the formation of stars and the emergence of agrarian-based societies.

• In the early universe there were molecular clouds composed of low density hydrogen and helium gas. According to current conceptions of star formation, regions where density was high enough (as a result of random fluctuations) became gravitationally unstable and began to collapse. The gravitational collapse released a lot of heat and energy in the same way as a stretched spring releases its potential energy as it resumes its equilibrium length. Part of the energy was radiated but the remainder increased the temperature of the core until fusion ignition occurred. This marked the birth of a new star. If its mass was greater than five solar masses it had a fairly short life time. After several million years its core began to contract and as it shrank it grew hotter; this triggered a new series of nuclear reactions leading to the formation of heavier elements up to iron. Eventually, through a mechanism which is not yet clearly understood, the star exploded. It is known that elements heavier than iron were formed during this supernova explosion. Through its disintegration, all elements contained in the star were released into space to serve as raw material for new stars, planets and living creatures.

• […]

It would be a waste of time and energy to analyze and try to make sense of flawed or unreliable data. Ensuring the reliability of experimental data is a cornerstone of the natural sciences. In physics there is a proven procedure through which the validity of experimental results is established. Every time researchers claim to have observed a new physical phenomenon other teams around the world try to replicate the results. If their observations are consistent with those obtained by the first team, the phenomenon will be given the status of a new physical effect, often named after its discoverers. The Foucault pendulum (1851), the Zeeman effect (1902), and the Aharonov–Bohm effect (1959) are a few examples among many. On the contrary, if the claimed result cannot be replicated, the physics community comes to the conclusion that the first observation was spurious. A case in point was the flawed discovery of cold fusion in 1989, which will be described in some detail below. In what follows, this phase of verification will be referred to as the replication procedure. It provides an efficient way through which flawed observations can be eliminated. Unfortunately, there is no similar procedure in the social sciences even in cases where replication would be possible. In a previous chapter we mentioned the fact that thirty years after its discovery the reality of the Werther effect is still not clearly established.

Whenever the signal to noise ratio is smaller than one, identification is likely to be unclear, unconvincing and open to discussion. For instance, it is because the signal in the Werther effect is of the same magnitude as the noise that the very existence of this effect is still a matter of debate thirty years after Phillips's pioneering paper. Similarly, many variables which are of central importance in economics, e.g. the elasticities of commodity prices with respect to supply or demand, are not known with a precision better than 30% or 50%. Other figures regarding the accuracy of economic data can be found in Morgenstern (1950). Raising the signal to noise ratio is a crucial challenge for the social sciences. In this chapter we describe three methods for improving signal identification and we illustrate them through specific social phenomena.

Before we begin, an additional remark is in order. Signal detection is an important topic in mathematical statistics. Here however, we propose upstream solutions to be used in the design phase of an experiment. Once the data have been recorded the fate of the battle against noise is largely settled. Using one statistical technique rather than another will improve matters only marginally.

The extreme value technique

Suppose we wish to study how the period of a pendulum depends upon the initial angular deviation θ0. If we try initial amplitudes of 5, 10, 15, 20, 25 and 30 degrees the corresponding periods will differ by less than 5%.

In any society, whether human or non-human, there are links and interactions between individual units. A possible method for investigating these links is to consider what happens when they are severed. For instance, what will happen to an ant which is removed from its nest and kept in isolation in a place which offers the same conditions as the nest in terms of food, temperature, humidity and other factors? Will it have a shorter or longer life than the ants which remain in the nest? If, as may be expected, its life is shortened, one would like to know how much it is shortened. Before answering this question (which we do later on in this chapter) it is important to realize its implications. This point can be illustrated through the following gedankenexperiment. Suppose you are slicing bread and the knife slips. Your finger is cut and starts to bleed. You put on a plaster and think nothing more of it. However, some skin cells have been displaced down into the muscle tissue; if they survive and divide they will produce skin cells in a location where no skin cells should exist. Fortunately, these displaced skin cells undergo self-destruction, a mechanism known as apoptosis. Other expressions such as programmed cell death (PCD) or cell suicide (Raff 1998) are also used to designate the mechanism of self-destruction.

In order to establish a connection between our previous question about ants kept in isolation and the phenomenon of apoptosis one may wonder what happens when a skin cell is removed and kept in a test tube.

The chapters in Part II consider different kinds of macro-interactions, by which we mean interactions which take place between states or which act at the global level of societies. What makes the study of macro-interactions comparatively easier than the study of interpersonal micro-interactions is the fact that they are strong, persistent and their effects are often clearly visible. On the contrary, microinteractions are much weaker, mostly of shorter duration and their effects are often masked by those of macro-interactions.

The motivation and rationale for first studying macro-interactions can best be explained by way of a parallel with physics. It is well known that by sorting molecules according to their velocities, a device known as a Maxwell demon would be able to make a system work in violation of the second law of thermodynamics. Naturally, it would be easy to imagine other kinds of Maxwell demons whose intervention would result in the violation of other physical laws. With such demons at work in our laboratories it would be impossible to study physical systems in a meaningful way, at least until their intervention could be identified and discounted. Fortunately, Maxwell demons do not exist in physics, but in the study of social phenomena we must indeed face this kind of difficulty. In our societies there are numerous Maxwell demons who distort and influence social phenomena; if for some reason they cannot tamper with the phenomena themselves, they will try to alter and misrepresent their historical accounts.

In the previous chapter we studied the excess-suicides that arise when a substantial fraction of a population cannot get married. In this and the next chapters we study other mechanisms through which the number of unmarried people is inflated, for instance:

• An increase in the mean age at marriage which results in a greater proportion of unmarried men and women, especially in the 20–30 age group.

• An increase in the rates of divorce or widowhood which results in more people living without a partner.

In sociological studies postponed marriage, divorce and widowhood are usually considered separately. However, seen from a network perspective these are three facets of the same phenomenon, namely the non-establishment or severance of marital bonds. If m, d, w denote the marriage, divorce and widowhood rates respectively, it is possible to define a generalized marriage rate as mg = m – d – w; this rate encapsulates both the establishment of new links and their dissolution through divorce or widowhood. Our objective is to see to what extent these rates are connected with suicide rates. In accordance with the extreme value technique defined in earlier methodological guidelines we first examine three fairly extreme situations: (i) the suicide of young widowers; (ii) the effect of a 200% decline in marriages rates; (iii) the effect of a huge and sudden increase in marriage rate.

Suicide rate of young widowers

It has been known since Emile Durkheim that widows and widowers have a higher suicide rate than married people. Similar observations have been made in many time intervals and countries.

In this chapter and in the next we discuss what we believe is the main difference between physics and the social sciences, namely the high level of noise which spoils and often altogether forbids the measurement of many social variables. This might seem a fairly unconventional view. Usually many other differences are underscored, for instance the role of individual freedom, the diversity and complexity of human behavior and so on. More specifically it is often stated that (i) social systems are inherently more complicated in the sense that societies comprise many different kinds of agents and interactions; (ii) physical phenomena can be observed in the laboratory through experiments which can be repeated whereas social observations, it is argued, cannot be repeated. We do not mean that these differences do not exist but we believe that they are not as fundamental as may be the impression at first sight.

The first point can be readily dealt with by noting that the level of complexity depends on how much detail one wants to take into account. Seen at the molecular level, the dissolution of a piece of sugar in a cup of tea is a phenomenon of horrendous complexity in the sense that it involves many kinds of molecules and a great diversity of interactions. In contrast, for many social phenomena the precise nature of the agents is unimportant. For instance, the field of demography focuses on the numbers of birth and deaths, a perspective in which most other social features become irrelevant.