Take a large piece of material and measure some of its macroscopic properties, for example its density, compressibility or magnetisation. Now divide it into two roughly equal halves, keeping the external variables like pressure and temperature the same. The macroscopic properties of each piece will then be the same as those of the whole. The same holds true if the process is repeated. But eventually, after many iterations, something different must happen, because we know that matter is made up of atoms and molecules whose individual properties are quite different from those of the matter which they constitute. The length scale at which the overall properties of the pieces begin to differ markedly from those of the original gives a measure of what is termed the correlation length of the material. It is the distance over which the fluctuations of the microscopic degrees of freedom (the positions of the atoms and suchlike) are significantly correlated with each other. The fluctuations in two parts of the material much further apart than the correlation length are effectively disconnected from each other. Therefore it makes no appreciable difference to the macroscopic properties if the connection is completely severed.

Usually the correlation length is of the order of a few inter-atomic spacings. This means that we may consider really quite small collections of atoms to get a very good idea of the macroscopic behaviour of the material.

Quenched and annealed disorder

The systems discussed so far have been assumed to be homogeneous. Any real system will inevitably contain impurities. In most circumstances, one tries to eliminate them, but, under well controlled conditions, it is also interesting to study their effect on critical behaviour. In general, one would expect any kind of random in homogeneities to tend to disorder the system, and thus to lower the critical temperature. In fact, under certain circumstances, randomness may completely eliminate the ordered phase. Under other conditions, it is still possible for the system to order, but the universality class of the critical behaviour may be modified.

The first point to be made concerns the important distinction between annealed and quenched disorder. As a concrete example, suppose that we substitute some non-magnetic impurity atoms into a lattice of magnetic ions. The way we might do this is to mix some fraction of impurities into the melt, and let the system crystallise by cooling. If we allow this to happen very slowly, the impurities and the magnetic ions will remain in thermal equilibrium with each other, and the resulting distribution of impurities will be Gibbsian, governed by the final temperature and the various interactions between the different kinds of atom. Such a distribution of impurities is called annealed.

In this chapter the basic concepts of the modern approach to equilibrium critical behaviour, conventionally grouped under the title ‘renormalization group’, are introduced. This terminology is rather unfortunate. The mathematical structure of the procedure, in the sense that it may be said to have any rigorous underpinnings, is certainly not that of a group. Neither is renormalization in quantum field theory an essential element, although it has an intimate connection with some formulations of the renormalization group. In fact, the renormalization group framework may be applied to problems quite unrelated to field theory. The origins of the name may be traced to the particle physics of the 1960s, when it was optimistically hoped that everything in fundamental physics might be explained in terms of symmetries and group theory, rather than dynamics. One of the earliest applications of renormalization group ideas, in fact, was to the rather esoteric subject of the high energy behaviour of renormalized quantum electrodynamics. It took the vision of K. Wilson to realise that these methods had a far wider field of application in the scaling theory of critical phenomena that was being formulated by Fisher, Kadanoff and others in the latter part of the decade. By then, however, the name had become firmly attached to the subject.

Not only are the words ‘renormalization’ and ‘group’ examples of unfortunate terminology, the use of the definite article ‘the’ which usually precedes them is even more confusing.

Bounded and unbounded potentials

A molecular system is one in which the atoms are bound together by chemical bonds between specific pairs of atoms. To study the stability of molecular systems we have to study the stability of these individual chemical bonds. Often the bonds are not represented as a complex of electron orbitals but, rather, represented as attractive potentials between pairs of atoms that are functions only of interatom distance. All realistic models of these potentials are bounded. That is, if the two atoms involved are separated to large enough distances the interaction energy goes to zero, or depending on where one puts the origin in potential, to some finite value. Atoms are assumed to become independent of one another when separated by a large enough distance. In particular a bounded interaction potential does not go to infinity as the separation increases; if that happened atoms would be bound together the way quarks are and could never be found separated.

If one applies standard statistical mechanics to a molecule bound by such bounded potentials one always gets the result that the equilibrium state of the system is the dissociated state. In other words, the molecule is always melted. This is true at all temperatures, even infinitesimal temperatures, and is true for both classical and quantum analysis. In studying melting by statistical mechanics of systems with realistic potentials one is dealing with the contradictory situation of trying to study the melting of a molecule that is already melted.

Dynamics of conformation change

Biological macromolecules often undergo changes in shape carrying out their biological function. For example the action of muscle fibers involves a relative rotation of a section of a macromolecule. An important change in shape in DNA is the B to A conformation change that is caused by changes in the level of hydration of the helix. The A conformation is the low hydration form. In that transition no valence bonds, and likely no H-bonds, are even transiently disrupted. The change is mostly to the helicity of the helix, going from a complete turn in 10 base pairs to one in 11 base pairs. In the process the width of the helix increases, the length decreases, and the base pairs tilt away from the perpendicular to the helix axis (Saenger, 1984). Analysis of this transition has been carried out in the context of a soft mode displacive change transition (Eyster and Prohofsky, 1977; Lindsay et al., 1985). Displacive change is useful for transitions that involve continuous movement of atoms from one conformation to another, i.e. where the transition is second order. The mode that goes soft, in soft mode analysis, is related to the softening of the free energy barrier that separates the free energy minima as a function of order parameter, as discussed in a previous chapter.

Choice of methods

As discussed in earlier chapters, several advantages point to the use of an effective phonon approach to study dynamics in a large nonlinear system such as the DNA double helix. The first advantage is that in the effective phonon formulation unbounded interactions replace bounded interactions and this change allows one to apply statistical mechanical ensemble theory to the system without running into problems associated with infinite thermodynamic integrals. The second advantage is that the selfconsistent phonon approach divides the large nonlinear dynamics problem into two parts, 1) the incorporation of nonlinearities into an effective force constant, and 2) the solution of the large number of degrees of freedom by normal mode means. A third implicit advantage is related to the built in harmonic approximation, i.e. the advantage of not having to solve for the ground state conformation before being able to proceed to the dynamic solutions. All these theoretical and calculational advantages are present in the approach developed.

In addition to the theoretical advantages discussed above there is a very practical reason to formulate the dynamics as an effective phonon problem. Direct experimental observation of the vibrational excitations of double helices by Raman scattering and infrared absorption indicate that they are resonant rather than relaxational. The lines are broad but not so broad as to indicate that one is observing relaxational modes.

Biologically significant DNA

Most biologically significant DNA helices are very long, so long that it is a useful approximation to assume infinite length. Helical lattice dynamics can then be used for analysis without having to resort to large dimensional calculations if the systems have a repeating base sequence symmetry. The studies in earlier chapters dealt only with DNA which had repeating symmetry. One can learn much about the dynamics of native DNA from a study of repeating DNA because the polymer DNAs share much of the dynamics, and often bracket the behavior of native DNAs. For example, the melting temperature of native DNA falls between that of poly(dG)– poly(dC) and poly (dA–dT)–poly(dA–dT). The study of repeating DNA is, however, a study of the material science of the material DNA. It deals with ‘perfect DNA’ and not with the complex broken symmetry material of biological significance.

The departures from symmetry are of great importance as biological information is contained in them. A biologically oriented study then requires methods that can deal with departures from repeating symmetry but can still be applied to very long DNA. In this chapter we develop methods useful with symmetry breaking structures. We get around the difficulties of dealing with large or infinite systems by starting with initial repeat polymer solutions and applying the new methods to achieve appropriate solutions. The approach was initially developed to deal with defects in crystals, the mathematical method is a Green function approach that is detailed in Appendix 3.

Greater helix

The study of the dynamics of a helix with elements, such as a drug, attached to it is easiest when there is an excess of the element so that it forms a repeating pattern of attachment. This allows the use of lattice methods to solve the large dimensional problem associated with the long helix. When only one element (or a few elements) attaches to a long helix the study is best handled by methods developed in Chapters 11 and 13. One of the more extensively studied attaching drugs is daunomycin, probably because it is an important antitumor agent. It is one of a number of drugs that intercalate into the helix, i.e. part of it has a planar structure that enters between base pairs of the helix. The specific example studied is the repeating unit of daunomycin–poly(GCAT)–poly(ATGC) (Chen and Prohofsky, 1994). The calculation determines the probability of the drug dissociating from the DNA helix which can be converted to give the binding constant of the drug to the helix. The choice of the particular DNA sequence was dictated by the availability of X-ray conformational information (Wang et al., 1987). The helix is somewhat distorted by the intercalation and the calculation uses the distorted conformation and the correct daunomycin position. The bases are numbered in the unit cell by calling the first guanine G1 then proceeding down one strand in the 3′ to 5′ direction so that the next cytosine is C2, then comes A3 and T4.

Different time scales in bond dynamics

The difference in mass between electrons and atoms makes changes in atom displacements occur on a longer time scale than electronic transitions. If the atoms move, the electron orbitals adjust to the new distances on a time scale fast compared to that of the atom motion. If bonded at a particular separation where the bonded orbitals are not energetically favorable, the electrons will undergo a transition and the bond will dissociate. If not bonded at a particular separation where the orbitals are energetically favorable the bond will form. When the atom motion separates the atoms to the point where the bonded orbitals are unfavored the bond will break on a time scale fast compared to atom motion. The lasting change in bond status is on the slowest time scale and is determined by interatom distances.

The hydrogen bond involves three atoms, the hydrogen atom and the two more massive end atoms which in DNA are attached by valence bonds to the rest of the bases. This further increases their effective inertia. Because the hydrogen atom has much less mass than the end atoms there are three distinct time scales involved in H-bond dynamics.

Separate element approach

Biological processes on the helix are often carried out by specialized molecules that attach to the helix. The attachment is often soft, by nonbonded and H-bond interactions rather than by valence bonding. The nonbonded interactions by themselves tend to cause a less specific attachment to the helix, i.e. they allow movement along the helix and are not necessarily to particular positions on the helix. The more specific interactions to particular sites seem to involve specific H-bond interactions. To a large extent the attached molecules and the helix retain their individual identities in the sense that they are strongly bound internally by valence bonds but more loosely coupled to each other by H-bonds and nonbonded interactions. The dynamics of such a system can be efficiently studied by the splice methods developed in the previous chapter. No calculations of the interaction of a whole molecule with a helix have been carried out as yet except for the case developed in Chapter 10. This discussion will describe a partial calculation of greater generality which explores the effect of some parts of a whole molecule interacting with a helix and displays how calculations with a more complex whole molecule may be done.

The method discussed here is complimentary to the calculations in Chapter 10. There one assumed that a large number of the molecules were attached to the helix in a way that retained overall helical symmetry.

Source of energy

There is a direction to both replication and transcription that is not random. Both processes involve a complex of enzymes that move along the helix generating a copy of the information contained in the helix base sequence. One would expect, based on our understanding of the second law of thermodynamics, that these operations would require some kind of thermodynamic engine that would irreversibly burn energy and create entropy to allow the directed work to occur. The most common analysis of second law engines is in terms of Carnot engines that take in energy, do work, and exhaust energy which is the source of the increase in entropy. One question that could be asked is, to what extent can these local biological processes carried out by specific molecular complexes be described in terms of Carnot engines? Whether or not the process is Carnot-like, the investigation of processes driven by a flow of energy is an important problem and very little studied to date.

The complexes that do the work are made up only of macromolecules and a molecular engine has to be described differently than a macroscopic engine. Macro engines are classical systems that allow continuous heat flows into and out of the engine. Molecular systems are quantum systems in which the nearest equivalent to heat inflow is the absorption of particular excitations. The heat storage in a molecule is in terms of excitation levels of quantized modes of the system.

First and second order transitions

First order transitions occur between two distinct phases of a system. One phase is observed until a critical value of control parameters, such as temperature or pressure etc., at which the two phases coexist, and the system converts to the other phase for further changes in the control parameter. In some cases the system can display superheating or supercooling where the system can be coaxed into remaining in the wrong (nonequilibrium) phase past the value of the control parameter where the transition should have occurred. At the point where the two phases coexist one can experimentally observe an interface between the two regions of different phase. The change in phase is obvious; one can tell which side of the interface has which phase. Some intrinsic parameters of the system, therefore, have to undergo a discontinuous change. For example in the liquid–gas transition, the liquid is compact whereas the gas fills the available space. The density changes drastically and discontinuously with phase. A parameter, like the density for the liquid–gas transition, that unambiguously defines a phase, is called an order parameter. The name comes from studying ferromagnetic transitions where the alignment or order of the spins defines the phase.

Second order transitions don't have discontinuities in a primary parameter. They only have discontinuities in derivatives of primary observables, such as the specific heat, and that leads to the name second order.

Dealing with large macromolecules

Many problems of biological interest have to do with large macromolecules binding to other large macromolecules, giving rise to even larger systems. It would be an advantage if previous solutions of the dynamics of the separate macromolecules could be used in finding the dynamics of the combined macromolecules, rather than having to start each solution from scratch. This chapter discusses ways to use Green functions to determine the dynamics of a system that is made up of parts. Each of the parts can be analyzed separately and its spectrum compared to infrared and Raman observations allowing a refinement of the smaller problem. The dynamics of large macromolecules can then be constructed by combining dynamics of smaller molecules in much the same way as the actual molecules could be formed by chemically joining separate parts.

The scheme can also work for infinite systems whose separate parts have a symmetry that the combined system doesn't have. An example is the fork calculation introduced in the last chapter. The fork is the place where a section of double helical DNA is split into two single strands. Symmetry is broken by the fact that one half is double helical and the other half is single strands; the problem can't be reduced to block diagonalized finite secular matrices. Each separate part, extended in both directions, does have the proper symmetry.