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.

The harmonic approximation

Condensed matter physics made great progress by concentrating on excitations and avoiding the problem of ab initio solution of the conformation of crystals. The conformation or ground state problem is the determination of the static equilibrium structure of a system and is a difficult problem. For most crystals the ground state problem is still an unsolved theoretical problem. An example of progress without first solving for the ground state was the development of phonon theories for determining the motion of the atoms of a solid. Phonon is the term used to describe quantized propagating normal mode vibrational excitations and is used in analogy to photon, which describes the quantized propagating excitations of the electromagnetic field. Propagation of excitations is an essential element of large systems. Phonon theories apply not only to equilibrium phenomena such as the study of internal energy and specific heats but also to dynamic problems such as electrical and thermal conductivity. The study of excitations with the ability to propagate is an improvement over simple vibrational mode studies that only deal with static equilibrated systems. As we will discuss in later chapters, the propagation of energy along the helix is relevant to a number of biological processes and the distinction between propagating excitations and static excitations will be of importance.

Phonons are the principal excitations found in insulators and are important excitations in all types of condensed matter.

Anomalous stability of poly(dA)–poly(dT)

The four DNA bases come in two sizes. The larger bases, adenine and guanine, are variants of the purine structure which is a double ring structure (Saenger, 1984). The smaller two bases, thymine and cytosine, are pyrimidine variants with a single ring. The base pairing scheme requires one purine and one pyrimidine in each pair making the total span the same for all complementary base pairs. The total molecular mass of the two pairs is almost identical as well. The difference in size makes the overlap of neighbor bases different in homopolymers compared to copolymers. In homopolymers a purine is stacked above another purine and pyrimidines above pyrimidines. The interbase gap undergoes helical twist but tends to be continuous, as seen in Figure 9–6 in Saenger (1984). In the copolymers the gap moves from one side of the twisting center line to the other depending on which side is the purine or which the pyrimidine. This shifting coupled with the helical twist in the copolymers causes considerable overlap of atoms of the large bases from one level to the next in every second base pair. The overlap is between bases on different strands in the copolymer and the van der Waals stacking interactions at the overlap becomes a cross strand interaction. The copolymer has these large interstrand interactions but the homopolymer does not. One would therefore expect that copolymers with this added cross strand interaction would be more stable against strand separation melting than homopolymers.

Dynamics of macromolecules

Biological macromolecules carry out functions that require them to have very complex physical and chemical dynamics. Theoretical analysis of that dynamics has proven very difficult because the macromolecules are large systems whose dynamics is very nonlinear. The large size arises because the systems have many atoms which cannot be simply related by symmetry operations and the dynamics has to be analyzed down to motions on the atomic level to fully appreciate what is going on. This microscopic approach is important because macromolecules often function by changing their conformation on the atomic level. The nonlinearity results because many bonds are weak relative to physiological temperatures, in many biological processes bonds are broken and bonding rearranged. In this book we detail a method for studying macromolecular dynamics that is particularly well suited to large systems that are also highly nonlinear. With an additional operator included the method is particularly useful in studying the dissociation of chemical bonds in large nonlinear systems. We know of no other approach that can efficiently study the melting or bond disruption problem in such large systems on a microscopic scale. To date the majority of the applications of this method, and those presented in this book, are to problems of base pair separation in the DNA double helix. The advantages and disadvantages of the method and how it may be extended to other systems can, however, be seen in these applications.

Significant understanding of biological processes has been made by studying the dynamics of macromolecules at the microscopic level. The bulk of the researchers interested in the results are biochemists, chemists, pharmacists, and biologists. The MSPA approach developed in this book is based on methods used in condensed matter physics which are not familiar outside that discipline. The method is useful in solving long time scale problems in molecular biology that should be of interest to the biochemists etc. working in the field. This book is therefore aimed at presenting both a coherent development of the MSPA methodology and some of the background needed to understand it by persons not coming from a physics background. Biophysicists may find the results and the physics background of interest. To increase the usefulness of the book to the different readers I have, to the extent possible, concentrated on concepts and description in the main text and kept the mathematical formalism in appendices. On the other hand physicists may be interested in the way that physics methodologies have to be altered to be applied to be useful in this new situation. The complexity of biological systems is greater than that usually dealt with in condensed matter investigation and changes in approach are necessary. Physicists may also be interested in the development of a new approach to cooperative transitions that seems to work very well, particularly in large complex systems.

Changes in the ground state problem

As pointed out in Chapter 3, there is a calculational advantage to working in an effective harmonic approximation. All the work discussed so far has used this approximation, i.e. assumed a ground state for each polymer determined from experimental observation. Control variables that could alter that ground state have not been changed, the only control variable allowed to vary is the temperature. Other factors affect DNA melting, such as salt concentration, hydration level, and hydrostatic pressure, and it would be useful to extend the methods developed to study effects of changes in these other control variables. This chapter is about ways to introduce changes in control variables that would normally be thought of as elements affecting the ground state solution. We show that one can selectively put back into the problem particular static or ground state elements, without carrying out a full ground state solution, and determine their effects on the dynamics and melting.

For example, consider increasing the hydrostatic pressure on a system containing double helices. The atoms would be pushed closer together, altering the interatom distances in the ground state. Another example would be changes in salt concentration in the environment of the helix. Altering salt concentration changes the shielding of the Coulomb interaction between the highly charged phosphate groups, which would alter the static equilibrium positions in the helix.

The main focus of our previous discussion of MBE has been the identification of various universality classes. The models we discussed are expected to be valid on a coarse-grained level, at which the exact structure and form of an island does not matter. However, with the perfection of experimental tools, it is possible to observe the interface morphology at the atomic scale – leading to the discovery of rich island morphologies. In this chapter we focus on this early-time morphology, for which the coverage is less than one monolayer; this regime is usually referred to as submonolayer epitaxy.

The phenomenology is quite simple. Start with a flat interface, and deposit atoms with a constant flux. The deposited atoms diffuse on the surface until they meet another atom or the edge of an island, whereupon they stick. Thus if at a given moment we would photograph the surface, we would observe a number of clusters – called islands – with monomers diffusing between them. What is the typical size and number of the islands? What is their morphology? How do these quantities change with the coverage and with the flux? These are among the questions we address.

Model

Let us consider in more detail the deposition process outlined above. Consider a perfectly flat crystal surface with no atoms on it. At time zero we begin to deposit atoms with a constant flux F. Atoms arrive on the surface and diffuse (the deposition and diffusion processes take place simultaneously).

In Chapters 12–15, we discussed in detail the properties of growth processes dominated by deposition and surface diffusion. We saw that one origin of randomness is the stochastic nature of the deposition flux, which generates a nonconservative noise in the growth equations. A second component of randomness on a crystal surface comes from the activated character of the diffusion process. As we show in this chapter, this type of randomness generates a conservative component to the noise, leading to different exponents and universality classes.

While diffusive noise is certainly expected to be present in MBE, there is no experimental evidence for the universality class generated by it. For this reason, we separate it from the discussion of the simple MBE models. This chapter can be considered to be a theoretical undertaking of interest in its own right, investigating the effect of the conservation laws on the universality class.

Conservative noise

An important contribution to randomness on a crystal surface arises from the activated character of the diffusion process. Equation (13.5) results from a deterministic current that contributes to interface smoothing – i.e., only particle motion that aids smoothing is included. But, as discussed in Chapter 12, particle diffusion is an activated process in which all possible moves – each with its own probability – are allowed. Because the nature of the diffusion process is probabilistic, an inherent randomness is always present.