Table of Contents
Life and persistence
Function and Metabolism
The Thought Experiment
Life as a dynamic system
What is catalysis?
What are solitons?
Solitons in biology
Scale invariance in biology
Structure, energy, unity and resonance
Application of catalysis 1
Application of catalysis 2
Life as catalysis
Ontology of consciousness
Fractal catalysis and autopoiesis 1
Fractal catalysis and autopoiesis 2
What are soliton waves?
The reason for examining solitons in greater detail is that they are not only thought to play a role in enzyme catalysis, but are also thought to play a role in catalysis at the level of cell membranes (Sataric, Zakula, Ivik & Tuszynski, 1991), muscle function (Davydov, 1982) and, as will be discussed, in numerous other biological processes from DNA (Yakushevich 2001) to the action potential of a nerve cell (Aslanidi & Mornev 1996). The diverse biological processes that are thought to involve solitons lends weight to the proposition that there may be a common theme to all living processes.
Soliton waves (originally identified by J. Scott Russell in 1884) are non-dissipative waves that occur at the boundary between differing tendencies of waves. Thus, in water, the mathematics used to describe low amplitude waves are relatively simple -- linear in fact. Low amplitude waves have a tendency to dissipate, as when we drop a pebble in a pond and observe the waves spreading out. Also, the various frequencies that might comprise a low amplitude wave will gradually separate as a result of the different speeds at which they travel. High amplitude waves, on the other hand, behave non-linearly. They have a tendency to compress and cause criticality, white water, turbulence. However, just like low amplitude waves, the result is a rapid dissipation of energy and structure.
Right at the boundary between these two tendencies we find soliton waves. Examples of soliton waves include tsunami and certain types of vortices; Jupiter's Red Spot may well be a soliton wave. What makes these waves so interesting is their robustness. A tsunami may travel the length of the Pacific Ocean with relatively little dissipation. Soliton waves are practically frictionless. It seems that at the boundary the non-dissipative (or compressive) tendency of high amplitude waves exactly cancels out the dissipative tendency of low amplitude waves. Also, solitons can be found in a number of mediums. If physicists exactly balance two properties of light - refraction and diffraction, for example, solitons may be formed.
The robustness of solitons cannot be solely attributed to the nature of the waves themselves. There is the problem of context to be considered. A soliton was first observed in a canal. A canal is highly ordered - it has a constant width and depth. This feature of canals makes them highly favorable for the possibility of solitons occurring. Soliton waves are themselves highly structured, they embody a great deal of symmetry. Because the bottom and edges of the canal form the boundary conditions for the soliton, it is necessary that they embody a high degree of symmetry also. Thus, there is a strong relationship between the robustness of solitons and the degree of order or symmetry in the medium and boundary conditions that form their environment. In the case of solitons on membranes, there is a necessary relationship between the possibility of solitons existing and types of topological transformations which leave key geometric features unchanged (Rogers & Shadwick, 1982, p 107-110, also Rogers & Schief, 2002). Also, Lie group transformations are often used to find soliton solutions and are characterised by the fact that they involve invariance under transformation. Incidentally, Lie group transformations are used by computer scientists to design computer 'vision' systems.
The regular crystalline structure of DNA and the uniform diameter of microtubules are examples of invariance and they determine the particular solutions of solitons that can exist. As we shall see, the persistence of solitons within structured environments may well help us to understand the relationship between 'life' and the 'order' or symmetry in its environment.
This may be extremely significant. If we consider a biological structure such as DNA and consider the problem of providing energy to do work at particular locations, it would seem, at first glance, that we would require a complex external system to achieve this. However, if solitons utilise the 'order' or symmetry in a biological structures, then the structure itself may be providing the means to deliver the energy where it is required. So, the evolution of biological structures such as DNA or microtubules cannot be considered in isolation from the solitonic wave forms that they support. Conversely, if we choose to consider the evolution of life as the evolution of solitons then these cannot be considered in isolation from the material structures that support them. Life, I suggest, involves a necessary relationship between matter and energy in the form of soliton waves, such that the stability or robustness of one is dependent upon the other. As we shall see later, the brain may be a beautiful example of this synthesis.
The basic principles of cell chemistry may be significant in terms of the possibility of solitonic mechanisms also. As has been discussed, solitons may be formed at the boundary between linear and non-linear behaviour. Enzymes in the cell fall into two main groups - allestoric and non-allestoric. In non-allestoric enzyme catalysis the rate of the catalysed reaction goes up linearly with the concentration of the reagents. Conversely, allestoric enzyme catalysis gives rise to a non-linear relationship between the rate of the catalysed reaction and the concentration of reagents. The cell may indeed embody a fine balance between linear and non-linear tendencies necessary for soliton formation. Additionally, there are two further sub-groups of catalysed reaction in cell chemistry - catabolic and anabolic. Catabolic chemistry breaks large molecules into smaller molecules and anabolic chemistry builds larger molecules from smaller ones. These two aspects of cell chemistry must be finely balanced in the healthy cell. The question is - 'Might the balance between these two tendencies also give rise to solitons?'
An intriguing question arises about the possible relationship between the robust qualities of the enzyme and the robust qualities of the soliton. The soliton is not independent of the enzyme; it is expressed as movement amongst the atoms that comprise it. If solitons are indeed a principle agent of the process of catalysis then we have discovered an interesting concurrence of robustness in two seemingly different physical domains. On the one hand we have the robustness of the catalyst when considered as a physical entity, and on the other, we have the robustness of an energy wave form. I think it highly unlikely that this is mere coincidence.