The use of relational systems theory in chemistry and biology past present and future

When Kedem and Katchalsky [140, 141] introduced non-equilibrium thermodynamics into biology in the late 1950's they were among a handful of others who saw the need for this formalism in the study of living systems. Prior to this, equilibrium thermodynamics had some usefulness in trying to understand the energetics of living systems, but since their homeostatic nature made them much more like stationary states away from equilibrium, the theories for equilibrium were not very helpful.

Linear non-equilibrium thermodynamics itself had more of an impact in the realm of rethinking the conceptual framework for asking and answering energetic questions about living systems. From the very beginning of this rethinking it was clear to people like Katchalsky and Prigogine that the real breakthroughs would come when the formalism could be extended to nonlinear systems. This may be part of the reason why the conceptual insights gained using linear non-equilibrium thermodynamics never became as widely understood as they needed to be. As a result much time and energy goes into our modern discussions of complex systems in order to fill in gaps in the overall picture.

What were some of the conceptual changes that resulted from the non-equilibrium formalism? First and foremost, the entire set of rich conclusions about the nature of coupled systems and the appropriateness of this model for understanding how life was a direct result of the requirements imposed by the second law of thermodynamics. The highly interactive nature of living systems arises because of circumstances that require a response to a steady input of solar energy. The way the system responded to this energy throughput was by organizing better and better ways to keep this energy from becoming accumulated heat. Through coupled processes, the heat flow was channeled through biomass and a cooling effect was achieved which was part of the stabilization of the atmosphere. That atmosphere in turn provided a milieu that could sustain life.

A second realm of advances came for more technical reasons. Kedem and Katchalsky chose the realm of membrane transport for their first round of applications and, in particular, began by showing that without modeling the osmotic transient with a set of coupled equations, there was no way to fit the experimental curve describing the swelling and shrinking back to normal volume of a cell exposed to a slightly hypotonic solution due to permeable solutes. The role of the coupling coefficient, the reflection coefficient, in explaining some of the more difficult osmotic effects in tissue then followed from that.

Today, this information has become an integral part of every physiology text that deals with osmosis in a careful way.

A very useful result was in the application of Curie's Principle to the "relational" explanation for active transport. The simplest representation of active transport using the linear non-equilibrium formalism is as follows:

The linear domain of the system can be modeled phenomenologically by the equations,

where Jr is the flow (rate) f the ATPase reaction, A is the reaction affinity, A/s is the chemical potential difference across the membrane for the solute being actively transported, Js is the solute flow, and the L's are the coupling coefficients. The bold symbols are vectors having both magnitude and direction, while the others are scalars. The chemical reaction flow and force are scalars while the mass transport flow of solute through the membrane and its thermodynamic driving force, the chemical potential difference across the membrane, are vectors. In order for the equations to make mathematical and physical sense, the coupling coefficients must be vectors as well. The dot in Lrs • A/s is the vector dot product that results in a scalar. These "vectorial" coupling coefficients represent something about the structure of the space in which all this is happening. Either that space is asymmetrical so the coupling coefficient may be non- zero or the space is symmetrical and there is no coupling between mass flow and reaction, or, in other words, no active transport.

Curie's original statement could be translated and paraphrased as "Without the breaking of symmetry, nothing happens." It has a rigorous physical manifestation in non-equilibrium thermodynamics in that flows and forces of different tensorial character do not couple in an isotropic space (deGroot and Mazur [115], p. 31-33). In this case we can apply it by stating that the necessary and sufficient condition for active transport is that a chemical reaction takes place in an asymmetric space. The link between the two versions of Curie's principle was solidified by DeSimone and Caplan [142, 143] although it's meaning was understood by Kedem and Katchalsky in the late 1950's. To this day it has had little impact on biologists. Reduc-tionism has dictated a need to see a molecular mechanism to "explain" the phenomenon before they believe that they understand it.

Let us consider a relatively simple man-made experimental system to see what one mechanistic realization of this general principle can look like. Consider two cases of the system consisting of an enzyme imbedded in a membrane separating two solutions. In case a the enzyme is evenly distributed throughout the membrane while in case b it is only in one half. If the enzyme catalyzes the reaction A ^ B and each of the two solutions contains equal concentrations of A and B such that the ratio of concentrations is not the equilibrium ratio, the membrane bound enzyme will catalyze the reaction as A diffuses to enzyme sites in the membrane and B diffuses away. In the symmetric case, the diffusion paths are equal from either bathing solution so that the concentrations of A and B on both sides remain equal to each other even as the concentration of A diminishes and that of B is increased.

In the asymmetric case, the diffusion path through the membrane is much greater on one side than it is on the other. In that case the conversion of A to B one side will lag that on the other side and a gradient of A in one direction and B in the other will be created. Hence a gradient can be created iff the reaction is in an asymmetric space. (The same effect could be achieved by distributing the enzyme evenly throughout the membrane and shrinking the pores on one side). A close examination of every known scheme for active transport, either in theoretical models or in experiments obeys this symmetry/asymmetry principle.

This example shows more than an application of the non-equilibrium thermodynamic formalism. It shows the alternative way of approaching modeling in systems. It demonstrates that this type model can lead to principles which determine the conditions for important processes like active transport to exist. Yet, due to its non-mechanistic, phenomenological character, it does not get the status of an "explanation" among biologists. The new relational biology (Rosen [15-19]) develops this line of thinking further and points the way to a way of understanding the complexity of living systems as distinct from the simple mechanisms that have given a shadow of their magnificent reality.

Complexity theory teaches that real systems have other descriptions that are equally valid along with the classical, reductionist analysis. Network thermodynamics allows us to see some concrete example of what this means. By unifying the non-mechanistic thinking characteristic of thermo-dynamic reasoning with specific mechanistic realizations of very complicated systems it allows us to make the necessary transition from formalisms centered on mechanism to the complimentary formalisms which forsake that mechanistic detail for a way of understanding self-referential context dependent relationships which are at the heart of the real system. Network thermodynamics along with all the other mechanistic tools used in complexity research can never go beyond the scope of the largest model of reductionism, the dynamic system.

This introduction to the formal aspects of Network Thermodynamics has given a few examples of ways in which understanding of systems can be enhanced using Network Thermodynamics as an example of combining some non-mechanistic ideas with specific mechanistic models. The use of Network Thermodynamics was deliberate, but other ways of combining these two distinct ways of looking at systems are now well known in complexity research. What is almost universally ignored is that all these mechanistic approaches do not really deal with system complexity, rather they are a way of handling complication.

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