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by Homology Technology FTWAugust 18th, 2024

**Authors:**

(1) STUART KAUFFMAN;

(2) ANDREA ROL.

Part II. The first Miracle: The emergence of life is an expected phase transition – TAP and RAF.

Part IV. New Observations and Experiments: Is There Life in the Cosmos?

Conclusion and Acknowledgments

In this Part III we show that the emergence of collectively autocatalytic sets is an expected phase transition in chemical reaction networks as the diversity of molecular species in the system increases, and the diversity of the reactions among them increases even faster. We discuss first the phase transition to collective autocatalysis, RAF, in sufficiently rich chemical reaction networks. Then we marry the RAF theory to the TAP theory that yields the increasing diversity of chemical species in which the RAF phase transition must eventually occur. This TAP RAF union is new.

The molecules of life are combinatorial objects made of a diversity of atoms, CHNOPS, Carbon, Hydrogen, Nitrogen, Oxygen, Phosphorus, Sulfur, (24). A molecule comprised of, say 10 atoms, each bonded to one or two other atoms, can be constructed from a diverse set of fragments of that molecule. This diverse set is the set of substrates to reactions by which it can be formed from its components.

A simple example is a linear polymer of two building blocks A and B. An example is (ABBABAABBA). This polymer has ten building blocks. This single polymer can be constructed in 9 different ways, as is easily seen by breaking any one of the 9 bonds between adjacent building blocks. As the combinatorial complexity of molecules in a system increases, the ratio of reactions, R, to molecules, M, R/M, in the total reaction system increases.

A chemical reaction graph is a “bipartite graph” consisting of dots representing the species of molecules, and boxes representing the different reactions by which molecules transform into each other. Arrows point from the dots representing the substrates of a reaction into the reaction box. Arrows point from the reaction box to the products of the reaction. This graph represents the structure of the chemical reaction network, not the thermodynamic direction of flow that depends upon the displacement from chemical equilibrium, (3, 15).

There is a critical next step: If we knew which molecules catalyzed which reaction, we could represent this by a dotted arrow from the relevant catalyst molecule to the reaction it catalyzed. This structure is called a bipartite hypergraph, see figure 1b. Given a specific bipartite hypergraph it can be examined to see if it contains a collectively autocatalytic set. The reaction system in Figure 1b is an example of such a bipartite hypergraph that contains a collectively autocatalytic set, (3).

In general, we do not know which molecule catalyzes which reaction. In the absence of specific knowledge, theory and insight can proceed by the naïve assumption that each molecule has some probability, Pcat, to catalyze each reaction. This simple assumption already leads to remarkable results. Collectively autocatalytic sets arise as a first order phase transition as the diversity of molecules and reactions among them increase. We see why next, (3, 15,25).

**Random Graph Theory**

In 1959, two mathematicians, Erdos and Renyi, published a seminal paper on the properties of random graphs, (26). A graph is a set of dots or nodes or vertices. Each dot may be connected by a line to no other dots, one other dot, or some number of other dots.

Erdos and Renyi asked a wonderful question: Start with N nodes. Randomly pick up a pair of nodes and connect them by a line. Iteratively keep picking up random pairs of nodes and connecting them with lines.

Let N be the number of Nodes. At any step in this process let the number of lines connecting nodes be L. Consider the ratio: L/N. What happens to the graph as L increases for fixed N?

Magic happens when the ratio of L/N increases to 0.5. Suddenly a giant connected component, or web, arises. In such a giant component, each node is directly or indirectly connected to all the other nodes in that giant component. The ratio L/N = 0.5 is a first order phase transition. Given a fixed number of nodes, N, as L increases and passes L/N = 0.5, almost certainly a giant connected component arises.

**The Emergence of Molecular Reproduction as a First Order Phase Transition**

The emergence of collectively autocatalytic sets, also called RAFs, arises as the same phase transition in bipartite chemical reaction hypergraphs. Consider a given bipartite chemical reaction graph. Consider increasing the probability, Pcat, that any molecule catalyzes any given reaction. For each value Pcat assign at random according to Pcat which molecules catalyze which reactions. Does the system contain a collectively autocatalytic set? At some value of Pcat so many reactions are catalyzed that they form a giant component that is now collectively autocatalytic, (3,15,26).

This is precisely the first-order phase transition to molecular reproduction in sufficiently rich non-equilibrium chemical reaction systems.

More importantly, keep Pcat constant and increase the number and atomic complexity of the molecules in the system. The ratio of reactions to molecules, R/M, must increase. At some complexity of the molecules in the system, and the ratio R/M, a collectively autocatalytic set, an RAF, will emerge with probability approaching 1.0, (3). This is the essential first-order phase transition by which self-reproducing molecular systems such as the small molecule collectively autocatalytic sets in all 6700 prokaryotes can have arisen.

**The History of an Idea**

One of us, (14), created the first model demonstrating the spontaneous emergence of collectively autocatalytic sets in 1971. Two publications emerged in 1986, one by Kauffman alone with theorems demonstrating that such an emergence was expected, (15). The second by Farmer, Packard and Kauffman, (3), implemented a code based on a “binary polymer model”. Each model polymer was comprised of two monomers, A and B. The maximum length of a polymer, L, was fixed. All 2^L possible polymers could be part of the system. The reaction system was sustained by a “food set” consisting in a subset of the possible polymers. Only cleavage and condensation reactions were allowed. Each polymer had the same fixed probability, Pcat, to catalyze each reaction.

The results were convincing: For a fixed Pcat, as the length of the longest polymer allowed, L, increased, so the diversity of potential polymers increases as did the R/M ratio, collectively autocatalytic sets arose with probability approaching 1.0. Figure 1b is one such collectively autocatalytic set.

Extensive work over the next decades, (27,28, 29), has demonstrated: i. The rules assigning at random which polymer catalyzes which reaction can be uniform or power law. This makes little difference. ii. Models in which sequence recognition is included make little difference. iii. The number of reactions each polymer must catalyze is between 1 and 2. This is chemically plausible. iv. Importantly, Hordijk and Steel found that such collectively autocatalytic sets contain “irreducible autocatalytic sets” that jointly form more complex systems of many such irreducible sets, (28, 29). v. Vassas et al. pointed out that such irreducible autocatalytic sets, each able to function as an independent replicator, could function as “genes” that could be inherited and partitioned to daughter sets. Thus, such simple systems with no “genome” can evolve, (29). Such an exchange inheritance of irreducible autocatalytic sets almost surely played a large role in the early evolution and phylogeny of the small molecule autocatalytic sets now found in all 6700 prokaryotes, (11,12). Irreducible autocatalytic sets can be identified computationally. Therefore, such an evolution of metabolism is now open to detailed study

**The Expected Emergence of Molecular Reproduction in the Evolving Universe**

A first-order phase transition to molecular reproduction is expected in the chemical evolution of the universe where the diversity and complexity of molecules increased. At the earliest stage there were the fundamental particles, quarks, gluons, electrons, positrons. As the universe cooled, hadrons formed. Then the first elements, hydrogen, beryllium, formed. Later, in supernovae, the rest of the 98 stable atoms formed, (30).

The emergence of simple then ever-more complex molecules followed the same pattern from simple molecules and low diversity upward. The diversity, atomic complexity of the molecules, and the potential reactions among them increased, (31). The Murchison meteorite, formed five billion years ago with our solar system, has hundreds of thousands of molecular species and potential reactions among them, (32).

The theory we discuss here predicts that at some sufficient diversity of molecular species, M, and reactions among them, R, hence a sufficiently large R/M ratio, collectively autocatalytic sets will emerge as a first order phase transition.

**The New Mathematical Theory – TAP and RAF**

The theory of the emergence of collectively autocatalytic sets, RAFs, is well established, (15,16,25,27,28,29). We here marry that RAF theory to an independent theory, TAP, (33,34,35), that can explain the increasing diversity of molecular species in the evolving universe.

**The TAP Equation**

The TAP equation and its behavior are shown in Figure 4.

The dynamical properties of this simple equation are now well studied numerically and with theorems, (33,34,35). The properties are remarkable. The system is a discrete dynamical system in which molecules can combine with each other to form new molecules. If the current number of molecules is Mt, then for each subset of size i chosen among the Mt, (Mt choose i), the equation combines them to create a new molecule with a probability alpha ^ i, for 0 < alpha < 1.0.

If the system starts with a small number of beginning molecules, each of which can combine with copies of itself or other molecules to make new molecules, over time the number of kinds of molecules increases slowly then explodes upward hyperbolically and reaches infinity in a finite time, (33,34 35), Figure 4.

The TAP process is a crude model of the increasing chemical diversity of the universe, (31,32).

We now unite TAP and RAF. This union hopes to explain the expected emergence of collectively autocatalytic systems as a first order phase transition in the evolving universe. The simple step is to allow each molecule in TAP to catalyze each reaction in TAP at random with a fixed probability Pcat. The first results with respect to technological evolution were just published, (36). As time passes, the diversity of entities increases, then the first order phase transition to the emergence of collectively autocatalytic sets arises with probability 1.0, (36).

This united TAP-RAF theory demonstrates a basic truth. In the chemical evolution of the universe, molecular diversity increased by some analogue of the TAP process. As this occurred, the complexity of molecules increased, thus the number of reactions increased as did the ratio of reactions, R to molecules, M, R/M. But the same set molecules, M, are candidates to catalyze the same set of reactions, R, among the set of M molecules. Therefore, with any rough probability of catalysis, assigned among the molecules uniformly, as a power law, or otherwise, at some point the first order phase transition arises. Molecular reproduction is expected to arise in the evolving universe.

**The Spontaneous Emergence of Élan Vital – Evolving Life**

Any such non-equilibrium reproducing molecular reaction system is a Kantian Whole that achieves Catalytic Closure. In addition, the molecules that serve as catalysts are, themselves, boundary condition constraints of the release of energy in the specific reactions they catalyze. Thus, the system constructs its own boundary condition constraints on the release of energy that constructs the same boundary condition constraints. The system achieves Constraint Closure. We add some form of enclosure, for example, in a tiny pocket in a hydrothermal vent, or better, in a liposome whose lipids are synthesized by the same system, (37,38).

This newly recognized union of four closures, Kantian Whole, Catalytic Closure, Constraint Closure, and Spatial Closure is Bergson’s mysterious élan vital, (39), here rendered entirely non-mysterious. These four conditions constitute life. Life arises as an expected phase transition in the evolving universe.

We note parallels to recent work by R. Hazen and colleagues suggesting a new law for the emergence of functional diversity in the abiotic world, (40). The authors consider a set of stable objects, such as a diversity of molecules, and stable transformations among them, such as chemical reactions among the molecules, and point out that complex systems such as a high diversity of abiotic minerals form. “Function” is defined for features of minerals. These authors do not consider the emergence of Kantian Wholes that are collectively autocatalytic sets achieving catalytic closure and constraint closure, the emergence of evolving life. Here, “function” is defined in terms of how Parts evolve to sustain the evolving Whole.

This paper is available on arxiv under CC BY 4.0 DEED license.

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