Are complexity and homochirality co-emergent phenomena?

How did life begin? Why are amino acids left-handed (homochiral) in living organisms?

This paper suggests that these questions can be approached by analyzing computational constraints in complex dynamical systems (chemical or not) that are not in equilibrium, using the analogy of dynamic billiards on asymmetry graphs ¹ based on Wolfram ² cellular automata #30 and #110. It is proposed that complexity and homochirality are co-emergent phenomena based on the formulation:

ASYMMETRY + ENTROPY à COMPLEXITY ¹

where:

-‘Asymmetry’ = any scale-free irregularity or non-homogeneity, e.g., pH, chemical concentration, temperature or any other parameter that has a gradient (∇ Φ).
-‘Entropy’ = mixing, randomization or multiplicative noise.
-‘Complexity’ = symmetry breaking, typified computationally by Wolfram cellular automaton #110 and dynamic billiards on its A-graph, #110.

Miller’s ³ 1952 in vitro experiment demonstrated the abiotic production of amino acids in a putative early-Earth environment. Twenty amino acids in a 50:50, left and right-handed racemic mixture of stereoisomers (enantiomers), were produced by subjecting an oxygen-free mixture of water, methane, hydrogen, and ammonia to an electrical discharge. This was an important result but failed to explain why only Left-handed amino acids are associated with life. Miller’s experiment resulted in racemic mixtures, indicating that the chemical reactions in his setup were symmetric, i.e. taking place in a system in thermodynamic equilibrium. Because Gibbs free energy, Δ G = Δ H – T Δ S , ( G—change in Gibbs free energy, H—change in enthalpy, T—temperature, S— change in entropy) is the same for both left and right enantiomers, only racemic mixtures are possible.

Frank’s ⁴ amplification theory proposed that homochirality might have resulted from autocatalysis plus cross-chiral inhibition with an initial slight excess of one enantiomer (due to circularly polarized light, beta-decay, etc.). However, it has been suggested that natural racemization of these asymmetries occurs faster than the time required for the development of life, and the early prebiotic existence of both autocatalysis and cross-chiral inhibition is unlikely.

Several enantioselective chemical schemes have been developed to produce one stereoisomer for use as a bioactive product or drug. Phocomelia and other fetal abnormalities due to Thalidomide made it clear that one or the other enantiomer could be helpful or toxic. None of these clever methods has clarified the problem of the origin of life or why amino acids are left homochiral. During the past 50 years or so ^{5, 6}, it has become apparent that symmetry breaking associated with the development of complexity depends on systems that are not in thermodynamic equilibrium—‘far-from-equilibrium thermodynamically.’ It has been shown that homochirality can be produced in a chemical system maintained far-from-equilibrium thermodynamically if there is a continual external energy input to the system and a continual decay of that energy. Generalized, abstract mathematical theories known as artificial chemistry have been helpful. This paper proposes an abstract geometrical theory known as geometric chemistry, based on dynamical billiards on asymmetry graphs derived from Wolfram cellular automata.

It has been suggested that ⁶ homochirality can occur in a ‘noisy’ thermodynamic system in which there is a:

- constant energy input into the system and;
- an equal energy drain from the system, so that;
- the system is maintained in a far-from-equilibrium state.

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Figure 2

Figure 1 illustrates that the same amino acid has two enantiomers (stereoisomers), represented by left and right hands. The orientation of the molecules about the carbon center determines its chirality (handedness). Figure 2 shows a simplified graphic form of Figure 1. Figure 2 relates the chirality to the order (clockwise or counterclockwise) of the arrangement of the atomic numbers of the molecules covalently bonded to the central carbon atom. Curl the fingers of the left hand with the thumb pointing straight up. Then the atomic numbers of the molecules added to the apices of a carbon-centered tetrahedral molecule starting with the highest atomic number at the knuckle (metacarpophalangeal joint), then going clockwise as the atomic number of the molecules decreases towards the fingertip, with the lowest atomic number at the thumb. The right-hand orientation is defined by the same procedure using the curled fingers of the right hand. For amino acids, we have the set, S = [COOH, R, NH2, and H], with COOH (the carboxyl group) the highest atomic number and H (the hydrogen atom) as the lowest atomic number. The arrangement, from highest atomic number to lowest atomic number, defines a left or right carbon-centered tetrahedral molecule, following the Cahn-Ingold-Prelog system ⁷. An alpha- helix with left-handed amino acids is a right-handed helix. The chirality of amino acids determines the chirality of the Alpha helix polymer (catalyst in autocatalytic set). Dashed lines represent peptide bonds. In Figure 2 note that Left-handed amino acids fit together to make a right-handed helix. The dashed line connects X=NH2 of each amino acid to the Z=COOH of the next amino acid in a polymer chain. Reading left to right, following the arrows, traces out a right-handed polymer helix. A di-peptide covalent bond is formed between the carbon of COOH of one amino acid with NH2 of another amino acid. A water molecule is removed to form the dipeptide bond.

Figure 3 shows a 2-dimensional top view of a carbon-centered (grey circle) tetrahedral molecule.

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Figure 3

Figure 3 illustrates a 3-D representation of left, and also 2-D top views of left and right carbon-centered tetrahedral molecules. The carbon atom is in the center of the tetrahedron and is covalently bonded to other molecules at the apices of the tetrahedron. The apical molecule with the lowest atomic number is placed in the center of the 2-D figure. For a set of four atomic numbers, where d > c > b > a, the left side of the figure depicts a Left-handed molecule, and the right side of the figure depicts a Right-handed molecule. If two or more of the numbers in the set are the same, then the molecule is non-chiral or symmetric.

Figure 4 shows the asymmetry graph #110 (A-graph #110) derived from Wolfram cellular automaton #110, and the A-graph #30 derived from Wolfram cellular automaton #30. Recall from a previous paper ¹ that cellular automaton #110 and its A-graph represent a complex dynamic system governed by the Principle of Computational Equivalence (PCE) and the Principle of Computational Irreducibility (PCI) ². Cellular automaton #30 and its A-graph characterize a random system. Asymmetry-graph #110 represents a complex small-world network, while A-graph #30 represents a random network. The two graphs are shown together to point out how they handle information flow differently. The dashed lines indicate that a random network can be transformed into a small-world network by increasing information flow at small and large distances. Increasing information flow between distant nodes is equivalent to having ‘shortcuts’ between those distant nodes. Decreasing information flow at intermediate distances is equivalent to reducing connections between nodes that are medium distances apart. These, are the very changes required in a random network to make it isomorphic with a small-world network ⁸.

COVALENT BOND FORMATION—REDOX POTENTIAL- information flow

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Figure 4

Figure 4 illustrates A-graph #110. It is a rational, concave polygon. Its internal angles are π/n, where n is a rational number. Note the V-notch which forms a partial obstruction to a dynamic billiard ball passing either way from H1 to H2 or from H2 to H1. A-graph #30 is also shown to illustrate the differences between the two graphs. A-graph #30 is also a rational polygon, and mainly convex, thereby allowing, by contrast, free movement of a dynamic billiard ball from one side of the polygon to the other. It is because of this difference in billiard ball freedom of movement that the random dynamics of A-graph #30 computes equal clockwise and counterclockwise (racemic) tetrahedral structures, while the computational consequence of the partial obstruction in the complex A-graph #110 means Left-handed tetrahedral structures are sequestered in H2, and separated from racemic structures in H1. Thus, Left-homochirality is a computational consequence of complexity, and generally independent of any particular chemistry. The numbers and primed numbers represent relative atomic numbers; the ordinate represents information flow and covalent bond formation. The computational model is based on Figure 4:

- Dynamic billiards ^{9, 10} on a billiard table (defined by random A-graph #30), models thermodynamic systems at equilibrium, and computes both left and right (racemic) carbon-centered tetrahedral molecules.
- Dynamic billiards on a billiard table (defined by complex A-graph #110), models a far-from-equilibrium thermodynamic system, and computes Left homochirality.

Referring to Figure 5, in dynamic billiards the billiard ball is assumed to be infinitely small, travel in straight lines without spin or ‘English’, and have its energy undiminished by collision with the ‘wall’ of the A-graph billiard table, i.e., the collisions are ‘perfectly elastic.’ meaning that the energy of the ball is restored with each collision—a analogy with the idea of constant energy input to a thermodynamic system. Moreover, the energy of the collision is used to supply the activation energy required to form a covalent bond at an apex of a carbon-centered tetrahedral molecule, i.e., there is a constant removal or decay of energy. The energy released (exergonic) from the billiard ball when it strikes a billiard table wall is coupled to the energy required (endergonic) for the formation of a covalent bond—a redox reaction. The angle of incidence of the ball with the wall is equal to the angle of reflection of the billiard ball from the wall, just as in real billiards. By analogy with a chemical system, the computational system is maintained in a ‘far-from-equilibrium’ state.

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Figure 5

Figure 5 illustrates the ‘mirror law’—the angle of incidence ϴ = the angle of reflection ϴ, and the direction of the path of the ball before it hits the wall relative to the direction of the path after it bounces off the wall is either clockwise or counterclockwise. The horizontal dashed line is the 90o ‘normal’ to the wall. For example, if the ball starts at ‘A’, strikes the wall and bounces off along path B, then the path rotates in a clockwise direction. If the ball were to start at ‘B’, strike the wall, and bounce off along path A, then the path rotates counterclockwise.

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