# Molecular Diversity Required for the Formation of Autocatalytic Sets

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background: Autocatalytic Sets

- Reflexively autocatalytic (RA): each reaction $r\in \mathcal{R}$ is catalyzed by at least one molecule type that is either a product of $\mathcal{R}$ or is present in the food set F; and
- F-generated (F): all reactants involved in reactions in $\mathcal{R}$ can be created from the food set F by using a series of reactions only from $\mathcal{R}$ itself.

## 3. Results: Required Molecular Diversity for RAF Sets

#### 3.1. Binary Polymer Model

- (1)
- The E-R result is specifically for undirected graphs, while the catalysis graph is a directed graph. Getting a connected component in an undirected graph is much easier than getting a (strongly) connected component in a directed graph.
- (2)
- In the E-R setting the undirected graphs are simple (i.e., there is just a single edge between any two vertices) while in the setting described above there may be more than one edge.
- (3)
- In the E-R setting there is an equal probability p that each pair of nodes has an (undirected) edge. In other words, E-R random graphs are isotropic, but the catalysis graph is non-isotropic.
- (4)
- RAFs are required to be F-generated, and a giant connected component need not be (e.g., if the food set is a subset of the molecules not in the giant component).
- (5)
- RAFs might form well before a giant connected component does (i.e., a RAF is not required, a priori, to be large).

#### 3.2. Jain–Krishna Model

**Theorem**

**1.**

**Proof.**

## 4. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Kauffman, S.A. Question 1: Origin of life and the living state. Orig. Life Evolut. Biosph.
**2007**, 37, 315–322. [Google Scholar] [CrossRef] [PubMed] - Hordijk, W.; Hein, J.; Steel, M. Autocatalytic sets and the origin of life. Entropy
**2010**, 12, 1733–1742. [Google Scholar] [CrossRef] - Nghe, P.; Hordijk, W.; Kauffman, S.A.; Walker, S.I.; Schmidt, F.J.; Kemble, H.; Yeates, J.A.M.; Lehman, N. Prebiotic network evolution: Six key parameters. Mol. BioSyst.
**2015**, 11, 3206–3217. [Google Scholar] [CrossRef] [PubMed] - Sousa, F.L.; Hordijk, W.; Steel, M.; Martin, W.F. Autocatalytic sets in E. coli metabolism. J. Syst. Chem.
**2015**, 6, 4. [Google Scholar] [CrossRef] [PubMed] - Cazzolla Gatti, R.; Hordijk, W.; Kauffman, S. Biodiversity is autocatalytic. Ecol. Model.
**2017**, 346, 70–76. [Google Scholar] [CrossRef] - Cazzolla Gatti, R.; Fath, B.; Hordijk, W.; Kauffman, S.; Ulanowicz, R. Niche emergence as an autocatalytic process in the evolution of ecosystems. J. Theor. Biol.
**2018**, 454, 110–117. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Sievers, D.; von Kiedrowski, G. Self-replication of complementary nucleotide-based oligomers. Nature
**1994**, 369, 221–224. [Google Scholar] [CrossRef] [PubMed] - Kim, D.E.; Joyce, G.F. Cross-catalytic replication of an RNA ligase ribozyme. Chem. Biol.
**2004**, 11, 1505–1512. [Google Scholar] [CrossRef] [PubMed] - Ashkenasy, G.; Jegasia, R.; Yadav, M.; Ghadiri, M.R. Design of a directed molecular network. PNAS
**2004**, 101, 10872–10877. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Vaidya, N.; Manapat, M.L.; Chen, I.A.; Xulvi-Brunet, R.; Hayden, E.J.; Lehman, N. Spontaneous network formation among cooperative RNA replicators. Nature
**2012**, 491, 72–77. [Google Scholar] [CrossRef] [PubMed] - Arsène, S.; Ameta, S.; Lehman, N.; Griffiths, A.D.; Nghe, P. Coupled catabolism and anabolism in autocatalytic RNA sets. Nucleic Acids Res.
**2018**, 46, 9660–9666. [Google Scholar] [CrossRef] [PubMed] - Hordijk, W.; Steel, M. Chasing the tail: The emergence of autocatalytic networks. BioSystems
**2017**, 152, 1–10. [Google Scholar] [CrossRef] [PubMed] - Hordijk, W.; Steel, M. A formal model of autocatalytic sets emerging in an RNA replicator system. J. Syst. Chem.
**2013**, 4, 3. [Google Scholar] [CrossRef] [Green Version] - Hordijk, W.; Vaidya, N.; Lehman, N. Serial transfer can aid the evolution of autocatalytic sets. J. Syst. Chem.
**2014**, 5, 4. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hordijk, W.; Shichor, S.; Ashkenasy, G. The influence of modularity, seeding, and product inhibition on peptide autocatalytic network dynamics. ChemPhysChem
**2018**, 19, 2437–2444. [Google Scholar] [CrossRef] [PubMed] - Damer, B.; Deamer, D. Coupled phases and combinatorial selection in fluctuating hydrothermal pools: A scenario to guide experimental approaches to the origin of cellular life. Life
**2015**, 5, 872–887. [Google Scholar] [CrossRef] [PubMed] - Deamer, D.W. Assembling Life: How Can Life Begin on Earth and Other Habitable Planets? Oxford University Press: Oxford, UK, 2019. [Google Scholar]
- Watanabe, M.; Arai, S. The plastein reaction: Fundamentals and applications. In Biochemistry of Food Proteins; Springer: Boston, MA, USA, 1992; pp. 271–305. [Google Scholar]
- Kauffman, S.A. Cellular homeostasis, epigenesis and replication in randomly aggregated macromolecular systems. J. Cybern.
**1971**, 1, 71–96. [Google Scholar] [CrossRef] - Kauffman, S.A. Autocatalytic sets of proteins. J. Theor. Biol.
**1986**, 119, 1–24. [Google Scholar] [CrossRef] - Kauffman, S.A. The Origins of Order; Oxford University Press: Oxford, UK, 1993. [Google Scholar]
- Hordijk, W.; Steel, M. Detecting autocatalytic, self-sustaining sets in chemical reaction systems. J. Theor. Biol.
**2004**, 227, 451–461. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hordijk, W.; Kauffman, S.A.; Steel, M. Required levels of catalysis for emergence of autocatalytic sets in models of chemical reaction systems. Int. J. Mol. Sci.
**2011**, 12, 3085–3101. [Google Scholar] [CrossRef] [PubMed] - Hordijk, W.; Steel, M.; Kauffman, S. The structure of autocatalytic sets: Evolvability, enablement, and emergence. Acta Biotheor.
**2012**, 60, 379–392. [Google Scholar] [CrossRef] [PubMed] - Steel, M. The emergence of a self-catalysing structure in abstract origin-of-life models. Appl. Math. Lett.
**2000**, 3, 91–95. [Google Scholar] [CrossRef] - Mossel, E.; Steel, M. Random biochemical networks: The probability of self-sustaining autocatalysis. J. Theor. Biol.
**2005**, 233, 327–336. [Google Scholar] [CrossRef] [PubMed] - Hordijk, W.; Steel, M. Predicting template-based catalysis rates in a simple catalytic reaction model. J. Theor. Biol.
**2012**, 295. [Google Scholar] [CrossRef] [PubMed] - Hordijk, W.; Hasenclever, L.; Gao, J.; Mincheva, D.; Hein, J. An investigation into irreducible autocatalytic sets and power law distributed catalysis. Nat. Comput.
**2014**, 13, 287–296. [Google Scholar] [CrossRef] - Hordijk, W.; Wills, P.R.; Steel, M. Autocatalytic sets and biological specificity. Bull. Math. Biol.
**2014**, 76, 201–224. [Google Scholar] [CrossRef] [PubMed] - Smith, J.; Steel, M.; Hordijk, W. Autocatalytic sets in a partitioned biochemical network. J. Syst. Chem.
**2014**, 5, 2. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hordijk, W.; Steel, M. Autocatalytic sets in polymer networks with variable catalysis distributions. J. Math. Chem.
**2016**, 54, 1997–2021. [Google Scholar] [CrossRef] [Green Version] - Hordijk, W.; Smith, J.I.; Steel, M. Algorithms for detecting and analysing autocatalytic sets. Algorithms Mol. Biol.
**2015**, 10, 15. [Google Scholar] [CrossRef] [PubMed] - Vasas, V.; Fernando, C.; Santos, M.; Kauffman, S.; Sathmáry, E. Evolution before genes. Biol. Direct
**2012**, 7, 1. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hordijk, W.; Steel, M. Conditions for evolvability of autocatalytic sets: A formal example and analysis. Orig. Life Evolut. Biosph.
**2014**, 44, 111–124. [Google Scholar] [CrossRef] [PubMed] - Hordijk, W. Evolution of autocatalytic sets in computational models of chemical reaction networks. Orig. Life Evolut. Biosph.
**2016**, 46, 233–245. [Google Scholar] [CrossRef] [PubMed] - Hordijk, W.; Naylor, J.; Krasnogor, N.; Fellermann, H. Population dynamics of autocatalytic sets in a compartmentalized spatial world. Life
**2018**, 8, 33. [Google Scholar] [CrossRef] [PubMed] - Erdős, P.; Rényi, A. On random graphs. Publ. Math.
**1959**, 6, 290–297. [Google Scholar] - Erdős, P.; Rényi, A. On the evolution of random graphs. Publ. Math. Inst. Hung. Acad. Sci.
**1960**, 5, 17–61. [Google Scholar] - Steel, M.; Hordijk, W.; Smith, J. Minimal autocatalytic networks. J. Theor. Biol.
**2013**, 332, 96–107. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Steel, M.; Hordijk, W.; Xavier, J.C. Autocatalytic networks in biology: Structural theory and algorithms. J. R. Soc. Interface
**2019**, 16, 20180808. [Google Scholar] [CrossRef] - Jain, S.; Krishna, S. Autocatalytic sets and the growth of complexity in an evolutionary model. Phys. Rev. Lett.
**1998**, 81, 5684–5687. [Google Scholar] [CrossRef] - Jain, S.; Krishna, S. A model for the emergence of cooperation, interdependence, and structure in evolving networks. PNAS
**2001**, 98, 543–547. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Jain, S.; Krishna, S. Large extinctions in an evolutionary model: The role of innovation and keystone species. PNAS
**2002**, 99, 2055–2060. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bollobás, B.; Rasmussen, S. First cycles in random directed graph processes. Discrete Math.
**1989**, 75, 55–68. [Google Scholar] [CrossRef] [Green Version] - Scott, J.K.; Smith, G.P. Searching for peptide ligands with an epitope library. Science
**1990**, 249, 286–390. [Google Scholar] [CrossRef] - Quintarelli, A. Systems Optimization for the Selection of Phage Display Random Peptide Libraries. Ph.D. Thesis, University of Rome III, Roma, Italy, 2011. [Google Scholar]
- Tramontano, A.; Janda, K.D.; Lerner, R.A. Catalytic antibodies. Science
**1986**, 234, 1566–1570. [Google Scholar] [CrossRef] [PubMed] - Schwartz, A.W.; de Graaf, R.M. The prebiotic synthesis of carbohydrates: A reassessment. J. Mol. Evol.
**1993**, 36, 101–106. [Google Scholar] [CrossRef] - Zhang, X.V.; Martin, S.T. Driving parts of Krebs cycle in reverse through mineral photochemistry. J. Am. Chem. Soc.
**2006**, 128, 16032–16033. [Google Scholar] [CrossRef] [PubMed] - Muchowska, K.B.; Varma, S.J.; Chevallot-Beroux, E.; Lethuillier-Karl, L.; Li, G.; Moran, J. Metals promote sequences of the reverse Krebs cycle. Nat. Ecol. Evolut.
**2017**, 1, 1716–1721. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Varma, S.J.; Muchowska, K.B.; Chatelain, P.; Moran, J. Native iron reduces CO
_{2}to intermediates and end-products of the acetyl-CoA pathway. Nat. Ecol. Evolut.**2018**, 2, 1019–1024. [Google Scholar] [CrossRef] [PubMed] - Christen, P.; Mehta, P.K. From cofactor to enzymes: The molecular evolution of pyridoxal-5’-phosphate-dependent enzymes. Chem. Rec.
**2001**, 1, 436–447. [Google Scholar] [CrossRef] [PubMed] - Rees, D.C.; Howard, J.B. The interface between the biological and inorganic worlds: Iron-sulfur metalloclusters. Science
**2003**, 300, 929–931. [Google Scholar] [CrossRef] [PubMed] - Wieczorek, R.; Adamala, K.; Gasperi, T.; Polticelli, F.; Stano, P. Small and random peptides: An unexplored reservoir of potentially functional primitive organocatalysts. The case of Seryl-Histidine. Life
**2017**, 7, 19. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**An example of a reflexively autocatalytic and food-generated (RAF) set that appeared in a simple binary polymer model where molecules are “bit string polymers” that can be ligated together into longer ones (see next section). Solid dots represent molecule types (labeled by bit strings); open squares represent reactions (ligations). Solid arrows indicate molecule types going into (reactants) and coming out of (products) a reaction; dashed grey arrows indicate which molecule types catalyze which reactions. The food set F consists of the monomers and dimers (i.e., bit strings of lengths one and two). Adapted from [24].

**Figure 2.**The theoretically calculated (solid line) and the simulation-based (dashed line) minimal values for n to get autocatalytic sets with high probability.

**Figure 3.**The probability of finding RAF sets (Pr[RAF]) for increasing values of the maximum polymer length n for a fixed probability of catalysis $p={10}^{-5}$ in the binary polymer model.

**Figure 4.**The average maxRAF (solid line) and irreducible RAF (irrRAF) (dashed line) sizes (in number of polymer types) for various values of p and corresponding minimally required n (as taken from Figure 2).

**Figure 5.**An example of an autocatalytic (RAF) set in the Jain–Krishna model. The four molecule types at the top of the reaction network are directly created from an implicitly assumed “generic” food set, and they mutually catalyze each others formation.

**Figure 6.**The theoretically calculated bounds (solid lines) and simulation-based values (dashed line) for the minimal value of N to get autocatalytic sets with probability at least $P=0.50$ in the elementary model.

**Table 1.**The required polymer diversity for the existence of autocatalytic sets for various values of the probability of catalysis p, for both the binary polymer model (BPM) and the Jain–Krishna model (JKM).

p | ${10}^{-6}$ | $5\times {10}^{-6}$ | ${10}^{-5}$ | $5\times {10}^{-5}$ | ${10}^{-4}$ | $5\times {10}^{-4}$ |
---|---|---|---|---|---|---|

BPM | 131,070 | 32,766 | 16,382 | 4094 | 2046 | 510 |

JKM | 500,000 | 100,000 | 50,000 | 10,000 | 5000 | 1000 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Hordijk, W.; Steel, M.; Kauffman, S.A.
Molecular Diversity Required for the Formation of Autocatalytic Sets. *Life* **2019**, *9*, 23.
https://doi.org/10.3390/life9010023

**AMA Style**

Hordijk W, Steel M, Kauffman SA.
Molecular Diversity Required for the Formation of Autocatalytic Sets. *Life*. 2019; 9(1):23.
https://doi.org/10.3390/life9010023

**Chicago/Turabian Style**

Hordijk, Wim, Mike Steel, and Stuart A. Kauffman.
2019. "Molecular Diversity Required for the Formation of Autocatalytic Sets" *Life* 9, no. 1: 23.
https://doi.org/10.3390/life9010023