# Hitting Times of Some Critical Events in RNA Origins of Life

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Core Model

#### 2.1.1. System

#### 2.1.2. High-Fidelity Set

#### 2.1.3. Distance

#### 2.1.4. Fitness

#### 2.1.5. Similarity

#### 2.1.6. Fidelity

#### 2.1.7. Counting Representation

#### 2.1.8. Reaction Rates

#### 2.2. Hitting Times

#### 2.2.1. Functional Structure

#### 2.2.2. Statistical Structure

#### 2.3. Surface Chemistries

#### 2.4. Reactions as Measure-Kernel-Functions

#### 2.4.1. RNA Polymerization

**Definition**

**1**

**Theorem**

**1**

**Corollary**

**1**

**Proposition**

**1**

#### 2.4.2. Non-RNA Polymerization

#### 2.5. Decay

#### 2.6. Compartmentalization

#### 2.7. Metabolism

#### 2.8. Reaction Overview

## 3. Results

#### 3.1. Stability: ODEs

**Theorem**

**2.**

**Proof.**

**Corollary**

**2**

#### 3.2. Simulation Reaction State

#### 3.2.1. Core Model with “Tent” Functions, Probable Hitting $\mathbb{P}({\tau}_{v}\left(\theta \right)<\infty )\sim 1$

#### 3.2.2. Core Model with “Tent” Functions, Improbable Hitting $\mathbb{P}({\tau}_{v}\left(\theta \right)<\infty )\sim 0$

#### 3.2.3. Core Model with Linear Functions, Improbable Hitting $\mathbb{P}({\tau}_{v}\left(\theta \right)<\infty )\sim 0$

#### 3.2.4. Expanded Model with “Tent” Functions, Probable Hitting $\mathbb{P}({\tau}_{v}\left(\theta \right)<\infty )\sim 1$

#### 3.3. Hitting Times: Functional and Survival Analysis

#### 3.3.1. Core Model, ${\tau}_{v}\left(\theta \right)$ for $v=0.1$ with $\theta =(n,k)$ and “Tent” Functions

#### 3.3.2. Clay and Decay Model, ${\tau}_{v}\left(\theta \right)$ for $v=0.1$ with $\theta =(n,k,{k}_{\u2300},{k}_{clay-p},p)$ and “Tent” Functions

#### 3.4. Compartmentalization

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Discrete Probability Space

## Appendix B. Other Fitness Functions

## Appendix C. Measure-Kernel-Function

#### Appendix C.1. Reactions as Measure-Kernel-Functions

#### Appendix C.2. Deterministic Model

## Appendix D. Hitting Cardinality

#### Reaction Cardinality

**Table A1.**HDMR sensitivity indices of ${\varpi}_{v}\left(\theta \right)<\infty $ for core model and $v=0.1$.

$\mathit{\theta}$ | ${\mathbb{S}}_{\mathit{\theta}}$ |
---|---|

Sequence dimension n | 0.1619 |

Curvature k | 0.1464 |

∑ | 0.3083 |

## Appendix E. Approximate Reaction Rates

## Appendix F. High Dimensional Model Representation

## Appendix G. Reliability Definitions

## Appendix H. Additional Figures

#### Appendix H.1. Linear Landscape

**Figure A1.**Linear landscape: Measures of system population ${X}_{t}$ until hitting time ${\tau}_{v}$ for high-fidelity replicator volume fraction $v=0.25$ with sequence dimension $n=3$, fitness/similarity curvature $k=l=-log\left(0.01\right)/n$, initial population size, $I=|{X}_{0}|=10$, singleton high-fidelity replicator $R=\left\{\right\{x\left\}\right\}$, with linear fitness and similarity functions. (

**a**) Concentration of RNA sequences by Hamming distance to high-fidelity replicator; (

**b**) population size of RNA sequences by Hamming distance to high-fidelity replicator; (

**c**) polymerase RNA sequence output by Hamming distance to high-fidelity replicator.

#### Appendix H.2. Core Model with Clay

**Figure A2.**Core model with clay: Measures of system population ${X}_{t}$ until hitting time ${\tau}_{v}\left(\theta \right)$ for high-fidelity replicator volume fraction $v=0.25$ with sequence dimension $n=3$, fitness/similarity curvature $k=-log\left(0.01\right)/n$, initial population size $I=|{X}_{0}|=10$, singleton high-fidelity replicator $R=\left\{\right\{x\left\}\right\}$, double strand separation and formation rates reaction rate ${k}_{ss}={k}_{ds}=1$, clay replication fidelity probability $p=0.9$, and RNA polymerization rate ${k}_{rep}$ and clay polymerization rate ${k}_{clay-p}$ chosen such that the replicative mass of each is 10. (

**a**) Concentration of RNA sequences by Hamming distance to high-fidelity replicator; (

**b**) population size of RNA sequences by Hamming distance to high-fidelity replicator; (

**c**) polymerase RNA sequence output by Hamming distance to high-fidelity replicator; (

**d**) probability of reactions over time.

#### Appendix H.3. Hitting/Survival Analysis

**Figure A3.**Survival analysis of hitting time ${\tau}_{v}$ for high-fidelity replicator volume fraction $v=0.1$ for core model. (

**a**) Coefficients of the Cox proportional hazard survival model; (

**b**) survival curves in sequence dimension $n=L$ and landscape curvature $k=l$; (

**c**) cumulative hazard in sequence dimension $n=L$ and landscape curvature $k=l$.

**Figure A4.**First-order HDMR analysis of hitting time ${\tau}_{v}\left(\theta \right)<\infty $ for core model. (

**a**) Hexagonal-bin truth plot; (

**b**) HDMR component function for sequence length $n=L$, ${f}_{n}\left(n\right)$, in sequence length for hitting time, of hitting time; (

**c**) HDMR component function for landscape curvature, ${f}_{k}\left(k\right)$, in landscape curvature, of hitting time. The color function is from blue (negative) to white (zero) to red (positive). The black dots represent standard deviation of the error.

**Figure A5.**First-order HDMR analysis of hitting probability $\mathbb{P}({\tau}_{v}\left(\theta \right)<\infty )$ for core model. (

**a**) Hexagonal-bin truth plot; (

**b**) HDMR component function for sequence length, ${f}_{n}\left(n\right)$, in sequence length, of hitting probability; (

**c**) HDMR component function for landscape curvature, ${f}_{k}\left(k\right)$, in landscape curvature, of hitting probability. The color function is from blue (negative) to white (zero) to red (positive). The black dots represent standard deviation of the error.

**Figure A6.**Survival analysis of hitting time ${\tau}_{v}$ for volume fraction $v=0.1$ for expanded model (clay and decay). Coefficients of the Cox proportional hazards survival model.

**Figure A7.**First-order HDMR analysis of hitting probability $\mathbb{P}({\tau}_{v}\left(\theta \right)<\infty )$ for expanded model (clay and decay). (

**a**) Hexagonal-bin truth plot; (

**b**) HDMR component function for sequence length $n=L$, ${f}_{n}\left(n\right)$, in sequence length, of hitting probability; (

**c**) HDMR component function for curvature, ${f}_{k}\left(k\right)$, in curvature, of hitting probability; (

**d**) HDMR component function for clay fitness, ${f}_{{f}_{clay}}\left({f}_{clay}\right)$, in clay fitness, of hitting probability. The color function is from blue (negative) to white (zero) to red (positive). The black dots represent standard deviation of the error.

**Figure A8.**First-order HDMR analysis of hitting time ${\tau}_{v}\left(\theta \right)<\infty $ for expanded model (clay and decay). (

**a**) Hexagonal-bin truth plot; (

**b**) HDMR component function for sequence length, ${f}_{n}\left(n\right)$, in sequence length, of hitting time; (

**c**) HDMR component function for curvature, ${f}_{k}\left(k\right)$, in curvature, of hitting time; (

**d**) HDMR component function for clay-fraction, ${f}_{{f}_{clay}}\left({f}_{clay}\right)$, in clay fraction, of hitting time; (

**e**) HDMR component function for decay rate, ${f}_{{k}_{\u2300}}\left({k}_{\u2300}\right)$, in decay rate, of hitting time. The color function is from blue (negative) to white (zero) to red (positive). The black dots represent standard deviation of the error.

**Figure A9.**First-order HDMR analysis of hitting cardinality ${\varpi}_{v}\left(\theta \right)<\infty $ for core model and volume fraction $v=0.1$. (

**a**) Hexagonal-bin truth plot; (

**b**) HDMR component function for sequence dimension, ${f}_{n}\left(n\right)$, in sequence dimension, of hitting cardinality; (

**c**) HDMR component function for curvature, ${f}_{k}\left(k\right)$, in curvature, of hitting cardinality. The color function is from blue (negative) to white (zero) to red (positive). The black dots represent standard deviation of the error.

## References

- Ganti, T. The Principles of Life; Oxford University Press: Oxford, UK, 2003. [Google Scholar]
- Ganti, T. Chemoton Theory; Kluwer Academic/Plenum Publishers: Dordrecht, The Netherlands, 2003. [Google Scholar]
- Orgel, L.E. Evolution of the genetic apparatus. J. Mol. Biol.
**1968**, 38, 381–393. [Google Scholar] [CrossRef] - Gilbert, W. Origin of life: The RNA world. Nature
**1986**, 319, 618. [Google Scholar] [CrossRef] - Joyce, G.F. The antiquity of RNA-based evolution. Nature
**2002**, 418, 214–221. [Google Scholar] [CrossRef] [PubMed] - White, H.B. Coenzymes as fossils of an earlier metabolic state. J. Mol. Evol.
**1976**, 7, 101–104. [Google Scholar] [CrossRef] - Eigen, M. Selforganization of matter and the evolution of biological macromolecules. Naturwissenschaften
**1971**, 58, 465–523. [Google Scholar] [CrossRef] [PubMed] - Eigen, M.; Schuster, P. A principle of natural self-organization. Naturwissenschaften
**1977**, 64, 541–565. [Google Scholar] [CrossRef] - Cech, T.R. The Ribosome Is a Ribozyme. Science
**2000**, 289, 878. [Google Scholar] [CrossRef] - Diener, T.O. Potato spindle tuber “virus”. IV. A replicating, low molecular weight RNA. Virology
**1971**, 45, 411–428. [Google Scholar] [CrossRef] - Tupper, A.S.; Higgs, P.G. Rolling-circle and strand-displacement mechanisms for non-enzymatic RNA replication at the time of the origin of life. J. Theor. Biol.
**2021**, 527, 110822. [Google Scholar] [CrossRef] - 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] - de Farias, S.T.; dos Santos Junior, A.P.; Rêgo, T.G.; José, M.V. Origin and Evolution of RNA-Dependent RNA Polymerase. Front. Genet.
**2017**, 8, 125. [Google Scholar] [CrossRef] [PubMed] - Koonin, E.V.; Krupovic, M.; Ishino, S.; Ishino, Y. The replication machinery of LUCA: Common origin of DNA replication and transcription. BMC Biol.
**2020**, 18, 61. [Google Scholar] [CrossRef] - Ghadessy, F.J.; Ong, J.L.; Holliger, P. Directed evolution of polymerase function by compartmentalized self-replication. Proc. Natl. Acad. Sci. USA
**2001**, 98, 4552. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Tjhung, K.F.; Shokhirev, M.N.; Horning, D.P.; Joyce, G.F. An RNA polymerase ribozyme that synthesizes its own ancestor. Proc. Natl. Acad. Sci. USA
**2020**, 117, 2906. [Google Scholar] [CrossRef] - Ertem, G.; Ferris, J.P. Template-Directed Synthesis Using the Heterogeneous Templates Produced by Montmorillonite Catalysis. A Possible Bridge Between the Prebiotic and RNA Worlds. J. Am. Chem. Soc.
**1997**, 119, 7197–7201. [Google Scholar] [CrossRef] - Acevedo, O.L.; Orgel, L.E. Non-enzymatic transcription of an oligodeoxynucleotide 14 residues long. J. Mol. Biol.
**1987**, 197, 187–193. [Google Scholar] [CrossRef] - Szostak, J.W. The eightfold path to non-enzymatic RNA replication. J. Syst. Chem.
**2012**, 3, 2. [Google Scholar] [CrossRef] [Green Version] - Cairns-Smith, A.G. Genetic Takeover and the Mineral Origins of Life; Cambridge University Press: Cambridge, UK, 1987. [Google Scholar]
- Dyson, F. Origins of Life; Cambridge University Press: Cambridge, UK, 1999. [Google Scholar] [CrossRef] [Green Version]
- Sakuma, Y.; Imai, M. From vesicles to protocells: The roles of amphiphilic molecules. Life
**2015**, 5, 651–675. [Google Scholar] [CrossRef] - Szostak, J.W.; Bartel, D.P.; Luisi, P.L. Synthesizing life. Nature
**2001**, 409, 387–390. [Google Scholar] [CrossRef] - Segré, D.; Ben-Eli, D.; Deamer, D.W.; Lancet, D. The Lipid World. Orig. Life Evol. Biosph.
**2001**, 31, 119–145. [Google Scholar] [CrossRef] - Martin, W.; Baross, J.; Kelley, D.; Russell, M.J. Hydrothermal vents and the origin of life. Nat. Rev. Microbiol.
**2008**, 6, 805–814. [Google Scholar] [CrossRef] [PubMed] - Damer, B.; Deamer, D. The Hot Spring Hypothesis for an Origin of Life. Astrobiology
**2019**, 20, 429–452. [Google Scholar] [CrossRef] [Green Version] - 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] [Green Version] - Deamer, D.; Damer, B.; Kompanichenko, V. Hydrothermal Chemistry and the Origin of Cellular Life. Astrobiology
**2019**, 19, 1523–1537. [Google Scholar] [CrossRef] - Kvenvolden, K.; Lawless, J.; Pering, K.; Peterson, E.; Flores, J.; Ponnamperuma, C.; Kaplan, I.R.; Moore, C. Evidence for Extraterrestrial Amino-acids and Hydrocarbons in the Murchison Meteorite. Nature
**1970**, 228, 923–926. [Google Scholar] [CrossRef] [PubMed] - Miller, S.L.; Urey, H.C. Organic Compound Synthesis on the Primitive Earth. Science
**1959**, 130, 245. [Google Scholar] [CrossRef] - Pino, S.; Sponer, J.E.; Costanzo, G.; Saladino, R.; Mauro, E.D. From formamide to RNA, the path is tenuous but continuous. Life
**2015**, 5, 372–384. [Google Scholar] [CrossRef] [Green Version] - Powner, M.W.; Sutherland, J.D. Prebiotic chemistry: A new modus operandi. Phil. Trans. R. Soc. B Biol. Sci.
**2011**, 366, 2870–2877. [Google Scholar] [CrossRef] [Green Version] - Powner, M.W.; Gerland, B.; Sutherland, J.D. Synthesis of activated pyrimidine ribonucleotides in prebiotically plausible conditions. Nature
**2009**, 459, 239–242. [Google Scholar] [CrossRef] - Attwater, J.; Wochner, A.; Holliger, P. In-ice evolution of RNA polymerase ribozyme activity. Nat. Chem.
**2013**, 5, 1011–1018. [Google Scholar] [CrossRef] [Green Version] - Hays, L. NASA Astrobiology Strategy; Technical report; National Aeronautics and Space Administration: Washington, DC, USA, 2015.
- Coveney, P.V.; Swadling, J.B.; Wattis, J.A.D.; Greenwell, H.C. Theory, modelling and simulation in origins of life studies. Chem. Soc. Rev.
**2012**, 41, 5430–5446. [Google Scholar] [CrossRef] - Lanier, K.A.; Williams, L.D. The Origin of Life: Models and Data. J. Mol. Evol.
**2017**, 84, 85–92. [Google Scholar] [CrossRef] [Green Version] - Wu, M.; Higgs, P.G. The origin of life is a spatially localized stochastic transition. Biol. Direct
**2012**, 7, 42. [Google Scholar] [CrossRef] [Green Version] - Gillespie, D.T. Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem.
**1977**, 81, 2340–2361. [Google Scholar] [CrossRef] - Hordijk, W.; Steel, M. Detecting autocatalytic, self-sustaining sets in chemical reaction systems. J. Theor. Biol.
**2004**, 227, 451–461. [Google Scholar] [CrossRef] [Green Version] - Walker, S.I. Origins of life: A problem for physics, a key issues review. Rep. Prog. Phys.
**2017**, 80, 092601. [Google Scholar] [CrossRef] - Wattis, J.A.D.; Coveney, P.V. The Origin of the RNA World: A Kinetic Model. J. Phys. Chem. B
**1999**, 103, 4231–4250. [Google Scholar] [CrossRef] [Green Version] - Kun, Á.; Szilágyi, A.; Könnyű, B.; Boza, G.; Zachar, I.; Szathmáry, E. The dynamics of the RNA world: Insights and challenges. Ann. N. Y. Acad. Sci.
**2015**, 1341, 75–95. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Szilágyi, A.; Zachar, I.; Scheuring, I.; Kun, Á.; Könnyű, B.; Czárán, T. Ecology and Evolution in the RNA World Dynamics and Stability of Prebiotic Replicator Systems. Life
**2017**, 7, 48. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Scheuring, I.; Szilágyi, A. Diversity, stability, and evolvability in models of early evolution. Curr. Opin. Syst. Biol.
**2019**, 13, 115–121. [Google Scholar] [CrossRef] [Green Version] - Takeuchi, N.; Hogeweg, P. Evolutionary dynamics of RNA-like replicator systems: A bioinformatic approach to the origin of life. Phys. Life Rev.
**2012**, 9, 219–263. [Google Scholar] [CrossRef] [Green Version] - Orgel, L. A Simpler Nucleic Acid. Science
**2000**, 290, 1306. [Google Scholar] [CrossRef] [PubMed] - Maury, C.P.J. Origin of life. Primordial genetics: Information transfer in a pre-RNA world based on self-replicating beta-sheet amyloid conformers. J. Theor. Biol.
**2015**, 382, 292–297. [Google Scholar] [CrossRef] [Green Version] - Ehrenfreund, P.; Rasmussen, S.; Cleaves, J.; Chen, L. Experimentally Tracing the Key Steps in the Origin of Life: The Aromatic World. Astrobiology
**2006**, 6, 490–520. [Google Scholar] [CrossRef] - Kunin, V. A System of Two Polymerases—A Model for the Origin of Life. Orig. Life Evol. Biosph.
**2000**, 30, 459–466. [Google Scholar] [CrossRef] [PubMed] - Wright, S. Evolution and the Genetics of Populations, Volume 1; The University of Chicago Press: Chicago, IL, USA, 1984. [Google Scholar]
- Feng, X.; Pechen, A.; Jha, A.; Wu, R.; Rabitz, H. Global optimality of fitness landscapes in evolution. Chem. Sci.
**2012**, 3, 900–906. [Google Scholar] [CrossRef] - Davidson-Pilon, C. lifelines: Survival analysis in Python. J. Open Source Softw.
**2019**, 4, 1317. [Google Scholar] [CrossRef] [Green Version] - Bornholt, J.; Lopez, R.; Carmean, D.M.; Ceze, L.; Seelig, G.; Strauss, K. A DNA-Based Archival Storage System. In Proceedings of the Twenty-First International Conference on Architectural Support for Programming Languages and Operating Systems, Association for Computing Machinery, New York, NY, USA, 19–23 April 2016; pp. 637–649. [Google Scholar] [CrossRef] [Green Version]
- Kitadai, N.; Maruyama, S. Origins of building blocks of life: A review. Geosci. Front.
**2018**, 9, 1117–1153. [Google Scholar] [CrossRef] - Li, G.; Rabitz, H. General formulation of HDMR component functions with independent and correlated variables. J. Math. Chem.
**2012**, 50, 99–130. [Google Scholar] [CrossRef]

**Figure 1.**Measures of system population ${X}_{t}$ until hitting time ${\tau}_{v}$ for high-fidelity replicator volume fraction $v=0.25$ with sequence dimension $n=3$, fitness/similarity curvature $l=k=-log\left(0.01\right)/n$, initial population size $I=|{X}_{0}|=10$, singleton high-fidelity replicator $R=\left\{\right\{x\left\}\right\}$, with “tent” fitness and similarity functions. (

**a**) concentration of RNA sequences by Hamming distance to high-fidelity replicator; (

**b**) population size of RNA sequences by Hamming distance to high-fidelity replicator; (

**c**) polymerase RNA sequence output by Hamming distance to high-fidelity replicator.

**Figure 2.**Measures of system population ${X}_{t}$ until hitting time ${\tau}_{v}$ for high-fidelity replicator volume fraction $v=0.25$ with sequence dimension $n=3$, fitness/similarity curvature $l=k=-log\left(0.1\right)/n$, initial population size $I=|{X}_{0}|=10$, singleton high-fidelity replicator $R=\left\{\right\{x\left\}\right\}$, with “tent” fitness and similarity functions. (

**a**) concentration of RNA sequences by Hamming distance to high-fidelity replicator; (

**b**) population size of RNA sequences by Hamming distance to high-fidelity replicator; (

**c**) polymerase RNA sequence output by Hamming distance to high-fidelity replicator.

Name | Quantity | Baseline | Quantity | Proportional Hazards |
---|---|---|---|---|

Failure density | $f\left(t\right|\alpha ,\beta )$ | $\frac{\alpha}{t}{\left(\frac{t}{\beta}\right)}^{\alpha}{e}^{-{\left(\frac{t}{\beta}\right)}^{\alpha}}$ | $f(t,\theta |\alpha ,\beta ,\gamma )$ | $\frac{\alpha}{t}{\left(\frac{t}{\beta}\right)}^{\alpha}{e}^{\gamma \xb7\theta}{e}^{-{\left(\frac{t}{\beta}\right)}^{\alpha}{e}^{\gamma \xb7\theta}}$ |

Failure distribution | $F\left(t\right|\alpha ,\beta )$ | $1-{e}^{-{\left(\frac{t}{\beta}\right)}^{\alpha}}$ | $F(t,\theta |\alpha ,\beta ,\gamma )$ | $1-{e}^{-{\left(\frac{t}{\beta}\right)}^{\alpha}{e}^{\gamma \xb7\theta}}$ |

Reliability distribution | $R\left(t\right|\alpha ,\beta )$ | ${e}^{-{\left(\frac{t}{\beta}\right)}^{\alpha}}$ | $R(t,\theta |\alpha ,\beta ,\gamma )$ | ${e}^{-{\left(\frac{t}{\beta}\right)}^{\alpha}{e}^{\gamma \xb7\theta}}$ |

Cumulative hazard | $H\left(t\right|\alpha ,\beta )$ | ${\left(\frac{t}{\beta}\right)}^{\alpha}$ | $H(t,\theta |\alpha ,\beta ,\gamma )$ | ${\left(\frac{t}{\beta}\right)}^{\alpha}{e}^{\gamma \xb7\theta}$ |

Hazard rate | $h\left(t\right|\alpha ,\beta )$ | $\frac{\alpha}{t}{\left(\frac{t}{\beta}\right)}^{\alpha}$ | $h(t,\theta |\alpha ,\beta ,\gamma )$ | $\frac{\alpha}{t}{\left(\frac{t}{\beta}\right)}^{\alpha}{e}^{\gamma \xb7\theta}$ |

Reaction | Order |
---|---|

RNA double strand formation | 2 |

RNA double strand dissociation | 1 |

RNA polymerization | 2 |

RNA decay | 1 |

Clay polymerization | 1 |

Clay oligomerization | 0 |

Compartmentalization | 1 |

Metabolism to replication | 1 |

Metabolism & replication to metabolism | 2 |

$\mathit{\theta}$ | Name | Domain | Value(s) |
---|---|---|---|

n | sequence dimension | ${\mathbb{N}}_{>0}$ | $\{3,4,5\}$ |

k | RNA fitness parameter | ${\mathbb{R}}_{\ge 0}$ | $\{-log(i)/n:i=0.1,0.05,0.01,0.005,0.001\}$ |

l | RNA similarity parameter | ${\mathbb{R}}_{\ge 0}$ | $\{-log(i)/n:i=0.1,0.05,0.01,0.005,0.001\}$ |

m | RNA fidelity parameter | ${\mathbb{R}}_{\ge 0}$ | $-log\left(0.25\right)/n$ |

p | clay fidelity probability | $(0,1]$ | 0.9 |

${k}_{ss}$ | double-strand dissociation rate | ${\mathbb{R}}_{\ge 0}$ | 1 |

${k}_{ds}$ | double-strand formation rate | ${\mathbb{R}}_{\ge 0}$ | 1 |

${k}_{rep}$ | RNA replication rate | ${\mathbb{R}}_{\ge 0}$ | 10 |

${k}_{\u2300}$ | RNA decay rate | ${\mathbb{R}}_{\ge 0}$ | (0, 1) |

${k}_{clay-o}$ | clay RNA oligomerization rate | ${\mathbb{R}}_{\ge 0}$ | 1 |

${k}_{clay-p}$ | clay RNA polymerization rate | ${\mathbb{R}}_{\ge 0}$ | (0, 20) |

**Table 4.**HDMR sensitivity indices of hitting probability $\mathbb{P}({\tau}_{v}\left(\theta \right)<\infty )$ for the core model.

$\mathit{\theta}$ | ${\mathbb{S}}_{\mathit{\theta}}$ |
---|---|

Sequence length n | 0.06 |

Curvature k | 0.69 |

∑ | 0.75 |

**Table 5.**HDMR sensitivity indices of hitting time ${\tau}_{v}\left(\theta \right)$ for the core model.

$\mathit{\theta}$ | ${\mathbb{S}}_{\mathit{\theta}}$ |
---|---|

Sequence length n | 0.57 |

Curvature k | 0.04 |

∑ | 0.61 |

$\mathit{\theta}$ | Name | Coefficient ${\mathit{\gamma}}_{\mathit{\theta}}$ | p-Value |
---|---|---|---|

n | sequence dimension | 0.54 | <0.005 |

$k=l$ | RNA fitness parameter | 27.87 | <0.005 |

p | clay fidelity probability | −0.08 | 0.35 |

${k}_{\u2300}$ | RNA decay rate | 0.80 | <0.005 |

${f}_{clay}$ | fraction clay RNA polymerization rate | 0.96 | <0.005 |

**Table 7.**HDMR sensitivity indices of hitting probability $\mathbb{P}({\tau}_{v}\left(\theta \right)<\infty )$ for expanded model (clay and decay).

$\mathit{\theta}$ | ${\mathbb{S}}_{\mathit{\theta}}$ |
---|---|

Sequence length n | 0.0213 |

Curvature k | 0.6732 |

Decay rate ${k}_{\u2300}$ | 0.0120 |

Clay fidelity p | 0.0114 |

Fraction clay RNA polymerization rate ${f}_{clay}$ | 0.0219 |

∑ | 0.7399 |

**Table 8.**HDMR sensitivity indices of ${\tau}_{v}\left(\theta \right)<\infty $ for expanded model (clay and decay).

$\mathit{\theta}$ | ${\mathbb{S}}_{\mathit{\theta}}$ |
---|---|

Sequence dimension n | 0.0013 |

Curvature k | 0.0030 |

Decay ${k}_{\u2300}$ | 0.1129 |

Clay fidelity p | 0.0152 |

Fraction clay RNA polymerization rate ${f}_{clay}$ | 0.2014 |

∑ | 0.3339 |

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Bastian, C.D.; Rabitz, H.
Hitting Times of Some Critical Events in RNA Origins of Life. *Life* **2021**, *11*, 1419.
https://doi.org/10.3390/life11121419

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Bastian CD, Rabitz H.
Hitting Times of Some Critical Events in RNA Origins of Life. *Life*. 2021; 11(12):1419.
https://doi.org/10.3390/life11121419

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Bastian, Caleb Deen, and Hershel Rabitz.
2021. "Hitting Times of Some Critical Events in RNA Origins of Life" *Life* 11, no. 12: 1419.
https://doi.org/10.3390/life11121419