# A Polyaddition Model for the Prebiotic Polymerization of RNA and RNA-Like Polymers

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

**by truncation and compare this lower bound to the actual error found in our simulation. Finally**, we suggest methods to connect these theoretical predictions to experimental results.

## 1. Introduction

_{+}, and bonded A − B pairs separate from each other at a rate k

_{−}. These are assumed to be independent of the configuration of the reacting monomer units; that is, the bonding rate k

_{+}does not depend on whether each A and B terminus is the endpoint of a long polymer or of a free monomer, nor is the unbonding rate k

_{−}affected by the position of the A − B bond within a polymer. Under these conditions, the reactions affecting each bonding site take the following simple form:

## 2. Flory-Schulz Polymer Length Distribution

#### 2.1. Steady-State Bonding Probability from Reaction Rates

#### 2.2. Thermodynamics of Bonding

## 3. Dynamics

_{i}, P

_{j}, and P

_{i+j}. The chemical equations in this family are of the form:

#### 3.1. Continuous Dynamics

_{k}. This is fundamentally a stochastic jump process describing discrete numbers of k-mers, but in the thermodynamic limit as the number of reactants grows very large, we can concern ourselves with the deterministic, continuous evolution of the expected concentration $n\left(k\right)$ of k-mers.

_{k}occurs in the system of chemical equations: if it is on the left-hand side, a negative contribution is made to $\frac{\mathrm{d}}{\mathrm{d}t}n\left(k\right)$, and if on the right, the contribution is positive.

_{k}can appear in all three positions in the chemical Equation (8). For each equation where P

_{k}appears as the first term on the left side (i.e., for each possible synthesis partner $j\in \mathbb{N}$), we lose P

_{k}at a rate ${k}_{+}\left[{\mathrm{P}}_{k}\right]\left[{\mathrm{P}}_{j}\right]$, but gain it at a rate ${k}_{-}\left[{\mathrm{P}}_{k+j}\right]$. Each of those contributions should also be doubled to handle the functionally identical case where P

_{k}appears as the second term on the left side. Finally, when P

_{k}appears on the right side, for each possible split point $j\in \{1\dots k-1\}$, we gain P

_{k}at a rate ${k}_{+}\left[{\mathrm{P}}_{k-j}\right]\left[{\mathrm{P}}_{j}\right]$ and lose it at a rate ${k}_{-}\left[{\mathrm{P}}_{k}\right]$. The facts above can be consolidated into a single differential equation describing the evolution of $n\left(k\right)=\left[{\mathrm{P}}_{k}\right]$ as follows:

#### 3.2. Reduction to One Dimension

#### 3.3. Closed-Form Solution

## 4. Numerical Treatments

#### 4.1. Choice of Parameter Values

#### 4.2. Truncation

#### 4.3. Simulations

#### 4.4. Error Bound

#### 4.5. Applying the Error Bound

## 5. Comparison to Experiment

#### 5.1. Critical Concentration

#### 5.2. Polymer Yield

#### 5.3. Mass Distribution

#### 5.4. Degree of Polymerization

## 6. Discussion

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Gilbert, W. Origin of Life: The RNA World. Nature
**1986**, 319, 618. [Google Scholar] [CrossRef] - Neveu, M.; Kim, H.J.; Benner, S.A. The “Strong” RNA World Hypothesis: Fifty Years Old. Astrobiology
**2013**, 13, 391–403. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Deamer, D.; Damer, B.; Kompanichenko, V. Hydrothermal Chemistry and the Origin of Cellular Life. Astrobiology
**2019**, 19. [Google Scholar] [CrossRef] [PubMed] - Kruger, K.; Grabowski, P.J.; Zaug, A.J.; Sands, J.; Gottschling, D.E.; Cech, T.R. Self-Splicing RNA: Autoexcision and Autocyclization of the Ribosomal RNA Intervening Sequence of Tetrahymena. Cell
**1982**, 31, 147–157. [Google Scholar] [CrossRef] - Fedor, M.J.; Williamson, J.R. The Catalytic Diversity of RNAs. Nat. Rev. Mol. Cell Biol.
**2005**, 6, 399–412. [Google Scholar] [CrossRef] - Bartel, D.; Szostak, J. Isolation of New Ribozymes from a Large Pool of Random Sequences. Science
**1993**, 261, 1411–1418. [Google Scholar] [CrossRef] [Green Version] - Johnston, W.K.; Unrau, P.J.; Lawrence, M.S.; Glasner, M.E.; Bartel, D.P. RNA-Catalyzed RNA Polymerization: Accurate and General RNA-Templated Primer Extension. Science
**2001**, 292, 1319–1325. [Google Scholar] [CrossRef] [Green Version] - Wochner, A.; Attwater, J.; Coulson, A.; Holliger, P. Ribozyme-Catalyzed Transcription of an Active Ribozyme. Science
**2011**, 332, 209–212. [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] - Kauffman, S.A. The Origins of Order: Self-Organization and Selection in Evolution; Oxford University Press: Oxford, UK, 1993. [Google Scholar]
- Lancet, D.; Kedem, O.; Pilpel, Y. Emergence of Order in Small Autocatalytic Sets Maintained Far from Equilibrium: Application of a Probabilistic Receptor Affinity Distribution (RAD) Model. Berichte der Bunsengesellschaft für Physikalische Chemie
**1994**, 98, 1166–1169. [Google Scholar] [CrossRef] - Vasas, V.; Fernando, C.; Santos, M.; Kauffman, S.; Szathmáry, 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 Evol. Biosph.
**2014**, 44, 111–124. [Google Scholar] [CrossRef] [PubMed] - Orgel, L.E. Prebiotic Chemistry and the Origin of the RNA World. Crit. Rev. Biochem. Mol. Biol.
**2004**, 39, 99–123. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Higgs, P.G. The Effect of Limited Diffusion and Wet–Dry Cycling on Reversible Polymerization Reactions: Implications for Prebiotic Synthesis of Nucleic Acids. Life
**2016**, 6, 24. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Ross, D.S.; Deamer, D. Dry/Wet Cycling and the Thermodynamics and Kinetics of Prebiotic Polymer Synthesis. Life
**2016**, 6, 28. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hargrave, M.; Thompson, S.K.; Deamer, D. Computational Models of Polymer Synthesis Driven by Dehydration/Rehydration Cycles: Repurination in Simulated Hydrothermal Fields. J. Mol. Evol.
**2018**, 86, 501–510. [Google Scholar] [CrossRef] - Rajamani, S.; Vlassov, A.; Benner, S.; Coombs, A.; Olasagasti, F.; Deamer, D. Lipid-Assisted Synthesis of RNA-like Polymers from Mononucleotides. Orig. Life Evol. Biosph.
**2008**, 38, 57–74. [Google Scholar] [CrossRef] - Da Silva, L.; Maurel, M.C.; Deamer, D. Salt-Promoted Synthesis of RNA-like Molecules in Simulated Hydrothermal Conditions. J. Mol. Evol.
**2015**, 80, 86–97. [Google Scholar] [CrossRef] - DeGuzman, V.; Vercoutere, W.; Shenasa, H.; Deamer, D. Generation of Oligonucleotides under Hydrothermal Conditions by Non-Enzymatic Polymerization. J. Mol. Evol.
**2014**, 78, 251–262. [Google Scholar] [CrossRef] - Flory, P.J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, USA, 1953. [Google Scholar]
- Gupta, S.K.; Kumar, A. Reaction Engineering of Step Growth Polymerization.; Plenum Chemical Engineering Series; Plenum Press: New York, NY, USA, 1987. [Google Scholar]
- Voet, D.; Voet, J.G. Fundamentals of Biochemistry, 4th ed.; Wiley: Hoboken, NJ, USA, 2011. [Google Scholar]
- Gao, H.; Ma, X.; Lin, J.; Wang, L.; Cai, C.; Zhang, L.; Tian, X. Synthesis of Nanowires via Temperature-Induced Supramolecular Step-Growth Polymerization. Macromolecules
**2019**, 52, 7731–7739. [Google Scholar] [CrossRef] - Nelson, D.L.; Cox, M.M. Lehninger Principles of Biochemistry, 6th ed.; W. H. Freeman and Company: New York, NY, USA, 2013. [Google Scholar]
- Nam, I.; Lee, J.K.; Nam, H.G.; Zare, R.N. Abiotic Production of Sugar Phosphates and Uridine Ribonucleoside in Aqueous Microdroplets. Proc. Natl. Acad. Sci. USA
**2017**, 114, 12396–12400. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Orgel, L.E. Polymerization on the Rocks: Theoretical Introduction. Orig. Life Evol. Biosph.
**1998**, 28, 227–234. [Google Scholar] [CrossRef] - Monnard, P.A.; Kanavarioti, A.; Deamer, D.W. Eutectic Phase Polymerization of Activated Ribonucleotide Mixtures Yields Quasi-Equimolar Incorporation of Purine and Pyrimidine Nucleobases. J. Am. Chem. Soc.
**2003**, 125, 13734–13740. [Google Scholar] [CrossRef] [PubMed] - Ellis, R.J. Macromolecular Crowding: Obvious but Underappreciated. Trends Biochem. Sci.
**2001**, 26, 597–604. [Google Scholar] [CrossRef] - Costanzo, G.; Pino, S.; Ciciriello, F.; Mauro, E.D. Generation of Long RNA Chains in Water. J. Biol. Chem.
**2009**, 284, 33206–33216. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Flory, P.J. Thermodynamics of Heterogeneous Polymers and Their Solutions. J. Chem. Phys.
**1944**, 12, 425–438. [Google Scholar] [CrossRef] - Luo, B.; Smith, J.W.; Wu, Z.; Kim, J.; Ou, Z.; Chen, Q. Polymerization-like Co-Assembly of Silver Nanoplates and Patchy Spheres. ACS Nano
**2017**, 11, 7626–7633. [Google Scholar] [CrossRef] - Yang, C.; Ma, X.; Lin, J.; Wang, L.; Lu, Y.; Zhang, L.; Cai, C.; Gao, L. Supramolecular “Step Polymerization” of Preassembled Micelles: A Study of “Polymerization” Kinetics. Macromol. Rapid Commun.
**2018**, 39, 1700701. [Google Scholar] [CrossRef] - Gu, M.; Ma, X.; Zhang, L.; Lin, J. Reversible Polymerization-like Kinetics for Programmable Self-Assembly of DNA-Encoded Nanoparticles with Limited Valence. J. Am. Chem. Soc.
**2019**, 141, 16408–16415. [Google Scholar] [CrossRef] - Oivanen, M.; Kuusela, S.; Lönnberg, H. Kinetics and Mechanisms for the Cleavage and Isomerization of the Phosphodiester Bonds of RNA by Brønsted Acids and Bases. Chem. Rev.
**1998**, 98, 961–990. [Google Scholar] [CrossRef] - Rackauckas, C.; Nie, Q. DifferentialEquations.Jl – A Performant and Feature-Rich Ecosystem for Solving Differential Equations in Julia. J. Open Res. Softw.
**2017**, 5, 15. [Google Scholar] [CrossRef] [Green Version] - Oosawa, F.; Asakura, S. Thermodynamics of the Polymerization of Protein; Molecular Biology; Academic Press: London, UK, 1975. [Google Scholar]
- Flory, P.J. Molecular Size Distribution in Linear Condensation Polymers. J. Am. Chem. Soc.
**1936**, 58, 1877–1885. [Google Scholar] [CrossRef] - Lu, Y.; Gao, L.; Lin, J.; Wang, L.; Zhang, L.; Cai, C. Supramolecular Step-Growth Polymerization Kinetics of Pre-Assembled Triblock Copolymer Micelles. Polym. Chem.
**2019**, 10, 3461–3468. [Google Scholar] [CrossRef] - Xing, J.Y.; Xue, Y.H.; Lu, Z.Y.; Liu, H. In-Depth Analysis of Supramolecular Interfacial Polymerization via a Computer Simulation Strategy. Macromolecules
**2019**, 52, 6393–6404. [Google Scholar] [CrossRef]

**Figure 1.**A comparison of the four different cases described in Table 1 for the signs of $\mathsf{\Delta}{H}_{b}$ and $\mathsf{\Delta}{S}_{b}$. When $\mathsf{\Delta}{H}_{b}$ and $\mathsf{\Delta}{S}_{b}$ have the same sign, there is a critical temperature ${T}_{c}=\frac{\mathsf{\Delta}{H}_{b}}{\mathsf{\Delta}{S}_{b}}$ at which $\mathsf{\Delta}{G}_{b}=0$, so ${P}_{b}=50\%$ and polymerization changes between being favorable and unfavorable. When the signs differ, however, polymerization is either favorable or unfavorable regardless of temperature.

**Figure 2.**Closed-form solution to the dynamics of the Flory-Schulz rate parameter p starting from an initial condition $p=0$, corresponding to an all-monomer solution. The parameter itself is shown on the left, and the resulting concentrations of k-mers for k from 1 (blue) to 10 (cyan) are shown on the right.

**Figure 3.**Concentration of k-mers for k from 1 (blue) to 10 (cyan), for truncation lengths $d=100$ (

**left**) and $d=10$ (

**right**). The $d=100$ case, visually identical to the results shown in Figure 2, reaches the correct geometric distribution, whereas the $d=10$ case goes through a nonphysical inversion near time $t={10}^{5}$.

**Figure 4.**The steady state concentration distribution for $d=10$, $d=25$, and $d=100$ compared to the closed-form solution. By $d=100$, the numerical and analytical solutions are indistinguishable.

**Figure 5.**Comparison between the error bound (16) and the actual error in the results of our simulation. The left figure depicts the steady-state error and the theoretical lower bound $E\left({P}_{b}\right)$ as a function of d, and on the right is the time evolution of the error in a single simulation for $d=100$, compared to the bound $E\left(p\right)$ computed from the instantaneous analytical value of p as in Figure 2.

**Table 1.**The temperature dependence of the equilibrium probability of bond formation ${P}_{b}$ varies depending on the signs of two key thermodynamic quantities of interest: the free enthalpy change $\mathsf{\Delta}{H}_{b}$ and the corresponding entropy change associated with bond formation. Compare to Figure 1, which displays ${P}_{b}$ as a function of temperature in these four cases.

$\mathsf{\Delta}{\mathit{H}}_{\mathit{b}}$ | $\mathsf{\Delta}{\mathit{S}}_{\mathit{b}}$ | Effect on ${P}_{b}$ |
---|---|---|

+ | + | ${P}_{b}>0.5$ above $\frac{\mathsf{\Delta}{H}_{b}}{\mathsf{\Delta}{S}_{b}}$ |

+ | - | ${P}_{b}<0.5$ at all T |

- | + | ${P}_{b}>0.5$ at all T |

- | - | ${P}_{b}>0.5$ below $\frac{\mathsf{\Delta}{H}_{b}}{\mathsf{\Delta}{S}_{b}}$ |

Parameter | Description | Value |
---|---|---|

[U] | initial monomer concentration | 1 M |

∆G | Gibbs free energy of bonding | −1.5 kcal/mol |

k_{−} | unbonding rate | 10^{−6} s^{−1} |

k_{+} | bonding rate constant | 7.4 × 10^{−5} s^{−1}mol^{−1} |

P_{b} | steady-state bonding probability | 89% |

© 2020 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**

Spaeth, A.; Hargrave, M.
A Polyaddition Model for the Prebiotic Polymerization of RNA and RNA-Like Polymers. *Life* **2020**, *10*, 12.
https://doi.org/10.3390/life10020012

**AMA Style**

Spaeth A, Hargrave M.
A Polyaddition Model for the Prebiotic Polymerization of RNA and RNA-Like Polymers. *Life*. 2020; 10(2):12.
https://doi.org/10.3390/life10020012

**Chicago/Turabian Style**

Spaeth, Alex, and Mason Hargrave.
2020. "A Polyaddition Model for the Prebiotic Polymerization of RNA and RNA-Like Polymers" *Life* 10, no. 2: 12.
https://doi.org/10.3390/life10020012