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

^{1}

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## 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

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**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% |

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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