# On the Growth Rate of Non-Enzymatic Molecular Replicators

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## Abstract

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## 1. Introduction

**Figure 1.**Minimal template directed replicator: two complementary oligomers hybridize to a template strand (upper part). An irreversible ligation reaction transforms the oligomers into the complementary copy of the template. The newly obtained double strand can dehybridize (lower part) thus allowing for iteration of the process. We assume that ligation is rate limiting, which implies that hybridization and dehybridization are in local equilibrium.

## 2. Parabolic Growth and Replication Rate

**Figure 2.**Effective replication rate k (given by Equation 15) as a function of strand length and temperature. For strands below a critical length ${N}^{*}$ (here 10) the rate increases with temperature, for strands longer than ${N}^{*}$, the replication rate grows with decreasing temperature. The value of ${N}^{*}$ is determined through Equation (16). Note the saddle point of the surface where ${T}^{*}$ and ${N}^{*}$ intersect (Equation (12) ($\Delta {H}_{\text{base}}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}-1.5{k}_{\text{B}}{T}^{\prime}$, $\Delta {S}_{\text{base}}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}-1{k}_{\text{B}}$, $\Delta {H}_{\text{init}}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}0.50{k}_{\text{B}}{T}^{\prime}$, $\Delta {S}_{\text{init}}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}1.25{k}_{\text{B}}$, $\Delta {H}_{\text{L}}^{\u2021}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}5.25{k}_{\text{B}}{T}^{\prime}$, $A={10}^{3}$).

## 3. Spatially Resolved Replicator Model

**Figure 3.**Geometry of the nucleotide strands. The figure shows the angles that define inner- and intermolecular interactions for one nucleobase (shaded in grey).

## 4. Simulation Results

Parameter | Value | Comment | Equations |
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m | 1 | particle mass | (22)–(24) |

γ | 3 | friction coefficient | (22), (24) |

${k}_{\text{B}}{T}_{0}$ | 1 | equilibrium temperature | (25), (26) |

$\Delta t$ | $0.05$ | numerical time step | |

r | $0.25$ | particle radius | (28) |

${r}_{\text{bond}}$ | $0.45$ | bond length | (27) |

${r}_{\text{c}}$ | 1 | force cutoff radius | (33)–(34) |

${a}_{\text{bend}}$ | 5 | strand stiffness | (30) |

${\widehat{a}}_{\text{ortho}}$ | $2.5$ | angular stretching | (31) |

${\widehat{a}}_{\text{parallel}}$ | 1 | angular alignment | (32) |

${\widehat{a}}_{\text{hybrid}}$ | 10 | angular hybridization | (34) |

${a}_{\text{hybrid}}$ | 1 | complementary attraction | (33) |

#### 4.1. Diffusion

**Figure 4.**Diffusion coefficients measured for different strand lengths and temperatures (symbols) fitted to the prediction of the Einstein-Stokes relation (solid lines). For each parameter pair, 40 simulation runs over $1000\tau $ have been averaged.

#### 4.2. Radius of Gyration

**Figure 5.**Radius of gyration measured for different strand lengths and bending potentials (symbols) fitted to the prediction of the Flory mean field theory (solid lines). For each parameter pair, 40 simulation runs over $400\tau $ have been averaged. The upper panel shows results for homopolymers (e.g., poly-C), the lower panel compares those to radii of self-complementary strands. The plots also show the boundaries for maximally stretched chains ($\nu =1$—upper dotted line) and the expectation value of an ideal chain ($\nu \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}3/5$—lower dotted line).

#### 4.3. Melting Behavior

**Figure 6.**Systems of size ${10}^{3}$ are initialized with two complementary strands of length N. The sequence information is taken from the N central nucleotides of the master sequence denoted in each panel (e.g., $N=6$ implies sequence CABACD in the first panel). Each system is simulated over $50000\tau $, and the average fraction χ of hybridized nucleobases is determined. Error bars show the average and standard deviation of 40 measurements. Solid lines show the theoretical prediction $\chi \left(T\right)={\left(\right)}^{1}-1$ fitted individually to each data set via $\Delta S$ and $\Delta H$. Melting temperatures ${T}_{\text{m}}$ are obtained from the relation $\chi \left({T}_{\text{m}}\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.5$, and their scaling as a function of strand length is depicted in the inlays for the cases where enough melting points had been observed.

**Figure 7.**Melting curves for an oligomer that hybridizes to the left hand side of the master sequence in the presence of the right hand side oligomer. Data is obtained with the procedure described in Figure 6. For the analyzed master sequence, the results are comparable to those of two complementary strands of length $N/2$ (dotted lines).

#### 4.4. Effective Replication Rate

**Figure 8.**Hybridization energy changes $\Delta {G}_{\text{T}}$ obtained from the measurements of Section 4.3, sequence ACDCABACDCABACDCABAC (symbols), fitted to the analytical model of Equation (13) via the parameters $\Delta {H}_{\text{base}}=-1.81$, $\Delta S=-0.756$, $\Delta {H}_{\text{init}}=0.470$, and $\Delta {S}^{\prime}=-5.58$. Since ${\left[X\right]}_{\text{total}}=0.001$, we can estimate $\Delta {S}_{\text{init}}=1.33$.

**Figure 10.**Final replication rate k as a function of template length and temperature. The figure is produced by superposing the data from Figure 9 with the Arrhenius equation for the ligation reaction following Equation (15) with $A={10}^{3},\Delta {H}_{\text{L}}^{\u2021}=6.52{k}_{\text{B}}{T}^{\prime}$. For this parametrization, the critical strand length ${N}^{*}$ above which the temperature dependence of the reaction inverts is 8.

## 5. Discussion

## Acknowledgements

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Fellermann, H.; Rasmussen, S.
On the Growth Rate of Non-Enzymatic Molecular Replicators. *Entropy* **2011**, *13*, 1882-1903.
https://doi.org/10.3390/e13101882

**AMA Style**

Fellermann H, Rasmussen S.
On the Growth Rate of Non-Enzymatic Molecular Replicators. *Entropy*. 2011; 13(10):1882-1903.
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**Chicago/Turabian Style**

Fellermann, Harold, and Steen Rasmussen.
2011. "On the Growth Rate of Non-Enzymatic Molecular Replicators" *Entropy* 13, no. 10: 1882-1903.
https://doi.org/10.3390/e13101882