# On the Growth Rate of Non-Enzymatic Molecular Replicators

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 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 |
---|---|---|---|

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 $\chi \left(T\right)={\left(1+{e}^{\frac{\Delta H-T\Delta S}{RT}}\right)}^{-1}$ 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

## References and Notes

- Gilbert, W. The RNA world. Nature
**1986**, 319, 618. [Google Scholar] [CrossRef] - Monnard, P.A. The dawn of the RNA world: RNA Polymerization from monoribonucleotides under prebiotically plausible conditions. In Prebiotic Evolution and Astrobiology; Wong, J.T.F., Lazcano, A., Eds.; Landes Bioscience: Austin, TX, USA, 2008. [Google Scholar]
- Cleaves, J.H. Prebiotic chemistry, the premordial replicator and modern protocells. In Protocells: Bridging Nonliving and Living Matter; Rasmussen, S., Bedau, M., Chen, L., Deamer, D., Krakauer, D., Packard, N., Stadler, P., Eds.; MIT Press: Cambridge, MA, USA, 2009; p. 583. [Google Scholar]
- Rasmussen, S.; Chen, L.; Deamer, D.; Krakauer, D.C.; Packard, N.H.; Stadler, P.F.; Bedau, M.A. Transitions from nonliving to living matter. Science
**2004**, 303, 963–965. [Google Scholar] [CrossRef] [PubMed] - Rasmussen, S.; Chen, L.; Stadler, B.M.R.; Stadler, P.F. Proto-organism kinetics: Evolutionary dynamics of lipid aggregates with genes and metabolism. Orig. Life Evol. Biosph.
**2004**, 34, 171–180. [Google Scholar] [CrossRef] - Rasmussen, S.; Bailey, J.; Boncella, J.; Chen, L.; Collis, G.; Colgate, S.; DeClue, M.; Fellermann, H.; Goranovic, G.; Jiang, Y.; et al. Assembly of a minimal protocell. In Protocells: Bridging Nonliving and Living Matter; Rasmussen, S., Bedau, M., Chen, L., Deamer, D., Krakauer, D., Packard, N., Stadler, P., Eds.; MIT Press: Cambridge, MA, USA, 2008; pp. 125–156. [Google Scholar]
- Szostack, W.; Bartel, D.P.; Luisi, P.L. Synthesizing life. Nature
**2001**, 409, 387–390. [Google Scholar] [CrossRef] [PubMed] - Mansy, S.S.; Schrum, J.P.; Krishnamurthy, M.; Tobé, S.; Treco, D.A.; Szostak, J.W. Template-directed synthesis of a genetic polymer in a model protocell. Nature
**2008**, 454, 122–125. [Google Scholar] [CrossRef] [PubMed] - Hanczyc, M. Steps towards creating a synthetic protocell. In Protocells: Bridging Nonliving and Living Matter; Rasmussen, S., Bedau, M., Chen, L., Deamer, D., Krakauer, D., Packard, N., Stadler, P., Eds.; MIT Press: Cambridge, MA, USA, 2009; p. 107. [Google Scholar]
- The European Commission funded projects MatchIT. Available online: http://www.fp7-matchit.eu/ (access on 19 October 2011).
- ECCell. Available online: http://homepage.ruhr-uni-bochum.de/john.mccaskill/ECCell/ (access on 19 October 2011).
- Wu, T.; Orgel, L.E. Nonenzymic template-directed synthesis on oligodeoxycytidylate sequences in hairpin oligonucleotides. J. Am. Chem. Soc.
**1992**, 114, 317–322. [Google Scholar] [CrossRef] [PubMed] - Wu, T.; Orgel, L. Nonenzymatic template-directed synthesis on hairpin oligonucleotides. 3. Incorporation of adenosine and uridine residues. J. Am. Chem. Soc.
**1992**, 114, 7963–7969. [Google Scholar] [CrossRef] [PubMed] - Fernando, C.; Kiedrowski, G.v.; Szathmáry, E. A stochastic model of nonenzymatic nucleic acid replication: “Elongators” sequester replicators. J. Mol. Evol.
**2007**, 64, 572–585. [Google Scholar] [CrossRef] [PubMed] - Monnard, P.A.; Dörr, M.; Löffler, P. Possible role of ice in the synthesis of polymeric compounds. In Presented at the 38th COSPAR Scientific Assembly, Bremen, Germany, 15–18 July 2010.
- Kiedrowski, G.V. A Self-replicating hexadeoxynucleotide. Angew. Chem. Int. Ed.
**1986**, 25, 932–935. [Google Scholar] [CrossRef] - Sievers, D.; Kiedrowski, G.V. Self-replication of complementary nucleotide-based oligomers. Nature
**1994**, 369, 221–224. [Google Scholar] [CrossRef] [PubMed] - Bag, B.G.; Kiedrowski, G.V. Templates, autocatalysis and molecular replication. Pure App. Chem.
**1996**, 68. [Google Scholar] [CrossRef] - Joyce, G.F. Non-enzyme template-directed synthesis of RNA copolymers. Orig. Life Evol. Biosph.
**1984**, 14, 613–620. [Google Scholar] [CrossRef] - Lincoln, T.A.; Joyce, G.F. Self-sustained replication of an RNA enzyme. Science
**2009**, 323, 1229–1232. [Google Scholar] [CrossRef] [PubMed] - Wills, P.; Kauffman, S.; Stadler, B.; Stadler, P. Selection dynamics in autocatalytic systems: Templates replicating through binary ligation. Bull. Math. Biol.
**1998**, 60, 1073–1098. [Google Scholar] [CrossRef] - Rocheleau, T.; Rasmussen, S.; Nielson, P.E.; Jacobi, M.N.; Ziock, H. Emergence of protocellular growth laws. Philos. Trans. R. Soc. B
**2007**, 362, 1841–1845. [Google Scholar] [CrossRef] [PubMed] - Száthmary, E.; Gladkih, I. Sub-exponential growth and coexistence of non-enzymatically replicating templates. J. Theor. Biol.
**1989**, 138, 55–58. [Google Scholar] [CrossRef] - Kiedrowski, G.v.; Wlotzka, B.; Helbing, J.; Matzen, M.; Jordan, S. Parabolic growth of a self-replicating hexadeoxynucleotide bearing a 3’-5’-phosphoamidate linkage. Angew. Chem. Int. Ed.
**1991**, 30, 423–426. [Google Scholar] [CrossRef] - In particular, it has been shown that under parabolic growth conditions, competing replicators X
_{i}grow when sufficiently rare:$$\left[{X}_{i}\right]<{\left(\frac{{k}_{i}}{{k}_{\text{base}}}\frac{{\sum}_{j}\left[{X}_{j}\right]}{{\sum}_{j}{\left[{X}_{j}\right]}^{1/2}}\right)}^{2}\u27f9\frac{d}{dt}\left[{X}_{i}\right]>0$$_{i}and its selective pressure, such that replicator species with a high growth rate are also assigned a high evolutionary fitness. See [23] for the derivation. - Luther, A.; Brandsch, R.; Kiedrowski, G.V. Surface-promoted replication and exponential amplifcation of DNA analogues. Nature
**1998**, 396, 245–248. [Google Scholar] [PubMed] - Zhang, D.Y.; Yurke, B. A DNA superstructure-based replicator without product inhibition. Nat. Comput.
**2006**, 5, 183–202. [Google Scholar] [CrossRef] - Owczarzy, R.; Vallone, P.M.; Gallo, F.J.; Paner, T.M.; Lane, M.J.; Benight, A.S. Predicting sequence-dependent melting stability of short duplex DNA oligomers. Biopolymers
**1998**, 44, 217–239. [Google Scholar] [CrossRef] - Bloomfield, V.A.; Crothers, D.M.; Tinoco, I. Nucleic Acids; University Science Books: Sausalitos, CA, USA, 2000. [Google Scholar]
- Poland, D.; Scheraga, H.A. Occurrence of a phase transition in nucleic acid models. J. Chem. Phys.
**1966**, 45, 1456–1463. [Google Scholar] [CrossRef] [PubMed] - Hutton, T.J. Evolvable self-replicating molecules in an artificial chemistry. Artif. Life
**2002**, 8, 341–356. [Google Scholar] [CrossRef] [PubMed] - Smith, A.; Turney, P.; Ewaschuk, R. Self-replicating machines in continuous space with virtual physics. Artif. Life
**2003**, 9, 21–40. [Google Scholar] [CrossRef] - Klenin, K.; Merlitz, H.; Langowski, J. A Brownian Dynamics program for the simulation of linear and circular DNA and other wormlike chain polyelectrolytes. Biophys. J.
**1998**, 74, 780–788. [Google Scholar] [CrossRef] - Tepper, H.L.; Voth, G.A. A coarse-grained model for double-helix molecules in solution: Spontaneous helix formation and equilibrium properties. J. Chem. Phys.
**2005**, 122, 124906. [Google Scholar] [CrossRef] [PubMed] - Drukker, K.; Schatz, G.C. A model for simulating dynamics of DNA denaturation. J. Chem. Phys. B
**2000**, 104, 6108–6111. [Google Scholar] [CrossRef] - Fellermann, H.; Rasmussen, S.; Ziock, H.J.; Solé, R. Life-cycle of a minimal protocell: A dissipative particle dynamics (DPD) study. Artif. Life
**2007**, 13, 319–345. [Google Scholar] [CrossRef] [PubMed] - Kubo, R. The fluctuation-dissipation theorem. Rep. Prog. Phys.
**1966**, 29, 255. [Google Scholar] [CrossRef] - Ryckaert, J.P.; Ciccotti, G.; Berendsen, H.J.C. Numerical integration of the Cartesian equations of motion of a system with constraints: Molecular dynamics of n-Alkanes. J. Comp. Phys.
**1977**, 23, 327. [Google Scholar] [CrossRef] - Note that our approach would not work in the absence of a thermostat: to describe rotational motion properly, one would need to define orientations and angular momenta in a local reference frame that moves with the extended object to which the oriented point particle belongs. In this manner, rotational motion of the extended object gets propagated down to the angular momenta of the particles it consists of (A QShake algorithm would in addition be needed to properly conserve angular momenta in the constraints). While this approach is computationally significantly more cumbersome, we expect the result to be similar for the above model, in which rotation of extended objects is propagated down to its constituting particles through angular potentials and an overdamped thermostat.
- Tinland, B.; Pluen, A.; Sturm, J.; Weill, G. Peristence length of single stranded DNA. Macromolecules
**1997**, 30, 5763–5765. [Google Scholar] [CrossRef] - SantaLucia, J., Jr.; Allawi, H.T.; Seneviratne, A. Improved nearest-neighbor parameters for predicting DNA duplex stability. Biochemistry
**1996**, 35, 3555–3562. [Google Scholar] [CrossRef] [PubMed] - Teraoka, I. Polymer Solutions—An Introduction to Physical Properties; Wiley Interscience: New York, NY, USA, 2002. [Google Scholar]

© 2011 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 license (http://creativecommons.org/licenses/by/3.0/.)

## Share and Cite

**MDPI and ACS Style**

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.
https://doi.org/10.3390/e13101882

**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