# Entropic Analysis of Mirror Symmetry Breaking in Chiral Hypercycles

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## Abstract

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## 1. Introduction

## 2. Enantioselective Replicators in Open-Flow Reactors

## 3. Entropy Production

## 4. Entropy Production of the Extreme Flux Modes

## 5. Role of the Chemical Forces: The General Evolution Criterion

## 6. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

SMSB | Spontaneous mirror symmetry breaking |

NESS | Non-equilibrium stationary state |

SNA | Stoichiometric network analysis |

EFM | Extreme flux mode |

GEC | General evolution criterion |

CSTR | Continuously stirred tank reactor |

## Appendix A. GEC for Chemical Reactions in Open-Flow

## References

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**Figure 2.**Spontaneous mirror symmetry breaking (SMSB) in a simple two-hypercycle network Equations (1)–(6) in an open-flow reactor of volume $V=1$ L. The two homochiral cross-catalyzed enantioselective replicators ${}^{1}R$ and ${}^{2}R$ are fed by a common achiral resource A. The reaction rate constants are ${k}_{a}=1\times {10}^{4}$, ${k}_{-a}=1\times 10$, ${k}_{b}=1\times {10}^{3}$, ${k}_{-b}=5\times {10}^{-1}$, see text. Initial resource concentration in the reactor and in the constant input volume (f = 0.2 $\mathsf{\mu}$L/s): ${\left[A\right]}_{in}=1\times {10}^{-4}$ mol/L. Initial replicator concentrations in the reactor ${\left[{}^{1}{R}_{L}\right]}_{0}={\left[{}^{1}{R}_{D}\right]}_{0}={\left[{}^{2}{R}_{L}\right]}_{0}={\left[{}^{2}{R}_{D}\right]}_{0}=$$1\times {10}^{-6}$ mol/L and the initial chiral fluctuation is simulated by an incremental concentration of $\delta {\left[{}^{2}{R}_{L}\right]}_{0}=1\times {10}^{-23}$ mol/L in the ${}^{2}{R}_{L}$ enantiomer.

**Top row**: formation of the enantiomers from the initial input concentration of resource and the symmetry breaking bifurcation. Black dashed curves are $\left[A\right]$, blue curves are $\left[{R}_{L}\right]$, and the red curves are $\left[{R}_{D}\right]$ for each replicator. The SMSB event occurs at approximately $t\approx 2\times {10}^{8}$ s.

**Bottom row**: percent enantiomeric excess ($ee\%$) for each replicator reaches 100% homochirality. The rise in $ee$ initiates approximately at $t\approx 2\times {10}^{8}$ s. Qualitatively similar behavior is obtained when direct production and autocatalysis of the replicators are included in this scheme, as well as for other hypercyclic networks involving more replicators, and also for other system architectures [17].

**Figure 3.**The entropy production Equation (11) over the full time range of the simulation of Figure 2. The first pronounced peak at $t\approx {10}^{6}$ s corresponds to the almost complete conversion of A into both the enantiomers of each replicator. The production then decreases and levels off to a narrow plateau corresponding to the unstable racemic state, and then subsequently decreases once more to a final minimum value, with respect to this former unstable plateau, immediately after the symmetry breaking bifurcation occurring at $t\approx 2\times {10}^{8}$ s. Compare the time scales here to those in the top row of Figure 2.

**Figure 4.**Bifurcation in the partial entropy productions $\sigma \left({\mathit{E}}_{i}\right)$, in units of $\mathrm{J}\phantom{\rule{0.166667em}{0ex}}{\mathrm{K}}^{-1}{\mathrm{s}}^{-1}{\mathrm{L}}^{-1}$, associated with the enantiomeric pairs of extreme flux modes.

**Left**: upper (red) branch $\sigma \left({\mathit{E}}_{1}\right)$, lower (blue) branch $\sigma \left({\mathit{E}}_{4}\right)$.

**Right**: upper (red) branch $\sigma \left({\mathit{E}}_{2}\right)$, lower (blue) branch $\sigma \left({\mathit{E}}_{3}\right)$. See Table 1 and Equations (14)–(17).

**Figure 5.**Bifurcation in the entropy productions $\sigma \left({\mathit{E}}_{i}\right)$, in units of $\mathrm{J}\phantom{\rule{0.166667em}{0ex}}{\mathrm{K}}^{-1}{\mathrm{s}}^{-1}{\mathrm{L}}^{-1}$, associated with the enantiomeric pairs of extreme flux modes.

**Left**: upper (red) branch $\sigma \left({\mathit{E}}_{6}\right)$, lower (blue) branch $\sigma \left({\mathit{E}}_{9}\right)$.

**Right**: upper (red) branch $\sigma \left({\mathit{E}}_{7}\right)$, lower (blue) branch $\sigma \left({\mathit{E}}_{8}\right)$. See Table 1 and Equations (19)–(22).

**Figure 6.**The sum of the partial entropies over all the extreme flux modes, in units of $\mathrm{J}\phantom{\rule{0.166667em}{0ex}}{\mathrm{K}}^{-1}{\mathrm{s}}^{-1}{\mathrm{L}}^{-1}$. See Table 1 and Equations (14)–(22). Same parameters as employed in Figure 2. This sum vanishes at the final stable chiral non-equilibrium stationary state (NESS), as well as at the prior metastable racemic state, indicating that the entropy production exactly balances the exchange entropy $\sigma ={\sigma}_{e}$: see Equation (23). See text for behavior of the minuscule “spike” at $t\approx 2\times {10}^{8}$ s.

**Figure 7.**

**Left**: the temporal derivative of the entropy production starts off positive, increasing in the vicinity of the production peak, and subsequently goes negative and then to zero during the metastable racemic phase and is also zero after SMSB as the system approaches the final stable chiral NESS. The small negative “spike” at approximately $2\times {10}^{8}$ s (

**right hand graph**) shows the derivative at the SMSB transition itself. Compare this derivative to the entropy production curve in Figure 3. We can resolve the total derivative into two independent contributions [10]: $\frac{d\sigma}{dt}=\frac{{d}_{F}\sigma}{dt}+\frac{{d}_{J}\sigma}{dt}$, Equation (26), to test validity of the GEC: see Figure 8.

**Figure 8.**Blue curve: the change in the entropy production with respect to changes in the chemical forces F (the affinities), which is negative definite thoughout the entire time course and reaches zero at the final stable chiral NESS, and thus obeys the general evolution criterion (GEC) [9,10]; and Equation (A18). Red curve: the change in the entropy production with respect to changes in the flows J starts off positive then becomes negative after SMSB and then reaches zero from below on the approach to the final stable chiral NESS.

**Table 1.**Elementary flux modes ${\mathit{E}}_{i}$, the individual transformations they involve as enumerated in Equations (1)–(6), and their corresponding reaction subnetworks or pathways. The parity operation $\mathcal{P}$, which acts on enantiomers in three dimensional space, induces symmetries on these vectors by relating pairs of enantiomeric extreme flux modes (EFMs) [21]. These pairs or doublets are ${\mathit{E}}_{1}\iff {\mathit{E}}_{4}$, ${\mathit{E}}_{2}\iff {\mathit{E}}_{3}$, ${\mathit{E}}_{6}\iff {\mathit{E}}_{9}$, and ${\mathit{E}}_{7}\iff {\mathit{E}}_{8}$. There is one singlet: ${\mathit{E}}_{5}$.

EFM: | Reactions | Subnetwork: Reaction Pathway |
---|---|---|

${\mathit{E}}_{1}$ | (10),(9) | $2\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{}^{2}{R}_{L}+{}^{1}{R}_{L}\to A+{}^{2}{R}_{L}+{}^{1}{R}_{L}$, $A+{}^{2}{R}_{L}+{}^{1}{R}_{L}\to 2\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{}^{2}{R}_{L}+{}^{1}{R}_{L}$ |

${\mathit{E}}_{2}$ | (8),(7) | $2\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{}^{1}{R}_{L}+{}^{2}{R}_{L}\to A+{}^{1}{R}_{L}+{}^{2}{R}_{L}$ $A+{}^{1}{R}_{L}+{}^{2}{R}_{L}\to 2\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{}^{1}{R}_{L}+{}^{2}{R}_{L}$ |

${\mathit{E}}_{3}$ | (2),(1) | $2\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{}^{1}{R}_{D}+{}^{2}{R}_{D}\to A+{}^{1}{R}_{D}+{}^{2}{R}_{D}$ $A+{}^{1}{R}_{D}+{}^{2}{R}_{D}\to 2\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{}^{1}{R}_{D}+{}^{2}{R}_{D}$ |

${\mathit{E}}_{4}$ | (4),(3) | $2\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{}^{2}{R}_{D}+{}^{1}{R}_{D}\to A+{}^{2}{R}_{D}+{}^{1}{R}_{D}$ $A+{}^{2}{R}_{D}+{}^{1}{R}_{D}\to 2\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{}^{2}{R}_{D}+{}^{1}{R}_{D}$ |

${\mathit{E}}_{5}$ | (14),(13) | $A\to \varnothing $, $\overline{\varnothing}\to A$ |

${\mathit{E}}_{6}$ | (13),(12),(9) | $\overline{\varnothing}\to A$ ${}^{2}{R}_{L}\to \varnothing $ $A+{}^{2}{R}_{L}+{}^{1}{R}_{L}\to 2\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{}^{2}{R}_{L}+{}^{1}{R}_{L}$ |

${\mathit{E}}_{7}$ | (13),(11),(7) | $\overline{\varnothing}\to A$ ${}^{1}{R}_{L}\to \varnothing $ $A+{}^{1}{R}_{L}+{}^{2}{R}_{L}\to 2\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{}^{1}{R}_{L}+{}^{2}{R}_{L}$ |

${\mathit{E}}_{8}$ | (13),(5),(1) | $\overline{\varnothing}\to A$ ${}^{1}{R}_{D}\to \varnothing $ $A+{}^{1}{R}_{D}+{}^{2}{R}_{D}\to 2\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{}^{1}{R}_{D}+{}^{2}{R}_{D}$ |

${\mathit{E}}_{9}$ | (13),(6),(3) | $\overline{\varnothing}\to A$ ${}^{2}{R}_{D}\to \varnothing $ $A+{}^{2}{R}_{D}+{}^{1}{R}_{D}\to 2\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{}^{2}{R}_{D}+{}^{1}{R}_{D}$ |

**Table 2.**The partial entropy productions per unit volume $\sigma $ for chemical transformations of the form $A+B+C\rightleftharpoons 2B+C$ and for the irreversible pseudo-reactions $\to X$ and $Y\to $, see [22]. The reference equilibrium concentrations ${\left[X\right]}_{eq},{\left[Y\right]}_{eq}$, are those established when the reactor, and with the chemical mass of the final non-equilibrium stationary state (NESS), is isolated from the open flow; see text.

Transformation | Partial Entropy Production/Exchange |
---|---|

$A+B+C\to 2B+C$ | $\sigma (A+B+C\to 2B+C)=R\phantom{\rule{0.166667em}{0ex}}{k}_{+}\left[A\right]\left[B\right]\left[C\right]\mathrm{ln}\left(\frac{{k}_{+}\left[A\right]}{{k}_{-}\left[B\right]}\right)$ |

$2B+C\to A+B+C$ | $\sigma (2B+C\to A+B+C)=R\phantom{\rule{0.166667em}{0ex}}{k}_{-}{\left[B\right]}^{2}\left[C\right]\mathrm{ln}\left(\frac{{k}_{-}\left[B\right]}{{k}_{+}\left[A\right]}\right)$ |

$\to X$ | $\sigma (\to X)=R\phantom{\rule{0.166667em}{0ex}}f{\left[X\right]}_{in}\mathrm{ln}\left(\frac{{\left[X\right]}_{eq}}{\left[X\right]}\right)$ |

$Y\to $ | $\sigma (Y\to )=R\phantom{\rule{0.166667em}{0ex}}f\left[Y\right]\mathrm{ln}\left(\frac{\left[Y\right]}{{\left[Y\right]}_{eq}}\right)$ |

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Hochberg, D.; Ribó, J.M. Entropic Analysis of Mirror Symmetry Breaking in Chiral Hypercycles. *Life* **2019**, *9*, 28.
https://doi.org/10.3390/life9010028

**AMA Style**

Hochberg D, Ribó JM. Entropic Analysis of Mirror Symmetry Breaking in Chiral Hypercycles. *Life*. 2019; 9(1):28.
https://doi.org/10.3390/life9010028

**Chicago/Turabian Style**

Hochberg, David, and Josep M. Ribó. 2019. "Entropic Analysis of Mirror Symmetry Breaking in Chiral Hypercycles" *Life* 9, no. 1: 28.
https://doi.org/10.3390/life9010028