# Nonequilibrium Entropic Bounds for Darwinian Replicators

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## Abstract

**:**

## 1. Introduction

## 2. Entropic Bounds for Replicators

#### 2.1. The Extended Second Law

#### 2.2. Replicators & Reproducers

#### 2.3. Coarse-Grained Dynamics of Replicators

- $\mathcal{A}$—state in which the system contains a total number of $n-1$ active chains.
- $\mathcal{B}$—state in which the system contains a total amount of n active chains.

## 3. Discussion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

MDPI | Multidisciplinary Digital Publishing Institute |

DOAJ | Directory of open access journals |

ESL | Extended Second Law |

LEB | Lower Entropic Bound |

LHS | Left Hand Side |

RHS | Right Hand Side |

## Appendix A

## Appendix B

#### Appendix B.1

- Pick an element of the urn at random.
- If active, with probability g, pick a second element at random and (if not active) activate.
- Pick an element at random again.
- If active, with probability $\delta $, deactivate.

**Figure A1.**A summary of the rules of replication in an urn model. Active chains are drawn as filled balls and inactive chains are white balls. (

**a**) represents the action of selecting an active chain replicating following the simple replicator mechanism; (

**b**) shows the replicating process of a hypercyclic replicator; (

**c**) corresponds to the decay which, for the purpose of this work, is supposed to act equivalently in each replicator-type.

#### Appendix B.2

- Pick an element of the urn at random.
- If active, pick a second element at random.
- If active, with probability h, pick a third element at random and (if not active) activate.
- Pick an element at random again.
- If active, with probability $\delta $, deactivate.

#### Appendix B.3

- Pick an element of the urn at random. If active, then: (i) if in associated state ($AA$) then, with probability a, dissociate and iterate; (ii) if dissociated, pick a second element and, if active, with probability b, associate. Iterate this process until equilibrium is reached for association/dissociation reaction.
- Pick an element of the urn at random. If active, pick a second element at random, if empty, with probability c, replicate.
- Pick an element of the urn at random. If active, with probability $\delta $, deactivate.

**Figure A2.**This diagram shows how the urn model of parabolic replicators is implemented. (

**a**,

**b**) correspond to the rapid association/dissociation reactions, which are supposed to equilibrate in much shorter time-scales than the replicating process, which is shown in (

**c**), i.e., ${\tau}_{0}\ll {\tau}_{1}$. The process of equilibration (left box) is iterated a large number of times before the loop goes into the replication process (right box).

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**Figure 1.**Scheme of the formal approach to expressions (1)–(4). (

**a**) A time-discretization is implemented in order to characterize the microscopical reversibility condition; (

**b**) A qualitative scheme of possible trajectories between macrostates on the global phase space. The macroscopic coarse-grained states, $\mathcal{A}$ (dark shaded region) and $\mathcal{B}$ (light shaded region) are defined as disjoint ($\mathcal{A}\cap \mathcal{B}=\varnothing $) sections on the phase state $\mathsf{\Omega}$. The set of forward paths of duration $\tau $ constrained to start in $\mathcal{A}$ and finish in $\mathcal{B}$ is denoted by ${\mathbf{x}}_{\tau}$.

**Figure 2.**Hyperbolic and parabolic replicators. In (

**a**) we display a simplified scheme of an experimental implementation of a catalytic set of ribozymes forming a cooperative loop. Here each component of the system helps the next to replicate. Dashed lines indicate weaker catalytic links (modified from [37]). The parabolic system outlined in (

**b**) is based on complementary (template) peptide chains involving a ligation mechanism (adapted from [38]).

**Figure 3.**The central diagram corresponds to the space spanning the reduced variables $(\overline{h},\overline{c})$. We distinguish six phases depending on the dominance of the LEB of each replicator type, $\{\mathbf{S},\mathbf{H},\mathbf{P}\}$. (

**a**) $\mathbf{S}$-dominant (the simple replicator LEB exceeds that of both parabolic and hyperbolic); (

**b**) $\mathbf{P}/\mathbf{S}$ i.e., at low densities, it is $\mathbf{S}$-dominant, while, for $x>{x}_{ps}$ we observe $\mathbf{S}$ dominance; (

**c**) $\mathbf{P}$-dominant at all density values; (

**d**) $\mathbf{P}/\mathbf{H}$ $\mathbf{P}$ at low densities and $\mathbf{H}$-dominant for $x>{x}_{ph}$; (

**e**) $\mathbf{P}/\mathbf{S}/\mathbf{H}$ where the three replicators share dominance at some point, jumping orderedly at density values ${x}_{ps}<{x}_{sh}$; (

**f**) here simple replicators have a higher LEB at low densities than parabolic ones, but hyperbolic ones take over at high densities, $x>{x}_{sh}$. Numerical values of $(\overline{h},\overline{c})$ for each plot are: (

**a**) (0.8, 0.8); (

**b**) (1.42, 0.8); (

**c**) (1.82, 0.8); (

**d**) (1.5, 2); (

**e**) (1.125, 1.5); (

**f**) (0.75, 1.5), while $\alpha =0.5$ for all graphs.

**Table 1.**Summary of the minimal expressions for the kinetics of the three replicator classes discussed above. We have denoted as x the gross concentration of replicating molecules A, independently of the configuration.

Replicator Class | Reaction Scheme | Effective Dynamics |
---|---|---|

Simple | $A+E\to A+A$ | $\dot{x}=gx$ |

Hyperbolic | $A+A+E\to A+A+A$ | $\dot{x}=h{x}^{2}$ |

Parabolic | $A+E\to AA\leftrightarrow A+A$ | $\dot{x}=\rho {x}^{1/2}$ |

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Piñero, J.; Solé, R.
Nonequilibrium Entropic Bounds for Darwinian Replicators. *Entropy* **2018**, *20*, 98.
https://doi.org/10.3390/e20020098

**AMA Style**

Piñero J, Solé R.
Nonequilibrium Entropic Bounds for Darwinian Replicators. *Entropy*. 2018; 20(2):98.
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**Chicago/Turabian Style**

Piñero, Jordi, and Ricard Solé.
2018. "Nonequilibrium Entropic Bounds for Darwinian Replicators" *Entropy* 20, no. 2: 98.
https://doi.org/10.3390/e20020098