# Survival of Self-Replicating Molecules under Transient Compartmentalization with Natural Selection

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Transient Compartmentalization with a Fixed Inoculum Size

- Inoculate the compartments.
- Maturate the compartments.
- Pool compartment contents.

#### 2.2. Transient Compartmentalization with Variable Inoculum Size

## 3. Results

#### 3.1. Fixed Inoculum Size

#### 3.2. Variable Inoculum Size

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Exact Solution of the Maturation Equations

## Appendix B. Derivation of the Equations in the Λ ≫ 1 Limit

## Appendix C. Analysis of The Bifurcation

**Figure A1.**Maximal modulus and maximal imaginary part of eigenvalues of the Jacobian corresponding to Equation (6) for a carrying capacity $K=60$.

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**Figure 1.**(

**a**) Transient compartmentalization at fixed average number of molecules per compartment, and (

**b**) with a variable average number of molecules. In (

**a**), the cycle splits into steps of inoculation, with a fixed average number of molecules per compartment $\lambda $, maturation and then pooling, while in (

**b**) the inoculation step is done with a variable average number of molecules per compartment $\lambda \left(t\right)$ because the cycle contains in addition a dilution step. The green and red circles represent the replicators and their parasites, respectively.

**Figure 2.**Contour map of the fraction x of replicators as a function of $(\lambda ,\Lambda )$, for a carrying capacity $K=100$, where $\lambda $ denotes the average number of molecules per compartment and $\Lambda $ the relative growth rates of the parasites with respect to the host. The dotted line is the contour of $x=1$, which marks the border of the pure replicators phase (the red region). Above this line, a coexistence region exists between the two species at a fraction of replicators indicated by the color scale.

**Figure 3.**Oscillations in the average amount of self-replicating and parasite molecules per compartment as a function of the round number for $d=19$, $K=60$, and $\Lambda =5$. (

**a**) Average population size $\u2329\overline{m}\u232a$ of replicators and $\u2329\overline{n}-\overline{m}\u232a$ of parasites after the growth step plotted vs. round number. (

**b**) Fraction x of replicators and average $\lambda $ of inoculum size. Notice that the oscillations rebound close to the line $\lambda =1$.

**Figure 4.**Oscillations in the average amount of self-replicating and parasites molecules per compartment as a function of the round number for $K=60$ and $\Lambda \gg 1$. (

**a**) Steady oscillations at $d=18$ (unstable spirals), and (

**b**) damped oscillations at $d=22$ (stable spirals). Note the beating pattern in the oscillations visible in (

**a**).

**Figure 5.**(

**a**) Phase diagram in the plane (K, d) in the limit $\Lambda \gg 1$ containing three regions: unstable spirals (with an inset representing steady oscillations), stable spirals (with an inset representing damped oscillations) and stable node (with an inset representing a curve with no oscillations); and (

**b**) evolution of the fixed point coordinates (${x}^{*}$, ${\lambda}^{*}$) as a function of K, on the Hopf bifurcation (solid line) and on the transition line between the stable node and stable spirals (dashed line).

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**MDPI and ACS Style**

Laurent, G.; Peliti, L.; Lacoste, D.
Survival of Self-Replicating Molecules under Transient Compartmentalization with Natural Selection. *Life* **2019**, *9*, 78.
https://doi.org/10.3390/life9040078

**AMA Style**

Laurent G, Peliti L, Lacoste D.
Survival of Self-Replicating Molecules under Transient Compartmentalization with Natural Selection. *Life*. 2019; 9(4):78.
https://doi.org/10.3390/life9040078

**Chicago/Turabian Style**

Laurent, Gabin, Luca Peliti, and David Lacoste.
2019. "Survival of Self-Replicating Molecules under Transient Compartmentalization with Natural Selection" *Life* 9, no. 4: 78.
https://doi.org/10.3390/life9040078