# Multistable Protocells Can Aid the Evolution of Prebiotic Autocatalytic Sets

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## Abstract

**:**

## 1. Introduction

## 2. The Model

## 3. Results

#### 3.1. Deterministic Dynamics: Bistability with Two Distinct Growth Rates

#### 3.2. Stochastic Dynamics of a Single Protocell: Transitions between States of Different Growth Rates

#### 3.3. Protocell Population Dynamics: Dominance of the Autocatalytic State

## 4. Discussion And Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ACS | Autocatalytic set |

## Appendix A. Reaction Probabilities Used in Gillespie Algorithm

**Table A1.**List of unidirectional reactions in the model and their reaction probabilities per unit time. V appearing in the table is given by the right-hand side of Equation (9) in the main paper text. Deterministic rates of reactions in the last column are given as in Equations (5)–(7) of the main paper.

Reaction | Reaction Type | Reaction Probability per Unit Time | Deterministic Rate of Reaction |
---|---|---|---|

$A{\left(1\right)}_{ext}+A\left(2\right)\stackrel{\alpha {X}_{2}}{\u27f6}A\left(1\right)+A\left(2\right)$ | transport | $\alpha {X}_{2}$ | $\alpha {X}_{2}$ |

$A\left(1\right)+A\left(1\right)\stackrel{{k}_{F}}{\u27f6}A\left(2\right)$ | spontaneous | ${k}_{F}{X}_{1}({X}_{1}-1){V}^{-1}$ | ${k}_{F}{X}_{1}^{2}{V}^{-1}$ |

$A\left(1\right)+A\left(1\right)+A\left(4\right)\stackrel{\kappa {k}_{F}}{\u27f6}A\left(2\right)+A\left(4\right)$ | catalyzed | $\kappa {k}_{F}{X}_{4}{V}^{-1}{X}_{1}({X}_{1}-1){V}^{-1}$ | $\kappa {k}_{F}{X}_{4}{V}^{-1}{X}_{1}^{2}{V}^{-1}$ |

$A\left(2\right)+A\left(2\right)\stackrel{{k}_{F}}{\u27f6}A\left(4\right)$ | spontaneous | ${k}_{F}{X}_{2}({X}_{2}-1){V}^{-1}$ | ${k}_{F}{X}_{2}^{2}$ |

$A\left(2\right)+A\left(2\right)+A\left(4\right)\stackrel{\kappa {k}_{F}}{\u27f6}A\left(4\right)+A\left(4\right)$ | catalyzed | $\kappa {k}_{F}{X}_{4}{V}^{-1}{X}_{2}({X}_{2}-1){V}^{-1}$ | $\kappa {k}_{F}{X}_{4}{V}^{-1}{X}_{2}^{2}$ |

$A\left(2\right)\stackrel{{k}_{R}}{\u27f6}A\left(1\right)+A\left(1\right)$ | spontaneous | ${k}_{R}{X}_{2}$ | ${k}_{R}{X}_{2}$ |

$A\left(2\right)+A\left(4\right)\stackrel{\kappa {k}_{R}}{\u27f6}A\left(1\right)+A\left(1\right)+A\left(4\right)$ | catalyzed | $\kappa {k}_{R}{X}_{4}{V}^{-1}{X}_{2}$ | $\kappa {k}_{R}{X}_{4}{V}^{-1}{X}_{2}$ |

$A\left(4\right)\stackrel{{k}_{R}}{\u27f6}A\left(2\right)+A\left(2\right)$ | spontaneous | ${k}_{R}{X}_{4}$ | ${k}_{R}{X}_{4}$ |

$A\left(4\right)+A\left(4\right)\stackrel{\kappa {k}_{R}}{\u27f6}A\left(2\right)+A\left(2\right)+A\left(4\right)$ | catalyzed | $\kappa {k}_{R}{X}_{4}{V}^{-1}({X}_{4}-1)$ | $\kappa {k}_{R}{X}_{4}^{2}{V}^{-1}$ |

$A\left(2\right)\stackrel{\varphi}{\u27f6}\xd8$ | degradation | $\varphi {X}_{2}$ | $\varphi {X}_{2}$ |

$A\left(4\right)\stackrel{\varphi}{\u27f6}\xd8$ | degradation | $\varphi {X}_{4}$ | $\varphi {X}_{4}$ |

## Appendix B. Single-Cell Residence Time and Interdivision Time Distributions for Active/Inactive States of the Protocell

#### Appendix B.1. Definition of an Active/Inactive State of a Protocell

#### Appendix B.2. Definition of Residence Time and Interdivision Time

**Figure A1.**Distribution of residence times (time spent) and interdivision time in active and inactive states. Data were collected by simulating a single lineage of growing and dividing protocells over 2000 division cycles. Parameter values: $\kappa =2400,\phantom{\rule{0.277778em}{0ex}}{k}_{F}=1,\phantom{\rule{0.277778em}{0ex}}\varphi =20,\phantom{\rule{0.277778em}{0ex}}\alpha =100$. The average values of the interdivision times are $\langle {\tau}_{1}\rangle =0.2947$ and $\langle {\tau}_{2}\rangle =0.0772$, while the average residence times in the two states are $\langle {T}_{1}\rangle =3.413$ and $\langle {T}_{2}\rangle =1.916$.

## Appendix C. The Steady State Fraction of ACS-Active Protocells (f) in the Protocell Population

#### Appendix C.1. Calculation of f for a System with Finite-Ceiling K on the Total Population

#### Appendix C.2. Condition for Active Protocells to Dominate the Population

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**Figure 1.**An illustration of a protocell inside an aqueous medium buffered with monomeric food molecules, $A{\left(1\right)}_{ext}$. The protocell membrane is composed of dimer molecules $A\left(2\right)$.

**Figure 2.**Bifurcation diagram for the model: steady state concentration, ${x}_{4}$, of the catalyst versus catalytic efficiency, $\kappa $. The region between ${\kappa}^{I}(=1840)$ and ${\kappa}^{II}(=3580)$ is the region having three fixed points, two of which are stable (solid black curves) and one is unstable (red dotted curve).

**Inset**: Growth rate, $\mu $, of the protocell, versus $\kappa $. Parameters: hereafter, ${k}_{R}$ and v have been set to unity without loss of generality after non-dimensionalizing the model. ${k}_{F}=1$, $\varphi =20$, $\alpha =100$.

**Figure 3.**Deterministic trajectories in the bistable region of the model. $\kappa =2400$; other parameters are as in Figure 2. The time, t, is in dimensionless units in this figure and subsequent figures (effectively t is measured in units of 1/${k}_{R}$). (

**A**) Phase portrait projected onto the ${x}_{2}-{x}_{4}$ plane. Several trajectories starting with different initial conditions are shown; they reach one of two stable fixed points denoted by blue closed dots. All the solid curve trajectories end at the stable fixed point on the top right (ACS-active), while all the dotted trajectories end at the stable fixed point on the bottom left of the plot (ACS-inactive). The red open dot represents an unstable fixed point. The dashed curve is a schematic of the basin boundary between the two stable fixed-point attractors. (

**B**) Deterministic trajectories of populations (in log scale) of species $A\left(1\right)$, $A\left(2\right)$, and $A\left(4\right)$ and the protocell volume as functions of time for two initial conditions. ${V}_{c}=1000$. Initial conditions: IC1 (lower panel; dotted curves): ${X}_{1}=952,{X}_{2}=20,{X}_{4}=2$. IC2 (upper panel; solid curves): ${X}_{1}=944,{X}_{2}=20,{X}_{4}=4$. Protocell starting with IC1 ends up in the inactive state, in which the population of the catalyst $A\left(4\right)$ is less than one, as seen in dotted red curve in the lower panel. Protocell starting with IC2 ends up in the active state, in which the population of the catalyst is high (approximately between 10 and 20). The interdivision times in the inactive and active states are, respectively, ${\tau}_{1}=0.269$, ${\tau}_{2}=0.075$.

**Figure 4.**Stochastic simulation of the populations of species A(1), A(2), and A(4) for a single protocell lineage in the model. Parameter values are as in Figure 3, ${V}_{c}=1000$. Initial condition: ${X}_{1}=480,\phantom{\rule{0.277778em}{0ex}}{X}_{2}=10,\phantom{\rule{0.277778em}{0ex}}{X}_{4}=0$. Note the transitions of the protocell between the inactive and active states. From a long such simulation we find that the average interdivision times in the inactive and active states are, respectively, $\langle {\tau}_{1}\rangle =0.295$, and $\langle {\tau}_{2}\rangle =0.077$, while the average residence times in the two states are $\langle {T}_{1}\rangle =3.413$, and $\langle {T}_{2}\rangle =1.916$.

**Figure 5.**Time evolution of a population of protocells starting from a single protocell in the inactive state. Shown is the number of protocells in the inactive state (green), active state (orange), and their sum (blue). As inactive protocells grow and divide, their population increases. The orange curve departs from zero when one of the inactive protocells makes a stochastic transition to the active state. The two populations have different growth rates. After the total population reaches an externally imposed ceiling K (=100 in this figure), upon each cell division a randomly chosen protocell is removed from the population. The population eventually settles down in a stochastic steady state dominated by the active protocells. This is the natural selection of an autocatalytic state. Parameter values are as in Figure 4, $K=100$.

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**MDPI and ACS Style**

Singh, A.Y.; Jain, S.
Multistable Protocells Can Aid the Evolution of Prebiotic Autocatalytic Sets. *Life* **2023**, *13*, 2327.
https://doi.org/10.3390/life13122327

**AMA Style**

Singh AY, Jain S.
Multistable Protocells Can Aid the Evolution of Prebiotic Autocatalytic Sets. *Life*. 2023; 13(12):2327.
https://doi.org/10.3390/life13122327

**Chicago/Turabian Style**

Singh, Angad Yuvraj, and Sanjay Jain.
2023. "Multistable Protocells Can Aid the Evolution of Prebiotic Autocatalytic Sets" *Life* 13, no. 12: 2327.
https://doi.org/10.3390/life13122327