# Population Dynamics of Autocatalytic Sets in a Compartmentalized Spatial World

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## Abstract

**:**

## 1. Introduction

## 2. Background

- Reflexively autocatalytic (RA): each reaction in $\mathcal{R}$ is catalysed by at least one molecule from $\mathcal{R}$ itself; and
- F-generated (F): all reactants in $\mathcal{R}$ can be created from some food set F by using a sequence of reactions from $\mathcal{R}$ itself.

## 3. Methods

## 4. Results

#### 4.1. Dynamics of a Single Compartment

#### 4.2. Dynamics of a Population of Compartments

#### 4.3. The Influence of a Toxic Element

#### 4.4. The Influence of a Permeable Inducer

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Simulation Parameters

Parameter | Value |
---|---|

Figure 4, Figure 5 and Figure 6 | |

Spontaneous rates | |

c${}_{100}$,c${}_{110}$ | 0.02 |

c${}_{11100}$,c${}_{111}$,c${}_{010}$ | 0.003 |

c${}_{1010}$,c${}_{0111}$ | 0.005 |

c${}_{00100}$ | 0.001 |

Catalysed rates | |

c${}_{11100}$,c${}_{111}$,c${}_{010}$ | 0.005 |

c${}_{1010}$ | 0.03 |

c${}_{00100}$ | 0.02 |

c${}_{0111}$ | 0.01 |

Figure 6 | |

Catalysed rates | |

c${}_{01111}$ | 0.005 |

Figure 7 | |

Spontaneous rates | |

c${}_{110}$,c${}_{011}$ | 0.0001 |

Catalysed rates | |

c${}_{110}$,c${}_{011}$ | 0.05 |

c${}_{[011+110\phantom{\rule{4.pt}{0ex}}->\phantom{\rule{4.pt}{0ex}}01+1+110]}$ | 0.05 |

c${}_{[0+11+01\phantom{\rule{4.pt}{0ex}}->\phantom{\rule{4.pt}{0ex}}011+01]}$ | 0.5 |

Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9 | |

World size | 20 × 20 |

Compartments | 100 |

Compartment radius | 0.5 |

Voxel size | 2.5 × 2.5 |

Numerical time step | 0.01 |

Figure 5 | |

Diffusion coefficients | |

D${}_{0}$,D${}_{1}$,D${}_{00}$,D${}_{01}$,D${}_{10}$,D${}_{11}$ | 20 |

D${}_{010}$,D${}_{100}$,D${}_{110}$,D${}_{111}$,D${}_{1010}$,D${}_{00100}$,D${}_{01111}$,D${}_{11100}$ | 10 |

Decay rate | |

K${}_{0}$,K${}_{1}$,K${}_{00}$,K${}_{01}$,K${}_{10}$,K${}_{11}$ | 0.013 |

K${}_{010}$,K${}_{100}$,K${}_{110}$,K${}_{111}$,K${}_{1010}$,K${}_{00100}$,K${}_{01111}$,K${}_{11100}$ | 0 |

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**Figure 1.**RAF set example.

**Left**: an example of an RAF set as found in an instance of the binary polymer model. Black dots (labeled with bit strings) represent the molecule types, and white boxes represent reactions. Solid black arrows indicate molecules going into and coming out of a reaction, while dashed gray arrows indicate catalysis. Coloured polygones indicate some of the RAF subsets (see text).

**Right**: the six closed RAFs (colour coded) and their mutual subset relationships.

**Figure 2.**Flow diagram of the simulation algorithm. After initializing the model with N compartments, the outer loop iterates the system through time. In each iteration, the algorithm first solves for diffusion and decay of molecules in the environment using a finite difference approximation to the Fick equation. Then, for each compartment in the simulation, intracompartment molecular counts are updated by first solving permeation processes, and then running a Gillespie algorithm within each compartment. The algorithm is stopped after T time units have been simulated.

**Figure 3.**The basic simulation setup. A 16 × 16 grid with 100 randomly distributed compartments (black spheres) and concentration of food molecules (blue spheres) throughout the grid.

**Left**: shortly after starting the simulation.

**Right**: after an equilibrium distribution of food molecules has been reached.

**Figure 4.**A single compartment.

**Left**: a simulation run with a single compartment where the red RAF subset appears first.

**Right**: a simulation run with a single compartment where the blue RAF subset appears first. Insets: the sequence of compartment colour changes in the simulations. Numbers indicate at which time steps during the simulation the respective colour changes happened.

**Figure 5.**A population of compartments. Four snapshots over time from a simulation with 100 compartments.

**Figure 6.**The influence of a toxic element. The production of a toxic element by the red RAF subset can cause the blue RAF subset to be lost again, making a compartment change from blue to purple to red (indicated by the white circle).

**Figure 7.**A reaction network with an inducer. The reaction network used to show the influence of an inducer (molecule type 01). As before, dashed arrows indicate catalysis.

**Figure 8.**The influence of a permeable inducer. Three snapshots over time from a simulation where the RAF set produces an permeable inducer that can diffuse through the lattice. The blue spheres indicate the concentration of this inducer in the different grid locations.

**Figure 9.**Compartment counts.

**Left**: a comparison of compartment type counts between the base case and the influence of a toxic element.

**Right**: a comparison of yellow compartment counts with or without the inducer.

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**MDPI and ACS Style**

Hordijk, W.; Naylor, J.; Krasnogor, N.; Fellermann, H.
Population Dynamics of Autocatalytic Sets in a Compartmentalized Spatial World. *Life* **2018**, *8*, 33.
https://doi.org/10.3390/life8030033

**AMA Style**

Hordijk W, Naylor J, Krasnogor N, Fellermann H.
Population Dynamics of Autocatalytic Sets in a Compartmentalized Spatial World. *Life*. 2018; 8(3):33.
https://doi.org/10.3390/life8030033

**Chicago/Turabian Style**

Hordijk, Wim, Jonathan Naylor, Natalio Krasnogor, and Harold Fellermann.
2018. "Population Dynamics of Autocatalytic Sets in a Compartmentalized Spatial World" *Life* 8, no. 3: 33.
https://doi.org/10.3390/life8030033