# RNA World Modeling: A Comparison of Two Complementary Approaches

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## Abstract

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## Simple Summary

## Abstract

## 1. Introduction

## 2. Materials and Methods

#### 2.1. RP Model

#### 2.2. Modeling with Partial Differential Equations (PDE)

#### 2.2.1. Replication

#### 2.2.2. Decay

#### 2.2.3. Resource Formation

#### 2.2.4. Diffusion

#### 2.2.5. PDE Model and Its Assumptions

#### 2.2.6. The Solutions for a PDE Model

#### Well-Mixed Solution

**Special cases**There are two special cases that have to be analyzed separately. They will be described briefly.

- $m=0$: replicases and parasites can coexist in any proportion, but only if ${a}_{R}{d}_{P}={a}_{P}{d}_{R}$;
- The denominator in Equation (22) is equal to 0. Unless the previous special case is also fulfilled, there is no solution.

#### Linear Stability

#### Evolution of a Well-Mixed Solution

- $P\left(y\right)$ has to have a root—an equilibrium point. Moreover, it has to be negative for any y bigger than it and positive for anything smaller;
- $Q\left(y\right)$ cannot be positive.

#### Mutation

#### Partial Differential Equations Model Robustness

- Diffusion: The presence of diffusion allows spatial interactions to occur. If there exists a stable, local equilibrium, this will be reached at every point and the system will become homogenous. However, for more complicated scenarios (such as the presence of mutation and the lack of a stable, local equilibrium), more complex structures can emerge and lead to the system’s survival. These cases are analyzed using computer simulations in the latter part of this article;
- Complex formation: In this article, the replication of RNA molecules is treated as an instantaneous reaction, whereas, in reality, it takes time. In order to account for this, a new type of molecule can be added to the model—a complex of replicase with the template representing the ongoing replication—just as in [41]. However, the results obtained in the aforementioned work and this article show very similar behavior; hence, the addition of complexes does not seem to have much influence;
- Different chemical reaction dynamics: reactions are assumed to follow the law of mass action. Due to the very complicated nature of complex molecules, especially in biology, replicases can behave differently depending on the density of resources and potential templates. The simplest case was assumed due to the fact that no functional replicase has been created in vitro as of yet; thus, their exact properties remain unknown. Changing the way the chemical reactions in the model work can radically change the outcome; for instance, replicases “programmed” to ignore parasites and only copy other replicases (but only if there are plenty of resources present) would be able to ensure the system’s survival without any additional mechanisms.

#### 2.3. Description of the Partial Differential Equation Simulation Algorithm

- p: the density of parasites;
- r: the density of replicases;
- a: the average ${a}_{P}$ of parasites;
- n: the density of resources necessary for replication.

Algorithm 1 Differential equation simulation |

Initialize two 2D arrays—$current$ and $next$ |

for all field f of $current$ do |

Set $f.p$, $f.r$, $f.k$ and $f.n$ to 1. |

end for |

for all simulation step $step$ do |

for all field f of $current$ do |

$newreplicases={a}_{R}\ast \Delta t\ast f.r\ast f.r\ast (1-m)\ast f.n$ |

$newparasites=\Delta t\ast f.r\ast ({a}_{R}\ast f.r\ast m+f.a\ast f.p)\ast f.n$ |

$resourcesused=newreplicases+newparasites$ |

if $resourcesused>0$ and $resourcesused>n$ then |

$newreplicases=newreplicases\ast n/resourcesused$ |

$newparasites=newparasites\ast n/resourcesused$ |

end if |

Compute average values for f’s adjacent neighbors: $avg\_r$, $avg\_p$, $avg\_n$ and $avg\_a$. |

Find the corresponing field of f in $next$ - $f2$. |

$f2.r=f.r+newreplicases-d\ast \Delta t\ast f.r+D\ast \Delta t\ast (avg\_r-f.r)$ |

$f2.p=f.p+newparasites-d\ast \Delta t\ast f.p+D\ast \Delta t\ast (avg\_p-f.p)$ |

$new\_a=f.a\ast f.p+D\ast \Delta t\ast (avg\_a-f.a\ast f.p)$ |

if $f.p>0$then |

$f2.a=new\_a/f.p$ |

else |

$f2.a=f.a$ |

end if |

$f2.n=f.n+{n}_{0}\ast \Delta t-newreplicases-newparasites+{D}_{n}\ast \Delta t\ast (avg\_n-f.n)$ |

Randomly change $f2.a$ according to mutation probability distribution. |

Set all negative variables of $f2$ to 0. |

end for |

Save results of the current simulation step. |

Swap $current$ and $next$. |

end for |

#### 2.4. Description of the Multi-Agent Approach and Simulation Algorithm (MAS)

#### 2.4.1. MAS Parameters

- Position;
- Type (replicase or parasite);
- ${k}_{P}$ (parasites only): the probability of being replicated.

#### 2.4.2. First-Order Reactions

#### 2.4.3. Second Order Reactions

- Two parasites: no reaction;
- Parasite and replicase: parasite becomes a template for replication and its ${k}_{P}$ becomes the probability of the reaction;
- Two replicases: the reaction occurs with the probability of ${k}_{R}$ (global parameter).

#### 2.4.4. Mutation

#### 2.4.5. Diffusion

Algorithm 2 Multi-agent algorithm |

Initialize the agents’ positions randomly. |

Initialize all parasites with equal ${k}_{P}$ values. |

while (simulation time < time limit) and (there are both parasites and replicases present) do |

Decrease the RLT time for each agent. |

for all agents with the RLT = 0 do |

Remove the agent. |

end for |

Randomize the order of all agents. |

for all agent ${x}_{i}$ do |

Initialize an empty set of neighbors N. |

for all agent that overlaps ${x}_{i}$ do |

Add it to the set of neighbors N |

end for |

if $\left|N\right|>{N}_{max}$ then |

Remove agent ${x}_{i}$ from the simulation. |

end if |

Randomize the order of N. |

Move ${x}_{i}$ by a random vector (Gaussian distribution with variance $2D\Delta t$). |

for all ${n}_{j}\in N$ do |

if (${n}_{j}$ or ${x}_{i}$ is a replicase) and reaction occurred (probability ${k}_{P}$/${k}_{R}$) then |

Create a copy of the template and place it in the same position. |

Initialize new agents’ RLT value. |

if New agent is a parasite then |

Mutate new agent’s ${k}_{P}$. |

end if |

end if |

end for |

end for |

end while |

## 3. Results

#### 3.1. Scenarios 1–8

Diffusion Constant | Mutation Rate | Result |
---|---|---|

5 | 0.1 | Alive (Figure 4) |

5 | 0.2 | Alive (Figure 5) |

10 | 0.1 | Alive (Figure 6) |

10 | 0.2 | Extinction |

15 | 0.1 | Extinction |

15 | 0.2 | Extinction |

20 | 0.1 | Extinction |

20 | 0.2 | Extinction |

#### 3.2. Scenario 9

#### 3.3. Scenario 10

#### 3.4. Scenario 11

#### 3.5. Scenarios 12 and 13

#### 3.6. The Summary of Scenarios 1–13

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**For y < yC, the system behaves like a pendulum with friction; it oscillates around the equilibrium, but these oscillations become weaker over time. Eventually, y becomes stable.

**Figure 2.**With y being past the critical point, the Q function becomes positive, thus increasing the velocity. y diverges to infinity, which means that the replicases die off (y is the inverse density of replicases).

**Figure 3.**The waves formed in multi-agent systems. They represent various populations in different phases of the cycle. Replicases are gray, whereas parasites have colors denoting their reaction rate, with replicases ranging from red (very high) to yellow, green, blue, and purple (very low) [41].

**Figure 4.**Waves for the diffusion constant equal to 5 and mutation rate equal to 0.1. Replicases are gray; parasites are purple.

**Figure 5.**Waves for the diffusion constant equal to 5 and mutation rate equal to 0.2. Replicases are gray; parasites are purple.

**Figure 6.**Waves for the diffusion constant equal to 10 and mutation rate equal to 0.1. Replicases are gray; parasites are purple.

**Figure 7.**A well-mixed system, where replicases and parasites peacefully coexist. Replicases are gray; parasites are blue.

**Figure 8.**The evolution of y (the inverse of the number of replicases multiplied by 10,000 for readability) over time is very similar to the results obtained from Equation (46). The value of y begins to oscillate, but this oscillation is quickly slowed down.

**Figure 9.**The evolution of aP at one particular point in space. This oscillated between values of $0.4$ and $0.6$.

**Figure 10.**The waves formed in differential equation simulation, when inequality (27) is not fulfilled. Dark areas are devoid of replicases, whereas white areas have a high density of them.

**Figure 11.**Scenario 12: the density of replicases and parasites. Replicases dominated the system because parasites had a lower reaction rate.

**Figure 12.**Scenario 13: the densities of replicases and parasites were equal, so the two graphs actually overlap.

Parameter Name | Default Value | Description |
---|---|---|

$sizeX$, $sizeY$ | 1000 | Simulation area size |

d | 0.01 | Decay rate |

${k}_{R}$ | 1.0 | Probability of replicases reaction with replicases |

$\Delta t$ | 0.1 | Single step length ($\Delta t$ in equations) |

D | 5.0 | Diffusion constant |

${D}_{n}$ | 10.0 | Resources diffusion constant |

${n}_{0}$ | 1.0 | Resources production rate |

m | 0.0 | Mutation of a replicase into a parasite probability |

Parameter Name | Default Value | Description |
---|---|---|

$sizeX$, $sizeY$ | 800 | Simulation area size |

$agent\_size$ | 3.0 | Radius of the circle representing an agent |

${N}_{max}$ | 4 | Maximum number of neighbors |

d | 0.1 | Decay rate |

${a}_{R}$ | 1.0 | Affinity of replicases towards replicases |

$\Delta t$ | 1.0 | Single step length |

D | 15.0 | Diffusion constant |

$\delta $ | 0.1 | Parasite mutation speed |

${m}_{P}$ | 0.1 | Parasite mutation probability |

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Synak, J.; Rybarczyk, A.; Blazewicz, J. RNA World Modeling: A Comparison of Two Complementary Approaches. *Entropy* **2022**, *24*, 536.
https://doi.org/10.3390/e24040536

**AMA Style**

Synak J, Rybarczyk A, Blazewicz J. RNA World Modeling: A Comparison of Two Complementary Approaches. *Entropy*. 2022; 24(4):536.
https://doi.org/10.3390/e24040536

**Chicago/Turabian Style**

Synak, Jaroslaw, Agnieszka Rybarczyk, and Jacek Blazewicz. 2022. "RNA World Modeling: A Comparison of Two Complementary Approaches" *Entropy* 24, no. 4: 536.
https://doi.org/10.3390/e24040536