# Compartmentalization and Cell Division through Molecular Discreteness and Crowding in a Catalytic Reaction Network

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model

_{x}× L

_{y}with periodic boundary conditions. Each site is empty or occupied by no more than one molecule to ensure appropriate representation of the exclusion volume of molecules. The species identity is represented by a “color” of the molecule, and replications of molecules occur based on the catalytic relationship between the species as described below. We adopt discrete simulation step, and update the system at each step by three processes: replication, degradation and diffusion. Within each of the processes, the following procedure is applied to every molecule in a random order.

**Figure 1.**Three update processes of our model. The left column shows the configuration before each process, and the right column displays the outcome of the process. (

**a**) Replication: If two catalytically related molecules are located next to each other, replication can occur, and the product molecule is added to one of the six neighboring sites (gray) if an empty site is available. (

**b**) Degradation: In each step, each molecule is removed from the system with a fixed probability. (

**c**) Diffusion: In each step, every molecule moves to one of the four nearest neighboring sites (gray) if a site is empty.

## 3. Simulation Results

#### 3.1. Two Mutually Catalyzing Molecules

_{X}and γ

_{Y}= 1 − γ

_{X}, respectively. Here we denote p

_{Y}= pγ

_{Y}and p

_{X}= pγ

_{X }. Degradations occur as X → 0 or Y → 0 with degradation probabilities a

_{X}and a

_{Y}, respectively.

_{ini}× L

_{ini}. Here, the value of L

_{ini}is fixed to 10; however, the exact dimensions are not important if they are sufficiently large.

_{Y}is much smaller than a

_{X}, the two Y molecules diffuse over a larger distance than the size of the cluster, forming a dumbbell-like cluster around the Y molecules and resulting in division of the cluster. Through division after the replication of Y , the number of X molecules also doubles (Figure 2b), and recursive growth by successive replications of Y molecules is possible. Here, we also note that this lattice model allows for a case in which the distance between the two Y molecules decreases again through diffusion, leading to a decrease in the number of X molecules as the effective replication rate of X decreases.

**Figure 2.**Division of a compartmentalized structure for the model of two mutually catalytic molecules X (green) and Y (red). Parameters are p

_{Y}= pγ

_{Y}= 5 × 10

^{−}

^{6}, p

_{X}= 1 − p

_{Y}, a

_{X}= 0.01, a

_{Y}= 0, and L

_{x}= L

_{y}= 1000. (

**a**) Snapshots of the system. Y molecules are highlighted by arrows. The time steps increase from top left to top right, and then from bottom left to bottom right. Snapshots are shown for every 500 steps from 25,000 to 27,500. (

**b**) Time evolution of the number of X and Y molecules corresponding to the data of (a).

_{X}, in the four nearest-neighboring sites of the single Y is shown in Figure 3, as a function of p and a

_{X}corrected for the change in number of Y molecules, i.e., replication/degradation of Y are absent in computation of the average crowdedness. There are two distinct planes n

_{X}= 0 and n

_{X}> 3.5, against the parameters p and a

_{X }. In the region n

_{X}= 0, no replication occurs and degradation of X leads to extinction. In the region n

_{X}> 3.5, localized structure appears. We note that, in simulations with parameters near the boundary between the two regions, the behavior of the system depends on samples: the localized stationary structure or extinction. As long as the system maintains the structure, the value of n

_{X}is approximately the fully-occupied value 4. These results suggest that the discreteness of Y molecules and the exclusion volume effect set an upper limit for the effective replication rate of X. If the degradation rate of X is smaller than the replication rate, replication of X results in the crowding of X around the single Y . If the degradation rate of X is greater than the replication rate, extinction of X is observed.

**Figure 3.**The average number of X, n

_{X}, in the four nearest-neighboring sites around a single Y molecule, plotted as a function of p and a

_{X}. Here, γ

_{Y}= a

_{Y}= 0.

_{X}, p

_{X}, a

_{Y}and p

_{Y}, we observe three types of behavior for the p

_{Y}-a

_{Y}plane: extinction, division and explosion (Figure 4). The division process is observed for smaller p

_{Y}and a

_{Y}in the region of Figure 4 designated by red circles. For larger p

_{Y}(explosion region; green squares), both molecules grow and maintain a single cluster with mixed configuration (Figure 4, right). For larger a

_{Y}(extinction; blue diamonds), Y degradation is followed by the degradation of all X molecules, resulting in the extinction of all molecules.

**Figure 4.**Three types of behavior from the initial condition in the p

_{Y}-a

_{Y}space. Here, p

_{Y}= pγ

_{Y}is changed by γ

_{Y}< 0.5 with p fixed to one. p

_{X}= 1 − p

_{Y}and a

_{X}= 0.01. The plane is divided into three regions: extinction (blue diamonds), division (red circles) and explosion (green squares). Typical snapshots are also shown for the division and explosion regions. In the left snapshot, the two existing Y molecules (red) are highlighted by arrows.

#### 3.2. Three Cyclically Catalyzing Molecules

_{X}, p

_{Y}, and p

_{Z}, respectively. Degradations also occur as X → 0, Y → 0 and Z → 0 with degradation probabilities a

_{X}, a

_{Y}and a

_{Z}, respectively.

_{ini}× L

_{ini}with randomly located X and Z molecules.

**Figure 5.**Snapshots of the system. The X, Y and Z molecules are respectively denoted by green, red and blue dots. The Y molecules are highlighted with arrows. The time steps increase from top left to top right, and then from bottom left to bottom right. Snapshots for every 1000 steps are shown.

**Figure 6.**Division in the three-species hypercycle. Parameters are p

_{X}= 1, p

_{Y}= 0.0001, p

_{Z}= 0.01, a

_{X}= 0.01, a

_{Y}= 0, a

_{Z}= 0.001. (

**a**) Time evolution of the number of X(green) and of Y (red) molecules. (

**b**) Time evolution of the number of Z(blue) and of Y (red) molecules.

_{Z}and a

_{Z}in which the localized structure appears when a single Y molecule is fixed is indicated by red circles in Figure 7. For smaller p

_{Z}or greater a

_{Z}values (blue diamonds), extinction of Z occurs and prevents the growth of the structure. On the other hand, greater p

_{Z}values (green squares) yield crowded Z molecules, effectively preventing replication of X and resulting in the extinction of X and subsequently extinction of Z. This result indicates that formation of a dividing cluster crucially depends on the relative replication and degradation rates between Y and the other species as in the two-species case, but also between X and Z molecules in this three-species case. In the parameter space of Figure 7, for example, p

_{Z}should be smaller than p

_{X}for the emergence of the cluster.

**Figure 7.**Three types of generated structures. A single Y molecule is fixed by suppressing changes in Y molecule (i.e., p

_{Y}= a

_{Y}= 0). (1) A cluster of Z molecules surrounding a cluster of X molecules around a single Y (red). (2) Extinction of both X and Z: crowding of Z results in extinction of X, and subsequent extinction of Z. (3) Extinction of Z occurs. Here, p

_{X}= 1, p

_{Y}= 0, a

_{X}= 0.01 and a

_{Y}= 0.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Kamimura, A.; Kaneko, K.
Compartmentalization and Cell Division through Molecular Discreteness and Crowding in a Catalytic Reaction Network. *Life* **2014**, *4*, 586-597.
https://doi.org/10.3390/life4040586

**AMA Style**

Kamimura A, Kaneko K.
Compartmentalization and Cell Division through Molecular Discreteness and Crowding in a Catalytic Reaction Network. *Life*. 2014; 4(4):586-597.
https://doi.org/10.3390/life4040586

**Chicago/Turabian Style**

Kamimura, Atsushi, and Kunihiko Kaneko.
2014. "Compartmentalization and Cell Division through Molecular Discreteness and Crowding in a Catalytic Reaction Network" *Life* 4, no. 4: 586-597.
https://doi.org/10.3390/life4040586