Horizontal transfer between loose compartments stabilizes replication of fragmented ribozymes

Abstract

The emergence of replicases that can replicate themselves is a central issue in the origin of life. Recent experiments suggest that such replicases can be realized if an RNA polymerase ribozyme is divided into fragments short enough to be replicable by the ribozyme and if these fragments self-assemble into a functional ribozyme. However, the continued self-replication of such replicases requires that the production of every essential fragment be balanced and sustained. Here, we use mathematical modeling to investigate whether and under what conditions fragmented replicases achieve continued self-replication. We first show that under a simple batch condition, the replicases fail to display continued self-replication owing to positive feedback inherent in these replicases. This positive feedback inevitably biases replication toward a subset of fragments, so that the replicases eventually fail to sustain the production of all essential fragments. We then show that this inherent instability can be resolved by small rates of random content exchange between loose compartments (i.e., horizontal transfer). In this case, the balanced production of all fragments is achieved through negative frequency-dependent selection operating in the population dynamics of compartments. The horizontal transfer also ensures the presence of all essential fragments in each compartment, sustaining self-replication. Taken together, our results underline compartmentalization and horizontal transfer in the origin of the first self-replicating replicases.

Fig 1. Schematic of the model.

(A) The fragments X, Y, and their catalysts C are encapsulated into N_cell cells, and they undergo the reactions (1) to (3) to increase their volume. When the volume of a cell exceeds a threshold V_Div, the cell divides and the components inside are randomly partitioned into two daughter cells. At the same time, one randomly-chosen cell is removed to fix the total number of cells. (B) With the transfer, the components are transferred among the cells [denoted by blue arrows] with a transfer constant D through the time-course of the model shown in (A).

Fig 2. Sets of division threshold V_Div and the number of cells N_cell with which the unstable replication of reactions (1)–(3) is avoided by (A) compartmentalization alone and (B) that with horizontal transfer of the transfer constant D = 0.01.

For the sets shown as stable [red circles], the system can continuously have cells with both fragments in the simulations up to 4 × 10⁵ division events from an initial condition where V_Div/4 copies of each X and Y are in each cell. For the sets shown as unstable [blue squares], all cells with both fragments are lost from the system and it cannot continue growth. For the sets located at the boundary of stable and unstable area [shown in red triangles], the outcome depends on simulation runs.

Fig 3. (i) The number of fragments X_tot and Y_tot of dividing cells and (ii) the number of X-dominant and Y-dominant cells at corresponding time for the transfer constants (A) D = 0 (B) D = 0.001 (C) D = 0.01 and (D) D = 0.02.

Initially, the numbers of X_tot and Y_tot are approximately equal and, as time goes on, cells are differentiated into either of X-dominant or Y-dominant compositions. For D = 0 (A), the system is unstable: only X-dominant cells (for this run) dominate (ii) and finally, cells cannot continue growth. For D = 0.001 (B) and 0.01 (C), the system is stable; X and Y fragments coexist in each cell with unequal population (i). Here, the asymmetry between the major and minor fragments gets smaller as D increases. In addition, the two types of cells, X-dominant and Y-dominant cells coexist with the equal population (ii). As D increases further [D = 0.02 (D)], the system gets unstable and only either of X or Y remains (ii). The parameters are V_Div = 1000, N_cell = 100, k^f = k^b = 1, and k_x = k_y = 1.

Fig 4. The concentrations of minor fragments x and x_tot at division events for Y-dominant cells as a function of D.

Free [red] and Total [green] indicate x and x_tot, respectively. For the free fragments [x], the results of simulations [red curves with points for V_Div = 10³ and 10⁴] agree well with the solution $x^2=x_{tot}^2-c^2$ from Eq (13) [red curve]. For the total fragments [x_tot], the simulations [green curves with points for V_Div = 10³ and 10⁴] agree with the solution $x_{tot}^2$ of Eq (13) [green curve] for larger D, but shift to larger values for smaller D. This is because cells must possess at least one catalyst to divide, therefore, the total fragments including c shift to larger values as it approaches the minimum requirement. Note that $x_{tot}^2<1/2$, and the concentration of major fragments is obtained by $x_{tot}^1=1-x_{tot}^2$. For reference, the values of x_tot = 1/V_Div at which the number of c is equal to one for V_Div = 10³ and 10⁴ are shown by horizontal dotted lines, respectively. The other parameters are N_cell = 100, k^f = k^b = 1, and k_x = k_y = k = 1.

Fig 5. Flow diagram of Eq (10).

As schematically indicated in the left-top panel, the nullclines are shown for ${\dot{x}}_{tot}^1=0$ and ${\dot{x}}_{tot}^2=0$ in blue and orange, respectively, and the crossing points of them are solutions. The directions of v₁ = (1, 1) and v₂ = (1, −1) are also indicated. For the solutions, stable fixed points are shown in red: those with stable growth [i.e., both fragments are in each subsystem] are in red circles, and those without growth [either of fragments is lost from subsystems or whole systems] are in red triangles. Unstable solutions are in light-blue squares, and neutral solutions in the v₁-direction are in green stars at D = 0.02 (E). For D = 0 (A), the solution exists at $(x_{tot}^1,x_{tot}^2)=(1/2,1/2)$ but it is unstable. For small values of D (B to D), the stable fixed points with growth [red circles] appear in addition to fixed points without growth. At D = 0.02 (E), the fixed points with growth get unstable [shown in green stars] in v₁-directions. As D increases further (F), the two fixed points are still stable in v₂-directions, while the solution at $(x_{tot}^1,x_{tot}^2)=(1/2,1/2)$ is unstable in the direction. At D = 0.03125 (G), the system transits from the three fixed points to one fixed point.