Coexistence vs collapse in transposon populations

Abstract

Transposons are small, self-replicating DNA sequences found in every branch of life. Often, one transposon will parasitize another, forming a tiny intracellular ecosystem. In some species these ecosystems thrive, while in others they go extinct, yet little is known about when or why this occurs. Here, we present a stochastic model for these ecosystems and discover a transition from stable coexistence to population collapse when the propensity for a transposon to replicate comes to exceed that of its parasites. Our model also predicts that replication rates should be low in equilibrium, which appears to be true of many transposons in nature.

FIG. 1:

Helitron replication mechanism. (a) The transposase binds to one end of the helitron and begins peeling one strand. (b) The transposase excises and circularizes the single strand of helitron DNA. The cell repairs the missing material. (c) The helitron is converted into double stranded form and moved to a new location in the genome. (d) The helitron is inserted into a new genomic location. Note that some details of this process remain unknown. Please see Refs. [–23] for more details.

FIG. 2:

Simulation of a stable system of 3 autonomous and 3 non-autonomous strains, each starting at ${\lambda }_i\left(0\right)=1$, with $t$ measured in arbitrary units. ${\alpha }_i,{\omega }_i,q$, and $r_i$ were randomly chosen for each strain. In this instance, ${\alpha }_i=\{.75,.36,.50,.51,.58,.56\}$, ${\omega }_i=\{0,0,0,.61,.77,.42\}$, $q=.00076,r_i=.0099$. Note that of the six initial strains, only two persist to $t=\mathrm{\infty }$, one autonomous and one non-autonomous.

FIG. 3:

Total transposon counts after long simulations. At each point, ${\lambda }_a+{\lambda }_n$ was evaluated at $t=10000$, by which time the system had usually equilibrated. Parameters satisfied ${\alpha }_a{\alpha }_n=1,r_a{r}_n={10}^{-4}$, and $q={10}^{-2}$. The dotted curves denote the phase boundaries ${\alpha }_a={\alpha }_n$ and $\frac{q+r_a}{{\alpha }_a}-\frac{r_n}{{\alpha }_n}$.

FIG. 4:

Simulation of a finite population of autonomous and non-autonomous elements. Parameters used: ${\lambda }_a\left(0\right)={\lambda }_n\left(0\right)=1,{\alpha }_a=1,{\alpha }_n=2,q=r_a=r_n={10}^{-2},N=4096$. (left) Dynamics of ${\lambda }_a$ and ${\lambda }_n$. The infinite population equilibrium values ${\lambda }_a^0$ and ${\lambda }_n^0$ are indicated by dashed lines, while the dotted lines denote fluctuations of size $\sqrt{4{\mathrm{\Sigma }}_{aa}}$ and $\sqrt{4{\mathrm{\Sigma }}_{nn}}$. (right) Orbit of ${\lambda }_a$ and ${\lambda }_n$ with fixed point and $4\mathrm{\Sigma }$ ellipse.