How long do clades last? Can they last forever? In 1d, we can answer these questions by focusing on a clade’s endpoints.

Cells born between the endpoints will always descend from the clade. Cells born just outside the endpoints have equal odds of descending from the clade or not. As a result, the endpoints each follow a random walk forward in time.

Each generation, the clade shrinks if both endpoints move inward, and grows if both endpoints move outward.

In the continuum limit, the endpoints follow Brownian motions with variance 1/4 per unit time. Hence, the size of the clade follows a Brownian motion with variance 1/2 per unit time.

At each moment, the clade’s future persistence time has the same distribution as the endpoints’ coalescence time—the hitting time distribution \(\rho^\text{hit}_r\) for the clade’s current width \(r\). Keep in mind, though, that the clade will often be much older than its endpoints’ last common ancestor!