Punctuated model

In the driver model, each of the two daughter cells acquires kd ∼ Pois(md/2) driver mutations in each cell division, and can accumulate Nd driver mutations at maximum. Each driver mutation increases the cell division probability by f fold: g = g0fnd, where d0 is the base cell division probability and nd is the number of driver mutations accumulated in a cell.

Here, we incorporate punctuated evolution into the driver model to build the “punctuated” model. For the models considered so far, we assumed that cells can infinitely grows without a decrease of growth speed. However, it is more natural to assume that there exists a limit of population because of the resource limitation and the growth speed gradually slows down as the population size approaches the limit. The limit of population sizes called the carrying capacity is employed in the logistic equation. By mimicking the logistic equation, we modified the division probabilit as g = g0fnd(1 − p/pc), where pc is the carrying capacity. To reproduce punctuated evolution, we additionally employ an “explosive" driver mutation, which negates the effect of the carrying capacity. After a cell accumulates driver mutations up to the maximum Nd, the explosive driver mutation is introduced at a probability me. For a cell that has the explosive driver mutation, the carrying capacity pc is set to infinite; that is, it is assumed that the explosive driver mutation rapidly evolves the cell so that it can conquer the growth limit and attain infinite proliferation ability.

Information of variables and parameters are listed in Tables 1 and 2. In MASSIVE, we converted md,me, pc and P to log scale, i.e., md' = log10 md, me' = log10 me, pc' = log10 pc, and P' = log10 P, and then tested every combination of md' ∈ {-2, -1.5, -1}, me' ∈ {-6, -5.5, -5,...,-3}, pe' ∈ {3, 3.5, 4} and P' ∈ {3, 4, 5, 6}. All results are explorable in the focused and comparative view modes of the MASSIVE viewer.

Table 1. a list of the variabes

symbol description
kd number of driver mutations obtained in a cell division
nd number of driver mutations accumulated in a cell
p population size
t number of time steps
g cell division probability

Table 1. a list of the parameters

symbol description value
md expected number of driver mutations generated per cell division {10-2, 10-1.5, 10-1}
me probability of acquiring an explosive mutation {10-6, 10-5.5, 10-5,...,10-3}
Nd maximum number of driver mutations accumulated in a cell 3
f increase of the cell division probability per driver mutation 100.3
g0 base cell division probability 10-4
pc carrying capacity {103, 103.5, 104}
P maximum population size {103, 104, 105, 106}
T maximum number of time steps 106