Neutral-s model

Next, we examined the neutral evolution mode by analyzing the “neutral” model, where we considered only neutral mutations that do not affect cell division and death. In a unit time, a cell divides into two daughter cells with a constant probability g0 without dying. Similarly to driver mutations in the driver model, in each cell division, each of the two daughter cells acquires kn ∼ Pois(mn/2) neutral mutations. We assumed that neutral mutations acquired by different division events occur at different genomic positions and each cell can accumulate Nn mutations at maximum. In this study, we set Nn = 1000, which is sufficiently large that no cell reaches at the upper limit, except in a few exceptional cases. The simulation started from one cell without mutations and ended when population size p reached P or time t reached T.

Furthermore, the neutral model is exteneded to the “neutral-s model", which assumes that two types of cell exist: stem and differentiated. Stem cells divide with a probability go without dying. For each cell division of stem cells, a symmetrical division generating two stem cells occurs with a probability s, while an asymmetrical division generating one stem cell and one differentiated cell occurs with a probability 1 − s. A differentiated cell symmetrically divides to generate two differentiated cells with a probability g0 but dies with a probability d0d. The means of accumulating neutral mutations in the two types of cell is the same as that in the original neutral model, which means that the neutral-s model is equal to the original neutral model when s = 0 or d0d=0.

Information of variables and parameters are listed in Tables 1 and 2. For MASSIVE, we defined δ = log10(d0d/g0) and converted mn, s and P to log scale, i.e., mn' = log10 mn', s' = log10 s and P' = log10 P. We then tested every combination of δ ∈ {-0.5, -0.25, 0, 0.25, 0.5}, mn' ∈ {-2, -1, 0, 1}, s' ∈ {-2, -1.5, -1, -0.5, 1} and P' ∈ {3, 4, 5}. All results are explorable in the focused and comparative view modes of the MASSIVE viewer.

Table 1. a list of the variabes

symbol description
kn number of neutral mutations obtained in a cell division
nn number of neutral mutations accumulated in a cell
p population size
t number of time steps

Table 1. a list of the parameters

symbol description value
mn expected number of neutral mutations generated per cell division {10-2, 10-1.5, 10-0.5,...,101}
Nn maximum number of neutral mutations accumulated in a cell 1000
g0 base cell division probability 10-2
d0d base cell death probability for differentiated cells {10-2.5, 10-2.25, 10-2,10-1.75, 10-1.5}
s symmetric division probability {10-2, 10-1.5, 10-1.0,10-0.5, 100}
P maximum population size {103, 104, 105}
T maximum number of time steps 106