From: rusin@vesuvius.math.niu.edu (Dave Rusin) Newsgroups: sci.math Subject: Re: probability and cells Date: 2 Oct 1997 05:31:57 GMT In article , Il Gabibbo wrote: >A cell has probability 2/3 to reproduce itself, and the remaining 1/3 >to die before reproducing. When the cell reproduces it splits in two >young new cells that have again the possibility to reproduse or to die >(the probabilities are the same). > >Will the cell have an endless descent? Will the descent terminate >sooner or later? How many probabilities are there this to happen? I'm not entirely convinced the premise is sufficiently well-grounded to allow us to answer the probability questions precisely. Some of the questions one might pose can be answered easily if we _assume_ there is an answer! For example, suppose the population P of the descendants of single cell attains a limiting expected value as time progresses; then this population is a weighted average of zero (for the unfortunate third of the cells which die) and P + P (for the others, who have two children) giving the equation P = (1/3)(0) + (2/3)(P+P), whose only solution is P=0. Thus, if there is an expected descendant-pool size, it has to be zero; but this reasoning does not rule out the possibility that the descendants are expected to be infinite, for example. Likewise, if there is a well-defined probability p that the family line will eventually die out, then this number is a weighted average of 1 (for the 1/3 of the population which dies out during this generation) and p^2 (for the rest of the population, whose family lines die out iff both of the children's lines do). The equation p = (1/3)(1) + (2/3)(p^2) has _two_ solutions, p=1 and p=1/2. So, either: almost every family dies out, or half of them do, or else there is no such well-defined probability p. (Note that this argument assumes that the events of family die-out for two siblings are independent, a premise which is difficult to defend on biological grounds!) The situation is, to me, a little clearer if there are well-defined generations. In that case, the population counts in each generation can be modelled as a stochastic process. After a cell has (died or) split once, there is an even number of cells; so we compute vectors v_n whose i-th coordinate is the probability that the population is 2i at the end of the n-th generation. Assuming independence of all events, the next vector v_(n+1) can be computed as a matrix product M*v_n, where the (i,j)-entry of M is the probability of changing from a population of 2j to a population of 2i at the transition between generations. (These are all infinite matrices -- with row- and column-indices starting at 0 -- but almost all entries in each column are zero, so there is no problem with convergence.) (The matrix M is readily computed: the first column is (1, 0, 0, ...)^t, since extinction is forever! Other columns are computed in turn using M_(i,j) = binomial(2j,i)* 2^i/3^(2j) = [ M(i,j-1) + 4 M(i-1,j-1) + 4 M(i-2,j-1) ]/9 . Then we compute v_1 = (1/3, 2/3, 0, 0, ...)^t v_2 = (11/27, 8/27, 8/27, 0, 0, ...)^t and so on.) The expected population of the descendant pool after n generations is the product (0, 1, 2, 3, ...) * v_n; it's easy to show this is (4/3)^n, so that in particular, we expect the descendant pool to grow without bound with the passage of time. Experiments with a few small values of n suggest that the i-th entry of v_n converges to 0 as n -> oo, except when i=0; the 0-th entry of v_n surely increases with n and is bounded by 1, so this sequence has a limit p; as noted above, this number must be either p=1/2 or p=1, and the data for small n certainly suggest the former is the correct value. If the v_n converged in any useful sense to a vector v we could conclude M*v=v. Unfortunately, pointwise convergence is _not_ useful and indeed M*v=v is impossible for any distribution vector v except v = (1, 0, 0, ...)^t . So my best answers to the original questions are: > Will the cell have an endless descent? Maybe (a 50-50 chance). > Will the descent terminate sooner or later? Later (that is, the expected no. of generations is infinite, since for half the population the descent never ends). I'm not sure how to measure expected time-to-extinction. > How many probabilities are there this to happen? I don't understand the question. dave