A Hierarchical Probability Model of Colon Cancer

We consider a model of fixed size $N = 2^l$ in which there are $l$ generations of daughter cells and a stem cell. In each generation $i$ there are $2^{i-1}$ daughter cells. At each integral time unit the cells split so that the stem cell splits into a stem cell and generation 1 daughter cell and the generation $i$ daughter cells become two cells of generation $i+1$. The last generation is removed from the population. The stem cell gets first and second mutations at rates $u_1$ and $u_2$ and the daughter cells get first and second mutations at rates $v_1$ and $v_2$. We find the distribution for the time it takes to get two mutations as $N$ goes to infinity and the mutation rates go to 0. We also find the distribution for the location of the mutations. Several outcomes are possible depending on how fast the rates go to 0. The model considered has been proposed by Komarova (2007) as a model for colon cancer

Grief: Putting the Past before Us

Grief research in philosophy agrees that one who grieves grieves over the irreversible loss of someone whom the griever loved deeply, and that someone thus factored centrally into the griever’s sense of purpose and meaning in the world. The analytic literature in general tends to focus its treatments on the paradigm case of grief as the death of a loved one. I want to restrict my account to the paradigm case because the paradigm case most persuades the mind that grief is a past-directed emotion. The phenomenological move I propose will enable us to respect the paradigm case of grief and a broader but still legitimate set of grief-generating states of affairs, liberate grief from the view that grief is past directed or about the past, and thus account for grief in a way that separates it from its closest emotion-neighbor, sorrow, without having to rely on the affective quality of those two emotions.If the passing of the beloved causes the grief but is not what the grief is about, then we can get at the nature of grief by saying its temporal orientation is in the past, but its temporal meaning is the present and future—the new significance of a world with the pervasive absence that is the world without the beloved. The no-longer of grief is a no-longer oriented by a past that is referred a present and future. Looking at the griever’s relation to time can tell us much about the pain and the object of grief, then. As the griever puts the past before himself with a certainty about this world “henceforth,” a look at the griever’s lived sense of the fi nality of the irreversibly lost liberates grief from the tendency in the literature to be reduced to a past-directed emotion, accounts for grief ’s intensity, its affective force or poignancy, and thus enables us to separate grief from sorrow according to its intentionalobject in light of the temporal meaning of these emotions