Each daughter cell can, in principle, take a different cell fate that contributes differently to the distribution of replicative ages (figure 1a cell self-renews symmetrically, both daughter cells stay in the same compartment and increase their cellular age by one (). (ii)?With probability a cell differentiates symmetrically, effectively removing it from the compartment of differentiated cells . (iii)?With probability 1 ? ? that might differ for each cellular age into the progenitor compartment to be constant over time. Using the above, we can formulate differential equations for the change of the number of cells in each age class = 1 + ? to be the self-renewal parameter which critically determines the most relevant results of our model. with little added variation. In the limiting case of a strict binary differentiation tree without self-renewal, the shape of the output distribution becomes indistinguishable from that of the input distribution. Our results suggest that a comparison of cellular age distributions between healthy and cancerous tissues may inform about dynamical changes within the hierarchical tissue structure, i.e. an acquired increased self-renewal capacity in certain tumours. Furthermore, we compare our theoretical results to telomere length distributions in granulocyte populations of 10 healthy individuals across different ages, highlighting that our theoretical expectations agree with experimental observations. cells of each replicative age class and after each division the replicative age of both daughter cells increases by one . Each daughter cell can, in principle, take a different cell fate that contributes differently to the distribution of replicative ages (figure 1a cell self-renews symmetrically, both daughter cells stay in the same compartment and increase their cellular age by one (). (ii)?With probability a cell differentiates symmetrically, effectively removing it from the compartment of differentiated cells . (iii)?With probability 1 ? ? that might differ for each cellular age into the progenitor compartment to be constant over time. Using the above, we can formulate differential equations for the change of the number of cells in each age class = 1 + ? to UNC569 be the self-renewal parameter which critically determines the most relevant results of our model. As and are probabilities with + 1, the self-renewal parameter can be in the range 0 2. However, as we are interested in homeostasis and not an exponentially growing tissue, the symmetric division probability in our case must be smaller than the symmetric differentiation probability and therefore 0 1. The above system of ordinary differential equations can be solved analytically (see appendix?E). However, as we assume that the dynamics on the level of stem cells is much slower compared to progenitor compartments, we can investigate the equilibrium LENG8 antibody solutions to equation?(2.1) for each age class UNC569 = 0 (see appendix?A). The general solution is 2.2 which is equivalent to a convolution sum of the influx and between zero and or by asymmetric division with probability 1 ? ? and go into the next downstream compartment. The compartment number is shown as superscript, the UNC569 total number of compartments is = 4. (Online version in colour.) To allow for multiple compartments, we can identify the output distribution UNC569 of a compartment and the input distribution of the next downstream compartment + 1, 2.3 2.1.1. Total cell outflux For our purpose, it is desirable to compare the effect of different tissue structures, that is a different number of total compartments and the self-renewal parameter such that the total output of cells remains constant, i.e. assuring certain replenishing needs of a UNC569 specific tissue. For this, we formulate differential equations for the change of the total number of cells in each of the compartments with a compartment-specific proliferation rate for each cell is the total influx into the first compartment (= 0) (i.e. the sum of all direct stem cell derived progenitors per time unit). The total outflux is related to the number of cells in the last compartment (see appendix?B): 2.4 This allows us to adjust the self-renewal parameter such that the outflux remains constant given an influx for any number of compartments 1 (see above section), the minimum amplification of cell production is given by corresponding to = 0. 2.2. Properties of the replicative age distribution 2.2.1. Mean and variance The mean and variance of the replicative age distribution under steady-state conditions can be calculated analytically, see appendix?C. The mean of the replicative age distribution in the progenitor compartment increases compared to the influx based on the self-renewal to where ?= is the average replicative age of the influx. Note that the average replicative age of the outflux = ?is increased by one to account for the extra differentiation step 2 2.5 The minimal increase of the mean between influx and outflux for no self-renewal (= 0) is therefore equal to one. The variance denotes the variance of the replicative age distribution of the influx. Generally, also the higher moments ?with in equation?(2.2) is and will therefore vanish for on replicative age, such that for all holds ? is not declining fast enough and is in the same order as.