Influence of telomerase activity and initial distribution on human follicular aging: Moving from a discrete to a continuum model.

A discrete model is proposed for the temporal evolution of a population of cells sorted according to their telomeric length. This model assumes that, during cell division, the distribution of the genetic material to daughter cells is asymmetric, i.e. chromosomes of one daughter cell have the same telomere length as the mother, while in the other daughter cell telomeres are shorter. Telomerase activity and cell death are also taken into account. The continuous model is derived from the discrete model by introducing the generational age as a continuous variable in [0,h], being h the Hayflick limit, i.e. the number of times that a cell can divide before reaching the senescent state. A partial differential equation with boundary conditions is obtained. The solution to this equation depends on the initial telomere length distribution. The initial and boundary value problem is solved exactly when the initial distribution is of exponential type. For other types of initial distributions, a numerical solution is proposed. The model is applied to the human follicular growth from preantral to preovulatory follicle as a case study and the aging rate is studied as a function of telomerase activity, the initial distribution and the Hayflick limit. Young, middle and old cell-aged initial normal distributions are considered. In all cases, when telomerase activity decreases, the population ages and the smaller the h value, the higher the aging rate becomes. However, the influence of these two parameters is different depending on the initial distribution. In conclusion, the worst-case scenario corresponds to an aged initial telomere distribution.

Mathematical modelling of ageing acceleration of the human follicle due to oxidative stress and other factors.

There is a gradual telomere shortening due to the inability of the replication machinery to copy the very ends of chromosomes. Additionally, other factors such as high levels of oxidation (free radicals or reactive oxygen species (ROS)), e.g. due to cumulated stress, inflammation or tobacco smoke, accelerate telomere shortening. In humans, the average telomere length is about 10-15 kb at birth and telomeres shorten at a pace of 70 bp per year. However, when cells are exposed to ROS, telomere attrition happens at a faster pace, generating a wide variety of telomere size distribution in different length percentiles, which are different to what is expected just by age. In this work, the generational age of a cell is associated with its telomere length (TL), from certain maximum to the minimal TL that allows replication. In order to study the accumulation of aged granulosa cells in human follicles, from preantral to preovulatory size, a mathematical model is proposed, regarding different degrees of accelerated telomere shortening, which reflect the action of ROS in addition to the telomere shortening that happens after cell division. In cases of cells with TL shorter than cells with average TL, with low telomerase activity and accelerated telomere shortening, the mathematical model predicts an aged outcome in preovulatory follicles. The model provides a plausible explanation for what has been observed in oocytes from older women, which have been exposed to ROS for a longer period of time and have bad outcomes after in vitro fertilization.

Mathematical model to study the aging of the human follicle according to the telomerase activity.

The aim of this work is to study the aging rate at which human follicles reach the preovulatory state. To this end, both telomere length and telomerase activity effects on granulosa cells (GCs) aging has been studied. GCs are somatic cells which determine the development of the oocyte. A human preantral follicle takes approximately 85 days to achieve the preovulatory size, going through several stages (Gougeon, 1996). The telomere length of GCs of each class of follicles, during folliculogenesis, are modelled using a chemical master equation formalism similar to the one in Wesch et al. (2016). Seven differential ordinary systems of equations, corresponding to seven stages of the follicule maturation, concatenated in time, are considered. The mitotic and death rates are approximated by using the mean number of GCs in each class of follicles and the time they remain on each stage. The influence of different telomerase activity rates and the telomere shortening of the preovulatory follicle is studied. Some cases of infertility are associated with low levels of telomerase activity and short telomeres in GCs. The method aims at understanding how low levels of telomerase activity in preovulatory stages lead to the accumulation of aged GCs. In the case of higher telomerase activities, the mathematical model predicts a more juvenile outcome in preovulatory follicles. Juvenile GCs, could be critical for embryo development if the oocyte were fertilized, since GCs, transformed in corpus luteum, must divide and increase their size (Alila and Hansel, 1984) to sustain early pregnancy (Csapo et al., 1972).