Mathematical modelling of telomere length dynamics

On Discretely Structured Growth Models and Their Moments

The logistic equation is ubiquitous in applied mathematics as a minimal model of saturating growth. Here, we examine a broad generalisation of the logistic growth model to discretely structured populations, motivated by examples that range from the ageing of individuals in a species to immune cell exhaustion by cancerous tissue. Through exploration of a range of concrete examples and a general analysis of polynomial kinetics, we derive necessary and sufficient conditions for the dependence of the kinetics on structure to result in closed, low-dimensional moment equations that are exact. Further, we showcase how coarse-grained moment information can be used to elucidate the details of structured dynamics, with immediate potential for model selection and hypothesis testing. This paper belongs to the special collection: Problems, Progress and Perspectives in Mathematical and Computational Biology.

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 telomere length dynamics

Telomeres are repetitive DNA sequences located at the ends of chromosomes. During cell division, an incomplete copy of each chromosome's DNA is made, causing telomeres to shorten on successive generations. When a threshold length is reached replication ceases and the cell becomes 'senescent'. In this paper, we consider populations of telomeres and, from discrete models, we derive partial differential equations which describe how the distribution of telomere lengths evolves over many generations. We initially consider a population of cells each containing just a single telomere. We use continuum models to compare the effects of various mechanisms of telomere shortening and rates of cell division during normal ageing. For example, the rate (or probability) of cell replication may be fixed or it may decrease as the telomeres shorten. Furthermore, the length of telomere lost on each replication may be constant, or may decrease as the telomeres shorten. Where possible, explicit solutions for the evolution of the distribution of telomere lengths are presented. In other cases, expressions for the mean of the distribution are derived. We extend the models to describe cell populations in which each cell contains a distinct subpopulation of chromosomes. As for the simpler models, constant telomere shortening leads to a linear reduction in telomere length over time, whereas length-dependent shortening results in initially rapid telomere length reduction, slowing at later times. Our analysis also reveals that constant telomere loss leads to a Gaussian (normal) distribution of telomere lengths, whereas length-dependent loss leads to a log-normal distribution. We show that stochastic models, which include a replication probability, also lead to telomere length distributions which are skewed.

Stochastic simulations of normal aging and Werner's syndrome

Human cells typically consist of 23 pairs of chromosomes. Telomeres are repetitive sequences of DNA located at the ends of chromosomes. During cell replication, a number of basepairs are lost from the end of the chromosome and this shortening restricts the number of divisions that a cell can complete before it becomes senescent, or non-replicative. In this paper, we use Monte Carlo simulations to form a stochastic model of telomere shortening to investigate how telomere shortening affects normal aging. Using this model, we study various hypotheses for the way in which shortening occurs by comparing their impact on aging at the chromosome and cell levels. We consider different types of length-dependent loss and replication probabilities to describe these processes. After analyzing a simple model for a population of independent chromosomes, we simulate a population of cells in which each cell has 46 chromosomes and the shortest telomere governs the replicative potential of the cell. We generalize these simulations to Werner's syndrome, a condition in which large sections of DNA are removed during cell division and, amongst other conditions, results in rapid aging. Since the mechanisms governing the loss of additional basepairs are not known, we use our model to simulate a variety of possible forms for the rate at which additional telomeres are lost per replication and several expressions for how the probability of cell division depends on telomere length. As well as the evolution of the mean telomere length, we consider the standard deviation and the shape of the distribution. We compare our results with a variety of data from the literature, covering both experimental data and previous models. We find good agreement for the evolution of telomere length when plotted against population doubling.