Elsevier

Biosystems

Volume 91, Issue 1, January 2008, Pages 262-267
Biosystems

A computational model for telomere-dependent cell-replicative aging

https://doi.org/10.1016/j.biosystems.2007.10.003Get rights and content

Abstract

Telomere shortening provides a molecular basis for the Hayflick limit. Recent data suggest that telomere shortening also influence mitotic rate. We propose a stochastic growth model of this phenomena, assuming that cell division in each time interval is a random process which probability decreases linearly with telomere shortening.

Computer simulations of the proposed stochastic telomere-regulated model provides good approximation of the qualitative growth of cultured human mesenchymal stem cells.

Introduction

Telomere are specialized nucleoproteins involved in protection and stabilization of chromosomes ends. They contain tandem repetitive arrays of DNA which in vertebrates is the sequence (TTAGGG)n. The overall human telomere sizes range from 15 kb at birth to less than 5 kb at chronic disease states (Shay and Wright, 2005). With cell division, telomere lose TTAGGG repeats mainly by incomplete replication of linear chromosomes by conventional DNA polymerases (Greider, 1996). So, most of somatic cells experience progressive telomeric shortening with cell division.

There are compelling evidences that cells count past divisions, and that maturation and aging depends on the number of such divisions. The most likely counting mechanisms appears to be telomere shortening. There are also evidences that bellow a certain telomere-length cells cease to divide (replicative senescence) (Shin et al., 2006). This mechanism provides a good explanation for the Hayflick limit (Hayflick, 1965).

Recent in vitro experiments has shown that telomere shortening correlates negatively with cell proliferation in cultured mesenchymal cells. In this setting, there is a gradual decrease both in telomere length and population-doubling rate Baxter et al., 2004, Bonab et al., 2006. Furthermore, reduction in telomeric length was found to be linearly correlated with the proliferative capacity of cells (Gupta et al., 2007).

In vivo research also indicates that telomere shortening negatively affects cellular proliferation. This phenomena has been verified in recent study on mice tumoral growth. Moreover, this effect was also observed under apoptosis inhibition, as described in Feldser and Greider (2007). These results were long awaited and may have important implications on future cancer therapy Zimmermann and Martens, 2007, Sedivy, 2007.

Our main purpose is to corroborate telomere-dependent replicative aging by a stochastic model and computer simulations. Cell division is regarded as a random process which probability decreases linearly with telomere shortening, yielding a stochastic telomere-regulated growth model. Computer simulations of this model match recent data from cultured human mesenchymal stem cells (hMSCs). Human MSCs can be readily isolated from bone marrow and expanded in culture (Caplan and Bruder, 2001). They have attracted much attention because of their regenerative potential, since these cells can be differentiated into chondrogenic, osteogenic, adipogenic and miogenic lineages (Pansky et al., 2007).

Deterministic Gompertzian growth provides a very good fitness for somatic Laird, 1965, Laird et al., 1965, Begall, 1997 and tumoral growth Laird, 1969, Norton, 1988, Sullivan and Salmon, 1972, Demicheli, 1980. The stochastic telomere-regulated model presented in this work also provides a good approximation to Gompertzian growth, both in computer simulations and theoretical ground.

Section snippets

A stochastic telomere-regulated growth model

It is well known (Lin and Yan, 2005) that a cell can not divide if its telomere is shorter than some minimal length. Very short telomere length correspond to crisis cell (Greider, 1996). It is also known that cells lose 30–150 bases of telomere in each cell division (Lin and Yan, 2005). In view of this facts, we will build our model considering the following assumptions:

A1

There is an initial cell population of N0 cells, all of them having the same telomere length L0.

A2

At each cell division, the

Gompertzian growth as an ODE approximation of the stochastic model

In a continuous time model, assumption A4 must be replaced by an exponential distribution model:

A4c

For a cell of telomere length L[0,L0] at time t, division at τt is a random process with an exponential distribution:p(τ)=λ(L)expλ(L)(τt),τ[t,)whereλ(L)=α(LLmin).

For a cell with with telomere length L>Lmin, the mean time to divide isT¯=1λ(L)=α(LLmin)1.

Hence, for a large cell population N with telomere length L>Lmin at t0:dNdt=λ(L)N=α(LLmin)N.

After k synchronized division cycles

From the deterministic to the stochastic model

In the stochastic model, the inverse of the expected time interval between successive mitosis is the probability of mitosis in a time unit. Therefore, it is relevant to investigate this magnitude in a discrete deterministic model which follows a Gompertzian dynamics. Here, discrete is related to the size of cell population, which shall be integer. This is the aim of this section. We will show that for such a deterministic dynamics, the inverse of the time interval (between successive mitosis)

Discussion

In the current work we presented a model of how telomere shortening correlates with growth and replicative senescence. Somatic growth is a complex biological phenomenon that is controlled by many mechanisms. Culture of cell offers a simplified setting for the study of growth and replicative aging. The growth pattern of cultured human stromal cells was qualitatively similar to the process generated by our stochastic model. We assume that telomere shorten at each cell division and no telomerase

Acknowledgements

We thanks the anonymous referee for the suggestions and criticism which improved this work. We also thanks Sergio Telles, MD for his enlightening discussions on telomere shortening and aging process.

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    Partially supported by CNPq grants 300755/2005-8, 475647/2006-8 and by PRONEX-Optimization.

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