Computing the Length of the Shortest Telomere in the Nucleus

The telomere length can either be shortened or elongated by an enzyme called telomerase after each cell division. Interestingly, the shortest telomere is involved in controlling the ability of a cell to divide. Yet, its dynamics remains elusive. We present here a stochastic approach where we model this dynamics using a Markov jump process. We solve the forward Fokker-Planck equation to obtain the steady state distribution and the statistical moments of telomere lengths. We focus specifically on the shortest one and we estimate its length difference with the second shortest telomere. After extracting key parameters such as elongation and shortening dynamics from experimental data, we compute the length of telomeres in yeast and obtain as a possible prediction the minimum concentration of telomerase required to ensure a proper cell division.

Figure 1

Telomere length in the yeast S. cerevisiae. (a) CDF of nucleotides added per elongation, fitted to Ref. 5. (b) Probability of elongation as a function of the telomere length computed from Eq. (2) and fitted to Ref. 5 (dots). (c) Equilibrium telomere length distribution simulated from Eq. (1) ($n=500$, 100 000 runs) compared with the analytical stationary distribution (red) computed from Eq. (7). (d) Average dynamics (10 000 runs) of short (150 bp) and long (600 bp) telomere return to steady state.

Figure 2

Stationary distribution of telomere length for various values of $\theta$ and $B$. Histograms of $\theta (X_n+1/B)$ using Eq. (1) (500 steps, 50 000 runs), compared with the scaled Gamma pdf of parameter $1+(1/B)$.

Figure 3

Shortest telomeres. (a) Mean shortest $X_{(1:2n)}=\theta (L_1+1/B)$ (circle) and second shortest telomere $X_{(2:2n)}=\theta (L_2+1/B)$ scaled lengths (square) as a function of $B$ ($\theta =0.02$, $n=50$, $\mathrm{\text{runs}}=10000$), compared with formula (12) (continuous line) and formula (16) (dotted line). (b) Simulated distribution of the two shortest telomere lengths $L_1$ and $L_2$ among 32 (100 000 runs) with the distribution of Fig. 1c (dotted line). (c) Comparing the ratio ${\overline{X}}_{(1:2n)}$ ${\overline{X}}_{(2:2n)}$ computed from formula (16) (continuous line) and numerically (square).

Figure 4

Probability to maintain the shortest telomere over a critical length. (a): We plot the length $L_q$ for which $P(L_{1:n}<L_q)=q$, as a function of $B$, with $q=0.01$. We indicate the critical value $B_c$ for which $L_q=L_c=120\mathrm{bp}$. (b) Frequency of elongation for normal value of $B$ and $B_c$ obtained using Eq. (17).