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An inverse problem in cell dynamics: Recovering an initial distribution of telomere lengths from measurements of senescence times

Jules Olayé

TL;DR

The paper tackles the inverse problem of recovering the initial telomere-length distribution from observed senescence times. It develops a pair of deterministic telomere-shortening models (one- and multi-telomere) and shows that, under a large-scale shortening regime, these nonlocal integro-differential equations can be approximated by transport equations. The authors derive estimators for the initial distribution using the method of characteristics and establish pointwise and Lebesgue-space error bounds that decay exponentially in time, with explicit dependence on the dimension and scaling. They extend the analysis to the multidimensional telomere setting, discuss the curse of dimensionality, and validate the approach with simulations and experimental data, while outlining limitations and directions for improving observability and extending to diffusion-augmented or extreme-value-informed models. Overall, the work provides rigorous justifications for transport-based inversions in telomere-shortening dynamics and offers practical estimation procedures with quantified guarantees and highlighted caveats.

Abstract

Telomeres are repetitive sequences situated at both ends of the chromosomes of eukaryotic cells. At each cell division, they are eroded until they reach a critical length that triggers a state in which the cell stops to divide: the senescent state. In this work, we are interested in the link between the initial distribution of telomere lengths and the distribution of senescence times. We propose a method to retrieve the initial distribution of telomere lengths, using only measurements of senescence times. Our approach relies on approximating our models with transport equations, which provide natural estimators for the initial telomere lengths distribution. We investigate this method from a theoretical point of view by providing bounds on the errors of our estimators, pointwise and in all Lebesgue spaces. We also illustrate it with estimations on simulations, and discuss its limitations related to the curse of dimensionality.

An inverse problem in cell dynamics: Recovering an initial distribution of telomere lengths from measurements of senescence times

TL;DR

The paper tackles the inverse problem of recovering the initial telomere-length distribution from observed senescence times. It develops a pair of deterministic telomere-shortening models (one- and multi-telomere) and shows that, under a large-scale shortening regime, these nonlocal integro-differential equations can be approximated by transport equations. The authors derive estimators for the initial distribution using the method of characteristics and establish pointwise and Lebesgue-space error bounds that decay exponentially in time, with explicit dependence on the dimension and scaling. They extend the analysis to the multidimensional telomere setting, discuss the curse of dimensionality, and validate the approach with simulations and experimental data, while outlining limitations and directions for improving observability and extending to diffusion-augmented or extreme-value-informed models. Overall, the work provides rigorous justifications for transport-based inversions in telomere-shortening dynamics and offers practical estimation procedures with quantified guarantees and highlighted caveats.

Abstract

Telomeres are repetitive sequences situated at both ends of the chromosomes of eukaryotic cells. At each cell division, they are eroded until they reach a critical length that triggers a state in which the cell stops to divide: the senescent state. In this work, we are interested in the link between the initial distribution of telomere lengths and the distribution of senescence times. We propose a method to retrieve the initial distribution of telomere lengths, using only measurements of senescence times. Our approach relies on approximating our models with transport equations, which provide natural estimators for the initial telomere lengths distribution. We investigate this method from a theoretical point of view by providing bounds on the errors of our estimators, pointwise and in all Lebesgue spaces. We also illustrate it with estimations on simulations, and discuss its limitations related to the curse of dimensionality.
Paper Structure (63 sections, 27 theorems, 176 equations, 7 figures)

This paper contains 63 sections, 27 theorems, 176 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Biological motivation.
  3. Presentation of the problem.
  4. Informal description of the models.
  5. Difficulty.
  6. Estimation strategy.
  7. Review of the literature.
  8. Main contributions.
  9. Organisation of the paper.
  10. Presentation of the models and the main result
  11. The models
  12. Model with one telomere.
  13. Model with $2k$ telomeres.
  14. Assumptions and main result
  15. Assumptions.
  16. ...and 48 more sections

Key Result

Theorem 2.6

We recall the constants $\lambda_N$ and $\lambda'_N$ defined in eq:approximation_eigenvalues. The following statements hold.

Figures (7)

  • Figure 1: Estimation results in the single-telomere model for different values of $N$, when $b = 1$, $g = 1_{[0,1]}$ and $n_0\in\left\{h_{1,4},h_{2,1.5}\right\}$.
  • Figure 2: Estimation results in the single-telomere model when $b = 1$, $g = 1_{[0,1]}$, $N = 40$, and $n_0\in\{\mathcal{H}_{1/2}, \mathcal{H}_{1/3}, \mathcal{H}_{1/5}, \mathcal{H}_{1/7}\}$.
  • Figure 3: Estimation results in the model with several telomeres when $b = 1$, $g = 1_{[0,1]}$, $N = 40$, $n_0=h_{1,4}$, and $k\in\{1,3,5,15,30,50\}$.
  • Figure 4: Estimation results for the estimator $\overline{n}_0^{(1,\alpha_{0.1})}$ defined in \ref{['eq:estimators_simulations_onetelo']}, when $b = 1$, $g = 1_{[0,1]}$, $N = 40$, $n_0 \in \left\{h_{1,4},h_{2,1.5}\right\}$, and for different values of $n_s$.
  • Figure 5: Estimation results for the estimator $\overline{n}_0^{(2k,\alpha)}$ defined in \ref{['eq:estimators_simulations_several_telos']}, when $b = 1$, $g = 1_{[0,1]}$, $N = 40$, $n_s = 3000$, $\alpha = 0.275$ and $\left(n_0,k\right) \in\left\{\left(h_{1,4},5\right),\left(h_{2,1.5},16\right)\right\}$.
  • ...and 2 more figures

Theorems & Definitions (50)

  • Remark 2.1
  • Remark 2.2
  • Example 2.3
  • Remark 2.4
  • Remark 2.5
  • Theorem 2.6: Main result
  • Remark 2.7
  • Corollary 2.8: Estimation errors in Lebesgue spaces
  • Remark 3.1
  • Proposition 3.2: Pointwise approximation errors, one telomere
  • ...and 40 more