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. 2014 Mar 11;111(10):E880-7.
doi: 10.1073/pnas.1324267111. Epub 2014 Feb 5.

Mathematical model of adult stem cell regeneration with cross-talk between genetic and epigenetic regulation

Jinzhi Lei  1 , Simon A Levin, Qing Nie
Affiliations

Mathematical model of adult stem cell regeneration with cross-talk between genetic and epigenetic regulation

Jinzhi Lei et al. Proc Natl Acad Sci U S A. .

Abstract

Adult stem cells, which exist throughout the body, multiply by cell division to replenish dying cells or to promote regeneration to repair damaged tissues. To perform these functions during the lifetime of organs or tissues, stem cells need to maintain their populations in a faithful distribution of their epigenetic states, which are susceptible to stochastic fluctuations during each cell division, unexpected injury, and potential genetic mutations that occur during many cell divisions. However, it remains unclear how the three processes of differentiation, proliferation, and apoptosis in regulating stem cells collectively manage these challenging tasks. Here, without considering molecular details, we propose a genetic optimal control model for adult stem cell regeneration that includes the three fundamental processes, along with cell division and adaptation based on differential fitnesses of phenotypes. In the model, stem cells with a distribution of epigenetic states are required to maximize expected performance after each cell division. We show that heterogeneous proliferation that depends on the epigenetic states of stem cells can improve the maintenance of stem cell distributions to create balanced populations. A control strategy during each cell division leads to a feedback mechanism involving heterogeneous proliferation that can accelerate regeneration with less fluctuation in the stem cell population. When mutation is allowed, apoptosis evolves to maximize the performance during homeostasis after multiple cell divisions. The overall results highlight the importance of cross-talk between genetic and epigenetic regulation and the performance objectives during homeostasis in shaping a desirable heterogeneous distribution of stem cells in epigenetic states.

Keywords: dynamic programming; fitness function; optimization; robustness; systems biology.

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Conflict of interest statement

The authors declare no conflict of interest.

Figures

Fig. 1.
Fig. 1.
Model Illustration. (A) At the tth cell cycle, cells in the resting phase either enter the proliferating phase with the probability of formula image, or differentiate into other cell types with the probability of formula image. The proliferating cells undergo apoptosis with the probability of formula image. Resting phase cells occasionally migrate to the quiescent phase and vice versa under stress. (B) The performance function formula image quantifies how well the tissue fits to its physiological properties. The changes in the tissue state formula image at each cell cycle are determined by the three quantities formula image chosen to maximize the performance at the next cycle to give formula image. An evolutionary fitness function at homeostasis, denoted by W, is the limit of formula image when formula image.
Fig. 2.
Fig. 2.
Distribution of cells at homeostasis under three different combinations of the epigenetic regulation. (A) Both formula image and formula image are independent of x, and formula image changes with x. (Inset) The performance function formula image is shown. (B) formula image is independent of x, and formula image and formula image change with x. (C) Both formula image and formula image are independent of x, and formula image changes with x. Shadow regions formula image represent defective states. (D) Time course of Nt under the three conditions (red, green, and blue for conditions A–C, respectively). (See SI Text, section S5 for details on simulations.)
Fig. 3.
Fig. 3.
Recovery of the cell population and distribution of epigenetic states after a sudden loss of half of the total population of cells. (A) Cell population time courses. (B) The function formula image at formula image cell cycles after the sudden loss. (C) The function formula image at formula image cell cycles after the sudden loss. Three different controls are shown: strategy A (red), B (greed), and C (blue). For strategy C, the Hill coefficient m varies from 1 to 10 (from bottom to top in A). The cell populations at homeostasis are normalized to their maximum levels. See SI Text, section S5 for other parameters used in simulations.
Fig. 4.
Fig. 4.
Tissue response to temporal changes in differentiation. (A) Cell population time courses under three different strategies for proliferation. The red dashed line represents strategy A, the green solid line strategy B, and the blue dashed/dotted line strategy C with the Hill coefficient formula image. (B) Time course of the number of differentiated cells formula image. (C) Time course of average differentiation formula image. Shadows indicate the time window of increasing differentiation. (D) Cell distributions (strategy A) at three time points (marked with arrows in C), before (filled circles, formula image), during (dashed line, formula image), and after (solid line, formula image) the temporal change of differentiation. (E) Same as D but using strategy B.
Fig. 5.
Fig. 5.
Evolution dynamics for different control strategies. Time is measured by the number of mutations. Results for strategies A (green), B (black) and C (magenta) are shown using the average of 10 independent sample evolution dynamics. (A) Fitness during evolution. (B) The cell performance function formula image. Inset shows formula image based on strategy B. (C) Cell population N. (D) The cell population when formula image is used as the evolutionary fitness. Inset shows the dynamics for strategy B in which the population size markedly increases after 1,000 mutations. (E) Time course of formula image when formula image is used as the evolutionary fitness.
Fig. 6.
Fig. 6.
An example of evolutionary dynamics of the apoptosis formula image and the proliferation formula image following strategy B. (A) The evolution of formula image, with the initial formula image for each x. (B) The proliferation formula image that is initiated from formula image. (C) The density formula image during homeostasis at two time points of mutations, indicated by dashed lines in B. Evolutionary fitness W, population number N, cell performance formula image, and proliferation probability formula image are also given in each case.
See this image and copyright information in PMC

Comment in

  • Collective dynamics of stem cell populations.
    MacArthur BD. MacArthur BD. Proc Natl Acad Sci U S A. 2014 Mar 11;111(10):3653-4. doi: 10.1073/pnas.1401030111. Epub 2014 Mar 3. Proc Natl Acad Sci U S A. 2014. PMID: 24591612 Free PMC article. No abstract available.

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