October 2017, 14(5&6): 1379-1397. doi: 10.3934/mbe.2017071

A mathematical model of stem cell regeneration with epigenetic state transitions

Zhou Pei-Yuan Center for Applied Mathematics, Tsinghua University, Beijing 100084, China

* Corresponding author: Jinzhi Lei

Received  May 18, 2016 Accepted  January 15, 2017 Published  May 2017

Fund Project: This work is supported by National Natural Science Foundation of China (91430101 and 11272169)

In this paper, we study a mathematical model of stem cell regeneration with epigenetic state transitions. In the model, the heterogeneity of stem cells is considered through the epigenetic state of each cell, and each epigenetic state defines a subpopulation of stem cells. The dynamics of the subpopulations are modeled by a set of ordinary differential equations in which epigenetic state transition in cell division is given by the transition probability. We present analysis for the existence and linear stability of the equilibrium state. As an example, we apply the model to study the dynamics of state transition in breast cancer stem cells.

Citation: Qiaojun Situ, Jinzhi Lei. A mathematical model of stem cell regeneration with epigenetic state transitions. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1379-1397. doi: 10.3934/mbe.2017071
References:
[1]

R. C. AdamH. YangS. RockowitzS. B. LarsenM. NikolovaD. S. OristianL. PolakM. KadajaA. AsareD. Zheng and E. Fuchs, Pioneer factors govern super-enhancer dynamics in stem cell plasticity and lineage choice, Nature, 521 (2015), 366-370. doi: 10.1038/nature14289.

[2]

M. AdimyF. Crauste and S. Ruan, A mathematical study of the hematopoiesis process with applications to chronic myelogenous leukemia, SIAM journal on applied mathematics, 65 (2005), 1328-1352. doi: 10.1137/040604698.

[3]

M. AdimyF. Crauste and S. Ruan, Stability and Hopf bifurcation in a mathematical model of pluripotent stem cell dynamics, Nonlinear Analysis: Real World Applications, 6 (2005), 651-670. doi: 10.1016/j.nonrwa.2004.12.010.

[4]

M. AdimyF. Crauste and S. Ruan, Periodic oscillations in leukopoiesis models with two delays, J Theor Biol, 242 (2006), 288-299. doi: 10.1016/j.jtbi.2006.02.020.

[5]

S. BernardJ. Bélair and M. C. Mackey, Oscillations in cyclical neutropenia: New evidence based on mathematical modeling, J Theor Biol, 223 (2003), 283-298. doi: 10.1016/S0022-5193(03)00090-0.

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F. J. Burns and I. F. Tannock, On the existence of a G0-phase in the cell cycle, Cell Proliferation, 3 (1970), 321-334.

[7]

H. H. ChangM. HembergM. BarahonaD. E. Ingber and S. Huang, Transcriptome-wide noise controls lineage choice in mammalian progenitor cells, Nature, 453 (2008), 544-547. doi: 10.1038/nature06965.

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S. J. Corey, M. Kimmel and J. N. Leonard (eds.), A Systems Biology Approach to Blood, vol. 844 of Advances in Experimental Medicine and Biology, Springer, London, 2014.

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D. C. Dale and M. C. Mackey, Understanding, treating and avoiding hematological disease: better medicine through mathematics?, Bull Math Biol, 77 (2015), 739-757. doi: 10.1007/s11538-014-9995-x.

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D. DingliA. Traulsen and J. M. Pacheco, Stochastic dynamics of hematopoietic tumor stem cells, Cell Cycle (Georgetown, Tex), 6 (2007), 461-466. doi: 10.4161/cc.6.4.3853.

[11]

B. DykstraD. KentM. BowieL. McCaffreyM. HamiltonK. LyonsS.-J. LeeR. Brinkman and C. Eaves, Long-term propagation of distinct hematopoietic differentiation programs in vivo, Stem Cell, 1 (2007), 218-229. doi: 10.1016/j.stem.2007.05.015.

[12]

T. M. Gibson and C. A. Gersbach, Single-molecule analysis of myocyte differentiation reveals bimodal lineage commitment, Integr Biol (Camb), 7 (2015), 663-671. doi: 10.1039/C5IB00057B.

[13]

P. B. GuptaC. M. FillmoreG. JiangS. D. ShapiraK. TaoC. Kuperwasser and E. S. Lander, Stochastic State Transitions Give Rise to Phenotypic Equilibrium in Populations of Cancer Cells, Cell, 146 (2010), 633-644.

[14]

K. HayashiS. M. C. de Sousa LopesF. Tang and M. A. Surani, Dynamic equilibrium and heterogeneity of mouse pluripotent stem cells with distinct functional and epigenetic states, Stem Cell, 3 (2008), 391-401. doi: 10.1016/j.stem.2008.07.027.

[15]

G. M. HuC. Y. LeeY.-Y. ChenN. N. Pang and W. J. Tzeng, Mathematical model of heterogeneous cancer growth with an autocrine signalling pathway, Cell Prolif, 45 (2012), 445-455. doi: 10.1111/j.1365-2184.2012.00835.x.

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D. Huh and J. Paulsson, Non-genetic heterogeneity from stochastic partitioning at cell division, Nat Genet, 43 (2011), 95-100. doi: 10.1038/ng.729.

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A. D. LanderK. K. GokoffskiF. Y. M. WanQ. Nie and A. L. Calof, Cell lineages and the logic of proliferative control, PLoS biology, 7 (2009), e15-e15.

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J. LeiS. A. Levin and Q. Nie, Mathematical model of adult stem cell regeneration with cross-talk between genetic and epigenetic regulation, Proc Natl Acad Sci USA, 111 (2014), E880-E887. doi: 10.1073/pnas.1324267111.

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J. Lei and M. C. Mackey, Stochastic differential delay equation, moment stability, and application to hematopoietic stem cell regulation system, SIAM journal on applied mathematics, 67 (2007), 387-407. doi: 10.1137/060650234.

[20]

J. Lei and M. C. Mackey, Multistability in an age-structured model of hematopoiesis: Cyclical neutropenia, J Theor Biol, 270 (2011), 143-153. doi: 10.1016/j.jtbi.2010.11.024.

[21]

J. Lei and C. Wang, On the reducibility of compartmental matrices, Comput Biol Med, 38 (2008), 881-885. doi: 10.1016/j.compbiomed.2008.05.004.

[22]

B. D. MacArthur, Collective dynamics of stem cell populations, Proc Natl Acad Sci USA, 111 (2014), 3653-3654. doi: 10.1073/pnas.1401030111.

[23]

B. D. MacArthur and I. R. Lemischka, Statistical mechanics of pluripotency, Cell, 154 (2013), 484-489. doi: 10.1016/j.cell.2013.07.024.

[24]

M. C. Mackey, Unified hypothesis for the origin of aplastic anemia and periodic hematopoiesis, Blood, 51 (1978), 941-956.

[25]

M. C. Mackey, Cell kinetic status of haematopoietic stem cells, Cell Prolif, 34 (2001), 71-83. doi: 10.1046/j.1365-2184.2001.00195.x.

[26]

M. Mangel and M. B. Bonsall, Phenotypic evolutionary models in stem cell biology: Replacement, quiescence, and variability, PLoS ONE, 3 (2008), e1591. doi: 10.1371/journal.pone.0001591.

[27]

M. Mangel and M. B. Bonsall, Stem cell biology is population biology: Differentiation of hematopoietic multipotent progenitors to common lymphoid and myeloid progenitors, Theor Biol Med Model, 10 (2012), 5-5. doi: 10.1186/1742-4682-10-5.

[28]

C. S. Potten and M. Loeffler, Stem cells: Attributes, cycles, spirals, pitfalls and uncertainties. Lessons for and from the crypt, Development, 110 (1990), 1001-1020.

[29]

A. V. ProbstE. Dunleavy and G. Almouzni, Epigenetic inheritance during the cell cycle, Nat Rev Mol Cell Biol, 10 (2009), 192-206. doi: 10.1038/nrm2640.

[30]

J. E. PurvisK. W. KarhohsC. MockE. BatchelorA. Loewer and G. Lahav, p53 dynamics control cell fate, Science, 336 (2012), 1440-1444. doi: 10.1126/science.1218351.

[31]

J. E. Purvis and G. Lahav, Encoding and decoding cellular information through signaling dynamics, Cell, 152 (2013), 945-956. doi: 10.1016/j.cell.2013.02.005.

[32]

A. RezzaZ. WangR. SennettW. QiaoD. WangN. HeitmanK. W. MokC. ClavelR. YiP. ZandstraA. Ma'ayan and M. Rendl, Signaling networks among stem cell precursors, transit-amplifying progenitors, and their niche in developing hair follicles, Cell Rep, 14 (2016), 3001-3018. doi: 10.1016/j.celrep.2016.02.078.

[33]

I. Rodriguez-BrenesN. Komarova and D. Wodarz, Evolutionary dynamics of feedback escape and the development of stem-cell–driven cancers, Proc Natl Acad Sci USA, 108 (2011), 18983-18988.

[34]

P. Rué and A. Martinez-Arias, Cell dynamics and gene expression control in tissue homeostasis and development, Mol Syst Biol, 11 (2015), 792-792.

[35]

T. SchepelerM. E. Page and K. B. Jensen, Heterogeneity and plasticity of epidermal stem cells, Development, 141 (2014), 2559-2567. doi: 10.1242/dev.104588.

[36]

Z. S. SingerJ. YongJ. TischlerJ. A. HackettA. AltinokM. A. SuraniL. Cai and M. B. Elowitz, Dynamic heterogeneity and DNA methylation in embryonic stem cells, Mol Cell, 55 (2014), 319-331. doi: 10.1016/j.molcel.2014.06.029.

[37]

K. Takaoka and H. Hamada, Origin of cellular asymmetries in the pre-implantation mouse embryo: A hypothesis, Philos Trans R Soc Lond B Biol Sci, 369 (2014). doi: 10.1098/rstb.2013.0536.

[38]

J. E. Till, E. A. McCulloch and L. Siminovitch, A Stochastic Model of Stem Cell Proliferation, Based on the Growth of Spleen Colony-Forming Cells, in Proceedings of the National Academy of Sciences of the United States of America, 1963, 29–36. doi: 10.1073/pnas.51.1.29.

[39]

A. TraulsenT. LenaertsJ. M. Pacheco and D. Dingli, On the dynamics of neutral mutations in a mathematical model for a homogeneous stem cell population, Journal of the Royal Society, Interface / the Royal Society, 10 (2013), 20120810-20120810. doi: 10.1098/rsif.2012.0810.

[40]

H. Wu and Y. Zhang, Reversing DNA methylation: Mechanisms, genomics, and biological functions, Cell, 156 (2014), 45-68. doi: 10.1016/j.cell.2013.12.019.

[41]

M. Zernicka-GoetzS. A. Morris and A. W. Bruce, Making a firm decision: Multifaceted regulation of cell fate in the early mouse embryo, Nat Rev Genet, 10 (2009), 467-477. doi: 10.1038/nrg2564.

[42]

X.-P. ZhangF. LiuZ. Cheng and W. Wang, Cell fate decision mediated by p53 pulses, Proc Natl Acad Sci USA, 106 (2009), 12245-12250. doi: 10.1073/pnas.0813088106.

[43]

D. ZhouD. WuZ. LiM. Qian and M. Q. Zhang, Population dynamics of cancer cells with cell state conversions, Quant Biol, 1 (2013), 201-208. doi: 10.1007/s40484-013-0014-2.

[44]

C. ZhugeX. Sun and J. Lei, On positive solutions and the Omega limit set for a class of delay differential equations, DCDS-B, 18 (2013), 2487-2503. doi: 10.3934/dcdsb.2013.18.2487.

show all references

References:
[1]

R. C. AdamH. YangS. RockowitzS. B. LarsenM. NikolovaD. S. OristianL. PolakM. KadajaA. AsareD. Zheng and E. Fuchs, Pioneer factors govern super-enhancer dynamics in stem cell plasticity and lineage choice, Nature, 521 (2015), 366-370. doi: 10.1038/nature14289.

[2]

M. AdimyF. Crauste and S. Ruan, A mathematical study of the hematopoiesis process with applications to chronic myelogenous leukemia, SIAM journal on applied mathematics, 65 (2005), 1328-1352. doi: 10.1137/040604698.

[3]

M. AdimyF. Crauste and S. Ruan, Stability and Hopf bifurcation in a mathematical model of pluripotent stem cell dynamics, Nonlinear Analysis: Real World Applications, 6 (2005), 651-670. doi: 10.1016/j.nonrwa.2004.12.010.

[4]

M. AdimyF. Crauste and S. Ruan, Periodic oscillations in leukopoiesis models with two delays, J Theor Biol, 242 (2006), 288-299. doi: 10.1016/j.jtbi.2006.02.020.

[5]

S. BernardJ. Bélair and M. C. Mackey, Oscillations in cyclical neutropenia: New evidence based on mathematical modeling, J Theor Biol, 223 (2003), 283-298. doi: 10.1016/S0022-5193(03)00090-0.

[6]

F. J. Burns and I. F. Tannock, On the existence of a G0-phase in the cell cycle, Cell Proliferation, 3 (1970), 321-334.

[7]

H. H. ChangM. HembergM. BarahonaD. E. Ingber and S. Huang, Transcriptome-wide noise controls lineage choice in mammalian progenitor cells, Nature, 453 (2008), 544-547. doi: 10.1038/nature06965.

[8]

S. J. Corey, M. Kimmel and J. N. Leonard (eds.), A Systems Biology Approach to Blood, vol. 844 of Advances in Experimental Medicine and Biology, Springer, London, 2014.

[9]

D. C. Dale and M. C. Mackey, Understanding, treating and avoiding hematological disease: better medicine through mathematics?, Bull Math Biol, 77 (2015), 739-757. doi: 10.1007/s11538-014-9995-x.

[10]

D. DingliA. Traulsen and J. M. Pacheco, Stochastic dynamics of hematopoietic tumor stem cells, Cell Cycle (Georgetown, Tex), 6 (2007), 461-466. doi: 10.4161/cc.6.4.3853.

[11]

B. DykstraD. KentM. BowieL. McCaffreyM. HamiltonK. LyonsS.-J. LeeR. Brinkman and C. Eaves, Long-term propagation of distinct hematopoietic differentiation programs in vivo, Stem Cell, 1 (2007), 218-229. doi: 10.1016/j.stem.2007.05.015.

[12]

T. M. Gibson and C. A. Gersbach, Single-molecule analysis of myocyte differentiation reveals bimodal lineage commitment, Integr Biol (Camb), 7 (2015), 663-671. doi: 10.1039/C5IB00057B.

[13]

P. B. GuptaC. M. FillmoreG. JiangS. D. ShapiraK. TaoC. Kuperwasser and E. S. Lander, Stochastic State Transitions Give Rise to Phenotypic Equilibrium in Populations of Cancer Cells, Cell, 146 (2010), 633-644.

[14]

K. HayashiS. M. C. de Sousa LopesF. Tang and M. A. Surani, Dynamic equilibrium and heterogeneity of mouse pluripotent stem cells with distinct functional and epigenetic states, Stem Cell, 3 (2008), 391-401. doi: 10.1016/j.stem.2008.07.027.

[15]

G. M. HuC. Y. LeeY.-Y. ChenN. N. Pang and W. J. Tzeng, Mathematical model of heterogeneous cancer growth with an autocrine signalling pathway, Cell Prolif, 45 (2012), 445-455. doi: 10.1111/j.1365-2184.2012.00835.x.

[16]

D. Huh and J. Paulsson, Non-genetic heterogeneity from stochastic partitioning at cell division, Nat Genet, 43 (2011), 95-100. doi: 10.1038/ng.729.

[17]

A. D. LanderK. K. GokoffskiF. Y. M. WanQ. Nie and A. L. Calof, Cell lineages and the logic of proliferative control, PLoS biology, 7 (2009), e15-e15.

[18]

J. LeiS. A. Levin and Q. Nie, Mathematical model of adult stem cell regeneration with cross-talk between genetic and epigenetic regulation, Proc Natl Acad Sci USA, 111 (2014), E880-E887. doi: 10.1073/pnas.1324267111.

[19]

J. Lei and M. C. Mackey, Stochastic differential delay equation, moment stability, and application to hematopoietic stem cell regulation system, SIAM journal on applied mathematics, 67 (2007), 387-407. doi: 10.1137/060650234.

[20]

J. Lei and M. C. Mackey, Multistability in an age-structured model of hematopoiesis: Cyclical neutropenia, J Theor Biol, 270 (2011), 143-153. doi: 10.1016/j.jtbi.2010.11.024.

[21]

J. Lei and C. Wang, On the reducibility of compartmental matrices, Comput Biol Med, 38 (2008), 881-885. doi: 10.1016/j.compbiomed.2008.05.004.

[22]

B. D. MacArthur, Collective dynamics of stem cell populations, Proc Natl Acad Sci USA, 111 (2014), 3653-3654. doi: 10.1073/pnas.1401030111.

[23]

B. D. MacArthur and I. R. Lemischka, Statistical mechanics of pluripotency, Cell, 154 (2013), 484-489. doi: 10.1016/j.cell.2013.07.024.

[24]

M. C. Mackey, Unified hypothesis for the origin of aplastic anemia and periodic hematopoiesis, Blood, 51 (1978), 941-956.

[25]

M. C. Mackey, Cell kinetic status of haematopoietic stem cells, Cell Prolif, 34 (2001), 71-83. doi: 10.1046/j.1365-2184.2001.00195.x.

[26]

M. Mangel and M. B. Bonsall, Phenotypic evolutionary models in stem cell biology: Replacement, quiescence, and variability, PLoS ONE, 3 (2008), e1591. doi: 10.1371/journal.pone.0001591.

[27]

M. Mangel and M. B. Bonsall, Stem cell biology is population biology: Differentiation of hematopoietic multipotent progenitors to common lymphoid and myeloid progenitors, Theor Biol Med Model, 10 (2012), 5-5. doi: 10.1186/1742-4682-10-5.

[28]

C. S. Potten and M. Loeffler, Stem cells: Attributes, cycles, spirals, pitfalls and uncertainties. Lessons for and from the crypt, Development, 110 (1990), 1001-1020.

[29]

A. V. ProbstE. Dunleavy and G. Almouzni, Epigenetic inheritance during the cell cycle, Nat Rev Mol Cell Biol, 10 (2009), 192-206. doi: 10.1038/nrm2640.

[30]

J. E. PurvisK. W. KarhohsC. MockE. BatchelorA. Loewer and G. Lahav, p53 dynamics control cell fate, Science, 336 (2012), 1440-1444. doi: 10.1126/science.1218351.

[31]

J. E. Purvis and G. Lahav, Encoding and decoding cellular information through signaling dynamics, Cell, 152 (2013), 945-956. doi: 10.1016/j.cell.2013.02.005.

[32]

A. RezzaZ. WangR. SennettW. QiaoD. WangN. HeitmanK. W. MokC. ClavelR. YiP. ZandstraA. Ma'ayan and M. Rendl, Signaling networks among stem cell precursors, transit-amplifying progenitors, and their niche in developing hair follicles, Cell Rep, 14 (2016), 3001-3018. doi: 10.1016/j.celrep.2016.02.078.

[33]

I. Rodriguez-BrenesN. Komarova and D. Wodarz, Evolutionary dynamics of feedback escape and the development of stem-cell–driven cancers, Proc Natl Acad Sci USA, 108 (2011), 18983-18988.

[34]

P. Rué and A. Martinez-Arias, Cell dynamics and gene expression control in tissue homeostasis and development, Mol Syst Biol, 11 (2015), 792-792.

[35]

T. SchepelerM. E. Page and K. B. Jensen, Heterogeneity and plasticity of epidermal stem cells, Development, 141 (2014), 2559-2567. doi: 10.1242/dev.104588.

[36]

Z. S. SingerJ. YongJ. TischlerJ. A. HackettA. AltinokM. A. SuraniL. Cai and M. B. Elowitz, Dynamic heterogeneity and DNA methylation in embryonic stem cells, Mol Cell, 55 (2014), 319-331. doi: 10.1016/j.molcel.2014.06.029.

[37]

K. Takaoka and H. Hamada, Origin of cellular asymmetries in the pre-implantation mouse embryo: A hypothesis, Philos Trans R Soc Lond B Biol Sci, 369 (2014). doi: 10.1098/rstb.2013.0536.

[38]

J. E. Till, E. A. McCulloch and L. Siminovitch, A Stochastic Model of Stem Cell Proliferation, Based on the Growth of Spleen Colony-Forming Cells, in Proceedings of the National Academy of Sciences of the United States of America, 1963, 29–36. doi: 10.1073/pnas.51.1.29.

[39]

A. TraulsenT. LenaertsJ. M. Pacheco and D. Dingli, On the dynamics of neutral mutations in a mathematical model for a homogeneous stem cell population, Journal of the Royal Society, Interface / the Royal Society, 10 (2013), 20120810-20120810. doi: 10.1098/rsif.2012.0810.

[40]

H. Wu and Y. Zhang, Reversing DNA methylation: Mechanisms, genomics, and biological functions, Cell, 156 (2014), 45-68. doi: 10.1016/j.cell.2013.12.019.

[41]

M. Zernicka-GoetzS. A. Morris and A. W. Bruce, Making a firm decision: Multifaceted regulation of cell fate in the early mouse embryo, Nat Rev Genet, 10 (2009), 467-477. doi: 10.1038/nrg2564.

[42]

X.-P. ZhangF. LiuZ. Cheng and W. Wang, Cell fate decision mediated by p53 pulses, Proc Natl Acad Sci USA, 106 (2009), 12245-12250. doi: 10.1073/pnas.0813088106.

[43]

D. ZhouD. WuZ. LiM. Qian and M. Q. Zhang, Population dynamics of cancer cells with cell state conversions, Quant Biol, 1 (2013), 201-208. doi: 10.1007/s40484-013-0014-2.

[44]

C. ZhugeX. Sun and J. Lei, On positive solutions and the Omega limit set for a class of delay differential equations, DCDS-B, 18 (2013), 2487-2503. doi: 10.3934/dcdsb.2013.18.2487.

Figure 1.  Model illustration. During stem cell regeneration, cells in the resting phase either enter the proliferating phase with a rate $\beta$, or be removed from the resting pool with a rate $\gamma$. The proliferating cells undergo apoptosis with a probability $\mu$. Each daughter cell generated from mitosis is either a differentiated cell (with a probability $\kappa$) or a stem cell (with a probability $(1-\kappa)$)
Figure 2.  Transition dynamics. (A) The cell population dynamics. (B) The percentage of epigenetic-state cells at the equilibrium state. Here, results of $\nu = 0$ (green), $20$ (blue), and $200$ (red) are shown. In simulations, the initial cell population is taken as $N(0) = 300$, and $N(0, X_i)=1, (i=1, \cdots, 300)$
Figure 3.  Simulation of cell-state dynamics. Dynamics of cell-state proportion with different initial states (left: $(S, B, L)=(99.9, 0.05, 0.05)$, middle: $(S, B, L)=(0.05, 0.05, 99.9)$, right: $(S, B, L)=(0.05, 99.9, 0.05)$). Markers are data taken from [13]. Parameters are listed in Table 1
Table 1.  Parameter values in the model of cancer cell state transition. Left: the probabilities $\gamma, \kappa, \mu$ for cells of these three states. Right: the transition matrix $p(X, Y), X, Y\in \Omega$
Parameter S B L S B L
$\gamma$ 0.95 0.7 0.65 S 0.58 0.04 0.01
$\kappa$ 0.02 0.03 0 B 0.07 0.47 0
$\mu$ 0.1 0.1 0.1 L 0.35 0.49 0.09
Parameter S B L S B L
$\gamma$ 0.95 0.7 0.65 S 0.58 0.04 0.01
$\kappa$ 0.02 0.03 0 B 0.07 0.47 0
$\mu$ 0.1 0.1 0.1 L 0.35 0.49 0.09
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