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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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
[1]

Ebraheem O. Alzahrani, Muhammad Altaf Khan. Androgen driven evolutionary population dynamics in prostate cancer growth. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020426

[2]

A. M. Elaiw, N. H. AlShamrani, A. Abdel-Aty, H. Dutta. Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020441

[3]

Laurent Di Menza, Virginie Joanne-Fabre. An age group model for the study of a population of trees. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020464

[4]

H. M. Srivastava, H. I. Abdel-Gawad, Khaled Mohammed Saad. Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020433

[5]

Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020103

[6]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432

[7]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450

[8]

Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020316

[9]

Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045

[10]

Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020339

[11]

Shao-Xia Qiao, Li-Jun Du. Propagation dynamics of nonlocal dispersal equations with inhomogeneous bistable nonlinearity. Electronic Research Archive, , () : -. doi: 10.3934/era.2020116

[12]

Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074

[13]

Chao Xing, Jiaojiao Pan, Hong Luo. Stability and dynamic transition of a toxin-producing phytoplankton-zooplankton model with additional food. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020275

[14]

José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020376

2018 Impact Factor: 1.313

Metrics

  • PDF downloads (100)
  • HTML views (166)
  • Cited by (0)

Other articles
by authors

[Back to Top]