August  2019, 24(8): 3971-3994. doi: 10.3934/dcdsb.2018339

Stochastic dynamics of cell lineage in tissue homeostasis

1. 

Department of Mathematics, University of California, Irvine, Irvine, CA 92697, USA

2. 

Department of Mathematics, University of California, Riverside, Riverside, CA 92507, USA

3. 

Department of Mathematics, Department of Developmental and Cell Biology, University of California, Irvine, Irvine, CA 92697, USA

* Corresponding author: Qing Nie

Contributed in honor of Peter Kloeden on the occasion of his 70th birthday

Received  June 2018 Revised  August 2018 Published  January 2019

Fund Project: This work is supported by the NIH grants U01AR073159, R01GM107264, and R01NS095355; a grant from the Simons Foundation (594598, QN), and the NSF grant DMS1763272, DMS1562176, and DMS1762063.

During epithelium tissue maintenance, lineages of cells differentiate and proliferate in a coordinated way to provide the desirable size and spatial organization of different types of cells. While mathematical models through deterministic description have been used to dissect role of feedback regulations on tissue layer size and stratification, how the stochastic effects influence tissue maintenance remains largely unknown. Here we present a stochastic continuum model for cell lineages to investigate how both layer thickness and layer stratification are affected by noise. We find that the cell-intrinsic noise often causes reduction and oscillation of layer size whereas the cell-extrinsic noise increases the thickness, and sometimes, leads to uncontrollable growth of the tissue layer. The layer stratification usually deteriorates as the noise level increases in the cell lineage systems. Interestingly, the morphogen noise, which mixes both cell-intrinsic noise and cell-extrinsic noise, can lead to larger size of layer with little impact on the layer stratification. By investigating different combinations of the three types of noise, we find the layer thickness variability is reduced when cell-extrinsic noise level is high or morphogen noise level is low. Interestingly, there exists a tradeoff between low thickness variability and strong layer stratification due to competition among the three types of noise, suggesting robust layer homeostasis requires balanced levels of different types of noise in the cell lineage systems.

Citation: Yuchi Qiu, Weitao Chen, Qing Nie. Stochastic dynamics of cell lineage in tissue homeostasis. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3971-3994. doi: 10.3934/dcdsb.2018339
References:
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M. Acar, J. T. Mettetal and A. Van Oudenaarden, Stochastic switching as a survival strategy in fluctuating environments, Nature genetics, 40 (2008), 471. Google Scholar

[2]

D. Austin, M. Allen, J. McCollum, R. Dar, J. Wilgus, G. Sayler, N. Samatova, C. Cox and M. Simpson, Gene network shaping of inherent noise spectra, Nature, 439(2006), 608. Google Scholar

[3]

S. V. Avery, Microbial cell individuality and the underlying sources of heterogeneity, Nature Reviews Microbiology, 4 (2006), 577. Google Scholar

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A. Becskei and L. Serrano, Engineering stability in gene networks by autoregulation, Nature, 405 (2000), 590. Google Scholar

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W. J. BlakeG. BalaázsiM. A. KohanskiF. J. IsaacsK. F. MurphyY. KuangC. R. CantorD. R. Walt and J. J. Collins, Phenotypic consequences of promoter-mediated transcriptional noise, Molecular Cell, 24 (2006), 853-865.   Google Scholar

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T. BorovskiE. M. Felipe De SousaL. Vermeulen and J. P. Medema, Cancer stem cell niche: The place to be, Cancer Research, 71 (2011), 634-639.   Google Scholar

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C.-S. ChouW.-C. LoK. K. GokoffskiY.-T. ZhangF. Y. WanA. D. LanderA. L. Calof and Q. Nie, Spatial dynamics of multistage cell lineages in tissue stratification, Biophysical Journal, 99 (2010), 3145-3154.   Google Scholar

[8]

F. Doetsch, A niche for adult neural stem cells., Development, 13 (2003), 543-550.   Google Scholar

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H. Du, Y. Wang, D. Haensel, B. Lee, X. Dai and Q. Nie, Multiscale modeling of layer formation in epidermis, PLoS Computational Biology, 14 (2018), e1006006. Google Scholar

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A. D. Economou, A. Ohazama, T. Porntaveetus, P. T. Sharpe, S. Kondo, M. A. Basson, A. Gritli-Linde, M. T. Cobourne and J. B. Green, Periodic stripe formation by a Turing mechanism operating at growth zones in the mammalian palate, Nature Genetics, 44 (2012), 348. Google Scholar

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M. B. ElowitzA. J. LevineE. D. Siggia and P. S. Swain, Stochastic gene expression in a single cell, Science, 297 (2002), 1183-1186.   Google Scholar

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L. Gammaitoni, P. Haänggi, P. Jung and F. Marchesoni, Stochastic resonance, Reviews of Modern Physics, 70 (1998), 223. Google Scholar

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H. Ge, H. Qian and X. S. Xie, Stochastic phenotype transition of a single cell in an intermediate region of gene state switching, Physical Review Letters, 114 (2015), 078101. Google Scholar

[14]

J. HastyJ. PradinesM. Dolnik and J. J. Collins, Noise-based switches and amplifiers for gene expression., Proceedings of the National Academy of Sciences, 97 (2000), 2075-2080.   Google Scholar

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D. Huh and J. Paulsson, Non-genetic heterogeneity from stochastic partitioning at cell division, Nature Genetics, 43 (2011), 95. Google Scholar

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M. Kærn, T. C. Elston, W. J. Blake and J. J. Collins, Stochasticity in gene expression: From theories to phenotypes, Nature Reviews Genetics, 6 (2005), 451. Google Scholar

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D. C. Kirouac, G. J. Madlambayan, M. Yu, E. A. Sykes, C. Ito and P. W. Zandstra, Cell-cell interaction networks regulate blood stem and progenitor cell fate, Molecular Systems Biology, 5 (2009), 293. Google Scholar

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A. D. Lander, Pattern, growth, and control, Cell, 144 (2011), 955-969.   Google Scholar

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A. D. Lander, K. K. Gokoffski, F. Y. Wan, Q. Nie and A. L. Calof, Cell lineages and the logic of proliferative control, PLoS Biology, 7 (2009), e1000015. Google Scholar

[22]

A. D. Lander, J. Kimble, H. Clevers, E. Fuchs, D. Montarras, M. Buckingham, A. L. Calof, A. Trumpp and T. Oskarsson, What does the concept of the stem cell niche really mean today?, BMC Biology, 10 (2012), 19. Google Scholar

[23]

A. Li, S. Figueroa, T.-X. Jiang, P. Wu, R. Widelitz, Q. Nie and C.-M. Chuong, Diverse feather shape evolution enabled by coupling anisotropic signalling modules with self organizing branching programme, Nature Communications, 8 (2017), ncomms14139. Google Scholar

[24]

L. Li and T. Xie, Stem cell niche: Structure and function, Annu. Rev. Cell Dev. Biol., 21 (2005), 605-631.   Google Scholar

[25]

C.-M. LinT. X. JiangR. E. BakerP. K. MainiR. B. Widelitz and C.-M. Chuong, Spots and stripes: pleomorphic patterning of stem cells via p-ERK-dependent cell chemotaxis shown by feather morphogenesis and mathematical simulation, Developmental Biology, 334 (2009), 369-382.   Google Scholar

[26]

W.-C. LoC.-S. ChouK. K. GokoffskiF. Y.-M. WanA. D. LanderA. L. Calof and Q. Nie, Feedback regulation in multistage cell lineages, Mathematical Biosciences and Engineering: MBE, 6 (2009), 59-82.  doi: 10.3934/mbe.2009.6.59.  Google Scholar

[27]

F. LucianiD. ChampevalA. HerbetteL. DenatB. AylajS. MartinozziR. BallottiR. KemlerC. R. Goding and F. De Vuyst, Biological and mathematical modeling of melanocyte development, Development, 138 (2011), 3943-3954.   Google Scholar

[28]

A. Marciniak-CzochraT. StiehlA. D. HoW. Jaäger and W. Wagner, Modeling of asymmetric cell division in hematopoietic stem cells-regulation of self-renewal is essential for efficient repopulation, Stem Cells and Development, 18 (2009), 377-386.   Google Scholar

[29]

H. H. McAdams and A. Arkin, Stochastic mechanisms in gene expression, Proceedings of the National Academy of Sciences, 94 (1997), 814-819.   Google Scholar

[30]

S. McCroskeryM. ThomasL. MaxwellM. Sharma and R. Kambadur, Myostatin negatively regulates satellite cell activation and self-renewal, The Journal of Cell Biology, 162 (2003), 1135-1147.   Google Scholar

[31]

M. D. McDonnell and D. Abbott, What is stochastic resonance? Definitions, misconceptions, debates, and its relevance to biology, PLoS Computational Biology, 5 (2009), e1000348, 9pp. doi: 10.1371/journal.pcbi.1000348.  Google Scholar

[32]

F. L. Moolten and N. L. Bucher, Regeneration of rat liver: Transfer of humoral agent by cross circulation, Science, 158 (1967), 272-274.   Google Scholar

[33]

K. A. Moore and I. R. Lemischka, Stem cells and their niches, Science, 311 (2006), 1880-1885.   Google Scholar

[34]

J. Ovadia and Q. Nie, Numerical Methods for Two-Dimensional Stem Cell Tissue Growth, Journal of Scientific Computing, 58 (2014), 149-175.  doi: 10.1007/s10915-013-9728-6.  Google Scholar

[35]

J. Ovadia and Q. Nie, Stem cell niche structure as an inherent cause of undulating epithelial morphologies., Biophysical Journal, 104 (2013), 237-246.   Google Scholar

[36]

C. RackauckasT. Schilling and Q. Nie, Mean-independent noise control of cell fates via intermediate states, iScience, 3 (2018), 11-20.   Google Scholar

[37]

C. V. Rao, D. M. Wolf and A. P. Arkin, Control, exploitation and tolerance of intracellular noise, Nature, 420 (2002), 231. Google Scholar

[38]

J. RaspopovicL. MarconL. Russo and J. Sharpe, Digit patterning is controlled by a Bmp-Sox9-Wnt Turing network modulated by morphogen gradients, Science, 345 (2014), 566-570.   Google Scholar

[39]

T. Ruiz-HerreroK. AlessandriB. V. GurchenkovP. Nassoy and L. Mahadevan, Organ size control via hydraulically gated oscillations, Development, 144 (2017), 4422-4427.   Google Scholar

[40]

M. L. SimpsonC. D. CoxM. S. AllenJ. M. McCollumR. D. DarD. K. Karig and J. F. Cooke, Noise in biological circuits, Wiley Interdisciplinary Reviews: Nanomedicine and Nanobiotechnology, 1 (2009), 214-225.   Google Scholar

[41]

C. L. StokesD. A. Lauffenburger and S. K. Williams, Migration of individual microvessel endothelial cells: stochastic model and parameter measurement, Journal of Cell Science, 99 (1991), 419-430.   Google Scholar

[42]

M. Thattai and A. Van Oudenaarden, Stochastic gene expression in fluctuating environments, Genetics, 167 (2004), 523-530.   Google Scholar

[43]

T. TumbarG. GuaschV. GrecoC. BlanpainW.E. LowryM. Rendl and E. Fuchs, Defining the epithelial stem cell niche in skin, Science, 303 (2004), 359-363.   Google Scholar

[44]

L. Wang, J. Xin and Q. Nie, A critical quantity for noise attenuation in feedback systems, PLoS Computational Biology, 6 (2010), e1000764, 17pp. doi: 10.1371/journal.pcbi.1000764.  Google Scholar

[45]

Q. Wang, W. R. Holmes, J. Sosnik, T. Schilling and Q. Nie, Cell sorting and noise-induced cell plasticity coordinate to sharpen boundaries between gene expression domains, PLoS Computational Biology, 13 (2017), e1005307. Google Scholar

[46]

H.-H. WuS. IvkovicR. C. MurrayS. JaramilloK. M. LyonsJ. E. Johnson and A. L. Calof, Autoregulation of neurogenesis by GDF11, Neuron, 37 (2003), 197-207.   Google Scholar

[47]

T.-H. Yen and N. A. Wright, The gastrointestinal tract stem cell niche, Stem Cell Reviews, 2 (2006), 203-212.   Google Scholar

[48]

J. Zhang, C. Niu, L. Ye, H. Huang, X. He, W.-G. Tong, J. Ross, J. Haug, T. Johnson and J. Q. Feng, Identification of the haematopoietic stem cell niche and control of the niche size., Nature, 425(2003), 836. Google Scholar

[49]

L. Zhang, K. Radtke, L. Zheng, A. Q. Cai, T. F. Schilling and Q. Nie, Noise drives sharpening of gene expression boundaries in the zebrafish hindbrain, Molecular Systems Biology, 8 (2012), 613. Google Scholar

show all references

References:
[1]

M. Acar, J. T. Mettetal and A. Van Oudenaarden, Stochastic switching as a survival strategy in fluctuating environments, Nature genetics, 40 (2008), 471. Google Scholar

[2]

D. Austin, M. Allen, J. McCollum, R. Dar, J. Wilgus, G. Sayler, N. Samatova, C. Cox and M. Simpson, Gene network shaping of inherent noise spectra, Nature, 439(2006), 608. Google Scholar

[3]

S. V. Avery, Microbial cell individuality and the underlying sources of heterogeneity, Nature Reviews Microbiology, 4 (2006), 577. Google Scholar

[4]

A. Becskei and L. Serrano, Engineering stability in gene networks by autoregulation, Nature, 405 (2000), 590. Google Scholar

[5]

W. J. BlakeG. BalaázsiM. A. KohanskiF. J. IsaacsK. F. MurphyY. KuangC. R. CantorD. R. Walt and J. J. Collins, Phenotypic consequences of promoter-mediated transcriptional noise, Molecular Cell, 24 (2006), 853-865.   Google Scholar

[6]

T. BorovskiE. M. Felipe De SousaL. Vermeulen and J. P. Medema, Cancer stem cell niche: The place to be, Cancer Research, 71 (2011), 634-639.   Google Scholar

[7]

C.-S. ChouW.-C. LoK. K. GokoffskiY.-T. ZhangF. Y. WanA. D. LanderA. L. Calof and Q. Nie, Spatial dynamics of multistage cell lineages in tissue stratification, Biophysical Journal, 99 (2010), 3145-3154.   Google Scholar

[8]

F. Doetsch, A niche for adult neural stem cells., Development, 13 (2003), 543-550.   Google Scholar

[9]

H. Du, Y. Wang, D. Haensel, B. Lee, X. Dai and Q. Nie, Multiscale modeling of layer formation in epidermis, PLoS Computational Biology, 14 (2018), e1006006. Google Scholar

[10]

A. D. Economou, A. Ohazama, T. Porntaveetus, P. T. Sharpe, S. Kondo, M. A. Basson, A. Gritli-Linde, M. T. Cobourne and J. B. Green, Periodic stripe formation by a Turing mechanism operating at growth zones in the mammalian palate, Nature Genetics, 44 (2012), 348. Google Scholar

[11]

M. B. ElowitzA. J. LevineE. D. Siggia and P. S. Swain, Stochastic gene expression in a single cell, Science, 297 (2002), 1183-1186.   Google Scholar

[12]

L. Gammaitoni, P. Haänggi, P. Jung and F. Marchesoni, Stochastic resonance, Reviews of Modern Physics, 70 (1998), 223. Google Scholar

[13]

H. Ge, H. Qian and X. S. Xie, Stochastic phenotype transition of a single cell in an intermediate region of gene state switching, Physical Review Letters, 114 (2015), 078101. Google Scholar

[14]

J. HastyJ. PradinesM. Dolnik and J. J. Collins, Noise-based switches and amplifiers for gene expression., Proceedings of the National Academy of Sciences, 97 (2000), 2075-2080.   Google Scholar

[15]

D. Huh and J. Paulsson, Non-genetic heterogeneity from stochastic partitioning at cell division, Nature Genetics, 43 (2011), 95. Google Scholar

[16]

A. Jentzen and P. E. Kloeden, Taylor Approximations for Stochastic Partial Differential Equations, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9781611972016.  Google Scholar

[17]

M. Kærn, T. C. Elston, W. J. Blake and J. J. Collins, Stochasticity in gene expression: From theories to phenotypes, Nature Reviews Genetics, 6 (2005), 451. Google Scholar

[18]

D. C. Kirouac, G. J. Madlambayan, M. Yu, E. A. Sykes, C. Ito and P. W. Zandstra, Cell-cell interaction networks regulate blood stem and progenitor cell fate, Molecular Systems Biology, 5 (2009), 293. Google Scholar

[19]

P. E. Kloeden, The Numerical Solution of Stochastic Differenttial Equations, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-12616-5.  Google Scholar

[20]

A. D. Lander, Pattern, growth, and control, Cell, 144 (2011), 955-969.   Google Scholar

[21]

A. D. Lander, K. K. Gokoffski, F. Y. Wan, Q. Nie and A. L. Calof, Cell lineages and the logic of proliferative control, PLoS Biology, 7 (2009), e1000015. Google Scholar

[22]

A. D. Lander, J. Kimble, H. Clevers, E. Fuchs, D. Montarras, M. Buckingham, A. L. Calof, A. Trumpp and T. Oskarsson, What does the concept of the stem cell niche really mean today?, BMC Biology, 10 (2012), 19. Google Scholar

[23]

A. Li, S. Figueroa, T.-X. Jiang, P. Wu, R. Widelitz, Q. Nie and C.-M. Chuong, Diverse feather shape evolution enabled by coupling anisotropic signalling modules with self organizing branching programme, Nature Communications, 8 (2017), ncomms14139. Google Scholar

[24]

L. Li and T. Xie, Stem cell niche: Structure and function, Annu. Rev. Cell Dev. Biol., 21 (2005), 605-631.   Google Scholar

[25]

C.-M. LinT. X. JiangR. E. BakerP. K. MainiR. B. Widelitz and C.-M. Chuong, Spots and stripes: pleomorphic patterning of stem cells via p-ERK-dependent cell chemotaxis shown by feather morphogenesis and mathematical simulation, Developmental Biology, 334 (2009), 369-382.   Google Scholar

[26]

W.-C. LoC.-S. ChouK. K. GokoffskiF. Y.-M. WanA. D. LanderA. L. Calof and Q. Nie, Feedback regulation in multistage cell lineages, Mathematical Biosciences and Engineering: MBE, 6 (2009), 59-82.  doi: 10.3934/mbe.2009.6.59.  Google Scholar

[27]

F. LucianiD. ChampevalA. HerbetteL. DenatB. AylajS. MartinozziR. BallottiR. KemlerC. R. Goding and F. De Vuyst, Biological and mathematical modeling of melanocyte development, Development, 138 (2011), 3943-3954.   Google Scholar

[28]

A. Marciniak-CzochraT. StiehlA. D. HoW. Jaäger and W. Wagner, Modeling of asymmetric cell division in hematopoietic stem cells-regulation of self-renewal is essential for efficient repopulation, Stem Cells and Development, 18 (2009), 377-386.   Google Scholar

[29]

H. H. McAdams and A. Arkin, Stochastic mechanisms in gene expression, Proceedings of the National Academy of Sciences, 94 (1997), 814-819.   Google Scholar

[30]

S. McCroskeryM. ThomasL. MaxwellM. Sharma and R. Kambadur, Myostatin negatively regulates satellite cell activation and self-renewal, The Journal of Cell Biology, 162 (2003), 1135-1147.   Google Scholar

[31]

M. D. McDonnell and D. Abbott, What is stochastic resonance? Definitions, misconceptions, debates, and its relevance to biology, PLoS Computational Biology, 5 (2009), e1000348, 9pp. doi: 10.1371/journal.pcbi.1000348.  Google Scholar

[32]

F. L. Moolten and N. L. Bucher, Regeneration of rat liver: Transfer of humoral agent by cross circulation, Science, 158 (1967), 272-274.   Google Scholar

[33]

K. A. Moore and I. R. Lemischka, Stem cells and their niches, Science, 311 (2006), 1880-1885.   Google Scholar

[34]

J. Ovadia and Q. Nie, Numerical Methods for Two-Dimensional Stem Cell Tissue Growth, Journal of Scientific Computing, 58 (2014), 149-175.  doi: 10.1007/s10915-013-9728-6.  Google Scholar

[35]

J. Ovadia and Q. Nie, Stem cell niche structure as an inherent cause of undulating epithelial morphologies., Biophysical Journal, 104 (2013), 237-246.   Google Scholar

[36]

C. RackauckasT. Schilling and Q. Nie, Mean-independent noise control of cell fates via intermediate states, iScience, 3 (2018), 11-20.   Google Scholar

[37]

C. V. Rao, D. M. Wolf and A. P. Arkin, Control, exploitation and tolerance of intracellular noise, Nature, 420 (2002), 231. Google Scholar

[38]

J. RaspopovicL. MarconL. Russo and J. Sharpe, Digit patterning is controlled by a Bmp-Sox9-Wnt Turing network modulated by morphogen gradients, Science, 345 (2014), 566-570.   Google Scholar

[39]

T. Ruiz-HerreroK. AlessandriB. V. GurchenkovP. Nassoy and L. Mahadevan, Organ size control via hydraulically gated oscillations, Development, 144 (2017), 4422-4427.   Google Scholar

[40]

M. L. SimpsonC. D. CoxM. S. AllenJ. M. McCollumR. D. DarD. K. Karig and J. F. Cooke, Noise in biological circuits, Wiley Interdisciplinary Reviews: Nanomedicine and Nanobiotechnology, 1 (2009), 214-225.   Google Scholar

[41]

C. L. StokesD. A. Lauffenburger and S. K. Williams, Migration of individual microvessel endothelial cells: stochastic model and parameter measurement, Journal of Cell Science, 99 (1991), 419-430.   Google Scholar

[42]

M. Thattai and A. Van Oudenaarden, Stochastic gene expression in fluctuating environments, Genetics, 167 (2004), 523-530.   Google Scholar

[43]

T. TumbarG. GuaschV. GrecoC. BlanpainW.E. LowryM. Rendl and E. Fuchs, Defining the epithelial stem cell niche in skin, Science, 303 (2004), 359-363.   Google Scholar

[44]

L. Wang, J. Xin and Q. Nie, A critical quantity for noise attenuation in feedback systems, PLoS Computational Biology, 6 (2010), e1000764, 17pp. doi: 10.1371/journal.pcbi.1000764.  Google Scholar

[45]

Q. Wang, W. R. Holmes, J. Sosnik, T. Schilling and Q. Nie, Cell sorting and noise-induced cell plasticity coordinate to sharpen boundaries between gene expression domains, PLoS Computational Biology, 13 (2017), e1005307. Google Scholar

[46]

H.-H. WuS. IvkovicR. C. MurrayS. JaramilloK. M. LyonsJ. E. Johnson and A. L. Calof, Autoregulation of neurogenesis by GDF11, Neuron, 37 (2003), 197-207.   Google Scholar

[47]

T.-H. Yen and N. A. Wright, The gastrointestinal tract stem cell niche, Stem Cell Reviews, 2 (2006), 203-212.   Google Scholar

[48]

J. Zhang, C. Niu, L. Ye, H. Huang, X. He, W.-G. Tong, J. Ross, J. Haug, T. Johnson and J. Q. Feng, Identification of the haematopoietic stem cell niche and control of the niche size., Nature, 425(2003), 836. Google Scholar

[49]

L. Zhang, K. Radtke, L. Zheng, A. Q. Cai, T. F. Schilling and Q. Nie, Noise drives sharpening of gene expression boundaries in the zebrafish hindbrain, Molecular Systems Biology, 8 (2012), 613. Google Scholar

Figure 1.  A schematic diagram of a main cell lineage in epithelium. Stem cells and TA cells proliferate with probabilities $ p_0 $ and $ p_1 $ and differentiate with probabilities $ 1-p_0 $ and $ 1-p_1 $. TD cells undergo cell death with rate $ d_2 $. All three types of cells can secrete molecule A that inhibits self-renewal probability $ p_0 $. TD and TA cells secrete molecule G that inhibits self-renewal probability $ p_1 $. Molecules A and G are diffusive in the epithelium. The apical surface is moving with the dynamic position $ z_{\max} $ and no-flux boundary condition is imposed. On the other hand, leaky boundary condition is imposed at the basal lamina with its position fixed.
Figure 2.  A baseline simulation for the system containing all three kinds of noise. The spatial distribution of three types of cells and different mophogens at four different time points: A. t = 0; B. t = 330; C. t = 860; D. t = 1200. E. Layer thickness in one particular stochastic simulation. F. Stratification factor of stem cells ($ sf(C_0) $). G. Stratification factor of TA cells ($ sf(C_1) $). In E-G, the black dash line is the steady-state value for corresponding quantities in the deterministic system. The noise levels used are $ \varepsilon_0 = \varepsilon_1 = 0.6 $, $ \sigma_0 = \sigma_1 = 10^{-4} $, and $ \omega_0 = \omega_1 = 0.58 $.
Figure 3.  Simulations with only cell-intrinsic noise. Dash lines represent the corresponding quantities at homeostasis. A. Layer thickness in three simulations with $ \varepsilon = 0.2 $, $ 0.6 $ and $ 1 $. B. The mean $ TH $. The error bars show the standard deviation. C. The mean $ CV $. The error bars show the standard deviation of $ CV $. The mean $ SF $ of D. stem cells and E. TA cells. The error bars show the standard deviation. F. Distribution of cells and morphogens in a specific simulation with $ \varepsilon = 0.6 $ at time $ t = 400 $. In (B-E), all statistical quantities are captured based on $ 20 $ simulations, and the standard deviations (error bars) are negligible compared to the means.
Figure 4.  Simulations with only cell-extrinsic noise. Dash lines represent the corresponding quantities at homeostasis. A. Layer thickness in three simulations with $ \sigma = 1\times 10^{-3} $, $ 2\times 10^{-3} $ and $ 4\times 10^{-3} $. B. The mean $ TH $. The error bars show the standard deviation. C. The mean $ CV $. The error bars show the standard deviation of $ CV $. The mean $ SF $ of D. stem cells and E. TA cells. The error bars show the standard deviation. F. Distribution of cells and morphogens in a specific simulation with $ \sigma = 3\times 10^{-3} $ at time $ t = 400 $. In (B-E), all statistical quantities are captured based on $ 20 $ simulations, and the standard deviations (error bars) are negligible compared to the means.
Figure 5.  Simulations with only morphogens noise. Dash lines represent the corresponding quantities at homeostasis. A. Layer thickness in three simulations with $ \omega = 0.4 $, $ 0.6 $ and $ 1 $. B. The mean $ TH $. The error bars show the standard deviation. C. The mean $ CV $. The error bars show the standard deviation of $ CV $. The mean $ SF $ of D. stem cells and E. TA cells. The error bars show the standard deviation. F. Distribution of cells and morphogens in a specific simulation with $ \omega = 0.6 $ at time $ t = 400 $. In (B-E), all statistical quantities are captured based on $ 20 $ simulations, and the standard deviations (error bars) are negligible compared to the means.
Figure 6.  Simulations with both cell-intrinsic noise and cell-extrinsic noise. Simulations with different noise levels are shown in (A-I). In each subfigure, the panel on the top shows the dynamics of layer thickness, the panel on the bottom shows the dynamics of layer stratification of stem cells ($ sf(C_0) $). The dash line represents for the corresponding quantity at homeostasis. Three different levels are chosen for each type of noise. For cell-intrinsic noise level $ \varepsilon $: $ 0.2 $ (Low), $ 0.6 $ (Medium), $ 1 $ (High). For cell-extrinsic noise level $ \sigma $: $ 5\times 10^{-4} $ (Low), $ 1\times 10^{-3} $ (Medium), $ 2\times 10^{-3} $ (High).
Figure 7.  Simulations with both cell-intrinsic noise and morphogen noise. Simulations with different noise levels are shown in (A-I). In each subfigure, the panel on the top shows the dynamics of layer thickness, the panel on the bottom shows the dynamics of layer stratification of stem cells ($ sf(C_0) $). The dash line represents for the corresponding quantity at homeostasis. Three different levels are chosen for each type of noise. For cell-intrinsic noise level $ \varepsilon $: $ 0.2 $ (Low), $ 0.6 $ (Medium), $ 1 $ (High). For morphogen noise level $ \omega $: $ 0.2 $ (Low), $ 0.6 $ (Medium), $ 1 $ (High).
Figure 8.  Simulations for maintaining homeostasis ($ SS $ = 0.49mm) with different combinations of three types of noise. Points with the same color and the same marker represent for simulations with the same cell-intrinsic noise level $ \varepsilon $, where $ \varepsilon = 0.2 $, $ 0.4 $, $ 0.6 $, $ 0.8 $ and $ 1 $ respectively. The strips, filled with color gradient, roughly divide the plane into several regions. Data points located in the region next to dark/light color of an individual strip have more/less desirable properties. A. The relation between the cell-extrinsic noise level $ \sigma $ and the morphogen noise level $ \omega $. The blue strip sketches the green points with maximal cell-intrinsic noise level $ \varepsilon = 1 $. It divides this plane into stabilized region (region Ⅰ-Ⅳ) and non-stabilized region (region Ⅴ). The stabilized region is divided into four parts (region Ⅰ-Ⅳ) by a red strip and a green strip. These regions will be introduced next. B. The relation between layer thickness variability ($ CV $) and layer stratification factor of stem cells ($ SF(C_0) $). The red strip with $ CV = 20% $ divides this plane into two regions with low $ CV $ or high $ CV $. Also the green strip with $ SF(C_0) = 0.4 $ divides the plane into two regions with high $ SF $ or low $ SF $. The red and the green strips together divide the stabilized region into four regions (Region Ⅰ: low $ CV $ and high $ SF $; Region Ⅱ: high $ CV $ and high $ SF $; Region Ⅲ: low $ CV $ and low $ SF $; Region Ⅳ: high $ CV $ and low $ SF $). C. The relation between $ \sigma $ and $ CV $. D. The relation between $ \omega $ and $ CV $. E. The relation between $ \sigma $ and $ SF(C_0) $. F. The relation between $ \omega $ and $ SF(C_0) $.
Table 1.  The statistics of $ TH $, $ CV $ and $ SF(C_0) $ with combined cell-intrinsic ($ \varepsilon $) and cell-extrinsic ($ \sigma $) noise. All quantities are captured based on $ 20 $ simulations.
$ 0 $ $ 5\times10^{-4} $ $ 1\times10^{-3} $ $ 2\times10^{-3} $
$ 0 $ $ TH $ $ 0.49 $mm $ 0.53 $mm $ 0.58 $mm $ 0.75 $mm
$ CV $ $ 0% $ $ 1% $ $ 3% $ $ 7% $
$ SF $ $ 0.91 $ $ 0.90 $ $ 0.88 $ $ 0.40 $
$ 0.2 $ $ TH $ $ 0.45 $mm $ 0.49 $mm $ 0.54 $mm $ 0.69 $mm
$ CV $ $ 7% $ $ 7% $ $ 7% $ $ 8% $
$ SF $ $ 0.91 $ $ 0.90 $ $ 0.82 $ $ 0.39 $
$ 0.6 $ $ TH $ $ 0.24 $mm $ 0.28 $mm $ 0.30 $mm $ 0.44 $mm
$ CV $ $ 30% $ $ 25% $ $ 23% $ $ 17% $
$ SF $ $ 0.84 $ $ 0.69 $ $ 0.52 $ $ 0.27 $
$ 1 $ $ TH $ $ 0.12 $mm $ 0.19 $mm $ 0.22 $mm $ 0.38 $mm
$ CV $ $ 89% $ $ 36% $ $ 32% $ $ 19% $
$ SF $ $ 0.88 $ $ 0.42 $ $ 0.32 $ $ 0.18 $
$ 0 $ $ 5\times10^{-4} $ $ 1\times10^{-3} $ $ 2\times10^{-3} $
$ 0 $ $ TH $ $ 0.49 $mm $ 0.53 $mm $ 0.58 $mm $ 0.75 $mm
$ CV $ $ 0% $ $ 1% $ $ 3% $ $ 7% $
$ SF $ $ 0.91 $ $ 0.90 $ $ 0.88 $ $ 0.40 $
$ 0.2 $ $ TH $ $ 0.45 $mm $ 0.49 $mm $ 0.54 $mm $ 0.69 $mm
$ CV $ $ 7% $ $ 7% $ $ 7% $ $ 8% $
$ SF $ $ 0.91 $ $ 0.90 $ $ 0.82 $ $ 0.39 $
$ 0.6 $ $ TH $ $ 0.24 $mm $ 0.28 $mm $ 0.30 $mm $ 0.44 $mm
$ CV $ $ 30% $ $ 25% $ $ 23% $ $ 17% $
$ SF $ $ 0.84 $ $ 0.69 $ $ 0.52 $ $ 0.27 $
$ 1 $ $ TH $ $ 0.12 $mm $ 0.19 $mm $ 0.22 $mm $ 0.38 $mm
$ CV $ $ 89% $ $ 36% $ $ 32% $ $ 19% $
$ SF $ $ 0.88 $ $ 0.42 $ $ 0.32 $ $ 0.18 $
Table 2.  The statistics of $ TH $, $ CV $ and $ SF(C_0) $ with combined cell-intrinsic ($ \varepsilon $) and morphogen ($ \omega $) noise. All quantities are captured based on $ 20 $ simulations.
$ 0 $ $ 0.2 $ $ 0.6 $ $ 1 $
$ 0 $ $ TH $ $ 0.49 $mm $ 0.58 $mm $ 1.06 $mm $ 1.33 $mm
$ CV $ $ 0% $ $ 3% $ $ 9% $ $ 11% $
$ SF $ $ 0.91 $ $ 0.92 $ $ 0.92 $ $ 0.91 $
$ 0.2 $ $ TH $ $ 0.45 $mm $ 0.54 $mm $ 0.98 $mm $ 1.23 $mm
$ CV $ $ 7% $ $ 7% $ $ 11% $ $ 13% $
$ SF $ $ 0.91 $ $ 0.91 $ $ 0.92 $ $ 0.90 $
$ 0.6 $ $ TH $ $ 0.24 $mm $ 0.26 $mm $ 0.43 $mm $ 0.52 $mm
$ CV $ $ 30% $ $ 29% $ $ 30% $ $ 33% $
$ SF $ $ 0.84 $ $ 0.83 $ $ 0.84 $ $ 0.81 $
$ 1 $ $ TH $ $ 0.12 $mm $ 0.13 $mm $ 0.15 $mm $ 0.16 $mm
$ CV $ $ 89% $ $ 87% $ $ 97% $ $ 108% $
$ SF $ $ 0.88 $ $ 0.88 $ $ 0.87 $ $ 0.84 $
$ 0 $ $ 0.2 $ $ 0.6 $ $ 1 $
$ 0 $ $ TH $ $ 0.49 $mm $ 0.58 $mm $ 1.06 $mm $ 1.33 $mm
$ CV $ $ 0% $ $ 3% $ $ 9% $ $ 11% $
$ SF $ $ 0.91 $ $ 0.92 $ $ 0.92 $ $ 0.91 $
$ 0.2 $ $ TH $ $ 0.45 $mm $ 0.54 $mm $ 0.98 $mm $ 1.23 $mm
$ CV $ $ 7% $ $ 7% $ $ 11% $ $ 13% $
$ SF $ $ 0.91 $ $ 0.91 $ $ 0.92 $ $ 0.90 $
$ 0.6 $ $ TH $ $ 0.24 $mm $ 0.26 $mm $ 0.43 $mm $ 0.52 $mm
$ CV $ $ 30% $ $ 29% $ $ 30% $ $ 33% $
$ SF $ $ 0.84 $ $ 0.83 $ $ 0.84 $ $ 0.81 $
$ 1 $ $ TH $ $ 0.12 $mm $ 0.13 $mm $ 0.15 $mm $ 0.16 $mm
$ CV $ $ 89% $ $ 87% $ $ 97% $ $ 108% $
$ SF $ $ 0.88 $ $ 0.88 $ $ 0.87 $ $ 0.84 $
Table 3.  Parameters used in Eq. (2) to Eq. (7).
Parameters Values Units
$ \nu_0 $, $ \nu_1 $ $ 1 $ $ \ln 2* $(cell cycle)$ ^{-1} $
$ d_2 $ $ 0.01 $ $ \ln 2* $(cell cycle)$ ^{-1} $
$ D_A $, $ D_G $ $ 10^{-5} $ mm$ ^2 $s$ ^{-1} $
$ \mu_0 $, $ \mu_1 $, $ \mu_2 $, $ \eta_1 $, $ \eta_2 $ $ 10^{-3} $ s$ ^{-1}\mu M $
$ a_{\deg} $, $ g_{\deg} $ $ 10^{-3} $ s$ ^{-1} $
$ \alpha_A $, $ \alpha_G $ $ 10 $ mm$ ^{-1} $
$ \bar{p}_0 $ $ 0.95 $ -
$ \bar{p}_1 $ $ 0.5 $ -
$ \gamma_A $ $ 1.6 $ $ \mu M^{-1} $
$ \gamma_G $ $ 2 $ $ \mu M^{-1} $
Parameters Values Units
$ \nu_0 $, $ \nu_1 $ $ 1 $ $ \ln 2* $(cell cycle)$ ^{-1} $
$ d_2 $ $ 0.01 $ $ \ln 2* $(cell cycle)$ ^{-1} $
$ D_A $, $ D_G $ $ 10^{-5} $ mm$ ^2 $s$ ^{-1} $
$ \mu_0 $, $ \mu_1 $, $ \mu_2 $, $ \eta_1 $, $ \eta_2 $ $ 10^{-3} $ s$ ^{-1}\mu M $
$ a_{\deg} $, $ g_{\deg} $ $ 10^{-3} $ s$ ^{-1} $
$ \alpha_A $, $ \alpha_G $ $ 10 $ mm$ ^{-1} $
$ \bar{p}_0 $ $ 0.95 $ -
$ \bar{p}_1 $ $ 0.5 $ -
$ \gamma_A $ $ 1.6 $ $ \mu M^{-1} $
$ \gamma_G $ $ 2 $ $ \mu M^{-1} $
Table 4.  Noise levels used in Eq. (7) and (8) in different figures.
$ \varepsilon_0 $, $ \varepsilon_1 $ $ \sigma_0 $, $ \sigma_1 $ $ \omega_0 $, $ \omega_1 $
Figure 2 $ 0.6 $ $ 10^{-4} $ 0.58
Figure 3F $ 0.6 $ $ 0 $ $ 0 $
Figure 4F $ 0 $ $ 3\times10^{-3} $ $ 0 $
Figure 5F $ 0 $ $ 0 $ $ 0.6 $
Figure 6 Low; $ 0.2 $ Low: $ 5\times 10^{-4} $ $ 0 $
Medium: $ 0.6 $ Medium: $ 1\times10^{-3} $
High: $ 1 $ High: $ 2\times10^{-3} $
Figure 7 Low: $ 0.2 $ $ 0 $ Low: $ 0.2 $
Medium: $ 0.6 $ Medium: $ 0.6 $
High: $ 1 $ High: $ 1 $
$ \varepsilon_0 $, $ \varepsilon_1 $ $ \sigma_0 $, $ \sigma_1 $ $ \omega_0 $, $ \omega_1 $
Figure 2 $ 0.6 $ $ 10^{-4} $ 0.58
Figure 3F $ 0.6 $ $ 0 $ $ 0 $
Figure 4F $ 0 $ $ 3\times10^{-3} $ $ 0 $
Figure 5F $ 0 $ $ 0 $ $ 0.6 $
Figure 6 Low; $ 0.2 $ Low: $ 5\times 10^{-4} $ $ 0 $
Medium: $ 0.6 $ Medium: $ 1\times10^{-3} $
High: $ 1 $ High: $ 2\times10^{-3} $
Figure 7 Low: $ 0.2 $ $ 0 $ Low: $ 0.2 $
Medium: $ 0.6 $ Medium: $ 0.6 $
High: $ 1 $ High: $ 1 $
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