doi: 10.3934/dcdsb.2019074

Cell-type switches induced by stochastic histone modification inheritance

Zhou Pei-Yuan Center for Applied Mathematics, MOE Key Laboratory of Bioinformatics, Tsinghua University, Beijing China, 100084

* Corresponding author: Jinzhi Lei

Received  September 2018 Revised  November 2018 Published  April 2019

Fund Project: This work was supported by the National Natural Science Foundation of China (NSFC 917030301)

Cell plasticity is important for tissue developments during which somatic cells may switch between distinct states. Genetic networks to yield multistability are usually required to yield multiple states, and either external stimuli or noise in gene expressions are trigger signals to induce cell-type switches. In many biological systems, cells show highly plasticity and can switch between different states spontaneously, but maintaining the dynamic equilibrium of the cell population. Here, we considered a mechanism of spontaneous cell-type switches through the combination between gene regulation network and stochastic epigenetic state transitions. We presented a mathematical model that consists of a standard positive feedback loop with changes of histone modifications during cell cycling. Based on the model, nucleosome state of an associated gene is a random process during cell cycling, and hence introduces an inherent noise to gene expression, which can automatically induce cell-type switches in cell cycling. Our model reveals a simple mechanism of spontaneous cell-type switches through a stochastic histone modification inheritance during cell cycle. This mechanism is inherent to the normal cell cycle process, and is independent to the external signals.

Citation: Rongsheng Huang, Jinzhi Lei. Cell-type switches induced by stochastic histone modification inheritance. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019074
References:
[1]

U. Alon, An Introduction to Systems Biology, Boca Raton, FL, 2007. Google Scholar

[2]

M. R. AtkinsonM. A. SavageauJ. T. Myers and A. J. Ninfa, Development of genetic circuitry exhibiting toggle switch or oscillatory behavior in escherichia coli, Cell, 113 (2003), 597-607. doi: 10.1016/S0092-8674(03)00346-5. Google Scholar

[3]

D. W. Austin, M. S. Allen, J. M. Mccollum, R. D. Dar, J. R. Wilgus, G. S. Sayler, N. F. Samatova, C. D. Cox and M. L. Simpson, Gene network shaping of inherent noise spectra, in Bio MICRO and Nanosystems Conference, 2007,608. doi: 10.1109/BMN.2006.330897. Google Scholar

[4]

J. S. BeckerD. Nicetto and K. S. Zaret, H3K9me3-dependent heterochromatin: Barrier to cell fate changes, Trends Genet., 32 (2015), 29-41. doi: 10.1016/j.tig.2015.11.001. Google Scholar

[5]

L. BintuJ. YongY. E. AntebiK. McCueY. KazukiN. UnoM. Oshimura and M. B. Elowitz, Dynamics of epigenetic regulation at the single-cell level, Science, 351 (2016), 720-724. Google Scholar

[6]

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

[7]

J. L. Cherry and F. R. Adler, How to make a biological switch, J. Theor. Biol., 203 (2000), 117-133. doi: 10.1006/jtbi.2000.1068. Google Scholar

[8]

P. J. ChoiL. CaiK. Frieda and X. S. Xie, A stochastic single-molecule event triggers phenotype switching of a bacterial cell, Science, 322 (2008), 442-446. doi: 10.1126/science.1161427. Google Scholar

[9]

K. CuiC. ZangT.-Y. RohD. E. SchonesR. W. ChildsW. Peng and K. Zhao, Chromatin signatures in multipotent human hematopoietic stem cells indicate the fate of bivalent genes during differentiation, Stem Cell, 4 (2009), 80-93. doi: 10.1016/j.stem.2008.11.011. Google Scholar

[10]

I. B. DoddM. A. MicheelsenK. Sneppen and G. Thon, Theoretical analysis of epigenetic cell memory by nucleosome modification, Cell, 129 (2007), 813-822. doi: 10.1016/j.cell.2007.02.053. Google Scholar

[11]

S.-J. DunnG. MartelloB. YordanovS. Emmott and A. G. Smith, Defining an essential transcription factor program for naïve pluripotency, Science, 344 (2014), 1156-1160. Google Scholar

[12]

H. EaswaranH.-C. Tsai and S. B. Baylin, Cancer epigenetics: Tumor heterogeneity, plasticity of stem-like states, and drug resistance, Mol. Cell, 54 (2014), 716-727. doi: 10.1016/j.molcel.2014.05.015. Google Scholar

[13]

M. B. ElowitzA. J. LevineE. D. Siggia and P. S. Swain, Stochastic gene expression in a single cell, Science, 297 (2002), 1183-1186. doi: 10.1126/science.1070919. Google Scholar

[14]

T. EnverM. PeraC. Peterson and P. W. Andrews, Stem cell states, fates, and the rules of attraction, Stem Cell, 4 (2008), 387-397. doi: 10.1016/j.stem.2009.04.011. Google Scholar

[15]

M. FarlikF. HalbritterF. MüllerF. A. ChoudryP. EbertJ. KlughammerS. FarrowA. SantoroV. CiaurroA. MathurR. UppalH. G. StunnenbergW. H. OuwehandE. LaurentiT. LengauerM. Frontini and C. Bock, DNA methylation dynamics of human hematopoietic stem cell differentiation, Cell Stem Cell, 19 (2016), 808-822. doi: 10.1016/j.stem.2016.10.019. Google Scholar

[16]

J. E. Ferrell, Bistability, bifurcations, and Waddington's epigenetic landscape, Curr. Biol., 22 (2012), R458-R466. doi: 10.1016/j.cub.2012.03.045. Google Scholar

[17]

J. E. Ferrell Jr, Self-perpetuating states in signal transduction: Positive feedback, double-negative feedback and bistability, Curr. Opin. Cell Biol., 14 (2002), 140. Google Scholar

[18]

J. E. Ferrell Jr. and W. Xiong, Bistability in cell signaling: How to make continuous processes discontinuous, and reversible processes irreversible, Chaos, 11 (2001), 227-236. Google Scholar

[19]

W. A. FlavahanE. Gaskell and B. E. Bernstein, Epigenetic plasticity and the hallmarks of cancer, Science, 357 (2017), eaal2380. doi: 10.1126/science.aal2380. Google Scholar

[20]

C. Furusawa and K. Kaneko, A dynamical-systems view of stem cell biology, Science, 338 (2012), 215-217. doi: 10.1126/science.1224311. Google Scholar

[21]

N. G. van Kampen, Stochastic processes in physics and chemistry, Physics Today, 36 (1983), p78. doi: 10.1063/1.2915501. Google Scholar

[22]

D. J. GaffneyG. McvickerA. A. PaiY. N. FondufemittendorfN. LewellenK. MicheliniJ. WidomY. Gilad and J. K. Pritchard, Controls of nucleosome positioning in the human genome, PloS Genetics, 8 (2012), e1003036. doi: 10.1371/journal.pgen.1003036. Google Scholar

[23]

T. S. GardnerC. R. Cantor and J. J. Collins, Construction of a genetic toggle switch in escherichia coli, Nature, 403 (2000), 339-342. doi: 10.1038/35002131. Google Scholar

[24]

W. GuoZ. KeckesovaJ. L. DonaherT. ShibueV. TischlerF. ReinhardtS. ItzkovitzA. NoskeU. Zürrer-HärdiG. BellW. L. TamS. A. ManiA. van Oudenaarden and R. A. Weinberg, Slug and Sox9 cooperatively determine the mammary stem cell state, Cell, 148 (2012), 1015-1028. doi: 10.1016/j.cell.2012.02.008. Google Scholar

[25]

P. GuptaG. U. GuruduttaD. Saluja and R. P. Tripathi, PU.1 and partners: Regulation of haematopoietic stem cell fate in normal and malignant haematopoiesis, J. Cell Mol. Med., 13 (2009), 4349-4363. doi: 10.1111/j.1582-4934.2009.00757.x. Google Scholar

[26]

J. HastyJ. PradinesM. Dolnik and J. J. Collins, Noise-based switches and amplifiers for gene expression, Proc. Nat. Acad. Sci. USA, 97 (2000), 2075-2080. doi: 10.1073/pnas.040411297. Google Scholar

[27]

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

[28]

M. HembergerW. Dean and W. Reik, Epigenetic dynamics of stem cells and cell lineage commitment: Digging Waddington's canal, Nat. Rev. Mol. Cell Biol., 10 (2009), 526-537. doi: 10.1038/nrm2727. Google Scholar

[29]

R. Huang and J. Lei, Dynamics of gene expression with positive feedback to histone modifications at bivalent domains, Int. J. Mod. Phys. B, 32 (2018), 1850075, 22pp. doi: 10.1142/S0217979218500753. Google Scholar

[30]

S. HuangI. Ernberg and S. Kauffman, Cancer attractors: A systems view of tumors from a gene network dynamics and developmental perspective, Semin. Cell Dev. Biol., 20 (2009), 869-876. doi: 10.1016/j.semcdb.2009.07.003. Google Scholar

[31]

R. Jaenisch and A. Bird, Epigenetic regulation of gene expression: How the genome integrates intrinsic and environmental signals, Nat. Genet., 33 (2003), 245-254. doi: 10.1038/ng1089. Google Scholar

[32]

X. Jiao and J. Lei, Dynamics of gene expression based on epigenetic modifications, Communications in Information and Systems, 18 (2018), 125-148. doi: 10.4310/CIS.2018.v18.n3.a1. Google Scholar

[33]

M. KaernT. C. ElstonW. J. Blake and J. J. Collins, Stochasticity in gene expression: From theories to phenotypes, Nat. Rev. Genet., 6 (2005), 451-64. doi: 10.1038/nrg1615. Google Scholar

[34]

S. KanjiV. J. Pompili and H. Das, Plasticity and maintenance of hematopoietic stem cells during development, Recent Patents on Biotechnology, 5 (2011), 40-53. doi: 10.2174/187220811795655896. Google Scholar

[35]

B. B. Kaufmann and O. A. Van, Stochastic gene expression: From single molecules to the proteome, Curr. Opin. Genet. Dev., 17 (2007), 107-112. doi: 10.1016/j.gde.2007.02.007. Google Scholar

[36]

B. B. KaufmannQ. YangJ. T. Mettetal and V. O. Alexander, Heritable stochastic switching revealed by single-cell genealogy, PloS Biol., 5 (2007), e239. doi: 10.1371/journal.pbio.0050239. Google Scholar

[37]

E. Laurenti and B. Göttgens, From haematopoietic stem cells to complex differentiation landscapes, Nature, 553 (2018), 418-426. Google Scholar

[38]

J. Lei, Stochasticity in single gene expression with both intrinsic noise and fluctuation in kinetic parameters, J. Theor. Biol., 256 (2009), 485-492. doi: 10.1016/j.jtbi.2008.10.028. Google Scholar

[39]

J. LeiG. HeH. Liu and Q. Nie, A delay model for noise-induced bi-directional switching, Nonlinearity, 22 (2009), 2845-2859. doi: 10.1088/0951-7715/22/12/003. Google Scholar

[40]

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

[41]

C. Li and J. Wang, Quantifying waddington landscapes and paths of non-adiabatic cell fate decisions for differentiation, reprogramming and transdifferentiation, Journal of the Royal Society Interface, 10 (2013), 20130787. doi: 10.1098/rsif.2013.0787. Google Scholar

[42]

Q. LiA. WennborgE. AurellE. DekelJ.-Z. ZouY. XuS. Huang and I. Ernberg, Dynamics inside the cancer cell attractor reveal cell heterogeneity, limits of stability, and escape, Proc. Natl. Acad. Sci. USA, 113 (2016), 2672-2677. doi: 10.1073/pnas.1519210113. Google Scholar

[43]

E. LimpertW. Stahel and M. Abbt, Log-normal distributions across the sciences: Keys and clues, BioScience, 51 (2001), 341-352. Google Scholar

[44]

H. Maamar and D. Dubnau, Bistability in the Bacillus subtilis K-state (competence) system requires a positive feedback loop, Mol. Microbio., 56 (2005), 615-624. doi: 10.1111/j.1365-2958.2005.04592.x. Google Scholar

[45]

B. D. MacArthurA. SevillaM. LenzF.-J. MüllerB. M. SchuldtA. A. SchuppertS. J. RiddenP. S. StumpfM. FidalgoA. Ma'ayanJ. Wang and I. R. Lemischka, Nanog-dependent feedback loops regulate murine embryonic stem cell heterogeneity, Nat. Cell Biol., 14 (2012), 1139-1147. doi: 10.1038/ncb2603. Google Scholar

[46]

I. C. MacaulayV. SvenssonC. LabaletteL. FerreiraF. HameyT. VoetS. A. Teichmann and A. Cvejic, Single-cell RNA-sequencing reveals a continuous spectrum of differentiation in hematopoietic cells, Cell Rep., 14 (2016), 966-977. doi: 10.1016/j.celrep.2015.12.082. Google Scholar

[47]

A. L. MacLeanT. Hong and Q. Nie, Exploring intermediate cell states through the lens of single cells, Curr. Opin. Syst. Biol., 9 (2018), 32-41. doi: 10.1016/j.coisb.2018.02.009. Google Scholar

[48]

M. A. Nieto, Epithelial plasticity: A common theme in embryonic and cancer cells, Science, 342 (2013), 1234850. doi: 10.1126/science.1234850. Google Scholar

[49]

B. Øksendal, Stochastic Differential Equations–An Introdcution with Applications, Sixth edition, Universitext, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-642-14394-6. Google Scholar

[50]

E. M. OzbudakM. ThattaiH. N. LimB. I. Shraiman and O. A. Van, Multistability in the lactose utilization network of escherichia coli, Nature, 427 (2004), 737-740. doi: 10.1038/nature02298. Google Scholar

[51]

J. Paulsson, Summing up the noise in gene networks, Nature, 427 (2004), 415-418. doi: 10.1038/nature02257. Google Scholar

[52]

L. PinelloJ. XuS. H. Orkin and G. C. Yuan, Analysis of chromatin-state plasticity identifies cell-type-specific regulators of H3K27me3 patterns, Proc. Natl. Acad. Sci. USA, 111 (2014), E344-E353. doi: 10.1073/pnas.1322570111. Google Scholar

[53]

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

[54]

C. RackauckasT. Schilling and Q. Nie, Mean-independent noise control of cell fates via intermediate states, iScience, 3 (2018), 11-20. doi: 10.1016/j.isci.2018.04.002. Google Scholar

[55]

A. Raj and A. V. Oudenaarden, Nature, nurture, or chance: Stochastic gene expression and its consequences, Cell, 135 (2008), 216-226. doi: 10.1016/j.cell.2008.09.050. Google Scholar

[56]

A. RegevS. A. TeichmannE. S. LanderI. AmitC. BenoistE. BirneyB. BodenmillerP. CampbellP. CarninciM. ClatworthyH. CleversB. DeplanckeI. DunhamJ. EberwineR. EilsW. EnardA. FarmerL. FuggerB. GöttgensN. HacohenM. HaniffaM. HembergS. KimP. KlenermanA. KriegsteinE. LeinS. LinnarssonE. LundbergJ. LundebergP. MajumderJ. C. MarioniM. MeradM. MhlangaM. NawijnM. NeteaG. NolanD. Pe'erA. PhillipakisC. P. PontingS. QuakeW. ReikO. Rozenblatt-RosenJ. SanesR. SatijaT. N. SchumacherA. ShalekE. ShapiroP. SharmaJ. W. ShinO. StegleM. StrattonM. J. T. StubbingtonF. J. TheisM. UhlenA. van OudenaardenA. WagnerF. WattJ. WeissmanB. WoldR. XavierN. Yosef and Hu man Cell Atlas Meeting Participants, The human cell atlas, Elife, 6 (2017), 503. doi: 10.7554/eLife.27041. Google Scholar

[57]

P. RiemkeM. CzehJ. FischerC. WalterS. GhaniM. ZepperK. AgelopoulosS. LettermannM. L. GebhardtN. MahA. WeilemannM. GrauV. GröningT. HaferlachD. LenzeR. DelwelM. PrinzM. A. Andrade-NavarroG. LenzM. DugasC. Müller-Tidow and F. Rosenbauer, Myeloid leukemia with transdifferentiation plasticity developing from T-cell progenitors, EMBO J., 35 (2016), 2399-2416. doi: 10.15252/embj.201693927. Google Scholar

[58]

M. S. SamoilovG. Price and A. P. Arkin, From fluctuations to phenotypes: The physiology of noise, Sci. STKE, 2006 (2006), re17. doi: 10.1126/stke.3662006re17. Google Scholar

[59]

V. ShahrezaeiJ. F. Ollivier and P. S. Swain, Colored extrinsic fluctuations and stochastic gene expression, Molecular Systems Biology, 4 (2008), 196. doi: 10.1038/msb.2008.31. Google Scholar

[60]

V. Shahrezaei and P. S. Swain, The stochastic nature of biochemical networks, Current Opinion in Biotechnology, 19 (2008), 369-374. doi: 10.1016/j.copbio.2008.06.011. Google Scholar

[61]

X. Shen and S. H. Orkin, Glimpses of the epigenetic landscape, Stem Cell, 4 (2008), 1-2. doi: 10.1016/j.stem.2008.12.006. Google Scholar

[62]

K. SneppenM. A. Micheelsen and I. B. Dodd, Ultrasensitive gene regulation by positive feedback loops in nucleosome modification, Mol. Syst. Biol., 4 (2008), 182-182. doi: 10.1038/msb.2008.21. Google Scholar

[63]

S. S. Sommer and N. A. Rin, The lognormal distribution fits the decay profile of eukaryotic mRNA, Biochem. Biophys. Res. Commun., 90 (1979), 135-141. doi: 10.1016/0006-291X(79)91600-0. Google Scholar

[64]

P. S. SwainM. B. Elowitz and E. D. Siggia, Intrinsic and extrinsic contributions to stochasticity in gene expression, Proc. Nat. Acad. Sci. USA, 99 (2002), 12795-12800. doi: 10.1073/pnas.162041399. Google Scholar

[65]

N. TakadaH. J. PiV. H. SousaJ. WateresG. FishellA. Kepecs and P. Osten, A developmental cell-type switch in cortical interneurons leads to a selective defect in cortical oscillations, Nat. Commun., 5 (2014), 5333. doi: 10.1038/ncomms6333. Google Scholar

[66]

W. L. Tam and R. A. Weinberg, The epigenetics of epithelial-mesenchymal plasticity in cancer, Nat. Med., 19 (2013), 1438-1449. doi: 10.1038/nm.3336. Google Scholar

[67]

T. Tian and K. Burrage, Stochastic models for regulatory networks of the genetic toggle switch, Proc. Nat. Acad. Sci. USA, 103 (2006), 8372-8377. doi: 10.1073/pnas.0507818103. Google Scholar

[68]

J. TsoiL. RobertK. ParaisoC. GalvanK. M. SheuJ. LayD. J. L. WongM. AtefiR. ShiraziX. WangD. BraasC. S. GrassoN. PalaskasA. Ribas and T. G. Graeber, Multi-stage differentiation defines melanoma subtypes with differential vulnerability to drug-induced iron-dependent oxidative stress, Cancer Cell, 33 (2018), 890-904, e5. doi: 10.1016/j.ccell.2018.03.017. Google Scholar

[69]

L. WangB. L. WalkerS. IannacconeD. BhattP. J. Kennedy and W. T. Tse, Bistable switches control memory and plasticity in cellular differentiation, Proc. Natl. Acad. Sci. USA, 106 (2009), 6638-6643. doi: 10.1073/pnas.0806137106. Google Scholar

[70]

L. D. Wang and A. J. Wagers, Dynamic niches in the origination and differentiation of haematopoietic stem cells, Nat. Rev. Mol. Cell Biol., 12 (2011), 643-655. doi: 10.1038/nrm3184. Google Scholar

[71]

W. WestonJ. ZayasR. PerezJ. George and R. Jurecic, Dynamic equilibrium of heterogeneous and interconvertible multipotent hematopoietic cell subsets, Sci. Rep., 4 (2014), 5199-5199. doi: 10.1038/srep05199. Google Scholar

[72]

W. Xia and J. Lei, Formulation of the protein synthesis rate with sequence information, MBE, 15 (2018), 507-522. doi: 10.3934/mbe.2018023. Google Scholar

[73]

H. ZhangX.-J. TianA. MukhopadhyayK. S. Kim and J. Xing, Statistical mechanics model for the dynamics of collective epigenetic histone modification, Phys. Rev. Lett., 112 (2014), 068101. doi: 10.1103/PhysRevLett.112.068101. Google Scholar

[74]

J. ZhangX.-J. TianH. ZhangY. TengR. LiF. BaiS. Elankumaran and J. Xing, TGF-β-induced epithelial-to-mesenchymal transition proceeds through stepwise activation of multiple feedback loops, Sci. Signal, 7 (2014), ra91-ra91. Google Scholar

[75]

J. X. Zhou and S. Huang, Understanding gene circuits at cell-fate branch points for rational cell reprogramming, Trends Genet., 27 (2011), 55-62. doi: 10.1016/j.tig.2010.11.002. Google Scholar

[76]

Y. ZhouJ. KimX. Yuan and T. Braun, Epigenetic modifications of stem cells: A paradigm for the control of cardiac progenitor cells, Circ. Res., 109 (2011), 1067-1081. doi: 10.1161/CIRCRESAHA.111.243709. Google Scholar

show all references

References:
[1]

U. Alon, An Introduction to Systems Biology, Boca Raton, FL, 2007. Google Scholar

[2]

M. R. AtkinsonM. A. SavageauJ. T. Myers and A. J. Ninfa, Development of genetic circuitry exhibiting toggle switch or oscillatory behavior in escherichia coli, Cell, 113 (2003), 597-607. doi: 10.1016/S0092-8674(03)00346-5. Google Scholar

[3]

D. W. Austin, M. S. Allen, J. M. Mccollum, R. D. Dar, J. R. Wilgus, G. S. Sayler, N. F. Samatova, C. D. Cox and M. L. Simpson, Gene network shaping of inherent noise spectra, in Bio MICRO and Nanosystems Conference, 2007,608. doi: 10.1109/BMN.2006.330897. Google Scholar

[4]

J. S. BeckerD. Nicetto and K. S. Zaret, H3K9me3-dependent heterochromatin: Barrier to cell fate changes, Trends Genet., 32 (2015), 29-41. doi: 10.1016/j.tig.2015.11.001. Google Scholar

[5]

L. BintuJ. YongY. E. AntebiK. McCueY. KazukiN. UnoM. Oshimura and M. B. Elowitz, Dynamics of epigenetic regulation at the single-cell level, Science, 351 (2016), 720-724. Google Scholar

[6]

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

[7]

J. L. Cherry and F. R. Adler, How to make a biological switch, J. Theor. Biol., 203 (2000), 117-133. doi: 10.1006/jtbi.2000.1068. Google Scholar

[8]

P. J. ChoiL. CaiK. Frieda and X. S. Xie, A stochastic single-molecule event triggers phenotype switching of a bacterial cell, Science, 322 (2008), 442-446. doi: 10.1126/science.1161427. Google Scholar

[9]

K. CuiC. ZangT.-Y. RohD. E. SchonesR. W. ChildsW. Peng and K. Zhao, Chromatin signatures in multipotent human hematopoietic stem cells indicate the fate of bivalent genes during differentiation, Stem Cell, 4 (2009), 80-93. doi: 10.1016/j.stem.2008.11.011. Google Scholar

[10]

I. B. DoddM. A. MicheelsenK. Sneppen and G. Thon, Theoretical analysis of epigenetic cell memory by nucleosome modification, Cell, 129 (2007), 813-822. doi: 10.1016/j.cell.2007.02.053. Google Scholar

[11]

S.-J. DunnG. MartelloB. YordanovS. Emmott and A. G. Smith, Defining an essential transcription factor program for naïve pluripotency, Science, 344 (2014), 1156-1160. Google Scholar

[12]

H. EaswaranH.-C. Tsai and S. B. Baylin, Cancer epigenetics: Tumor heterogeneity, plasticity of stem-like states, and drug resistance, Mol. Cell, 54 (2014), 716-727. doi: 10.1016/j.molcel.2014.05.015. Google Scholar

[13]

M. B. ElowitzA. J. LevineE. D. Siggia and P. S. Swain, Stochastic gene expression in a single cell, Science, 297 (2002), 1183-1186. doi: 10.1126/science.1070919. Google Scholar

[14]

T. EnverM. PeraC. Peterson and P. W. Andrews, Stem cell states, fates, and the rules of attraction, Stem Cell, 4 (2008), 387-397. doi: 10.1016/j.stem.2009.04.011. Google Scholar

[15]

M. FarlikF. HalbritterF. MüllerF. A. ChoudryP. EbertJ. KlughammerS. FarrowA. SantoroV. CiaurroA. MathurR. UppalH. G. StunnenbergW. H. OuwehandE. LaurentiT. LengauerM. Frontini and C. Bock, DNA methylation dynamics of human hematopoietic stem cell differentiation, Cell Stem Cell, 19 (2016), 808-822. doi: 10.1016/j.stem.2016.10.019. Google Scholar

[16]

J. E. Ferrell, Bistability, bifurcations, and Waddington's epigenetic landscape, Curr. Biol., 22 (2012), R458-R466. doi: 10.1016/j.cub.2012.03.045. Google Scholar

[17]

J. E. Ferrell Jr, Self-perpetuating states in signal transduction: Positive feedback, double-negative feedback and bistability, Curr. Opin. Cell Biol., 14 (2002), 140. Google Scholar

[18]

J. E. Ferrell Jr. and W. Xiong, Bistability in cell signaling: How to make continuous processes discontinuous, and reversible processes irreversible, Chaos, 11 (2001), 227-236. Google Scholar

[19]

W. A. FlavahanE. Gaskell and B. E. Bernstein, Epigenetic plasticity and the hallmarks of cancer, Science, 357 (2017), eaal2380. doi: 10.1126/science.aal2380. Google Scholar

[20]

C. Furusawa and K. Kaneko, A dynamical-systems view of stem cell biology, Science, 338 (2012), 215-217. doi: 10.1126/science.1224311. Google Scholar

[21]

N. G. van Kampen, Stochastic processes in physics and chemistry, Physics Today, 36 (1983), p78. doi: 10.1063/1.2915501. Google Scholar

[22]

D. J. GaffneyG. McvickerA. A. PaiY. N. FondufemittendorfN. LewellenK. MicheliniJ. WidomY. Gilad and J. K. Pritchard, Controls of nucleosome positioning in the human genome, PloS Genetics, 8 (2012), e1003036. doi: 10.1371/journal.pgen.1003036. Google Scholar

[23]

T. S. GardnerC. R. Cantor and J. J. Collins, Construction of a genetic toggle switch in escherichia coli, Nature, 403 (2000), 339-342. doi: 10.1038/35002131. Google Scholar

[24]

W. GuoZ. KeckesovaJ. L. DonaherT. ShibueV. TischlerF. ReinhardtS. ItzkovitzA. NoskeU. Zürrer-HärdiG. BellW. L. TamS. A. ManiA. van Oudenaarden and R. A. Weinberg, Slug and Sox9 cooperatively determine the mammary stem cell state, Cell, 148 (2012), 1015-1028. doi: 10.1016/j.cell.2012.02.008. Google Scholar

[25]

P. GuptaG. U. GuruduttaD. Saluja and R. P. Tripathi, PU.1 and partners: Regulation of haematopoietic stem cell fate in normal and malignant haematopoiesis, J. Cell Mol. Med., 13 (2009), 4349-4363. doi: 10.1111/j.1582-4934.2009.00757.x. Google Scholar

[26]

J. HastyJ. PradinesM. Dolnik and J. J. Collins, Noise-based switches and amplifiers for gene expression, Proc. Nat. Acad. Sci. USA, 97 (2000), 2075-2080. doi: 10.1073/pnas.040411297. Google Scholar

[27]

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

[28]

M. HembergerW. Dean and W. Reik, Epigenetic dynamics of stem cells and cell lineage commitment: Digging Waddington's canal, Nat. Rev. Mol. Cell Biol., 10 (2009), 526-537. doi: 10.1038/nrm2727. Google Scholar

[29]

R. Huang and J. Lei, Dynamics of gene expression with positive feedback to histone modifications at bivalent domains, Int. J. Mod. Phys. B, 32 (2018), 1850075, 22pp. doi: 10.1142/S0217979218500753. Google Scholar

[30]

S. HuangI. Ernberg and S. Kauffman, Cancer attractors: A systems view of tumors from a gene network dynamics and developmental perspective, Semin. Cell Dev. Biol., 20 (2009), 869-876. doi: 10.1016/j.semcdb.2009.07.003. Google Scholar

[31]

R. Jaenisch and A. Bird, Epigenetic regulation of gene expression: How the genome integrates intrinsic and environmental signals, Nat. Genet., 33 (2003), 245-254. doi: 10.1038/ng1089. Google Scholar

[32]

X. Jiao and J. Lei, Dynamics of gene expression based on epigenetic modifications, Communications in Information and Systems, 18 (2018), 125-148. doi: 10.4310/CIS.2018.v18.n3.a1. Google Scholar

[33]

M. KaernT. C. ElstonW. J. Blake and J. J. Collins, Stochasticity in gene expression: From theories to phenotypes, Nat. Rev. Genet., 6 (2005), 451-64. doi: 10.1038/nrg1615. Google Scholar

[34]

S. KanjiV. J. Pompili and H. Das, Plasticity and maintenance of hematopoietic stem cells during development, Recent Patents on Biotechnology, 5 (2011), 40-53. doi: 10.2174/187220811795655896. Google Scholar

[35]

B. B. Kaufmann and O. A. Van, Stochastic gene expression: From single molecules to the proteome, Curr. Opin. Genet. Dev., 17 (2007), 107-112. doi: 10.1016/j.gde.2007.02.007. Google Scholar

[36]

B. B. KaufmannQ. YangJ. T. Mettetal and V. O. Alexander, Heritable stochastic switching revealed by single-cell genealogy, PloS Biol., 5 (2007), e239. doi: 10.1371/journal.pbio.0050239. Google Scholar

[37]

E. Laurenti and B. Göttgens, From haematopoietic stem cells to complex differentiation landscapes, Nature, 553 (2018), 418-426. Google Scholar

[38]

J. Lei, Stochasticity in single gene expression with both intrinsic noise and fluctuation in kinetic parameters, J. Theor. Biol., 256 (2009), 485-492. doi: 10.1016/j.jtbi.2008.10.028. Google Scholar

[39]

J. LeiG. HeH. Liu and Q. Nie, A delay model for noise-induced bi-directional switching, Nonlinearity, 22 (2009), 2845-2859. doi: 10.1088/0951-7715/22/12/003. Google Scholar

[40]

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

[41]

C. Li and J. Wang, Quantifying waddington landscapes and paths of non-adiabatic cell fate decisions for differentiation, reprogramming and transdifferentiation, Journal of the Royal Society Interface, 10 (2013), 20130787. doi: 10.1098/rsif.2013.0787. Google Scholar

[42]

Q. LiA. WennborgE. AurellE. DekelJ.-Z. ZouY. XuS. Huang and I. Ernberg, Dynamics inside the cancer cell attractor reveal cell heterogeneity, limits of stability, and escape, Proc. Natl. Acad. Sci. USA, 113 (2016), 2672-2677. doi: 10.1073/pnas.1519210113. Google Scholar

[43]

E. LimpertW. Stahel and M. Abbt, Log-normal distributions across the sciences: Keys and clues, BioScience, 51 (2001), 341-352. Google Scholar

[44]

H. Maamar and D. Dubnau, Bistability in the Bacillus subtilis K-state (competence) system requires a positive feedback loop, Mol. Microbio., 56 (2005), 615-624. doi: 10.1111/j.1365-2958.2005.04592.x. Google Scholar

[45]

B. D. MacArthurA. SevillaM. LenzF.-J. MüllerB. M. SchuldtA. A. SchuppertS. J. RiddenP. S. StumpfM. FidalgoA. Ma'ayanJ. Wang and I. R. Lemischka, Nanog-dependent feedback loops regulate murine embryonic stem cell heterogeneity, Nat. Cell Biol., 14 (2012), 1139-1147. doi: 10.1038/ncb2603. Google Scholar

[46]

I. C. MacaulayV. SvenssonC. LabaletteL. FerreiraF. HameyT. VoetS. A. Teichmann and A. Cvejic, Single-cell RNA-sequencing reveals a continuous spectrum of differentiation in hematopoietic cells, Cell Rep., 14 (2016), 966-977. doi: 10.1016/j.celrep.2015.12.082. Google Scholar

[47]

A. L. MacLeanT. Hong and Q. Nie, Exploring intermediate cell states through the lens of single cells, Curr. Opin. Syst. Biol., 9 (2018), 32-41. doi: 10.1016/j.coisb.2018.02.009. Google Scholar

[48]

M. A. Nieto, Epithelial plasticity: A common theme in embryonic and cancer cells, Science, 342 (2013), 1234850. doi: 10.1126/science.1234850. Google Scholar

[49]

B. Øksendal, Stochastic Differential Equations–An Introdcution with Applications, Sixth edition, Universitext, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-642-14394-6. Google Scholar

[50]

E. M. OzbudakM. ThattaiH. N. LimB. I. Shraiman and O. A. Van, Multistability in the lactose utilization network of escherichia coli, Nature, 427 (2004), 737-740. doi: 10.1038/nature02298. Google Scholar

[51]

J. Paulsson, Summing up the noise in gene networks, Nature, 427 (2004), 415-418. doi: 10.1038/nature02257. Google Scholar

[52]

L. PinelloJ. XuS. H. Orkin and G. C. Yuan, Analysis of chromatin-state plasticity identifies cell-type-specific regulators of H3K27me3 patterns, Proc. Natl. Acad. Sci. USA, 111 (2014), E344-E353. doi: 10.1073/pnas.1322570111. Google Scholar

[53]

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

[54]

C. RackauckasT. Schilling and Q. Nie, Mean-independent noise control of cell fates via intermediate states, iScience, 3 (2018), 11-20. doi: 10.1016/j.isci.2018.04.002. Google Scholar

[55]

A. Raj and A. V. Oudenaarden, Nature, nurture, or chance: Stochastic gene expression and its consequences, Cell, 135 (2008), 216-226. doi: 10.1016/j.cell.2008.09.050. Google Scholar

[56]

A. RegevS. A. TeichmannE. S. LanderI. AmitC. BenoistE. BirneyB. BodenmillerP. CampbellP. CarninciM. ClatworthyH. CleversB. DeplanckeI. DunhamJ. EberwineR. EilsW. EnardA. FarmerL. FuggerB. GöttgensN. HacohenM. HaniffaM. HembergS. KimP. KlenermanA. KriegsteinE. LeinS. LinnarssonE. LundbergJ. LundebergP. MajumderJ. C. MarioniM. MeradM. MhlangaM. NawijnM. NeteaG. NolanD. Pe'erA. PhillipakisC. P. PontingS. QuakeW. ReikO. Rozenblatt-RosenJ. SanesR. SatijaT. N. SchumacherA. ShalekE. ShapiroP. SharmaJ. W. ShinO. StegleM. StrattonM. J. T. StubbingtonF. J. TheisM. UhlenA. van OudenaardenA. WagnerF. WattJ. WeissmanB. WoldR. XavierN. Yosef and Hu man Cell Atlas Meeting Participants, The human cell atlas, Elife, 6 (2017), 503. doi: 10.7554/eLife.27041. Google Scholar

[57]

P. RiemkeM. CzehJ. FischerC. WalterS. GhaniM. ZepperK. AgelopoulosS. LettermannM. L. GebhardtN. MahA. WeilemannM. GrauV. GröningT. HaferlachD. LenzeR. DelwelM. PrinzM. A. Andrade-NavarroG. LenzM. DugasC. Müller-Tidow and F. Rosenbauer, Myeloid leukemia with transdifferentiation plasticity developing from T-cell progenitors, EMBO J., 35 (2016), 2399-2416. doi: 10.15252/embj.201693927. Google Scholar

[58]

M. S. SamoilovG. Price and A. P. Arkin, From fluctuations to phenotypes: The physiology of noise, Sci. STKE, 2006 (2006), re17. doi: 10.1126/stke.3662006re17. Google Scholar

[59]

V. ShahrezaeiJ. F. Ollivier and P. S. Swain, Colored extrinsic fluctuations and stochastic gene expression, Molecular Systems Biology, 4 (2008), 196. doi: 10.1038/msb.2008.31. Google Scholar

[60]

V. Shahrezaei and P. S. Swain, The stochastic nature of biochemical networks, Current Opinion in Biotechnology, 19 (2008), 369-374. doi: 10.1016/j.copbio.2008.06.011. Google Scholar

[61]

X. Shen and S. H. Orkin, Glimpses of the epigenetic landscape, Stem Cell, 4 (2008), 1-2. doi: 10.1016/j.stem.2008.12.006. Google Scholar

[62]

K. SneppenM. A. Micheelsen and I. B. Dodd, Ultrasensitive gene regulation by positive feedback loops in nucleosome modification, Mol. Syst. Biol., 4 (2008), 182-182. doi: 10.1038/msb.2008.21. Google Scholar

[63]

S. S. Sommer and N. A. Rin, The lognormal distribution fits the decay profile of eukaryotic mRNA, Biochem. Biophys. Res. Commun., 90 (1979), 135-141. doi: 10.1016/0006-291X(79)91600-0. Google Scholar

[64]

P. S. SwainM. B. Elowitz and E. D. Siggia, Intrinsic and extrinsic contributions to stochasticity in gene expression, Proc. Nat. Acad. Sci. USA, 99 (2002), 12795-12800. doi: 10.1073/pnas.162041399. Google Scholar

[65]

N. TakadaH. J. PiV. H. SousaJ. WateresG. FishellA. Kepecs and P. Osten, A developmental cell-type switch in cortical interneurons leads to a selective defect in cortical oscillations, Nat. Commun., 5 (2014), 5333. doi: 10.1038/ncomms6333. Google Scholar

[66]

W. L. Tam and R. A. Weinberg, The epigenetics of epithelial-mesenchymal plasticity in cancer, Nat. Med., 19 (2013), 1438-1449. doi: 10.1038/nm.3336. Google Scholar

[67]

T. Tian and K. Burrage, Stochastic models for regulatory networks of the genetic toggle switch, Proc. Nat. Acad. Sci. USA, 103 (2006), 8372-8377. doi: 10.1073/pnas.0507818103. Google Scholar

[68]

J. TsoiL. RobertK. ParaisoC. GalvanK. M. SheuJ. LayD. J. L. WongM. AtefiR. ShiraziX. WangD. BraasC. S. GrassoN. PalaskasA. Ribas and T. G. Graeber, Multi-stage differentiation defines melanoma subtypes with differential vulnerability to drug-induced iron-dependent oxidative stress, Cancer Cell, 33 (2018), 890-904, e5. doi: 10.1016/j.ccell.2018.03.017. Google Scholar

[69]

L. WangB. L. WalkerS. IannacconeD. BhattP. J. Kennedy and W. T. Tse, Bistable switches control memory and plasticity in cellular differentiation, Proc. Natl. Acad. Sci. USA, 106 (2009), 6638-6643. doi: 10.1073/pnas.0806137106. Google Scholar

[70]

L. D. Wang and A. J. Wagers, Dynamic niches in the origination and differentiation of haematopoietic stem cells, Nat. Rev. Mol. Cell Biol., 12 (2011), 643-655. doi: 10.1038/nrm3184. Google Scholar

[71]

W. WestonJ. ZayasR. PerezJ. George and R. Jurecic, Dynamic equilibrium of heterogeneous and interconvertible multipotent hematopoietic cell subsets, Sci. Rep., 4 (2014), 5199-5199. doi: 10.1038/srep05199. Google Scholar

[72]

W. Xia and J. Lei, Formulation of the protein synthesis rate with sequence information, MBE, 15 (2018), 507-522. doi: 10.3934/mbe.2018023. Google Scholar

[73]

H. ZhangX.-J. TianA. MukhopadhyayK. S. Kim and J. Xing, Statistical mechanics model for the dynamics of collective epigenetic histone modification, Phys. Rev. Lett., 112 (2014), 068101. doi: 10.1103/PhysRevLett.112.068101. Google Scholar

[74]

J. ZhangX.-J. TianH. ZhangY. TengR. LiF. BaiS. Elankumaran and J. Xing, TGF-β-induced epithelial-to-mesenchymal transition proceeds through stepwise activation of multiple feedback loops, Sci. Signal, 7 (2014), ra91-ra91. Google Scholar

[75]

J. X. Zhou and S. Huang, Understanding gene circuits at cell-fate branch points for rational cell reprogramming, Trends Genet., 27 (2011), 55-62. doi: 10.1016/j.tig.2010.11.002. Google Scholar

[76]

Y. ZhouJ. KimX. Yuan and T. Braun, Epigenetic modifications of stem cells: A paradigm for the control of cardiac progenitor cells, Circ. Res., 109 (2011), 1067-1081. doi: 10.1161/CIRCRESAHA.111.243709. Google Scholar

Figure 1.  Illustration of gene regulation combines a positive feedback with stochastic histone modification at the enhancer region. Histone modifications alter the 50% effective concentration (EC50) of the toggle switch to regulate gene expression (see Formulations for details)
Figure 2.  The bifurcation diagrams with respect to $ K $ and $ u $. (A) The protein level at steady state as a function of $ K $. Blue dashed lines show the saddle-node bifurcations ($ K^* $ and $ K^{**} $). (B) The heat map representation of joint density distribution of the protein level and $ u $ for a typical trajectory under random inheritance of histone modifications and without the external noise. In the simulation, $ \varphi(u_k) = a + b (u_k - a) $ and $ a = 0.5, b = 0.8 $ (to be detailed in Section 3.2)
Figure 3.  Cell-type switches induced by external noise. (A) Sample dynamics of protein level in $ 10^5 $ cycles. In the simulation, $ K = 450 $, $ \sigma = 1.8 $. (B) The heat map representation of the switching frequency with respect to $ K $ and $ \sigma $. (C) A sample dynamics of $ \lambda_p $ with $ \sigma = 1.5 $. Inset is the probability distribution of $ \ln(\lambda_p) $. (D) The probability distribution of the steady state lifetimes with respect to the trajectory in (A)
Figure 4.  Dynamics of nucleosome states under random histone modification inheritance. (A) A sample dynamics of the nucleosome state along with cell cycles. (B) The distribution (blue) of the sequence $ \{u_k\} $ with three sets of parameters of $ a $ and $ b $, and the Beta distribution density function (red) (16) with shape parameters $ (\alpha, \beta) $ given by (17)
Figure 5.  Frequencies of cell-type switches with histone modification to affect the transcription activities. (A) The heat map representation of the switching frequency with respect to $ a $ and $ b $ without the external noise. (B) The switching frequency as a function of $ a $ with varying parameter $ b $. (C) The heat map representation of the switching frequency with respect to $ b $ and the noise strength $ \sigma $ (here $ a = 0.5 $). (D) The heat map representation of the switching frequency with respect to $ a $ and the noise strength $ \sigma $ (here $ b = 0.7 $)
Figure 6.  Dynamic equilibrium of cell regeneration. Figures show the probability distribution of the protein level at the end of cycle $ 0 $, $ 5 $, $ 10 $, $ 15 $, and $ 20 $. Results for 4 sample simulations ((A)-(D)) are shown, each with different initial conditions. Parameters are $ a = 0.5, b = 0.7 $
Figure 7.  Stationary distribution of proteins in a cell colony. (A)The probability distribution of the protein level at the end of $ 20 $ cycles with fixed $ b = 0.7 $ and varied $ a $: $ a = 0.4 $, $ a = 0.5 $, and $ a = 0.6 $. (B) The probability distribution of the protein level at the end of $ 20 $ cycles with fixed $ a = 0.5 $ and varied $ b $: $ b = 0.6 $, $ b = 0.7 $, and $ b = 0.8 $. We take $ \sigma = 0.5 $ in simulations
Figure 8.  Potential landscapes. (A)The potential landscape (23) as a function of protein level with $ K = 200 $, $ K = 450 $, and $ K = 750 $. In the simulation, $ \sigma = 0.1 $. (B)The Waddington landscape of colony evolution from single cells. The potential was obtained from simulations results of 30 independent trajectories. Parameters are $ a = 0.5, b = 0.7 $
Figure 9.  Dynamics of intermediate cell states. (A) A sample dynamics of transitions between three cell states based on the assumption (24). (B) The distribution of nucleosome state $ u_k $. (C) The distribution of the protein levels along the trajectory. In the simulation, we took $ m = 6 $, $ c = 0.25 $, $ d = 0.5 $, $ u_0 = 0.488 $, $ n = 2.5 $, $ \gamma_1 = 0.499 $, $ \gamma_2 = 6.471 $, $ \sigma = 0.5 $, and the other parameters are the same as those in Table 1
Table 1.  Model parameters used in simulations
Parameters Definitions Values Source$ ^* $
$ \lambda_{m,1} $ basal transcriptional rate $ 2 h^{-1} $ [1]
$ \lambda_{m,2} $ maximum promotion in transcription $ 18 h^{-1} $ [1]
due to the protein regulation
$ \lambda_p $ translational efficiency $ 2 h^{-1} $ [1]
$ d_m $ mRNA degradation rate $ 0.5 h^{-1} $ [1]
$ d_p $ protein degradation rate $ 0.08 h^{-1} $ [1]
$ N $ number of nucleosomes $ 60 $ a.a.
$ n $ Hill coefficient $ 4 $ a.a.
$ T $ duration of one cell cycle $ 20h $ a.a.
$ \gamma_1 $ constant coefficient $ 6.780 $ adjust
$ \gamma_2 $ constant coefficient $ 9.433 $ adjust
* Here a.a. means arbitrary assigned, adjust means that we adjust the parameters to yield bistability.
Parameters Definitions Values Source$ ^* $
$ \lambda_{m,1} $ basal transcriptional rate $ 2 h^{-1} $ [1]
$ \lambda_{m,2} $ maximum promotion in transcription $ 18 h^{-1} $ [1]
due to the protein regulation
$ \lambda_p $ translational efficiency $ 2 h^{-1} $ [1]
$ d_m $ mRNA degradation rate $ 0.5 h^{-1} $ [1]
$ d_p $ protein degradation rate $ 0.08 h^{-1} $ [1]
$ N $ number of nucleosomes $ 60 $ a.a.
$ n $ Hill coefficient $ 4 $ a.a.
$ T $ duration of one cell cycle $ 20h $ a.a.
$ \gamma_1 $ constant coefficient $ 6.780 $ adjust
$ \gamma_2 $ constant coefficient $ 9.433 $ adjust
* Here a.a. means arbitrary assigned, adjust means that we adjust the parameters to yield bistability.
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