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The Mathematics of Phenotypic State Transition: Paths and Potential

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Journal of the Indian Institute of Science Aims and scope

Abstract

Change in the phenotype of a cell is considered as a transition of a cell from one cellular state to another. Cellular state transition can be driven by an external cue or by the noise in molecular processes. Over the years, generalized physical principles, and associated mathematical models have been developed to understand phenotypic state transition. Starting with Waddington’s epigenetic landscape, phenotypic state transition is seen as a movement of cells on a potential landscape. Though the landscape model is close to the thermodynamic principles of state change, it is difficult to envisage it from experimental observations. Therefore, phenotypic state transition is often considered as a discrete state jump process. This approach is particularly useful to estimate the paths of state transition from experimental observations. In this review, we discuss both of these approaches and the associated mathematical formulations. Furthermore, we explore the opportunities to connect these two approaches and the limitations of our current understanding and mathematical methods.

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References

  1. Hartl DL, Jones EW (1998) Genetics principles and analysis, 4th edn. Jones and Barlett Publishers, Sudbury

    Google Scholar 

  2. Tyler S (2003) Epithelium–the primary building block for metazoan complexity. Integr Comp Biol 43:55–63

    Google Scholar 

  3. Kim DH, Xing T, Yang Z, Dudek R, Lu Q, Chen YH (2017) Epithelial mesenchymal transition in embryonic development, tissue repair and cancer: a comprehensive overview. J Clin Med 7:1

    Google Scholar 

  4. Stumpf PS, Smith RCG, Lenz M, Schuppert A, Muller FJ, Babtie A, Chan TE, Stumpf MPH, Please CP, Howison SD, Arai F, MacArthur BD (2017) Stem cell differentiation as a non-markov stochastic process. Cell Syst 5(268–282):e267

    Google Scholar 

  5. Pisco AO, Brock A, Zhou J, Moor A, Mojtahedi M, Jackson D, Huang S (2013) Non-Darwinian dynamics in therapy-induced cancer drug resistance. Nat Commun 4:2467

    Google Scholar 

  6. Kumar N, Cramer GM, Dahaj SAZ, Sundaram B, Celli JP, Kulkarni RV (2019) Stochastic modeling of phenotypic switching and chemoresistance in cancer cell populations. Sci Rep 9:10845

    Google Scholar 

  7. Wang W, Quan Y, Fu Q, Liu Y, Liang Y, Wu J, Yang G, Luo C, Ouyang Q, Wang Y (2014) Dynamics between cancer cell subpopulations reveals a model coordinating with both hierarchical and stochastic concepts. PLoS ONE 9:e84654

    Google Scholar 

  8. Gupta PB, Fillmore CM, Jiang G, Shapira SD, Tao K, Kuperwasser C, Lander ES (2011) Stochastic state transitions give rise to phenotypic equilibrium in populations of cancer cells. Cell 146:633–644

    CAS  Google Scholar 

  9. Yang G, Quan Y, Wang W, Fu Q, Wu J, Mei T, Li J, Tang Y, Luo C, Ouyang Q, Chen S, Wu L, Hei TK, Wang Y (2012) Dynamic equilibrium between cancer stem cells and non-stem cancer cells in human SW620 and MCF-7 cancer cell populations. Br J Cancer 106:1512–1519

    CAS  Google Scholar 

  10. Tam WL, Weinberg RA (2013) The epigenetics of epithelial-mesenchymal plasticity in cancer. Nat Med 19:1438–1449

    CAS  Google Scholar 

  11. Murke F, Castro SVC, Giebel B, Görgens AJS (2015) Concise review: asymmetric cell divisions in stem. Cell Biol 7:2025–2037

    CAS  Google Scholar 

  12. Su Y, Wei W, Robert L, Xue M, Tsoi J, Garcia-Diaz A, Homet Moreno B, Kim J, Ng RH, Lee JW, Koya RC, Comin-Anduix B, Graeber TG, Ribas A, Heath JR (2017) Single-cell analysis resolves the cell state transition and signaling dynamics associated with melanoma drug-induced resistance. Proc Natl Acad Sci USA 114:13679–13684

    CAS  Google Scholar 

  13. Steinestel K, Eder S, Schrader AJ, Steinestel J (2014) Clinical significance of epithelial-mesenchymal transition. Clin Transl Med 3:17

    Google Scholar 

  14. Pastushenko I, Blanpain C (2019) EMT transition states during tumor progression and metastasis. Trends Cell Biol 29:212–226

    CAS  Google Scholar 

  15. Calloni R, Cordero EA, Henriques JA, Bonatto D (2013) Reviewing and updating the major molecular markers for stem cells. Stem Cells Dev 22:1455–1476

    CAS  Google Scholar 

  16. Armond JW, Saha K, Rana AA, Oates CJ, Jaenisch R, Nicodemi M, Mukherjee S (2014) A stochastic model dissects cell states in biological transition processes. Sci Rep 4:3692

    Google Scholar 

  17. Trapnell C (2015) Defining cell types and states with single-cell genomics. Genome Res 25:1491–1498

    CAS  Google Scholar 

  18. Devaraj V, Bose B (2019) Morphological state transition dynamics in EGF-induced epithelial to mesenchymal transition. J Clin Med 8:911

    CAS  Google Scholar 

  19. Mandal M, Ghosh B, Anura A, Mitra P, Pathak T, Chatterjee J (2016) Modeling continuum of epithelial mesenchymal transition plasticity. Integr Biol (Camb) 8:167–176

    CAS  Google Scholar 

  20. Sommer C, Hoefler R, Samwer M, Gerlich DW (2017) A deep learning and novelty detection framework for rapid phenotyping in high-content screening. Mol Biol Cell 28:3428–3436

    CAS  Google Scholar 

  21. Buggenthin F, Buettner F, Hoppe PS, Endele M, Kroiss M, Strasser M, Schwarzfischer M, Loeffler D, Kokkaliaris KD, Hilsenbeck O, Schroeder T, Theis FJ, Marr C (2017) Prospective identification of hematopoietic lineage choice by deep learning. Nat Methods 14:403–406

    CAS  Google Scholar 

  22. Carpenter AE, Jones TR, Lamprecht MR, Clarke C, Kang IH, Friman O, Guertin DA, Chang JH, Lindquist RA, Moffat J, Golland P, Sabatini DM (2006) Cell Profiler: image analysis software for identifying and quantifying cell phenotypes. Genome Biol 7:R100

    Google Scholar 

  23. Wang W, Douglas D, Zhang J, Chen YJ, Cheng YY, Kumari S, Enuameh MS, Dai Y, Wallace CT, Watkins SC, Shu W, Xing J (2019) M-TRACK: a platform for live cell multiplex imaging reveals cell phenotypic transition dynamics inherently missing in snapshot data. bioRxiv. https://doi.org/10.1101/2019.12.12.874248

    Article  Google Scholar 

  24. Kimmel JC, Chang AY, Brack AS, Marshall WF (2018) Inferring cell state by quantitative motility analysis reveals a dynamic state system and broken detailed balance. PLoS Comput Biol 14:e1005927

    Google Scholar 

  25. Rimchala T, Kamm RD, Lauffenburger DA (2013) Endothelial cell phenotypic behaviors cluster into dynamic state transition programs modulated by angiogenic and angiostatic cytokines. Integr Biol (Camb) 5:510–522

    CAS  Google Scholar 

  26. Waddington CH (1957) The strategy ofthe genes: a discussion of some aspects of theoretical biology. George Allen & Unwin Ltd, London

    Google Scholar 

  27. Strogatz SH (2018) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. CRC Press, Boca Raton

    Google Scholar 

  28. Lu M, Jolly MK, Levine H, Onuchic JN, Ben-Jacob E (2013) MicroRNA-based regulation of epithelial-hybrid-mesenchymal fate determination. Proc Natl Acad Sci USA 110:18144–18149

    CAS  Google Scholar 

  29. Jolly MK, Boareto M, Huang B, Jia D, Lu M, Ben-Jacob E, Onuchic JN, Levine H (2015) Implications of the hybrid epithelial/mesenchymal phenotype in metastasis. Front Oncol 5:155

    Google Scholar 

  30. Zhang J, Tian XJ, Zhang H, Teng Y, Li R, Bai F, Elankumaran S, Xing J (2014) TGF-beta-induced epithelial-to-mesenchymal transition proceeds through stepwise activation of multiple feedback loops. Sci Signal 7:91

    Google Scholar 

  31. Tian XJ, Zhang H, Xing J (2013) Coupled reversible and irreversible bistable switches underlying TGFbeta-induced epithelial to mesenchymal transition. Biophys J 105:1079–1089

    CAS  Google Scholar 

  32. Huang S, Guo YP, May G, Enver T (2007) Bifurcation dynamics in lineage-commitment in bipotent progenitor cells. Dev Biol 305:695–713

    CAS  Google Scholar 

  33. Chickarmane V, Troein C, Nuber UA, Sauro HM, Peterson C (2006) Transcriptional dynamics of the embryonic stem cell switch. PLoS Comput Biol 2:e123

    Google Scholar 

  34. Chickarmane V, Peterson C (2008) A computational model for understanding stem cell, trophectoderm and endoderm lineage determination. PLoS ONE 3:e3478

    Google Scholar 

  35. Ferrell JE Jr (2012) Bistability, bifurcations, and Waddington's epigenetic landscape. Curr Biol 22:R458–466

    CAS  Google Scholar 

  36. Tripathi S, Xing J, Levine H, Jolly MK (2019) Mathematical modeling of plasticity and heterogeneity in EMT

  37. Bose I, Pal M (2017) Criticality in cell differentiation. J Biosci 42:683–693

    Google Scholar 

  38. Schiesser WE (2014) Stem cell differentiation. Differential equation analysis in biomedical science and engineering. John Wiley & Sons Inc, Hoboken, pp 217–239

    Google Scholar 

  39. Bhattacharya S, Zhang Q, Andersen ME (2011) A deterministic map of Waddington's epigenetic landscape for cell fate specification. BMC Syst Biol 5:85

    Google Scholar 

  40. Dill KA, Bromberg S, Stigter D (2003) Molecular driving forces: statistical thermodynamics in chemistry and biology. Garland Sci. https://doi.org/10.1002/macp.200390113

    Article  Google Scholar 

  41. Ao P (2004) Potential in stochastic differential equations: novel construction. J Phys A Math Gen 37:L25–L30

    Google Scholar 

  42. Wang J (2015) Landscape and flux theory of non-equilibrium dynamical systems with application to biology. Adv Phys 64:1–137

    Google Scholar 

  43. Biswas K, Jolly MK, Ghosh A (2019) Stability and mean residence times for hybrid epithelial/mesenchymal phenotype. Phys Biol 16:025003

    CAS  Google Scholar 

  44. Li C, Wang J (2013) Quantifying Waddington landscapes and paths of non-adiabatic cell fate decisions for differentiation, reprogramming and transdifferentiation. J R Soc Interface 10:20130787

    Google Scholar 

  45. Li C, Hong T, Nie Q (2016) Quantifying the landscape and kinetic paths for epithelial-mesenchymal transition from a core circuit. Phys Chem Chem Phys 18:17949–17956

    CAS  Google Scholar 

  46. Qiu K, Gao KF, Yang LJ, Zhang ZK, Wang R, Ma HS, Jia Y (2017) A kinetic model of multiple phenotypic states for breast cancer cells. Sci Rep 7:9890

    Google Scholar 

  47. Zhao L, Wang J (2016) Uncovering the mechanisms of Caenorhabditis elegans ageing from global quantification of the underlying landscape. J R Soc Interface 13:20160421

    Google Scholar 

  48. Li C, Wang J (2014) Quantifying the underlying landscape and paths of cancer. J R Soc Interface 11:20140774

    Google Scholar 

  49. Guo J, Lin F, Zhang X, Tanavde V, Zheng J (2017) NetLand: quantitative modeling and visualization of Waddington's epigenetic landscape using probabilistic potential. Bioinformatics 33:1583–1585

    CAS  Google Scholar 

  50. Li C, Wang J (2013) Quantifying cell fate decisions for differentiation and reprogramming of a human stem cell network: landscape and biological paths. PLoS Comput Biol 9:e1003165

    CAS  Google Scholar 

  51. Zhou JX, Aliyu MD, Aurell E, Huang S (2012) Quasi-potential landscape in complex multi-stable systems. J R Soc Interface 9:3539–3553

    Google Scholar 

  52. Wang J, Zhang K, Xu L, Wang E (2011) Quantifying the Waddington landscape and biological paths for development and differentiation. Proc Natl Acad Sci USA 108:8257–8262

    CAS  Google Scholar 

  53. Wang J, Xu L, Wang E, Huang S (2010) The potential landscape of genetic circuits imposes the arrow of time in stem cell differentiation. Biophys J 99:29–39

    CAS  Google Scholar 

  54. Yu P, Nie Q, Tang C, Zhang L (2018) Nanog induced intermediate state in regulating stem cell differentiation and reprogramming. BMC Syst Biol 12:22

    Google Scholar 

  55. Lv C, Li X, Li F, Li T (2015) Energy landscape reveals that the budding yeast cell cycle is a robust and adaptive multi-stage process. PLoS Comput Biol 11:e1004156

    Google Scholar 

  56. Allen LJS (2010) An introduction to stochastic processes with applications to biology. Chapman and Hall/CRC, New York

    Google Scholar 

  57. Kalbfleisch JD (1984) Least-squares estimation of transition probabilities from aggregate data. Can J Stat 12:169–182

    Google Scholar 

  58. Lee TC, Judge GG, Zellner A (1970) Estimating the parameters of the Markov probability model from aggregate time series data. North-Holland Pub Co., Amsterdam

    Google Scholar 

  59. Dent W, Ballintine R (1971) A review of the estimation of transition probabilities in Markov chains. J Aust J Agric Econ 15:69–81

    Google Scholar 

  60. Kaur I, Rajarshi MJC, Computation S (2012) Ridge regression for estimation of transition probabilities from aggregate data. Commun Stat 41:524–530

    Google Scholar 

  61. Buder T, Deutsch A, Seifert M, Voss-Bohme A (2017) Cell trans: an R package to quantify stochastic cell state transitions. Bioinform Biol Insights 11:1177932217712241

    Google Scholar 

  62. Farahat WA, Asada HH (2012) Estimation of state transition probabilities in asynchronous vector markov processes. J Dyn Syst Meas Control 134:6

    Google Scholar 

  63. Aster RC, Borchers B, Thurber CH (2018) Parameter estimation and inverse problems. Elsevier, Amsterdam

    Google Scholar 

  64. Chou IC, Voit EO (2009) Recent developments in parameter estimation and structure identification of biochemical and genomic systems. Math Biosci 219:57–83

    CAS  Google Scholar 

  65. Goetz H, Melendez-Alvarez JR, Chen L, Tian XJ (2019) A plausible accelerating function of intermediate states in cancer metastasis. bioRxiv. https://doi.org/10.1101/828343

    Article  Google Scholar 

  66. Sisan DR, Halter M, Hubbard JB, Plant AL (2012) Predicting rates of cell state change caused by stochastic fluctuations using a data-driven landscape model. Proc Natl Acad Sci USA 109:19262–19267

    CAS  Google Scholar 

  67. Atkins P, de Paula J (2006) Atkin's physical chemistry, 8th edn. W. H Freeman and Company, New York

    Google Scholar 

  68. Moris N, Arias AM (2017) The hidden memory of differentiating cells. Cell Syst 5:163–164

    CAS  Google Scholar 

  69. Takahashi K, Yamanaka S (2006) Induction of pluripotent stem cells from mouse embryonic and adult fibroblast cultures by defined factors. Cell 126:663–676

    CAS  Google Scholar 

  70. Malik N, Rao MS (2013) A review of the methods for human iPSC derivation. Methods Mol Biol 997:23–33

    CAS  Google Scholar 

  71. Xie X, Fu Y, Liu J (2017) Chemical reprogramming and transdifferentiation. Curr Opin Genet Dev 46:104–113

    CAS  Google Scholar 

  72. Cieslar-Pobuda A, Knoflach V, Ringh MV, Stark J, Likus W, Siemianowicz K, Ghavami S, Hudecki A, Green JL, Los MJ (2017) Transdifferentiation and reprogramming: Overview of the processes, their similarities and differences. Biochim Biophys Acta Mol Cell Res 1864:1359–1369

    CAS  Google Scholar 

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Acknowledgements

This work is supported by the Department of Biotechnology, Government of India (Project No. BT/PR13560/COE/34/44/2015). V. D. is supported by Department of Biotechnology, Ministry of Science and Technology, Government of India (Project No. BT/PR13560/COE/34/44/2015).

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  1. Department of Biosciences and Bioengineering, Indian Institute of Technology Guwahati, Guwahati, 781039, India

    Vimalathithan Devaraj & Biplab Bose

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  1. Vimalathithan Devaraj
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Correspondence to Biplab Bose.

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Devaraj, V., Bose, B. The Mathematics of Phenotypic State Transition: Paths and Potential. J Indian Inst Sci 100, 451–464 (2020). https://doi.org/10.1007/s41745-020-00173-6

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