Skip to main content
SpringerLink
Log in
Book cover

Mammary Stem Cells pp 247–266Cite as

Mathematical Modelling as a Tool to Understand Cell Self-renewal and Differentiation

  • Protocol

Part of the book series: Methods in Molecular Biology ((MIMB,volume 1293))

Abstract

Mathematical modeling is a powerful technique to address key questions and paradigms in a variety of complex biological systems and can provide quantitative insights into cell kinetics, fate determination and development of cell populations. The chapter is devoted to a review of modeling of the dynamics of stem cell-initiated systems using mathematical methods of ordinary differential equations. Some basic concepts and tools for cell population dynamics are summarized and presented as a gentle introduction to non-mathematicians. The models take into account different plausible mechanisms regulating homeostasis. Two mathematical frameworks are proposed reflecting, respectively, a discrete (punctuated by division events) and a continuous character of transitions between differentiation stages. Advantages and constraints of the mathematical approaches are presented on examples of models of blood systems and compared to patients data on healthy hematopoiesis.

This is a preview of subscription content, log in via an institution.

Protocol
USD   49.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   139.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Springer Nature is developing a new tool to find and evaluate Protocols. Learn more

References

  1. Adimy M, Crauste F (2012) Delay differential equations and autonomous oscillations in hematopoietic stem cell dynamics modeling. Math Model Nat Phenom 7(06):1–22

    Article  Google Scholar 

  2. Alarcón T, Getto Ph, Marciniak-Czochra A, Vivanco MdM (2011) A model for stem cell population dynamics with regulated maturation delay. Discrete Continuous Dyn Syst B (Suppl):32–43

    Google Scholar 

  3. Diekmann O, Getto Ph (2005) Boundedness, global existence and continuous dependence for nonlinear dynamical systems describing physiologically structured populations. J Differ Equ 215:268–319

    Article  Google Scholar 

  4. Diekmann O, van Gils S, Verduyn Lunel SM, Walther H-O (1995) Delay equations, functional-, complex-, and nonlinear analysis. Springer, New York

    Google Scholar 

  5. Diekmann O, Gyllenberg M, Huang H, Kirkilionis M, Metz JAJ, Thieme HR (2001) On the formulation and analysis of general deterministic structured population models II. Nonlinear theory J Math Biol 43:157–189

    Article  CAS  Google Scholar 

  6. Diekmann O, Getto Ph, Gyllenberg M (2007) Stability and bifurcation analysis of Volterra functional equations in the light of suns and stars. SIAM J Math Anal 39(4):1023–1069

    Article  Google Scholar 

  7. Doumic M, Marciniak-Czochra A, Perthame B, Zubelli J (2011) Structured population model of stem cell differentiation. SIAM J Appl Math 71:1918–1940

    Article  Google Scholar 

  8. Getto Ph, Marciniak-Czochra A, Nakata Y, Vivanco MdM (2013) Global dynamics of two-compartment models for cell production systems with regulatory mechanisms. Math Biosci 245:258–268

    Article  PubMed  Google Scholar 

  9. Hale JK, Verduyn Lunel SM (1991) Introduction to functional differential equations. Springer, New York

    Google Scholar 

  10. Handin RI, Lux SE, Stossel TP (2003) Blood: principles and practice of hematology, 2nd edn. Lippincott, Phiadelphia

    Google Scholar 

  11. Hartung F, Krisztin T, Walther H-O, Wu J, Functional differential equations with state dependent delays: theory and applications. In: Handbook of differential equations: ordinary differential equations, vol 4. Elsevier, Amsterdam

    Google Scholar 

  12. Jandl JH (1996) Blood cell formation. Little, Brown and Company, Boston

    Google Scholar 

  13. Kaushansky K, Lichtman AM, Beutler LE, Kipps TJ, Prchal J, Seligsohn U (2010) Williams hematology, 8th edn. Mcgraw-Hill Professional, New York

    Google Scholar 

  14. Kimmel M, Axelrod DE (2002) Branching processes in biology. Springer, New York

    Book  Google Scholar 

  15. Lajtha LG et al (1969) Kinetic properties of haemopoietic stem cells. Cell Prolif 2:39–49

    Article  Google Scholar 

  16. Lander A, Gokoffski K, Wan F, Nie Q, Calof A (2009) Cell lineages and the logic of proliferative control. PLoS Biol 7:e1000015

    Article  PubMed Central  Google Scholar 

  17. Lansdorp PM (1998) Stem cell biology for the transfusionist. Vox Sang 74(Suppl. 2):91–94

    Article  CAS  PubMed  Google Scholar 

  18. Lo W, Chou C, Gokoffski K, Wan F, Lander A, Calof A, Nie Q (2009) Feedback regulation in multistage cell lineages. Math Biosci Eng 6:59–82

    Article  PubMed Central  PubMed  Google Scholar 

  19. Loeffler M, Roeder I (2002) Tissue stem cells: definition, plasticity, heterogeneity, self-organization and models - a conceptual approach. Cells Tissues Organs 171:8–26

    Article  PubMed  Google Scholar 

  20. Marciniak-Czochra A, Stiehl T (2011) Mathematical models of hematopoietic reconstitution after stem cell transplantation. In: Bock HG, Carraro T, Jaeger W, Koerkel S, Rannacher R, Schloeder JP (eds) Model based parameter estimation: theory and applications. Springer, Heidelberg

    Google Scholar 

  21. Marciniak-Czochra A, Stiehl T, Ho AD, Jäger W, Wagner W (2009) Modeling asymmetric cell division in hematopoietic stem cells—regulation of self-renewal is essential for efficient repopulation. Stem Cells Dev 18:377–386

    Article  CAS  PubMed  Google Scholar 

  22. Marciniak-Czochra A, Stiehl T, Wagner W (2009) Modeling of replicative senescence in hematopoietic development. Aging (Albany NY) 1:723–32

    CAS  Google Scholar 

  23. Metcalf D (2008) Hematopoietic cytokines. Blood 111:485–491

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  24. Metz JAJ, Diekmann O (1986) The dynamics of physiologically structured populations. In: LNB, vol 68. Springer, New York

    Google Scholar 

  25. Michor F, Hughes TP, Iwasa Y, Branford S, Shah NP, Sawyers CL, Nowak MA (2005) Dynamics of chronic myeloid leukaemia. Nature 435:1267–1270

    Article  CAS  PubMed  Google Scholar 

  26. Nakata Y, Getto P, Marciniak-Czochra A, Alarcon T (2011) Stability analysis of multi-compartment models for cell production systems. J Biol Dyn. http://dx.doi.org/10.1080/17513758.2011.558214 [published online]

  27. Rodriguez-Brenes IA, Wodarz D, Komarova NL (2013) Stem cell control, oscillations, and tissue regeneration in spatial and non-spatial models. Front Oncol 3:82

    Article  PubMed Central  PubMed  Google Scholar 

  28. Roeder I, Kamminga LM, Braesel K, Dontje B, de Haan G, Loeffler M (2005) Competitive clonal hematopoiesis in mouse chimeras explained by a stochastic model of stem cell organization. Blood 105:609–616

    Article  CAS  PubMed  Google Scholar 

  29. Roeder I, Horn K, Sieburg HB, Cho R, Muller-Sieburg C, Loeffler M (2008) Characterization and quantification of clonal heterogeneity among hematopoietic stem cells: a model-based approach. Blood 112:4874–4883

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  30. Roeder I, Herberg M, Horn M (2009) An age structured model of hematopoietic stem cell organization with application to chronic myeloid leukemia. Bull Math Biol 71:602

    Article  PubMed  Google Scholar 

  31. Shinjo K, Takeshita A, Ohnishi K, Ohno R (1997) Granulocyte colony-stimulating factor receptor at various differentiation stages of normal and leukemic hematopoietic cells. Leuk Lymphoma 25:37–46

    CAS  PubMed  Google Scholar 

  32. Stiehl T, Marciniak-Czochra A (2011) Characterization of stem cells using mathematical models of multistage cell lineages. Math Comput Model 53:1505–1517

    Article  Google Scholar 

  33. Stiehl T, Marciniak-Czochra A (2012) Mathematical modelling of leukemogenesis and cancer stem cell dynamics. Math Model Nat Phenom 7:166–202

    Article  Google Scholar 

  34. Stiehl T, Ho AD, Marciniak-Czochra A (2013) The impact of CD34+ cell dose on engraftment after stem cell transplantations: personalized estimates based on mathematical modeling. Bone Marrow Transplant. doi:10.1038/bmt.2013.138 [published online]

  35. Stiehl T, Baran N, Ho AD, Marciniak-Czochra A (2014) Clonal selection and therapy resistance in acute leukemias: mathematical modelling explains different proliferation patterns at diagnosis and relapse. J R Soc Interface 11:20140079. http://dx.doi.org/10.1098/rsif.2014.0079

  36. Walenda T, Stiehl T, Braun H, Fröbel J, Ho AD, Schroeder T, Goecke T, Germing U, Marciniak-Czochra A, Wagner W (2014) Feedback signals in myelodysplastic syndromes: increased self-renewal of the malignant clone suppresses normal hematopoiesis. PLOS Comput Biol doi:10.1371/journal.pcbi.1003599

    PubMed Central  PubMed  Google Scholar 

  37. Wazewska-Czyzewska M (1984) Erythrokinetics radioisotopic methods of investigation and mathematical approach. Foreign Scientific Publications Dept. of the National Center for Scientific, Technical, and Economic Information, Springfield

    Google Scholar 

  38. Whichard ZL et al (2010) Hematopoiesis and its disorders: a systems biology approach. Blood 115:2339–2347

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  39. Youssefpour H, Li X, Lander AD, Lowengrub JS (2012) Multispecies model of cell lineages and feedback control in solid tumors. J Theor Biol 204:39–59

    Article  Google Scholar 

  40. Ziebell F, Martin-Villalba A, Marciniak-Czochra A (2014) Mathematical modelling of adult hippocampal neurogenesis: effects of altered stem cell dynamics on cell counts and BrdU-labelled cells. J R Soc Interface 11:20140144

    Article  PubMed Central  PubMed  Google Scholar 

Download references

Acknowledgements

PG was supported by the German Research Council (DFG) and the Spanish Ministry of Economy and Competitiveness under project MTM 2010-18318. AM-C was supported by the Collaborative Research Center, SFB 873 “Maintenance and Differentiation of Stem Cells in Development and Disease.” The authors would like to thank Thomas Stiehl and Marcel Mohr for help in preparation of figures.

Author information

Authors and Affiliations

  1. TU Dresden, Fachrichtung Mathematik, Institut für Analysis, 01062, Dresden, Germany

    Philipp Getto

  2. Basque Center for Applied Mathematics (BCAM), 48009, Bilbao, Spain

    Philipp Getto

  3. Heidelberg University, Institute of Applied Mathematics, Interdisciplinary Center for Scientific Computing and BIOQUANT, Heidelberg, Germany

    Anna Marciniak-Czochra

Authors
  1. Philipp Getto
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Anna Marciniak-Czochra
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Philipp Getto .

Editor information

Editors and Affiliations

  1. CIC bioGUNE, Technological Park of Bizkaia, Derio, Spain

    Maria del Mar Vivanco

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer Science+Business Media LLC New York

About this protocol

Cite this protocol

Getto, P., Marciniak-Czochra, A. (2015). Mathematical Modelling as a Tool to Understand Cell Self-renewal and Differentiation. In: Vivanco, M. (eds) Mammary Stem Cells. Methods in Molecular Biology, vol 1293. Humana Press, New York, NY. https://doi.org/10.1007/978-1-4939-2519-3_15

Download citation

  • DOI: https://doi.org/10.1007/978-1-4939-2519-3_15

  • Publisher Name: Humana Press, New York, NY

  • Print ISBN: 978-1-4939-2518-6

  • Online ISBN: 978-1-4939-2519-3

  • eBook Packages: Springer Protocols

Publish with us

Policies and ethics

Search

Navigation