Acta Biotheoretica

, Volume 58, Issue 4, pp 405–413 | Cite as

On the Calibration of a Size-Structured Population Model from Experimental Data

  • Marie DoumicEmail author
  • Pedro Maia
  • Jorge P. Zubelli
Regular Article


The aim of this work is twofold. First, we survey the techniques developed in Perthame and Zubelli (Inverse Probl 23(3):1037–1052, 2007), Doumic et al. (Inverse Probl 25, 2009) to reconstruct the division (birth) rate from the cell volume distribution data in certain structured population structured population models. Secondly, we implement such techniques on experimental cell volume distributions available in the literature so as to validate the theoretical and numerical results. As a proof of concept, we use the experimental data experimental data reported in the classical work of Kubitschek (Biophys J 9(6):792–809, 1969) concerning Escherichia coli in vitro experiments measured by means of a Coulter transducer-multichannel analyzer system (Coulter Electronics, Inc., Hialeah, FL, USA). Despite the rather old measurement technology, the reconstructed division rates still display potentially useful biological features.


Structured populations Inverse problems Experimental data Biological applications 



The authors were supported by the CNPq-INRIA agreement INVEBIO. JPZ was supported by CNPq under grants 302161/2003-1 and 474085/2003-1. JPZ is thankful to the RICAM special semester and to the International Cooperation Agreement Brazil-France. A substantial part of this work was developed during a 3-month international internship of PM at INRIA Rocquencourt during the Spring of 2008 and supported by INRIA. The authors thank very much S. Boatto (UFRJ) for facilitating this visit and for helpful discussions.


  1. Bauer F, Kindermann S (2008) The quasi-optimality criterion for classical inverse problems. Inverse Probl 24Google Scholar
  2. Baumeister J, Leitão A (2005) Topics in inverse problems. Publicações Matemáticas do IMPA. [IMPA Mathematical Publications]. Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro. 25° Colóquio Brasileiro de Matemática. [25th Brazilian Mathematics Colloquium]Google Scholar
  3. Cooper S (2006) Distinguishing between linear and exponential cell growth during the division cycle: single-cell studies, cell-culture studies, and the object of cell-cycle research. Theor Biol Med Model 3:10CrossRefGoogle Scholar
  4. Doumic JM, Gabriel P (2010) Eigenelements of a general aggregation-fragmentation model. Math Model Method Appl Sci 20(5):757–783Google Scholar
  5. Doumic M, Perthame B, Zubelli JP (2009) Numerical solution of an inverse problem in size-structured population dynamics. Inverse Probl 25Google Scholar
  6. Engl HW, Hanke M, Neubauer A (1996) Regularization of inverse problems, vol 375 of Mathematics and its Applications. Kluwer, DordrechtGoogle Scholar
  7. Harvey RJ, Marr AG, Painter PR (1967) Kinetics of growth of individual cells of Escherichia coli and Azobacter agilis. J Bacteriol 93:605–617Google Scholar
  8. Hatzis C, Porro D (2006) Morphologically-strucutred models of growing budding yeast populations. J Biotechnol 124:420–438CrossRefGoogle Scholar
  9. Koch AL (1993) Biomass growth rate during the prokaryote cell cycle. Crit Rev Microbiol 19(1):17–42CrossRefGoogle Scholar
  10. Kubitschek HE (1969) Growth during the bacterial cell cycle: analysis of cell size distribution. Biophys J 9(6):792–809CrossRefGoogle Scholar
  11. Maia P (2009) Tópicos em teoria da homogeneização e equações de populações estruturadas. Master’s thesis, UFRJ, BrazilGoogle Scholar
  12. Metz JAJ, Diekmann O (1986) Formulating models for structured populations. In: The dynamics of physiologically structured populations (Amsterdam, 1983), vol 68 of Lecture Notes in Biomath. Springer, Berlin, pp 78–135Google Scholar
  13. Michel P (2006) Existence of a solution to the cell division eigenproblem. Model Math Meth Appl Sci 16(suppl. issue 1):1125–1153CrossRefGoogle Scholar
  14. Michel P, Mischler S, Perthame B (2005) General relative entropy inequality: an illustration on growth models. J Math Pures Appl 84(9):1235–1260Google Scholar
  15. Mitchison J (2005) Single cell studies of the cell cycle and some models. Theor Biol Med Model 2(1):4CrossRefGoogle Scholar
  16. Perthame B (2007) Transport equations arising in biology. In: Frontiers in Mathematics. Frontiers in Mathematics, BirkhauserGoogle Scholar
  17. Perthame B, Ryzhik L (2005) Exponential decay for the fragmentation or cell-division equation. J Differ Equ 210(1):155–177CrossRefGoogle Scholar
  18. Perthame B, Zubelli JP (2007) On the inverse problem for a size-structured population model. Inverse Probl 23(3):1037–1052CrossRefGoogle Scholar
  19. Prescott LM, Klein DA, Harley JP (2002) Microbiology. McGraw-Hill, New YorkGoogle Scholar
  20. Trueba F (1981) A morphometric analysis of Escherichia coli and other rod-shaped bacteria. PhD thesis, University of AmsterdamGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.BANG Project-TeamINRIA RocquencourtRocquencourtFrance
  2. 2.UFRJ, Cidade UniversitariaRio de JaneiroBrazil
  3. 3.IMPARio de JaneiroBrazil

Personalised recommendations