Abstract
A computational cell cycle model that can describe three state properties, cell maturation age, specific mRNA and protein content, has been developed. Cell cycle propression is monitored by maturation age, and population heterogeneity is generated by the introduction of a probable random event embedded in the G1 phase. Specific mRNA is generated with a constant transcription rate at the single-cell level, and its turnover is governed by a first-order decay. Translation is modelled as a first-order dependence on the transcripts, and the protein product is subsequently exported. Dynamic chemostat simulations are used to demonstrate the ability of the model to track evolving parent and daughter subpopulations in maturation and cellular contents. The cell subpopulations eventually converge to an equilibrium distribution corresponding to the steady state of a chemostat, and halving of cellular content at cell division is the dominant driving force leading towards the population equilibrium state.
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Abbreviations
- a :
-
maturation age, age
- a B :
-
maturation age in the B-phase, age
- D :
-
dilution rate, (volumetric flow rate)/(culture volume) time−1
- k d :
-
first-order cell death rate, time−1
- K m :
-
Monod constant, mole·volume−1
- k T :
-
transition rate, time−1
- K T :
-
transition saturation constant, mole·volume−1
- k x :
-
first-order export rate, time−1
- k y :
-
first-order turnover rate, time−1
- m :
-
normalised messenger transcripts
- m c :
-
exogenous maintenance coefficient, mole·cell−1 time−1
- n A :
-
cell density in the A-state, cell·(transcripts/cell)−1 (chains/cell)−1 volume−1
- n B :
-
cell density in the B-phase, cell·age−1 (transcripts/cell)−1 (chains/cell)−1 volume−1
- N V :
-
total cell density, cell·volume−1
- p :
-
normalised protein chains
- S :
-
substrate concentration, mole·volume−1
- S o :
-
inlet substrate concentration, mole·volume−1
- t :
-
time
- T B :
-
B-phase duration, age
- v B :
-
maturation velocity in the B-phase, age·time−1
- v x :
-
rate of protein chain accumulation, (chains/cell)·time−1
- v 0 x :
-
translation rate, (chains/cell)·(transcripts/cell)−1 time−1
- v y :
-
rate of mRNA accumulation, (transcripts/cell)·time−1
- v 0 y :
-
zero-order transcription rate, (transcripts/cell)·time−1
- x :
-
protein chains in a cell, (chains/cell)
- y :
-
specific mRNA in a cell, (transcripts/cell)
- Y N/S :
-
cell number yield coefficient, cell·mole−1
- μ:
-
specific growth rate, time−1
References
Aarnæs, E., Clausen, O. P., Kirkhus, B., andde Angelis, P. (1993): ‘Heterogeneity in the mouse epidermal cell cycle analysed by computer simulations’,Cell Prolif.,26, pp. 205–219
Adams, J., Rothman, E. D., andBeran, K. (1981): ‘The age structure of populations of Saccharomyces cerevisiae’,Math. Biosci.,53, pp. 249–263
Alberghina, L., Mariani, L., andMartegani, E. (1981): ‘Cell cycle variability: Modeling and simulation’, inRotenberg, M. (Ed.), ‘Biomathematics and cell kinetics’ (Elsevier/North-Holland, Amsterdam, 1981), pp. 295–309
Bélair, J., Mackey, M. C., andMahaffy, J. M. (1995): ‘Age-structured and two-delay models for erythropoiesis’,Math. Biosci.,128, pp. 317–346
Bélair, J., andMahaffy, J. M. (2001): ‘Variable maturation velocity and parameter sensitivity in a model of haematopoiesis’,IMA J. Math. Appl. Med. Biol.,18, pp. 193–211
Bertuzzi, A., Gandolfi, A., andVitelli, R. (1986): ‘A regularization procedure for estimating cell kinetic parameters from flow cytometry data’,Math. Biosci.,82, pp. 63–85
Bertuzzi, A., Faretta, M., Gandolfi, A., Sinisgalli, C., Starace, G., Valoti, G., andUbezio, P. (2002): ‘Kinetic heterogeneity of an experimental tumour revealed by BrdUrd incorporation and mathematical modelling’,Bull. Math. Biol.,64, pp. 355–384
Bibila, T. A., andFlickinger, M. C. (1991): ‘A structured model for monoclonal antibody synthesis in exponentially growing and stationary phase hybridoma cells’,Biotechnol. Bioeng.,37, pp. 210–226
Cain, S. J., andChau, P. C. (1997a): ‘A transition probability cell cycle model simulation of bivariate DNA/bromodeoxyuridine distributions’,Cytometry,27, pp. 239–249
Cain, S. J., andChau, P. C. (1997b): ‘Transition probability cell cycle model. Part I. Balanced growth’,J. Theor. Biol.,185, pp. 55–67
Cain, S. J., andChau, P. C. (1997c): ‘Transition probability cell cycle model. Part II. Non-balanced growth’,J. Theor. Biol.,185, pp. 69–79
Cain, S. J., andChau, P. C. (1998): ‘Transition probability cell cycle model with product formation’,Biotechnol. Bioeng.,58, pp. 387–399
Cazzador, L., andMariani, L. (1993): ‘Growth and production modeling in hybridoma continuous cultures’,Biotechnol. Bioeng.,42, pp. 1322–1330
Fredrickson, A. G., Ramkrishna, D., andTsuchiya, H. M. (1967): ‘Statistics and dynamics of procaryotic cell populations’,Math. Biosci. 1, pp. 327–374
Fredrickson, A. G., andTsuchiya, H. M. (1977): ‘Microbial kinetics and dynamics’, inLapidus, L. andAmundson, N. R. (Eds): ‘Chemical reactor theory: a review’ (Prentice-Hall, Englewood Cliffs, NJ, 1977)
Grasman, J. (1990): ‘A deterministic model of the cell cycle’,Bull. Math. Biol.,52, pp. 535–547
Gyllenberg, M., andWebb, G. F. (1987): ‘Age-size structure in populations with quiescence’,Math. Biosci.,86, pp. 67–95
Hannsgen, K. B., Tyson, J. J., andWatson, L. T. (1985): ‘Steady-state size distributions in probabilistic models of the cell division cycle’,SIAM J. Appl. Math.,45, pp. 523–540
Hirsch, H. R. (1983): ‘Influence of the existence of a resting state on the probability of cell division in culture’,J. Theor. Biol.,100, pp. 399–410
Hirsch, H. R. (1984): ‘Influence of the existence of a resting state on the decay of synchronization in cell culture’,J. Theor. Biol.,111, pp. 61–79
Keasling, J. D., Kuo, H., andVahanian, G. (1995): ‘A Monte Carlo simulation of the Escherichia coli cell cycle’,J. Theor. Biol.,176, pp. 411–430
Liou, J.-J., Srienc, F., andFredrickson, A. G. (1997): ‘Solutions of population balance models based on a successive generations approach’,Chem. Eng. Sci.,52, pp. 1529–1540
Mahaffy, J. M., Belair, J., andMackey, M. C. (1998): ‘Hematopoietic model with moving boundary condition and state dependent delay: applications in erythropoiesis’,J. Theor. Biol.,190, pp. 135–146
Mc Adams, H. H., andArkin, A. (1999): ‘It's a noisy business! Genetic regulation at the nanomolar scale’,Trends Genet.,15, pp. 65–69
Nielsen, L. K., Reid, S., andGreenfield, P. F. (1997): ‘Cell cycle model to describe animal cell size variation and lag between cell number and biomass dynamics’,Biotech. Bioeng.,56, pp. 372–379
Rotenberg, M. (1982): ‘Theory of distributed quiescent state in the cell cycle’,J. Theor. Biol.,96, pp. 495–509
Rotenberg, M. (1983): ‘Transport theory for growing cell populations’,J. Theor. Biol.,103, pp. 181–199
Rounseville, K. J., andChau, P. C. (2004): ‘Three-dimensional cell cycle model supplementary material’, available at http://pcclab.ucsd.edu/ccm/
Schiesser, W. E. (1991): ‘The numerical method of lines: integration of partial differential equations’ (Academic Press, San Diego, 1991)
Smith, J. A., andMartin, L. (1973): ‘Do cells cycle?’,Proc. Natl. Acad. Sci. USA,70, pp. 1263–1267
Smith, J. A., andMartin, L. (1974): ‘Regulation of cell proliferation’, inPadilla, G. M., Cameron, I. L., andZimmerman, A. (Eds): ‘Cell cycle controls’, (Academic Press, New York, 1974), pp. 43–60
Sulsky, D. (1993): ‘Numerical solution of structured population models. I. Age structure’,J. Math. Biol.,31, pp. 817–839
Watts, D. J., andStrogatz, S. H. (1998): ‘Collective dynamics of ‘small-world’ networks’,Nature,393, pp. 440–442
Webb, G. F., andGrabosch, A. (1987): ‘Asynchronous exponential growth in transition probability models of the cell cycle’,SIAM J. Math. Anal.,18, pp. 897–908
Yanagisawa, M., Dolbeare, F., Todoroki, T., andGray, J. W. (1985): ‘Cell cycle analysis using numerical simulation of bivariate DNA/bromodeoxyuridine distributions’,Cytometry,6, pp. 550–562
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Rounseville, K.J., Chau, P.C. Three-dimensional cell cycle model with distributed transcription and translation. Med. Biol. Eng. Comput. 43, 155–161 (2005). https://doi.org/10.1007/BF02345138
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DOI: https://doi.org/10.1007/BF02345138