Model example: the cell cycle

The eukaryotic cell cycle is very complex and is one of the most studied topics, since its misregulation leads to cancers. It is possibly a good example of a mathematical model as it deals with simple calculus but gives valid results. Two research groups [40] [41] have produced several models of the cell cycle simulating several organisms. They have recently produced a generic eukaryotic cell cycle model which can represent a particular eukaryote depending on the values of the parameters, demonstrating that the idiosyncrasies of the individual cell cycles are due to different protein concentrations and affinities, while the underlying mechanisms are conserved (Csikasz-Nagy et al., 2006).

By means of a system of ordinary differential equations these models show the change in time (dynamical system) of the protein inside a single typical cell; this type of model is called a deterministic process (whereas a model describing a statistical distribution of protein concentrations in a population of cells is called a stochastic process).

To obtain these equations an iterative series of steps must be done: first the several models and observations are combined to form a consensus diagram and the appropriate kinetic laws are chosen to write the differential equations, such as rate kinetics for stoichiometric

reactions, Michaelis-Menten kinetics for enzyme substrate reactions and

Goldbeter–Koshland kinetics for ultrasensitive transcription factors, afterwards the parameters of the equations (rate constants, enzyme efficiency coefficients and Michealis

Mathematical biology 103 constants) must be fitted to match observations; when they cannot be fitted the kinetic equation is revised and when that is not possible the wiring diagram is modified. The parameters are fitted and validated using observations of both wild type and mutants, such as protein half-life and cell size.

In order to fit the parameters the differential equations need to be studied. This can be done either by simulation or by analysis.

In a simulation, given a starting vector (list of the values of the variables), the progression of the system is calculated by solving the equations at each time-frame in small increments. In analysis, the proprieties of

the equations are used to investigate the behavior of the system depending of the values of the parameters and variables. A system of differential equations can be represented as a vector field, where each vector described the change (in concentration of two or more protein) determining where and how

fast the trajectory (simulation) is heading. Vector fields can have several special points: a stable point, called a sink, that attracts in all directions (forcing the concentrations to be at a certain value), an unstable point, either a source or a saddle point which repels (forcing the concentrations to change away from a certain value), and a limit cycle, a closed trajectory towards which several trajectories spiral towards (making the concentrations oscillate).

A better representation which can handle the large number of variables and parameters is called a bifurcation diagram(Bifurcation theory): the presence of these special steady-state points at certain values of a parameter (e.g. mass) is represented by a point and once the parameter passes a certain value, a qualitative change occurs, called a bifurcation, in which the nature of the space changes, with profound consequences for the protein concentrations: the cell cycle has phases (partially corresponding to G1 and G2) in which mass, via a stable point, controls cyclin levels, and phases (S and M phases) in which the concentrations change independently, but once the phase has changed at a bifurcation event (Cell cycle checkpoint), the system cannot go back to the previous levels since at the current mass the vector field is profoundly different and the mass cannot be reversed back through the bifurcation event, making a checkpoint irreversible. In particular the S and M checkpoints are regulated by means of special bifurcations called a Hopf bifurcationand an infinite period bifurcation.

Mathematical biology 104

Mathematical/theoretical biologists

• Pere Alberch • Anthony F. Bartholomay • J. T. Bonner • Jack Cowan • Gerd B. Müller • Walter M. Elsasser • Claus Emmeche • Andree Ehresmann • Marc Feldman • Ronald A. Fisher • Brian Goodwin • Bryan Grenfell • J. B. S. Haldane • William D. Hamilton • Lionel G. Harrison • Michael Hassell

• Sven Erik Jørgensen

• George Karreman • Stuart Kauffman • Kalevi Kull • Herbert D. Landahl • Richard Lewontin • Humberto Maturana • Robert May

• John Maynard Smith

• Howard Pattee

• George R. Price

• Erik Rauch

• Nicolas Rashevsky

• Ronald Brown (mathematician)

• Johannes Reinke

• Robert Rosen

• Rene Thom

• Jakob von Uexküll

• Robert Ulanowicz • Francisco Varela • C. H. Waddington • Arthur Winfree • Lewis Wolpert • Sewall Wright • Christopher Zeeman

Mathematical biology 105

Mathematical, theoretical and computational biophysicists

• Nicolas Rashevsky

• Ludwig von Bertalanffy

• Francis Crick • Manfred Eigen • Walter Elsasser • Herbert Frohlich, FRS • Francois Jacob • Martin Karplus • George Karreman • Herbert D. Landahl

• Ilya, Viscount Prigogine

• SirJohn Randall • James D. Murray • Bernard Pullman • Alberte Pullman • Erwin Schrodinger • Klaus Schulten • Peter Schuster • Zeno Simon • D'Arcy Thompson • Murray Gell-Mann

References

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• Robert Rosen, Dynamical system theory in biology. New York, Wiley-Interscience (1970) ISBN 0471735507 [67]

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Writings in Western Mathematics. Elsevier: 936-44.

• Israel, G (1988), "On the contribution of Volterra and Lotka to the development of modern biomathematics.[68]", History and philosophy of the life sciences10(1): 37-49, PMID:3045853, http://www.ncbi.nlm.nih.gov/pubmed/3045853

• Scudo, F M (1971), "Vito Volterra and theoretical ecology.[69]", Theoretical population

biology2(1): 1-23, 1971 Mar, PMID:4950157, http://www.ncbi.nlm.nih.gov/pubmed/ 4950157

• S.H. Strogatz, Nonlinear dynamics and Chaos: Applications to Physics, Biology,

Chemistry, and Engineering.Perseus, 2001, ISBN 0-7382-0453-6

• N.G. van Kampen, Stochastic Processes in Physics and Chemistry, North Holland., 3rd ed. 2001, ISBN 0-444-89349-0

Mathematical biology 107

• I. C. Baianu., Computer Models and Automata Theory in Biology and Medicine.,

Monograph, Ch.11 in M. Witten (Editor), Mathematical Models in Medicine, vol. 7., Vol.

7: 1513-1577 (1987),Pergamon Press:New York, (updated by Hsiao Chen Lin in 2004[70]

,[71],[72]ISBN 0080363776 [73].

• P.G. Drazin, Nonlinear systems. C.U.P., 1992. ISBN 0-521-40668-4

• L. Edelstein-Keshet, Mathematical Models in Biology. SIAM, 2004. ISBN 0-07-554950-6

• G. Forgacs and S. A. Newman, Biological Physics of the Developing Embryo. C.U.P., 2005. ISBN 0-521-78337-2

• A. Goldbeter, Biochemical oscillations and cellular rhythms. C.U.P., 1996. ISBN 0-521-59946-6

• L.G. Harrison, Kinetic theory of living pattern. C.U.P., 1993. ISBN 0-521-30691-4

• F. Hoppensteadt, Mathematical theories of populations: demographics, genetics and

epidemics. SIAM, Philadelphia, 1975 (reprinted 1993). ISBN 0-89871-017-0

• D.W. Jordan and P. Smith, Nonlinear ordinary differential equations, 2nd ed. O.U.P., 1987. ISBN 0-19-856562-3

• J.D. Murray, Mathematical Biology. Springer-Verlag, 3rd ed. in 2 vols.: Mathematical

Biology: I. An Introduction, 2002 ISBN 0-387-95223-3; Mathematical Biology: II. Spatial Models and Biomedical Applications, 2003 ISBN 0-387-95228-4.

• E. Renshaw, Modelling biological populations in space and time. C.U.P., 1991. ISBN 0-521-44855-7

• S.I. Rubinow, Introduction to mathematical biology. John Wiley, 1975. ISBN 0-471-74446-8

• L.A. Segel, Modeling dynamic phenomena in molecular and cellular biology. C.U.P., 1984. ISBN 0-521-27477-X

• L. Preziosi, Cancer Modelling and Simulation. Chapman Hall/CRC Press, 2003. ISBN 1-58488-361-8.

In document Complexity, Emergent Systems and Complex Biological Systems: Complex Systems Theory and Biodynamics. [Edited book by I.C. Baianu, with listed contributors (2011)] (Page 102-107)

Mathematical/theoretical biologists

Mathematical, theoretical and computational biophysicists

See also

References