Definitions

# Mathematical biology

For use of basic artimethics in Biology, see relevant topic, such as Serial dilution.

Mathematical biology, biological mathematical modeling, biomathematics or computational biomodeling is an interdisciplinary field of academic study which aims at modeling natural, biological processes using applied mathematical techniques and tools. It has both practical and theoretical applications in biological research: In cell biology, protein interactions are typically expressed as “cartoon” models, which, although easy to visualize, do not fully describe the systems studied: to do this, mathematical models are required, which, by describing the systems in a quantitative manner, can better simulate their behavior and hence predict unseen properties.

## Importance

Applying mathematics to biology has a long history, but only recently has there been an explosion of interest in the field. Some reasons for this include:

• the explosion of data-rich information sets, due to the genomics revolution, which are difficult to understand without the use of analytical tools,
• recent development of mathematical tools such as chaos theory to help understand complex, nonlinear mechanisms in biology,
• an increase in computing power which enables calculations and simulations to be performed that were not previously possible, and
• an increasing interest in in silico experimentation due to ethical considerations, risk, unreliability and other complications involved in human and animal research.

## Areas of research

Below is a list of some areas of research in mathematical biology and links to related projects in various universities. These examples are characterised by complex, nonlinear mechanisms and it is being increasingly recognised that the result of such interactions may only be understood through mathematical and computational models. Due to the wide diversity of specific knowledge involved, biomathematical research is often done in collaboration between mathematicians, physicists, engineers, biologists, physicians, zoologists, chemists etc.

### Population dynamics

Population dynamics has traditionally been the dominant field of mathematical biology. Work in this area dates back to the 19th century. The Lotka–Volterra predator-prey equations are a famous example. In the past 30 years, population dynamics has been complemented by evolutionary game theory, developed first by John Maynard Smith. Under these dynamics, evolutionary biology concepts may take a deterministic mathematical form. Population dynamics overlap with another active area of research in mathematical biology: mathematical epidemiology, the study of infectious disease affecting populations. Various models of viral spread have been proposed and analysed, and provide important results that may be applied to health policy decisions.

### Modeling cell and molecular biology

This area has received a boost due to the growing importance of molecular biology.

• Mechanics of biological tissues
• Theoretical enzymology and enzyme kinetics
• Cancer modelling and simulation
• Modelling the movement of interacting cell populations
• Mathematical modelling of scar tissue formation
• Mathematical modelling of intracellular dynamics
• Mathematical modelling of the cell cycle

### Modelling physiological systems

• Modelling of arterial disease
• Multi-scale modelling of the heart

## Mathematical methods

A model of a biological system is converted into a system of equations, although the word 'model' is often used synonymously with the system of corresponding equations. The solution of the equations, by either analytical or numerical means, describes how the biological system behaves either over time or at equilibrium. There are many different types of equations and the type of behavior that can occur is dependent on both the model and the equations used. The model often makes assumptions about the system. The equations may also make assumptions about the nature of what may occur.

The following is a list of mathematical descriptions and their assumptions.

### Deterministic processes (dynamical systems)

A fixed mapping between an initial state and a final state. Starting from an initial condition and moving forward in time, a deterministic process will always generate the same trajectory and no two trajectories cross in state space.

### Stochastic processes (random dynamical systems)

A random mapping between an initial state and a final state, making the state of the system a random variable with a corresponding probability distribution.

### Spatial modelling

One classic work in this area is Alan Turing's paper on morphogenesis entitled The Chemical Basis of Morphogenesis, published in 1952 in the Philosophical Transactions of the Royal Society.

• Travelling waves in a wound-healing assay
• Swarming behaviour
• The mechanochemical theory of morphogenesis
• Biological pattern formation
• Spatial distribution modeling using plot samples

## Example of a model: The Cell Cycle

The eukaryotic cell cycle is very complex and is one of the most studied topics, since its misregulation results in cancer. It is possibly a good example of a mathematical model as it deals with simple calculus but gives valid results. and the Bela Novak lab (Oxford University) 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 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 bifurcation and an infinite period bifurcation.

## References

### Bibliographical

• 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
• 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

### External

• F. Hoppensteadt, . Notices of American Mathematical Society, Sept. 1995.
• M. C. Reed, Notices of American Mathematical Society, March, 2004.
• R. M. May, . Science, February 6, 2004.
• J. D. Murray, Scientific American, 258(3): 80-87, 1988.
• S. Schnell, R. Grima, P. K. Maini, , American Scientist, Vol 95, pages 134-142, March-April 2007.
• Chen KC et al. Integrative analysis of cell cycle control in budding yeast. Mol Biol Cell. 2004 Aug;15(8):3841-62.
• Csikász-Nagy A et al. Analysis of a generic model of eukaryotic cell-cycle regulation. Biophys J. 2006 Jun 15;90(12):4361-79.
• Fuss H, et al. Mathematical models of cell cycle regulation. Brief Bioinform. 2005 Jun;6(2):163-77.
• Lovrics A et al. BMC Bioinform. 2006 Nov 9;7:494.