Mathematical biology or biomathematics is an interdisciplinary field of academic study which aims at modeling natural, biological processes using mathematical techniques and tools. It has both practical and theoretical applications in biological research. Interdisciplinary work is that which integrates concepts across different disciplines. ... Biology studies the variety of life (clockwise from top-left) E. coli, tree fern, gazelle, Goliath beetle Biology (from Greek: βίος, bio, life; and λόγος, logos, knowledge), also referred to as the biological sciences, is the study of living organisms utilizing the scientific method. ... For other meanings of mathematics or math, see Mathematics (disambiguation) and Math (disambiguation). ...

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 the complications involved in human and animal research.

Genomics is the study of an organisms entire genome; Rathore et al, . Investigation of single genes, their functions and roles is something very common in todays medical and biological research, and cannot be said to be genomics but rather the most typical feature of molecular biology. ... For other uses, see Chaos Theory (disambiguation). ... This article is about the machine. ... This article is about the general term. ... In Silico is a full length artist album by Deepsky View From a Stairway Jareths Church The Mansion World (Deepskys Trippin In Unknown Territory Mix) Ride Three Sheets to the Wind Atia Metro Smile Cosmic Dancer (2002 remix) Until the End of the World Let Me Live Categories...

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, biologists, physicians, zoologists, chemists etc. A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... This article or section does not cite any references or sources. ... Zoology (from Greek: ζῴον, zoion, animal; and λόγος, logos, knowledge) is the biological discipline which involves the study of animals. ... For other uses, see Chemistry (disambiguation). ...

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. Population dynamics is the study of marginal and long-term changes in the numbers, individual weights and age composition of individuals in one or several populations, and biological and environmental processes influencing those changes. ... Alternative meaning: Nineteenth Century (periodical) (18th century — 19th century — 20th century — more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801-1900 in the sense of the Gregorian calendar. ... The Lotka-Volterra equations, also known as the predator-prey equations, are a pair of first order, non-linear, differential equations frequently used to describe the dynamics of biological systems in which two species interact, one a predator and one its prey. ... Evolutionary game theory (EGT) is the application of population genetics-inspired models of change in gene frequency in populations to game theory. ... Professor John Maynard Smith[1], F.R.S. (6 January 1920 – 19 April 2004) was a British evolutionary biologist and geneticist. ... It is possible to model mathematically the progress of most infectious diseases to discover the likely outcome of an epidemic or to help manage them by vaccination. ...

Modelling cell and molecular biology

This area has received a boost due to the growing importance of molecular biology. Molecular biology is the study of biology at a molecular level. ...

• Modelling of neurons and carcinogenesis [1]
• Mechanics of biological tissues [2]
• Theoretical enzymology and enzyme kinetics [3]
• Cancer modelling and simulation [4]
• Modelling the movement of interacting cell populations [5]
• Mathematical modelling of scar tissue formation [6]
• Mathematical modelling of intracellular dynamics [7]

Drawing by Santiago Ramón y Cajal of neurons in the pigeon cerebellum. ... The hazard symbol for carcinogenic chemicals in the Globally Harmonized System. ... Dihydrofolate reductase from with its two substrates, dihydrofolate (right) and NADPH (left), bound in the active site. ... Cancer is a class of diseases or disorders characterized by uncontrolled division of cells and the ability of these to spread, either by direct growth into adjacent tissue through invasion, or by implantation into distant sites by metastasis (where cancer cells are transported through the bloodstream or lymphatic system). ...

Modelling physiological systems

• Modelling of arterial disease [8]
• Multi-scale modelling of the heart [9]

Section of an artery For other uses, see Artery (disambiguation). ... The heart and lungs, from an older edition of Grays Anatomy. ...

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. In mathematics, the point is an equilibrium point for the differential equation if for all . ...

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. The Lorenz attractor is an example of a non-linear dynamical system. ...

• Ordinary differential equations (Continuous time. Continuous state space. No spatial derivatives.) See also Numerical ordinary differential equations.
• Partial differential equations (Continuous time. Continuous state space. Spatial derivatives.) See also Numerical partial differential equations.
• Maps (Discrete time. Continuous state space)

In mathematics, a differential equation is an equation that describes a prescribed relationship between a set of unknowns which are to be regarded as an unknown function and its (ordinary or partial) derivatives. ... Numerical ordinary differential equations is the part of numerical analysis which studies the numerical solution of ordinary differential equations (ODEs). ... In mathematics, and in particular analysis, a partial differential equation (PDE) is an equation involving partial derivatives of an unknown function. ... Numerical partial differential equations is the branch of numerical analysis that studies the numerical solution of partial differential equations. ...

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. In the mathematics of probability, a stochastic process is a random function. ... In probability theory, a random variable is a quantity whose values are random and to which a probability distribution is assigned. ... In mathematics and statistics, a probability distribution is a function of the probabilities of a mutually exclusive and exhaustive set of events. ...

• Non-Markovian processes -- Generalized master equation (Continuous time with memory of past events. Discrete state space. Waiting times of events (or transitions between states) discretely occur and have a generalized probability distribution.)
• Jump Markov process -- Master equation (Continuous time with no memory of past events. Discrete state space. Waiting times between events discretely occur and are exponentially distributed.) See also Monte Carlo method for numerical simulation methods, specifically Continuous-time Monte Carlo which is also called kinetic Monte Carlo or the stochastic simulation algorithm.
• Continuous Markov process -- stochastic differential equations or a Fokker-Planck equation (Continuous time. Continuous state space. Events occur continuously according to a random Wiener process.)

In physics, a master equation is a phenomenological first-order differential equation describing the time-evolution of the probability of a system to occupy each one of a discrete set of states: where Pk is the probability for the system to be in the state k, while the matrix is... In mathematics and statistics, a probability distribution is a function of the probabilities of a mutually exclusive and exhaustive set of events. ... In physics, a master equation is a phenomenological first-order differential equation describing the time-evolution of the probability of a system to occupy each one of a discrete set of states: where Pk is the probability for the system to be in the state k, while the matrix is... Monte Carlo methods are a widely used class of computational algorithms for simulating the behavior of various physical and mathematical systems, and for other computations. ... It has been suggested that this article or section be merged with Markov property. ... A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, thus resulting in a solution which is itself a stochastic process. ... The Fokker-Planck equation (named after Adriaan Fokker and Max Planck; also known as the Kolmogorov Forward equation) describes the time evolution of the probability density function of position and velocity of a particle. ... A single realization of a one-dimensional Wiener process A single realization of a three-dimensional Wiener process In mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. ...

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. Alan Mathison Turing, OBE, FRS (23 June 1912 – 7 June 1954) was an English mathematician, logician, and cryptographer. ... Morphogenesis (from the Greek morphê shape and genesis creation) is one of three fundamental aspects of developmental biology along with the control of cell growth and cellular differentiation. ... Cover of Cover the first volume of , published in 1665 The Philosophical Transactions of the Royal Society, or Phil. ...

• Travelling waves in a wound-healing assay [10]
• Swarming behaviour [11]
• The mechanochemical theory of morphogenesis [12]
• Biological pattern formation [13]

This article is about swarms in biology. ... Morphogenesis (from the Greek morphê shape and genesis creation) is one of three fundamental aspects of developmental biology along with the control of cell growth and cellular differentiation. ...

Bibliographical references

• 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
• 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 references

• F. Hoppensteadt, Getting Started in Mathematical Biology. Notices of American Mathematical Society, Sept. 1995.
• M. C. Reed, Why Is Mathematical Biology So Hard? Notices of American Mathematical Society, March, 2004.
• R. M. May, Uses and Abuses of Mathematics in Biology. Science, February 6, 2004.
• J. D. Murray, How the leopard gets its spots? Scientific American, 258(3): 80-87, 1988.
• S. Schnell, R. Grima, P. K. Maini, Multiscale Modeling in Biology, American Scientist, Vol 95, pages 134-142, March-April 2007.

See also

• Bioinformatics, Systems biology, biologically-inspired computing, biostatistics, cellular automata, excitable medium, Ewens's sampling formula, mathematical model, morphometrics, population genetics, theoretical biology, D'Arcy Thompson, Neighbour-sensing model, Coalescent theory, Software for molecular modeling

Map of the human X chromosome (from the NCBI website). ... Systems biology is a term used very widely in the biosciences, particularly from the year 2000 onwards, and in a variety of contexts. ... Biologically-inspired computing (also bio-inspired computing) is a field of study that loosely knits together subfields related to the topics of connectionism, social behaviour and emergence. ... Biostatistics or biometry is the application of statistics to a wide range of topics in biology. ... A cellular automaton (plural: cellular automata) is a discrete model studied in computability theory and mathematics. ... An excitable medium is a nonlinear dynamical system which has the capacity to propagate a wave of some description, and which cannot support the passing of another wave until a certain amount of time has passed (known as the refractory time). ... In population genetics, Ewenss sampling formula, introduced by Warren Ewens, states that under certain conditions (specified below), if a random sample of n gametes is taken from a population and classified according to the gene at a particular locus then the probability that there are a1 alleles represented once... A mathematical model is an abstract model that uses mathematical language to describe the behaviour of a system. ... Generally, morphometrics (from the Greek: morph, meaning shape or form, and metron”, meaning measurement) comprises methods of extracting measurements from shapes. ... Population genetics is the study of the distribution of and change in allele frequencies under the influence of the four evolutionary forces: natural selection, genetic drift, mutation, and migration. ... Theoretical biology is an interdisciplinary field of academic study and research that involves the use of quantitative tools in biology. ... DArcy Wentworth Thompson (May 2, 1860- June 21, 1948) was a biologist and mathematician and the author of the 1917 book, On Growth and Form, an influential work of striking originality. ... Development of the cone like structure (view above, slice below) Neighbour-Sensing model - the proposed hypothesis of the fungal morphogenesis. ... In genetics, coalescent theory is a retrospective model of population genetics that traces all alleles of a gene in a sample from a population to a single ancestral copy shared by all members of the population, known as the most recent common ancestor (MRCA; sometimes also termed the coancestor to... Min - Optimization, MD - Molecular Dynamics, MC - Monte Carlo, QM - Quantum mechanics. ...

External links

• The Collection of Biostatistics Research Archive
• Statistical Applications in Genetics and Molecular Biology
• The International Journal of Biostatistics
• Society for Mathematical Biology
• European Society for Mathematical and Theoretical Biology
• Biomathematics Research Centre at University of Canterbury
• Centre for Mathematical Biology at Oxford University
• Mathematical Biology at the National Institute for Medical Research
• Institute for Medical BioMathematics
• Mathematical Biology Systems of Differential Equations from EqWorld: The World of Mathematical Equations
• Systems Biology Workbench - a set of tools for modelling biochemical networks