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Information about Biomathematics

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.

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.

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.

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.

Modelling cell and molecular biology

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

Modelling physiological systems

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.
  • 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.)

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.

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

See also

External links

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General subfields within biology
Anatomy - Astrobiology - Biochemistry - Bioinformatics - Botany - Cell biology - Ecology - Developmental biology - Evolutionary biology - Genetics - Genomics - Marine biology - Human biology - Microbiology - Molecular biology - Origin of life - Paleontology - Parasitology - Pathology - Physiology - Taxonomy - Zoology
Interdisciplinarity is the act of drawing from and integrating two or more academic disciplines, professions, technologies, departments, their methods and insights, in the pursuit of a common goal.
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Biology (from Greek: βίος, bio, "life"; and λόγος, logos, "knowledge"), also referred to as the biological sciences, is the scientific study of life.
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Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
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Genomics is the study of an organism's entire genome. The field includes intensive efforts to determine the entire DNA sequence of organisms and fine-scale genetic mapping efforts.
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chaos theory describes the behavior of certain nonlinear dynamical systems that under specific conditions exhibit dynamics that are sensitive to initial conditions (popularly referred to as the butterfly effect).
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computer is a machine which manipulates data according to a list of instructions.

Computers take numerous physical forms. The first devices that resemble modern computers date to the mid-20th century (around 1940 - 1941), although the computer concept and various machines
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simulation is an imitation of some real thing, state of affairs, or process. The act of simulating something generally entails representing certain key characteristics or behaviours of a selected physical or abstract system.
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In Silico is a full length artist album by Deepsky

In Silico

Studio album by Deepsky
Released February 5, 2002
Genre Progressive electronic music
Label Kinetic Records

Track listing:

  1. "View From a Stairway"

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Physics is the science of matter[1] and its motion[2][3], as well as space and time[4][5] —the science that deals with concepts such as force, energy, mass, and charge.
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Physiology (from Greek: φυσις, physis, “nature, origin”; and λόγος, logos, "knowledge") is the study of the mechanical, physical, and biochemical functions of living organisms.
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Zoology (from Greek: ζῴον, zoion, "animal"; and λόγος, logos, "knowledge") is the biological discipline which involves the study of animals.
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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.
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For the periodical, see .
The 19th Century (also written XIX century) lasted from 1801 through 1900 in the Gregorian calendar. It is often referred to as the "1800s.
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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.
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Evolutionary game theory (EGT) is the application of population genetics-inspired models of change in gene frequency in populations to game theory. It differs from classical game theory by focusing on the dynamics of strategy change more than the properties of strategy equilibria.
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Professor John Maynard Smith,[1] F.R.S. (6 January 1920 – 19 April 2004) was a British evolutionary biologist and geneticist. Originally an aeronautical engineer during the Second World War, he then took a second degree in genetics under the well-known biologist J.
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It is possible to mathematically model the progress of most infectious diseases to discover the likely outcome of an epidemic or to help manage them by vaccination. This article uses some basic assumptions and some simple mathematics to find parameters for various infectious
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Molecular biology is the study of biology at a molecular level. The field overlaps with other areas of biology and chemistry, particularly genetics and biochemistry. Molecular biology chiefly concerns itself with understanding the interactions between the various systems of a cell,
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Neurons (also known as neurones and nerve cells) are electrically excitable cells in the nervous system that process and transmit information. In vertebrate animals, neurons are the core components of the brain, spinal cord and peripheral nerves.
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The term carcinogen refers to any substance, radionuclide or radiation which is an agent directly involved in the promotion of cancer or in the facilitation
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Enzyme kinetics is the study of the rates of chemical reactions that are catalysed by enzymes. The study of an enzyme's kinetics provides insights into the catalytic mechanism of this enzyme, its role in metabolism, how its activity is controlled in the cell and how drugs and
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Arteries are muscular blood vessels that carry blood away from the heart.[1] All arteries, with the exception of the pulmonary and umbilical arteries, carry oxygenated blood.

The circulatory system is extremely important for sustaining life.
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heart is a muscular organ responsible for pumping blood through the blood vessels by repeated, rhythmic contractions, or a similar structure in the annelids, mollusks, and arthropods.
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In mathematics, the point is an equilibrium point for the differential equation



if for all .

Similarly, the point is an equilibrium point (or fixed point) for the difference equation



if for .
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dynamical system concept is a mathematical formalization for any fixed "rule" which describes the time dependence of a point's position in its ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and
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In mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable.
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Numerical ordinary differential equations is the part of numerical analysis which studies the numerical solution of ordinary differential equations (ODEs). This field is also known under the name numerical integration,
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In mathematics, a partial differential equation (PDE) is a type of differential equation, i. e. a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables.
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