Mathematical biology 662088 223633683 2008-07-05T00:14:43Z Lightbot 7178666 Units/dates/other {{for|use of statistics in Biology|Biostatistics}} :''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, [[biology|biological]] processes using [[Applied mathematics|applied mathematical]] techniques and tools. It has both practical and theoretical applications in biological research: In cell biology protein interactions are typically expressed as a “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 [[computer|computing]] power which enables calculations and [[simulation]]s 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, [[physics|physicists]], engineers, biologists, [[physiology|physicians]], [[zoology|zoologists]], [[chemistry|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 equation|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 modelling in epidemiology|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 of [[neuron]]s and [[carcinogen]]esis [http://www.maths.gla.ac.uk/research/groups/biology/kal.htm] *Mechanics of biological tissues [http://www.maths.gla.ac.uk/~rwo/research_areas.htm] *Theoretical enzymology and [[enzyme kinetics]] [http://www.informatics.indiana.edu/schnell/research/enzymology.asp] *[[Cancer]] modelling and simulation [http://calvino.polito.it/~mcrtn/] *Modelling the movement of interacting cell populations [http://www.ma.hw.ac.uk/~jas/researchinterests/index.html] *Mathematical modelling of scar tissue formation [http://www.ma.hw.ac.uk/~jas/researchinterests/scartissueformation.html] *Mathematical modelling of intracellular dynamics [http://www.sbi.uni-rostock.de/dokumente/p_gilles_paper.pdf] *Mathematical modelling of the cell cycle [http://mpf.biol.vt.edu/Research.html] ===Modelling physiological systems=== *Modelling of [[artery|arterial]] disease [http://www.maths.gla.ac.uk/~nah/research_interests.html] *Multi-scale modelling of the [[heart]] [http://www.integrativebiology.ox.ac.uk/heartmodel.html] ==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 point|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 system]]s)=== 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. *[[Difference equations]] Discrete time, continuous state space. *[[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) ===[[Stochastic process]]es (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 -- [[Master equation|Generalized master equation]] (Continuous time with memory of past events. Discrete state space. [[Waiting time]]s of events (or transitions between states) discretely occur and have a generalized [[probability distribution]].) * Jump [[Continuous-time Markov process|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 equation]]s 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]]. *Travelling waves in a wound-healing assay [http://www.maths.ox.ac.uk/~maini/public/gallery/twwha.htm] *[[Swarming]] behaviour [http://www.math.ubc.ca/people/faculty/keshet/research.html] *The mechanochemical theory of [[morphogenesis]] [http://www.maths.ox.ac.uk/~maini/public/gallery/mctom.htm] *Biological pattern formation [http://www.maths.ox.ac.uk/~maini/public/gallery/bpf.htm] *Spatial distribution modeling using plot samples [http://links.jstor.org/sici?sici=0030-1299%28199008%2958%3A3%3C257%3ASDOTMU%3E2.0.CO%3B2-S&size=LARGE&origin=JSTOR-enlargePage] ==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. The JJ Tyson lab (Virginia Tech)[http://mpf.biol.vt.edu/Tyson%20Lab.html] and the Bela Novak lab (Budapest University of Technology)[http://cellcycle.mkt.bme.hu/] 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).<br /> By means of a system of [[ordinary differential equation]]s 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 system|deterministic process]] (whereas a model describing a statistical distribution of protein concentrations in a population of cells is called a [[stochastic process]]).<br /> 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 [[Reaction rate|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.<br /> In order to fit the parameters the differential equations need to be studied. This can be done either by simulation or by analysis. <br /> 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.<br /> [[Image:Cell cycle bifurcation diagram.jpg|thumb|500px]] 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 [[Fixed point (mathematics)|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).<br /> 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]]. ==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 *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 references== *F. Hoppensteadt, ''[http://www.ams.org/notices/199509/hoppensteadt.pdf Getting Started in Mathematical Biology]''. Notices of American Mathematical Society, Sept. 1995. *[[Michael C Reed (mathematician)|M. C. Reed]], ''[http://www.resnet.wm.edu/~jxshix/math490/reed.pdf Why Is Mathematical Biology So Hard?]'' Notices of American Mathematical Society, March, 2004. *R. M. May, ''[http://www.resnet.wm.edu/~jxshix/math490/may.pdf Uses and Abuses of Mathematics in Biology]''. Science, February 6, 2004. *J. D. Murray, ''[http://www.resnet.wm.edu/~jxshix/math490/murray.doc How the leopard gets its spots?]'' Scientific American, 258(3): 80-87, 1988. * S. Schnell, R. Grima, P. K. Maini, ''[http://eprints.maths.ox.ac.uk/567/01/224.pdf Multiscale Modeling in Biology]'', 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. ''[http://www.biomedcentral.com/content/pdf/1471-2105-7-494.pdf Time scale and dimension analysis of a budding yeast cell cycle model.]'' BMC Bioinform. 2006 Nov 9;7:494. ==See also== * [[Molecular modelling]] * [[Bioinformatics]] * [[Systems biology]] * [[Computational biology]] * [[Biocybernetics]] * [[biologically-inspired computing]] * [[biostatistics]] * [[cellular automata]] * [[Charles S. Peskin]] * [[epidemiology]], * [[excitable medium]] * [[Ewens's sampling formula]] * [[mathematical model]] * [[morphometrics]] * [[population dynamics of fisheries]] * [[population genetics]] * [[theoretical biology]] * [[D'Arcy Thompson]] * [[Neighbour-sensing model]] * [[Coalescent theory]] * [[Software for molecular mechanics modeling|Software for molecular modeling]] * [[protein folding]],[[BlueGene]] and [[Folding@home]]. ==External links== *[http://biocyb.cs.ucla.edu/research.html UCLA Biocybernetics Laboratory] *[http://www.tucs.fi/research/labs/combio.php TUCS Computational Biomodelling Laboratory] *[http://www.agr.nagoya-u.ac.jp/english/e3senko-1.html Nagoya University Division of Biomodeling] *[http://www.bmi2.bmt.tue.nl/Biomedinf/ Technische Universiteit Biomodeling and Informatics] *[http://wiki.biological-cybernetics.de BioCybernetics Wiki, a vertical wiki on biomedical cybernetics and systems biology] ==External links== * [http://www.biostatsresearch.com/repository/ The Collection of Biostatistics Research Archive] * [http://www.bepress.com/sagmb/ Statistical Applications in Genetics and Molecular Biology] * [http://www.bepress.com/ijb/ The International Journal of Biostatistics] *[http://www.smb.org/ Society for Mathematical Biology] *[http://www.esmtb.org/ European Society for Mathematical and Theoretical Biology] *[http://www.math.canterbury.ac.nz/bio/ Biomathematics Research Centre at University of Canterbury] *[http://www.maths.ox.ac.uk/cmb/ Centre for Mathematical Biology at Oxford University] *[http://mathbio.nimr.mrc.ac.uk/ Mathematical Biology at the National Institute for Medical Research] *[http://www.imbm.org/ Institute for Medical BioMathematics] * [http://eqworld.ipmnet.ru/en/solutions/syspde/spde-toc2.pdf ''Mathematical Biology Systems of Differential Equations''] from EqWorld: The World of Mathematical Equations *[http://sbw.kgi.edu Systems Biology Workbench - a set of tools for modelling biochemical networks] {{Biology-footer}} [[Category:Bioinformatics]] [[Category:Epidemiology]] [[Category:Interdisciplinary fields]] [[Category:Biostatistics]] [[Category:Mathematical biology| ]] [[es:Biología matemática]] [[fr:Biomathématique]] [[hu:Biomatematika]] [[ja:数理生物学]] [[pl:Biomatematyka]] [[ro:Biomatematică]] [[sk:Biomatematika]] [[fi:Matemaattinen biologia]] [[sv:Matematisk biologi]] [[vi:Toán sinh học]] [[uk:Математична біологія]] [[zh:生物数学]]