 Mathematical and theoretical biology

Mathematical and theoretical biology is an interdisciplinary scientific research field with a range of applications in biology, medicine and biotechnology.^{[1]} The field may be referred to as mathematical biology or biomathematics to stress the mathematical side, or as theoretical biology to stress the biological side.^{[2]} It includes at least four major subfields: biological mathematical modeling, relational biology/complex systems biology (CSB), bioinformatics and computational biomodeling/biocomputing.^{[3]}^{[4]} Mathematical biology aims at the mathematical representation, treatment and modeling of biological processes, using a variety of applied mathematical techniques and tools. It has both theoretical and practical applications in biological, biomedical and biotechnology research. For example, in cell biology, protein interactions are often represented as "cartoon" models, which, although easy to visualize, do not accurately describe the systems studied. In order to do this, precise mathematical models are required. By describing the systems in a quantitative manner, their behavior can be better simulated, and hence properties can be predicted that might not be evident to the experimenter.
Such mathematicial areas as calculus, probability theory, statistics, linear algebra, abstract algebra, graph theory, combinatorics, algebraic geometry, topology, dynamical systems, differential equations and coding theory are now being applied in biology.^{[5]}
Contents
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 datarich 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
Several areas of specialized research in mathematical and theoretical biology^{[6]}^{[7]}^{[8]}^{[9]}^{[10]} as well as external links to related projects in various universities are concisely presented in the following subsections, including also a large number of appropriate validating references from a list of several thousands of published authors contributing to this field. Many of the included examples are characterised by highly complex, nonlinear, and supercomplex mechanisms, as it is being increasingly recognised that the result of such interactions may only be understood through a combination of mathematical, logical, physical/chemical, molecular and computational models. Due to the wide diversity of specific knowledge involved, biomathematical research is often done in collaboration between mathematicians, biomathematicians, theoretical biologists, physicists, biophysicists, biochemists, bioengineers, engineers, biologists, physiologists, research physicians, biomedical researchers, oncologists, molecular biologists, geneticists, embryologists, zoologists, chemists, etc.
Computer models and automata theory
Main article: Modelling biological systemsA monograph on this topic summarizes an extensive amount of published research in this area up to 1986^{[11]}^{[12]}^{[13]}, including subsections in the following areas: computer modeling in biology and medicine, arterial system models, neuron models, biochemical and oscillation networks, quantum automata, quantum computers in molecular biology and genetics^{[14]}, cancer modelling^{[15]}, neural nets, genetic networks, abstract categories in relational biology^{[16]}, metabolicreplication systems, category theory^{[17]} applications in biology and medicine,^{[18]} automata theory, cellular automata, tessallation models^{[19]}^{[20]} and complete selfreproduction, chaotic systems in organisms, relational biology and organismic theories.^{[21]}^{[22]} This published report also includes 390 references to peerreviewed articles by a large number of authors.^{[6]}^{[23]}^{[24]}
Modeling cell and molecular biology
This area has received a boost due to the growing importance of molecular biology.^{[9]}
 Mechanics of biological tissues^{[25]}
 Theoretical enzymology and enzyme kinetics
 Cancer modelling and simulation^{[26]}^{[27]}
 Modelling the movement of interacting cell populations^{[28]}
 Mathematical modelling of scar tissue formation^{[29]}
 Mathematical modelling of intracellular dynamics^{[30]}
 Mathematical modelling of the cell cycle^{[31]}
Modelling physiological systems
Molecular set theory
Molecular set theory was introduced by Anthony Bartholomay, and its applications were developed in mathematical biology and especially in Mathematical Medicine.^{[34]} Molecular set theory (MST) is a mathematical formulation of the widesense chemical kinetics of biomolecular reactions in terms of sets of molecules and their chemical transformations represented by settheoretical mappings between molecular sets. In a more general sense, MST is the theory of molecular categories defined as categories of molecular sets and their chemical transformations represented as settheoretical mappings of molecular sets. The theory has also contributed to biostatistics and the formulation of clinical biochemistry problems in mathematical formulations of pathological, biochemical changes of interest to Physiology, Clinical Biochemistry and Medicine.^{[34]}^{[35]}
Population dynamics
Population dynamics has traditionally been the dominant field of mathematical biology. Work in this area dates back to the 19th century, and even as far as 1798 when Thomas Malthus formulated the first principle of population dynamics, which later became known as the Malthusian growth model. The Lotka–Volterra predatorprey equations are another 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 the spread of infections have been proposed and analyzed, and provide important results that may be applied to health policy decisions.
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.
Mathematical biophysics
The earlier stages of mathematical biology were dominated by mathematical biophysics, described as the application of mathematics in biophysics, often involving specific physical/mathematical models of biosystems and their components or compartments.
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.
 Difference equations/Maps – 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.
 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.
 NonMarkovian 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 dynamic Monte Carlo method and Gillespie algorithm.
 Continuous Markov process – stochastic differential equations or a FokkerPlanck 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 woundhealing assay^{[36]}
 Swarming behaviour^{[37]}
 A mechanochemical theory of morphogenesis^{[38]}
 Biological pattern formation^{[39]}
 Spatial distribution modeling using plot samples^{[40]}
 Relational biology
Abstract Relational Biology (ARB)^{[41]} is concerned with the study of general, relational models of complex biological systems, usually abstracting out specific morphological, or anatomical, structures. Some of the simplest models in ARB are the MetabolicReplication, or (M,R)systems introduced by Robert Rosen in 19571958 as abstract, relational models of cellular and organismal organization.^{[citation needed]}
Phylogenetics
Phylogenetics is an area that deals with the reconstruction and analysis of phylogenetic (evolutionary) trees and networks based on inherited characteristics^{[42]}
Model example: the cell cycle
Main article: Cellular modelThe 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 ^{[43]}^{[44]} 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 (CsikaszNagy 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, MichaelisMenten 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 halflife 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 timeframe 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 steadystate 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.^{[citation needed]}See also
 Abstract relational biology^{[45]}^{[46]}
 Artificial life
 Biocybernetics
 Biologically inspired computing
 Biosemiotics
 Cellular automata^{[6]}
 Coalescent theory
 Complex systems biology^{[47]}^{[48]}^{[49]}
 Computational biology
 Digital morphogenesis
 Dynamical systems in biology^{[48]}^{[49]}^{[50]}^{[51]}^{[52]}
 Epidemiology
 Evolution theories and Population Genetics
 Ewens's sampling formula
 Journal of Theoretical Biology
 Mathematical models
 Molecular modelling
 Molecular modelling on GPU
 Software for molecular modeling
 MetabolicReplication Systems^{[53]}
 Models of Growth and Form
 Neighboursensing model
 Morphometrics
 Organismic systems (OS)^{[54]}
 Organismic supercategories and Complex Systems^{[49]}^{[55]}
 Population dynamics of fisheries
 Protein folding, also blue Gene and folding@home
 Quantum computers
 Quantum genetics
 Relational biology or Abstract Relational Biology (ARB)^{[56]}
 Selfreproduction^{[57]} (also called selfreplication in a more general context).
 Computational gene models
 Simulated reality
 Systems biology^{[58]}
 Theoretical biology^{[59]}
 Theoretical ecology
 Topological models of morphogenesis
For use of statistics in biology, see Biostatistics. For use of basic arithmetics in biology, see relevant topic, such as Serial dilution.
Societies and Institutes
 Division of Mathematical Biology at NIMR
 National Institute for Mathematical and Biological Synthesis (NIMBioS)
 Society for Mathematical Biology
 European Society for Mathematical and Theoretical Biology
Notes
 ^ Mathematical and Theoretical Biology: A European Perspective
 ^ "There is a subtle difference between mathematical biologists and theoretical biologists. Mathematical biologists tend to be employed in mathematical departments and to be a bit more interested in math inspired by biology than in the biological problems themselves, and vice versa." Careers in theoretical biology
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 ^ http://library.bjcancer.org/ebook/109.pdf L. Preziosi, Cancer Modelling and Simulation. Chapman Hall/CRC Press, 2003. ISBN 1584883618.
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 ^ ^{a} ^{b} ^{c} Baianu, I. C.; Brown, R.; Georgescu, G.; Glazebrook, J. F. (2006). "Complex Nonlinear Biodynamics in Categories, Higher Dimensional Algebra and Łukasiewicz–Moisil Topos: Transformations of Neuronal, Genetic and Neoplastic Networks". Axiomathes 16: 65. doi:10.1007/s1051600539738.
 ^ ŁukasiewiczTopos Models of Neural Networks, Cell Genome and Interactome Nonlinear Dynamic Models (2004) http://cogprints.org/3701/01/ANeuralGenNetworkLuknTopos_oknu4.pdf/
 ^ Complex Systems Analysis of Arrested Neural Cell Differentiation during Development and Analogous Cell Cycling Models in Carcinogenesis (2004) http://cogprints.org/3687/
 ^ ^{a} ^{b} "Research in Mathematical Biology". Maths.gla.ac.uk. http://www.maths.gla.ac.uk/research/groups/biology/kal.htm. Retrieved 20080910.
 ^ J. R. Junck. Ten Equations that Changed Biology: Mathematics in ProblemSolving Biology Curricula, Bioscene, (1997), 23(1):1136^{[dead link]} New Link (Aug 2010)
 ^ "Computer Models and Automata Theory in Biology and Medicine" (1986). In:Mathematical Modeling: Mathematical Models in Medicine, volume 7:15131577, M. Witten, Ed., Pergamon Press: New York. http://cdsweb.cern.ch/record/746663/files/COMPUTER_MODEL_AND_AUTOMATA_THEORY_IN_BIOLOGY2p.pdf
 ^ Lin, H.C. 2004. "Computer Simulations and the Question of Computability of Biological Systems": 115,doi=10.1.1.108.5072. https://tspace.library.utoronto.ca/bitstream/1807/2951/2/compauto.pdf
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 ^ "Quantum Interactomics and Cancer Mechanisms" (2004): 116, Research Report communicated to the Institute of Genomic Biology, University of Illinois at Urbana https://tspace.library.utoronto.ca/retrieve/4969/QuantumInteractomicsInCancer_Sept13k4E_cuteprt.pdf
 ^ Kainen,P.C. 2005."Category Theory and Living Systems", In: Charles Ehresmann's Centennial Conference Proceedings: 15,University of Amiens, France, October 79th, 2005, A. Ehresmann, Organizer and Editor. http://vbmehr.pagespersoorange.fr/ChEh/articles/Kainen.pdf
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 ^ "Computer models and automata theory in biology and medicine  KLI Theory Lab". Theorylab.org. 20090526. http://theorylab.org/node/56690. Retrieved 20100317.
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 ^ http://www.sbi.unirostock.de/dokumente/p_gilles_paper.pdf
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 ^ "Integrative Biology  Heart Modelling". Integrativebiology.ox.ac.uk. http://www.integrativebiology.ox.ac.uk/heartmodel.html. Retrieved 20100317.
 ^ ^{a} ^{b} "molecular set category". PlanetPhysics. http://planetphysics.org/encyclopedia/CategoryOfMolecularSets2.html. Retrieved 20100317.
 ^ Representation of Unimolecular and Multimolecular Biochemical Reactions in terms of Molecular Set Transformations http://planetmath.org/?op=getobj&from=objects&id=10770^{[unreliable medical source?]}
 ^ "Travelling waves in a wound". Maths.ox.ac.uk. http://www.maths.ox.ac.uk/~maini/public/gallery/twwha.htm. Retrieved 20100317.
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 ^ "The mechanochemical theory of morphogenesis". Maths.ox.ac.uk. http://www.maths.ox.ac.uk/~maini/public/gallery/mctom.htm. Retrieved 20100317.
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 ^ Hurlbert, Stuart H. (1990). "Spatial Distribution of the Montane Unicorn". Oikos 58 (3): 257–271. JSTOR 3545216.
 ^ Abstract Relational Biology (ARB)
 ^ Charles Semple (2003), Phylogenetics, Oxford University Press, ISBN 9780198509424
 ^ "The JJ Tyson Lab". Virginia Tech. Archived from the original on March 8, 2008. http://web.archive.org/web/20080308120536/http://mpf.biol.vt.edu/Tyson+Lab.html. Retrieved 20080910.
 ^ "The Molecular Network Dynamics Research Group". Budapest University of Technology and Economics. http://cellcycle.mkt.bme.hu/.
 ^ "abstract relational biology (ARB)". PlanetPhysics. http://planetphysics.org/encyclopedia/AbstractRelationalBiologyARB.html. Retrieved 20100317.
 ^ "Molecular Evolution and Protobiology  KLI Theory Lab". Theorylab.org. 20090526. http://theorylab.org/node/52354. Retrieved 20100317.
 ^ Baianu, I. C.; Brown, R.; Glazebrook, J. F. (2007). "Categorical Ontology of Complex Spacetime Structures: the Emergence of Life and Human Consciousness". Axiomathes 17: 223. doi:10.1007/s1051600790112.
 ^ ^{a} ^{b} Brown, R.; Glazebrook, J. F.; Baianu, I. C. (2007). "A Conceptual Construction of Complexity Levels Theory in Spacetime Categorical Ontology: NonAbelian Algebraic Topology, ManyValued Logics and Dynamic Systems". Axiomathes 17: 409. doi:10.1007/s1051600790103.
 ^ ^{a} ^{b} ^{c} Baianu, I. (1970). "Organismic supercategores: II. On multistable systems". The Bulletin of Mathematical Biophysics 32 (4): 539. doi:10.1007/BF02476770. PMID 4327361.
 ^ Robert Rosen, Dynamical system theory in biology. New York, WileyInterscience (1970) ISBN 0471735507 http://www.worldcat.org/oclc/101642
 ^ Organismic supercategories and qualitative dynamics of systems
 ^ Organismic supercategories. II. On multistable systemsB�ianu, I (1970). "Organismic supercategories. II. On multistable systems.". The Bulletin of Mathematical Biophysics 32 (4): 539–61. doi:10.1007/BF02476770. PMID 4327361.
 ^ "category of $(M,R)$ systems". PlanetPhysics. http://planetphysics.org/encyclopedia/RSystemsCategoryOfM.html. Retrieved 20100317.
 ^ Organisms as Supercomplex Systems http://planetmath.org/?op=getobj&from=objects&id=10890
 ^ Supercategorical Approach to Complex MetaSystems and Ontology MultiLevels
 ^ Abstract Relational Biology (ARB) and MetabolicReplication (MR) Systems
 ^ Natural Transformations of Organismic Structures http://planetphysics.org/encyclopedia/NaturalTransformationsOfOrganismicStructures.html
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References
 D. Barnes, D. Chu, (2010). Introduction to Modelling for Biosciences. Springer Verlag. ISBN 1849963258.
 Israel G (1988). "On the contribution of Volterra and Lotka to the development of modern biomathematics". History and Philosophy of the Life Sciences 10 (1): 37–49. PMID 3045853.
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 N.G. van Kampen, Stochastic Processes in Physics and Chemistry, North Holland., 3rd ed. 2001, ISBN 0444893490
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 Theoretical biology
 Bonner, J. T. 1988. The Evolution of Complexity by Means of Natural Selection. Princeton: Princeton University Press.
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Further reading
 Hoppensteadt, F. (September 1995). "Getting Started in Mathematical Biology". Notices of American Mathematical Society. http://www.ams.org/notices/199509/hoppensteadt.pdf.
 Reed, M. C. (March 2004). "Why Is Mathematical Biology So Hard?". Notices of American Mathematical Society. http://www.resnet.wm.edu/~jxshix/math490/reed.pdf.
 May, R. M. (2004). "Uses and Abuses of Mathematics in Biology". Science 303 (5659): 790–793. doi:10.1126/science.1094442. PMID 14764866.
 Murray, J. D. (1988). "How the leopard gets its spots?". Scientific American 258 (3): 80–87. doi:10.1038/scientificamerican038880. http://www.resnet.wm.edu/~jxshix/math490/murray.doc.
 Schnell, S.; Grima, R.; Maini, P. K. (2007). "Multiscale Modeling in Biology". American Scientist 95: 134–142. http://eprints.maths.ox.ac.uk/567/01/224.pdf.
 Chen, Katherine C.; Calzone, Laurence; CsikaszNagy, Attila; Cross, FR; Cross, Frederick R.; Novak, Bela; Tyson, John J. (2004). "Integrative analysis of cell cycle control in budding yeast". Mol Biol Cell 15 (8): 3841–3862. doi:10.1091/mbc.E03110794. PMC 491841. PMID 15169868. http://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=491841.
 CsikászNagy, Attila; Battogtokh, Dorjsuren; Chen, Katherine C.; Novák, Béla; Tyson, John J. (2006). "Analysis of a generic model of eukaryotic cellcycle regulation". Biophys J. 90 (12): 4361–4379. doi:10.1529/biophysj.106.081240. PMC 1471857. PMID 16581849. http://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=1471857.
 Fuss, H.; Dubitzky, Werner; Downes, C. Stephen; Kurth, Mary Jo (2005). "Mathematical models of cell cycle regulation". Brief Bioinform. 6 (2): 163–177. doi:10.1093/bib/6.2.163. PMID 15975225.
 Lovrics, Anna; CsikászNagy, Attila; Zsély1, István Gy; Zádor, Judit; Turányi, Tamás; Novák, Béla (2006). "Time scale and dimension analysis of a budding yeast cell cycle model". BMC Bioinform. 9 (7): 494. doi:10.1186/147121057494.
External links
 The Society for Mathematical Biology
 Theoretical and mathematical biology website
 Complexity Discussion Group
 UCLA Biocybernetics Laboratory
 TUCS Computational Biomodelling Laboratory
 Nagoya University Division of Biomodeling
 Technische Universiteit Biomodeling and Informatics
 BioCybernetics Wiki, a vertical wiki on biomedical cybernetics and systems biology
 Bulletin of Mathematical Biology
 European Society for Mathematical and Theoretical Biology
 Journal of Mathematical 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
 The Collection of Biostatistics Research Archive
 Statistical Applications in Genetics and Molecular Biology
 The International Journal of Biostatistics
 Theoretical Modeling of Cellular Physiology at Ecole Normale Superieure, Paris
 Lists of references
 A general list of Theoretical biology/Mathematical biology references, including an updated list of actively contributing authors.
 A list of references for applications of category theory in relational biology.^{[unreliable medical source?]}
 An updated list of publications of theoretical biologist Robert Rosen
 Theory of Biological Anthropology (Documents No. 9 and 10 in English)
 Drawing the Line Between Theoretical and Basic Biology (a forum article by Isidro T. Savillo)
Related journals
 Acta Biotheoretica
 Bioinformatics
 Biological Theory
 BioSystems
 Bulletin of Mathematical Biology
 Ecological Modelling
 Journal of Mathematical Biology
 Journal of Theoretical Biology
 Journal of the Royal Society Interface
 Mathematical Biosciences
 Medical Hypotheses
 Rivista di BiologiaBiology Forum
 Theoretical and Applied Genetics
 Theoretical Biology and Medical Modelling
 Theoretical Population Biology
 Theory in Biosciences (formerly: Biologisches Zentralblatt)
Related societies
 ESMTB: European Society for Mathematical and Theoretical Biology
 The Israeli Society for Theoretical and Mathematical Biology
 Société Francophone de Biologie Théorique
 International Society for Biosemiotic Studies
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 Mathematical and theoretical biology
 Applied mathematics
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