• 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.

  • Some reasons for this include: • The rapid growth of data-rich information sets, due to the genomics revolution, which are difficult to understand without the use of analytical
    tools[9] • Recent development of mathematical tools such as chaos theory to help understand complex, non-linear mechanisms in biology • An increase in computing power, which facilitates calculations and simulations not previously possible
    • 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[10][11][12][13][14] 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.

  • 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).

  • • Travelling waves in a wound-healing assay[46] • Swarming behaviour[47] • A mechanochemical theory of morphogenesis[48] • Biological pattern formation[49] • Spatial distribution
    modeling using plot samples[50] • Turing patterns[51] Mathematical methods[edit] 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

  • [2] Theoretical biology focuses more on the development of theoretical principles for biology while mathematical biology focuses on the use of mathematical tools to study
    biological systems, even though the two terms are sometimes interchanged.

  • [citation needed] Fritz Müller described the evolutionary benefits of what is now called Müllerian mimicry in 1879, in an account notable for being the first use of a mathematical
    argument in evolutionary ecology to show how powerful the effect of natural selection would be, unless one includes Malthus’s discussion of the effects of population growth that influenced Charles Darwin: Malthus argued that growth would be
    exponential (he uses the word “geometric”) while resources (the environment’s carrying capacity) could only grow arithmetically.

  • The solution of the equations, by either analytical or numerical means, describes how the biological system behaves either over time or at equilibrium.

  • [16][17][18] Complex systems biology[edit] An elaboration of systems biology to understand the more complex life processes was developed since 1970 in connection with molecular
    set theory, relational biology and algebraic biology.

  • 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.

  • Mathematical and theoretical biology, or biomathematics, is a branch of biology which employs theoretical analysis, mathematical models and abstractions of the living organisms
    to investigate the principles that govern the structure, development and behavior of the systems, as opposed to experimental biology which deals with the conduction of experiments to prove and validate the scientific theories.

  • 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.

  • [15] Algebraic biology[edit] Algebraic biology (also known as symbolic systems biology) applies the algebraic methods of symbolic computation to the study of biological problems,
    especially in genomics, proteomics, analysis of molecular structures and study of genes.

  • 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.

  • For example, abstract relational biology (ARB)[53] is concerned with the study of general, relational models of complex biological systems, usually abstracting out specific
    morphological, or anatomical, structures.

  • Molecular set theory[edit] Molecular set theory (MST) is a mathematical formulation of the wide-sense chemical kinetics of biomolecular reactions in terms of sets of molecules
    and their chemical transformations represented by set-theoretical mappings between molecular sets.

  • [52] In a more general sense, MST is the theory of molecular categories defined as categories of molecular sets and their chemical transformations represented as set-theoretical
    mappings of molecular sets.

  • Computer models and automata theory[edit] A monograph on this topic summarizes an extensive amount of published research in this area up to 1986, 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,[22] cancer modelling,[23] neural
    nets, genetic networks, abstract categories in relational biology,[24] metabolic-replication systems, category theory[25] applications in biology and medicine,[26] automata theory, cellular automata,[27] tessellation models[28][29] and complete
    self-reproduction, chaotic systems in organisms, relational biology and organismic theories.

  • [1] The field is sometimes called mathematical biology or biomathematics to stress the mathematical side, or theoretical biology to stress the biological side.

  • Because of the complexity of the living systems, theoretical biology employs several fields of mathematics,[5] and has contributed to the development of new techniques.

  • 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.

  • In analysis, the properties of the equations are used to investigate the behavior of the system depending on the values of the parameters and variables.

  • Describing systems in a quantitative manner means their behavior can be better simulated, and hence properties can be predicted that might not be evident to the experimenter.

  • They have recently produced a generic eukaryotic cell cycle model that 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).

  • It is possibly a good example of a mathematical model as it deals with simple calculus but gives valid results.

  • • 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.

  • Phylogenetics is an area that deals with the reconstruction and analysis of phylogenetic (evolutionary) trees and networks based on inherited characteristics[45] Traditional
    population genetic models deal with alleles and genotypes, and are frequently stochastic.

  • Ronald Fisher made fundamental advances in statistics, such as analysis of variance, via his work on quantitative genetics.

  • [54] Model example: the cell cycle The eukaryotic cell cycle is very complex and is one of the most studied topics, since its misregulation leads to cancers.

  • [16][30] Modeling cell and molecular biology This area has received a boost due to the growing importance of molecular biology.

  • A better representation, which handles the large number of variables and parameters, is a bifurcation diagram using bifurcation theory.

  • Stochastic processes (random dynamical systems)[edit] 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.

  • The equations may also make assumptions about the nature of what may occur.

  • This can be done either by simulation or by analysis.

  • Spatial modelling[edit] 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.

  • When infinitesimal effects at a large number of gene loci are considered, together with the assumption of linkage equilibrium or quasi-linkage equilibrium, one derives quantitative

  • • Difference equations/Maps – discrete time, continuous state space.


Works Cited

[‘o “What is mathematical biology | Centre for Mathematical Biology | University of Bath”. Archived from the original on 2018-09-23. Retrieved 2018-06-07.
o ^ “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 Archived 2019-09-14
at the Wayback Machine
o ^ Longo G, Soto AM (October 2016). “Why do we need theories?” (PDF). Progress in Biophysics and Molecular Biology. From the Century of the Genome to the Century of the Organism: New Theoretical Approaches. 122 (1): 4–10.
doi:10.1016/j.pbiomolbio.2016.06.005. PMC 5501401. PMID 27390105.
o ^ Montévil M, Speroni L, Sonnenschein C, Soto AM (October 2016). “Modeling mammary organogenesis from biological first principles: Cells and their physical constraints”. Progress
in Biophysics and Molecular Biology. From the Century of the Genome to the Century of the Organism: New Theoretical Approaches. 122 (1): 58–69. arXiv:1702.03337. doi:10.1016/j.pbiomolbio.2016.08.004. PMC 5563449. PMID 27544910.
o ^ Robeva R, Davies
R, Hodge T, Enyedi A (Fall 2010). “Mathematical biology modules based on modern molecular biology and modern discrete mathematics”. CBE: Life Sciences Education. The American Society for Cell Biology. 9 (3): 227–40. doi:10.1187/cbe.10-03-0019. PMC
2931670. PMID 20810955.
o ^ Mallet J (July 2001). “Mimicry: an interface between psychology and evolution”. Proceedings of the National Academy of Sciences of the United States of America. 98 (16): 8928–30. Bibcode:2001PNAS…98.8928M. doi:10.1073/pnas.171326298.
PMC 55348. PMID 11481461.
o ^ Ian Stewart (1998), Life’s Other Secret: The New Mathematics of the Living World, New York: John Wiley, ISBN 978-0471158455
o ^ Keller EF (2002). Making Sense of Life: Explaining Biological Development with Models,
Metaphors and Machines. Harvard University Press. ISBN 978-0674012509.
o ^ Reed M (November 2015). “Mathematical Biology is Good for Mathematics”. Notices of the AMS. 62 (10): 1172–1176. doi:10.1090/noti1288.
o ^ Baianu IC, Brown R, Georgescu
G, Glazebrook JF (2006). “Complex Non-linear Biodynamics in Categories, Higher Dimensional Algebra and Łukasiewicz–Moisil Topos: Transformations of Neuronal, Genetic and Neoplastic Networks”. Axiomathes. 16 (1–2): 65–122. doi:10.1007/s10516-005-3973-8.
S2CID 9907900.
o ^ Baianu IC (2004). “Łukasiewicz-Topos Models of Neural Networks, Cell Genome and Interactome Nonlinear Dynamic Models” (PDF). Archived from the original on 2007-07-13. Retrieved 2011-08-07.
o ^ Baianu I, Prisecaru V (April 2012).
“Complex Systems Analysis of Arrested Neural Cell Differentiation during Development and Analogous Cell Cycling Models in Carcinogenesis”. Nature Precedings. doi:10.1038/npre.2012.7101.1.
o ^ Jump up to:a b “Research in Mathematical Biology”.
Retrieved 2008-09-10.
o ^ Jungck JR (May 1997). “Ten equations that changed biology: mathematics in problem-solving biology curricula” (PDF). Bioscene. 23 (1): 11–36. Archived from the original (PDF) on 2009-03-26.
o ^ Montévil M, Mossio M (May
2015). “Biological organisation as closure of constraints” (PDF). Journal of Theoretical Biology. 372: 179–91. Bibcode:2015JThBi.372..179M. doi:10.1016/j.jtbi.2015.02.029. PMID 25752259. S2CID 4654439.
o ^ Jump up to:a b Baianu IC (1987). “Computer
Models and Automata Theory in Biology and Medicine”. In Witten M (ed.). Mathematical Models in Medicine. Vol. 7. New York: Pergamon Press. pp. 1513–1577.
o ^ Barnett MP (2006). “Symbolic calculation in the life sciences: trends and prospects” (PDF).
In Anai H, Horimoto K (eds.). Algebraic Biology 2005. Computer Algebra in Biology. Tokyo: Universal Academy Press. Archived from the original (PDF) on 2006-06-16.
o ^ Preziosi L (2003). Cancer Modelling and Simulation (PDF). Chapman Hall/CRC Press.
ISBN 1-58488-361-8. Archived from the original (PDF) on March 10, 2012.
o ^ Witten M, ed. (1986). “Computer Models and Automata Theory in Biology and Medicine” (PDF). Mathematical Modeling : Mathematical Models in Medicine. Vol. 7. New York: Pergamon
Press. pp. 1513–1577.
o ^ Lin HC (2004). “Computer Simulations and the Question of Computability of Biological Systems” (PDF).
o ^ Computer Models and Automata Theory in Biology and Medicine. 1986.
o ^ “Natural Transformations Models in Molecular
Biology”. SIAM and Society of Mathematical Biology, National Meeting. Bethesda, MD. N/A: 230–232. 1983.
o ^ Baianu IC (2004). “Quantum Interactomics and Cancer Mechanisms” (PDF). Research Report Communicated to the Institute of Genomic Biology,
University of Illinois at Urbana.
o ^ Kainen PC (2005). “Category Theory and Living Systems” (PDF). In Ehresmann A (ed.). Charles Ehresmann’s Centennial Conference Proceedings. University of Amiens, France, October 7–9th, 2005. pp. 1–5.
o ^ “bibliography
for category theory/algebraic topology applications in physics”. PlanetPhysics. Archived from the original on 2016-01-07. Retrieved 2010-03-17.
o ^ “bibliography for mathematical biophysics and mathematical medicine”. PlanetPhysics. 2009-01-24.
Archived from the original on 2016-01-07. Retrieved 2010-03-17.
o ^ “Cellular Automata”. Los Alamos Science. Fall 1983.
o ^ Preston K, Duff MJ (1985-02-28). Modern Cellular Automata. ISBN 9780306417375.
o ^ “Dual Tessellation – from Wolfram
MathWorld”. 2010-03-03. Retrieved 2010-03-17.
o ^ “Computer models and automata theory in biology and medicine | KLI Theory Lab”. 2009-05-26. Archived from the original on 2011-07-28. Retrieved 2010-03-17.
o ^
Ogden R (2004-07-02). “rwo_research_details”. Archived from the original on 2009-02-02. Retrieved 2010-03-17.
o ^ Wang Y, Brodin E, Nishii K, Frieboes HB, Mumenthaler SM, Sparks JL, Macklin P (January 2021). “Impact of tumor-parenchyma
biomechanics on liver metastatic progression: a multi-model approach”. Scientific Reports. 11 (1): 1710. Bibcode:2021NatSR..11.1710W. doi:10.1038/s41598-020-78780-7. PMC 7813881. PMID 33462259.
o ^ Oprisan SA, Oprisan A (2006). “A Computational
Model of Oncogenesis using the Systemic Approach”. Axiomathes. 16 (1–2): 155–163. doi:10.1007/s10516-005-4943-x. S2CID 119637285.
o ^ “MCRTN – About tumour modelling project”. Retrieved 2010-03-17.
o ^ “Jonathan Sherratt’s
Research Interests”. Retrieved 2010-03-17.
o ^ “Jonathan Sherratt’s Research: Scar Formation”. Retrieved 2010-03-17.
o ^ Kuznetsov AV, Avramenko AA (April 2009). “A macroscopic model of traffic jams in axons”. Mathematical
Biosciences. 218 (2): 142–52. doi:10.1016/j.mbs.2009.01.005. PMID 19563741.
o ^ Wolkenhauer O, Ullah M, Kolch W, Cho KH (September 2004). “Modeling and simulation of intracellular dynamics: choosing an appropriate framework”. IEEE Transactions on
NanoBioscience. 3 (3): 200–7. doi:10.1109/TNB.2004.833694. PMID 15473072. S2CID 1829220.
o ^ “Tyson Lab”. Archived from the original on July 28, 2007.
o ^ Fussenegger M, Bailey JE, Varner J (July 2000). “A mathematical model of caspase function
in apoptosis”. Nature Biotechnology. 18 (2): 768–74. doi:10.1038/77589. PMID 10888847. S2CID 52802267.
o ^ Noè U, Chen WW, Filippone M, Hill N, Husmeier D (2017). “Inference in a Partial Differential Equations Model of Pulmonary Arterial and Venous
Blood Circulation using Statistical Emulation” (PDF). 13th International Conference on Computational Intelligence Methods for Bioinformatics and Biostatistics, Stirling, UK, 1–3 Sep 2016. Lecture Notes in Computer Science. Vol. 10477. pp. 184–198.
doi:10.1007/978-3-319-67834-4_15. ISBN 9783319678337.
o ^ “Integrative Biology – Heart Modelling”. Archived from the original on 2009-01-13. Retrieved 2010-03-17.
o ^ Trappenberg TP (2002). Fundamentals of Computational
Neuroscience. United States: Oxford University Press Inc. pp. 1. ISBN 978-0-19-851582-1.
o ^ Churchland PS, Koch C, Sejnowski TJ (March 1994). “What Is Computational Neuroscience?”. In Gutfreund H, Toulouse G (eds.). Biology And Computation: A Physicist’s
Choice. Vol. 3. pp. 25–34. ISBN 9789814504140.
o ^ Semple C (2003). SAC Phylogenetics. Oxford University Press. ISBN 978-0-19-850942-4.
o ^ “Travelling waves in a wound”. Archived from the original on 2008-06-06. Retrieved 2010-03-17.
o ^
“Leah Edelstein-Keshet: Research Interests f”. Archived from the original on 2007-06-12. Retrieved 2005-02-26.
o ^ “The mechanochemical theory of morphogenesis”. Archived from the original on 2008-06-06. Retrieved 2010-03-17.
o ^
“Biological pattern formation”. Archived from the original on 2004-11-12. Retrieved 2010-03-17.
o ^ Hurlbert SH (1990). “Spatial Distribution of the Montane Unicorn”. Oikos. 58 (3): 257–271. doi:10.2307/3545216. JSTOR 3545216.
o ^
Wooley TE, Baker RE, Maini PK (2017). “Chapter 34: Turing’s theory of morphogenesis”. In Copeland BJ, Bowen JP, Wilson R, Sprevak M (eds.). The Turing Guide. Oxford University Press. ISBN 978-0198747826.
o ^ Jump up to:a b “molecular set category”.
PlanetPhysics. Archived from the original on 2016-01-07. Retrieved 2010-03-17.
o ^ “Abstract Relational Biology (ARB)”. Archived from the original on 2016-01-07.
o ^ Rosen R (2005-07-13). Life Itself: A Comprehensive Inquiry Into the Nature, Origin,
and Fabrication of Life. Columbia University Press. ISBN 9780231075657.
o ^ “The JJ Tyson Lab”. Virginia Tech. Archived from the original on 2007-07-28. Retrieved 2008-09-10.
o ^ “The Molecular Network Dynamics Research Group”. Budapest University
of Technology and Economics. Archived from the original on 2012-02-10.
• Edelstein-Keshet L (2004). Mathematical Models in Biology. SIAM. ISBN 0-07-554950-6.
• Hoppensteadt F (1993) [1975]. Mathematical Theories of Populations: Demographics, Genetics
and Epidemics (Reprinted ed.). Philadelphia: SIAM. ISBN 0-89871-017-0.
• Renshaw E (1991). Modelling Biological Populations in Space and Time. C.U.P. ISBN 0-521-44855-7.
• Rubinow SI (1975). Introduction to Mathematical Biology. John Wiley. ISBN
• Strogatz SH (2001). Nonlinear Dynamics and Chaos: Applications to Physics, Biology, Chemistry, and Engineering. Perseus. ISBN 0-7382-0453-6.
• “Biologist Salary | Payscale”. Payscale.Com, 2021, Biologist Salary | PayScale. Accessed
3 May 2021.
Theoretical biology
• Bonner JT (1988). The Evolution of Complexity by Means of Natural Selection. Princeton: Princeton University Press. ISBN 0-691-08493-9.
• Mangel M (2006). The Theoretical Biologist’s Toolbox. Quantitative Methods
for Ecology and Evolutionary Biology. Cambridge University Press. ISBN 0-521-53748-7.
Photo credit:’]