Mathematical Biology

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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.[1] The field is sometimes called mathematical biology or biomathematics to stress the mathematical side, or theoretical biology to stress the biological side.[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.[3][4]

Mathematical biology aims at the mathematical representation and modeling of biological processes, using techniques and tools of applied mathematics. It can be useful in both theoretical and practical research. 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. This requires precise mathematical models.

Mathematics has been used in biology as early as the 13th century, when Fibonacci used the famous Fibonacci series to describe a growing population of rabbits. In the 18th century, Daniel Bernoulli applied mathematics to describe the effect of smallpox on the human population. Thomas Malthus' 1789 essay on the growth of the human population was based on the concept of exponential growth. Pierre François Verhulst formulated the logistic growth model in 1836.

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.[6]

The term \"theoretical biology\" was first used as a monograph title by Johannes Reinke in 1901, and soon after by Jakob von Uexküll in 1920. One founding text is considered to be On Growth and Form (1917) by D'Arcy Thompson,[7] and other early pioneers include Ronald Fisher, Hans Leo Przibram, Vito Volterra, Nicolas Rashevsky and Conrad Hal Waddington.[8]

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. 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.

Abstract relational biology (ARB) 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 Metabolic-Replication, or (M,R)--systems introduced by Robert Rosen in 1957-1958 as abstract, relational models of cellular and organismal organization.

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.[16][17][18]

A monograph on this topic summarizes an extensive amount of published research in this area up to 1986,[19][20][21] 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.[16][30]

Evolutionary biology has been the subject of extensive mathematical theorizing. The traditional approach in this area, which includes complications from genetics, is population genetics. Most population geneticists consider the appearance of new alleles by mutation, the appearance of new genotypes by recombination, and changes in the frequencies of existing alleles and genotypes at a small number of gene loci. 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 genetics. Ronald Fisher made fundamental advances in statistics, such as analysis of variance, via his work on quantitative genetics. Another important branch of population genetics that led to the extensive development of coalescent theory is phylogenetics. 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.

In evolutionary game theory, developed first by John Maynard Smith and George R. Price, selection acts directly on inherited phenotypes, without genetic complications. This approach has been mathematically refined to produce the field of adaptive dynamics.

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.

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. It was introduced by Anthony Bartholomay, and its applications were developed in mathematical biology and especially in mathematical medicine.[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. 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.[52]

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. Some of the simplest models in ARB are the Metabolic-Replication, or (M,R)--systems introduced by Robert Rosen in 1957-1958 as abstract, relational models of cellular and organismal organization.[54]

The 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 [55][56] have produced several models of the cell cycle simulating several organisms. 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).

The Mathematical Biology Program supports research in areas of applied and computational mathematics with relevance to the biological sciences. Successful proposals must demonstrate mathematical innovation, biological relevance and significance, and strong integration between mathematics and biology.

Some projects of interest to the Mathematical Biology Program may include development of mathematical concepts and tools traditionally seen in other disciplinary programs within the Division of Mathematical Sciences, e.g., topology, probability, statistics, computational mathematics, etc. In general, if a proposal is appropriate for review by more than one NSF program, it is advisable to contact the program officers handling each program to determine when and where the proposal should be submitted and to facilitate the review process.

The Society for Mathematical Biology was founded in 1973 to promote the development and dissemination of research and education at the interface between the mathematical and biological sciences. It does so through its meetings, awards, and publications. The Society serves a diverse community of researchers and educators in academia, in industry, and government agencies throughout the world.

The Society is governed by an Elected Board of Directors. Membership to the Society is open to scientists who share the stated purpose of the Society and who have educational, research, or practical experience in mathematical biology or in an allied scientific field. Categories of membership include student, regular, and life members.

The Journal of Mathematical Biology (JOMB) focuses on scientific advancements in mathematical modelling and analysis of biological systems. JOMB publishes the highest scientific quality peer-reviewed research with significant impact on the discipline. Submitted papers should provide new biological insights as a result of rigorous mathematical analysis or develop new mathematical concepts and tools relevant for the understanding of biological systems. Authors should provide a brief discussion of the main results to make them accessible to a wider audience, including readers with a background in biology.Biological topics include, but are not limited to, cell and developmental biology, physiology, neurobiology, genetics and population genetics, genomics, ecology, behavioural biology, evolution, epidemiology, immunology, molecular and structural biology, biofluids, biomechanics, cancer biology and medicine. Mathematical approaches cover a wide range of mathematical disciplines, such as dynamical systems, differential equations, stochastic processes, geometry and topology, logic, graph theory, game theory, continuum mechanics, as well as computational approaches.State-of-the-art survey papers, and perspective papers are welcome too. Proposals for topical collections within the journal are encouraged.All submissions to the journal, including to a topical collection, are single-blind peer-reviewed and a final decision is made by the Editors-in-Chief. 59ce067264

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