Mathematical Biology - Journal of Pure and Applied Mathematics

In contrast to experimental biology, which deals with conducting experiments to support and validate scientific theories, mathematical and theoretical biology, also known as biomathematics, is a branch of biology that uses theoretical analysis, mathematical models, and abstractions of living organisms to investigate the principles that govern the structure, development, and behaviour of the systems. The discipline is occasionally referred to as mathematical biology, biomathematics, or theoretical biology to emphasise the mathematical and biological aspects, respectively. Even though the terms are occasionally used interchangeably, theoretical biology is more concerned with the establishment of theoretical principles for biology whereas mathematical biology is more concerned with the application of mathematical methods to analyse biological processes. The goal of mathematical biology is to represent and model biological processes mathematically using methods and equipment from applied mathematics. Both theoretical and practical research can benefit from it. When systems are described quantitatively, it is easier to simulate their behaviour and predict properties that might not be obvious to the experimenter. To do this, precise mathematical models are needed.

Early History

Since the 13th century, when Fibonacci employed the renowned Fibonacci sequence to describe an expanding population of rabbits, mathematics has been utilised to explain biological phenomena. Daniel Bernoulli used mathematics to analyse the impact of smallpox on the human population in the 18th century. The idea of exponential growth served as the foundation for Thomas Malthus' 1789 essay on the expansion of the human population. The logistic growth model was developed by Pierre François Verhulst in 1836. Fritz Müller first used a mathematical argument in evolutionary ecology to demonstrate the effectiveness of natural selection in his 1879 description of the evolutionary advantages of what is now known as Müllerian mimicry. This is true even if one takes into account Thomas Malthus's discussion of the impacts of population growth.

Areas of Research

The following subsections succinctly present various areas of specialised research in mathematical and theoretical biology as well as external links to related projects in various universities, along with a significant amount of appropriate validating references from a list of many thousands of published authors who have contributed to this field. Since it is becoming increasingly clear that the outcome of such interactions may only be understood through a combination of mathematical, logical, physical/chemical, molecular, and computational models, many of the examples provided are characterised by highly complex, nonlinear, and supercomplex mechanisms.

Abstract Relational Biology

The study of generic, relational representations of intricate biological systems—often abstracting out particular morphological or anatomical structures—is the focus of abstract relational biology (ARB). The Metabolic-Replication, or (M, R)-systems, which Robert Rosen proposed in 1957–1958 as abstract, relational models of cellular and organismal organisation, are some of the most basic models in ARB. Other strategies include Kauffman's Work-Constraints cycles, the idea of closure of constraints, and the concept of autopoiesis created by Maturana and Varela. 

Evolutionary Biology 

Mathematical Theorising: There has been a great deal of mathematical theorising about evolutionary biology. Population genetics is the conventional method in this field, which has genetic problems. 

Phylogenetics: The majority of population geneticists take into account changes in the frequencies of existing alleles and genotypes at a select few gene loci, as well as the introduction of new alleles and genotypes through mutation and recombination, respectively. Quantitative genetics is derived when infinitesimal effects at several gene loci are taken into account together with the assumption of linkage equilibrium or quasi-linkage equilibrium.

Ronald Fisher's: Ronald Fisher's work on quantitative genetics led to important developments in statistics, such as the analysis of variance. Phylogenetics is a significant subfield of population genetics that greatly influenced the evolution of coalescent theory. 

Mathematical Methods

Although the term "model" is frequently used interchangeably with the system of corresponding equations, a model of a biological system is transformed into an equation system. The equations' solutions, whether obtained analytically or numerically, represent how a biological system acts throughout time or when it is in equilibrium. The kind of behaviour that can happen depends on both the model and the equations used, and there are many different types of equations. The model frequently presupposes things about the system. The equations might also make assumptions regarding the type of potential outcomes.

Molecular Set Theory

In terms of sets of molecules and their chemical transformations represented by set-theoretical mappings between molecular sets, Molecular Set Theory (MST) is a mathematical treatment of the wide-sense chemical kinetics of biomolecular reactions. Anthony Bartholomay introduced it, and its applications in mathematical biology and particularly in mathematical medicine were developed. In a broader sense, MST is the theory of molecular categories, which are represented as set-theoretical mappings of molecular sets, and their chemical transformations. In mathematical formulations of pathological, and biochemical changes of interest to Physiology, Clinical Biochemistry, and Medicine, the theory has also influenced biostatistics and the formulation of clinical biochemistry problems.

Several models and observations must be combined to create a consensus diagram before selecting the proper kinetic laws to use in the differential equations, such as rate kinetics for stoichiometric reactions, Michaelis-Menten kinetics for enzyme-substrate reactions, and Goldbeter-Koshland kinetics for ultrasensitive transcription factors. Then the parameters of the equations (rate constants, for example) must be determined. Observations of both wild-type and mutants, such as protein half-life and cell size, are used to fit and validate the parameters. The analysis uses the properties of the equations to look into how the system behaves in relation to the values of the parameters and variables. A set of differential equations can be seen as a vector field, where each vector represents a shift in protein concentration that determines the trajectory (or speed) of the simulation. Vector fields may contain a number of unique points, including a stable point, called a sink that attracts in all directions and forces concentrations to remain at a specific value, a source or a saddle point that repels and forces concentrations to change away from a specific value, and a limit cycle, a closed trajectory that several trajectories spiral towards and causes concentrations to remain at a specific value.