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Population dynamics has traditionally been the dominant field of mathematical biology. Work in this area dates back to the 19th century. The 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 epidemiology, the study of infectious disease affecting populations. Various models of viral spread have been proposed and analyzed, and provide important results that may be applied to health policy decisions.

Molecular set theory was introduced by Anthony Bartholomay, and its applications were developed in mathematical biology and especially in Mathematical Medicine.[30] 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. 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.[31][32]

A monograph on this topic summarizes an extensive amount of published research in this area up to 1987,[10] 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, cancer modelling, neural nets, genetic networks, abstract relational biology, metabolic-replication systems, category theory[11] applications in biology and medicine,[12] automata theory,cellular automata, tessallation models[13][14] and complete self-reproduction, chaotic systems in organisms, relational biology and organismic theories.[15][16] This published report also includes 390 references to peer-reviewed articles by a large number of authors.[17][18][19]

Modeling cell and molecular biology

This area has received a boost due to the growing importance of molecular biology.[20]
Mechanics of biological tissues[21]
Theoretical enzymology and enzyme kinetics
Cancer modelling and simulation [22][23]
Modelling the movement of interacting cell populations[24]
Mathematical modelling of scar tissue formation[25]
Mathematical modelling of intracellular dynamics[26]
Mathematical modelling of the cell cycle[27]

Modelling physiological systems
Modelling of arterial disease [28]
Multi-scale modelling of the heart [29]

Several areas of specialized research in mathematical and theoretical biology[4][5][6][7][8][9] 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.

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 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.
For use of statistics in biology, see Biostatistics.
For use of basic arithmetics in biology, see relevant topic, such as Serial dilution.

Mathematical biology is also called theoretical biology,[1] and sometimes biomathematics. It includes at least four major subfields: biological mathematical modeling, relational biology/complex systems biology (CSB), bioinformatics and computational biomodeling/biocomputing. It is an interdisciplinary academic research field with a wide range of applications in biology, medicine[2] and biotechnology.[3]

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.

In theoretical biology, kinetic logic is a kind of temporal logic that allows one to describe tendencies in a regulatory system to evolve based on its current state, and is particularly useful in the study of biological feedback, whether homeostatic or epigenetic. In general, kinetic logic avoids continuous descriptions that use differential equations, instead preferring symbolic descriptions where the elements of the state are approximated by Boolean variables and functions.

Kinetic logic was proposed by the French biologist René Thomas.

The Journal of Theoretical Biology is a scientific journal about theoretical biology; dealing with theoretical issues, as well as mathematical and computational aspects of biology. Some research areas covered by the papers published in the journal are population genetics, morphogenesis, evolutionary biology and immunology. It is published by Elsevier.

The most-cited paper in the journal is W.D. Hamilton’s 1964 paper on kin selection entitled The Genetical Evolution of Social Behavior.

Recent experiments have proven that alterations in metabolism, immune cell function, cell division, and cell attachment all occur in the hypogravity of space. For example, after a matter of days in microgravity (< 10-3 g), human immune cells were unable to differentiate into mature cells. One of the large implications of this is that if certain cells cannot differentiate in space, organisms may not be able to reproduce successfully after exposure to
Scientists believe that the stress associated with space flight is responsible for the inability of some cells to differentiate. These stresses can alter metabolic activities and can disturb the chemical processes in living organisms. A specific example would be that of bone cell growth. Microgravity impedes the development of bone cells. Bone cells must attach themselves to something shortly after development and will die if they cannot. Without the downward pull of a gravitational force on these bone cells, they float around randomly and eventually die off. This suggests that the direction of gravity may give the cells clues as to where to attach themselves.

Every day the realization of space habitation becomes closer, and even today space stations exist and are home to long-term, though not yet permanent residents. Because of this there is a growing scientific interest in how changes in the gravitational field influence different aspects of the physiology of living organisms, especially mammals since these results can normally be closely related to the expected effects on humans. All current research in this field can be classified into two groups.[3]

The first group consists of the experiments that involve gravitational fields of less than one g, termed hypogravity. All space travel is done in hypogravity, and effective gravitational fields on any space station without Artificial gravity are on the order of hypogravity, and therefore the understanding of the effects of hypogravity on the human body is necessary for prolonged space travel and colonization.

The second group consist of those involving gravitational fields of more than one g, termed hypergravity. For brief periods during take-off and landing of space craft astronauts are under the influence of hypergravity. Understanding the effects of hypergravity are also necessary if colonization of planets larger than the Earth is ever to take place.