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 • 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 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 • Swarming behaviour • A mechanochemical theory of morphogenesis • Biological pattern formation • Spatial distribution
modeling using plot samples • Turing patterns 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
 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.
 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.
 Complex systems biology 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
 Algebraic biology 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) is concerned with the study of general, relational models of complex biological systems, usually abstracting out specific
morphological, or anatomical, structures.
Molecular set theory 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.
Computer models and automata theory 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, cancer modelling, neural
nets, genetic networks, abstract categories in relational biology, metabolic-replication systems, category theory applications in biology and medicine, automata theory, cellular automata, tessellation models and complete
self-reproduction, chaotic systems in organisms, relational biology and organismic theories.
 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, 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 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.
 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.
 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) 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 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.
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