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56-629: Mathematical Biology is a two-part monograph on mathematical biology first published in 1989 by the applied mathematician James D. Murray . It is considered to be a classic in the field and sweeping in scope. Part I of Mathematical Biology covers population dynamics , reaction kinetics , oscillating reactions , and reaction-diffusion equations. Part II of Mathematical Biology focuses on pattern formation and applications of reaction-diffusion equations. Topics include: predator-prey interactions, chemotaxis , wound healing , epidemic models , and morphogenesis . Since its initial publication,

112-542: A system capable of producing and maintaining itself by creating its own parts. The term was introduced in the 1972 publication Autopoiesis and Cognition: The Realization of the Living by Chilean biologists Humberto Maturana and Francisco Varela to define the self-maintaining chemistry of living cells . The concept has since been applied to the fields of cognition , neurobiology , systems theory , architecture and sociology . Niklas Luhmann briefly introduced

168-522: A boost due to the growing importance of molecular biology . Modelling physiological systems Computational neuroscience (also known as theoretical neuroscience or mathematical neuroscience) is the theoretical study of the nervous system. Ecology and evolutionary biology have traditionally been the dominant fields of mathematical biology. Evolutionary biology has been the subject of extensive mathematical theorizing. The traditional approach in this area, which includes complications from genetics,

224-563: A car factory, which uses raw materials (components) to generate a car (an organized structure) which is something other than itself (the factory). However, if the system is extended from the factory to include components in the factory's "environment", such as supply chains, plant / equipment, workers, dealerships, customers, contracts, competitors, cars, spare parts, and so on, then as a total viable system it could be considered to be autopoietic. Of course, cells also require raw materials (nutrients), and produce numerous products -waste products,

280-468: A concrete autopoietic system, however, we project this system on the space of our manipulations and make a description of this projection." Autopoiesis was originally presented as a system description that was said to define and explain the nature of living systems . A canonical example of an autopoietic system is the biological cell . The eukaryotic cell, for example, is made of various biochemical components such as nucleic acids and proteins , and

336-500: A dynamic of changes that can be recalled as sensory-motor coupling . This continuous dynamic is considered as a rudimentary form of knowledge or cognition and can be observed throughout life-forms. An application of the concept of autopoiesis to sociology can be found in Niklas Luhmann's Systems Theory , which was subsequently adapted by Bob Jessop in his studies of the capitalist state system. Marjatta Maula adapted

392-495: A final state. Starting from an initial condition and moving forward in time, a deterministic process always generates the same trajectory, and no two trajectories cross in state space. 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 . One classic work in this area is Alan Turing 's paper on morphogenesis entitled The Chemical Basis of Morphogenesis , published in 1952 in

448-473: A living system. One question that arises is about the connection between cognition seen in this manner and consciousness. The separation of cognition and consciousness recognizes that the organism may be unaware of the substratum where decisions are made. What is the connection between these realms? Thompson refers to this issue as the " explanatory gap ", and one aspect of it is the hard problem of consciousness , how and why we have qualia . A second question

504-419: 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 . Because of the complexity of the living systems , theoretical biology employs several fields of mathematics, and has contributed to the development of new techniques. Mathematics has been used in biology as early as

560-421: 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 ). To obtain these equations an iterative series of steps must be done: first

616-651: 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

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672-475: Is awareness. There are multiple criticisms of the use of the term in both its original context, as an attempt to define and explain the living, and its various expanded usages, such as applying it to self-organizing systems in general or social systems in particular. Critics have argued that the concept and its theory fail to define or explain living systems and that, because of the extreme language of self-referentiality it uses without any external reference, it

728-406: Is organized into bounded structures such as the cell nucleus , various organelles , a cell membrane and cytoskeleton . These structures, based on an internal flow of molecules and energy, produce the components which, in turn, continue to maintain the organized bounded structure that gives rise to these components. An autopoietic system is to be contrasted with an allopoietic system, such as

784-406: Is really an attempt to give substantiation to Maturana's radical constructivist or solipsistic epistemology , or what Danilo Zolo has called instead a "desolate theology". An example is the assertion by Maturana and Varela that "We do not see what we do not see and what we do not see does not exist". According to Razeto-Barry, the influence of Autopoiesis and Cognition: The Realization of

840-459: Is whether autopoiesis can provide a bridge between these concepts. Thompson discusses this issue from the standpoint of enactivism . An autopoietic cell actively relates to its environment. Its sensory responses trigger motor behavior governed by autopoiesis, and this behavior (it is claimed) is a simplified version of a nervous system behavior. The further claim is that real-time interactions like this require attention, and an implication of attention

896-569: The Philosophical Transactions of the Royal Society . 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 equations. The solution of the equations, by either analytical or numerical means, describes how the biological system behaves either over time or at equilibrium . There are many different types of equations and

952-495: The autocatalytic sets of Stuart Kauffman , similar to an earlier proposal by Freeman Dyson . All of these (including autopoiesis) found their original inspiration in Erwin Schrödinger's book What is Life? but at first they appear to have little in common with one another, largely because the authors did not communicate with one another, and none of them made any reference in their principal publications to any of

1008-450: 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

1064-544: The absence of genetic variation, are treated by the field of population dynamics . Work in this area dates back to the 19th century, and even as far as 1798 when Thomas Malthus formulated the first principle of population dynamics, which later became known as the Malthusian growth model . The Lotka–Volterra predator-prey equations are another famous example. Population dynamics overlap with another active area of research in mathematical biology: mathematical epidemiology ,

1120-508: The algebraic methods of symbolic computation to the study of biological problems, especially in genomics , proteomics , analysis of molecular structures and study of genes . 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. A monograph on this topic summarizes an extensive amount of published research in this area up to 1986, including subsections in

1176-412: 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). By means of

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1232-400: The concentrations oscillate). A better representation, which handles the large number of variables and parameters, is a bifurcation diagram using bifurcation theory . The presence of these special steady-state points at certain values of a parameter (e.g. mass) is represented by a point and once the parameter passes a certain value, a qualitative change occurs, called a bifurcation, in which

1288-515: The concept of autopoiesis in a business context. The theory of autopoiesis has also been applied in the context of legal systems by not only Niklas Luhmann, but also Gunther Teubner. Patrik Schumacher has applied the term to refer to the 'discursive self-referential making of architecture.' Varela eventually further applied autopoesis to develop models of mind, brain, and behavior called non- representationalist , enactive , embodied cognitive neuroscience , culminating in neurophenomenology . In

1344-409: The concept of autopoiesis to organizational theory . In their 1972 book Autopoiesis and Cognition , Chilean biologists Maturana and Varela described how they invented the word autopoiesis. "It was in these circumstances ... in which he analyzed Don Quixote's dilemma of whether to follow the path of arms ( praxis , action) or the path of letters ( poiesis , creation, production), I understood for

1400-614: The context of textual studies, Jerome McGann argues that texts are "autopoietic mechanisms operating as self-generating feedback systems that cannot be separated from those who manipulate and use them". Citing Maturana and Varela, he defines an autopoietic system as "a closed topological space that 'continuously generates and specifies its own organization through its operation as a system of production of its own components, and does this in an endless turnover of components ' ", concluding that "Autopoietic systems are thus distinguished from allopoietic systems, which are Cartesian and which 'have as

1456-414: The environment is extended to include cognition. Initially, Maturana defined cognition as behavior of an organism "with relevance to the maintenance of itself". However, computer models that are self-maintaining but non-cognitive have been devised, so some additional restrictions are needed, and the suggestion is that the maintenance process, to be cognitive, involves readjustment of the internal workings of

1512-410: 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

1568-444: 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 Traditional population genetic models deal with alleles and genotypes, and are frequently stochastic . Many population genetics models assume that population sizes are constant. Variable population sizes, often in

1624-416: The extracellular matrix, intracellular messaging molecules, etc. Autopoiesis in biological systems can be viewed as a network of constraints that work to maintain themselves. This concept has been called organizational closure or constraint closure and is closely related to the study of autocatalytic chemical networks where constraints are reactions required to sustain life. Though others have often used

1680-448: The field has grown rapidly from the 1960s onwards. Some reasons for this include: 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. Many of

1736-413: 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. The following is a list of mathematical descriptions and their assumptions. A fixed mapping between an initial state and

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1792-519: The first time the power of the word "poiesis" and invented the word that we needed: autopoiesis . This was a word without a history, a word that could directly mean what takes place in the dynamics of the autonomy proper to living systems." They explained that, "An autopoietic machine is a machine organized (defined as a unity) as a network of processes of production (transformation and destruction) of components which: (i) through their interactions and transformations continuously regenerate and realize

1848-655: 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. Modeling cell and molecular biology This area has received

1904-508: 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

1960-417: The interdependence between the parts of organisms. They emphasize the circularities that these interdependences lead to. Theoretical biologists developed several concepts to formalize this idea. 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. Some of

2016-399: The kinetic equation is revised and when that is not possible the wiring diagram is modified. The parameters are fitted and validated using observations of both wild type and mutants, such as protein half-life and cell size. To fit the parameters, the differential equations must be studied. This can be done either by simulation or by analysis. In a simulation, given a starting vector (list of

2072-457: The living system. With the publication of The Embodied Mind in 1991, Varela, Thompson and Rosch applied autopoesis to make non- representationalist , and enactive models of mind, brain and behavior , which further developed embodied cognitive neuroscience , later culminating in neurophenomenology . The connection of autopoiesis to cognition, or if necessary, of living systems to cognition, is an objective assessment ascertainable by observation of

2128-610: The monograph has come to be seen as a highly influential work in the field of mathematical biology. It serves as the essential text for most high level mathematical biology courses around the world, and is credited with transforming the field from a niche subject into a standard research area of applied mathematics . Mathematical and theoretical biology 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

2184-403: 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

2240-437: The network of processes (relations) that produced them; and (ii) constitute it (the machine) as a concrete unity in space in which they (the components) exist by specifying the topological domain of its realization as such a network." They described the "space defined by an autopoietic system" as "self-contained", a space that "cannot be described by using dimensions that define another space. When we refer to our interactions with

2296-422: The one produced by their environment. Autopoiesis has been proposed as a potential mechanism of abiogenesis , by which molecules evolved into more complex cells that could support the development of life. Autopoiesis is just one of several current theories of life, including the chemoton of Tibor Gánti , the hypercycle of Manfred Eigen and Peter Schuster , the ( M,R ) systems of Robert Rosen , and

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2352-520: The other theories. Nonetheless, there are more similarities than may be obvious at first sight, for example between Gánti and Rosen. Until recently there have been almost no attempts to compare the different theories and discuss them together. An extensive discussion of the connection of autopoiesis to cognition is provided by Evan Thompson in his 2007 publication, Mind in Life . The basic notion of autopoiesis as involving constructive interaction with

2408-514: 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. In particular the S and M checkpoints are regulated by means of special bifurcations called a Hopf bifurcation and an infinite period bifurcation . Autopoiesis The term autopoiesis (from Greek αὐτo- (auto)  'self' and ποίησις ( poiesis )  'creation, production') refers to

2464-614: The process of the emergence of necessary features out of chaotic contingency, the thinker of contingency's gradual self-organisation, of the gradual rise of order out of chaos." Autopoiesis can be defined as the ratio between the complexity of a system and the complexity of its environment. This generalized view of autopoiesis considers systems as self-producing not in terms of their physical components, but in terms of its organization, which can be measured in terms of information and complexity. In other words, we can describe autopoietic systems as those producing more of their own complexity than

2520-425: The product of their functioning something different from themselves ' ". Coding and markup appear allopoietic ", McGann argues, but are generative parts of the system they serve to maintain, and thus language and print or electronic technology are autopoietic systems. The philosopher Slavoj Žižek , in his discussion of Hegel , argues: "Hegel is – to use today's terms – the ultimate thinker of autopoiesis, of

2576-527: The several models and observations are combined to form a consensus diagram and the appropriate kinetic laws are chosen to write 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, afterwards the parameters of the equations (rate constants, enzyme efficiency coefficients and Michaelis constants) must be fitted to match observations; when they cannot be fitted

2632-615: 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. The eukaryotic cell cycle is very complex and has been the subject of intense study, 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 have produced several models of

2688-444: 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. Other approaches include the notion of autopoiesis developed by Maturana and Varela , Kauffman 's Work-Constraints cycles, and more recently the notion of closure of constraints. Algebraic biology (also known as symbolic systems biology) applies

2744-443: The study of infectious disease affecting populations. Various models of the spread of infections have been proposed and analyzed, and provide important results that may be applied to health policy decisions. 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

2800-470: The system in some metabolic process . On this basis it is claimed that autopoiesis is a necessary but not a sufficient condition for cognition. Thompson wrote that this distinction may or may not be fruitful, but what matters is that living systems involve autopoiesis and (if it is necessary to add this point) cognition as well. It can be noted that this definition of 'cognition' is restricted, and does not necessarily entail any awareness or consciousness by

2856-467: The term as a synonym for self-organization , Maturana himself stated he would "[n]ever use the notion of self-organization ... Operationally it is impossible. That is, if the organization of a thing changes, the thing changes". Moreover, an autopoietic system is autonomous and operationally closed, in the sense that there are sufficient processes within it to maintain the whole. Autopoietic systems are "structurally coupled" with their medium, embedded in

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2912-440: The trajectory (simulation) is heading. Vector fields can have several special points: a stable point , called a sink, that attracts in all directions (forcing the concentrations to be at a certain value), an unstable point , either a source or a saddle point , which repels (forcing the concentrations to change away from a certain value), and a limit cycle, a closed trajectory towards which several trajectories spiral towards (making

2968-473: The type of behavior that can occur is dependent on both the model and the equations used. The model often makes assumptions about the system. The equations may also make assumptions about the nature of what may occur. 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

3024-481: The values of the variables), the progression of the system is calculated by solving the equations at each time-frame in small increments. 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. 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

3080-502: The word "geometric") while resources (the environment's carrying capacity ) could only grow arithmetically. 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 , and other early pioneers include Ronald Fisher , Hans Leo Przibram , Vito Volterra , Nicolas Rashevsky and Conrad Hal Waddington . Interest in

3136-646: Was introduced by Anthony Bartholomay , and its applications were developed in mathematical biology and especially in mathematical medicine. 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. Theoretical approaches to biological organization aim to understand

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