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Mathematical biology

 
Sci-Tech Dictionary: mathematical biology
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(¦math·ə¦mad·ə·kəl bī′äl·ə·jē)

(biology) A discipline that encompasses all applications of mathematics, computer technology, and quantitative theorizing to biological systems, and the underlying processes within the systems.

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Sci-Tech Encyclopedia: Mathematical biology
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The application of mathematics to biological systems. Mathematical biology spans all levels of biological organization and biological function, from the configuration of biological macromolecules to the entire ecosphere over the course of evolutionary time.

The influence of physics on mathematical biology has been twofold. On the one hand, organisms simply are material systems, and presumably can be analyzed in the same terms as any other material system. Reductionism, the theory that biological processes find their resolution in the particularities of physics, finds its practical embodiment in biophysics. Thus, one of the roots of mathematical biology is what was originally called mathematical biophysics. On the other hand, other early investigations in mathematical biology, such as population dynamics (mathematical ecology), exploited the form of such analyses, such as using differential rate equations, but they expressed their analyses in strictly biological terms. Such approaches were guided by analogy with mathematical physics rather than by reduction to physics and so rest on the form rather than the substance of physics. See also Biophysics; Physics.

Both of these approaches are important, especially since organisms possess characteristics that have no obvious counterpart in inorganic systems. As a result, mathematical biology has acquired an independent and unique character. In several important cases, these characteristics have required a reconsideration of physics itself, as in the impact of open systems on classical thermodynamics.

Surrogacy and models

The idea that something can be learned about a system by studying a different system, or surrogate, is central to all science. The relation between a system and its surrogates is embodied in the concept of a model. The basic idea of mathematical biology is that an appropriate formal or mathematical system may similarly be used as a surrogate for a biological system. The use of mathematical models offers possibilities that transcend what can be done on the basis of observation and experiment alone. See also Model theory.

For example, morphological differences between related species can be made to disappear by means of relatively simple coordinate transformations of the space in which the forms are embedded. Surrogacy explicitly becomes a matter of intertransformability, or similarity, and what is true for morphology also holds true for other functional relationships that are characteristic of organisms, whether they be chemical, physical, or evolutionary. These assertions of surrogacy and modeling can be restated: closely related implies similar. This is a nontrivial assertion: “closely related” is a metric relation pertaining to genotypes, whereas “similar” is an equivalence relation based on phenotypes. It is the similarity relation between phenotypes that provides the basis for surrogacy. Thus the question immediately arises: given a genotype, how far can it be varied or changed or mutated, and still preserve similarity

Such questions fall mathematically into the province of stability theory, particularly structural stability. Under very general conditions, there exist many genomes that are unstable (bifurcation points) in the sense that however high a degree of metric approximation is chosen, the associated phenotypes may be dissimilar, that is, not intertransformable. That observation by R. Thom provides the basis for his theory of catastrophes and demonstrates the complexity of the surrogacy relationship. The fundamental importance of such ideas for phenomena of development, for evolution (particularly for macroevolution), and for the extrapolation of data from one species to another, or the relation between health and disease, is evident. See also Catastrophe theory; Macroevolution.

Metaphor

A closely related group of ideas that are characteristic of mathematical biology may be described as metaphoric. One example of a metaphoric approach is the study of brain activities through the application of the properties of neural networks, that is, networks of interconnected boolean (binary-state) switches. Appropriately configured switching networks are known to exhibit behaviors that are analogous to those that characterize the brain, such as learning, memory, and discrimination. That is, networks of neuronlike units can automatically manifest brainlike behaviors and can be regarded as metaphorical brains. Such boolean neural nets also underlie digital computation, a relationship which is explored in the hybrid area of artificial intelligence. The same mathematical formulation of switching networks arises in genetic and developmental phenomena, such as the concept of operon, and in other physiological systems, such as the immune system. See also Artificial intelligence; Neural network; Operon.

Another important example of metaphor in biology is morphogenesis, or pattern generation, through the coupling of chemical reactions with physical diffusion. Chemical reactions tend to make systems heterogeneous, diffusion tends to smooth them out, and combining the two can lead to highly complex behaviors. Since reactions and diffusions typically occur together in biological systems, exploring the general properties of such systems can illuminate pattern generation in general.

Such ideas turn out to be closely related to those of bifurcation and catastrophe and have a profound impact on physics itself, since they are inherently associated with systems that are thermodynamically open and hence completely outside the realm of classical thermodynamics. The behavior of such open systems can be infinitely more complicated than those that are commonly explored in physics. Open systems may possess large numbers of stable and unstable steady states of various types, as well as more complicated oscillatory steady-state behaviors (limit cycles) and still more general behaviors collectively called chaotic. Changes in initial conditions or in environmental circumstances can result in dramatic switching (bifurcations) between these modes of behavior. See also Chaos.

Applications

Perhaps the biotechnology that has affected everyone most directly is medicine. Medicine can be regarded as a branch of control theory, geared to the maintenance or restoration of a state of health. It is unique in that the systems needed for control are themselves control systems that are far more intricate and complex than any that can be fabricated. In addition to the light it sheds on the processes needed for control, mathematical biology is indispensable for designing the controls themselves and for assessing their costs, benefits, safety, and efficacy.

In general, the object of any theory of control is to produce an algorithm, or protocol, that will achieve optimal results. Mathematical biology allows one to relate systems of different characters through the exploitation of their mathematical commonalities. Biology has many optimal designs and optimal controls, which are the products of biological evolution through natural selection. The design of optimal therapies in medicine is analogous to the generation of optimal organisms. Thus the mathematical theory appropriate for analyzing one discipline of biology, such as evolution, itself becomes transmuted into a theory of control in an entirely different realm. The same holds true for other biotechnologies, such as the efficient exploitation of biological populations. See also Mathematical ecology.

Wikipedia: Mathematical biology
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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.

Contents

Importance

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.

Areas of research

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.

Computer models and automata theory

Main article: Modelling biological systems

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

Molecular set theory

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]

Population dynamics

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.

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 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 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.

Mathematical biophysics

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.

Deterministic processes (dynamical systems)

A fixed mapping between an initial state and a final state. Starting from an initial condition and moving forward in time, a deterministic process will always generate the same trajectory and no two trajectories cross in state space.

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.

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.

Phylogenetics

Phylogenetics is an area that deals with the reconstruction and analysis of phylogenetic (evolutionary) trees and networks based on inherited characteristics[38]

Model example: the cell cycle

Main article: Cellular model

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 [39][40] have produced several models of the cell cycle simulating several organisms. They have recently produced a generic eukaryotic cell cycle model which 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 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 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 Michealis constants) must be fitted to match observations; when they cannot be fitted 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.
In order to fit the parameters the differential equations need to be studied. This can be done either by simulation or by analysis.
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 small increments.

Cell cycle bifurcation diagram.jpg

In analysis, the proprieties of the equations are used to investigate the behavior of the system depending of 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 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 the concentrations oscillate).
A better representation which can handle the large number of variables and parameters is called a bifurcation diagram(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 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. In particular the S and M checkpoints are regulated by means of special bifurcations called a Hopf bifurcation and an infinite period bifurcation.

Mathematical/theoretical biologists

Mathematical, theoretical and computational biophysicists

See also

For use of statistics in biology, see Biostatistics.
For use of basic arithmetics in biology, see relevant topic, such as Serial dilution.
Nuvola apps edu mathematics blue-p.svg Mathematics portal

Mathematical and Theoretical Biology Societies and Institutes

References

  • S.H. Strogatz, Nonlinear dynamics and Chaos: Applications to Physics, Biology, Chemistry, and Engineering. Perseus, 2001, ISBN 0-7382-0453-6
  • N.G. van Kampen, Stochastic Processes in Physics and Chemistry, North Holland., 3rd ed. 2001, ISBN 0-444-89349-0
  • I. C. Baianu., Computer Models and Automata Theory in Biology and Medicine., Monograph, Ch.11 in M. Witten (Editor), Mathematical Models in Medicine, vol. 7., Vol. 7: 1513-1577 (1987),Pergamon Press:New York, (updated by Hsiao Chen Lin in 2004[61],[62],[63] ISBN 0080363776 [64].
  • P.G. Drazin, Nonlinear systems. C.U.P., 1992. ISBN 0-521-40668-4
  • L. Edelstein-Keshet, Mathematical Models in Biology. SIAM, 2004. ISBN 0-07-554950-6
  • G. Forgacs and S. A. Newman, Biological Physics of the Developing Embryo. C.U.P., 2005. ISBN 0-521-78337-2
  • A. Goldbeter, Biochemical oscillations and cellular rhythms. C.U.P., 1996. ISBN 0-521-59946-6
  • L.G. Harrison, Kinetic theory of living pattern. C.U.P., 1993. ISBN 0-521-30691-4
  • F. Hoppensteadt, Mathematical theories of populations: demographics, genetics and epidemics. SIAM, Philadelphia, 1975 (reprinted 1993). ISBN 0-89871-017-0
  • D.W. Jordan and P. Smith, Nonlinear ordinary differential equations, 2nd ed. O.U.P., 1987. ISBN 0-19-856562-3
  • J.D. Murray, Mathematical Biology. Springer-Verlag, 3rd ed. in 2 vols.: Mathematical Biology: I. An Introduction, 2002 ISBN 0-387-95223-3; Mathematical Biology: II. Spatial Models and Biomedical Applications, 2003 ISBN 0-387-95228-4.
  • E. Renshaw, Modelling biological populations in space and time. C.U.P., 1991. ISBN 0-521-44855-7
  • S.I. Rubinow, Introduction to mathematical biology. John Wiley, 1975. ISBN 0-471-74446-8
  • L.A. Segel, Modeling dynamic phenomena in molecular and cellular biology. C.U.P., 1984. ISBN 0-521-27477-X
  • L. Preziosi, Cancer Modelling and Simulation. Chapman Hall/CRC Press, 2003. ISBN 1-58488-361-8.

Lists of references

  • A general list of Theoretical biology/Mathematical biology references, including an updated list of actively contributing authors[65].
  • A list of references for applications of category theory in relational biology[66].
  • An updated list of publications of theoretical biologist Robert Rosen[67]

External

Notes: inline and online

  1. ^ Mathematical Biology and Theoretical Biophysics-An Outline: What is Life? http://planetmath.org/?op=getobj&from=objects&id=10921
  2. ^ http://www.kli.ac.at/theorylab/EditedVol/W/WittenM1987a.html
  3. ^ http://en.scientificcommons.org/1857372
  4. ^ http://www.kli.ac.at/theorylab/index.html
  5. ^ http://www.springerlink.com/content/w2733h7280521632/
  6. ^ http://en.scientificcommons.org/1857371
  7. ^ http://cogprints.org/3687/
  8. ^ "Research in Mathematical Biology". Maths.gla.ac.uk. http://www.maths.gla.ac.uk/research/groups/biology/kal.htm. Retrieved 2008-09-10. 
  9. ^ http://acube.org/volume_23/v23-1p11-36.pdf J. R. Junck. Ten Equations that Changed Biology: Mathematics in Problem-Solving Biology Curricula, Bioscene, (1997), 1-36
  10. ^ http://en.scientificcommons.org/1857371
  11. ^ http://planetphysics.org/encyclopedia/BibliographyForCategoryTheoryAndAlgebraicTopologyApplicationsInTheoreticalPhysics.html
  12. ^ http://planetphysics.org/encyclopedia/BibliographyForMathematicalBiophysicsAndMathematicalMedicine.html
  13. ^ Modern Cellular Automata by Kendall Preston and M. J. B. Duff http://books.google.co.uk/books?id=l0_0q_e-u_UC&dq=cellular+automata+and+tessalation&pg=PP1&ots=ciXYCF3AYm&source=citation&sig=CtaUDhisM7MalS7rZfXvp689y-8&hl=en&sa=X&oi=book_result&resnum=12&ct=result
  14. ^ http://mathworld.wolfram.com/DualTessellation.html
  15. ^ Baianu, I. C. 1987, Computer Models and Automata Theory in Biology and Medicine., in M. Witten (ed.),Mathematical Models in Medicine, vol. 7., Ch.11 Pergamon Press, New York, 1513-1577. http://cogprints.org/3687/
  16. ^ http://www.kli.ac.at/theorylab/EditedVol/W/WittenM1987a.html
  17. ^ http://www.springerlink.com/content/w2733h7280521632/
  18. ^ Currently available for download as an updated PDF: http://cogprints.ecs.soton.ac.uk/archive/00003718/01/COMPUTER_SIMULATIONCOMPUTABILITYBIOSYSTEMSrefnew.pdf
  19. ^ http://planetphysics.org/encyclopedia/BibliographyForMathematicalBiophysics.html
  20. ^ "Research in Mathematical Biology". Maths.gla.ac.uk. http://www.maths.gla.ac.uk/research/groups/biology/kal.htm. Retrieved 2008-09-10. 
  21. ^ http://www.maths.gla.ac.uk/~rwo/research_areas.htm
  22. ^ http://www.springerlink.com/content/71958358k273622q/
  23. ^ http://calvino.polito.it/~mcrtn/
  24. ^ http://www.ma.hw.ac.uk/~jas/researchinterests/index.html
  25. ^ http://www.ma.hw.ac.uk/~jas/researchinterests/scartissueformation.html
  26. ^ http://www.sbi.uni-rostock.de/dokumente/p_gilles_paper.pdf
  27. ^ http://mpf.biol.vt.edu/Research.html
  28. ^ http://www.maths.gla.ac.uk/~nah/research_interests.html
  29. ^ http://www.integrativebiology.ox.ac.uk/heartmodel.html
  30. ^ http://planetphysics.org/encyclopedia/CategoryOfMolecularSets2.html
  31. ^ Representation of Uni-molecular and Multimolecular Biochemical Reactions in terms of Molecular Set Transformations http://planetmath.org/?op=getobj&from=objects&id=10770
  32. ^ http://planetphysics.org/encyclopedia/CategoryOfMolecularSets2.html
  33. ^ http://www.maths.ox.ac.uk/~maini/public/gallery/twwha.htm
  34. ^ http://www.math.ubc.ca/people/faculty/keshet/research.html
  35. ^ http://www.maths.ox.ac.uk/~maini/public/gallery/mctom.htm
  36. ^ http://www.maths.ox.ac.uk/~maini/public/gallery/bpf.htm
  37. ^ http://links.jstor.org/sici?sici=0030-1299%28199008%2958%3A3%3C257%3ASDOTMU%3E2.0.CO%3B2-S&size=LARGE&origin=JSTOR-enlargePage
  38. ^ Charles Semple (2003), Phylogenetics, Oxford University Press, ISBN 9780198509424
  39. ^ "The JJ Tyson Lab". Virginia Tech. http://mpf.biol.vt.edu/Tyson%20Lab.html. Retrieved 2008-09-10. 
  40. ^ "The Molecular Network Dynamics Research Group". Budapest University of Technology and Economics. http://cellcycle.mkt.bme.hu/. 
  41. ^ http://planetphysics.org/encyclopedia/AbstractRelationalBiologyARB.html
  42. ^ http://www.kli.ac.at/theorylab/EditedVol/M/MatsunoKDose_84.html
  43. ^ Baianu, I. C. 1987, Computer Models and Automata Theory in Biology and Medicine., in M. Witten (ed.),Mathematical Models in Medicine, vol. 7., Ch.11 Pergamon Press, New York, 1513-1577. http://www.springerlink.com/content/w2733h7280521632/
  44. ^ http://www.springerlink.com/content/v1rt05876h74v607/?p=2bd3993c33644512ba7069ed7fad0046&pi=1
  45. ^ http://www.springerlink.com/content/j7t56r530140r88p/?p=2bd3993c33644512ba7069ed7fad0046&pi=3
  46. ^ http://www.springerlink.com/content/98303486x3l07jx3/
  47. ^ Robert Rosen, Dynamical system theory in biology. New York, Wiley-Interscience (1970) ISBN 0471735507 http://www.worldcat.org/oclc/101642
  48. ^ http://www.springerlink.com/content/j7t56r530140r88p/?p=2bd3993c33644512ba7069ed7fad0046&pi=3
  49. ^ http://cogprints.org/3674/
  50. ^ http://cogprints.org/3829/
  51. ^ http://www.ncbi.nlm.nih.gov/pubmed/4327361
  52. ^ http://www.springerlink.com/content/98303486x3l07jx3/
  53. ^ http://www.kli.ac.at/theorylab/ALists/Authors_R.html
  54. ^ Organisms as Super-complex Systems http://planetmath.org/?op=getobj&from=objects&id=10890
  55. ^ http://www.springerlink.com/content/98303486x3l07jx3/
  56. ^ http://planetmath.org/encyclopedia/SupercategoriesOfComplexSystems.html
  57. ^ http://planetmath.org/?method=l2h&from=objects&name=NaturalTransformationsOfOrganismicStructures&op=getobj
  58. ^ http://www.kli.ac.at/theorylab/ALists/Authors_R.html
  59. ^ http://www.kli.ac.at/theorylab/index.html
  60. ^ http://www.worldcat.org/oclc/101642
  61. ^ http://cogprints.org/3718/1/COMPUTER_SIMULATIONCOMPUTABILITYBIOSYSTEMSrefnew.pdf
  62. ^ http://www.springerlink.com/content/w2733h7280521632/
  63. ^ http://www.springerlink.com/content/n8gw445012267381/
  64. ^ http://www.bookfinder.com/dir/i/Mathematical_Models_in_Medicine/0080363776/
  65. ^ http://www.kli.ac.at/theorylab/index.html
  66. ^ http://planetmath.org/?method=l2h&from=objects&id=10746&op=getobj
  67. ^ Publications list for Robert Rosen http://www.people.vcu.edu/~mikuleck/rosen.htm

External links

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