# mathematical biology and theoretical biophysics of DNA

## 1 Introduction

Mathematical biology (also known as theoretical biology) is the study of biological principles and laws, together with the formulation of mathematical models– and also the logical and mathematical representation [3]– of complex biological systems at all levels of biological organization, from the quantum/molecular level to the physiological, systemic and the whole organism levels.

## 2 Mathematical biophysics

### 2.1 History

Mathematical biophysics has dominated for over half a century developments in mathematical biology as theoretical or mathematical physicists have expanded their interests to applying mathematical and physical concepts to studying living organisms and in repeated attempts to ‘define life itself’ [1, 28].

A prominent early example was the famous Erwin $Schr\"{o}dinger$ ’s book (published in 1945 in Cambridge, UK) entitled suggestively “What is Life?”, and that was perhaps too critically re-evaluated a decade ago by Robert Rosen. This interesting and concise book appears to have inspired a decade later the discovery of the double helical, molecular structures of A- and B- DNA crystals/paracrystals ([26, 9, 12, 25]) by Maurice Wilkins, Rosalind Franklin, Francis Crick and James D. Watson, with the first two (bio)physical chemists working at that time with X-ray Diffraction of DNA crystals at http://www.kcl.ac.uk/schools/biohealth/graduate/taught/molbiophysics/King’s College in London, (see also the websites about http://www.kcl.ac.uk/college/history/people/franklin_wilkins.htmlRosalind Franklin and Maurice Wilkins), and the last two researchers working at http://www.phy.cam.ac.uk/The Cavendish Laboratory of the University of Cambridge(UK), (see also http://www.admin.cam.ac.uk/news/dp/2004072903related news at, and also the http://www.pom.cam.ac.uk/initiative.htmlnew Biology and Physics of Medicine Laboratory at http://www.pom.cam.ac.uk/initiative.htmlThe Cavendish). With the notable exception of Rosalind Franklin and Robert Rosen, the other three mathematical and experimental biophysicists became http://nobelprize.org/nobel_prizes/medicine/laureates/1962/Nobel Laureates in Physiology and Medicine.

Notably DNA configurations in vivo include a significant amount of dynamic, partial disorder and may be defined at best as paracrystals ([26, 9, 12]), a fact which has important consequences for functional biology and in vivo molecular genetics. Moreover, other structures (such as Z-DNA) were discovered in certain organisms, and other configurations were found under physiological conditions (see, for example, the excellent, DNA structure representations rendered by computers on pp. 852-854 in Voet and Voet, 1995 [27], as well as a recent update review [25] and earlier generalizations [14, 15]), such as the http://www.phy.cam.ac.uk/research/bss/molbiophysics.phpDNA G-quadruplexes that can control gene transcription and translation - especially in cancers.

Erwin $Schr\"{o}dinger$’s fundamental contribution to Quantum Mechanics preceded the others discussed in the previous paragraph by more than two decades when he formulated the fundamental equations of Quantum Mechanics which bear his name, and modestly called the operator appearing in the $Schr\"{o}dinger$ equations the “Hamiltonian operator” (http://planetmath.org/HamiltonianOperatorOfAQuantumSystem) - a term universally employed in the Theoretical and Mathematical Physics literature that now bears the name of the distinguished Irish physicist, Sir William Rowan Hamilton. Hamilton is now also considered to be one of the world’s greatest mathematicians (see for example, his introduction of the concept of quaternions in 1835), and he was also the first foreign Member to be elected to the US National Academy of Sciences in 1865. Subsequently, Schrödinger was awarded a Nobel Prize for his fundamental, theoretical (and mathematical) physics contribution by the Stockholm Nobel Committee, and soon thereafter in 1941 became the Director of the (Dublin) Institute for Advanced Studies (DIAS) in Ireland, instead of joining Albert Einstein on the staff at Princeton’s Institute for Advanced Studies.

## 3 Recent developments

Robert Rosen (1937-1998) was a prominent relational biologist who completed his PhD studies with Nicolas Rashevsky, the former Head of the Committee for Mathematical Biology at the University of Chicago, USA, with a Thesis on Relational Biology (Metabolic-Replication Systems, or $(M,R)$-systems). His publications (see bibliography) include an impressive number of volumes and textbooks on Theoretical Biology, Relational Biology, Anticipation, Ageing, Complex Dynamical Systems in Biology, (Bio) Chemical Morphogenesis and Quantum Genetics. He also reported in 1958 the first abstract representation of living organisms in special, small categories of sets called categories of metabolic–replication systems, or category of $(M,R)$-systems (http://planetmath.org/CategoryOfMRSystems3).

To quote Robert Rosen:

“Ironically, the idea that life requires an explanation is a relatively new one. To the ancients life simply was; it was a given; a first principle…”

One might add also that to most biologists “Life” is still a given, but something that might be ‘explained by reduction to genes, nucleic acids, enzymes and small biomolecules’, i.e. some sort of ordered ‘bag’ of biochemicals mostly filled with aqueous solutions inside selective biomembranes, etc. Robert Rosen’s viewpoint was quite different from this: he saw life as a dynamic, relational pattern in categories of metabolic-repair (open) systems characterized by flows–relational/material, energetic and informational processes– perhaps closer to the injunction by Heraclitus of “panta rhei”-everything flows, but with the very important addition that life flows in a uniquely complex relational pattern that is observed only in living systems, thus perhaps uniquely defining Life as a special, super-complex process ([8]. Once life stops– even though the material structure is still there– the essential relational flow (related to energetic, informational as well as material) patterns are gone forever, with the possible exceptions of the ‘raising from the dead in the Egyptian myths about Osiris’ , and also in certain well-known sections of the New Testament.

## References

• 1 Erwin Schroedinger.1945. What is Life?. Cambridge University Press: Cambridge (UK).
• 2 Nicolas Rashevsky.1954, Topology and life: In search of general mathematical principles in biology and sociology, Bull. Math. Biophys. 16: 317-348.
• 3 Nicolas Rashevsky. 1965. Models and Mathematical Principles in Biology. In: Waterman/Morowitz, Theoretical and Mathematical Biology, pp. 36-53.
• 4 Rosalind E. Franklin and R.G. Gosling. 1953. Evidence for 2-chain helix in crystalline structure of sodium deoxyribonucleate (DNA). Nature 177: 928-930.
• 5 Wilkins, M.H.F. et al. 1953. Helical structure of crystalline deoxypentose nucleic acid (DNA). Nature 172: 759-762.
• 6 Francis H.C. Crick. 1953. Fourier transform of a coiled coil. Acta Cryst. 6: 685-687
• 7 H. R. Wilson. 1966. Diffraction of X-rays by Proteins, Nucleic Acids and Viruses. London: Arnold.
• 8 I. C. Baianu, J. F. Glazebrook, R. Brown and G. Georgescu.: Complex Nonlinear Biodynamics in Categories, Higher dimensional Algebra and Łukasiewicz-Moisil Topos: Transformation of Neural, Genetic and Neoplastic Networks, Axiomathes, 16: 65-122(2006). http://www.bangor.ac.uk/ mas010/pdffiles/Axio7complx_Printedk7_v17p223_fulltext.pdfavailable here as PDF
• 9 I.C. Baianu. 1974. Ch.4 in Structural Studies by X-ray Diffraction and Electron Microscopy of Erythrocite and Bacterial Plasma Membranes. PhD Thesis, London: a University of London Library publication.
• 10 Baianu, I.C.: 1977, A Logical Model of Genetic Activities in Łukasiewicz Algebras: The Non-linear Theory. Bulletin of Mathematical Biology, 39: 249-258.
• 11 I.C. Baianu. 1978. X-ray Scattering by Partially Disordered Membrane Lattices. Acta Crystall. A34: 731-753. (paper contributed from The Cavendish Laboratory, Cambridge in 1979).
• 12 I.C. Baianu. 1980. Structural Order and Partial Disorder in Biological Systems. Bull. Math. Biol., 42: 186-191. (paper contributed from The Cavendish Laboratory, Cambridge in 1979).
• 13 I. C. Baianu, C. Critchley, Govindjee, AND H. S. Gutowsky. 1984. NMR study of chloride ion interactions with thylakoid membranes Proced. Natl. Acad. Sci. USA (Biophysics)., Vol. 81, pp. 3713–3717, June 1984, (Key words: photosynthesis/oxygen evolution/chloride binding/chlorine–35/halophytes).
• 14 Baianu, I. C.: 1986-1987a, 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; available downloads as: CERN Preprint No. EXT-2004-072- http://doe.cern.ch//archive/electronic/other/ext/ext-2004-072.pdfCERN Preprint as PDF, or http://en.scientificcommons.org/1857371as external html document .
• 15 Baianu, I. C.: 1987b, Molecular Models of Genetic and Organismic Structures, in Proceed. Relational Biology Symp. Argentina; http://doc.cern.ch/archive/electronic/other/ext/extusers/2004 67/MolecularModels-ICB3.docCERN Preprint No.EXT-2004-067.
• 16 Baianu, I. C.: 1983, Natural Transformation Models in Molecular Biology., in Proceedings of the SIAM Natl. Meet., Denver,CO.;
http://cogprints.org/3675/Eprint: and http://cogprints.org/3675/0l/Naturaltransfmolbionu6.pdfhtml document.
• 17 Baianu, I.C.: 1984, A Molecular-Set-Variable Model of Structural and Regulatory Activities in Metabolic and Genetic Networks, FASEB Proceedings 43, 917.
• 18 Baianu, I.C.: 2004a. Łukasiewicz-Topos Models of Neural Networks, Cell Genome and Interactome Nonlinear Dynamic Models (2004). Eprint. Cogprints–Sussex Univ.
• 19 Baianu, I.C.: 2004b Łukasiewicz-Topos Models of Neural Networks, Cell Genome and Interactome Nonlinear Dynamics). CERN Preprint EXT-2004-059. Health Physics and Radiation Effects (June 29, 2004).
• 20 Baianu, I. C., Glazebrook, J. F. and G. Georgescu: 2004, Categories of Quantum Automata and N-Valued Łukasiewicz Algebras in Relation to Dynamic Bionetworks, (M,R)–Systems and Their Higher Dimensional Algebra,
http://www.ag.uiuc.edu/fs401/QAuto.pdfAbstract and Preprint of Report as PDF or as http://www.medicalupapers.com/quantum+automata+math+categories+baianu/an html document.
• 21 Baianu, I. C.: 2004b, Quantum Interactomics and Cancer Mechanisms, http: bioline.utsc.utoronto.ca/archive/00001978/01/
Quantum Interactomics In Cancer--Sept13k4E-- cuteprt.pdf
Preprint No. 00001978.
• 22 Baianu, I. C.: 2006, Robert Rosen’s Work and Complex Systems Biology, Axiomathes 16(1–2):25–34.
• 23 Baianu I. C., Brown R., Georgescu G. and J. F. Glazebrook: 2006, Complex Nonlinear Biodynamics in Categories, Higher Dimensional Algebra and Łukasiewicz-Moisil Topos: Transformations of Neuronal, Genetic and Neoplastic Networks., Axiomathes, 16 Nos. 1-2: 65-122.
• 24 Baianu, I.C., R. Brown and J.F. Glazebrook. : 2007, Categorical Ontology of Complex Spacetime Structures: The Emergence of Life and Human Consciousness, Axiomathes, 17: 35-168.
• 25 Baianu, I.C. et al. 2009. http://planetphysics.org/?op=getobj&from=books&id=220“DNA Molecular Structure, Dynamics and Spectroscopy.” (Free GNUL download.)
• 26 R. Hosemann and S. N. Bagchi. 1962. Direct Analysis of Diffraction by Matter. Amsterdam: North Holland.
• 27 D. Voet and J.G. Voet. 1995. Biochemistry. 2nd Edition, New York, Chichester, Brisbone, Toronto, Singapore: J. Wiley and Sons, INC., 1,361 pages, over 3,000 high-resolution molecular models in color – (an excellently illustrated textbook)
• 28 Robert Rosen. 1997 and 2002. Essays on Life Itself.
• 29 Rosen, R.: 1958a, A Relational Theory of Biological Systems Bulletin of Mathematical Biophysics 20: 245-260.
• 30 Rosen, R.: 1958b, The Representation of Biological Systems from the Standpoint of the Theory of Categories., Bulletin of Mathematical Biophysics 20: 317-341.
• 31 Rosen, R. 1960. A quantum-theoretic approach to genetic problems. Bulletin of Mathematical Biophysics 22: 227-255.
• 32 Rosen, R.: 1987, On Complex Systems, European Journal of Operational Research 30, 129-134.
• 33 Rosen,R. 1970, Dynamical Systems Theory in Biology. New York: Wiley Interscience.
• 34 Rosen,R. 1970, Optimality Principles in Biology, New York and London: Academic Press.
• 35 Rosen,R. 1978, Fundamentals of Measurement and Representation of Natural Systems, Elsevier Science Ltd,
• 36 Rosen,R. 1985, Anticipatory Systems: Philosophical, Mathematical and Methodological Foundations. Pergamon Press.
• 37 Rosen,R. 1991, Life Itself: A Comprehensive Inquiry into the Nature, Origin, and Fabrication of Life, Columbia University Press
• 38 Ehresmann, C.: 1984, Oeuvres complètes et commentées: Amiens, 1980-84, edited and commented by Andrée Ehresmann.
• 39 Ehresmann, A. C. and J.-P. Vanbremersch: 2006, The Memory Evolutive Systems as a Model of Rosen’s Organisms, in Complex Systems Biology, I.C. Baianu, Editor, Axiomathes 16 (1–2), pp. 13-50.
• 40 Eilenberg, S. and Mac Lane, S.: 1942, Natural Isomorphisms in Group Theory., American Mathematical Society 43: 757-831.
• 41 Eilenberg, S. and Mac Lane, S.: 1945, The General Theory of Natural Equivalences, Transactions of the American Mathematical Society 58: 231-294.
• 42 Elsasser, M.W.: 1981, A Form of Logic Suited for Biology., In: Robert, Rosen, ed., Progress in Theoretical Biology, Volume 6, Academic Press, New York and London, pp 23-62.
 Title mathematical biology and theoretical biophysics of DNA Canonical name MathematicalBiologyAndTheoreticalBiophysicsOfDNA Date of creation 2013-03-22 18:18:09 Last modified on 2013-03-22 18:18:09 Owner bci1 (20947) Last modified by bci1 (20947) Numerical id 110 Author bci1 (20947) Entry type Topic Classification msc 93A05 Classification msc 92B20 Classification msc 92B05 Classification msc 18-00 Classification msc 18D35 Synonym theoretical biology Synonym mathematical biophysics Related topic ComplexSystemsBiology Related topic CategoryOfSets Related topic HamiltonianOperatorOfAQuantumSystem Related topic BibliographyForMathematicalBiophysics Related topic CategoryOfMRSystems3 Related topic RobertRosen Related topic NicolasRashevsky Related topic Quaternions Related topic SimilarityAndAnalogousSystemsDynamicAdjointness2 Related topic PhysicalMathematicsAndEng Defines A- and B- DNA paracrystals