Skip to main content

A reappraisal of the form – function problem. Theory and phenomenology

Abstract

This paper is aimed at demonstrating that some geometrical and topological transformations and operations serve not only as promoters of many specific genetic and cellular events in multicellular living organisms, but also as initiators of the organization and regulation of their functions. Thus, changes in the form and structure of macromolecular and cellular systems must be directly associated to their functions. There are specific classes of enzymes that manipulate the geometry and topology of complex DNA–protein structures, and thereby they perform many important cellular processes, including segregation of daughter chromosomes, gene regulation, and DNA repair. We argue that form has an organizing power, hence a causal action, in the sense that it enables to induce functional events during different biological processes, at the supramolecular, cellular, and organismal levels of organization. Clearly, topological forms must be matched with specific kinetic and dynamical parameters to have a functional effectiveness in living systems. This effectiveness is remarkably apparent, to give an example, in the regulation of the genome functions and in cell activity. In more general terms, we try to show that the conformational plasticity of biological systems depends on different kinds of topological manipulations performed by specific families of enzymes. In doing so, they catalyze all those spatial and dynamical changes of biological structures that are suitable for the functions to be acted by the organism.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17

Notes

  1. Recall that chromatin is achieved through the wrapping of DNA around a core of height histone proteins at regular intervals along the entire length of the chromosome, forming the basic building blocks of the chromatin fiber, the nucleosomes (McGinty and Tan 2015). The nucleosomes are further compacted into high-order chromatin architecture, and organized into condensed compartments or heterochromatin domain and open compartments or euchromatin domain. Within the nucleus, histones provide the energy (mainly in the form of electrostatic interactions) to fold DNA. As a result, chromatin can be packaged into a much smaller volume than DNA alone. Chromosome compaction is on the order of several thousand-fold, yet these chromosomes have to be unraveled every cell cycle to be replicated accurately and the daughter chromosomes must be topologically unlinked to allow their separation and segregation into the daughter cells. During mitosis, although most of the chromatin is tightly compacted, there are small regions that are not as tightly compacted. These regions often correspond to promoter regions of genes that were active in that cell type prior to chromatin formation. During interphase (1), chromatin is in its least condensed state and appears loosely distributed throughout the nucleus. Chromatin condensation begins during prophase (2) and chromosomes become visible. Chromosomes remain condensed throughout the various stages of mitosis (2–5). Condensing chromatin is necessary not only for structural and functional (which we describe accurately in the main text), but also for physical reasons. There are proper physical properties that the condensation of chromatin into sturdy chromosomes must realize. Chromosomes must be stiff, robust, and elastic enough to withstand forces coming from pulling microtubules and cytoplasmic drags during mitosis to prevent damage and breaks caused by external tensions (Durickovic et al. 2013). Compaction status of chromatin is regulated by structural (spatial) and chemical modifications upon DNA sequences and histone proteins, such as DNA methylation (Suzuki and Bird 2008), histone acetylation, and methylation. Chromatin compaction regulates transcription activities, and impacts many genomic functions such as DNA replication, damage, and repair. Therefore, our capacity to explore chromatin architecture and its epigenomics states at molecular and macromolecular scales is essential to our understanding of functional significance of chromatin compaction status and elucidate many biological and anomalous processes.

  2. This problem was addressed especially by Denis Nobel in the book The Music of Life. Biology beyond the genome, Oxford University Press, Oxford, 2006, and by Stuart Kauffman in its book The Origins of Order: Self-Organization and Selection in Evolution, Oxford University Press, Oxford, 1993.

  3. Topological information is information about a knot or link that does not depend upon the material from which it is made and is not changed by stretching or bending that material so long as it is not torn in the process. We do not want the knot to break up when the material undergoes some change in one or more of its physical parameters or to disappear in the course of such a stretching process by slipping over one of the ends of the rope. Precisely, topological information is invariant by deformation. Topological information about knots and links can be obtained from different sources. (1) From their diagrammatic representation and the associated Reidemeister moves. (2) From their numerical, algebraic, and topological invariants, starting with the most basic like the linking number to other more complete and powerful invariants like the Jones polynomial. (3) From quantum groups and quantum invariants of 3- and 4-manifolds. (4) From statistical mechanic models and critical phenomena. (5) From macroscopic physics, especially fluid mechanics and hydrodynamics. 6) From molecular biology, particularly from the replication and recombination processes.

  4. Condensins are large protein complexes that play a central role in chromosome assembly and segregation during mitosis and meiosis in the three domains of life. They display highly characteristics, rod-shaped structures with SMC (structural maintenance of chromosomes) ATPases as their core subunits and organize large-scale chromosome structure through active mechanisms. Most eukaryotic species have two distinct condensins’ complexes whose balanced usage is adapted flexibly to different organisms and cell types. One has observed both conserved features and rich variations of condensin-based chromosome organization. Cohesins are another representative class of eukaryotic SMC protein complexes. They play a central role in sister chromatid cohesion during mitosis and meiosis. Recent studies highlight their participation in gene regulation, in close collaboration with the insulator CTCF.

  5. Type IB topoisomerases can facilitate DNA rotation in either direction, and they can relax negative or positive supercoils.

  6. Recall that in a protein, individual amino acids constituting the primary sequence interact with one another to form secondary structures such as helices and like-sheets surfaces. Next, individual amino acids from distant parts of the primary sequence can intermingle via charge-charge, hydrophobic, disulfide, or other interactions, and the formation of these bonds and interactions will serve to change the shape of the overall protein; this typical and complex folded structure corresponds to its tertiary structure. In other words, tertiary structure is the three-dimensional structure of a protein. Precisely, the tertiary structure of proteins deals with how the local structures are put together and ordered in space following certain geometric and combinatorial rules and codes. For example, the -helices may be oriented parallel to each other or at right-angles. Therefore, the tertiary structure refers to the folding of the different segments of helices, sheets, turns, and the remainder of the protein into the native three-dimensional structure.

  7. By the terms of replication fork, one designs a site in double-stranded DNA at which the template strands are separated and addition of deoxyribonucleotides to each newly formed chain occurs. The notion of template denotes a molecular “mold” that dictates the structure of another molecule; most commonly, one strand of DNA that directs synthesis of a complementary DNA strand during DNA replication of an RNA during transcription.

  8. DNA gyrase is an essential bacterial enzyme that catalyzes the ATP-dependent negative supercoiling of double-stranded closed-circular DNA. Discovered in 1976, gyrase belongs to a class of enzymes known as topoisomerase of type IIA that are involved in the control of topological transitions of DNA. In contrast to other types II topoisomerases, DNA gyrase is the only enzyme that is capable of actively underwinding (i.e., negatively supercoiling) the double helix. It accomplishes underwinding by wrapping DNA around itself in a right-handed fashion (creating thus a positive supercoil) and carrying out its strand passage reaction in a unidirectional manner (thus converting a positive to a negative supercoil). The ability of gyrase to wrap DNA during its strand passage reaction allows it to remove positive supercoils that accumulate in front of replication forks and transcription complexes even faster than it can introduce negative supercoils into relaxed DNA. In other words, the negative supercoiling activity of DNA gyrase far exceeds the ability of the enzyme to remove knots and tangles from the genetic materials. Therefore, the major physiological roles of DNA gyrase stem directly from its ability to underwind (opening) the double helix. Therefore, gyrase maintains negative supercoiling of the genome, facilitating the initiation of transcription and replication. It also relaxes positive supercoils in front of elongating polymerases.

  9. The general definition is as follows. A framed knot (K, V) in S3 is a knot K equipped with a continuous non-vanishing vector filed V normal to the knot, called a framing. Similarly, a framed link in S3 is a link L where each component is equipped with a framing. A framed knot can be visualized as a tangled ribbon that has had its two ends glued after an even number of half-twists, so as to yield an orientable surface. Note that this means we exclude the cases in which the ribbon is glued together after an odd number of half-twists, i.e., a Möbius band. More precisely, the ribbon forms an embedded annulus, one of whose boundary components are identified with the specified knot K. For a given knot K, two framings on K are considered to be equivalent if one can be transformed into the other by a smooth deformation. This is indeed an equivalence relation on the set of framings, and as such, the term “framing” will be used to refer to either an equivalence class or a representative vector field.

  10. We can also give the following definition. Given a knot K in the 3-sphere S3, consider a singular disk D2 bounded by K and the intersections of K with the interior of the disk. The absolute number of intersections defines the framing function of the knot. One can show that the framing function is symmetric except at a finite number of points. The symmetric axis is a new knot invariant, called the natural framing of the knot. More formally: Let K: S1 S3 be an unoriented knot. Let D be the 2-disk. We define a compressing disk of K to be the map ƒ: D S3, such that ƒ|D = K and such that ƒ|int(D) is transverse to K. Then, ƒ|int(D) has only finitely many intersections with the knot. We call the intersections points the holes of the compressing disk, and denote their number by n(ƒ). So, n(ƒ) =|{ƒ–1(K) \(\cap\) int(D)}). The knottedness or linking coefficient Lk(K): = min{n(ƒ)| ƒ(D) a compressing disk} is a basic invariant of the knot K.

  11. In eukaryotes, genes can be broadly classified as TATA-containing and TATA-less based on the presence or absence of a TATA box in their promoter sequences. They have been studied in depth in yeast, and it is reported that TATA-containing genes are expressed at extremely high or low levels, are stress-induced, and are under evolutionary selective pressure, when compared to TATA-less genes. The two classes of genes also vary in their usage of transcription factors (SAGA vs. TFIID) in yeast. Furthermore, in yeast, TATA-containing genes prefer sub-telomeric location in the genome and have more duplicates. The structural features of TATA-containing TATA-less promoters are distinctly different in lower eukaryotes. The TATA-containing core promoters are less stable, more flexible, and more curved compared to TATA-less promoters in S. cerevisiae, C. elegans, and D. melanogaster. In mouse and human, stability and curvature are distinguishing features of TATA-containing and TATA-less promoters.

  12. Chromosomal and plasmid DNA molecules in bacterial cells are maintained under torsional tension and are therefore supercoiled. With the exception of extreme thermophiles, supercoiling has a negative sign, which means that the torsional tension diminishes the DNA helicity and facilitates strand separation.

  13. Linear DNA generally migrates between the nicked circle and the supercoiled forms. However, it may also migrate the same distance as nicked circle—it migrates as predicted by the length of the DNA.

  14. Historically, the theory of minimal surfaces was born with the optimization problem formulated by Lagrange: «Given a closed curve in tridimensional space, we have to found that surface which minimizes the area, among all those that have as boundary such a curve». In the 1850’s Plateau was the first to understand that each closed curve may be the boundary of a minimal surface. The conjecture, known as the Plateau’s problem, attracted many mathematicians, and the complete solution is due to Jesse Douglas in 1931 (Douglas 1931). The catenoid is a rotational surface bounded by two circles placed in two parallel planes. It was the first minimal surface know, which was discovered by Euler in 1744; the helicoid was discover by Lagrange in 1766. The minimal surface has equal surface tension in all their points, which means geometrically that the average curvature H is = 0. Hence, a minimal surface has, in every point, average curvature H = 0. Such a minimal surface needs not be minimizing for the area.

  15. A chord diagram is a finite trivalent undirected graph with an embedded oriented circle and all vertices on that circle, regarded modulo cyclic identification, if any. Equivalently, this is a pairing (by chords) of all elements in a cyclic order (the boundary vertices). Topologically, a chord diagram is an even number of distinct points on the circle, grouped in pairs, up to an orientation preserving homeomorphism of the circle. Such a diagram is pictured by a certain number of chords with distinct endpoints in a circle.

  16. Stated differently, unwinding of the helix during DNA replication (by the action of helicase) results in supercoiling of the DNA ahead of the replication fork. This supercoiling increases with the progression of the replication fork. If the replication supercoiling is not relieved, it will physically prevent the movement of helicase.

  17. Anfinsen’s experiments concern protein folding. In the 1950s, Christian Anfinsen conducted a series of experiments in which he determined that all the information needed to form the three-dimensional structure of the protein (polypeptide chain) is stored in the specific sequence of amino acids in that polypeptide. Later experiments confirmed this fact, i.e., that primary structure determines the final confirmation of the protein. In his first experiment, Anfinsen used some appropriate denaturing agents to break down the secondary and tertiary structure of ribonuclease. Precisely, he used urea agent to break down non-covalent bonds (also called disulfide bounds) such as hydrogen bonds holding the secondary structure, and then, he used the beta-mercaptoethanol to reduce and break down the disulfide bonds holding the tertiary structure together. The effect of the exposition of the native enzyme to these two agents was the complete denaturation of the protein. And when he removed the two agents simultaneously via dialysis, he found that the protein refolded back into its original biological active form. Then, in a second experiment, instead of removing the two agents at the same time, he first removed the beta-mercaptoethanol, and afterward, he removed the urea. What Anfinsen discovered was that the final protein refolded but became scrambled and was no longer biologically active. The hypothesis putted forward by Anfinsen was that this happened, because the non-covalent bonds could not form in the presence of urea, and so, disulfide bonds formed incorrectly. In a third experiment, he found that if he exposed the scrambled, inactive protein to trace amounts of beta-mercaptoethanol in the absence of urea, the biologically active native structure eventually reformed. This happens, because the tiny amount of beta-mercaptoethanol was enough to catalyze the breaking of the incorrect disulfide bonds. Finally, the protein formed the correct disulfide bridges and returned to its native form, because this was thermodynamically most stable and lowest in energy form.

  18. That is a number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Every integer is a rational number, for example, 5 = 5/1.

  19. Let us give this simple example. In \({\mathbb{R}}\) 3, the unknot (the circle S1) is not ambient isotopic to the trefoil knot, since one cannot be deformed into the other through a continuous map of homeomorphisms of the ambient space. Yet, they are ambient-isotopic in \({\mathbb{R}}\) 4.

References

  • Adams CC (2000) The knots book: an elementary introduction to the mathematical theory of knots. W. H. Freeman, New York

    Google Scholar 

  • Akutsu Y, Wadati M (1987) Knot invariants and critical statistical systems. J Phys Soc Jpn 56:839–842

    Article  Google Scholar 

  • Alberts B (2003) DNA replication and recombination. Nature 421:431–435

    PubMed  Article  CAS  Google Scholar 

  • Andersen JE, Penner RC, Reidys CM, Waterman MS (2013) Topologically classification and enumeration of RNA structures by genus. J Math Bio 67(5):1261–1278

    CAS  Article  Google Scholar 

  • Anfinsen CB (1973) Principles that govern the folding of protein chains. Science 181(4096):223–230

    CAS  PubMed  Article  Google Scholar 

  • Bates A, Maxwell A (2005) DNA topology, 2nd edn. Oxford University Press, Oxford

    Google Scholar 

  • Begun A, Liubimov S, Molochkov A, Niemi AJ (2021) On topology and knotted entanglements in protein folding. PLoS ONE 16(1):1–17

    Article  CAS  Google Scholar 

  • Bian Q, Belmont AS (2012) Revisiting higher order large-scale chromatin organization. Curr Opin Cell Biol 24(3):359–366

    CAS  PubMed  PubMed Central  Article  Google Scholar 

  • Birmarn JS (1974) Braids, links, and mapping class groups. Princeton University Press, Princeton

    Google Scholar 

  • Boi L (2005) Topological knots models in physics and biology. In: Boi L (ed) Geometries of nature, living systems and human cognition. New interactions of mathematics with natural sciences and humanities. World Scientific, Singapore, pp 203–278

  • Boi L (2006) Mathematical knot theory. In: Françoise J-P, Naber G, Sun TS (eds) Encyclopedia of mathematical physics, vol 3. Elsevier, Oxford, pp 399–406

    Chapter  Google Scholar 

  • Boi L (2007a) Geometrical topological modeling of supercoiling in supramolecular structures. Biophys Rev Lett 2(3):1–13

    Google Scholar 

  • Boi L (2007b) Modelling supercoiling in biological structures. In: Di Gesù V, Lo Bosco G, Maccarone MC (eds) Modelling and simulation in science. World Scientific, Singapore, pp 187–200

  • Boi L (2007c) Sur quelques propriétés géométriques globales des systèmes vivants. Bull D’histoire D’épistémol Sci Vie 14:71–113

    Article  Google Scholar 

  • Boi L (2009) Epigenetic phenomena, chromatin dynamics, and gene expression. New theoretical approaches in the study of living systems. Biol Forum 101(3):405–442

    Google Scholar 

  • Boi L (2011a) When topology meets biology ‘for life’. Remarks on the way in which topological form modulates biological function. In: New trends in geometry and its role in the natural and life sciences. Imperial College Press, London, pp 241–303

  • Boi L (2011b) Plasticity and complexity in biology: topological organization, regulatory protein networks and mechanism of gene expression. In: Terzis G, Arp R (eds) Information and living systems. Philosophical and Scientific Perspectives. The MIT Press, Cambridge, pp 205–250

  • Boi L (2017) The interlacing of upward and downward causation in complex living systems: on interactions, self-organization, emergence, and wholeness. In: Paolini Paoletti M, Orilia F (eds) Philosophical and scientific perspectives on Downward causation. Routledge, London, pp 180–203

  • Boi L (2021a) Geometrical modeling of DNA and the structural complexity of the chromosome. J Biophys (forthcoming)

  • Boi L (2021b) A topological and dynamical approach to the study of complex living systems. In: Albeverio S, Mastrogiacomo E (eds) Complexity and emergence. Springer, Heidelberg, pp 57–104

    Google Scholar 

  • Boi L (2021c) Knots, diagrams, and kid’s shoelaces: on spaces and theirs forms. In: Boi L, Lobo C (eds) When form becomes substance. Power of gesture, diagrammatical intuition and phenomenology of space. Birkhäuser, Basel, pp 137–208

  • Boles CT, White JH, Cozzarelli NR (1990) Structure of plectonemically supercoiled DNA. J Mol Biol 213(4):931–951

    CAS  PubMed  Article  Google Scholar 

  • Bon M, Vernizzi G, Orland H, Zee A (2008) Topological classification of RNA structures. J Mol Biol 379:900–911

    CAS  PubMed  Article  Google Scholar 

  • Brunello L, Levens D, Gupta A, Kouzine F (2012) The importance of being supercoiled: How DNA mechanic regulate dynamic processes. Biophys Acta (BBA) Gene Regul Mech 1819(7):632–638

    Article  CAS  Google Scholar 

  • Buck D (2009) DNA topology. Proc Symp Appl Math 66:1–33

    Article  Google Scholar 

  • Buck D, Valencia D (2011) Characterization of knots and links arising from site-specific recombination of twist knots. J Phys A 44(4):1–36

    Article  CAS  Google Scholar 

  • Burde G, Zieschang H (2003) Knots, 2nd edn. de Gruyter, Berlin

    Google Scholar 

  • Carbone A, Gromov M (2001) Mathematical slices of molecular biology, Gazette des Mathématiciens. Soc Math France 8:11–80

    Google Scholar 

  • Cavalli G, Heard E (2019) Advances in epigenetics link genetics to environment and disease. Nature 571:39–68

    Article  CAS  Google Scholar 

  • Conway JH (1970) An enumeration of knots and links, and some of their algebraic properties J. In: Leech (ed) Computational problems in abstract algebra. Pergamon Press, Oxford, pp 329–358

    Google Scholar 

  • Cozzarelli NR, Spengler SJ, Stasiak A (1985) The stereostructure of knots and catenanes produced by phase λ integrative recombination: implications for mechanism and DNA structure. Cell 42:325–334

    PubMed  Article  Google Scholar 

  • Cozzarelli NR (1992) Evolution of DNA topology: implications for its biological role. In: New scientific applications of geometry and topology, PSAM, vol 45, Amer. Math. Soc

  • Cremer T et al (2004) Higher order chromatin architecture in the cell nucleus: on the way from structure to function. Biol Cell 96:555–567

    CAS  PubMed  Article  Google Scholar 

  • Culler M, Gordon MCA, Leucke J, Shalen PB (1987) Dehn surgery on knots. Ann Math 125(2):237–300

    Article  Google Scholar 

  • Danchin A (1978) Ordre et dynamique du vivant. Éditions du Seuil, Paris

    Google Scholar 

  • Danchin E, Charmantier A (2011) Beyond DNA: Integrating inclusive inheritance into an extended theory of evolution. Nat Rev Gen 12:475–486

    CAS  Article  Google Scholar 

  • Darcy IK, Levene SD, Scharein RG (2014) Introduction to DNA topology. In: Jonoska N, Saito M (eds) Discrete and topological models in molecular biology. Springer, Heidelberg, pp 327–345

    Chapter  Google Scholar 

  • Dehn M (1910) Über die topologie des dreidimensionalen raumes. Math Ann 69(1):137–168

    Article  Google Scholar 

  • Dixon JR, Gorkin DV, Ren B (2016) Chromatin dynamics: the unit of chromosome organization. Mol Cell 62(5):668–680

    CAS  PubMed  PubMed Central  Article  Google Scholar 

  • Douglas J (1931) Solution of the problem of Plateau. Trans Am Math Soc 33(1):263–321

    Article  Google Scholar 

  • Durickovic B, Goriely A, Maddocks JH (2013) Twist and stretch of helices explained via the Kirchhoff-Love rod model of elastic filaments. Phys Rev Lett 111:108103–108105

    Article  CAS  Google Scholar 

  • Dyson F (1985) Origins of life. Cambridge University Press, Cambridge

    Google Scholar 

  • Elhamdadi M, Hajij M, Istvan K (2020) Framed knots. Math Intell 42:7–22

    Article  Google Scholar 

  • Ernst C, Sumners DW (1990) A calculus for rational tangles: applications to DNA recombination. Math Proc Cambr Math Soc 108(3):489–515

    Article  Google Scholar 

  • Felsenfeld G, Groudine M (2003) Controlling the double helix. Nature 421:448–453

    PubMed  Article  CAS  Google Scholar 

  • Flapan E, Grevet J, Li Q, Sun CD, Wong H (2014) Knotted and linked products of recombination on T(2, n)#T(2, m) substrates. J Korean Math Soc 51(4):817–836

    Article  Google Scholar 

  • Flapan E, He A, Wong A (2019) Topological description of protein folding. Proc Natl Acad Sci USA 116(19):9360–9369

    CAS  PubMed  PubMed Central  Article  Google Scholar 

  • Forterre P, Gribaldo S, Gadelle D, Serre M-C (2007) Origins and evolution of DNA topoisomerases. Biochimie 89(4):427–446

    CAS  PubMed  Article  Google Scholar 

  • Fuller FB (1978) Decomposition of the linking number of a closed ribbon: a problem from molecular biology. Proc Natl Acad Sci USA 75(8):3557–3561

    CAS  PubMed  PubMed Central  Article  Google Scholar 

  • Furlan-Margaril M, Recillas-Targa F (2011) Chromatin remodeling and epigenetic regulation during development. In: Chimal-Monroy J (ed) Topics in animals and plant development: from cell differentiation to morphogenesis, pp 221–247

  • Goldman JR, Kauffman LH (1997) Rational tangles. Adv Appl Math 18(3):300–332

    Article  Google Scholar 

  • Goodwin B, Webster G (1996) Form and transformation: generative and relational principles in biology. Cambridge University Press, Cambridge

    Google Scholar 

  • Gordon CM (2006) Some aspects of classical knot theory. In: Hausmann JC (ed) Knot theory, lecture notes in mathematics, vol 685. Springer, Heidelberg, Berlin, pp 1–60

    Google Scholar 

  • Gromov M (2011) Crystals, proteins, stability and isoperimetry. Bull Am Math Soc (NS) 48(2):229–257

    Article  Google Scholar 

  • Hinde E, Cardarelli F, Digman MA, Gratton E (2012) Changes in chromatin compaction during the cell cycles revealed by micrometer-scale measurement of molecular flow in the nucleus. Biophys J 102(3):691–697

    CAS  PubMed  PubMed Central  Article  Google Scholar 

  • Hirano T (2016) Condensin-based chromosome organization from bacteria to vertebrates. Cell 164(5):847–857

    CAS  PubMed  Article  Google Scholar 

  • Holliday R (1987) The inheritance of epigenetic defects. Science 238:163–170

    CAS  PubMed  Article  Google Scholar 

  • Huang FW, Reidys CM (2015) Shapes of topological RNA structures. Math Biosci 270:57–65

    CAS  PubMed  Article  Google Scholar 

  • Huang FW, Reidys CM (2016) Topological language for RNA. Math Biosci 282:109–120

    CAS  PubMed  Article  Google Scholar 

  • Jaenisch R, Bird A (2003) Epigenetic regulation of gene expression: how the genome integrates intrinsic and environmental signals. Nat Genet 33:245–254

    CAS  PubMed  Article  Google Scholar 

  • Jones VFR (1985) A polynomial invariant for knots via von Neumann algebras. Bull Am Math Soc 12:103–111

    Article  Google Scholar 

  • Jost J (2019) Biologie und mathematik. Springer, Berlin, Heidelberg

    Book  Google Scholar 

  • Jost D, Carrivain P, Cavalli G, Vaillant C (2014) Modeling epigenome folding: formation and dynamics of topologically associated chromatin domains. Nucleic Acids Res 42(15):9541–9549

    Article  CAS  Google Scholar 

  • Kauffman LH (1987) On knots. Princeton University Press, Princeton

    Google Scholar 

  • Kauffman LH (1990) An invariant of regular isotopy. Trans Am Math Soc 318(2):417–471

    Article  Google Scholar 

  • Kauffman S (1993) The origins of order: self-organization and selection in evolution. Oxford University Press, Oxford

    Google Scholar 

  • Kauffman LH (2001) Knots and physics, world scientific, series on knots and everything, vol 1. World Scientific, London

  • Kauffman LH (2005) Knots. In: Boi L (ed) Geometries of nature, living systems and human cognition. The new interactions of mathematics with natural sciences and the humanities. World Scientific, Singapore, pp 131–202

    Chapter  Google Scholar 

  • Kauffman LH, Lambropoulou S (2004) On the classification of rational tangles. Adv Appl Math 33(2):199–237

    Article  Google Scholar 

  • Képès F, Vaillant C (2003) Transcriptional-based solenoidal model of chromosomes. Complexus 1(4):171–180

    Article  Google Scholar 

  • Kervaire M (1965) Les nœuds de dimensions supérieures. Bull Soc Math France 93:225–271

    Article  Google Scholar 

  • Kimmins S, Sassoni-Corsi P (2005) Chromatin remodeling and epigenetic features of germ cells. Nature 434:583–589

    CAS  PubMed  Article  Google Scholar 

  • Kirby R (1978) A calculus for framed links in S3. Invent Math 45(1):35–56

    Article  Google Scholar 

  • Kitano H (2004) Biological robustness. Nat Rev Genet 5(11):826–837

    CAS  PubMed  Article  Google Scholar 

  • Lal A et al (2016) Genome scale patterns of supercoiling in a bacterial chromosome. Nat Commun 7(1):11055–11163

    CAS  PubMed  PubMed Central  Article  Google Scholar 

  • Lickorish WBR (1997) An introduction to knot theory, graduate texts in mathematics, vol 175. Springer, Heidelberg

    Book  Google Scholar 

  • Lodish H, Berk A, Zipursky A et al (2000) Molecular cell biology, 4th edn. W. H. Freeman, New York

    Google Scholar 

  • Mazur B (2004) Perturbations, deformations, and variations (and “near-misses”) in geometry, physics, and number theory. Bull Am Math Soc (NS) 41(3):307–336

    Article  Google Scholar 

  • McClintock M (1984) The significance and responses of the genome to challenge. Science 226:792–801

    CAS  PubMed  Article  Google Scholar 

  • McGinty RK, Tan S (2015) Nucleosome structure and function. Chem Rev 115:2255–2273

    CAS  PubMed  Article  Google Scholar 

  • Misteli T (2007) Beyond the sequence. Cellular organization of genome function. Cell 128(4):787–800

    CAS  PubMed  Article  Google Scholar 

  • Murasugi K (1996) Knot theory and its applications. Birkhäuser, Boston

    Google Scholar 

  • Muskhelishvili G, Travers A (2016) The regulatory role of DNA supercoiling in nucleoprotein complex assembly and genetic activity. Biophys Rev 8(Suppl. 1):5–22

    CAS  PubMed  PubMed Central  Article  Google Scholar 

  • Nicolas G, Prigogine I (1977) Self-organization in nonequilibrium systems: from dissipative structures to order through fluctuations. Wiley, New York

    Google Scholar 

  • Noble D (2006) The music of life. Biology beyond the genome. Oxford University Press, Oxford

    Google Scholar 

  • Noble D (2008) Genes and causation. Phil Trans R Soc Lond A 366(1878):3001–3015

    CAS  Google Scholar 

  • Ochs F et al (2019) Stabilization of chromatin topology safeguards genome integrity. Nature 574:571–574

    CAS  PubMed  Article  Google Scholar 

  • Ophl WF, Roberts GW (1978) Topological considerations in the theory of replication of DNA. J Math Biol 6:383–402

    Article  Google Scholar 

  • Penner RC (2016) Moduli spaces and macromolecules. Bull Am Math Soc 53:217–269

    Article  Google Scholar 

  • Penner RC, Waterman MS (1993) Spaces of RNA secondary structures. Adv Math 101(1):31–49

    Article  Google Scholar 

  • Peselis A, Serganov A (2014) Structure and function of pseudoknots involved in gene expression control. Wiley Interdisc Rev RNA 5(6):803–822

    CAS  Article  Google Scholar 

  • Porter LL, Looger LL (2018) Extant fold-switching proteins are widespread. Proc Natl Acad Sci USA 115(23):5968–5973

    CAS  PubMed  PubMed Central  Article  Google Scholar 

  • Ramam V, Shendure J, Duan Z (2016) Understanding Spatial Genome Organization: Methods and Insights. Genom Proteom Bioinform 14(1):7–20

    Article  Google Scholar 

  • Reidemeister K (1927) Elementare begründung der knotentheorie. Abh Math Sem Univ Hamburg 5(1):2432

    Google Scholar 

  • Reidemeister K (1932) Knotentheorie. Springer, Heidelberg/Berlin/New York

    Google Scholar 

  • Ricca RL, Nipoti B (2011) Gauss’s linking number revisited. J Knot Theory Ramific 20(10):1325–1343

    Article  Google Scholar 

  • Ridgway P, Almouzni G (2001) Chromatin assembly and organization. J Cell Sci 114:2711–2722

    CAS  PubMed  Article  Google Scholar 

  • Roca J (1998) Topoisomerases. Adv Genome Biol 5:463–485

    Article  Google Scholar 

  • Rolfsen D (1976) Knots and links, mathematical lecture series, vol 7. Publish or Perish, Huston

    Google Scholar 

  • Rosen R (1970) Dynamical systems theory in biology. Wiley, New York

    Google Scholar 

  • Scherrer K, Jost J (2007) Gene and genon concept: coding versus regulation. A conceptual and information-theoretic analysis of genetic storage and expression in the light of modern molecular biology. Theory Biosci 126(2):65–113

    CAS  PubMed  PubMed Central  Article  Google Scholar 

  • Seifert H (1935) Über die Geschlecht von Knoten. Math Ann 110(1):571–592

    Article  Google Scholar 

  • Sergei MM (2001) DNA topology: fundamentals, encyclopedia of life sciences. Nature Publishing Group, Berlin, pp 1–11

    Google Scholar 

  • Simondon G (2005) L’individuation à la lumière des notions de forme et d’information, Jérôme Million, Paris

  • Spera M (2006) A survey on the differential and symplectic geometry of linking numbers. Milan J Math 74:139–197

    Article  Google Scholar 

  • Strick TR, Allemand J-F, Bensimon D, Croquette V (1998) Behavior of Supercoiled DNA. Biophys J 74:2016–2028

    CAS  PubMed  PubMed Central  Article  Google Scholar 

  • Sumners DW (1990) Untangling DNA. Math Intell 12(3):71–80

    Article  Google Scholar 

  • Sumners DW (1992) Knot theory and DNA. In: New scientific applications of geometry and topology, PSAM, 45, Amer Math Soc, pp 39–72

  • Sutormin DA et al (2021) Diversity and Functions of Type II Topoisomerases. Acta Natur 13(1):59–75

    CAS  Article  Google Scholar 

  • Suzuki MM, Bird A (2008) DNA methylation landscapes: provocative insights from epigenomics. Nat Rev 9:465–476

    CAS  Article  Google Scholar 

  • Theimer CA, Blois CA, Feigon J (2005) Structure of the human telomerase RNA pseudoknot reveals conserved tertiary interactions essential for function. Mol Cell 17(5):671–682

    CAS  PubMed  Article  Google Scholar 

  • Thom R (1972) Stabilité structurelle et morphogenèse. Benjamin, New York

    Google Scholar 

  • Thom R (1989) Modèles mathématiques de la morphogenèse. Christian Bourgois, Paris

    Google Scholar 

  • Vazques M, Sumners DW (2004) Tangles analysis of Gin site-specific recombination. Math Proc Camb Phil Soc 136(565):565–582

    Article  Google Scholar 

  • Venkata RY, Bansal M (2017) DNA structural features of eukaryotic TATA-containing and TATA-less promoters. FEBS Open Bio 7(3):324–334

    Article  CAS  Google Scholar 

  • Villota-Salazar NA, Mendoza-Mendoza A, Gonzáles-Prieto JM (2016) Epigenetics: from the past to the present. Front Life Sci 9(4):347–370

    CAS  Article  Google Scholar 

  • Vologodskii AV (1992) The topology and physics of circular DNA. CRC Press, Boca Raton, FL

    Google Scholar 

  • Waddington CH (1957) The strategy of the genes. Routledge, London

    Google Scholar 

  • Waddington CH (ed) (1968) Toward a theoretical biology. Routledge, London, pp 1968–1969

    Google Scholar 

  • Wang JC (1996) DNA topoisomerases. Ann Rev Biochem 65:635–692

    CAS  PubMed  Article  Google Scholar 

  • Wang JC, Caron PR, Kim RA (1990) The role of DNA topoisomerases in recombination and genome stability: a double-edged sword. Cell 62:403–406

    CAS  PubMed  Article  Google Scholar 

  • White JH (1989) An introduction to the geometry and topology of DNA structures. CRC Press, Boca Raton

    Google Scholar 

  • White JH, Cozzarelli NR, Bauer WR (1988) Helical repeat and linking number of surface-wrapped DNA. Science 241:323–327

    CAS  PubMed  Article  Google Scholar 

  • Wu FY (1992) Knot theory and statistical mechanics. Rev Mod Phys 64(4):1099–1129

    Article  Google Scholar 

  • Zeeman EC (1960) Unknotting spheres. Ann Math 72:350–361

    Article  Google Scholar 

  • Zeeman EC (1965) Twisting spun knots. Trans Am Math Soc 115:471–495

    Article  Google Scholar 

  • Zhurkin VB, Norouzi D (2021) Topological polymorphism of nucleosome and folding of chromatin. Biophys J 120(4):577–585

    CAS  PubMed  PubMed Central  Article  Google Scholar 

Download references

Acknowledgements

We wish to thank the referees for very useful comments which allowed the revision of several inaccuracies concerning some mathematical and biological statements and hence the improvement of this article. We also would like to thank Andras Paldi, Moncef Ladjimi, Hans Liljenström, Jürgen Jost, and Carlos Lobo for helpful discussions.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Luciano Boi.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Boi, L. A reappraisal of the form – function problem. Theory and phenomenology. Theory Biosci. 141, 73–103 (2022). https://doi.org/10.1007/s12064-022-00368-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12064-022-00368-8

Keywords

  • Form
  • Function
  • Geometry
  • Topology
  • Recombination
  • Epigenome
  • Cell activity
  • Global metabolism