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

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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.

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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.

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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.

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  1. École des Hautes Études en Sciences Sociales, Centre de Mathématiques (CAMS), 54, bd Raspail, 75006, Paris, France

    Luciano Boi

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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

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