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15 August 2004 | Draft

DNA Supercoiling as a Pattern for Understanding Psycho-social Twistedness

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This is an annex to Engaging with Questions of Higher Order: cognitive vigilance required for higher degrees of twistedness (2004)

Introduction
Structure of DNA
Forms of DNA
-- Supercoiled (or "knotted")
-- Relaxed
Descriptive properties associated with supercoiling
-- Writhing
-- Twisting
-- Linking number
-- Density
-- Replication
-- Denaturation, melting, breathing and unzipping
-- DNA-knots
Energy associated with different structures
-- Minimum energy (stable)
-- Higher energy (unstable)
References


Introduction

The review here of twistedness in DNA provides a technical basis for the discussion in the main paper (Engaging with Questions of Higher Order: cognitive vigilance required for higher degrees of twistedness, 2004).

The insights in the main paper regarding "twistedness" reflect an intuitive understanding of complexity which calls for deeper insight to understand how twistedness works and why it may be vitally important in some psycho-social processes -- as well as being highly problematic in others. Part of the difficulty in approaching this matter is that "twistedness" is in most cases used unthinkingly as a pejorative term to characterize a pattern which is felt to inhibit right-thinking and clarity. The argument here is that, given its importance at every scale in nature, from the organization of nebula to the organization of the human cell, there is a case for distinguishing various forms of twistedness and understanding their function. This could be especially valuable to reconciling apparently irreconcilable understandings in society.

The merit of focusing on the nature and function of twisting in DNA is that it provides a rich natural template. It offers a sense of the degree of complexity that it may be required to master in order to comprehend how twistedness "works" in practice. It might also be argued that, as a process active in every human body and inherent to human life, humans may well have some kind of profound intuitive understanding of how it works and the "rightness" of such working. Some of the very explicit dynamics of this process may also offer patterns for understanding how the inhibiting effects of "twistedness" may be addressed when they are perceived to be a constraint on human development.

Understanding of how DNA works has been much enriched by concepts from topology -- as a branch of mathematics that deals with structural properties that are unchanged by deformations such as stretching and bending. This use of mathematics is especially important because there is no experimental way to observe the dynamics of enzymatic action directly, notably with respect to knotting and coiling of DNA (see De Witt Sumners. Lifting the Curtain: Using Topology to Probe the Hidden Action of Enzymes, 1995; Xiaoyan R. Bao, et al. Behavior of Complex Knots in Single DNA Molecules, 2003).

Chromosomal DNA molecules are very long and thin. There is over a metre of DNA in every human cell in a space of some 0.0006 centimetres diametre. If DNA were constrained to be linear it would not fit into a cell. It must therefore fold many times to fit within the confines of a cell. The DNA is composed of 10** base pairs. This density of packing results in tangles and knots in the DNA that are essential to enable the cell to divide (involving transcription and replication).

Structure of DNA

DNA is a double stranded molecule composed of two polarized strands (of deoxyribonucleotide polymers) which run in opposite directions (termed antiparallel) and wind around a central, common axis -- one is entwined about the other such that an overall helical shape results (known as a plectonemic helix). Both are wound in a right-handed manner. This structure is to be contrasted with a paranemic helix, in which a pair of coils lie side by side without interwinding. The strands are occasionally distinguished as the Watson strand and the Crick strand.

In the case of the molecular structure of eukaryotic chromosomes in each human cel, 2 meztres of DNA is packaged into the cell nucleus. To access the information, it must be unwound as a double helix and needs to be "spread out" in the nucleus. However during cell division (mitosis), in order to move them around, they are packaged as follows into dense bundles:

  • Nucleosome formation ("beads on a string"): 2.5 loops of DNA wrapped around core DNA
  • Solenoid Formation "beaded string is coiled"): 6 nucleosomes per solenoid coil
  • Supercoiling (coil of solenoids is itself coiled): coiled coil is then folded - as in mitotic chromosome, namely a 10,000 fold reduction in length

Each nucleotide base of one strand is paired with a nucleotide base on the other strand to create a stable structure of the two polymers. The pairing of the four types of bases (A, T, C, G) by hydrogen bonds is not random: an A pairs with a T and a G pairs with a C. The bases on the outside of the helix are exposed to solvent within two grooves along the helix, the "major groove" and the "minor groove". It is within these grooves that DNA interacts with other molecules. The three structural variation of these grooves ("A", "B" and "Z" DNA), which differ in the relationship between the bases and the helical axis, offer one mechanism by which reactivity of DNA is modulated:

  • B-DNA : Fully hydrated DNA, the most common encountered in vivo. Owing to the location of the helical axis in the center of the base pairs, the edges of the base pairs are about equally deep in the interior.
  • A-DNA : When B-DNA is dehydrated, there is a reversible structural change to A-DNA
  • Z-DNA : Unlike B-DNA and A-DNA, Z-DNA is a left-handed helix. The conformational change from B-DNA to Z-DNA is one mechanism for relief of the torsional strain found in B-DNA in vivo, and may serve as a switch mechanism to regulate gene expression.

In circular double helix DNA (closed circular ccDNA), both strands are covalently joined to form a circular duplex molecule. The geometry of such an assembly is such that its number of coils cannot be changed without first breaking one of its strands. This topological "dilemma" is resolved within the cell -- to ensure proper biological functioning -- by specialized enzymes that unknot, untwist and unwind the DNA to enable replication and then reform the compact mode thereafter.

Heptad repeats: The coiled coil is a ubiquitous protein-folding motif. The accepted hallmark of the coiled coil is the seven-residue heptad repeat..A coiled-coil protein consists of two identical strands of amino acid sequences that wrap around each other. The amino acids in a coiled-coil structure reside on seven different structural positions on the coil, forming a heptad repeat (see The Heptad Repeat of The Coiled-coil Structure). Heptad repeats are characteristic of certain proteins. (see also images of David Gossard. Coiled Coils. 2003). Most coiled-coil sequences contain heptad repeats, namely seven residue patterns -- denoted abcdefg -- in which the a and d residues (core positions) are generally hydrophobic. As there are 3.6 residues to each turn of the alpha-helix, these a and d residues form a hydrophobic seam, which, as each heptad is slightly under two turns, slowly twists around the helix. The coiled-coil is formed by component helices coming together to bury their hydrophobic seams. As the hydrophobic seams twist around each helix, so the helices also twist to coil around each other, burying the hydrophobic seams and forming a supercoil. It is the characteristic interdigitation of side chains between neighbouring helices, known as knobs-into-holes packing, that defines the structure as a coiled coil (see Jenny Shipway. An Introduction to Coiled Coils. 2000) [more | more].

Forms of DNA

  • Supercoiled (or "knotted"): Double stranded circular (or linear) DNA can have tertiary or higher order structure. Superhelicity is therefore sometimes referred to as DNA's tertiary structure. Supercoils refer to the DNA structure in which double-stranded circular DNA twists around each other. This is termed supercoiling, supertwisting or superhelicity -- meaning the coiling of a coil, also understood in terms of knots. Only topological closed domains (such as a covalently closed circle) can undergo supercoiling. A linear molecule can have topological domains as long as there is a region of the DNA bounded by constraints on the rotation of the DNA double helix. Eukaryotic DNAs in association with nuclear proteins acquire superhelical conformation in chromosomes.

    Adding a twist to the DNA (as catalyzed by an enzyme), imposes a strain. A DNA segment so strained that is closed into a circle would then contort into a figure of eight (or its topological equivalents) -- the simplest supercoil. This is the shape that a circular DNA assumes to accomodate one too many or one too few helical twists. For each additional helical twist that is accomodated, the lobes will show one more roation about their axis. Such superhelicity results in more compact structures. In any other naturally found geometry, the DNA is either under- or overwound. Its helical axis does not lie in a plane or on the surface of a sphere because of writhing and twisting of it. This is the physical solution to the potential (torsional) energy minimization problem. Supercoiling can therefore be :
    • negative (right-handed): Supercoils formed by deficit in link are called negative supercoils. They result from underwinding, unwinding or subtractive twisting of the DNA helix (due to a deficit in link). The two lobes of the figure of eight then appear rotated counterclockwise with respect to each other. All naturally occuring double stranded DNAs are negatively supercoiled. Negative supercoiling facilitates DNA-strand separation during replication, recombination and transcription. All the naturally occuring double stranded DNAs are negatively supercoiled (including bacterial and viral circular duplex DNAs).
    • positive (left-handed): Supercoils formed by an increase in link are called positive supercoils. They result from tighter winding or overwinding of the DNA helix (due to an increase in link) resulting in extra helical twists. The two lobes of the figure of eight then appear rotated clockwise with respect to each other. This would compact DNA as effectively as negative supercoiling, but would make strand separation much more difficult.
    • In non-dividing eukaryotic cells, chromosomal DNA is wrapped around a nucleosome core which consists of highly basic proteins called histones. The DNA is wrapped around the nucleosome in a left-handed solenoidal arrangement. This negative supercoiling is one of the forms taken up by underwound DNA.

  • Relaxed: Circular DNA without any superhelical twist is known as a relaxed molecule. DNA in its relaxed (ideal) state usually assumes the B configuration. In a relaxed double-helical segment of DNA, the two strands twist around the helical axis once every 10.6 base pairs of sequence. Relaxed, closed circular DNA, is defined as DNA which has no supercoils when constrained to lie flat in a plan. The following structures are consistent with the relaxed state: (a) Linear DNA (either straight or curved) (b) Closed circular DNA, provided its axis lies in a plane or on the surface of a sphere

Supercoiling is thus vital to two major functions. It helps pack large circular rings of DNA into a small space by making the rings highly compact. It also helps in the unwinding of DNA required for its replication and transcription. Supercoiled DNA is thus the biological active form. The normal biological functioning of DNA occurs only if it is in the proper topological state.

Descriptive properties associated with supercoiling

"Supercoiling" is an abstract mathematical property, and represents the sum of what are termed "twist" and "writhe". "Supercoil" is seldom used as a noun with reference to DNA topology. It is the combination of twists and writhes that impart the supercoiling, and these occur in response to a change in the linking number. A coiled structure is at a higher energy (less stable). When the linking number is reduced in closed circular DNA, the molecule supercoils by minimizing twisting and bending. To partially relieve the strain introduced by the change in linking number (a 'deficit' in the link), the DNA must distort in other ways -- compensating with a change in twist or writhe. These are, physically, the two ways that the DNA can do so. The relationship of twist, writhe and supercoiling is expressed by the equation S = T + W (known as White's formula). Twist and writhe are geometric quantities. Unusually, link as a topological property is equal to the sum of two geometric properties. Their values change if the ribbon is deformed in space. Link, twist and writhe can be either positive or negative. Link is always an integer, whereas twist and writhe can take any real values.

  • Writhing: Global contortions of circular DNA are described as "writhe". The writhing number describes the supertwisting or supercoiling of the helix in space. It is the number of turns that the duplex axis makes about the superhelix axis. Writhe describes the supercoiling, the coiling of the DNA coil. It is a measure of the DNA's superhelicity (supercoiling) and can be positive or negative. Writhe is a measure of the coiling, bending or non-planarity of the axis of the double helix. A right-handed coil is assigned a negative number (negative supercoiling) and a left-handed coil is assigned a positive number (positive supercoiling). When a molecule is relaxed and contains no supercoils, the linking number = the twist number since W= 0 The linking number of relaxed DNA is L 0 L 0 = N/10.5, where N is the number of base pairs in the DNA fragment.

  • Twisting: Twist is the number of helical turns in the DNA, i.e., the complete revolutions that one polynucleotide strand makes about the duplex axis in the particular conformation under consideration. Twist is normally the number of base pairs divided by 10.4, that is the number of bases per turn of the helix. Twist is altered by deformation and is a local phenomenon. The total twist is the sum of all of the local twists. Twist is a measure of deformation due to a twisting motion.
    Twist and writhe are interconvertable. In part because chromosomes may be very large, segments in the middle may act as if their ends are anchored. As a result, they may be unable to distribute excess twist to the rest of the chromosome or to absorb twist to recover from underwinding -- the segments may become supercoiled, in other words. In response to supercoiling, they will assume an amount of writhe, just as if their ends were joined.

  • Linking number: This is a topological property that determines the degree of supercoiling; It defines the number of times a strand of DNA winds in the right-handed direction around the helix axis when the axis is constrained to lie in a plane. It is the number of times that one DNA strand crosses about the other when the DNA is made to lie flat on a plane. If both strands are covalently intact, the linking number cannot change. Link is thus a topological invariant, remaining unaltered even if the two curves are deformed in space -- as long as neither is cut. Topology theory indicates that the sum of T and W equals to linking number: L=T+W. For example, in the circular DNA of 5400 basepairs, the linking number is 5400/10=540. When a molecule is relaxed and contains no supercoils, the linking number = the twist number since W= 0. Thus if there is no supercoiling, then W=0, T=L=540. If there is positive supercoiling, W=+20, T=L-W=520. In the special cases in which axis of the double helix remains in a plane or on the surface of a sphere, then twist equals the linking number, and there is no writhe, but all other cases are considerably more complex. Supercoiling can be caused even by an increase in the linking number (though this does not occur in nature).

  • Density: the density of supercoiling.is useful to define as a property that distinguishes DNAs varying significantly in size. Superhelical density is the number of supercoils per turn of helix. It is denoted by the Greek letter sigma. It is defined as the number of turns that have been added or subtracted in the supercoiled DNA, compared to the relaxed state, divided by the total number of turns in the DNA if it were relaxed (which would normally be bp/10.5). Typically, sigma is between -.05 and -.07 (5-7% underwinding) in isolated natural DNA

  • Link altering enzymes: The functionality of DNA is related to its topology which is maintained by enzymes that are capable of altering it. Nature has come up with particular enzymes that control the knottedness (as well as other topological states such as twist-induced supercoiling) of DNA. The exact ability of these enzymes to locate a knot in a circular DNA is an unresolved question in molecular biology. Known as Topoisomerases, these enzymes change the structure by altering the DNA link of a molecule. This is achieved by temporarily breaking one of the strands, passing the other strand through it, and then resealing the bonds. This effectively changes the linking number in the DNA. The enzymes are of two types:
    • Type-1: function by creating transient single-strand breaks in DNA, altering the link by one, by cutting one strand and passing the other strand through the break.
    • Type-2: alter the link by two, by breaking both the strands of the double helix at the same time and passing a segment of the double helix through the break.
    Many topoisomerase enzymes sense supercoiling and either generate or dissipate it as they change DNA topology.

  • Replication: Level of supercoiling is known to be important for initiation of replication. In DNA replication, the two strands of DNA have to be separated, which leads either to overwinding of surrounding regions of DNA or to supercoiling. During replication, only part of the DNA unwinds (200 bps) while the rest of the DNA still remain configured at 10 bps per turn. A specialized set of enzymes (gyrase, topoisomerases) is present to introduce supercoils that favor strand separation; The degree of supercoils can be quantitatively described. Because of the wound configuration of DNA, biochemical transactions requiring strand separation necessitate chromosome movement (spin) about the long axis of the DNA. DNA replication, recombination and transcription all require DNA rotation. During DNA synthesis the rotation speed approaches 6000 rpm.

  • DNA Replication, RNA transcription, and Gene expression: Negative supercoiling in cells is energetically unfavorable and must be introduced in some manner: Therefore transcription is supercoil dependent. Topological domains may thus alter local regulation of gene expression. The overall level of supercoiling in a cell could have a global effect on gene expression. DNA is transcribed into mRNA in the nucleus. There are particular codons to which the enzymes are begin the transcription. The mRNA travels to, and attaches itself to a ribosome. There, the nucleotides are read from the start codon, in three letter sections, each of which code for a particular amino acid (some amino acids have multiple codons that code for them). The tRNA brings amino acids from the surrounding cytosol to the ribosome, and attach them in the order coded by the mRNA. The amino acids are then bonded together by peptide bonds. They are in a long linear chain, which is the primary conformation. The conformation to secondary structure is usually spontaneous, based on the interactions between the amino acids within the polypeptide. However, when going to the tertiary and quarternary conformations, there are usually larger proteins called chaperones that assist in the folding.

  • Denaturation, melting, breathing and unzipping: A physical property cricual to the function of DNA in replication and trqnscription is the ease with which its component parts can separate and be rejoined. This process is sometimes referred to as "melting" and "reannealing", or "denaturation" and "renaturation". DNA denaturation (or "melting") is due to the breakage of the hydrogen bonds in the Watson-Crick base-pairs, and is therefore reversible. This "unzipping" can be brought about by several processes. Under physiological conditions, local DNA-breathing occurs spontaneously due to thermal fluctuations. This opens up transient bubbles of a few tens of base pairs. These breathing fluctuations may be supported by single-strand binding proteins, thereby lowering the DNA base pair stability. DNA breathing and the lability of local stretches of the DNA double-helix is essential for numerous physiological processes such as the association of single-strand binding proteins, and the initiation of replication and transcription. (see Ralf Metzler and Andreas Hanke. Knots, bubbles, untying, and breathing: probing the topology of DNA and other biomolecules. 2004). [more | more]

  • DNA-knots: DNA knots can arise in various biological reactions involving circular DNA molecules. Site specific recombination enzymes are known to produce specific families of knots. Identification of the knot types indicates the mechanism of action of a given enzyme and the overall shape of supercoiled circular DNA molecules at the moment of knotting [more | more]. Internal pairing in short single-stranded DNA can be utilised for the construction of different types of DNA knots [images]. The presence of knots inhibits the assembly of chromatin. Knotted chromosomes cannot be separated during mitosis, and knots in a chromosome may serve as topological barriers between different sections of chromosomes, such that the genomic structural organisation is altered.

Energy associated with different structures

The energy of the molecule changes if there is a change in pitch (that is, the number of bases per full turn) or bending of the double helix ring. Even a small change in the pitch of the DNA results in a large increase in energy

  • Minimum energy: Linear DNA assumes the B configuration because it is the one of minimum energy. Linear molecules of DNA assume a configuration known as the b-configuration. Deviation from this relaxed state increases the energy of the DNA molecule, although circular DNA of large diameter increases it least.

  • Higher energy: In the ring form too, the DNA double helix tries to attain the state of minimum energy. The DNA ring approximates the b-configuration of the linear molecule while trying to attain the state of minimum energy. This packaging of DNA deforms it physically, thereby increasing its energy. Such an increase in stored (potential) energy within the molecule is then available to drive reactions such as the unwinding events that occur during DNA replication and transcription. Too much stored energy is not necessarily a good thing, though. In nature, this problem is addressed by having DNA form supercoils, in which the helical axis of the DNA curves itself into a coil. Supercoiling or the formation of a superhelix structure minimizes the excess energy that builds up when DNA molecules are deformed during the packing process.

  • At this point, it's a good idea to mention that supercoiling is not necessarily the only solution to the problem of normalizing the number of base pairs per helix in an unwound piece of DNA. You could also separate the two strands by breaking the hydrogen bonds between complementary bases in contiguous base pairs until the remaining DNA has the correct number of base per per turn. In terms of energy needed, though, it requires a lot more energy to break the H-bonds than to supercoil. Nevertheless, strand separation does occur during replication and transcription and it turns out that it is the physics of the underwinding that facilitates the strand separation. Cruciform structures also require some unpairing of the base pairs and, again, it is the underwinding that maintains the required strand separation.

References

Ralf Metzler and Andreas Hanke. Knots, bubbles, untying, and breathing: probing the topology of DNA and other biomolecules. 2004 [text]

Jeremy Narby. The Cosmic Serpent: DNA and the Origins of Knowledge. New York, Jeremy P. Tarcher/Putnam, 1999

Daniele Focosi. Cell Cycles [text]

Rensselaer Polytechnic Institute. DNA Structure and Topology. [text]

N Patrick Higgins. Chromosome Structure .[text]


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