Mark E Curtis

The geometry of DNA: a structural revision

- critical reason applied to the Crick and Watson proposal - 

Mark E. Curtis: contact

I would like to suggest a modification to the structure for the salt of deoxyribose nucleic acid (D.N.A.) that was proposed by Francis Crick and James Watson in April 1953.

In 1995 I began an investigation into the structure of DNA with the intention of producing a series of drawings and paintings of the double helix. My interest stemmed from certain features of my work as an artist, specifically my inquiries into the nature and depiction of space. In the manner of renaissance perspectival artists such as Uccello, I embarked on scale drawings of the helical structure using the standard textbook dimensions that derive from x-ray diffraction data. In the course of this work, discrepancies emerged, and it became clear to me that the Crick and Watson structure does not conform to geometric principles. Indeed my attempts to translate their theory from two into three dimensions ran into considerable topological problems.

Since the article in nature was published, their proposal would appear to have been fully vindicated by almost all-available empirical evidence. However, I have since discovered that a growing minority do recognise flaws in Crick and Watsons conclusions, specifically in terms of its topology and thereby the ability of the structure to replicate itself. Although some research has even gone so far as to question the very existence of the double helical structure itself, my proposal retains the double helix as its fundamental basis.

Without compromising the essence of their structure, I propose a resolution of the geometrical inconsistencies by means of a simple change in the position of alignment between the purines and pyrimidines. This realignment is founded entirely upon geometric principles and further investigation revealed a series of mathematical equations that describe a three-dimensional geometric helical space that conforms to the known ratios of DNA. In this paper, I will demonstrate the mathematical basis for this re-reading of DNA and moreover the simplicity and purity it engenders, when applied to the molecular structure of the base-pairing.

The Double Helix

The structure of the salt of deoxyribose nucleic acid is undoubtedly a double helix, with ten bases to each turn and an individual rotation of 36 degrees. Moreover, the approximate dimensions of a complete turn of DNA’s helix are well known: the diameter of the helix is 20Å (angstroms), the base height is 3.4Å and thus the helix extension is 34Å. These data enabled a simplified and systematic perspective projection of the structure, see Fig 1 (below).

Fig 1

The Mathematics

Although, theoretically, it is possible to construct helices from almost any series of polygons, there happens to be only one polygonal formation that would fulfil all the necessary criteria: ten regular pentagons orientated about a decagon, see Fig 2 (below).

Fig 2

When translated into three-dimensional space, the pentagons become prisms with all lengths equal (ii).

A complete turn of the helix is formed by progressive rotation and extension of the ten regular prismatic pentagons (iii).

The diameter of the helix (HD) and the height of the helix extension (HE) have a direct and constant proportional ratio to the pentagon length (PL) and pentagon diameter (PD).

In order to establish the precise nature of this ratio we must devise some trigonometrical equations.

The values of a b c (iv) are ascertained as follows:

1/2a = c × sin 36°

b = c × cos 36°

c = 1/2a ÷ sin 36° or  b ÷ cos 36°

The values of PL and PD may be determined as follows:

PL = 2 × 1/2a = a

PD = b + c

It follows, therefore, that:

HD = 4 × (b + c)

HE = 10 × a

The above equations enable us to establish the mathematical constant ratios correct to ten decimal places:

PL = 1

PD = 1.5388417686

HD = 6.1553670744

HE = 10

From these ratios, it is possible to come by a set of interrelated equations that would determine any of the measurements of a helix constructed from regular prismatic pentagons:

PL  =  PD  ÷  1.5388417686  or  HD  ÷  6.1553670744  or  HE  ÷  10

PD  =  PL  ×  1.5388417686  or  HD  ÷ 4

HD  =  PL ×  6.1553670744  or  PD  × 4

HE  =  PL ×  10

We are now in a position to apply any one of the known dimensions of DNA to the above equations. For example, if the PL – i.e. the base height – is known to be approximately 3.4 Å:

PD  =  3.4 (PL)  x  1.5388417686  =  5.23206201324 Å

HD  =  3.4 (PL)  x  6.1553670744  =  20.928248053 Å

HE  =  3.4 (PL)  x  10  =  34Å

These figures would appear to cohere with the known dimensions of DNA as revealed by the x-ray diffraction data and we now find ourselves in a position to construct an exact dimensional composite of the DNA double helix. In fig 2(above) I used single prismatic pentagons to illustrate the 3-dimensional helix. However, to ensure stability in the structure as it would exist in reality, it is necessary to place adjacent pentagons on each plane and then stack accordingly, thereby creating the ‘double helix’. Fig 3 (below).

Fig 3

The application of geometry to the DNA molecular structure

The pairing of purines and pyrimidines in the Crick and Watson proposal locates the molecular pentagons that make up a part of the purines on the outside of the paired bases Fig 4 (below). However, it has been demonstrated that a more fluent and natural geometry would require ‘adjacent pentagons’ at the heart of the base pairing. It is therefore necessary to re-orientate the established molecular pairing, so that the pentagons of adenine (A) and guanine (G) form hydrogen bonds to the hexagonal structures of both thymine (T) and cytosine (C) Fig 5 (below). 

This spatial arrangement of the hydrogen bonds enables the necessary adjacent second pentagon to become viable. Moreover, it also accords with our understanding of molecular bonding and maintains Chargaff’s base pairing ratios - the specificity of G with C and A with T.

It would seem appropriate, at this point, to refer to the different formations that the molecules of guanine and thymine can take, namely the enol and keto states. Guanine and thymine may exist in either enol or keto configurations and are believed to pass through tautomeric shifts from one to the other – in other words, an indiscernible and spontaneous leap by the hydrogen atom from a specific oxygen atom to a specific nitrogen atom or vice-versa. The Crick and Watson proposal requires the keto formation, whereas this alternate proposal is viable in either formation.

In addition, it should be noted that presently the sugar-phosphate chains are depicted attached to N(1) of the pyrimidines and N(9) of the purines by covalent (glycosidic) bonds. However, both prior to 1953 and even today, when the base molecules are represented in isolation, hydrogen atoms are illustrated attached to these nitrogen atoms. For the purposes of this proposal, the four hydrogens are retained and one of them – guanine N(9) – now plays a practical role in the pairing of guanine with cytosine Fig 5 (above).

This alternative geometrical formulation reveals each base pairing to be an almost mirror image of the other, allowing a structure which, when fully assembled, exhibits both uniformity and stability. Most importantly perhaps, the alignment of the ‘spurs’ – that is to say, the NH2’s (G, C & A) or CH3 (T) – on each base pair is consistent. If either base pair were inverted, the spurs would naturally project in the opposite direction without upsetting the generative components of the helix. This ensures that the spurs attached to cytosine and adenine construct a constant, sequence specific, ‘secondary helix’ within the space generated by the primary helix; and that the spurs attached to guanine and thymine provide a sequence specific ‘capping’ of the ‘cavities’ generated by the conformation of the primary helix. These features not only display clear, predictable and simple mechanisms that may be utilised in the coding and articulation of amino acids but also a predictable braille like readability with which enzymes may be able to engage.

With respect to the sugar-phosphate chains, while it would be possible to incorporate covalent bonds into this proposal, to my mind, the nature of this revised structure does suggest a strong possibility that the sugar phosphate chains may be secured by means of multiple ‘hydrogen bonds’. The sheer quantities of accessible hydrogen and oxygen, both on the outside of this revised structure and also on the sugar-phosphate chains themselves, would potentially tie in with such a suggestion. Inevitably the first point of separation would continue to be the two hydrogen bonds that hold purine to pyrimidine, as those bonds would have considerably less resistance than the multiplicity of bonds that would hold the base-pair to the sugars and phosphates.

Conclusions

This proposed structure for DNA is wholly founded upon mathematical principles. Although the geometrical modification to the base pairings is relatively minor, the resulting double helix manifests a clarity altogether distinct from that offered by Crick and Watson and would appear to shed light upon a number of areas of continuing uncertainty.

The implications of these findings are inevitably far-reaching and would potentially affect many areas of research. However, these lie beyond my capacity to evaluate and I will restrict myself to reiterating the principal features of this proposal:

• Geometric equations predict the dimensions of DNA’s structure. Not only does the pentagonal geometry anticipate the helical dimensions but it also demonstrates a principle causation.

• The pentagonal geometry provides the architectural dynamics required to build a consistent, stable and uniform right or left handed helical structure and establishes why there should be consistently ten bases contained within a single turn of the helix. Incidentally, when converted to the molecular dimension I would certainly predict degrees of variation, certainly between 9.5 and 10.5 bases per turn, but perhaps even more.

• Both the hollow centre and side-by-side structural formation ensure instant access at any point within the helix. This would permit the DNA (even circular) to open and close during its replication functions without entangling itself.

• This modification to the base pairing would appear to be able to exist in either the enol or keto formation.

• While the sugar-phosphate backbones will undoubtedly prove integral to both the stability and perhaps the right handed orientation of the helical structure, it is the geometry of the base-pair molecules themselves that are ultimately responsible for the formation of the helix.

I have now set down a reinterpretation of DNA’s structure as I see it. It should be remembered that, by necessity, I publish this paper in my capacity as an artist and in the knowledge that nothing I have set down has yet been subjected to scientific scrutiny. It should be read as such, allowing the visual material to complement the text and vice-versa. This, after all, was the trigger for my initial interest. Aware of the further implications, I offer it in the hope that it may stimulate wider research and debate.

Mark E. Curtis 1997