An amateur's contribution to the design of Telford's Menai Suspension Bridge: a commentary on Gilbert (1826) 'On the mathematical theory of suspension bridges'.

Davies Gilbert's work on the catenary is notable on two counts. First, it influenced Thomas Telford in formulating his final design for the Menai Strait suspension bridge (1826); and second, it established for the first time the form of the 'catenary of equal strength'. The classical catenary is a uniform flexible chain or cable hanging freely under gravity between supports. The 'catenary of equal strength' is the form of a cable whose cross-sectional area is made proportional to the tension at each point, so that the tensile stress is uniform throughout. In this paper I provide a sketch of the lives and achievements of Gilbert and Telford, and of their interaction over the Menai Bridge. There follows a commentary on Gilbert's 1826 paper, and on his two related publications; and a brief sketch of the earlier history of the catenary. I then describe the development of the suspension bridge up to the present time. Finally, I discuss relations between mathematical analysts and practical engineers. This commentary was written to celebrate the 350th anniversary of the journal Philosophical Transactions of the Royal Society.

Geometric principles in the assembly of α-helical bundles.

α-Helical coiled coils are usually stabilized by hydrophobic interfaces between the two constituent α-helices, in the form of 'knobs-into-holes' packing of non-polar residues arranged in repeating heptad patterns. Here we examine the corresponding 'hydrophobic cores' that stabilize bundles of four α-helices. In particular, we study three different kinds of bundle, involving four α-helices of identical sequence: two pack in a parallel and one in an anti-parallel orientation. We point out that the simplest way of understanding the packing of these 4-helix bundles is to use Crick's original idea that the helices are held together by 'hydrophobic stripes', which are readily visualized on the cylindrical surface lattice of the α-helices; and that the 'helix-crossing angle'--which determines, in particular, whether supercoiling is left- or right-handed--is fixed by the slope of the lattice lines that contain the hydrophobic residues. In our three examples the constituent α-helices have hydrophobic repeat patterns of 7, 11 and 4 residues, respectively; and we associate the different overall conformations with 'knobs-into-holes' packing along the 7-, 11- and 4-start lines, respectively, of the cylindrical surface lattices of the constituent α-helices. For the first two examples, all four interfaces between adjacent helices are geometrically equivalent; but in the third, one of the four interfaces differs significantly from the others. We provide a geometrical explanation for this non-equivalence in terms of two different but equivalent ways of assembling this bundle, which may possibly constitute a bistable molecular 'switch' with a coaxial throw of about 12 Å. The geometrical ideas that we deploy in this paper provide the simplest and clearest description of the structure of helical bundles. In an appendix, we describe briefly a computer program that we have devised in order to search for 'knobs-into-holes' packing between α-helices in proteins.

A "mechanistic" explanation of the multiple helical forms adopted by bacterial flagellar filaments.

The corkscrew-like flagellar filaments emerging from the surface of bacteria such as Salmonella typhimurium propel the cells toward nutrient and away from repellents. This kind of motility depends upon the ability of the flagellar filaments to adopt a range of distinct helical forms. A filament is typically constructed from ~30,000 identical flagellin molecules, which self-assemble into a tubular structure containing 11 near-longitudinal protofilaments. A "mechanical" model, in which the flagellin building block has the capacity to switch between two principal interfacial states, predicts that the filament can assemble into a "canonical" family of 12 distinct helical forms, each having unique curvature and twist: these include two "extreme" straight forms having left- and right-handed twists, respectively, and 10 intermediate helical forms. Measured shapes of the filaments correspond well with predictions of the model. This report is concerned with two unanswered questions. First, what properties of the flagellin determine which of the 12 discrete forms is preferred? Second, how does the interfacial "switch" work, at a molecular level? Our proposed solution of these problems is based mainly on a detailed examination of differences between the available electron cryo-microscopy structures of the straight L and R filaments, respectively.

The buckling of spherical liposomes.

In the classical "first approximation" theory of thin-shell structures, the constitutive relations for a generic shell element--i.e. the elastic relations between the bending moments and membrane stresses and the corresponding changes in curvature and strain, respectively-are written as if an element of the shell is flat, although in reality it is curved. In this theory it is believed that discrepancies on account of the use of "flat" constitutive relations will be negligible provided the ratio shell-radius/thickness is of sufficiently large order. In the study of drawing of narrow, cylindrical "tethers" from liposomes it has been known for many years that it is necessary to use instead a constitutive law which explicitly describes a curved element in order to make sense of the mechanics; and indeed such tethers are generally of "thick-walled" proportions. In this paper we show that the proper constitutive relations for a curved element must also be used in the study, by means of shell equations, of the buckling of initially spherical thin-walled giant liposomes under exterior pressure: these involve the inclusion of what we call the "Mkappa" terms, which are not present in the standard "first-approximation" theory. We obtain analytical expressions for both the bifurcation buckling pressure and the slope of the post-buckling path, in terms of the dimensions and elastic constants of the lipid bi-layer, and also the initial state of bending moment in the vesicle. We explain physically how the initial bending moment can affect the bifurcation pressure, whereas it cannot in "first-approximation" theory. We use these results to map the conditions under which the vesicle buckles into an oblate, as distinct from a prolate ("rugby-ball") shape. Some of our results were obtained long ago by the use of energy methods; but our aim here has been to identify precisely what is lacking in "first-approximation" theory in relation to liposomes, and so to put the "shell equations" approach onto a firm footing in mechanics.

Mechanics of tether formation in liposomes.

It is well-known that a "tether" may be drawn out from a pressurized liposome by means of a suitably applied radial-outward force applied locally to the lipid bilayer. The tether is a narrow, uniform cylindrical tube, which joins the main vesicle in a short "transition region." A first-order energy analysis establishes the broad relationship between the force F needed to draw the tether, the radius R0 of the tether, the bending-stiffness constant B for the lipid bilayer and the membrane tension T in the pressurized liposome. The aim of the present paper is to study in detail the "transition region" between the tether and the main vesicle, by means of a careful application of the engineering theory of axisymmetric shell structures. It turns out that the well-known textbook "thin-shell" theory is inadequate for this purpose, because the tether is evidently an example of a thick-walled shell; and a novel ingredient of the present study is the introduction of elastic constitutive relations that are appropriate to the thick-shell situation. The governing equations are set up in dimensionless form, and are solved by means of a "shooting" technique, starting with a single disposable parameter at a point on the meridian in the tether, which can be adjusted until the boundary conditions at the far "equator" of the main vessel are satisfied. It turns out that the "transition region" between the tether and the main vessel is well characterized by only a few parameters, while the tether and main vessel themselves are described by very simple equations. Introduction of the thick-shell constitutive relation makes little difference to the conformation of and stress-resultants in, the main vessel; but it makes a great deal of difference in the tether itself Indeed, a kind of phase-change appears to take place in the "transition region" between these two zones of the liposome.

How to untwist an alpha-helix: structural principles of an alpha-helical barrel.

Recent crystallographic studies have revealed that 12 alpha-helices can pack in an anti-parallel fashion to form a hollow cylinder of nearly uniform radius. In this architecture, which we refer to as an alpha-barrel, the helices are inclined with respect to the cylindrical axis, and thus they curve and twist. As with conventional coiled-coils, the helices of the barrel associate via "knobs-into-holes" interactions; however, their packing is distinct in several important ways. First, the alpha-barrel helices untwist in comparison with the helices found in two-stranded coiled-coils and, as a consequence of this distortion, their knobs approach closely one end of the complementary holes. This effect defines a requirement for particular size and shape of the protruding residues, and it is associated with a relative axial translation of the paired helices. Second, as each helix packs laterally with two neighbours, the helices have two sequence patterns that are phased to match the two interfaces. The two types of interface are not equivalent and, as one travels around the circumference of the cylinder's interior, they alternate between one type where the knobs approach the holes straight-on, and a second type in which they are inclined. The choice of amino acid depends on the interface type, with small hydrophobic side-chains preferred for the direct contacts and larger aliphatic side-chains for the inclined contacts. Third, small residues are found preferentially on the inside of the tube, in order to make the "wedge" angle between helices compatible with a 12-member tube. Finally, hydrogen-bonding interactions of side-chains within and between helices support the assembly. Using these salient structural features, we present a sequence template that is compatible with some underlying rules for the packing of helices in the barrel, and which may have application to the design of higher-order assemblies from peptides, such as nano-tubes. We discuss the general implications of relative axial translation in coiled-coils and, in particular, the potential role that this movement could play in allosteric mechanisms.

Two distinct modes of protein-induced bending in DNA.

Crystallised "naked" DNA oligomers in the B form show significant conformational mobility, particularly at CA/TG and TA/TA steps: there is a range in Roll angle of some 15 degrees between consecutive base-pairs, and Slide and Twist are directly coupled to Roll. We call such motions "mode I". They are sufficient to enable DNA to curve gently around proteins such as histone octamers in the nucleosome particle. When DNA bends around other proteins, such as CAP and TBP, its distortion is much more severe. Although the DNA in close contact with these proteins includes the CA/TG and TA/TA steps, respectively, the mode I flexibility is not deployed: instead, a more severe "mode II" manoeuvre is observed in DNA/protein co-crystals. Mode II has several distinctive physical features. First, its range of Roll angle is much wider than for mode I. Second, the major-groove width remains more-or-less constant as Roll increases, whereas it decreases significantly as Roll increases in mode I; and this enables the major groove of the DNA to accommodate a protein moiety in its severely bent conformation. Third, the value of Slide remains more-or-less constant as Roll increases, whereas it decreases in mode I. In general, in both modes I and II, the major-groove width appears to be closely related to the Slide between base-pairs. In mode II there appears to be a definite "point pivot" on the major-groove side of the two base-pairs that constitute a dinucleotide step, formed either by the steric interlocking of propeller-twisted base-pairs or by a bifurcated hydrogen bond. Distortion of DNA in mode II seems to be an intrinsic property of the double-helical structure, since it occurs whether protein is bound on the major-groove side (e.g. CAP) or on the minor-groove side (e.g. TBP). Mode II distortion occurs in a wider range of steps than those that show the largest mode-I variation; nevertheless, "access" to mode II deformation appears to be gained via mode I distortion at particular steps CA/TG and TA/TA.

Aspects of the mechanics of lobed liposomes.

Hotani has studied, by means of dark-field light microscopy, morphological transformations which unilamellar liposomes undergo when their interior volume decreases steadily with time as a consequence of osmosis. In a previous paper, we made a theoretical study of the initial buckling of an originally spherical vesicle into the observed oblate spheroidal shape; and we argued that some in-plane shear elastic stiffness is required-in addition to the well-known flexural stiffness of the lipid bilayer-in order to explain the observed phenomena. In the present paper, we consider a later stage in the chain of morphological transitions observed by Hotani, when a series of cudgel-shaped lobes have sprung out of a previously axisymmetric, biconcave-shaped vesicle. Specifically, we compare the observed shapes of such lobes with half of a series of "peanut"-shaped vesicles that are an equilibrium conformation of an initially spherical liposome under reduced internal volume. We find that the shapes do not match well. On the other hand, the observed lobe forms do match satisfactorily portions of "undulating tube" shapes which evolve from a hypothetical cylindrical vesicle, according to some simple calculations. In view of this agreement, we are led to propose that the formation of cudgel-shaped lobes requires some sliding of one lipid monolayer over another. This conflicts, of course, with the Love-Kirchhoff hypothesis which is normally invoked at the outset of analyses of lipid vesicles by means of classical thin-shell theory; but it is in accord with previous suggestions in the context of more obviously severe distortion of the lipid bilayer.

Topological measurement of protein-induced DNA bend angles.

A new method measures independently the changes in the DNA bend and winding angles that occur when a protein binds to its specific site in DNA. The procedure requires an investigation of the change in DNA topology induced when the protein binds to tandemly repeated sites of varying repeat length in circular DNA. This new method is superior to currently used methods because topology permits the measurements to be derived absolutely from first principles, and no comparison standards are required. The method is used to determine the bend and winding angles induced when catabolite gene activator protein binds to its site. The bend value obtained (69 +/- 4 degrees) is intermediate to those reported for two crystal forms of the complex.

Propeller-twisting of base-pairs and the conformational mobility of dinucleotide steps in DNA.

When DNA is bent around a protein, it must distort. The distortion occurs by changes in the conformation of successive dinucleotide steps. Bending does not necessarily occur uniformly: some steps might remain particularly rigid, i.e. they might deform relatively little, while others might take more than their proportional share of deformation. We investigate here the deformational capacity of specific dinucleotide steps by examining a database of crystallized oligomers. Dividing the steps into ten types by sequence (AA( = TT), AC( = GT), AG( = CT), AT, CA( = TG), CG, GA( = TC), GC, GG( = CC) and TA), we find that some step types are practically rigid, while others have considerable internal mobility or conformational flexibility. Now in general base-pairs are not planar, but have Propeller-Twist. We find a clear empirical correlation between the level of Propeller-Twist in the base-pairs and the flexibility of the dinucleotide step which they constitute. Propeller-Twist in the base-pairs makes stacking into a dinucleotide step more awkward than in plane base-pairs. In particular, it provides a stereochemical "locking" effect which can make steps with highly Propeller-Twisted base-pairs rigid. Although the origins of Propeller-Twist are not yet clearly understood, this result provides a key to understanding the flexibility of DNA in bending around proteins.

A useful role for "static" models in elucidating the behaviour of DNA in solution.

Double-helical DNA is a long and flexible molecule that is in constant motion under thermal perturbations, more so in solution that in the crystal. Some workers, for example Olsen et al., have argued that the behaviour of this molecule in assays such as circularization or gel electrophoresis can only be understood properly by means of theories that take full account of its dynamical nature due to thermal motions. Other workers, per contra, have claimed success at explaining aspects of the behaviour of DNA in solution by means of "static" models that focus on "time-averaged" conformations. In these static models, the intrinsic curvature of DNA and its flexibility are both related to sequence-dependent base-stacking effects, that are susceptible to study by the inherently static tools of X-ray crystallography and electron microscopy. Here we examine the question of whether such static models can, in practice, provide a clear understanding of what are generally acknowledged to be dynamic phenomena. Our investigation discusses some general principles of scientific method, and how suitable conceptual models are chosen; it describes the basic concept of "persistence length", and argues that long, superhelical DNA may be regarded at once as locally stiff yet globally flexible; it cites experimental evidence on gel-running which suggests that the flexibility of the molecule is not a crucial factor in relation to its mobility in electrophoretic gels; and it summarizes many data from gel-running, X-ray crystallography and electron microscopy, all of which provide a similar picture of DNA in solution as a stable, sequence-dependent polymer. Therefore, our investigation clearly favours the use of static models to explain many important aspects of the behaviour of DNA in solution; while it accepts the use of "dynamic" models in certain specific cases, such as the kinetics of circularization, where the rate-limiting step is a high-energy thermal vibration away from the most-stable structure.

Structural mechanics of bent DNA.

The DNA molecule is a familiar object. It is often depicted in magazines and advertisements as a double helix, with the letters of the genetic code strung along the two spiral backbones and joined together in pairs. In such pictures the molecule is usually shown as straight; yet in the chromosomes of living organisms, DNA is curved and wound up into condensed packages. This article explains what is involved in such bending of DNA in the cell. It uses the ideas of structural mechanics--a tool of engineers--to show how the various components fit together when the molecule is bent.

The assessment of the geometry of dinucleotide steps in double-helical DNA; a new local calculation scheme.

In this paper, we develop a new local Euler-angle-based scheme for assessing the internal kinematics or geometry of a general dinucleotide step in double-helical DNA. The geometry of a dinucleotide step is completely defined by: (1) the base-pair parameters that describe the relative position and orientation of one base with respect to the other in a standard Watson-Crick base-pair, and (2) the step parameters that describe the relative position and orientation of the two base-pairs. The key feature of our scheme is that it makes use of the concept of a mid-step reference frame. In addition to ensuring that identical values of step parameters are obtained irrespective of the direction of reckoning of a dinucleotide step (in the 5'-->3' direction along either strand), this mid-step-triad concept leads to local definitions of the step parameters that render them independent of the overall global conformation of the oligomer in question. In addition to presenting our own calculation scheme we also examine critically the most widely used package for the calculation of some of the step and base-pair parameters, viz, the NEWHELIX suite of programmes by R.E. Dickerson. Finally, a dodecamer, a decamer and an octamer are arbitrarily selected from a public data-base (N.D.B at Rutgers), and their step parameters are calculated by using both NEWHELIX and the proposed scheme. A comparison of the results is given whereby it is shown that for the step parameters: Helical Twist and Slide, and the base-pair parameters Propeller and Buckle, NEWHELIX and our proposed scheme give rather similar values. Substantial differences are seen, however, in the case of Rise. Two alternative definitions are given by NEWHELIX for the calculation of Roll and Tilt. Whereas one definition agrees well with our proposed scheme, the other is substantially different.

The mechanics of axially symmetric liposomes.

Hotani has filmed morphological transformations in unilamellar liposomes, starting from a spherical shape, when the interior volume decreases steadily. Hotani's liposomes showed no evidence of general thermal fluctuations. We use a finite-deformation theory of axisymmetric, quasi-static thin shells to analyze theoretically bifurcations and changes of shape in liposomes under decreasing volume. The main structural action in a lipid bilayer is generally agreed to be its elastic resistance to bending, and it is usual to regard surface deformation as being like that of a two-dimensional liquid. We find, however, that some in-plane shear elasticity is also needed in order to produce the observed post-bifurcation behavior. Such an elasticity would be difficult to measure directly.

A study of electrophoretic mobility of DNA in agarose and polyacrylamide gels.

The aim of this paper is to clarify the mechanism of gel electrophoresis of DNA under constant-field conditions. We have conducted a large number of experiments on double-stranded DNA varying in length between approximately 10 and approximately 50,000 base-pairs, in both agarose and polyacrylamide gels ranging from 0.5% to 12% concentration, and with electric field strengths ranging from 0.5 to 8 V/cm. We have made (logarithmic) plots of velocity against length of DNA for all of the various test conditions. At the left-hand side of these plots, all of the empirical curves have a unique, standard shape. When the curves are normalized so that their left-hand parts coincide, a second feature emerges in that, while for any given test the curve follows the "master curve" up to a certain point, it then "breaks away" and becomes horizontal. We describe these two patterns of behaviour as "regions 1 and 2", respectively. We find simple yet comprehensive empirical formulae that fit the observations in the two regions of behaviour: these express the velocity in terms of length of DNA, electric field strength and gel concentration. We then construct two separate theories for the two regions of behaviour. The first theory involves the statistics of motion of an object through a random array of gel obstacles, with the instantaneous speed depending on the number of obstacles with which the object is currently in contact. The second theory is based on the mechanical hypothesis (for which there is other, independent support) that the DNA moves through the gel by piling up against a barrier, which eventually breaks or deforms under the resulting force, thereby allowing the DNA to move on to the next barrier. The statistical theory is an adaptation of existing work, while the mechanical one is new. We also describe experiments on the migration of repeated-sequence, curved DNA with length up to 1500 base-pairs, and we discuss its behaviour in terms of our two theories. Our studies by electron microscopy are consistent with the view that this repeated-sequence DNA adopts a superhelical configuration. Finally, we show that a very wide range of observations may be understood clearly by means of our two theoretical schemes.

The intrinsic curvature of DNA in solution.

We propose a detailed quantitative scheme for explaining the anomalous electrophoretic mobility in polyacrylamide gels of repeating sequence DNA. We assume that such DNA adopts a superhelical configuration in these circumstances, and migrates less quickly than straight DNA of the same length because it can only pass through larger holes. The retardation is maximal when the length of the DNA reaches one superhelical turn, but is less for shorter pieces. We attribute the curvature of the superhelix to different angles of roll at each kind of dinucleotide step, i.e. an opening up of an angle by an increased separation on the minor-groove side. The main effect is due to a difference of about 3 degrees in roll values between AA/TT and other steps, together with a difference of about 1 degree in the angle of helical twist: we deduce these values explicitly from some of the available data on gel-running. The scheme involves a simple calculation of the superhelical parameters for any given repeating sequence, and it gives a good correlation with all of the available data. We argue that these same base-step angular parameters are also consistent with observations from X-ray diffraction of crystallized oligomers, and particularly with the recent data on CGCA6GCG from Nelson et al. We are concerned here with the intrinsic curvature of unconstrained DNA, as distinct from the curvature of DNA in association with protein molecules; and this paper represents a first attempt at an absolute determination.

Sequence-specific positioning of core histones on an 860 base-pair DNA. Experiment and theory.

Previous experiments have shown that the locations of the histone octamer on DNA molecules of 140 to 240 base-pairs (bp) are influenced strongly by the nucleotide sequence. Here we have studied the locations of the histone octamer on a relatively long DNA molecule of 860 bp, using two different nucleases, micrococcal and DNAase I. Data were obtained from both the protein--DNA complexes and from the naked DNA at single-bond resolution, and then were analyzed by densitometry to yield plots of differential cleavage, which show clearly the changes in cutting due to the addition of protein. Our results show that the placement of core histones on the 860 bp molecule is definitely non-random. The digestion data provide evidence for five nucleosome cores, the centers of which lie in defined locations. In all but one of these protein--DNA complexes, the DNA adopts a unique, highly preferred rotational setting with respect to the protein surface. Another protein--DNA complex is unusual in that it protects 200 bp from digestion, yet is cut in its very center as if it were split into two parts. The apparent average twist of the DNA within all of these protein--DNA complexes is 10.2(+/- 0.1) bp, as measured by the periodicity of DNAase I digestion. This value is in excellent agreement with the twist of 10.21(+/- 0.05) bp deduced from the periodicity of sequence content in chicken nucleosome core DNA. In addition, we observe a discontinuity in the periodic cutting by DNAase I of about -1 to -3 bonds in going from any nucleosome core to the next. The most plausible interpretation of this discontinuity is that it reflects the angle by which adjacent protein--DNA complexes are aligned. Thus, any nucleosome may be related to its neighbor by a left-handed rotation in space of -1/10.2 to -3/10.2 helix turns, or -35 degrees to -105 degrees. Repeated many times, this operation would build a long, left-handed helix of nucleosomes similar to that described by many workers for the packing of nucleosomes in chromatin. In order to look for any long-range influences on the positioning of the histone octamer in the 860 bp molecule (as would be expected if the nucleosomes have to fit into some higher-order structure), we have examined the locations of the histone octamer on five different isolated short fragments of the 860-mer, all of nucleosomal length.(ABSTRACT TRUNCATED AT 400 WORDS)