C. R. Calladine

Mechanics of sequence-dependent stacking of bases in B-DNA

Abstract Dickerson and his colleagues have described the structure of the DNA dodecamer C-G-C-G-A-A-T-T-C-G-C-G in the B form at a level that shows clearly several aspects of some base sequence-dependent departures from the ideal, regular helical structure of B-DNA. I argue that the detailed conformation is a consequence of simple steric repulsive forces between purine bases in consecutive base-pairs but on opposite backbones. These repulsions are a consequence of the “propeller twist” of the base-pairs, together with the larger size of the purine bases, and they may occur in either the major or the minor groove. The argument is conducted in terms of the structural mechanics of a deformable elastic system. These repulsive forces between the base-pairs are resisted by stresses in the helical backbones, which may be studied quantitatively via the variation in torsion angles δ along the backbones, at the points where the sugar rings are connected. There is also a correlation between the cross-chain purine repulsions and the perturbations in helical twist angle between successive base-pairs. The work suggests some comments on the proposed “alternating B” form, the Z form and the A form of DNA.

Matrix analysis of statically and kinematically indeterminate frameworks

Abstract The paper is concerned with the structural mechanics of assemblies of bars and pinjoints, particularly where they are simultaneously statically and kinematically indeterminate. The physical significance of the four linear-algebraic vector subspaces of the equilibrium matrix is examined, and an algorithm is set up which determines the rank of the matrix and the bases for the four subspaces. In particular, this algorithm gives full details of any states of self-stress and modes of inextensional deformation which an assembly may possess. A scheme is devised for the segregation of inextensional modes into rigid-body modes (up to six of these may be allowed by the foundation constraints) and “internal” mechanisms. In some circumstances a state of self-stress may impart first-order stiffness to an inextensional mode. A matrix method for detecting this effect is devised, and it is shown that if there is no state of self-stress which imparts first-order stiffness to a given mode, then that mode can undergo rather large distortion which involves either zero change in length of the bars or, possibly, changes in length of third or higher order in the displacements. The significance of negative stiffness, as indicated by the matrix method, is discussed. The paper contains simple examples which illustrate all of the main points of the work.

Buckminster Fuller's “Tensegrity” structures and Clerk Maxwell's rules for the construction of stiff frames

Abstract Maxwell has shown that b bars assembled into a frame having j joints would, in general, be simply stiff if b = 3 j −6. Some of Buckminster Fullers “Tensegrity” structures have fewer bars than are necessary to satisfy Maxwells rule, and yet are not “mechanisms” as one might expect, but are actually stiff structures. Maxwell anticipates special cases of this sort, and states that their stiffness will “be of a low order”. In fact, the conditions under which Maxwells exceptional cases occur also permit at least one state of “self-stress” in the frame. Linear algebra enables us to find the number of “incipient” modes of low-order stiffness of the frame in terms of the numbers of bars, joints and independent states of self-stress. Self-stress in the frame has the effect of imparting first-order stiffness to the frame, and it seems from experiments that a single state of self-stress can stiffen a large number of modes. It is this factor which Fuller exploits to make satisfactory structures.

Strain-rate and inertia effects in the collapse of two types of energy-absorbing structure

Abstract The dynamic plastic collapse of energy-absorbing structures is more difficult to understand than the corresponding quasi-static collapse, on account of two effects which may be described as the “strain-rate factor” and the “inertia factor” respectively. The first of these is a material property whereby the yield stress is raised, while the second can affect the collapse mode, etc. It has recently been discovered [6,7]that structures whose load-deflection curve falls sharply after an initial “peak” are much more “velocity sensitive” than structures whose load-deflection curve is “flat-topped” (Fig. 1a); that is, when a given amount of energy is delivered by a moving mass, the final deflection depends more strongly on the impact velocity. In this paper we investigate strain-rate and inertia effects in these two types of structure by means of some simple experiments performed in a “drop hammer” testing machine, together with some simple analysis which enables us to give a satisfactory account of the experimental observations. The work is motivated partly by difficulties which occur in small-scale model testing of energy-absorbing structures, on account of the fact that the “strain-rate” and “inertia” factors not only scale differently in general, but also affect the two distinct types of structure differently.

Conformational characteristics of DNA: empirical classifications and a hypothesis for the conformational behaviour of dinucleotide steps

This paper is concerned with an investigation of the geometry and structure of DNA as revealed by X–ray diffraction of single crystal oligomeric structures. A database of atomic coordinates of 60 naked (i.e. not bound to any protein or drug) DNA oligomers (25 dodecamers, 18 decamers, 16 octamers and 1 tetramer) is set up and carefully described. An extensive empirical study of the geometries of DNA dinucleotide steps in the database, involving only unmodified Watson–Crick base pairs (A–T and G–C), is reported, and a number of new correlations and classifications are described in detail. The main conclusions include the kinematic classification of dinucleotide steps into two main classes: rigid and loose (or flexible or bistable). ‘Continuously flexible’ steps are shown to exercise their flexibility along a well–defined single–degree–of–freedom pattern, with roll, slide and twist all correlated linearly. The rigid steps are AA/TT, AT and GA/TC, and the loose (bistable) steps are GG/CC, GC, CG while the loose (continuously flexible) steps are CA/TG and TA. AC/GT is the least clear of all steps and it is perhaps best described as neither a rigid nor a loose step but rather an ‘intermediate’ step. The base–pair parameters are also carefully examined and the resulting pivotal correlation between the average propeller and the flexibility of the step (equals the standard deviation of slide), that we have recently described elsewhere (El Hassan and Calladine 1996), is examined in some detail. A simple two–parameter scheme for the description of the conformation of the sugar phosphate backbone is given and used to classify the sugar phosphate backbones in all entries of our database into A–backbone and B–backbone conformations. The role of the backbone in determining the conformational preferences of the dinucleotide steps is examined by demonstrating that whereas the B–backbone conformation permits a fairly narrow channel in the roll/slide/twist conformational space, with all three parameters linearly correlated, the A–backbone allows only a small ‘box” region near the high–roll, low–twist, low–slide end of the space.Finally, the empirically determined conformational characteristics of the various dinucleotide steps are accounted for in terms of (a) mechanical stacking effects associated with propeller–twisting of constituent base pairs (the propeller–flexibility correlation), (b) chemical stacking effects associated with the special electrostatic charge distributions and π–pi;effects in homogeneous G|C steps (Hunter 1993), and (c) backbone-dictated effects that govern in the absence of (a) and (b).

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.

First-order infinitesimal mechanisms

Abstract This paper discusses the analytical conditions under which a pin-jointed assembly, which has s independent states of self-stress and m independent mechanisms, tightens up when its mechanisms are excited. A matrix algorithm is set up to distinguish between first-order infinitesimal mechanisms (which are associated with second-order changes of bar length) and higher-order infinitesimal or finite mechanisms. It is shown that, in general, this analysis requires the computation of s quadratic forms in m variables, which can be easily computed from the states of self-stress and mechanisms of the assembly. If any linear combination of these quadratic forms is sign definite, then the mechanisms are first-order infinitesimal. An efficient and general algorithm to investigate these quadratic forms is given. The calculations required are illustrated for some simple examples. Many assemblies of practical relevance admit a single state of self-stress ( s = 1), and in this case the algorithm proposed is straightforward to implement. This work is relevant to the analysis and design of pre-stressed mechanisms, such as cable systems and tensegrity frameworks.

Principles of sequence-dependent flexure of DNA

The curvature of a bent rod may be defined in several different, but equivalent ways. The best way of describing the curvature of double-helical DNA is by an angle of turning per base step. Curvature comes mainly from the angle of roll between successive base-pairs, and this is defined as positive when the angle opens up on the minor groove side of the bases. DNA forms a plane curve if the roll angle values along the molecule alternate periodically between positive and negative, with a complete period equal to the helical repeat. It is known from studies of crystallized oligomers that the roll angles for particular dinucleotide steps have preferred values, or lie in preferred ranges of values. Therefore the formation of a plane curve will be easier with some base sequences of DNA than with others. We set up a computer algorithm for determining the ease with which DNA of given sequence will adopt a curved form. The algorithm has two different sets of constants: in model 1 the base step parameters come from an inspection of crystallized oligomers, and in model 2 data from a statistical survey of the incidence of dinucleotide steps in a large number of samples of chicken erythrocyte core DNA is incorporated. Both forms of the algorithm successfully locate the dyad of the nucleosome sequence (modulo 10) in a frog gene, and suggest strongly that sequence-dependent flexural properties of DNA play a part in the recognition of binding sites by nucleosome cores.

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)

Inertia and strain-rate effects in a simple plate-structure under impact loading

Summary Previous studies have shown that the way in which metal structures absorb energy by gross distortion under impact conditions depends on the generic type of structure. In particular they have shown that structures which respond to quasi-static testing by means of an initial peak load followed by a falling load as deformation proceeds (‘type II’ response, corresponding broadly to plates loaded endwise) exhibit both inertia and strain-rate effects under impact loading from moving strikers. This paper describes a detailed study of these phenomena by means of experiment and theory. Experiments were conducted in a drop-hammer rig on a large number of specimens having the same general geometry, but made in two different sizes and of two different materials (mild steel, aluminium alloy) chosen for their different strain-rate characteristics in the plastic range. The experiments involved the overall measurement of final distortion of the specimens in relation to a wide range of testing conditions with moderate velocity; strain gauge studies, high-speed photography and other investigations of the detailed behaviour. The main emphasis of the various assays was to discover the way in which the initial kinetic energy of the striker was dissipated within the structure. During the course of the work, Zhang and Yu proposed a simple analysis of the same phenomena by means of a model based on the ideas of classical inelastic impact theory. According to their theory, a significant fraction of the incident kinetic energy of the striker is absorbed during the initial impact event; and this fraction depends only on the ratio of the mass of the striker to the mass of the specimen and the initial crookedness, but not on the velocity of impact. Our experiments agreed with this analysis in some overall respects, but were irreconcilable with it in several others, for which we had amassed substantial data. We therefore produced a revised analysis, which was less austere than that of Zhang and Yu but which nevertheless remained essentially simple. We show in the paper that this new theory agrees satisfactorily with all aspects of the experimental observations. The analysis reveals clearly the roles of inertia and strain rate in impact conditions. It also produces two new dimensionless groups, which together provide a key to classification of the various patterns of behaviour which are possible in the impact response of ‘type II” specimens.