Theory of Phyllotaxis. I. A Geometric Model for Helical Forms of Consecutive Phyllotaxis

We have developed a geometric model for helical forms of consecutive phyllotaxis on the basis of an axiomatic approach. It follows from the model that rudiment growth and the movement of the cylindrical rudiment surface in the absence of a displacement in the direction along the rudiment axis leads to a repeating transition of tetragonal packaging of the rudiment into hexagonal packaging and vice versa. Under these conditions, sequences of rudiments produce left-handed and right-handed helices, the number of which at the circumference of the cylinder corresponds to adjacent numbers of the Fibonacci series. We demonstrate that the left-handed and right-handed isomers of helical forms of the consecutive phyllotaxis appear as a result of the transition of an unstable symmetric structure of the embryo at early developmental stages into stable left-handed or right-handed structures.     

Morphodynamics of phyllotaxis

A morphodynamic model for phyllotaxis based upon an axiomatic approach is presented. We show that the helical forms of alternate phyllotaxis can be derived from the assumption of the rudiment's growth and movement on the cylindrical embryo surface in the absence of a longitudinal displacement. This leads to the repeating transition of tetragonal packaging of the rudiments into hexagonal packaging and vice versa. Under these conditions, sequences of rudiments produce either left-handed or right-handed helices, the number of which at the circumference of the cylinder corresponds to adjacent numbers of the Fibonacci series. Cross-opposite phyllotaxis forms are defined as superior with respect to the alternate ones, and verticillate phyllotaxis forms as superior with respect to the cross-opposite ones. Different phyllotaxis forms can be interpreted as a result of stretching of crystalline structures of the embryo formed by dense packing of rudiments. The superior phyllotaxis forms can be considered as the additive summation of lower forms. Morphodynamic mechanisms underlying the formation of multiple forms of helical phyllotaxis are discussed.     

The theory of phyllotaxis. II. Crystallographic interpretation of the interrelation between lower and superior phyllotaxis forms

Cross-opposite phyllotaxis forms are defined as superior with respect to the alternate ones and verticillate phyllotaxis forms as superior with respect to the opposite ones. Different phyllotaxis forms can be interpreted as a result of stretching of crystal-like structures of the embryo formed by dense packing of rudiments. Based on hypothetical concepts of the properties of plant rudiments and embryos, possible mechanisms of the formation of superior phyllotaxis forms from the lower ones have been analyzed. It was shown that the superior phyllotaxis forms can be considered as the results of additive summation of the lower forms. The theoretical conclusions are confirmed by the examples of polymorphic phyllotaxis in conspecific plants and by the facts of accidental splitting of superior phyllotaxis forms into the corresponding lower forms in nature and in experiment. The mechanisms underlying the formation of multiple forms of helical phyllotaxis have been proposed. The concept of a new type of mixed hexagonal-tetragonal phyllotaxis has been formulated and the mechanism of its formation has been considered. The forms of corn grain packaging in the corncob and leaf arrangement on the strawberry tomato stem are given as examples of true hexagonal-tetragonal phyllotaxis in nature.     

Structural theory of phyllotaxis. II. Relationship between lower and higher forms of phyllotaxis

Opposite phyllotaxis forms are defined as superior ones in relation to alternate phyllotaxis forms, and verticillate phyllotaxis forms are defined as superior ones in relation to opposite phyllotaxis forms. On the basis of hypothetical notions about the properties of plant bumps and embryos, the probable mechanisms of creation of superior phyllotaxis forms from the lower ones are analyzed. It is shown that superior phyllotaxis forms can be considered to result from the combination of lower ones and that the superior forms can be split into the corresponding lower ones under artificial or natural influences.     

Structural study of poly(beta-benzyl-L-aspartate) monolayers at air-liquid interfaces.

As normally studied, in the solid state or in solution, poly(beta-benzyl-L-aspartate) (PBLA) differs from the other helical polyamino acids in that its alpha-helical conformation is most stable in the left-handed rather than in the right-handed form. The slightly lower energy per residue for the left-handed form in PBLA is easily perturbed, however. The helical screw sense can be inverted in a polar environment and, upon heating above 100 degrees C, a distorted left-handed helix or omega-helix is irreversibly formed. From external reflectance Fourier transform infrared measurements at the air-water interface, the conformation of PBLA in the monolayer state is directly established for the first time. The infrared frequencies of the amide bands suggest that right-handed alpha-helices are formed on the surface of water immediately after spreading the monolayers and independently of the polypeptide conformational distribution in the spreading solution. The right-handed helical form prevails throughout the slow compression of the Langmuir monolayers to collapsed films. The helical screw sense can be reversed by lowering the polarity of the aqueous phase. In addition, an alternate conformation similar to the omega-helix forms on addition of small amounts of isopropanol to the aqueous subphase, and appears to be an intermediate in the helix-helix transition.     

Conformational studies of polymers and copolymers of L-aspartate ester. II. Infrared studies and the factors involved in the formation of the omega-helix

It has already been show that the helix senses of poly(β-benzyl L -aspartate) and poly(β-methyl L -aspartate) are left-handed, while the poly esters of n-propyl, isopropyl, n-butyl, and phenethyl L -asparate are all right-handed. The effect of changes in helix sense from the left-handed to the right-handed α-helical form on the infrared spectra of copolymers of benzyl L -aspartate with ethyl, n-butyl, isopropyl, n-propyl, and phenethyl L -aspartate have been studied. Those show that for the right-handed helical form the amide band frequencies fall within the range given by Elliott,⁷ while for the left-handed form the frequencies are higher. The frequency ranges for the two helix senses are given and have been used to show that poly (β-n-propyl L -aspartate) in chloroform solution undergoes a transition from the right-handed to the left-handed helix form on heating. Polarized infrared studies of the different copolymers show that the disposition of the side chain ester groups is different for the two forms. Although methyl L -aspartate forms a left-handed α-helix similar to benzyl L -aspartate, the introduction of methyl L -aspartate residues into poly (β-benzyl L -aspartate) prevents the formation of the ω-helix. The factors involved in the formation of this helix form are discussed.     

DNA duplex with the potential to change handedness after every half a turn.

Polymorphic forms of the DNA duplex with long stretches of structural monotony are known. Several alternating purine-pyrimidine sequences have been shown to adopt left-handed Z-conformation. We report a DNA sequence d(CGCGCGATCGAT)n exhibiting alternating right-handed B and left-handed Z helical conformation after every half a turn. Further, this unusual conformation with change in handedness after every six base pairs was induced at physiological superhelical density.     

Flagellar filaments of the deep-sea bacteria Idiomarina loihiensis belong to a family different from those of Salmonella typhimurium

The helical filaments of the bacterial flagella so far studied seem to be universal in the bacterial kingdom. Despite the variation in flagellin molecular masses, which range from 24 kDa to 62 kDa in different species, there are only two forms: either the so-called Normal (left-handed) or the Curly (right-handed). The Normal and Curly helical forms are asymmetric; the two characteristic helical parameters, which are the pitch and diameter, of Normal filaments are twice those of Curly filaments. Both the universality of these two helical forms and their asymmetry are biological puzzles. We found that the marine bacteria Idiomarina loihiensis have flagella with left-handed Curly-like filaments. Analysis of the polymorphic forms under different pH conditions showed that the Curly-like filaments are actually Normal filaments having a smaller pitch and diameter than those of Salmonella typhimurium. A minor modification of Calladine's model for a filament lattice can explain the variant helical forms. Pseudomonas aeruginosa filaments also belong to the family of I.loihiensis filaments. Thus, there are at least two families of flagella filaments.     

Three-dimensional structure of complexes of single-stranded DNA-binding proteins with DNA. IKe and fd gene 5 proteins form left-handed helices with single-stranded DNA

Specimen-tilting in an electron microscope was used to determine the three-dimensional architecture of the helical complexes formed with DNA by the closely related single-stranded DNA binding proteins of fd and IKe filamentous viruses. The fd gene 5 protein is the only member of the DNA-helix-destabilizing class of proteins whose structure has been determined crystallographically, and yet a parameter essential to molecular modeling of the co-operative interaction of this protein with DNA, the helix handedness, has not been available prior to this work. We find that complexes formed by titrating fd viral DNA with either the fd or IKe gene 5 protein have a left-handed helical sense. Complexes isolated from Escherichia coli infected by fd virus are also found to be left-handed helical; hence, the left-handed fd helices are not an artefact of reconstitution in vitro. Because the proteins and nucleic acid of the complexes are composed of asymmetric units which cannot be fitted equivalently to right-handed and left-handed helices, these results rule out a previous computer graphics atomic model for the helical fd complexes: a right-handed helix had been assumed for the model. Our work provides a defined three-dimensional structural framework within which to model the protein-DNA and protein-protein interactions of two structurally related proteins that bind contiguously and co-operatively on single-stranded DNAs.     

Photomorphogenesis and phyllotaxis during vegetative growth in Sinapis alba and Xanthium strumarium

Abstract The effect of light on the rate of formation of leaf primordia was investigated at the apex of seedlings of Sinapis alba and Xanthium strumarium. It was found that light accelerates this rate. On the other hand, no significant light effect was found on the angles of divergence of successive leaves during the transition from the almost decussate leaf position of the cotyledons to the helical phyllotaxis of the stem leaves. In fact, light and dark grown plants use the same leaves for the transition from decussate to helical phyllotaxis. Thus, if time is plotted in ‘biological units’ (number of primordia) there is no difference between light and dark grown plants. Using scanning electron microscope techniques it was found that the ‘primordia free apical area’ enlarges during development. The rate of enlargement is accelerated by light. However, if time is expressed in biological units (number of primordia) no difference between light and dark grown plants exists. It is concluded that light accelerates the realization of the apical pattern without interfering with the specification of the pattern. In other words, light accelerates the development of an apex without affecting the temporal and spatial coordination of the events.     

Methylation of cytosine in the 5-position alters the structural and energetic properties of the supercoil-induced Z-helix and of B-Z junctions

The structural and energetic consequences of cytosine methylation in the 5-position on the supercoil-dependent B-Z equilibrium in alternating dC-dG sequences cloned into recombinant plasmids were investigated. The helical parameters determined with the band shift method for right-handed [10.7 base pairs (bp)/turn] and left-handed (12.8 bp/turn) 5MedC-dG inserts were different from the helical repeat values for unmethylated dC-dG inserts (10.5 bp/turn in the right-handed and 11.5 bp/turn in the left-handed form). We analyzed the thermodynamic parameters delta GBZ (free energy difference per base pair between right-handed and left-handed helix structure), delta Gjx (free energy for formation of one B-Z junction), and b (helix unwinding at a junction region) for varying lengths of dC-dG inserts by two-dimensional gel electrophoresis and application of a statistical mechanics model. A comparison of plasmids fully methylated in vitro with HhaI methylase and their unmethylated counterparts revealed that delta Gjx is not significantly changed by cytosine methylation. However, this base modification results in an approximate 3-fold decrease of delta GBZ and an approximate 2-fold decrease of the unwinding b at B-Z junction regions. Analysis of a pair of related plasmids, each containing two dC-dG blocks, revealed qualitatively different transition behaviors. When the two dC-dG blocks were separated by 95 bp of a mixed sequence, they underwent independent B to Z transitions with separate nucleation events and junction formations. When the two blocks were separated by only a 4 bp GATC sequence, only one nucleation event was necessary, and the Z-helix spread across the nonalternating GATC region.(ABSTRACT TRUNCATED AT 250 WORDS)     

Biphasic tissue Doppler waveforms during isovolumic phases are associated with asynchronous deformation of subendocardial and subepicardial layers.

Subendocardial and subepicardial layers of the left ventricle (LV) are characterized with right- and left-handed helical orientations of myocardial fibers. We investigated the origin of biphasic deformations of the LV wall during isovolumic contraction (IVC) and relaxation (IVR). In eight open-chest adult pigs, strain rates were measured along the right- and left-handed helical directions in the LV anterior wall by implanting 16 sonomicrometry crystals. Sonomicrometry strain rates were compared with the longitudinal subendocardial strain rates obtained by tissue Doppler imaging. During ejection and diastolic filling, shortening and lengthening occurred synchronously along the right- and left-handed helical directions. However, during IVC and IVR, the deformations were dissimilar in the two directions. Transmural shortening during IVC occurred along the right-handed helical direction and was accompanied with transient lengthening in the left-handed helical direction. Conversely, during IVR, the LV lengthened along the left-handed helical direction and shortened in the right-handed helical direction. Peak subendocardial strain rates obtained by tissue Doppler imaging during IVC and IVR correlated with corresponding sonomicrometry strain rate values obtained along the right- and left-handed helical directions (r = 0.81, P < 0.001 and r = 0.70, P = 0.001, respectively). Our data suggest that brief counterdirectional movements occur within the LV wall during IVC and IVR. Shortening along the right-handed helical direction is accompanied with reciprocal lengthening in the left-handed helical direction during IVC and vice versa during IVR. The results support an association between asynchronous deformation of subendocardial and subepicardial muscle fibers and the biphasic isovolumic movements observed with high-resolution tissue Doppler imaging.     

Mechanics of bacterial macrofiber initiation.

The twisting and writhing during growth of single-cell filaments of Bacillus subtilis which lead to macrofiber formation was studied in both left- and right-handed forms of strains FJ7 and RHX. Filament bending, touching, and loop formation (folding), followed by winding up into a double-strand fiber, were documented. Subsequent folds that produced multistrandedness were also examined. The rate of loop rotation during winding up was measured for 26 loops from 16 clones. In most cases, the first loop formed turned at a lower rate than those produced by the following cycles of folding. The sequence of folding topologies differed in FJ7 and RHX strains and in left- versus right-handed structures. Right-handed FJ7 routinely gave rise to four-stranded helices at the second fold, whereas left-handed FJ7 and both left-handed and right-handed forms of RHX made structures with predominantly two double-stranded helical regions. Left-handed RHX structures frequently produced second folds within the initial loop itself, resulting in T- or Y-shaped fibers. Sixteen cases in which the initial touch of a filament to itself produced a loop that snapped open before it could wind up into a double-strand fiber were found. The snap motions were used to obtain estimates of the forces generated by helical growth of single filaments and to investigate theoretical models involving the material properties of cell filaments. In general, the mechanical behavior of growing single-cell filaments and fibers consisting of two-, three-, or four-strand helices was similar to that described for larger, mature, multifilament macrofibers. The behavior of multicellular macrofibers can be understood, therefore, in terms of individual cell growth.     

Ising model forB-Z transition in supercoiled DNA

The possible existence of nucleic acids in right-handed and left-handed helical forms is considered. A statistical mechanical model is developed to obtain an expression for a change in twist during helical transformation in terms of corresponding free energies and linking for a supercoiled DNA. The theoretically predicted values are compared with those determined experimentally. The physico-chemical significance of the parameters is discussed.     

Noise and robustness in phyllotaxis

A striking feature of vascular plants is the regular arrangement of lateral organs on the stem, known as phyllotaxis. The most common phyllotactic patterns can be described using spirals, numbers from the Fibonacci sequence and the golden angle. This rich mathematical structure, along with the experimental reproduction of phyllotactic spirals in physical systems, has led to a view of phyllotaxis focusing on regularity. However all organisms are affected by natural stochastic variability, raising questions about the effect of this variability on phyllotaxis and the achievement of such regular patterns. Here we address these questions theoretically using a dynamical system of interacting sources of inhibitory field. Previous work has shown that phyllotaxis can emerge deterministically from the self-organization of such sources and that inhibition is primarily mediated by the depletion of the plant hormone auxin through polarized transport. We incorporated stochasticity in the model and found three main classes of defects in spiral phyllotaxis--the reversal of the handedness of spirals, the concomitant initiation of organs and the occurrence of distichous angles--and we investigated whether a secondary inhibitory field filters out defects. Our results are consistent with available experimental data and yield a prediction of the main source of stochasticity during organogenesis. Our model can be related to cellular parameters and thus provides a framework for the analysis of phyllotactic mutants at both cellular and tissular levels. We propose that secondary fields associated with organogenesis, such as other biochemical signals or mechanical forces, are important for the robustness of phyllotaxis. More generally, our work sheds light on how a target pattern can be achieved within a noisy background.     

Theory of Phyllotaxis. 2. Crystallographic Interpretation of the Interrelation between Lower and Superior Phyllotaxis Forms

Cross-opposite phyllotaxis forms are defined as superior with respect to the alternate ones and verticillate phyllotaxis forms as superior with respect to the opposite ones. Different phyllotaxis forms can be interpreted as a result of stretching of crystal-like structures of the embryo formed by dense packing of rudiments. Based on hypothetical concepts of the properties of plant rudiments and embryos, possible mechanisms of the formation of superior phyllotaxis forms from the lower ones have been analyzed. It was shown that the superior phyllotaxis forms can be considered as the results of additive summation of the lower forms. The theoretical conclusions are confirmed by the examples of polymorphic phyllotaxis in conspecific plants and by the facts of accidental splitting of superior phyllotaxis forms into the corresponding lower forms in nature and in experiment. The hexagonal-tetragonal type of phyllotaxis was theoretically predicted and found in nature. The mechanism underlying the formation of multiple forms of the helical phyllotaxis was considered.     

Conformational studies of polymers and copolymers of L-aspartate esters. I. Preparation and solution studies

An alcoholysis method is described for the modification of high molecular weight poly(β-benzyl L -asparatate); by this method the benzyl groups in the polypeptide have been replaced by methyl, ethyl, isopropyl, n-propyl, and phenethyl groups to give a series of copolymers of each of the corresponding aspartate esters with benzyl L -aspartate. By repeating the reactions, replacement of better than 99% has been achieved in some cases to give in effect the homopolymer. Optical rotatory dispersion studies show that of all the systems studied only poly(β-methyl L -aspartate) has the left-handed helix sense, the others are right-handed. It is shown further that the helix sense is not an intrinsic property of the nature of the aspartate side chain. Raising the temperature of chloroform solutions of the right-handed form of the copolymers of benzyl L -aspartate and ethyl L -aspartate results in a transition to the left-handed helix, the temperature of the transition being dependent on the composition of the copolymer. Also poly(β-n-propyl L -aspartate) undergoes a transition from the right- to the left-handed helix form at 59°C. These results suggest a general pattern of behavior of poly(aspartate esters) and that with suitable conditions of solvent and temperature they may be in either the right- or left-handed helical form.     

Micro-video study of moving bacterial flagellar filaments: III. Cyclic transformation induced by mechanical force

Dynamic images of isolated bacterial flagellar filaments undergoing cyclic transformations were recorded by dark-field light microscopy and an ultrasensitive video camera. Flagellar filaments derived from Salmonella SJ25 sometimes stick to a glass surface by short segments near one end. When such a filament, which is a left-handed helix, was subjected to a steady flow of a viscous solution of methylcellulose, its free portion was found to transform cyclically between left-handed (normal) and right-handed (curly or semi-coiled) helical forms. The transformations did not occur simultaneously throughout the whole length of a filament, but occurred at a transition point, which proceeded along the filament. Each transformation process consisted of three phases: initiation, growth and travel. The magnitudes of the mechanical forces, torque and tension, which were generated on a filament by the viscous flow, were obtained by quantitative hydrodynamic analyses. The torque was found responsible for initiating the transformation. The critical magnitude of torque required to induce the normal to semi-coiled transformation was ?11 × 10^?19 N m and that for the reverse transformation from the semi-coiled to the normal form was 4 × 10^?19 N m. Therefore, the filaments showed the characteristics of hysteresis during the cyclic transformation. New types of unstable right-handed helical forms (medium and large) were also induced by mechanical force.     

A temperature-dependent Z to B to single-strand transition in d(CGCG)

A thermally induced left- to right-handed helical interconversion for the self-complementary tetradeoxynucleotide d(CGCG) has been observed in the presence of NaCl at concentrations from 1.5 to 2.25M. Analysis of the Raman spectrum of d(CGCG) in aqueous solution as a function of temperature indicates that at low temperature a left-handed Z-helical form is the predominant species. An increase in temperature results in a decrease in the population of the left-handed form and an increase in the population of the right-handed form. Further elevation in temperature results in extensive disruption of base stacking and a loss of secondary structure. This unstacking presumably represents dissociation to a single-stranded structure. The data suggest that a temperature-dependent Z to B to single-strand transition occurs under the conditions employed. A direct Z to single-strand conversion was not identified. In contrast to previous examples, the thermally induced left- to right-handed helical conversion for d(CGCG) can occur in the absence of chemically modified nucleic acid bases, alcohol solutions, or divalent ions.     

A new model for DNA containing A.T and I.C base pairs.

DNA polymers containing exclusively A.T or I.C base pairs frequently exhibit D- or E-type X-ray diffraction patterns when dried. The distribution of intensities in fiber patterns appears to demand helical structures with 7 and 7.5 bp/turn, respectively, but it is not stereochemically possible to wind a right-handed antiparallel B-family helix this tightly. It is a simple matter, however, to build a left-handed helix with 7-7.5 bp/turn by incorporating Hoogsteen pairing into a Z helix framework. X-ray intensities calculated from this novel left-handed Hoogsteen model provide as reasonable a fit to the D-DNA diffraction pattern as do intensities calculated from previously proposed right-handed 8-fold models.