Gel electrophoretic analysis of the geometry of a DNA four-way junction

Branched DNA molecules (Holliday structures) are believed to be key intermediates in the process of homologous genetic recombination. However, despite the importance of such structures, their transient nature makes it difficult to analyze their physical properties. In an effort to evaluate several models for the geometry of such branched molecules, a stable, synthetic DNA four-way junction has been constructed. The geometry of the synthetic junction has been probed by gel electrophoresis, utilizing the fact that bent DNA molecules demonstrate reduced mobilities on polyacrylamide gels to an extent that varies with the degree of the bend angle. From the synthetic four-way junction, we have produced a set of molecules in which all combinations of two junction arms have been extended by 105 base-pairs. The electrophoretic mobilities of the extended junctions differ in a manner which indicates that the junction is not a completely flexible structure; nor is it tetrahedral or planar-tetragonal. Instead, the four strands that comprise the DNA four-way junction are structurally non-equivalent. The significance of these observations with regard to previous models for four-way junction geometry is discussed.     

Finding the optimal lengths for three branches at a junction

This paper presents an exact analytical solution to the problem of locating the junction point between three branches so that the sum of the total costs of the branches is minimized. When the cost per unit length of each branch is known the angles between each pair of branches can be deduced following reasoning first introduced to biology by Murray. Assuming the outer ends of each branch are fixed, the location of the junction and the length of each branch are then deduced using plane geometry and trigonometry. The model has applications in determining the optimal cost of a branch or branches at a junction. Comparing the optimal to the actual cost of a junction is a new way to compare cost models for goodness of fit to actual junction geometry. It is an unambiguous measure and is superior to comparing observed and optimal angles between each daughter and the parent branch. We present data for 199 junctions in the pulmonary arteries of two human lungs. For the branches at each junction we calculated the best fitting value of x from the relationship that flow ∞ (radius)^x. We found that the value of x determined whether a junction was best fitted by a surface, volume, drag or power minimization model. While economy of explanation casts doubt that four models operate simultaneously, we found that optimality may still operate, since the angle to the major daughter is less than the angle to the minor daughter. Perhaps optimality combined with a space filling branching pattern governs the branching geometry of the pulmonary artery.     

Modeling the large-scale geometry of human coronary arteries

Two principles suffice to model the large-scale geometry of normal human coronary arterial networks. The first principle states that artery diameters are set to minimize the power required to distribute blood through the network. The second principle states that arterial tree geometries are set to globally minimize the lumen volume. Given only the coordinates of an arterial tree's source and "leaves", the model predicts the nature of the network connecting the source to the leaves. Measurements were made of the actual geometries of arterial trees from postmortem healthy human coronary arteriograms. The tree geometries predicted by the model look qualitatively similar to the actual tree geometries and have volumes that are within a few percent of those of the actual tree geometries. Human coronary arteries are therefore within a few percent of perfect global volume optimality. A possible mechanism for this near-perfect global volume optimality is suggested. Also, the model performs best under the assumption that the flow is not entirely steady and laminar.     

The cost of departure from optimal radii in microvascular networks

In the Murray optimality model of branching vasculatures, the radii of vessels are related to blood viscosity, vascular metabolic rate, and blood flow rate, in such a way as to minimize the total work (hydraulic and metabolic) of the system. The model predicts that flow is proportional to the cube of a vessel radius, and that at junctions the cube of the radius of the parent vessel equals the sum of the cubes of the daughter radii. In comparing real vasculatures to the Murray model, we have previously had no expressions for evaluating the apparent energy cost for departures from the optimal junction exponent of 3. Such expressions are derived here. They show that junction exponents, from about 1.5 to large positive values, are within 5% of the energy minimum. With the new equations, observed individual junctions or entire vascular trees can be compared, energy-wise, with the Murray optimum. Junctions in the transverse arteriolar trees of cat sartorius muscle were compared to the Murray optimality model, using these new expressions. The junction exponents for these small pre-capillary vessels had a broad range, with a median value greater than the Murray optimum of 3. The exponents were restricted, however, to values requiring, at individual junctions, little increase in energy. The majority of junctions had energy costs less than 1% above the Murray minimum. For entire trees involving many junctions the departures from optimality averaged less than 10%. Thus, while the branching geometry for these microvascular trees deviates significantly from the Murray optimum in the direction of larger daughter to parent ratios, the departures are small in energy terms.     

Relation of branching angles to optimality for four cost principles

The literature has suggested that branching angles depend on some principle of optimality. Most often cited are the minimization of lumen surface, volume, power and drag. The predicted angles depend on the principle applied, chi and alpha. Assuming flow o r chi, chi can be determined from r chi 0 = r chi 1 + r chi 2 when the radii of the parent (r0) major (r1) and minor (r2) daughters are known. The term alpha = r2/r1. Using different values for chi and alpha, we present graphs for the major and minor branching angles theta 1 and theta 2 and psi = theta 1 + theta 2 for each of the four optimization principles. Because psi is almost independent of alpha for values of chi and alpha found in 198 junctions taken from a human pulmonary artery, we are able to produce a plot of psi versus chi for each of the four principles on one graph. A junction can be provisionally classified as optimizing for a given principle if, knowing chi, the psi obs - psi pred is least for that principle. We find that this nomographic classification agrees almost perfectly with a previous classification based on a more exacting measure, the percent cost index I, where I = observed cost/minimum cost. We explain why this is to be expected in most but not all cases. First we generate a contoured percent cost surface of c = I - 100 around the optimally located junction, J, and superimpose a surface of equal angular deviations a = psi pred-psi obs. We find that c increases and a usually increases with distance from J as the actual junction moves along a straight line away from J. We then produce a plot of c versus a for two competing principles. A comparison of the principles demonstrates that, for most cases, a is smaller for the principle which has the smaller c value.     

Cost analysis of arterial branching in the cardiovascular systems of man and animals

A new scheme is presented whereby data on arterial branching can be interpreted in terms of direct cost to the physiological system. The scheme makes it possible to assess, at a glance, the true degree of optimality of an arterial network. Departure from optimality is indicated in terms of cost, rather than in terms of the difference between theoretical and measured branching angles. The scheme is applied to several groups of biological data and new conclusions are reached with regard to their degrees of optimality.     

Dominance of geometric overelastic factors in pulse transmission through arterial branching

The branching characteristic of the arterial system is such that blood pressure pulses propagate with minimum loss. This characteristic depends on the geometric and elastic properties of branching vessels. In the current investigation, mathematical relations of branching geometry and elastic properties are formulated and their relative contributions to pulse reflection at an arterial junction are analyzed. Results show that alteration of pulse transmission through the junction is more significantly affected by changes in branching vessel radii and wall thickness than by corresponding percentage changes in vessel wall elastic moduli.     

The Mass-Longevity Triangle: Pareto Optimality and the Geometry of Life-History Trait Space

When organisms need to perform multiple tasks they face a fundamental tradeoff: no phenotype can be optimal at all tasks. This situation was recently analyzed using Pareto optimality, showing that tradeoffs between tasks lead to phenotypes distributed on low dimensional polygons in trait space. The vertices of these polygons are archetypes—phenotypes optimal at a single task. This theory was applied to examples from animal morphology and gene expression. Here we ask whether Pareto optimality theory can apply to life history traits, which include longevity, fecundity and mass. To comprehensively explore the geometry of life history trait space, we analyze a dataset of life history traits of 2105 endothermic species. We find that, to a first approximation, life history traits fall on a triangle in log-mass log-longevity space. The vertices of the triangle suggest three archetypal strategies, exemplified by bats, shrews and whales, with specialists near the vertices and generalists in the middle of the triangle. To a second approximation, the data lies in a tetrahedron, whose extra vertex above the mass-longevity triangle suggests a fourth strategy related to carnivory. Each animal species can thus be placed in a coordinate system according to its distance from the archetypes, which may be useful for genome-scale comparative studies of mammalian aging and other biological aspects. We further demonstrate that Pareto optimality can explain a range of previous studies which found animal and plant phenotypes which lie in triangles in trait space. This study demonstrates the applicability of multi-objective optimization principles to understand life history traits and to infer archetypal strategies that suggest why some mammalian species live much longer than others of similar mass.     

Dispatching and Conflict-Free Routing of Automated Guided Vehicles: An Exact Approach

This article presents an exact solution approach for the problem of the simultaneous dispatching and conflict-free routing of automated guided vehicles. The vehicles carry out material handling tasks in a flexible manufacturing system (FMS). The objective is to minimize the costs related to the production delays. The approach is based on a set partitioning formulation. The proposed model is solved to optimality by a column generation method, which is embedded in a branch-and-cut exploration tree. The proposed model and solution methodology are tested on several scenarios with up to four vehicles in the manufacturing system. The results show that most of these scenarios can be solved to optimality in less than three minutes of computational time.     

Optimality principles for model-based prediction of human gait

Although humans have a large repertoire of potential movements, gait patterns tend to be stereotypical and appear to be selected according to optimality principles such as minimal energy. When applied to dynamic musculoskeletal models such optimality principles might be used to predict how a patient's gait adapts to mechanical interventions such as prosthetic devices or surgery. In this paper we study the effects of different performance criteria on predicted gait patterns using a 2D musculoskeletal model. The associated optimal control problem for a family of different cost functions was solved utilizing the direct collocation method. It was found that fatigue-like cost functions produced realistic gait, with stance phase knee flexion, as opposed to energy-related cost functions which avoided knee flexion during the stance phase. We conclude that fatigue minimization may be one of the primary optimality principles governing human gait.     

Structure of Three-Way DNA Junctions 1. Non-Planar DNA Geometry

Abstract

Three-way junctions were obtained by annealing two synthetic DNA-oligomers. One of the strands contains a short palindrome sequence, leading to the formation of a hairpin with four base pairs in the stem and four bases in the loop. Another strand is complementary to the linear arms of the first hairpin-containing strand. Both strands were annealed to form a three-way branched structure with sticky ends on the linear arms. The branched molecules were ligated, and the ligation mixture was analysed on a two-dimensional gel in conditions which separated linear and circular molecules. Analysis of 2D-electrophoresis data shows that circular molecules with high mobility are formed. Formation of circular molecules is indicative of bends between linear arms. We estimate the magnitude of the angle between linear arms from the predominant size of the circular molecules formed. When the junction-to-junction distance is 20–21 bp, trimers and tetramers are formed predominately, giving an angle between linear arms as small as 60–90°. Rotation of the hairpin position in the three- way junction allowed us to measure angles between other arms, yielding similar values. These results led us to conclude that the three-way DNA junction possesses a non-planar pyramidal geometry with 60–90° between the arms. Computer modeling of the three-way junction with 60° pyramidal geometry showed a predominantly B-form structure with local distortions at the junction points that diminish towards the ends of the helices. The size distributions of circular molecules are rather broad indicating a dynamic flexibility of three-way DNA junctions.     

Fourth Rank Immobile Nucleic Acid Junctions

Abstract

An immobile nucleic acid junction composed of four dodecanucleotides has been designed according to principles of minimum symmetry aided by equilibrium calculations, and has been synthesized by automated phosphotriester techniques. We can demonstrate its tetrameric character and its 1:1:1:1 stoichiometry by gel electrophoresis. Thermal denaturation monitored by ultraviolet hyperchromism indicates that the complex is stable relative to its component arms. High resolution NMR spectroscopy suggests that this junction exists in more than one conformer at room temperature. The data from this junction are compared with the data from a similar junction composed of four hexadecanucleotides.     

The stereochemistry of a four-way DNA junction: a theoretical study.

The stereochemical conformation of the four-way helical junction in DNA (the Holliday junction; the postulated central intermediate of genetic recombination) has been analysed, using molecular mechanical computer modelling. A version of the AMBER program package was employed, that had been modified to include the influence of counterions and a global optimisation procedure. Starting from an extended planar structure, the conformation was varied in order to minimise the energy, and we discuss three structures obtained by this procedure. One structure is closely related to a square-planar cross, in which there is no stacking interaction between the four double helical stems. This structure is probably closely similar to that observed experimentally in the absence of cations. The remaining two structures are based on related, yet distinct, conformations, in which there is pairwise coaxial stacking of neighbouring stems. In these structures, the four DNA stems adopt the form of two quasi-continuous helices, in which base stacking is very similar to that found in standard B-DNA geometry. The two stacked helices so formed are not aligned parallel to each other, but subtend an angle of approximately 60 degrees. The strands that exchange between one stacked helix and the other are disposed about the smaller angle of the cross (i.e. 60 degrees rather than 120 degrees), generating an approximately antiparallel alignment of DNA sequences. This structure is precisely the stacked X-structure proposed on the basis of experimental data. The calculations indicate distortions from standard B-DNA conformation that are required to adopt the stacked X-structure; a widening of the minor groove at the junction, and reorientation of the central phosphate groups of the exchanging strands. An important feature of the stacked X-structure is that it presents two structurally distinct sides. These may be recognised differently by enzymes, providing a rationalisation for the points of cleavage by Holliday resolvases.     

Structures of helical junctions in nucleic acids

Our knowledge of the architectural principles of nucleic acid junctions has seen significant recent advances. The conformation of DNA junctions is now well understood, and this provides a new basis for the analysis of important structural elements in RNA. The most significant new data have come from X-ray crystallography of four-way DNA junctions; incidentally showing the great importance of serendipity in science, since none of the three groups had deliberately set out to crystallise a junction. Fortunately the results confirm, and of course extend, the earlier conformational studies of DNA junctions in almost every detail. This is important, because it means that these methods can be applied with greater confidence to new systems, especially in RNA. Methods like FRET, chemical probing and even the humble polyacrylamide gel can be rapid and very powerful, allowing the examination of a large number of sequence variants relatively quickly. Molecular modelling in conjunction with experiments is also a very important component of the general approach. Ultimately crystallography provides the gold standard for structural analysis, but the other, simple approaches have considerable value along the way. At the beginning of this review I suggested two simple folding principles for branched nucleic acids, and it is instructive to review these in the light of recent data. In brief, these were the tendency for pairwise coaxial stacking of helical arms, and the importance of metal ion interactions in the induction of folding. We see that both are important in a wide range of systems, both in DNA and RNA. The premier example is the four-way DNA junction, which undergoes metal ion-induced folding into the stacked X-structure that is based on coaxial stacking of arms. As in many systems, there are two alternative ways to achieve this depending on the choice of stacking partners. Recent data reveal that both forms often exist in a dynamic equilibrium, and that the relative stability of the two conformers depends upon base sequence extending a significant distance from the junction. The three-way junction has provided a good test of the folding principles. Perfect three-way (3H) DNA junctions seem to defy these principles in that they appear reluctant to undergo coaxial stacking of arms, and exhibit little change in conformation with addition of metal ions. Modelling suggests that such a junction is stereochemically constrained in an extended conformation. However, upon inclusion of a few additional base pairs at the centre (to create a 3HS2 junction for example) the additional stereochemical flexibility allows two arms to undergo coaxial stacking. Such a junction exhibits all the properties consistent with the general folding principles, with ion-induced folding into a form based on pairwise coaxial stacking of arms in one of two different conformers. The three-way junction is therefore very much the exception that proves the rule. It is instructive to compare the folding of corresponding species in DNA and RNA, where we find both similarities and differences. The RNA four-way junction can adopt a structure that is globally similar to the stacked X-structure (Duckett et al. 1995a), and the crystal structure of the DNAzyme shows that the stacked X-conformation can include one helical pair in the A-conformation (Nowakowski et al. 1999). However, modelling suggests that the juxtaposition of strands and grooves will be less satisfactory in RNA, and the higher magnesium ion concentration required to fold the RNA junction indicates a lower stability of the antiparallel form. Perhaps the biggest difference between the properties of the DNA and RNA four-way junctions is the lack of an unstacked structure at low salt concentrations for the RNA species. This clearly reflects a major difference in the electrostatic interactions in the RNA junction. In general the folding of branched DNA provides some good indications on the likely folding of the corresponding RNA species, but caution is required in making the extrapolation because the two polymers are significantly different. A number of studies point to the flexibility and malleability of branched nucleic acids, and this turns out to have particular significance in their interactions with proteins. Proteins such as the DNA junction-resolving enzymes exhibit considerable selectivity for the structure of their substrates, which is still not understood at a molecular level. Despite this, it appears to be universally true that these proteins distort the global, and in some cases at least the local, structure of the junctions. The somewhat perplexing result is that the proteins appear to distort the very property that they recognise. In general it seems that four-way DNA junctions are opened to one extent or another by interaction with proteins. (ABSTRACT TRUNCATED)     

Prediction of LDL concentration at the luminal surface of a vascular endothelium

To find out whether concentration polarization of low-density lipoprotein (LDL) occurs at the surface of a vascular endothelium or not, transport of LDL in flowing blood to an water-permeable endothelium was studied theoretically by means of CFD. Calculations were carried out for an endothelium exposed to a Couette flow by assuming that the surface geometry of the endothelium could be expressed by a cosine function. Two typical cases were considered for the permeability of endothelium to water; one was uniform permeability everywhere in the endothelium, and the other was uneven permeability which was augmented at the intercellular junction. It was found that, in both cases, the surface concentration of LDL increased in going distally from the entrance, taking locally high and low values at the valleys and hills of the endothelium, respectively, and the variation was larger in the case of endothelium with uneven permeability. These results clearly showed that concentration polarization of LDL which might affect the uptake of LDL by the arterial wall certainly occurs at the surface of the endothelium even if the flow is disturbed microscopically by the uneven surface of the endothelium.     

Time hierarchy in enzymatic reaction chains resulting from optimality principles

Several optimality principles reasonable for the evolution of enzyme reaction chains are formulated, considering the kinetic parameters of the enzymes to be variables. The solutions of these parametric programming problems are studied for the case of linear kinetics and turn out to be not unique in every case. The investigations are confined to stationary states. The net flux through the chain is considered a fixed parameter. The optimality criteria concern the osmotic pressure caused by the intermediates, various relaxation times, largest time scale, and controllability. They all yield distinct time hierarchies. The influence of a constraint concerning the sum of intermediate concentrations is studied. Various combinations of the single criteria are treated as multiobjective programming problems.     

Heuristics and general principles of learning

This research on decision-making heuristics is similar to research on animal learning in at least two ways. First, optimality modeling has not proven to be very useful for either research area. Second, both of these research areas seek to find general principles (or heuristics) that are applicable to different species in different settings. However, the basic principles of classical and operant conditioning seem to be more uniform across species and situations, whereas decision-making heuristics can vary for different species and different situations, even for tasks with very similar characteristics.     

Relationship between fitness and heterogeneity in exponentially growing microbial populations

Despite major environmental and genetic differences, microbial metabolic networks are known to generate consistent physiological outcomes across vastly different organisms. This remarkable robustness suggests that, at least in bacteria, metabolic activity may be guided by universal principles. The constrained optimization of evolutionarily motivated objective functions, such as the growth rate, has emerged as the key theoretical assumption for the study of bacterial metabolism. While conceptually and practically useful in many situations, the idea that certain functions are optimized is hard to validate in data. Moreover, it is not always clear how optimality can be reconciled with the high degree of single-cell variability observed in experiments within microbial populations. To shed light on these issues, we develop an inverse modeling framework that connects the fitness of a population of cells (represented by the mean single-cell growth rate) to the underlying metabolic variability through the maximum entropy inference of the distribution of metabolic phenotypes from data. While no clear objective function emerges, we find that, as the medium gets richer, the fitness and inferred variability for Escherichia coli populations follow and slowly approach the theoretically optimal bound defined by minimal reduction of variability at given fitness. These results suggest that bacterial metabolism may be crucially shaped by a population-level trade-off between growth and heterogeneity.     

A model of stress-induced geometrical remodeling of vessel segments adjacent to stents and artery/graft anastomoses

The mismatch between the elastic properties and initial geometry of a host artery and an implanted stent or graft cause significant stress concentration at the zones close to junctions. This may contribute to the often observed intimal hyperplasia, resulting in late lumen loss and eventual restenosis. This study proposes a mathematical model for stress-induced thickening of the arterial wall at the zones close to an implanted stent or graft. The host artery was considered initially as a cylindrical shell with constant thickness that was clamped to the stent or graft, which was assumed to be non-deformable in the circumferential direction. It was assumed that the abnormal circumferential and axial stresses due to the bending of the arterial wall cause wall thickening that tends to restore the stress state close to that existing far from the junction. The linear equations of a cylindrical shell with variable thickness were coupled to an evolution equation for the wall thickness. These equations were solved numerically and a parametric study was performed using finite difference method and explicit time step. The results show that the remodeling process is self-limiting and leads to local thickening that gradually decreases with distance from the edge of the stent/graft. Model predictions were tested against morphological findings existing in the literature. Recommendations on stent designs that reduce stress concentrations are discussed.     

Including metabolite concentrations into flux balance analysis: thermodynamic realizability as a constraint on flux distributions in metabolic networks

Background  

In recent years, constrained optimization – usually referred to as flux balance analysis (FBA) – has become a widely applied method for the computation of stationary fluxes in large-scale metabolic networks. The striking advantage of FBA as compared to kinetic modeling is that it basically requires only knowledge of the stoichiometry of the network. On the other hand, results of FBA are to a large degree hypothetical because the method relies on plausible but hardly provable optimality principles that are thought to govern metabolic flux distributions.