Supercoiled DNA is interwound in liquid crystalline solutions.

Two structures have been proposed for supercoiled DNA: it is idealized either as a toroidal ring or as a rod of two interwound duplex chains. The latter model is the most widely depicted but the evidence remains controversial. We have worked with monomers and dimers of two plasmids, pUC8 and pKS414, of similar size and natural superhelical density. pKS414 contains a bend promoting sequence whereas pUC8 does not. In concentrated solutions these plasmids form a partially ordered liquid crystalline phase which is found, using neutron diffraction, to consist of a hexagonally packed assembly of parallel rod-like particles. This shape strongly suggests an interwound conformation for which some structural parameters are deduced. The mass/unit length obtained by combining the area of the hexagonal lattice and the concentration is approximately 3.6 times that of linear DNA. This implies a shallow superhelical pitch angle approximately 36 degrees which, when combined with the known number of supercoil turns, yields the pitch approximately 360 A and radius approximately 80 A for the supercoil. Oriented X-ray fibre diffraction patterns at 92% relative humidity indicate a B type duplex structure. Nicked circular plasmids also form liquid crystals but their behaviour, as a function of concentration, differs from that of the superhelical plasmids.     

Configurational transitions in Fourier series-represented DNA supercoils.

A new Fourier series representation of supercoiled DNA is employed in Langevin dynamics simulations to study large-scale configurational motions of intermediate-length chains. The polymer is modeled as an ideal elastic rod subject to long-range van der Waals' interactions. The van der Waals' term prevents the self-contact of distant chain segments and also mimics attractive forces thought to stabilize the association of closely spaced charged rods. The finite Fourier series-derived polymer formulation is an alternative to the piecewise B-spline curves used in past work to describe the motion of smoothly deformed supercoiled DNA in terms of a limited number of independent variables. This study focuses on two large-scale configurational events: the interconversion between circular and figure-8 forms at a relatively low level of supercoiling, and the transformation between branched and interwound structures at a higher superhelical density.     

Specific interactions between DNA left-handed supercoils and actinomycin D

The interactions between the natural cyclopentapeptide antibiotic actinomycin D (ACT) and circular pBR322 DNA have been studied by freezing the topological state of the DNA in the complex by topoisomerase I reaction. Both supercoiled and relaxed DNAs, in the complexes at low antibiotic/DNA base-pair ratios, showed a dramatic decrease in linking number that cannot be explained by taking into account only the generally accepted unwinding of 28 degrees for each ACT molecule bound. Recent results derived from the crystallographic analysis of the complex between GpC and ACT suggest that ACT could mediate non-covalent cross-links between distant sections of DNA. Bridges between ACT and different sections of the pBR322 double helix could also explain our results. Two-dimensional gel electrophoresis of ACT-relaxed pBR322 DNA complexes reveals that all supercoils induced by ACT are negative. Two models of the complexes which correspond to the stabilization of DNA crossing by one or two molecules of ACT are proposed. In both cases the ability of ACT to stabilize only DNA left-handed supercoils is derived from the chirality of ACT, when it interacts with DNA.     

Modeling DNA supercoils and knots with B-spline functions

A method is offered to model the complex trajectories of closed circular DNA supercoils and knots. The trajectories are approximated by polygons and analytical expressions of the curves are generated from the polygons with B-spline functions. The resulting curves are used to evaluate the writhe and elastic energy of a series of interrelated supercoils, and to generate detailed atomic models of the deformed double helix.     

Competition between supercoils and toroids in single molecule DNA condensation

The condensation of free DNA into toroidal structures in the presence of multivalent ions and polypeptides is well known. Recent single molecule experiments have shown that condensation into toroids occurs even when the DNA molecule is subjected to tensile forces. Here we show that the combined tension and torsion of DNA in the presence of condensing agents dramatically modifies this picture by introducing supercoiled DNA as a competing structure in addition to toroids. We combine a fluctuating elastic rod model of DNA with phenomenological models for DNA interaction in the presence of condensing agents to compute the minimum energy configuration for given tension and end-rotations. We show that for each tension there is a critical number of end-rotations above which the supercoiled solution is preferred and below which toroids are the preferred state. Our results closely match recent extension rotation experiments on DNA in the presence of spermine and other condensing agents. Motivated by this, we construct a phase diagram for the preferred DNA states as a function of tension and applied end-rotations and identify a region where new experiments or simulations are needed to determine the preferred state.     

Mathematical modelling of metabolism

Mathematical models of the cellular metabolism have a special interest within biotechnology. Many different kinds of commercially important products are derived from the cell factory, and metabolic engineering can be applied to improve existing production processes, as well as to make new processes available. Both stoichiometric and kinetic models have been used to investigate the metabolism, which has resulted in defining the optimal fermentation conditions, as well as in directing the genetic changes to be introduced in order to obtain a good producer strain or cell line. With the increasing availability of genomic information and powerful analytical techniques, mathematical models also serve as a tool for understanding the cellular metabolism and physiology.     

The geometry of DNA supercoils modulates the DNA cleavage activity of human topoisomerase I

Human topoisomerase I plays an important role in removing positive DNA supercoils that accumulate ahead of replication forks. It also is the target for camptothecin-based anticancer drugs that act by increasing levels of topoisomerase I-mediated DNA scission. Evidence suggests that cleavage events most likely to generate permanent genomic damage are those that occur ahead of DNA tracking systems. Therefore, it is important to characterize the ability of topoisomerase I to cleave positively supercoiled DNA. Results confirm that the human enzyme maintains higher levels of cleavage with positively as opposed to negatively supercoiled substrates in the absence or presence of anticancer drugs. Enhanced drug efficacy on positively supercoiled DNA is due primarily to an increase in baseline levels of cleavage. Sites of topoisomerase I-mediated DNA cleavage do not appear to be affected by supercoil geometry. However, rates of ligation are slower with positively supercoiled substrates. Finally, intercalators enhance topoisomerase I-mediated cleavage of negatively supercoiled substrates but not positively supercoiled or linear DNA. We suggest that these compounds act by altering the perceived topological state of the double helix, making underwound DNA appear to be overwound to the enzyme, and propose that these compounds be referred to as ‘topological poisons of topoisomerase I’.     

Generation of supercoils in nicked and gapped DNA drives DNA unknotting and postreplicative decatenation

Due to the helical structure of DNA the process of DNA replication is topologically complex. Freshly replicated DNA molecules are catenated with each other and are frequently knotted. For proper functioning of DNA it is necessary to remove all of these entanglements. This is done by DNA topoisomerases that pass DNA segments through each other. However, it has been a riddle how DNA topoisomerases select the sites of their action. In highly crowded DNA in living cells random passages between contacting segments would only increase the extent of entanglement. Using molecular dynamics simulations we observed that in actively supercoiled DNA molecules the entanglements resulting from DNA knotting or catenation spontaneously approach sites of nicks and gaps in the DNA. Type I topoisomerases, that preferentially act at sites of nick and gaps, are thus naturally provided with DNA–DNA juxtapositions where a passage results in an error-free DNA unknotting or DNA decatenation.     

Asymmetric removal of supercoils suggests how topoisomerase II simplifies DNA topology

Type-IIA topoisomerases consume ATP as they catalyse the interconversion of DNA topoisomers by transporting one DNA segment through a transient break in another. It remains unclear how their activity simplifies the topology of DNA below equilibrium values. Here we report that eukaryotic topoisomerase II narrows the thermal distribution of DNA supercoils, by mainly removing negative DNA crossings. Surprisingly, this asymmetry in supercoil removal is not due to deformation of the DNA before strand passage. Topoisomerase II neither bends nor alters the helical conformation of the interacting DNA. Rather, it appears to interact with a third DNA segment, in addition to the gated and the transported segments. Remarkably, the simultaneous interaction with three DNA segments accounts for the asymmetric removal of supercoils in relaxed DNA and gives a clue to how topoisomerase II simplifies the topology of DNA against the thermal drive.     

Mathematical modelling in physiology

The article illustrates the method of mathematical modelling in physiology as a unique tool to study physiological processes. A number of demonstrated examples appear as a result of long-term experience in mathematical modelling of electrical and mechanical phenomena in the heart muscle. These examples are presented here to show that the modelling provides insight into mechanisms underlying these phenomena and is capable to predict new ones that were previously unknown. While potentialities of the mathematical modelling are analyzed with regard to the myocardium, they are quite universal to deal with any physiological processes.     

Mathematical modelling of biofilm structures

The morphology of biofilms received much attention in the last years. Several concepts to explain the development of biofilm structures have been proposed. We believe that biofilm structure formation depends on physical as well as general and specific biological factors. The physical factors (e.g. governing substrate transport) as well as general biological factors such as growth yield and substrate conversion rates are the basic factors governing structure formation. Specific strain dependent factors will modify these, giving a further variation between different biofilm systems. Biofilm formation seems to be primarily dependent on the interaction between mass transport and conversion processes. When a biofilm is strongly diffusion limited it will tend to become a heterogeneous and porous structure. When the conversion is the rate-limiting step, the biofilm will tend to become homogenous and compact. On top of these two processes, detachment processes play a significant role. In systems with a high detachment (or shear) force, detachment will be in the form of erosion, giving smoother biofilms. Systems with a low detachment force tend to give a more porous biofilm and detachment occurs mainly by sloughing. Biofilm structure results from the interplay between these interactions (mass transfer, conversion rates, detachment forces) making it difficult to study systems taking only one of these factors into account. This revised version was published online in August 2006 with corrections to the Cover Date.     

Mathematical modelling of tumour acidity

Acid-mediated tumour invasion is receiving increasing experimental and clinical attention. Previous models proposed to describe this phenomenon failed to capture key properties of the system, such as the existence of the benign steady state, or predicted incorrectly the size of the inter-tissue gap. Here we show that taking proper account of quiescence ameliorates these drawbacks as well as revealing novel behaviour. The simplicity of the model allows us to fully identify the key parameters controlling different aspects of behaviour.     

Mathematical modelling of corneal swelling

This paper presents a differential model of the corneal transport system capable of modelling thickness changes in response to osmotic perturbations applied to either limiting membrane. The work is directed towards understanding corneal behaviour in vivo. The model considers the coupled viscous flows within the corneal stroma and across the epithelial and endothelial membranes. The flows within the stroma are established based on transport theory in porous media, while the flows across the membranes are described using the phenomenological equations of irreversible thermodynamics. The ability of the numerical model to reproduce corneal thickness changes in response to endothelial perturbations was tested against available experimental data. The sensitivity of the model to changes in stromal and membrane transport coefficients was examined.     

Mathematical modelling of skeletal repair

Tissue engineering offers significant promise as a viable alternative to current clinical strategies for replacement of damaged tissue as a consequence of disease or trauma. Since mathematical modelling is a valuable tool in the analysis of complex systems, appropriate use of mathematical models has tremendous potential for advancing the understanding of the physical processes involved in such tissue reconstruction. In this review, the potential benefits, and limitations, of theoretical modelling in tissue engineering applications are examined with specific emphasis on tissue engineering of bone. A central tissue engineering approach is the in vivo implantation of a biomimetic scaffold seeded with an appropriate population of stem or progenitor cells. This review will therefore consider the theory behind a number of key factors affecting the success of such a strategy including: stem cell or progenitor population expansion and differentiation ex vivo; cell adhesion and migration, and the effective design of scaffolds; and delivery of nutrient to avascular structures. The focus will be on current work in this area, as well as on highlighting limitations and suggesting possible directions for future work to advance health-care for all.     

Mathematical modelling of juxtacrine patterning

Spatial pattern formation is one of the key issues in developmental biology. Some patterns arising in early development have a very small spatial scale and a natural explanation is that they arise by direct cell—cell signalling in epithelia. This necessitates the use of a spatially discrete model, in contrast to the continuum-based approach of the widely studied Turing and mechanochemical models. In this work, we consider the pattern-forming potential of a model for juxtacrine communication, in which signalling molecules anchored in the cell membrane bind to and activate receptors on the surface of immediately neighbouring cells. The key assumption is that ligand and receptor production are both up-regulated by binding. By linear analysis, we show that conditions for pattern formation are dependent on the feedback functions of the model. We investigate the form of the pattern: specifically, we look at how the range of unstable wavenumbers varies with the parameter regime and find an estimate for the wavenumber associated with the fastest growing mode. A previous juxtacrine model for Delta-Notch signalling studied by Collier et al. (1996, J. Theor. Biol. 183, 429–446) only gives rise to patterning with a length scale of one or two cells, consistent with the fine-grained patterns seen in a number of developmental processes. However, there is evidence of longer range patterns in early development of the fruit fly Drosophila. The analysis we carry out predicts that patterns longer than one or two cell lengths are possible with our positive feedback mechanism, and numerical simulations confirm this. Our work shows that juxtacrine signalling provides a novel and robust mechanism for the generation of spatial patterns.     

Mathematical modelling of tissue-engineered angiogenesis

We present a mathematical model for the vascularisation of a porous scaffold following implantation in vivo. The model is given as a set of coupled non-linear ordinary differential equations (ODEs) which describe the evolution in time of the amounts of the different tissue constituents inside the scaffold. Bifurcation analyses reveal how the extent of scaffold vascularisation changes as a function of the parameter values. For example, it is shown how the loss of seeded cells arising from slow infiltration of vascular tissue can be overcome using a prevascularisation strategy consisting of seeding the scaffold with vascular cells. Using certain assumptions it is shown how the system can be simplified to one which is partially tractable and for which some analysis is given. Limited comparison is also given of the model solutions with experimental data from the chick chorioallantoic membrane (CAM) assay.     

The geometry of DNA supercoils modulates topoisomerase-mediated DNA cleavage and enzyme response to anticancer drugs

Collisions with DNA tracking systems are critical for the conversion of transient topoisomerase-DNA cleavage complexes to permanent strand breaks. Since DNA is overwound ahead of tracking systems, cleavage complexes most likely to produce permanent strand breaks should be formed between topoisomerases and positively supercoiled molecules. Therefore, the ability of human topoisomerase IIalpha and IIbeta and topoisomerase I to cleave positively supercoiled DNA was assessed in the absence or presence of anticancer drugs. Topoisomerase IIalpha and IIbeta maintained approximately 4-fold lower levels of cleavage complexes with positively rather than negatively supercoiled DNA. Topoisomerase IIalpha also displayed lower levels of cleavage with overwound substrates in the presence of nonintercalative drugs. Decreased drug efficacy was due primarily to a drop in baseline (i.e., nondrug) cleavage, rather than an altered interaction with the enzyme-DNA complex. Similar results were seen for topoisomerase IIbeta, but the effects of DNA geometry on drug-induced scission were somewhat less pronounced. With both topoisomerase IIalpha and IIbeta, intercalative drugs displayed greater relative cleavage enhancement with positively supercoiled DNA. This appeared to result from negative effects of high concentrations of intercalative agents on underwound DNA. In contrast to the type II enzymes, topoisomerase I maintained approximately 3-fold higher levels of cleavage complexes with positively supercoiled substrates and displayed an even more dramatic increase in the presence of camptothecin. These findings suggest that the geometry of DNA supercoils has a profound influence on topoisomerase-mediated DNA scission and that topoisomerase I may be an intrinsically more lethal target for anticancer drugs than either topoisomerase IIalpha or IIbeta.