A topological invariant to predict the three-dimensional writhe of ideal configurations of knots and links.

We present herein a topological invariant of oriented alternating knots and links that predicts the three-dimensional (3D) writhe of the ideal geometrical configuration of the considered knot/link. The fact that we can correlate a geometrical property of a given configuration with a topological invariant supports the notion that the ideal configuration contains important information about knots and links. The importance of the concept of ideal configuration was already suggested by the good correlation between the 3D writhe of ideal knot configurations and the ensemble average of the 3D writhe of random configurations of the considered knots. The values of the new invariant are quantized: multiples of 4/7 for links with an odd number of components (including knots) and 2/7 plus multiples of 4/7 for links with an even number of components.

Linear relations between writhe and minimal crossing number in Conway families of ideal knots and links

We showed earlier how to predict the writhe of any rational knot or link in its ideal geometric configuration, or equivalently the average of the 3D writhe over statistical ensembles of random configurations of a given knot or link (Cerf and Stasiak 2000 Proc. Natl Acad. Sci. USA 97 3795). There is no general relation between the minimal crossing number of a knot and the writhe of its ideal geometric configuration. However, within individual families of knots linear relations between minimal crossing number and writhe were observed (Katritch et al 1996 Nature 384 142). Here we present a method that allows us to express the writhe as a linear function of the minimal crossing number within Conway families of knots and links in their ideal configuration. The slope of the lines and the shift between any two lines with the same

A partial ordering of knots and links through diagrammatic unknotting

In this paper we de. ne a partial ordering of knots and links using a special property derived from their minimal diagrams. A link K' is called a predecessor of a link K if Cr(K') < Cr(K) and a diagram of K' can be obtained from a minimal diagram D of K by a single crossing change. In such a case, we say that K' < K. We investigate the sets of links that can be obtained by single crossing changes over all minimal diagrams of a given link. We show that these sets are specific for different links and permit partial ordering of all links. Some interesting results are presented and many questions are raised.

Subknots in ideal knots, random knots, and knotted proteins.

We introduce disk matrices which encode the knotting of all subchains in circular knot configurations. The disk matrices allow us to dissect circular knots into their subknots, i.e. knot types formed by subchains of the global knot. The identification of subknots is based on the study of linear chains in which a knot type is associated to the chain by means of a spatially robust closure protocol. We characterize the sets of observed subknot types in global knots taking energy-minimized shapes such as KnotPlot configurations and ideal geometric configurations. We compare the sets of observed subknots to knot types obtained by changing crossings in the classical prime knot diagrams. Building upon this analysis, we study the sets of subknots in random configurations of corresponding knot types. In many of the knot types we analyzed, the sets of subknots from the ideal geometric configurations are found in each of the hundreds of random configurations of the same global knot type. We also compare the sets of subknots observed in open protein knots with the subknots observed in the ideal configurations of the corresponding knot type. This comparison enables us to explain the specific dispositions of subknots in the analyzed protein knots.

KnotProt: a database of proteins with knots and slipknots.

The protein topology database KnotProt, http://knotprot.cent.uw.edu.pl/, collects information about protein structures with open polypeptide chains forming knots or slipknots. The knotting complexity of the cataloged proteins is presented in the form of a matrix diagram that shows users the knot type of the entire polypeptide chain and of each of its subchains. The pattern visible in the matrix gives the knotting fingerprint of a given protein and permits users to determine, for example, the minimal length of the knotted regions (knot's core size) or the depth of a knot, i.e. how many amino acids can be removed from either end of the cataloged protein structure before converting it from a knot to a different type of knot. In addition, the database presents extensive information about the biological functions, families and fold types of proteins with non-trivial knotting. As an additional feature, the KnotProt database enables users to submit protein or polymer chains and generate their knotting fingerprints.

Linear random knots and their scaling behavior

We present here a nonbiased probabilistic method that allows us to consistently analyze knottedness of linear random walks with up to several hundred noncorrelated steps. The method consists of analyzing the spectrum of knots formed by multiple closures of the same open walk through random points on a sphere enclosing the walk. Knottedness of individual "frozen" configurations of linear chains is therefore defined by a characteristic spectrum of realizable knots. We show that in the great majority of cases this method clearly defines the dominant knot type of a walk, i.e., the strongest component of the spectrum. In such cases, direct end-to-end closure creates a knot that usually coincides with the knot type that dominates the random closure spectrum. Interestingly, in a very small proportion of linear random walks, the knot type is not clearly defined. Such walks can be considered as residing in a border zone of the configuration space of two or more knot types. We also characterize the scaling behavior of linear random knots.

Sedimentation and electrophoretic migration of DNA knots and catenanes.

Various site-specific recombination enzymes produce different types of knots or catenanes while acting on circular DNA in vitro and in vivo. By analysing the types of knots or links produced, it is possible to reconstruct the order of events during the reaction and to deduce the molecular "architecture" of the complexes that different enzymes form with DNA. Until recently it was necessary to use laborious electron microscopy methods to identify the types of knots or catenanes that migrate in different bands on the agarose gels used to analyse the products of the reaction. We reported recently that electrophoretic migration of different knots and catenanes formed on the same size DNA molecules is simply related to the average crossing number of the ideal representations of the corresponding knots and catenanes. Here we explain this relation by demonstrating that the expected sedimentation coefficient of randomly fluctuating knotted or catenated DNA molecules in solution shows approximately linear correlation with the average crossing number of ideal configurations of the corresponding knots or catenanes.

Construction and electrophoretic migration of single-stranded DNA knots and catenanes.

In recent years there has been growing interest in the question of how the particular topology of polymeric chains affects their overall dimensions and physical behavior. The majority of relevant studies are based on numerical simulation methods or analytical treatment; however, both these approaches depend on various assumptions and simplifications. Experimental verification is clearly needed but was hampered by practical difficulties in obtaining preparative amounts of knotted or catenated polymers with predefined topology and precisely set chain length. We introduce here an efficient method of production of various single-stranded DNA knots and catenanes that have the same global chain length. We also characterize electrophoretic migration of the produced single-stranded DNA knots and catenanes with increasing complexity.

DNA knots and DNA supercoiling.

Comment on: Witz G, et al. Proc Natl Acad Sci USA 2011; 108:3608-11.

Sedimentation of macroscopic rigid knots and its relation to gel electrophoretic mobility of DNA knots.

We address the general question of the extent to which the hydrodynamic behaviour of microscopic freely fluctuating objects can be reproduced by macrosopic rigid objects. In particular, we compare the sedimentation speeds of knotted DNA molecules undergoing gel electrophoresis to the sedimentation speeds of rigid stereolithographic models of ideal knots in both water and silicon oil. We find that the sedimentation speeds grow roughly linearly with the average crossing number of the ideal knot configurations, and that the correlation is stronger within classes of knots. This is consistent with previous observations with DNA knots in gel electrophoresis.

Metamorphism and kinetics in the Torres del Paine contact aureole

Contact aureoles provide an excellent geologic environment to study the mechanisms of metamorphic reactions in a natural system. The Torres del Paine (TP) intrusion is one of the most spectacular natural laboratories because of its excellent outcrop conditions. It formed in a period from 12.59 to 12.43 Ma and consists of three large granite and four smaller mafic batches. The oldest granite is on top, the youngest at the bottom of the granitic complex, and the granites overly the mafic laccolith. The TP intruded at a depth of 2-3 km into regional metamorphic anchizone to greenschist facies pelites, sandstones, and conglomerates of the Cerro Toro and Punta Barrosa formations. It formed a thin contact aureole of 150-400 m width. This thesis focuses on the reaction kinetics of the mineral cordierite in the contact aureole using quantitative textural analysis methods. First cordierite was formed from chlorite break¬down (zone I, ca. 480 °C, 750 bar). The second cordierite forming reaction was the muscovite break-down, which is accompanied by a modal decrease in biotite and the appearance of k- feldspar (zone II, 540-550 °C, 750 bar). Crystal sizes of the roundish, poikiloblastic cordierites were determined from microscope thin section images by manually marking each crystal. Images were then automatically processed with Matlab. The correction for the intersection probability of each crystal radius yields the crystal size distribution in the rock. Samples from zone I below the laccolith have the largest crystals (0.09 mm). Cordierites from zone II are smaller, with a maximum crystal radius of 0.057 mm. Rocks from zone II have a larger number of small cordierite crystals than rocks from zone I. A combination of these quantitative analysis with numerical modeling of nucleation and growth, is used to infer nucleation and growth parameters which are responsible for the observed mineral textures. For this, the temperature-time paths of the samples need to be known. The thermal history is complex because the main body of the intrusion was formed by several intrusive batches. The emplacement mechanism and duration of each batch can influence the thermal structure in the aureole. A possible subdivision of batches in smaller increments, so called pulses, will focus heat at the side of the intrusion. Focusing all pulses on one side increases the contact aureole size on that side, but decreases it on the other side. It forms a strongly asymmetric contact aureole. Detailed modeling shows that the relative thicknesses of the TP contact aureole above and below the intrusion (150 and 400 m) are best explained by a rapid emplacement of at least the oldest granite batch. Nevertheless, temperatures are significantly too low in all models, compared to observed mineral assemblages in the hornfelses. Hence, an other important thermal mechanisms needs to take place in the host rock. Clastic minerals in the immature sediments outside the contact aureole are hydrated due to small amounts of expelled fluids during contact metamorphism. This leads to a temperature increase of up to 50 °C. The origin of fluids can be traced by stable isotopes. Whole rock stable isotope data (6D and δ180) and chlorine concentrations in biotite document that the TP intrusion induced only very small amounts of fluid flow. Oxygen whole rock data show δ180 values between 9.0 and 10.0 %o within the first 5 m of the contact. Values increase to 13.0 - 15.0 %o further away from the intrusion. Whole rock 6D values display a more complex zoning. First, host rock values (-90 to -70 %o) smoothly decrease towards the contact by ca. 20 %o, up to a distance of ca. 150 m. This is followed by an increase of ca. 20 %o within the innermost 150 m of the aureole (-97.0 to -78 %o at the contact). The initial decrease in 6D values is interpreted to be due to Rayleigh fractionation accompanying the dehydration reactions forming cordierite, while the final increase reflects infiltration of water-rich fluids from the intrusion. An over-estimate on the quantity and the corresponding thermal effect yields a temperature increase of less than 30 °C. This suggests that fluid flow might have contributed only for a small amount to the thermal evolution of the system. A combination of the numerical growth model with the thermal model, including the hydration reaction enthalpies but neglecting fluid flow and incremental growth, can be used to numerically reproduce the observed cordierite textures in the contact aureole. This yields kinetic parameters which indicate fast cordierite crystallization before the thermal peak in the inner aureole, and continued reaction after the thermal peak in the outermost aureole. Only small temperature dependencies of the kinetic parameters seem to be needed to explain the obtained crystal size data. - Les auréoles de contact offrent un cadre géologique privilégié pour l'étude des mécanismes de réactions métamorphiques associés à la mise en place de magmas dans la croûte terrestre. Par ses conditions d'affleurements excellentes, l'intrusion de Torres del Paine représente un site exceptionnel pour améliorer nos connaissances de ces processus. La formation de cette intrusion composée de trois injections granitiques principales et de quatre injections mafiques de volume inférieur couvre une période allant de 12.50 à 12.43 Ma. Le plus vieux granite forme la partie sommitale de l'intrusion alors que l'injection la plus jeune s'observe à la base du complexe granitique; les granites recouvrent la partie mafique du laccolite. L'intrusion du Torres del Paine s'est mise en place a 2-3 km de profondeur dans un encaissant métamorphique. Cet encaissant est caractérisé par un métamorphisme régional de faciès anchizonal à schiste vert et est composé de pélites, de grès, et des conglomérats des formations du Cerro Toro et Punta Barrosa. La mise en place des différentes injections granitiques a généré une auréole de contact de 150-400 m d'épaisseur autour de l'intrusion. Cette thèse se concentre sur la cinétique de réaction associée à la formation de la cordiérite dans les auréoles de contact en utilisant des méthodes quantitatives d'analyses de texture. On observe plusieurs générations de cordiérite dans l'auréole de contact. La première cordiérite est formée par la décomposition de la chlorite (zone I, environ 480 °C, 750 bar), alors qu'une seconde génération de cordiérite est associée à la décomposition de la muscovite, laquelle est accompagnée par une diminution modale de la teneur en biotite et l'apparition de feldspath potassique (zone II, 540-550 °C, 750 bar). Les tailles des cristaux de cordiérites arrondies et blastic ont été déterminées en utilisant des images digitalisées des lames minces et en marquant individuellement chaque cristal. Les images sont ensuite traitées automatiquement à l'aide du programme Matlab. La correction de la probabilité d'intersection en fonction du rayon des cristaux permet de déterminer la distribution de la taille des cristaux dans la roche. Les échantillons de la zone I, en dessous du lacolite, sont caractérisés par de relativement grands cristaux (0.09 mm). Les cristaux de cordiérite de la zone II sont plus petits, avec un rayon maximal de 0.057 mm. Les roches de la zone II présentent un plus grand nombre de petits cristaux de cordiérite que les roches de la zone I. Une combinaison de ces analyses quantitatives avec un modèle numérique de nucléation et croissance a été utilisée pour déduire les paramètres de nucléation et croissance contrôlant les différentes textures minérales observées. Pour développer le modèle de nucléation et de croissance, il est nécessaire de connaître le chemin température - temps des échantillons. L'histoire thermique est complexe parce que l'intrusion est produite par plusieurs injections successives. En effet, le mécanisme d'emplace¬ment et la durée de chaque injection peuvent influencer la structure thermique dans l'auréole. Une subdivision des injections en plus petits incréments, appelés puises, permet de concentrer la chaleur dans les bords de l'intrusion. Une mise en place préférentielle de ces puises sur un côté de l'intrusion modifie l'apport thermique et influence la taille de l'auréole de contact produite, auréole qui devient asymétrique. Dans le cas de la première injection de granite, une modélisation détaillée montre que l'épaisseur relative de l'auréole de contact de Torres del Paine au-dessus et en dessous de l'intrusion (150 et 400 m) est mieux expliquée par un emplacement rapide du granite. Néanmoins, les températures calculées dans l'auréole de con¬tact sont trop basses pour que les modèles thermiques soient cohérants par rapport à la taille de cette auréole. Ainsi, un autre mecanisme exothermique est nécessaire pour permettre à la roche encais¬sante de produire les assemblages observés. L'observation des roches encaissantes entourant les granites montre que les minéraux clastiques dans les sédiments immatures au-dehors de l'auréole sont hydratés suite à la petite quantité de fluide expulsée durant le métamorphisme de contact et/ou la mise en place des granites. Les réactions d'hydratation peuvent permettre une augmentation de la température jusqu'à 50 °C. Afin de déterminer l'origine des fluides, une étude isotopique de roches de l'auréole de contact a été entreprise. Les isotopes stables d'oxygène et d'hydrogène sur la roche totale ainsi que la concentration en chlore dans la biotite indiquent que la mise en place des granites du Torres del Paine n'induit qu'une circulation de fluide limitée. Les données d'oxygène sur roche totale montrent des valeurs δ180 entre 9.0 et 10.0%o au sein des cinq premiers mètres du contact. Les valeurs augmentent jusqu'à 13.0 - 15.0 plus on s'éloigne de l'intrusion. Les valeurs 5D sur roche totale montrent une zonation plus complexe. Les valeurs de la roche encaissante (-90 à -70%o) diminuent progressivement d'environ 20%o depuis l'extérieur de l'auréole jusqu'à une distance d'environ 150 m du granite. Cette diminution est suivie par une augmentation d'environ 20%o au sein des 150 mètres les plus proches du contact (-97.0 à -78%o au contact). La diminution initiale des valeurs de 6D est interprétée comme la conséquence du fractionnement de Rayleigh qui accompagne les réactions de déshydratation formant la cordiérite, alors que l'augmentation finale reflète l'infiltration de fluide riche en eau venant de l'intrusion. A partir de ces résultats, le volume du fluide issu du granite ainsi que son effet thermique a pu être estimé. Ces résultats montrent que l'augmentation de température associée à ces fluides est limitée à un maximum de 30 °C. La contribution de ces fluides dans le bilan thermique est donc faible. Ces différents résultats nous ont permis de créer un modèle thermique associé à la for¬mation de l'auréole de contact qui intègre la mise en place rapide du granite et les réactions d'hydratation lors du métamorphisme. L'intégration de ce modèle thermique dans le modèle numérique de croissance minérale nous permet de calculer les textures des cordiérites. Cepen¬dant, ce modèle est dépendant de la vitesse de croissance et de nucléation de ces cordiérites. Nous avons obtenu ces paramètres en comparant les textures prédites par le modèle et les textures observées dans les roches de l'auréole de contact du Torres del Paine. Les paramètres cinétiques extraits du modèle optimisé indiquent une cristallisation rapide de la cordiérite avant le pic thermique dans la partie interne de l'auréole, et une réaction continue après le pic thermique dans la partie la plus externe de l'auréole. Seules de petites dépendances de température des paramètres de cinétique semblent être nécessaires pour expliquer les don¬nées obtenues sur la distribution des tailles de cristaux. Ces résultats apportent un éclairage nouveau sur la cinétique qui contrôle les réactions métamorphiques.

Cerebrospinal fluid PCR analysis and biochemistry in bodies with severe decomposition.

The aim of this study was to assess whether Neisseria meningitidis, Listeria monocytogenes, Streptococcus pneumoniae and Haemophilus influenzae can be identified using the polymerase chain reaction technique in the cerebrospinal fluid of severely decomposed bodies with known, noninfectious causes of death or whether postmortem changes can lead to false positive results and thus erroneous diagnostic information. Biochemical investigations, postmortem bacteriology and real-time polymerase chain reaction analysis in cerebrospinal fluid were performed in a series of medico-legal autopsies that included noninfectious causes of death with decomposition, bacterial meningitis without decomposition, bacterial meningitis with decomposition, low respiratory tract infections with decomposition and abdominal infections with decomposition. In noninfectious causes of death with decomposition, postmortem investigations failed to reveal results consistent with generalized inflammation or bacterial infections at the time of death. Real-time polymerase chain reaction analysis in cerebrospinal fluid did not identify the studied bacteria in any of these cases. The results of this study highlight the usefulness of molecular approaches in bacteriology as well as the use of alternative biological samples in postmortem biochemistry in order to obtain suitable information even in corpses with severe decompositional changes.