jeanflop/post_ocr_correction2 huggingface dataset:
of base pairs must be around a million. Dynamique de chaînes d'ADN enchevêtrées en présence de topoisomérases Résumé — Le comportement dynamique d'une solution semi-diluée de molécules linéaires d'ADN en présence de topoisomérases de type II, est discuté en utilisant les concepts de reptation et de relaxation des contraintes. Le taux de réaction des topoïsomérases étant connu, nous calculons la longueur minimale des chaînes d'ADN pour que la réaction catalysée par les topoisomérases modifie de manière sensible le comportement dynamique de la solution. Nous trouvons que le nombre de paires de base doit être d'environ un million. Version française abrégée Les ADN topoisomérases catalysent le franchissement d'obstacles topologiques dans les chaînes d'ADN, permettant à un segment d'ADN de passer au travers d'un autre segment. L'enzyme introduit une cassure transitoire dans l'une des chaînes, après le franchissement de l'obstacle, les segments cassés se recombinent pour redonner la chaîne initiale (Kornberg et Baker, 1991) (voir fig). Nous étudions ici les effets des topoisomérases de type II sur les propriétés dynamiques d'une solution de molécules linéaires d'ADN double brin, et plus particulièrement sur les temps maximaux de relaxation. Dans un régime de molécules linéaires enchevêtrées, l'action des topoisomérases modifie ce temps : le problème est de modéliser convenablement leur action et de déterminer les conditions dans lesquelles elle est observable par les techniques usuelles de la viscoélasticité. Pour ce faire nous examinons de très longues chaînes d'ADN en solution aqueuse, à une concentration C supérieure à c* la concentration de recouvrement. Les interactions électrostatiques sont écartées par du sel ajouté. Nous traitons la solution d'ADN comme une solution semi-diluée de chaînes flexibles neutres homopolymériques en bon solvant. D'une façon générale la dynamique de telles solutions peut être décrite comme la combinaison de deux processus: 1) la reptation d'une chaîne test dans un tube virtuel défini par les chaînes environnantes supposées fixées. Le temps nécessaire pour que la chaîne test sorte complètement du tube est noté T2 2) la réorganisation du tube due aux mouvements des chaînes environnantes et aux franchissements d'obstacles autorisés par les topoisomérases. La relaxation des contraintes associée à ce mécanisme
problème est de modéliser convenablement leur action et de déterminer les conditions dans lesquelles elle est observable par les techniques usuelles de la viscoélasticité. Pour ce faire nous examinons de très longues chaînes d'ADN en solution aqueuse, à une concentration C supérieure à c* la concentration de recouvrement. Les interactions électrostatiques sont écartées par du sel ajouté. Nous traitons la solution d'ADN comme une solution semi-diluée de chaînes flexibles neutres homopolymériques en bon solvant. D'une façon générale la dynamique de telles solutions peut être décrite comme la combinaison de deux processus: 1) la reptation d'une chaîne test dans un tube virtuel défini par les chaînes environnantes supposées fixées. Le temps nécessaire pour que la chaîne test sorte complètement du tube est noté T2 2) la réorganisation du tube due aux mouvements des chaînes environnantes et aux franchissements d'obstacles autorisés par les topoisomérases. La relaxation des contraintes associée à ce mécanisme est caractérisée par un temps maximal noté Treno. En absence d'enzyme, la réorganisation du tube est due exclusivement à la reptation des chaînes environnantes. La chaîne test suit alors un comportement analogue à celui de la reptation d'une chaîne autour d'obstacles fixes, à condition que les chaînes soient suffisamment longues et que la concentration soit suffisamment élevée (Klein, 1978). La concentration minimale est évaluée à 10 c*. En théorie, le temps de reptation est alors: Trep = T1 (C/C*)3/2,3-1, où xx est le temps maximal de relaxation de Zimm pour une chaîne isolée, et v = 0,588 (de Gennes, 1976). En présence de l'enzyme, les obstacles ont une structure labile, et l'on peut concevoir un régime où le temps caractéristique de réorganisation du tube, Treno, devient inférieur au temps caractéristique de reptation Trep. Dans ce cas on se ramènera à un problème déjà traité, qui est celui d'une chaîne test en présence d'obstacles mobiles (Viovy et al., 1991). Si l'on admet que la reptation de la chaîne et la réorganisation de l'environnement sont deux événements indépendants, le temps maximal effectif de relaxation de la chaîne, Tmax, est une moyenne harmonique des temps Treno et Trep. Nous évaluons Treno en fonction
et que la concentration soit suffisamment élevée (Klein, 1978). La concentration minimale est évaluée à 10 c*. En théorie, le temps de reptation est alors: Trep = T1 (C/C*)3/2,3-1, où xx est le temps maximal de relaxation de Zimm pour une chaîne isolée, et v = 0,588 (de Gennes, 1976). En présence de l'enzyme, les obstacles ont une structure labile, et l'on peut concevoir un régime où le temps caractéristique de réorganisation du tube, Treno, devient inférieur au temps caractéristique de reptation Trep. Dans ce cas on se ramènera à un problème déjà traité, qui est celui d'une chaîne test en présence d'obstacles mobiles (Viovy et al., 1991). Si l'on admet que la reptation de la chaîne et la réorganisation de l'environnement sont deux événements indépendants, le temps maximal effectif de relaxation de la chaîne, Tmax, est une moyenne harmonique des temps Treno et Trep. Nous évaluons Treno en fonction du temps Tcross requis par l'enzyme pour catalyser le franchissement d'un obstacle. Nous pourrons alors définir les conditions permettant de réaliser l'inégalité Treno < Trep. Le temps Tcross est tout d'abord identifié au temps Tobs de vie des obstacles constituant le tube. Celui-ci obéit à une loi de Rouse (Stockmayer, 1976). Le temps Treo est relié à Tobs comme le temps maximal d'un mouvement à la Rouse, Treno = Nc Tobs/3 r2, où Nc est le nombre — par chaîne — d'enchevêtrements définis par les contacts effectifs interchaînes. Exprimons Nc en fonction de la concentration: Nc = (C/C*)1/3 v-1), d'où Treno = (C/C*)2/3 V~1) Tobs/3 r2 D'après les données expérimentales disponibles (Osheroff et al., 1983), nous estimons Tcross à 2,4 sec. En prenant comme exemple l'ADN linéarisé du génome d'Escherichia coli qui contient 4,5. 106 paires de base, le temps xt dans (1) est de l'ordre de 70 s (Klotz et Zimm, 1972). Une décroissance de Tmax par rapport à xrep est alors observable dans le domaine 10 < r/r* < 860. En conclusion nous prédisons une décroissance significative de Tmax en présence de topoisomérases pour des molécules d'ADN ayant plus d'un million de paires de base. Une mesure de
1976). Le temps Treo est relié à Tobs comme le temps maximal d'un mouvement à la Rouse, Treno = Nc Tobs/3 r2, où Nc est le nombre — par chaîne — d'enchevêtrements définis par les contacts effectifs interchaînes. Exprimons Nc en fonction de la concentration: Nc = (C/C*)1/3 v-1), d'où Treno = (C/C*)2/3 V~1) Tobs/3 r2 D'après les données expérimentales disponibles (Osheroff et al., 1983), nous estimons Tcross à 2,4 sec. En prenant comme exemple l'ADN linéarisé du génome d'Escherichia coli qui contient 4,5. 106 paires de base, le temps xt dans (1) est de l'ordre de 70 s (Klotz et Zimm, 1972). Une décroissance de Tmax par rapport à xrep est alors observable dans le domaine 10 < r/r* < 860. En conclusion nous prédisons une décroissance significative de Tmax en présence de topoisomérases pour des molécules d'ADN ayant plus d'un million de paires de base. Une mesure de cette décroissance permettra d'évaluer l'effet des topoisomérases. Il existe une similarité entre le système ADN plus topoisomérases et les polymères vivants. Pour ces derniers, la recombinaison des chaînes cassées se fait au hasard. La similarité serait plus grande si l'on utilisait une combinaison d'endonucléase et de ligase à la place d'une topoisomérase. En mauvais solvant, les topoisomérases fonctionnent aussi efficacement qu'en bon solvant. Il sera intéressant d'étudier l'effet de ces catalyseurs dans la cinétique de condensation de ces molécules. The number of entanglement points can be enormous, since each turn of the helix (which occurs every tenth monomer) introduces one entanglement. These entanglements constitute a topological obstacle to the separation of the two strands of the double helix. DNA topoisomerases constitute a class of enzymes which catalyze the transient breakage and reunion of the strands of DNA molecules; the reversible breakage allows DNA segments to pass through one another. The mechanism of the reaction is not yet known, but it appears that following the break introduced by the enzyme the two broken segments are not allowed to move apart, being kept together through the intermediary of the enzyme. Type I DNA topoisomerases catalyze the breakage of one DNA strand, while
d'une topoisomérase. En mauvais solvant, les topoisomérases fonctionnent aussi efficacement qu'en bon solvant. Il sera intéressant d'étudier l'effet de ces catalyseurs dans la cinétique de condensation de ces molécules. The number of entanglement points can be enormous, since each turn of the helix (which occurs every tenth monomer) introduces one entanglement. These entanglements constitute a topological obstacle to the separation of the two strands of the double helix. DNA topoisomerases constitute a class of enzymes which catalyze the transient breakage and reunion of the strands of DNA molecules; the reversible breakage allows DNA segments to pass through one another. The mechanism of the reaction is not yet known, but it appears that following the break introduced by the enzyme the two broken segments are not allowed to move apart, being kept together through the intermediary of the enzyme. Type I DNA topoisomerases catalyze the breakage of one DNA strand, while type II topoisomerases catalyze the simultaneous breakage of the two strands of a DNA double helix. The two lines represent two DNA segments, AB and CD. A DNA topoisomerase permits segments AB and CD to pass through one another. A DNA type I topoisomerase can catalyze the reaction if at least one of the two segments is locally single-stranded. A type II topoisomerase is required if both segments are double-stranded (see also Roca and Wang, 1992). The reaction performed by a topoisomerase is schematically shown in the figure: in the absence of enzyme, the two DNA segments cannot cross each other due to the topological constraint described in (ii); the presence of a topoisomerase allows the crossing to take place with a characteristic time Tcross (to be discussed below). If the DNA segments belong to circular molecules, this reaction can modify their topological states, leading for instance to knotted structures, or catenanes. The catalytic properties of topoisomerases allow them to solve problems raised by both (i) and (ii) (see Kornberg and Baker, 1991). Here we investigate their potential role on problem (ii) only. We consider a solution of linear DNA molecules. In that case, the reaction performed by topoisomerases does
least one of the two segments is locally single-stranded. A type II topoisomerase is required if both segments are double-stranded (see also Roca and Wang, 1992). The reaction performed by a topoisomerase is schematically shown in the figure: in the absence of enzyme, the two DNA segments cannot cross each other due to the topological constraint described in (ii); the presence of a topoisomerase allows the crossing to take place with a characteristic time Tcross (to be discussed below). If the DNA segments belong to circular molecules, this reaction can modify their topological states, leading for instance to knotted structures, or catenanes. The catalytic properties of topoisomerases allow them to solve problems raised by both (i) and (ii) (see Kornberg and Baker, 1991). Here we investigate their potential role on problem (ii) only. We consider a solution of linear DNA molecules. In that case, the reaction performed by topoisomerases does not modify their static properties. Our aim is to see if it can affect their dynamic properties. We consider a solution of monodisperse chains above the overlap concentration c* and below the isotropic-anisotropic transition (the semidilute solution). We assume that the chain dynamics can be described as the combination of two independent processes, the reptation of the chain in a virtual tube defined by the surrounding chains, and the reorganization of the tube itself (called the constraint release process; see e.g. Viovy et al., 1991). The validity of the reptation hypothesis is being questioned for semidilute solutions, in particular because of the scarcity of entanglements (see section III. C in Lodge et al., 1990). However, this criticism does not apply to extremely long chains (Lodge et al., 1990), and this is the situation which we explore. The situation is then the following: in the absence of enzyme, tube reorganization is due to the reptation of the surrounding chains only. In presence of topoisomerase, we shall assume that each topological obstacle has a single lifetime (equal to Tcross): we face the problem of a chain trapped in a tube of mobile obstacles with a single lifetime, which is already solved (see
two independent processes, the reptation of the chain in a virtual tube defined by the surrounding chains, and the reorganization of the tube itself (called the constraint release process; see e.g. Viovy et al., 1991). The validity of the reptation hypothesis is being questioned for semidilute solutions, in particular because of the scarcity of entanglements (see section III. C in Lodge et al., 1990). However, this criticism does not apply to extremely long chains (Lodge et al., 1990), and this is the situation which we explore. The situation is then the following: in the absence of enzyme, tube reorganization is due to the reptation of the surrounding chains only. In presence of topoisomerase, we shall assume that each topological obstacle has a single lifetime (equal to Tcross): we face the problem of a chain trapped in a tube of mobile obstacles with a single lifetime, which is already solved (see Viovy et al., 1991). The system studied is apparently complex for two reasons: (a) DNA molecules are rather stiff polyelectrolytes. (b) Topoisomerases display some sequence specificity, i.e. perform the reaction shown in the figure with different rates for different DNA sequences. However, in practice (this will be justified below): — the DNA molecules considered are extremely long (much longer than the persistence length of the chain) and therefore flexible; — the enzymatic reaction requires salts: the description of a semidilute solution of polyelectrolyte with added salt can be obtained through a scaling procedure (Odijk, 1979); — the sequence specificity of the topoisomerase considered is low and can be neglected. We shall assume that each entanglement point can be crossed with the same time constant Tcross. Values of Tcross depend on the enzyme concentrations: we postulate that there is enough enzyme to saturate each entanglement point. In that case, Tcross is equal to the turnover number of the enzyme. For these reasons, we treat the DNA solution as a solution of flexible homopolymeric neutral chains under good solvent conditions. We consider a semidilute solution of monodisperse, linear, double-stranded DNA chains (concentration c > c* the overlap concentration). Each chain (made of
the DNA molecules considered are extremely long (much longer than the persistence length of the chain) and therefore flexible; — the enzymatic reaction requires salts: the description of a semidilute solution of polyelectrolyte with added salt can be obtained through a scaling procedure (Odijk, 1979); — the sequence specificity of the topoisomerase considered is low and can be neglected. We shall assume that each entanglement point can be crossed with the same time constant Tcross. Values of Tcross depend on the enzyme concentrations: we postulate that there is enough enzyme to saturate each entanglement point. In that case, Tcross is equal to the turnover number of the enzyme. For these reasons, we treat the DNA solution as a solution of flexible homopolymeric neutral chains under good solvent conditions. We consider a semidilute solution of monodisperse, linear, double-stranded DNA chains (concentration c > c* the overlap concentration). Each chain (made of N monomers) is enclosed in a virtual tube defined by the surrounding chains. The diameter of the tube, ξ(c), is also the mean separation between two entanglement points. We call Ne the number of entanglement points per chain, and g the number of monomers on the chain between two entanglement points. According to Daoud et al., 1975, we can express Ne as a function of the reduced concentration c/c*: Ne = (c/c*)1/(3 v-1), where v = 0.588 (giving Ne = (c/c*)1.31). For the reptation motion of the chain in the tube, the longest relaxation time (xrep) is predicted by de Gennes, 1976: xrep = (N t1) (ξ/c)2. The value of the exponent in (2) has been determined experimentally (Adam and Delsanti, 1983) and found to be equal to 2.05 rather than 1.6. Let xobst be the lifetime of the obstacles which constitute the tube. The dynamics of the tube can be described with the local-jump model (Stockmayer, 1976): the tube relaxes as a Rouse chain, and the longest relaxation time (named the reorganization time treo) for the tube is: there exists three possibilities: We consider first the free motion of the subchain (containing g monomers) between two successive entanglements. We
points. According to Daoud et al., 1975, we can express Ne as a function of the reduced concentration c/c*: Ne = (c/c*)1/(3 v-1), where v = 0.588 (giving Ne = (c/c*)1.31). For the reptation motion of the chain in the tube, the longest relaxation time (xrep) is predicted by de Gennes, 1976: xrep = (N t1) (ξ/c)2. The value of the exponent in (2) has been determined experimentally (Adam and Delsanti, 1983) and found to be equal to 2.05 rather than 1.6. Let xobst be the lifetime of the obstacles which constitute the tube. The dynamics of the tube can be described with the local-jump model (Stockmayer, 1976): the tube relaxes as a Rouse chain, and the longest relaxation time (named the reorganization time treo) for the tube is: there exists three possibilities: We consider first the free motion of the subchain (containing g monomers) between two successive entanglements. We compare this motion with the motion of the tube reorganizing itself as a consequence of the finite lifetime of the obstacles. If the motion of the tube is faster than the Rouse motion of the subchain (first possibility), then the chain does not feel the presence of the obstacles and is unentangled. The whole chain relaxes by Rouse relaxation. If the motion of the tube is slower, the chain is confined in the tube. The relaxation of the chain occurs through a combination of the two processes - reptation of the chain and tube reorganization. Since these are uncorrelated, the overall maximal relaxation time of the chain is given by: (4) (4) I/Tmax = l/Trep + l/Treo We introduce the dimensionless parameter: A = Treo/Trep = (c/c*) Tobs/(3 T, ) When a < 1 (second possibility), tube reorganization controls the maximal relaxation time; When a > 1 (third possibility) reptation controls the maximal relaxation time. In the absence of topoisomerases, Tobst is equal to xrep/6 (Klein, 1978). Therefore, a > 1, provided than N* > 18n2, which [according to (I)] corresponds roughly to c/c* > 10. Above this ratio, reptation controls the maximal relaxation time. Above c/c* > 10 and
is unentangled. The whole chain relaxes by Rouse relaxation. If the motion of the tube is slower, the chain is confined in the tube. The relaxation of the chain occurs through a combination of the two processes - reptation of the chain and tube reorganization. Since these are uncorrelated, the overall maximal relaxation time of the chain is given by: (4) (4) I/Tmax = l/Trep + l/Treo We introduce the dimensionless parameter: A = Treo/Trep = (c/c*) Tobs/(3 T, ) When a < 1 (second possibility), tube reorganization controls the maximal relaxation time; When a > 1 (third possibility) reptation controls the maximal relaxation time. In the absence of topoisomerases, Tobst is equal to xrep/6 (Klein, 1978). Therefore, a > 1, provided than N* > 18n2, which [according to (I)] corresponds roughly to c/c* > 10. Above this ratio, reptation controls the maximal relaxation time. Above c/c* > 10 and in the presence of (type II) topoisomerases, each obstacle will be destroyed and re-created ("on the other side of the chain") with a lifetime Tcross. We have a constraint release mechanism as described above, with Tobst = Tcross. This mechanism will lead to a significant effect (a decrease of the maximal relaxation time) if a < 1, which combined with c/c* > 10 gives the condition: (5) 10 < C/C* < 3n2 T1/Tcross The second inequality reflects the fact that the increase of Treo with concentration is faster than Trep, at fixed number N of base pairs. On the contrary, if the concentration is fixed, the increase of Treo with N is lower than Trep. We observe that (5) could be modified if the experimental exponent determined in (Adam and Delsanti, 1983) was used: this would enlarge the interval. NUMERICAL ESTIMATES. A remark on the meaning of Tcross: For a given enzyme, the observed turnover number strongly depends on the type of reaction monitored (for instance supercoiling of a single circular molecule, or catenation of multiple circles), and also on the experimental conditions. The direct measure of a turnover number for the reaction involving two linear DNA segments as shown
maximal relaxation time) if a < 1, which combined with c/c* > 10 gives the condition: (5) 10 < C/C* < 3n2 T1/Tcross The second inequality reflects the fact that the increase of Treo with concentration is faster than Trep, at fixed number N of base pairs. On the contrary, if the concentration is fixed, the increase of Treo with N is lower than Trep. We observe that (5) could be modified if the experimental exponent determined in (Adam and Delsanti, 1983) was used: this would enlarge the interval. NUMERICAL ESTIMATES. A remark on the meaning of Tcross: For a given enzyme, the observed turnover number strongly depends on the type of reaction monitored (for instance supercoiling of a single circular molecule, or catenation of multiple circles), and also on the experimental conditions. The direct measure of a turnover number for the reaction involving two linear DNA segments as shown in the figure has never been reported. For this reason, we must extrapolate from values determined in other reactions. E. coli DNA gyrase, the most studied type II topoisomerase, can perform 50 crossing events per minute (Higgins et al., 1978). However, this value is obtained under poor solvent conditions (DNA molecules being condensed by polyamines: see Krasnow and Cozzarelli, 1982), which are irrelevant for our purpose. The turnover number of gyrase under good solvent conditions has not been determined but is probably smaller (see Kreuzer and Cozzarelli, 1980). The fact that the highest values for the turnover number are obtained under poor solvent conditions appears rather general for topoisomerases (see Low et al., 1984). This can be understood from the reaction catalyzed by topoisomerases, which depends on the contact probability of DNA segments: this probability is increased with decreasing solvent qualities, or increasing DNA concentrations. We note here that estimates for the turnover number of gyrase in vivo are higher than the values obtained for poor solvent conditions in vitro (Cook et al., 1992). The turnover number of Drosophila type II DNA topoisomerase has been determined under good solvent conditions (Osheroff et al., 1983): this enzyme can introduce at least