jeanflop/post-ocr-correction huggingface dataset:
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
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.
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
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
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
(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
(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
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,
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
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
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
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
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 200 supercoils in relaxed circular DNA per minute and per
200 supercoils in relaxed circular DNA per minute and per molecule of enzyme, which corresponds to 100 crossing events, or a Tcross of 0.6 sec. For linear molecules, the reaction would be four times slower according to (Osheroff et al., 1983). The corresponding value of Tcross would then be 2.4 sec. Furthermore, the sequence specificity of the enzyme is low (Sander and Hsieh, 1985). Using the value of 2.4 sec., we see from (5) that x1 must be greater than 24/3 n2 or 0.8 sec. Experimental values of T1 (obtained from viscoelastic measurements) are given in Klotz and Zimm, 1972:
from viscoelastic measurements) are given in Klotz and Zimm, 1972: T2 DNA (N = 166,000) has an experimental T1 of 0.5 sec., which is too small. Linearized E. coli DNA (N = 4.5 x 106) has an experimental T1 of 70 sec., which gives an upper limit for c/c* of 860. Since c* is roughly equal to 2 µg/ml, the upper value of c is still in the semidilute regime (in particular much below the isotropic-anisotropic transition). We may also start from a known relaxation time T1, and ask what is the highest value of Tcross which would give detectable
is the highest value of Tcross which would give detectable effects: according to (5) it is given by: Tcross = 3n2 T1/10, or Tcross = 200 sec. for T1 = 70 sec. Thus it appears that even lower values of Tcross could be acceptable. CONCLUDING REMARKS. — 1. For sufficiently long (megabase size) DNA molecules, we predict a significant decrease of the longest relaxation time for a large concentration regime. This prediction relies on an estimate of the turnover number which may be too high; smaller values of the turnover number would shrink the concentration regime, but could still have
number would shrink the concentration regime, but could still have detectable effects. Experimentally, these effects could be investigated with the viscoelastic method described in Klotz and Zimm (1972). Our result applies directly to the interpretation of diffusion coefficients measurements. Concerning the viscoelastic properties, we need to specify the effect of topoisomerases on the long time-limit of the relaxation modulus G(t) (Doi and Edwards, 1986). 2. Investigations with poor solvents. Since, as observed above, topoisomerases work well under poor solvent conditions (where DNA molecules are aggregated), one can study the behaviour of DNA molecules condensed in the presence of topoisomerases. In
of DNA molecules condensed in the presence of topoisomerases. In particular, since condensation can be coupled to the isotropic-anisotropic (liquid crystalline) transition, one may ask if the presence of topoisomerases can effect the local ordering of the molecules. 3. Comparison with "living" polymers. "Living" polymers are linear chains that can break and recombine on experimental time-scales. According to a theoretical model (Cates, 1987), for a given chain, the two events breakage and recombination are uncorrelated: there is no higher probability for a chain end to link up with the chain from which it was recently detached than with any other
from which it was recently detached than with any other chain end in the vicinity. In the system considered here, breakage and reunion are correlated as noted above. An experimental system involving DNA molecules similar to the "living" polymer model could be obtained with a combination of a cleaving (a restriction endonuclease) and a rejoining (a ligase) enzyme (see chapters 9 and 13 in Romberg and Baker, 1991). Finally, as indicated to us by a referee, we note that breakage and reunion can also be correlated in certain "living" polymer systems (Kern et al., 1992). Note remise le 15 décembre
systems (Kern et al., 1992). Note remise le 15 décembre 1992, acceptée le 5 février 1993. REFERENCES M. ADAM and M. DELSANTI, Viscosity and longest relaxation time of semi-dilute polymer solution. I. Good solvent JPhysique, 44, 1983, pp. 1185-1193. M. E. CATES, Reptation of living polymers: dynamics of entangled polymers in the presence of reversible chain-scission reactions, Macromolecules, 20, 1987, pp. 2289-2296. D. N. COOK, D. MA, N. G. PON and J. E. HEARST, Dynamics of DNA supercoiling by transcription in Escherichia coli, Proc. Natl. Acad. Sci. USA, 89, 1992, pp. 10603-10607. M. DAOUD, J. P. COTTON, B. FARNOUX, G.
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reconstitution of enzymatic activity, Proc. Natl. Acad. Sci. USA, 75, 1978, pp. 1773-1777. F. KERN, P. MARECHAL, S. J. CANDAU and M. E. CATES, Rheological properties of semidilute and concentrated aqueous solutions of cetyltrimethylammonium bromide in the presence of potassium chloride, Langmuir, 8, 1992, pp. 437-440. J. KLEIN, The onset of entangled behavior in semidilute and concentrated polymer solutions, Macromolecules, 11, 1978, pp. 852-858. L. C. KLOTZ and B. H. ZIMM, Size of DNA determined by viscoelastic measurements: results on bacteriophages, Bacillus subtillis and Escherichia coli, J. Mol. Biol., 72, 1972, pp. 779-800. A. KORNBERG and T. A. BAKER, DNA
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Physics, 79, 1990, pp. 1-132. R. L. LOW, J. M. KAGUNI and A. KORNBERG, Potent catenation of supercoiled and gapped DNA circles by topoisomerase I in the presence of a hydrophilic polymer, J. Biol. Chem., 259, 1984, pp. 4576-4581. T. ODIJK, Possible scaling relations for semidilute polyelectrolyte solutions, Macromolecules, 12, 1979, pp. 688-693. N. OSHEROFF, E. R. SHELTON and D. BRUTLAG, DNA topoisomerase II from Drosophila melanogaster, J. Biol. Chem., 258, 1983, pp. 9536-9543. J. ROCA and J. C. WANG, The capture of a DNA double helix by an ATP-dependent protein clamp: a key step in DNA transport by type
protein clamp: a key step in DNA transport by type II DNA topoisomerases, Cell, 71, 1992, pp. 833-840. M. SANDER and T.-S. HSIEH, Drosophila topoisomerase II double-strand DNA cleavage: analysis of DNA sequence homology at the cleavage site, Nucl. Acids Res., 13, 1985, pp. 1057-1072. W. H. STOCKMAYER, in Molecular Fluids; R. BALIAN and G. WEILL Eds., Gordon and Breach, London, 1976. J.-L. VIOVY, M. RUBINSTEIN and R. H. COLBY, Constraint release in polymer melts: tube reorganization versus tube dilation, Macromolecules, 24, 1991, pp. 3587-3596. J.-L. S.: Service de Biochimie et de Genetique Moléculaire, Département de Biologie Cellulaire et Moléculaire;
et de Genetique Moléculaire, Département de Biologie Cellulaire et Moléculaire; G. J.: Laboratoire Leon Brillouin, CEA/CNRS, CE Saclay, 91191 Gif-sur-Yvette Cedex, France. Informatique/Computer Science Reconnaissance de caractères par le réseau neuronal aléatoire Myriam MOKHTARI Résumé Nous étudions le modèle neuronal aléatoire de Gelenbe (Gelenbe, 1989, 1991) comme mémoire auto-associative et présentons de nouvelles méthodes de stockage et de reconnaissance, que nous appliquons à la reconnaissance de chiffres. Nous constatons qu'une très bonne performance de reconnaissance est obtenue quelque soit la corrélation entre les exemples stockés et l'exemple à reconnaître. Les expériences montrent que plus le choix des paramètres est précis,
expériences montrent que plus le choix des paramètres est précis, meilleure est la performance de reconnaissance. Cependant, les méthodes deviennent lentes. Ici, nous présentons également des méthodes très rapides pouvant donner une bonne performance de reconnaissance. Character recognition with the random neural network Abstract — We study the random neural network model (Gelenbe, 1989, 1991) as an auto-associative memory and present new storage and recognition methods which we apply to digit recognition. Very good recognition performance is obtained whatever the correlation between the stored patterns and the pattern to be recognized. Experiments show that as the network parameter choice becomes
recognized. Experiments show that as the network parameter choice becomes more precise, the recognition performance improves. However, the learning and recognition methods become time consuming. Here, we also present very fast methods which can give a good recognition. 1. INTRODUCTION. Dans Gelenbe (1989, 1990), un modèle de réseau neuronal aléatoire a été introduit. Un neurone peut exciter ses voisins s'il émet des signaux positifs ou les inhiber par une émission de signaux négatifs. Le modèle a été étendu dans Gelenbe, Stafylopatis, Likas (1991) en donnant un signe à chaque neurone. Il y a accumulation de signaux positifs (respectivement négatifs) aux
Il y a accumulation de signaux positifs (respectivement négatifs) aux neurones positifs (respectivement négatifs). C'est ce modèle que nous considérons. Soient k(t) le vecteur des n potentiels à l'instant t et k = (k1, kn) une valeur particulière de k(t). D'après Gelenbe (1989), si toutes les probabilités d'excitation qt à l'état stable sont telles que O < qi < 1, la probabilité stationnaire p(k) = lim P [k(t) = k existe et est donnée par : p(k) = n (1 - qi) tf1. Les qi tendant à l'infini i = 1 sont calculés à partir du système d'équations non linéaires
1 sont calculés à partir du système d'équations non linéaires (l)-(3) où P (respectivement N) est l'ensemble des neurones positifs (respectivement négatifs) : où : A (i) (k (i)) : taux d'arrivées externes des signaux positifs (respectivamente négatifs). P + (i, j) : probabilité que le neurone j envoie un signal de même signe que j au neurone i. p (À i) : probabilité que le neurone j envoie un signal de signe opposé à j au neurone i. r(i) : taux d'émission de signaux par le neurone i. 2. LE RÉSEAU NEURONAL ALÉATOIRE COMME MÉMOIRE AUTO-ASSOCIATIVE. Une mémoire auto-associative
LE RÉSEAU NEURONAL ALÉATOIRE COMME MÉMOIRE AUTO-ASSOCIATIVE. Une mémoire auto-associative est un système qui apprend un ensemble d'exemples et les reconnaît même bruités. Soit S un ensemble de m exemples de dimension n : S = {x(1), ..., x(m)}, où X(k) = { X(k), ..., Xn }, avec Xjk) e { 1, 0}, et ~Xjk) est la k-ième entrée du neurone i. @ @ n t Nous utilisons une couche de n neurones tous interconnectés. Soit w (i, j) le poids de la connection entre les neurones i et j dans le réseau neuronal formel. On calcule les poids avec
dans le réseau neuronal formel. On calcule les poids avec la règle de Hebb : w (i, j) = L X[k)X[j] pour 1 < i, j < n et i #= j. En utilisant la correspondance vue dans Gelenbe (1989), on calcule r (i), p + (i, j), P (i, j) pour r(i)-^w(i,j), et si w (i, j) > 0 alors r(i)p+ (i,j) = w(i,j) sinon r (i) p (i, j) w (i, j) L'entrée d'un exemple X dans le réseau caractérise les neurones : si Xi =1', alors ieN, X(i) = X et A (i) = 0; si Xi
X(i) = X et A (i) = 0; si Xi = -1, alors i ∈ P, A (i) = A et X (i) = 0. Les définitions et théorèmes suivants réduisent l'espace pour choisir A et X : DÉFINITION 1 (hyper-stabilité) (Gelenbe, 1990). — Un réseau neuronal aléatoire est dit hyper-stable si : THÉORÈME 1 (Gelenbe, 1990). — Si un réseau est hyper-stable, le système d'équations (l)-(3) a une unique solution satisfaisant 0 < qi < 1 pour 1 < i < n. Par analogie : DÉFINITION 2 (hyper-instabilité) (Mokhtari, 1992 a). — Un réseau neuronal aléatoire est dit hyper-instable
1992 a). — Un réseau neuronal aléatoire est dit hyper-instable si 3 i : THÉORÈME 2 (Mokhtari, 1992 a). — Si un réseau est hyper-instable, le système d'équations (l)-(3) n'a pas de solution satisfaisant 0<qi<1 pour 1<i<n. Durant l'apprentissage, une heuristique (Mokhtari, 1992 a) déterminera les valeurs maximales de A et Â, où Ae[Amin, Ama] et Â. E [Â.min, Ama], avec Amin, Àmin (respectivement Amax, Xmax) calculés à partir de la condition d'hyper-stabilité (respectivement hyper-instabilité), et telles que le minimum des q, sur tous les neurones et tous les exemples appris (notée par a0) soit le plus grand et inférieur
appris (notée par a0) soit le plus grand et inférieur à 1. La reconnaissance est améliorée si à chaque neurone i positif (respectivement négatif) on associe une valeur A (i) (respectivement Â. (i)) telle que les qi des exemples appris soient élevés et du même ordre. Donc, après calcul des valeurs globales A et X, le processus suivant est répété jusqu'à ce que ot0 ne puisse plus être augmenté : incrémenter (respectivement décrémenter) de 1 le taux d'arrivées des neurones les moins excités (respectivement saturés) puis recalculer les qi des exemples à apprendre. Notons que les q, sont calculés par
exemples à apprendre. Notons que les q, sont calculés par l'équation du point fixe. Le comportement de notre mémoire est représenté ainsi : pendant l'apprentissage, pour chaque X(k) g S, tous les qi sont élevés (>> a0); pendant la reconnaissance d'un exemple X' bruité, on a : qi élevé si X,' est correct et qi faible si ~X; est incorrect. En appliquant un seuil discriminant a pour : si q > a alors Yi = X,i sinon Y, = ~Xj, Y sera la version corrigée de X'. La valeur appropriée de a est telle que X' soit parfaitement corrigé. Pour
de a est telle que X' soit parfaitement corrigé. Pour savoir donc si Y est la reconstruction exacte X, il suffit d'injecter Y au système et de vérifier que q, k > a0, 1 < i < n. Tant que cette condition n'est pas satisfaite, ce processus est répété avec un a plus faible. Si a devient inférieur à 0,5 (seuil raisonnable) l'exemple X' peut être corrigé en une seule passe. On commence donc par corriger les erreurs de X' correspondant aux neurones les moins excités : si qt, = min q alors X; = ~X sinon ~X; = y
min q alors X; = ~X sinon ~X; = y où min q est la valeur minimale des q, pour X'. On considère cette nouvelle entrée 4> en la réinjectant et en répétant le processus précédent. On arrête l'injection de X' lorsqu'un certain nombre de réinjections est supérieur à n par le taux de bruit maximal supposé, car il y a au pire 1 correction par passe. Cette méthode lente peut être accélérée car la plupart des calculs sont intrinsèquement parallèles et peuvent être exécutés sur une machine SIMD à mémoire locale ou partagée. La reconnaissance a été appliquée aux
mémoire locale ou partagée. La reconnaissance a été appliquée aux 10 chiffres donnés par la figure 1. Chaque chiffre codé sur 64 éléments (n = 64) est bruité jusqu'à 20 de bruit. L'exemple est considéré reconnu si le taux d'erreurs résiduel en sortie est inférieur à 6,25%. Le nombre de chiffres à apprendre m est représenté ainsi : Fig. 2. Performance de reconnaissance en fonction du nombre m de chiffres à apprendre et du taux de bruit d, obtenue par le modèle de Gelenbe. Fig. 2. Recognition performance as a function of the number m of stored digits and the
function of the number m of stored digits and the noise level d, obtained by Gelenbe's model. montre que pratiquement tous les exemples sont reconnus. La reconnaissance de chiffres non bruités (d= 0 %) est assurée par notre algorithme. 3. LE RÉSEAU AVEC NEURONES SATURÉS. Dans Gelenbe (1989), un théorème indique que si certains neurones sont saturés (Î±=-1), la solution au système d'équations (1)-(3) restreinte aux neurones non saturés s'exprime sous la forme produit précédente. Ce type de réseau permet un apprentissage rapide et simple puisqu'il suffit de poser pour tout i∈ N, à (i) = r (i) et A