Ghesquière

Louise Ghesquière
email: louise.ghesquiere@etu.univ-lille2.fr

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Quel est l'impact de la rupture des membranes amniotiques sur le rythme cardiaque foetal au cours du travail spontané? Does rupture of membranes have an impact on the fetal heart rate during spontaneous labor?

INTRODUCTION

Environ 10% des débuts de travail à terme sont précédés d'une rupture spontanée des membranes [START_REF] Dilbaz | Risk factors and perinatal outcomes associated with umbilical cord prolapse[END_REF][START_REF] Zlatnik | Management of premature rupture of membranes at term[END_REF]. À l'inverse, comme le révèle l'enquête nationale périnatale (ENP) publiée en 2016, l'amniotomie, ou rupture artificielle des membranes (RAM), serait réalisée dans 41,4% des naissances en France [START_REF] Blondel | Enquête Nationale Périnatale, rapport 2016: les naissances et les établissements, situation et évolution depuis 2010[END_REF]. Acte obstétrical, elle entre dans le cadre des procédures hospitalières selon la définition de la haute autorité de santé (HAS)(4). Elle permet de diriger le travail de manière active lors d'un déclenchement, en cas de dystocie, ou peut être réalisée en cas d'anomalie du rythme cardiaque foetal (ARCF). La revue de littérature de Smyth et al.

démontre en cas de RAM une diminution du temps de travail entre 0,8 et 2,3 heures, une diminution de l'utilisation de l'ocytocine, sans augmentation du nombre des césariennes [START_REF] Smyth | Amniotomy for shortening spontaneous labour[END_REF][START_REF] Sadler | A randomised controlled trial and metaanalysis of active management of labour[END_REF][START_REF] Ghafarzadeh | Effect of early amniotomy on dystocia risk and cesarean delivery in nulliparous women: a randomized clinical trial[END_REF][START_REF] Brisson-Carroll | The effect of routine early amniotomy on spontaneous labor: a meta-analysis[END_REF][START_REF] Shobeiri | Amniotomy in labor[END_REF][START_REF] Garite | The influence of elective amniotomy on fetal heart rate patterns and the course of labor in term patients: a randomized study[END_REF][START_REF] Verspyck | Abnormal fetal heart rate patterns associated with different labour managements and intrauterine resuscitation techniques[END_REF][START_REF] Brown | Package of care for active management in labour for reducing caesarean section rates in low-risk women[END_REF]. Cependant lorsque l'amniotomie seule est comparée à la rupture spontanée des membranes (RSM), la réduction du temps de travail n'est pas significative [START_REF] Smyth | Amniotomy for shortening spontaneous labour[END_REF]. L'Organisation Mondiale de la Santé (OMS) déclare qu'il n'existe « aucune preuve en faveur d'une politique d'amniotomie systématique en vue de réduire la durée du travail chez les femmes primipares ou multipares » [START_REF] Oms | WHO recommendation on routine amniotomy for prevention of delay in labour[END_REF].

La principale crainte dans les suites de la rupture des membranes (spontanée ou artificielle) est la survenue d'anomalies du rythme cardiaque foetal [START_REF] Brisson-Carroll | The effect of routine early amniotomy on spontaneous labor: a meta-analysis[END_REF][START_REF] Garite | The influence of elective amniotomy on fetal heart rate patterns and the course of labor in term patients: a randomized study[END_REF][START_REF] Verspyck | Abnormal fetal heart rate patterns associated with different labour managements and intrauterine resuscitation techniques[END_REF][START_REF] Goffinet | Early amniotomy increases the frequency of fetal heart rate abnormalities. Amniotomy Study Group[END_REF][START_REF] Herman | Risk factors for nonreassuring fetal heart rate tracings after artificial rupture of membranes in spontaneous labor[END_REF]. Goffinet et al. ont comparé 375 travails comprenant RAM précoces versus 377 travails expectatifs sans intervention et ont relevé la survenue de plus de ralentissements précoces, variables sévères et tardifs, ainsi qu'une répétition des ralentissements variables sévères en cas de RAM précoces.

Garite et al. quant à eux retrouvaient dans une série de 469 RAM la survenue de plus de ralentissements variables modérés [START_REF] Garite | The influence of elective amniotomy on fetal heart rate patterns and the course of labor in term patients: a randomized study[END_REF]. Récemment, sur 664 cas de RAM faites en phase active, Ganer et al. notaient 21,2% d'ARCF dans les suites d'une RAM. [START_REF] Herman | Risk factors for nonreassuring fetal heart rate tracings after artificial rupture of membranes in spontaneous labor[END_REF]. Dans leur série, les ARCF survenaient plus fréquemment en cas de multiparité, de petit poids pour l'âge gestationnel (PAG), d'une analgésie péridurale, de la présence d'ocytocine au moment de la rupture et de liquide amniotique méconial. Les ARCF étaient également étroitement liées à la dilatation pendant la rupture amniotique, ainsi qu'à la durée entre la RAM et l'accouchement.

Ainsi, les recommandations nationales ne préconisent pas de réaliser une amniotomie précoce et systématique en début de travail parce qu'elle induit plus fréquemment des ARCF [START_REF] Cngof | Recommandations pour la pratique -Modalités de surveillance pendant le travail[END_REF].

L'impact sur le RCF de la RSM en cours de travail est moins connu [START_REF] Vadivelu | Randomized controlled trial of the effect of amniotomy on the duration of spontaneous labor[END_REF][START_REF] Aladjem | Effects of spontaneous and artificial membrane rupture in labour upon fetal heart rate[END_REF] Le tableau 2 présente l'impact de la rupture sur le RCF, quel que soit son type (RSM et RAM). 44,6% d'ARCF (n=104, p<0,001) survenait dans l'heure suivant la rupture, avec une augmentation de l'incidence par rapport à la période avant la rupture des membranes des bradycardies (5,6% vs 0,4%, p=0,005), d'une tachycardie (4,3% vs 1,7%, p=0,0339), d'une variabilité anormale (11,6% vs 6,4%, p=0,0047), de ralentissements précoces (6% vs 1,7%, p=0,0039), variables typiques (6,9% vs 2,6%, p=0,0124), variables atypiques (24,9% vs 6 ,9%, p<0,001) et prolongés (17,6% VS 5,2%, p<0,001). Il n'y avait pas de différence quant à la survenue de ralentissements tardifs. Le RCF était classé selon FIGO plus suspect et Plusieurs mécanismes pourraient expliquer les ARCF après rupture des membranes amniotiques pendant le travail. En effet, lors de la contraction utérine, une hypoxie transitoire surviendrait à la fois par compression des vaisseaux utéroplacentaires, mais aussi par compression cordonale entraînant une réponse de type baroréflexe (ralentissements variables typiques, retrouvés dans la RSM) [START_REF] Lear | The myths and physiology surrounding intrapartum decelerations: the critical role of the peripheral chemoreflex[END_REF]. Lors de la rupture des membranes, un oligoamnios ou un anamnios se crée. On peut alors supposer que lors des contractions utérines, devant l'absence de liquide protecteur autour du cordon, les compressions cordonales seraient plus importantes devant l'absence du liquide amniotique autour du cordon. De même, en absence de liquide amniotique, la réponse vagale à la pression exercée par compression autour de la tête foetale lors de la contraction serait supérieure. Cela entraînerait la survenue notamment de ralentissement prolongé tel qu'observé dans le premier quart d'heure après RSM-RAM dans notre série.

En pratique clinique, il semble intéressant de pouvoir prédire les situations à risque d'ARCF avant amniotomie ou en cas de rupture spontanée des membranes. Dans notre série, le seul facteur prédictif était la présence d'ARCF avant la rupture. Ganer et al. retrouvaient que la parité et la dilatation lors de la RAM étaient des facteurs de risques associés à la survenue d'ARCF, données non retrouvées dans notre série [START_REF] Herman | Risk factors for nonreassuring fetal heart rate tracings after artificial rupture of membranes in spontaneous labor[END_REF]. Dans notre étude, la durée moyenne entre la rupture et l'accouchement était significativement plus courte en cas de RSM que de RAM s'expliquant par le fait que la rupture avait lieu plus fréquemment en phase active et à dilatation complète lors d'une RSM du fait du respect de la physiologie du travail telle que promue par les recommandations nationales du CNGOF [START_REF] Cngof | Recommandations pour la pratique -Modalités de surveillance pendant le travail[END_REF].

Une autre explication à cette différence de durée est probablement que les RAM ont eu lieu lors de travail plus long avec probablement stagnation au vue du taux plus élevé d'utilisation d'oxytocine dans ce groupe.

Cette étude comporte toutefois des limites, principalement l'interprétation du RCF par un seul observateur avec un biais possible du fait de la variabilité d'interprétation inter et intra individuelle. Le recueil des données a été réalisé de manière rétrospective dans les dossiers avec une possible erreur de retranscription de l'horaire exact de la rupture. L'effectif de notre échantillon engendre une puissance insuffisante avec certaines comparaisons proches de la significativité. La survenue d'ARCF en cas de RSM n'est statistiquement pas différente qu'en cas de RAM dans notre étude mais on retrouve presque 10% supplémentaires d'ARCF en cas de RAM. Une étude plus puissante devrait être menée pour savoir si le type de rupture peut modifier de manière significative la survenue d'ARCF.

CONCLUSION

J

  FIGO: International Federation of Gynecology and Obstetrics

  Le rythme cardiaque foetal était enregistré par monitorage externe ou interne. Le suivi du travail était reporté sur le partogramme. L'évènement RAM ou RSM était reporté sur celui-ci ou directement sur le RCF papier. Les anomalies du RCF étaient classées selon la classification nationale du CNGOF et internationale de la FIGO[START_REF] Carbonne | CNGOF classification of fetal heart rate: color code for obstetricians and midwives[END_REF][START_REF] Carbonne | Classification FIGO 2015 du rythme cardiaque foetal (RCF)[END_REF]. Les ralentissements étaient définis selon les critères du CNGOF et classés en précoce, variable typique ou atypique, tardif et prolongé. Les autres éléments relevés étaient la survenue d'une tachycardie (RCF > 160 bpm durant plus de 40 minutes), d'une bradycardie (RCF < 100 bpm durant plus de 10 minutes) et d'une anomalie de la variabilité (minime, réduite ou augmentée). Cinq périodes ont été définies pour l'analyse du RCF : H1 l'heure avant rupture des membranes amniotiques, puis un découpage en quart d'heure (Q1 à Q4) dans les suites de la rupture (spontanée ou artificielle). Les données relevées au niveau de la population étaient l'âge maternel, l'âge gestationnel, la parité, la présence pendant la grossesse d'un indice de masse corporelle (IMC) >25 kg/m2, d'un PAG (défini par une estimation du poids foetal inférieur au 10ème percentile), d'une grossesse à terme dépassée, et d'un tabagisme. En cours de travail, ont été relevés la présence d'une analgésie locorégionale, la dilatation lors de la rupture, l'existence de liquide amniotique méconial (LAM), le recours à l'ocytocine et enfin la durée entre la rupture des membranes et l'accouchement. Statistiques Les caractéristiques des deux groupes (ruptures spontanées ou artificielles des membranes) ont été comparées à l'aide d'un test du Chi-deux ou de Fisher exact pour les variables qualitatives, à l'aide d'un test t de Student pour les variables continues gaussiennes, et à l'aide d'un test U de Mann-Whitney pour les variables continues non gaussiennes. L'influence de la rupture sur le rythme cardiaque foetal a été évaluée en comparant les données avant la rupture aux données post-rupture à l'aide de test de Mc Nemar pour les variables qualitatives binaires et à l'aide de tests des rangs signés pour les variables qualitatives ordinales. L'évolution des différentes complications du rythme foetal au cours des quatre quarts d'heure d'enregistrement en post-rupture a été évaluée à l'aide d'un modèle linéaire mixte généralisé incluant le temps comme effet fixe et la patiente en effet aléatoire. Les comparaisons post-hoc deux à deux des différents temps d'enregistrement ont été réalisées en appliquant une correction de Bonferroni. Les facteurs influençant une anomalie du rythme cardiaque foetal en post-rupture ont été recherchés à l'aide du test du Chi-deux pour les variables binaires, à l'aide d'un test t de Student pour les variables continues gaussiennes, et à l'aide d'un test U de Mann-Whitney pour les variables continues non gaussiennes. Des tests bilatéraux ont été réalisés avec un niveau de significativité de 5%. Les analyses statistiques ont été effectuées à l'aide du logiciel SAS (SAS Institute version 9.4). RESULTATS 425 patientes à bas risque ont été prises en charge entre Janvier 2018 et Mars 2018. Parmi celles-ci, 324 (76,2%) patientes ont rompu en salle de naissance et 233 patientes (54,8%) ont été retenues au final (figure 1). 130 (55,7%) ont bénéficié d'une rupture artificielle, et 100 (44,3%) ont rompu spontanément.Le tableau 1 présente les caractéristiques des patientes selon le mode de rupture. Il n'existait pas de différence significative au niveau de l'âge maternel, du surpoids, du tabagisme et de la nulliparité. L'âge gestationnel à la naissance, la prévalence du post-terme et des PAG étaient également similaires. La durée moyenne entre la rupture et l'accouchement était significativement plus courte dans le groupe RSM à RAM (2,4 h vs 5 h, p<0,001). La rupture avait lieu en moyenne en phase de latence pour le groupe RAM (3cm (3-4cm)) et en phase active pour le groupe RSM ((7cm (4-9cm)) (p<0,001).

  pathologique post-rupture (p<0,001), et selon la classification CNGOF, plus à risque d'acidose (p<0,001).En cas de RAM, on retrouvait 47,7% (n=65) d'ARCF dans l'heure suivant celle-ci. En cas de RSM, on retrouvait 40,8% (n=39) d'ARCF. Cette différence n'était pas significative (p=0.2916), même après ajustement sur la dilatation en cm (p=0.1938).La figure 2 représente l'étude horaire du RCF après rupture des membranes amniotiques (RSM et RAM). Il n'existait pas de différence significative dans la survenue de bradycardie, de tachycardie, de ralentissement (précoce, tardif, variable typique, variable atypique) ni de variabilité anormale lorsqu'on comparait ces variables par quart d'heure. Les ralentissements prolongés étaient plus fréquents dans le premier quart d'heure par rapport au deuxième quart d'heure (11,2% vs 3,5%, p=0,012). Si l'on compare le type de rupture (RSM versus RAM) sur l'incidence horaire des ARCF (figure3), aucune différence significative n'a été retrouvée entre les quarts d'heure sur la même variable d'ARCF. Les bradycardies, les tachycardies, les ralentissements (précoces, tardifs, variables typiques et variables atypiques), les ralentissements prolongés ainsi que les variabilités anormales survenaient de la même manière dans les populations RSM et RAM.Le tableau 3 présente les facteurs de risque des ARCF après rupture des membranes amniotiques (RSM et RAM). La présence d'un RCF pathologique avant rupture (31,7% vs 3,1%, p<0,001) était un facteur favorisant l'apparition d'un RCF pathologique. Les caractéristiques maternelles et les autres données du travail n'étaient pas associées à un risque de survenue d'ARCF.DISCUSSIONLa rupture des membranes, spontanée ou provoquée, est un évènement fréquent pendant le travail. Peu de données sont toutefois disponibles sur les variations du RCF dans les suites de cet évènement et peu d'études à ce jour ont comparé le RCF en cas de RSM versus RAM(5- 12,14,15). Cette étude montre que l'incidence des ARCF après rupture des membranes amniotiques (RSM et RAM) était de 44,6% et que ce taux ne dépendait pas du type de rupture, des caractéristiques maternelles et du travail en dehors de la présence ou non d'un RCF pathologique avant rupture.Les données actuelles de la littérature sont rares pour la RSM. En 1997, l'étude randomisée de Goffinet et al comprenant 925 patientes de grossesses à bas risque à terme, et comparant la RAM précoce avec le travail expectatif retrouvait que la RAM, était responsable d'une augmentation des ralentissements variables sévères (1,4/h vs 0,7/h, p=0,011), des ralentissements précoces (3,0/h vs 2,1/h, p=0,048) et des ralentissements tardifs (3,3/h vs 2,3/h, p=0,021)[START_REF] Goffinet | Early amniotomy increases the frequency of fetal heart rate abnormalities. Amniotomy Study Group[END_REF]. Garite et al, soulignaient quant à eux, dans leur étude en 1993, une augmentation des ralentissements variables modérés[START_REF] Garite | The influence of elective amniotomy on fetal heart rate patterns and the course of labor in term patients: a randomized study[END_REF]. Dans la revue de littérature publiée en 2008 de Cochrane, la RAM associée à des doses élevées d'ocytocine, était mise en évidence comme à risque d'ARCF[START_REF] Smyth | Amniotomy for shortening spontaneous labour[END_REF][START_REF] Verspyck | Abnormal fetal heart rate patterns associated with different labour managements and intrauterine resuscitation techniques[END_REF]. Plus récemment, en 2018, Ganer et al. notaient 21,2% d'ARCF dans les suites d'une RAM[START_REF] Herman | Risk factors for nonreassuring fetal heart rate tracings after artificial rupture of membranes in spontaneous labor[END_REF]. Dans notre série, on retrouvait également plus d'ARCF après rupture des membranes amniotiques (RAM ou RSM) avec 44,6% d'ARCF. La survenue des ARCF ne dépendait pas de la nature artificielle ou spontanée de la rupture. Des ARCF étaient plus souvent décrit après RAM, 47,7% vs 40,8% en cas de RSM, cette différence n'était statistiquement pas significative. Dans l'étude horaire suivant la rupture, il y avait une nette différence entre le taux de ralentissements prolongés dans le premier quart d'heure (Q1) par rapport au deuxième (Q2) (p=0,0121). L'originalité de ce travail était d'étudier cette répartition horaire. Cela nous permet de mettre en évidence que la surveillance du RCF est importante et doit être vigilante dans les premières minutes suivant la rupture des membranes.

CNGOF Normal ou Faible risque A risque d'acidose Risque élevé d'acidose Risque majeur d'acidose

  La rupture des membranes est à risque de survenue d'anomalies du rythme cardiaque foetal, et ce quel qu'en soit le type. L'amniotomie n'entraîne pas plus d'ARCF que la rupture spontanée. 44,6% d'ARCF surviennent après rupture des membranes, soit presque une fois sur deux. Ce taux élevé suggère l'importance de la surveillance du RCF après rupture des membranes, de bien informer le couple lors de la RAM et de bien tracer l'évènement RSM/RAM dans le dossier. Figure 2 : Comparaison horaire de l'impact de la RSM/RSM sur le RCF : Résultats exprimés en N (%) ou en moyenne ± écart-type et médiane (Quartile 1, Quartile 3).Tableau 2: Impact de tous types de ruptures des membranes (RSM et RAM) sur le RCF

	Figure 1 -Flow Chart Tableau 1 : Caractéristiques de la population
	Figure 1 -Population de l'étude Analyse des ARCF RSM VS RAM (horaire) 425 patientes à bas risque Tous types de RAM RSM RCF 1h avant RCF 1h après Impact RAM+RSM sur le RCF: ruptures rupture rupture 35.00% N=233 N=130 N=103 N = 233 N= 233 comparaison par quart d'heure Age maternel moyen 29±5,6 29±5,3 29,9±5,3 Normal 197 (84,6%) 130 (55,8%) 30.00% AG 40,0 (39,0 ; 40,0) 40,0 (39,0 ; 41,0) 40,0 (39,0 ;40,0) Anormal	p 0,46 p <0,001 1
	25.00% Tabac Bradycardie 25.00% Nullipare Tachycardie		36 (15,5%) 112 (48,1%)	18 (13,8%) 1 (0,4%) 64 (49,2%) 4 (1,7%)	18 (17,5%) 13 (5,6%) 48 (46,6%) 10 (4,3%)	0,45 0,0005 0,69 0,0339
	20.00% IMC > à 25 kg/m2 Ralentissements (oui/non) 20.00% PAG Ralentissement(s) précoce(s)		84 (36,1%) 23 (9,9 %)	48 (36,9%) 29 (12,5%) 11 (8,5%) 4 (1,7%)	36 (35%) 93 (39,9%) 12 (11,7%) 14 (6,0%)	0,76 <0,001 0,42 0,0039
	101 exclusions Rupture des membranes avant 15.00% Post-terme Ralentissement(s) tardif(s) 15.00% Durée moyenne entre la Ralentissement(s) variable(s) typique(s) 41 (17,6%)	27 (20,8%) 10 (4,3%) 6 (2,6%)	P= 0,0121 14 (13,6%) 17 (7,3%) 16 (6,9%)	0,15 0,1266 0,0124
	travail n=90 Absence de rupture pendant le 10.00% rupture et l'accouchement Ralentissement(s) variable(s) atypique(s) 4,0 (1,5 ; 6,3) 10.00% (heures) Ralentissement(s) prolongé(s)	5,0 (3,0 ; 7,3) 16 (6,9%) 12 (5,2%)	2,4 (1,2 ; 4,4) 58 (24,9%) 41 (17,6%)	< 0,001 <0,001 <0,001
	travail n=1 Dossiers manquants n=10 5.00% Péridurale Variabilité anormale (oui/non) 227 (97,4%) 5.00% Dilatation lors de la rupture 4,0 (3,0 ; 7,0) FIGO	128 (98,5%) 15 (6,4%) 3,0 (3,0 ; 4,0)	99 (96,1%) 27 (11,6%) 7,0 (4,0 ; 9,0)	-0,0047 <0,001 <0,001
	0.00% Normal 0.00% Phase de latence (<5cm) Bradycardie Q1-RSM Q1-RAM Suspect	Ruptures des membranes amniotiques Tachycardie Ralentissements hors prolongés Ralentissements prolongés 197 (84,6%) 130 (55,8%) Variabilité anormale Q2-RSM Q2-RAM Q3-RSM Q3-RAM Q4-RSM 151 (64,8%) 112 (86,2%) 39 (37,9%) Q4-RAM 19 (8,2%) 48 (20,6%)
	Bradycardie Phase active (6-9cm)		n=324 (76,2%) Tachycardie 55 (23,6%) 11 (8,5%)	Ralentissements hors prolongés 44 (42,7%)
	Ralentissements prolongés Pathologique Dilatation complète	Q1 Q2 Q3 Q4 17 (7,5%) Variabilité anormale 27 (11,6%) 7 (5,4%)	55 (23,6%) 20 (19,4%)	<0,001
	91 exclusions Résultats exprimés en pourcentages (%). LAM 40 (17,2%)	19 (14,6%) 213 (91,4%)	21 (20,4%) 153 (65,7%)	0,25
	Ruptures pendant les efforts Données manquantes sur RCF RAM : Rupture artificielle des membranes expulsifs n=50 RSM : Rupture spontanée des membranes Ocytocine lors de la rupture 14 (6%)	12 (9,2%) 7 (3%)	2 (1,9%) 30 (12,9%)	0,020
	n=31 RCF : Rythme cardiaque foetal			13 (5,6%)	39 (16,7%)
	Inclusion n=233 (54,8%) 0 (0%) CNGOF = Comité national des gynécologues-obstétriciens Français. RCF < 1h avant rupture n=10 Q1 : Premier quart d'heure post rupture Q2 : Deuxième quart d'heure post rupture Q3 : Troisième quart d'heure post rupture AG= Age gestationnel IMC = Indice de masse corporelle Résultats exprimés en N (%) PAG= Petit poids pour l'âge gestationnel Q4 : Quatrième quart d'heure post rupture LAM= Liquide amniotique méconial. RCF= Rythme cardiaque foetal RAM = Rupture artificielle des membranes RSM = Rupture spontanée des membranes. FIGO= International Federation of Gynecology and Obstetrics	11 (4,7%)
	Rupture spontanée		Rupture artificielle
	N=103 (44,3%)		N=130 (55,7%)
	RCF : rythme cardiaque foetal

Figure 3 : Comparaison par quart d'heures des ARCF RSM versus RAM Résultats exprimés en N pourcentages (%). ARCF : Anomalie du rythme cardiaque foetal RAM : Rupture artificielle des membranes RSM : Rupture spontanée des membranes Q1 : Premier quart d'heure post rupture Q2 : Deuxième quart d'heure post rupture Q3 : Troisième quart d'heure post rupture Q4 : Quatrième quart d'heure post rupture


Tarek Khaled
email: tarek.khaled@univ-amu.fr

Belaid Benhamou
email: belaid.benhamou@univ-amu.fr

Dealing with Biology Systems in the Framework of Answer Set Programming

Keywords: Answer Set Programming, Logic Programming, Stable Models, Bioinformatics, Gene Networks, Knowledge Representation

Reasoning about gene networks is essential from various perspectives, such as predicting side effects of drugs or explaining unusual cellular behavior. Because of the massive size of these gene networks, a biologist can only work on a small part of the network. Thus, there is an essential requirement for logical representations and automated reasoning on such networks to help biologists to understand genetic interactions. However, the knowledge about gene networks is always incomplete and sometimes not accurate. Hence, knowledge has to be continuously revised and extended. In this work, we propose an approach based on non-monotonic logic programming, and the framework of Answer Set Programming(ASP), to represent and handle gene networks. We show how to model reasoning, predict events, and explain observations in gene networks. Finally, we show how our approach is applied to represent and resolve the DNA double-strand breaks, which is one of the most severe genomic lesions.

Introduction

Knowledge of biology is becoming increasingly important. Therefore, it is essential to use mathematical models to understand the dynamic behavior of a biological system. In this context, several quantitative and qualitative approaches have been proposed [START_REF] De | Modeling and simulation of genetic regulatory systems: A literature review[END_REF] to study gene networks. The quantitative models are often based on differential equations. The major limitation of these approaches is that numerical values in biology may be difficult to estimate, and the additional burden in the complexity of the model over the qualitative approach is not often compensated by greater accuracy. Modeling gene network helps understand the cell behavior. This modelization can lead to drug discovery and therapeutic procedures. The description of a gene network is often not complete, and this is due to the constant evolution of biology knowledge. The reactions of genes and the interactions between them are not fully discovered.

Consequently, the knowledge is not complete and could be inaccurate. Thus, we need to correct and complete it. Another feature in gene networks modeling is the integration of information coming from different sources that could be contradictory.

The construction of a knowledge base expressing molecular biology and in particular gene networks has always been at the center of research in artificial intelligence. There exist several works using logical approaches. For instance, the system HYPGENE [START_REF] Peter | Artificial intelligence methods for theory representation and hypothesis formation[END_REF][START_REF] Peter | Design methods for scientific hypothesis formation and their application to molecular biology[END_REF] where the knowledge base is implemented using a frame language 1 . This approach aims to resolve the mismatch between predictions calculated from the knowledge base and experimental observations. On the other hand, TRANSGENE [START_REF] Darden | Recent work in computational scientific discovery[END_REF][START_REF] Darden | Anomaly-driven theory redesign: computational philosophy of science experiments[END_REF] is an approach for the representation of knowledge based on a "functional representation" language. This language is chosen to overcome the shortcomings of systems using the frame language. The goal is to determine if a knowledge base can predict an observation. If not, then the knowledge base must contain faulty items that can be found and corrected. We also have the system GENEPATH [START_REF] Blaž Zupan | Genepath: a system for inference of genetic networks and proposal of genetic experiments[END_REF] which uses a set of inference rules which have been formalized and implemented using Prolog. The rules are applied on the basis of the initial knowledge to build a plausible network that explains the observations. These works share a common characteristic: they are all built on knowledge representation languages limited to monotonous reasoning.

In this paper, we represent gene networks and address all the features previously mentioned by using the Answer Set Programming (ASP) paradigm [START_REF] Victor | Stable models and an alternative logic programming paradigm[END_REF]. ASP is a knowledge representation and reasoning framework. It is used to express and handle a variety of combinatorial problems. Among them graph problems, planning, and model checking. It is a powerful modeling tool that can encode a significant number of problems. There is a growing number of practical applications of ASP in different fields [START_REF] Erdem | Applications of answer set programming[END_REF], especially those related to biology [START_REF] Gebser | Repair and prediction (under inconsistency) in large biological networks with answer set programming[END_REF][START_REF] Khaled | An asp-based approach for attractor enumeration in synchronous and asynchronous boolean networks[END_REF][START_REF] Khaled | An asp-based approach for boolean networks representation and attractor detection[END_REF]. Using ASP provides several advantages. First, it is appropriate to model biology information. It is also a robust declarative language for knowledge representation. It offers the possibility of reasoning on incomplete knowledge, and the last ASP solvers [START_REF] Janhunen | The answer set programming paradigm[END_REF] are very efficient and able to solve significant size problems.

We consider a qualitative approach to design gene networks. For instance, the representation of a protein concentration is assumed to have a finite number of states [START_REF] Kauffman | Understanding genetic regulatory networks[END_REF]. A protein is represented when it exceeds a certain threshold. Gene network is a collection of molecular regulators that interact with each other and with other substances in the cell to govern the gene expression levels. Biologists, usually express Gene Networks by graphs. It is, in general, the best way to represent interactions between genes. We use ASP to represent and manage the various interactions between genes. Gene networks are expressed as sets of rules of logic programs. Our primary concern is to reason on incomplete or partial information. In addition to that, our approach must be able to update the knowledge base without building a new model from scratch. We show the working principle of our method on some small biology problems then provide a biological regulatory network example to showcase that our approach is scalable.

In the remaining of this paper, we recall some notions on gene networks in Section 2. Section 3 gives some background on ASP and describes the ASP search method that we used to solve problems. We discuss the representation of a gene network in Section 4. We apply our approach to the DNA double-strand breaks in Section 5. Section 6 concludes the work.

Gene Network

The living cells are sources and receptors of chemical signals. An intern signal is initiated when a receptor (a protein molecule) receives an external one. Then, molecules inside the cell transpose the signal into cellular responses. The cellular response involves a change in gene expression, which leads the cell to produce a functional molecule, typically a protein. Almost all of the diseases could be described in terms of alteration in gene networks. An alteration in the signaling interactions and problems in cellular information processing are responsible for diseases like cancer. For instance, cancer is caused by the deregulation of networks regulating cell renewal.Each protein is synthesized from the pieces of information contained in its corresponding gene. For didactic purposes, we assume in this paper that a protein is present in the cell if the corresponding gene is being transcribed. A genetic network is represented by a graph where each node corresponds to a protein or a biological process. The arcs correspond to the influences of proteins from one to another. Biology systems have become an important research focus. The knowledge about the biology and cell interaction mechanisms is growing exponentially. Now, it is complicated for biologists to deal with the critical number of interactions in a gene network. One could use artificial intelligence techniques that are Knowledge representation and automated reasoning to help biologists. The proteins execute nearly all functions inside a cell [START_REF] Alberts | Molecular biology of the cell[END_REF]. The proteins receptors function is to sense the state of the extracellular environment and trigger corresponding intracellular reactions. Proteins also play the role of regulators of protein activities. They can promote the activity of others, inhibit reactions, or bind to another protein to create a compound. We illustrate a gene network by a simple example of the protein p53 signal network and describe the function of p53 during cancer in a cell. Moreover, the protein p53 plays a vital role as a tumor suppressor. This role is shown in the simplified network depicted in Figure 1, where its interactions with other proteins influence the activity of p53. The notations of Figure 1 are standard in biological representation. The symbol → expresses activation and inhibition. The main interactions are summarized in Figure 2. The line relaying two entities represent the binding between them. Figure 1 gives a summary example of interactions in a cell. Ultraviolet (UV) drives the cell to cancer, and an arrow indicates this. On the other hand, the UV activates the production of the protein p53. This protein blocks the growth of cancer. But in some conditions, p53 bounds to the protein mdm2 causing low concentration of mdm2 and p53, which blocks the functionality of p53. The main question for a biologist is how to block the binding between mdm2 and p53 to inhibit the tumor (cancer). Interactions between genes can be seen as a form of causality. In our approach, we use two binary relations that are active(A, B) and inhibit(C, B) to express the primary interactions between proteins. The first relation means that the protein A activate the production of the protein B, while the second relation means that C block the production of the protein B. This type of relationship gives a kind of causal link between both proteins. Such relations are important for our logical system, and they are expressed as follows:

1. if A activates B and A is true, then B is true 2. if C inhibits B and C is true, then B is false Inferences based on cause-and-effect relationships are very useful for reasoning about interactions in a gene network. The form of inference that has been best modeled is deductive reasoning. The paradigm of deductive reasoning is the modus ponens rule "If (if A then B) is true, and if A is true then, B is true." We can approximate this in the causal relation by "If A cause B and we observe A, then we must observe B." The first idea is to express both previous protein interactions as two classical logic inferences:

1. active(A, B), A → B 2. inhibit(C, B), C → ¬B But the causality in this context, could not be seen as a classical logic relation. A basic example is "If it rains, the grass gets wet." The formula Rain → wet means that if it rains, the grass is always wet. But this formula is too strong. Indeed, there may be exceptions to this rule. For instance, there could be parcels of grass that the rain could not reach. The formulation in the classical logic of genetic networks is problematic when there is a conflict. For example, if we know {A, C, active(A, B), inhibit(C, B)}. There is simultaneously activation and inhibition of the protein B, which results in a conflict that is due to biological reasons. Logically, we infer both B and ¬B at the same time leading to inconsistency. In classical logic, the existence of one inference rule is sufficient to legitimize the passage from a premise to a conclusion. However, it is necessary to consider all the rules as a whole because we need to take into account all the information already inferred. If new information comes to us, or a new event has been occurred, making conclusions already drawn to be invalidated. In this case, we have to revise the information and restore its consistency. To deal with these requirements, one could use non-monotonic reasoning, and more specifically, the ASP framework.

Answer Set Programming

The paradigm ASP [START_REF] Victor | Stable models and an alternative logic programming paradigm[END_REF] is fully declarative, it has a high level capacity of knowledge representation and efficient solvers. The basic idea of ASP is to represent the knowledge base as a set of rules forming a logic program and then reason on that program by computing its stable models [START_REF] Gelfond | The stable model semantics for logic programming[END_REF] or its answer sets [START_REF] Gelfond | Classical negation in logic programs and disjunctive databases[END_REF]. A logic program π is a finite set of rules of the form r:head(r) ← body(r). A positive logic program π is a set of rules of the form : r = A 0 ← A 1 , A 2 , ..., A m , with (m ≥ 0) and where A i∈{0,...,m} is an atom. A general logic program π is a set of rules of the form :

r = A 0 ← A 1 , A 2 , ..., A m , not A m+1 , ..., not A n , (0 ≤ m < n)
where A i∈{0...n} is an atom and not the symbol expressing the negation as failure. The positive body of r is body

+ (r) = {A 1 , A 2 , ..., A m } and the negative one is body -(r) = {A m+1 , ..., A n } . An extended logic program is a set of rules of the form r = L 0 ← L 1 , L 2 , ..., L m , not L m+1 , ..., not L n , (0 ≤ m < n)
where L i∈{0...n} is a literal (an atom A i or its negation ¬A i ). The reduct of a general program π with respect to a given set of literals X is the positive program π X . This program is obtained by deleting each rule containing a literal not L i in its negative body such that L i ∈ X, and by deleting all the atoms not L j in the remaining rules. The most known semantic for general logic programs is one of the stable models [START_REF] Gelfond | The stable model semantics for logic programming[END_REF]. A set X of atoms is a stable model of π iff X is identical to the minimal Herbrand model of the reduct π X obtained from π when considering the set of atoms X. The semantics of an extended logic program is represented by its answer sets [START_REF] Gelfond | Classical negation in logic programs and disjunctive databases[END_REF]. This latter can be seen as an extension of the stable model semantics defined for general logic programs [START_REF] Gelfond | The stable model semantics for logic programming[END_REF].

A new semantics is proposed in [START_REF] Benhamou | A new semantics for logic programs capturing and extending the stable model semantics[END_REF]. Among the advantages of this semantics is the easy characterization of the stable models and the extension that it provides for the stable model semantics. This semantics use a Horn clausal representation. It is based on a classical propositional language L composed by two types of variables, a subset of classical variables V = {A i : A i ∈ L} and an other subset nV = {not A i : not A i ∈ L}. For each variable A i ∈ V, there is a corresponding variable not A i ∈ nV designing the negation by failure of A i . A general logic program π={r : A 0 ← A 1 , A 2 , ..., A m ,not A m+1 , ..., not A n }, (0 ≤ m < n) is expressed in the propositional language L by a set of Horn clauses HC(π) = { r∈π (A 0 ∨ ¬A 1 ∨, ..., ¬A m ∨ ¬not A m+1 , ..., ¬not A n )} . The strong backdoor set (STB) of the logic program π is formed by the literals of the form not A i that occur in π. Formally, it is defined by S T B = {not A i ∈ nV : ∃r ∈ π, not A i ∈ body -(r)} ⊆ nV . Given a program π and its STB, an extension of HC(π) with respect to the STB or simply an extension of the pair (HC(π), S T B) is the set of consistent clauses derived from HC(π) when adding a maximal set of literals not A i ∈ S T B. Formally:

Definition 1 ([2]
). Let HC(π) be the Horn CNF encoding of a logic program π, S T B its strong backdoor and S ⊆ S T B. The set E = HC(π) ∪ S of clauses is then an extension of (HC(π), STB) if the following conditions hold

1. E is consistent, 2. ∀not A i ∈ STB -S , E ∪ {not A i } is inconsistent.
It is shown in [START_REF] Benhamou | A new semantics for logic programs capturing and extending the stable model semantics[END_REF], that each stable model of a logic program π corresponds to an extension E of HC(π) that satisfies the discriminant condition

(∀A i ∈ V, E | = ¬not A i ⇒ E | = A i ).
The main proved theoretical properties are the following:

Theorem 1 (See [START_REF] Benhamou | A new semantics for logic programs capturing and extending the stable model semantics[END_REF]). If X is a stable model of a logic program π, then there exists an extension E of (HC(π), S T B) satisfying the discriminant condition

(∀A i ∈ V, E | = ¬not A i ⇒ E | = A i ) such that X = {A i ∈ V : E | = A i }.
Theorem 2 (See [START_REF] Benhamou | A new semantics for logic programs capturing and extending the stable model semantics[END_REF]). If E is an extension of (HC(π), S T B), that verify the discriminant condition:

∀A i ∈ V, E | = ¬not A i ⇒ E | = A i , then X = {A i : E | = A i } is a stable model of π.
A positive logic program allows for modeling various problems. However, it turns out that many situations require the notion of negation. A program becomes general programs when negation as failure is introduced in the body of its rules. The intuitive meaning of the negation as failure, notL is that L cannot be proved using the rules. The definition of notL is different from the fact that L is false, which is expressed by the classical negation ¬L. A useful application of the classical negation is to express default rules [START_REF] Reiter | A logic for default reasoning[END_REF]. For example, we can express that bird flies by default with the rule f lies(X) ← bird(X), not¬ f lies(X). The meaning of this rules is: if bird(X) is true and we cannot prove that f lies(X) is false ( f lies(X) is possible), we can infer f lies(X). Extended logic programs admit both types of negations, classical negation, and negation as failure. In this work, we use the extended logic programs to represent gene networks. The semantics of an extended logic program can be defined by its reduction to a general program. Then, one can use the semantic summarized previously for general programs [START_REF] Benhamou | A new semantics for logic programs capturing and extending the stable model semantics[END_REF] to deduce the answer sets of the extended program. An extended program is composed by rules of the form r = L 0 ← L 1 , L 2 , ..., L m , not L m+1 , ..., not L n , (0 ≤ m < n) where L i∈{0...n} . To reduce an extended logic program into an equivalent general logic program, we generally replace any negative literals ¬L appearing in the representation with a new atom L . We add the clauses of integrity constraint ← L, L . This constraint prevents both L and ¬L to be simultaneously in the same stable model. The constraints prohibit that L and ¬L to be true at the same time. We compute the stable models of the resulting general program from which we can obtain the answer sets of the original extended program.

For the present work, we used the ASP solver presented in [START_REF] Khaled | A new method for computing stable models in logic programming[END_REF][START_REF] Khaled | Symmetry breaking in a new stable model search method[END_REF].

Representation of gene networks

In this section, we show how we express gene networks as a logic program that could be handled by ASP solvers. The use of such logical formalism could naturally express all the information that biologists want to represent. The advantage of such a logical framework is the possibility of representing knowledge in a declarative way that is close to a natural language. In our approach, genes and proteins are considered as the same objects. We assume that a protein is present only if a gene is activated. The protein interactions are described by using a propositional logic Programming language. This logical representation could be seen as an abstraction of protein interaction reality. We can use, for example, p53 to say that the protein is expressed. Indeed, the protein concentration is rarely precise, and in practice, the biologists' experiments show a qualitative interpretation of increasing or decreasing of the concentration.

The main disadvantage of classical logic is monotonicity. It's commonly known that they are not entirely adapted to manage dynamical changes and the evolution of a knowledge base. In monotonic logic, everything that is deduced from a knowledge base is always deduced when we add new information. That is, if we add a new formula f to a set of formulas F, then every formula g that is a logic consequence of F remains a logic consequence of F ∪ f . However, in real life, conclusions established before could turn invalid after the addition of new information. This situation is frequently met in the field of biology like gene networks where the knowledge increase exponentially. Classical logics are not appropriate in this case since they could not revise the knowledge regarding new information, could not deal with uncertain and incomplete information.

To deal with gene networks, one could use non-monotonic reasoning. Several non-monotonic logics are introduced in the literature ( [START_REF] Mcdermott | Non-monotonic logic i[END_REF][START_REF] Reiter | A logic for default reasoning[END_REF][START_REF] Schwind | A modal logic for hypothesis theory[END_REF]). These logics do not have the monotony property of classical logic. We can employ them to represent biological systems [START_REF] Schwind | A modal logic for hypothesis theory[END_REF]. They have great expressive power, but they face the problem of efficiency in practice. For example, the default logic [START_REF] Reiter | A logic for default reasoning[END_REF] or the hypotheses logic [START_REF] Schwind | A modal logic for hypothesis theory[END_REF] are two compelling non-monotonic frameworks for knowledge representation, but do not contain effective algorithmic tools to process this knowledge. A good compromise between the efficiency of tools and the power of expressiveness is to use the ASP framework. It is a rich non-monotonic formalism that has efficient solvers able to solve industrial problems.

Gene networks are expressed concisely and simply as extended logic programs. General logic programs provide an excellent language to express knowledge in the situation that justifies the use of the closed world assumption. This type of program does not represent incomplete and uncertain knowledge. The language should allow a third possibility for query answers that is the unknown value. The unknown value corresponds to the inability to produce a true or false answer. Now, we return to the formulas set out in section 3 to represent the "activate" and "inhibit" actions established for gene networks. To avoid conflicts, we can think of weakening them as follows:

1. if A activate B if A is true and we cannot prove that ¬B is true, then B is true 2. if C inhibit B if C is true and we cannot prove that B is true, then B is false We could therefore naturally express the two interactions in question in the form of rules of an extended logical program using both the default and the classical negations:

1. B ← active(A, B), A, not ¬B

¬B ← inhibit(C, B), C, not B

In this work, we are interested in formalizing several reasoning abilities. Especially, predicting the impact of a particular event, or explaining observations. Each of these abilities has a critical meaning to understand biological systems. For example, determining the effect of a drug on the production of a particular protein or detecting abnormal behavior of a cell (for example, a cell that multiplies instead of dying) are two essential facts in biology.

Predictive Reasoning

The predictive reasoning is the process of finding all the possible explanations derived from a knowledge base, i.e., given K a knowledge base and the results R, we want to infer R from K. R are logical consequences of K (K R). In ASP, the predictive reasoning consists in computing all the answer sets of a logic program. An answer set could be seen as a subgraph of the gene network that does not have conflicts. For instance, take K = {A, C, active(A, B), inhibit(C, B)} as a set of facts. By considering both previous rules, we find two answers set M 1 = {active(A, B), inhibit(C, B), A, B} and M 2 = {active(A, B), inhibit(C, B), C, ¬B} . We have two different explanations for the given knowledge. By using the ASP framework, the conflict is resolved, and we obtain two contradictory solutions.

In biological representations, we have to reason with incomplete, uncertain, revisable, and sometimes false information. Besides, we have to give all the explanations and possible contradictory conclusions for a given situation. The main goal of our approach is to handle these types of constraints. The expression of biological beliefs by nonmonotonic frameworks like ASP allows dealing with inherently incompatible configurations in gene networks. The rules of the form, Head ← Body containing negations as failure, express non-verified and revisable knowledge. In addition to these rules, we have some specific rules to express either verified knowledge of evident facts. We have two particular cases, and the first one is when the Head part of a given rule is empty (← Body). The corresponding rule represents an integrity constraint that could be a mutual exclusion between atoms. The second case is when the Body a rule is empty (Head ←). Such a rule represents a positive or negative elementary fact. Let us consider the interaction network presented in Figure 1. We give in the following the formalization of that gene network as a logic program. The literal Cancer expresses the fact that a tumor exists, and uv encodes the fact that the cell is stressed by ultraviolet.

uv ← (1) mdm2 ← (2) p53 ← uv (3) mdm2, p53, not ¬bind(p53, mdm2) (4)
bind(p53, mdm2) ← P53, not bind(p53, mdm2), not ¬inhibit(p53, cancer)

inhibit(p53, cancer) ← ¬inhibit(P53, cancer) ← bind(p53, mdm2)

cancer ← uv, not inhibit(p53, mdm2), not ¬cancer

x ← uv, not ¬x (8)

bind(x, mdm2) ← mdm2, x, not ¬bind(x, mdm2)

bind(x, p53) ← p53, x, not ¬bind(x, p53) (10)

¬bind(p53, mdm2) ← bind(x, mdm2) (11)
¬bind(p53, mdm2) ← bind(x, p53)

← bind(x, mdm2), bind(x, p53)

The gene network of Figure 1 is expressed by the rules 1 to 7. In Rule 3, uv induces a stress in the cell, which results in the expression of p53. The rule of line 4 states that p53 can bound to another protein mdm2 to create a new compound if we don't have any evidence that the binding between them is inhibited. The rule of line 5 gives the conditions under which the protein p53 could inhibit cancer. The rule given in line 6 states that the binding of P53 with mdm2 represses the cancer inhibition action of P53. Finally, Rule 7 expresses the conditions under which cancer is developed. We have constructed a knowledge base representing the interactions presented in Figure 1.

The knowledge base that is represented by an extended logic program that has one answer set M = {uv, mdm2, p53, bind(p53, mdm2), ¬inhibit(p53, cancer), cancer}. We can see, in M, the binding of both p53 and mdm2, which inhibit the capacity of p53 to suppress the tumor. The biological problem, in that case, is how to favor the cancer inhibition by p53. For instance, this could be done by preventing the binding between p53 and mdm2.

Biologists established that the level of a protein x is higher in cells subjected to stress. It is then important to give all the explanations about the influence of such protein x. Thus, we expand the existing knowledge base expressed previously (Rules 1 to 7) by the rules 8 to 13. Rule 8 expresses the influence of the uv stress on the production of x. Rule 9 says that if x and mdm2 have high concentrations, then the binding between them is triggered. Rule 10 encodes the conditions under which the protein x is bound with P53. Rules 11 and 12, express the influence of both compounds x -mdm2 and x -p53 on the binding between p53 and mdm2. The last rule expresses the fact that x could not bind to both mdm2 and p53.

The knowledge base expressed by the extended logic program formed by the rules (1 to 13) has three answer sets:

M 1 ={uv, mdm2, p53, bind(p53, mdm2), ¬inhibit(p53, cancer), cancer, x, ¬bind(x, mdm2), ¬bind(x, p53)} , M 2 ={uv, mdm2, p53, ¬bind(p53, mdm2), inhibit(p53, cancer), x, bind(x, p53), ¬bind(x, mdm2), ¬bind(p53, mdm2} and M 3 ={uv, mdm2, p53, ¬bind(p53, mdm2), inhibit(p53, cancer), x, bind(x, mdm2), ¬bind(x, P53)} . We can see in M 1 that p53 and mdm2 are binded. This inhibits the inhibition action of P53, thus the cancer is activated. The model M 2 is described in Figure 4. It particularly, expresses the binding between p53 and x which prevents the formation of the compound mdm2 -p53. This, leads P53 to inhibit the cancer. On the other hand, the formation of mdm2 -p53 is prevented in M 3 by the binding between mdm2 and x. Thus, P53 inhibits the cancer. M 3 is presented in Figure 3.

The non-monotonicity of our approach manifests itself in the different results obtained. The knowledge base formed by the rules 1 -8 predicts that cancer occurs. The cancer is due to the activation of uv and the binding of P53 and mdm2. After adding the new rules 8 -13, two solutions (M 2 and M 3 ) of the resulting knowledge base indicates that cancer may not occur, despite the activation of uv. We can then remark that our reasoning is not monotonic since cancer is not deduced when extending the knowledge base by new information.

Abductive Reasoning

The term abduction is defined as the process of inference consisting of finding hypothesis that explains observed phenomena [START_REF] Charles | Philosophical writings of Peirce[END_REF], formally : Definition 2. Let K = (T, H) be a knowledge system where T is a set of formulas representing the knowledge and H a subset of formulas representing the hypothesis. Let O be a formula representing the observations. A set E ⊆ H is an explanation of O (with respect to T ) if:

1. T ∪ E is consistent, and 2. O is derived from T ∪ E Several forms of abductions have been defined in logic programming [START_REF] Antonis | Abductive logic programming[END_REF]. We based our approach on the work presented in [START_REF] Eiter | Abduction from logic programs: Semantics and complexity[END_REF], formally :

Definition 3 ([6]

). A logic programming abduction problem is a tuple L, H, O, P , where L is a set of literals, H ⊆ L is a finite set of literals abducibles called hypothesis space; O ⊆ L is a finite set of literals called observations and P is the logic program. A solution of this problem is a set of literals E, such that:

1

. E ⊆ H 2. P ∪ E is consistent 3. P ∪ E | = O
The explanation E of the observations O is a subset of hypothesis literals that are consistent with the program P, and each stable model of P verifying the literals of E should verify the observations O. Here, we consider H as a set of ground literals expressed by the fact rules l ← and ¬l ←, where l ∈ H. Adding all the literals of H to the program, P would eventually result in an inconsistent program. To remove this inconsistency, one could block the application of a hypothesis by adding a formula not ¬l and not l to the body of the corresponding fact rule. Thus, we have a pair of rules for each l ∈ H, denoted by H r : l ← not ¬l and ¬l ← not l.

We can see that these rules have two answer sets, one in which l is false and another one in which l is true. It is important to note that we seek for all the explanations, not necessarily for the minimal ones. The basic idea is to explore all the combinations of hypotheses in H that imply O. To do this, we pose the observation O as a query to this extended logic program.

Exemple 1. Consider the logic program P = {p ← a, not q q ← a, b q ← c}, the set of hypothesis H = {a, b, c} and the observation O = {p}. We add to P the following hypothesis rules: H r = {a ← not ¬a; ¬a ← nota; b ← not ¬b; ¬b ← notb; c ← not ¬c; ¬c ← notc} Let π = P ∪ H r be the extended logic program formed by the rules of the logic program P and those of H r . We obtain the general logic program π from π by renaming each negated literal ¬A by a positive atom A . The Horn clausal representation of the logic program π is HC(π ) = {p ∨ ¬a ∨ ¬not q, q ∨ ¬a ∨ ¬b, q ∨ ¬c, a ∨ ¬not a , a ∨ ¬not a, b ∨ ¬not b , b ∨ ¬not b, c ∨ ¬not c , c ∨ ¬not c} and its strong backdoor is the set S T B = {not q, not a, not b, not c, not a , not b , not c }. The pair (HC(π ), S T B) admits one extension E 1 = HC(π ) ∪ {not q, not a }. Indeed, E 1 is maximally consistent with respect to the set S T B. Besides, the extension E 1 satisfies the discriminant condition. Thus, π has one stable model M 1 = {p, a, b , c } and the corresponding answer set of π is M 1 = {p, a, ¬b, ¬c}. We can see that the observation p is true in M 1 and its explanations are the other atoms E = {a, ¬b, ¬c} that are true in M 1 .

The explanation in our context consists of deriving from observed states O, suitable explanations E, which caused this observation. We can say that E is a possible cause that implied O. Let us now consider the Knowledge presented in Figure 1 that is expressed by the following rules:

p53 ← uv (14) mdm2, p53, not ¬bind(p53, mdm2) (15)
bind(p53, mdm2) ← uv, p53, bind(p53, mdm2), not ¬cancer

Exemple 2. We want to check if uv and mdm2 are the causes / explanation of cancer. We take then the set H = {uv, mdm2} as the hypothesis, and O = {cancer} as the observation. We add to the previous rules, the following hypothesis rules:

uv ← not ¬uv.

¬uv ← not uv.

) mdm2 ← not ¬mdm2. ( (18
) ¬mdm2 ← not mdm2. 19
Thus, we obtain for the resulting logic program the answer sets M = {uv, mdm2, p53, bind(p53, mdm2), cancer)}, where cancer is true. The explanation for the observation cancer are both uv and mdm2 that are true in M.

Application and Results

We have worked on the bibliographic data of the response to DNA double-strand break (DNA-DSBs) represented on a signaling pathway by Pommier [START_REF] Yvan | Targeting chk2 kinase: Molecular interaction maps and therapeutic rationale[END_REF]. DNA-DSBs are among the most severe genomic lesions. This representation is adequate for keeping the flow of information represented by gene expression, receptor, and protein structure through the apoptosis and cell cycle arrest. The primary mechanism of cancer could be considered as a complex system where the input is DNA-DSBs, and the output is either apoptosis (programmed cell death) or cancer. The response to DNA-DSBs is the process that decides the cell's survival or cell's death and the proliferation of metastasis (development of cancer). We first expressed the interaction map [START_REF] Yvan | Targeting chk2 kinase: Molecular interaction maps and therapeutic rationale[END_REF] into a logic program representation (202 rules represent the whole signaling pathway). This logic program contains two types of rules. The first type is used to express the elementary facts, where the dependencies are well understood. The second type is used to handle uncertain and imprecise information. The rules are first expressed as those of an extended logic program containing classical and default negations. Then, we implemented a parser that translate the extended logic program into an equivalent general logic program.

We established that the logic program formed by all the rules expressing the DNA DSBs map and the simulation of the DNA DSB expressed in one rule as a fact, contain 4 answer sets. Therefore, our main result is a new DNA DSB map containing only the main reactions that are depicted in Figure 5. This figure gives some valuable information for biologists, like the influences of proteins on cell death or their renewal. Figure 5 is composed of an input signal "DSB" shown at the top of the map and the outputs, "Cell Cycle Checkpoints" and "Apoptosis" that is shown at the bottom. Figure 5 confirms the well-established result in biology, which says that the loss of function of the p53 tumor suppressor protein is a significant step in the development of cancer. The loss of p53 function could be obtained either by a direct mutation of the p53 gene or by alterations in p53 regulators. It has been established that the mdm2 protein is an essential regulator of p53. The protein p53 plays a pivotal role in the regulation of multiple cellular mechanisms, including apoptosis. Figure 5 shows a map containing the main pathways that are expressed by the two most important answer sets of the corresponding logic program. One of them leads to the proliferation of metastasis (inhibition of cell cycle arrest), and the other one conducts cell apoptosis.

The main difference between the two pathways is the activation of chk2. This protein allows the inhibition of the cdc25 phosphatases that prevents the cell cycle arrest. Furthermore, the chk2 interacts with several other proteins including p53. The stabilization of p53 by chk2 leads to a cell cycle arrest. In the first pathway, the non-activation of chk2 leads to the inhibition of cell cycle arrest. In the second pathway, chk2 binds to the tumor suppressor p53 -bindingprotein1, also known as 53bp1. This binding represents a mandatory condition to the phosphorylation of pml, which enhances pml-dependent apoptosis. Figure 5 shows that chk2 plays an essential role in the regulation of cell division and apoptosis in response to DNA damage. Now, let us try to find an explanation for the observed behavior, for instance, the activation of chk2. One could retain two main explanations on the activation of the protein chk2. The first one results from the activation of atm and the second one from the activation of atr. Both atm and atr act near the top of the gene networks. DNA damage caused by DNA-DSB somehow activates atm, while atr is activated primarily in response to a different set of DNA lesions, including those caused by UV light. Once activated, atm and atr phosphorylate a set of proteins that lead to apoptosis. These include p53, chk2, nbs1 and brca1.

Conclusion

ASP is a known paradigm in logic programming that allows modeling systems exhibiting non-monotonic behavior. This framework is used to express gene regulatory networks. Answer sets for the gene networks -represented as logic programs-could be computed to show all possible reactions. We introduce a new approach to model gene networks as extended logic programs. Our logical representation has the advantage of better formalizing gene networks since all implicit assumptions are explicitly described by the rules forming the logic program. In conventional gene networks, this knowledge could be hidden, then could take place for some approximations. The approach remains, however, simple to apply. It does not require any prior knowledge of formal logic from biologists. It help biologists to discover contradictory information, and guide them during their experiments. A further significant advantage of our method is its ability to update the knowledge base quite easily. Such capacity is crucial due to the inherent incompleteness of the biology information. We used the study case of the DNA double strands break to check the effectiveness of our approach. The obtained results show information on some gene expression and protein structures of the DNA DSBs that are in apoptosis or cell cycle arrest. In this work, we only deal with synchronous updating. Consequently, all the events should happen within a one-time step. It is not possible to model interactions that have different speeds. As future work, we are looking to extend this approach to handle asynchronous updating.

Fig. 1 .

 1 Fig. 1. The p53 interaction network Fig. 2. Symbol definitions and map conventions (a) The double-arrowed line indicates that proteins A and B can bind to each other. The "node" placed on the line represents the complex A:B . (b) The representation of multi-molecular complexes: x is A:B and y is (A:B):C. This notation is extensible to any number of components in a complex. (c) Covalent modification of the protein A. The single-arrowed line indicates that A can exist in a phosphorylated state. The node represents the phosphorylated species. (d) Stoichiometric conversion of A into B. (e) Degradation of the protein A (i.e. conversion of A to its degradation product ). (f) Enzymatic stimulation of a reaction. (g) General symbol for stimulation. A bar behind the arrowhead signifies necessity. (h) General symbol for inhibition. (i) Shorthand symbol for transcriptional activation. (j) Shorthand symbol for transcription inhibition.

Fig. 3 .

 3 Fig. 3. p53 binds X Fig. 4. mdm2 binds X

Fig. 5 .

 5 Fig. 5. DNA DSBs map generated from our results

Tarek Khaled and Belaid Benhamou / Procedia Computer Science 00 (2020) 000-000

This language is used for knowledge representation in artificial intelligence , the frames are stored in the form of ontologies