E Bastidas-Arteaga

F Schoefs

M Sánchez-Silva

Probabilistic optimization of the management of corroding RC structures

This study presents a stochastic method to optimize the inspection of reinforced concrete structures subjected to chloride penetration. The strategy is oriented to find the time interval between two consecutive inspections that (1) minimize the costs of inspection, preventive and corrective repair; and (2) ensure a given level of safety. The optimal interval is therefore determined by using stochastic processes and decision theory. The chloride penetration phenomenon is represented by a Markov chain whose transition matrix is determined from simulations.

INTRODUCTION

Civil infrastructure deteriorates with time mainly as a result of environmental factors. Infrastructure deterioration may be progressive (e.g., as a result of chloride ingress), it may result from sudden events (shocks e.g., earthquakes) or as a combination of both. The cost-efficient design, management and operation of these systems depend, to a large extent, on the assumptions and quality of the deterioration models used for the analysis. Nowadays, design and management of infrastructure must consider economical, social and environmental constrains to reduce environmental impact, optimize resource allocation and decrease waste generation. This new tendency of design and management must also consider all the phenomena which affect the performance of structures. For RC structures, corrosion induced by chloride ingress generates important damage after 10 or 20 years of service (Rosquoet et al., 2006). Consequently, its inspection and maintenance are essential to ensure an optimal level of serviceability and safety during its lifespan.

The purpose of this paper is to develop a method to optimize the intervals of inspection of reinforced concrete structures subjected to chloride penetration. An optimal inspection interval minimizes the costs of inspection, preventive or corrective repair and failure. The method of optimization takes as starting point the work of [START_REF] Sheils | Development of a two stage inspection process for the assessment of deteriorating infrastructure[END_REF] and is presented in the following section. Afterwards, the discussion focuses on the estimation of the transition matrices from a comprehensive model of chloride penetration and the cost modeling. Finally, the proposed methodology is illustrated with a numerical example.

STRATEGY OF INSPECTION AND MAINTENANCE

General description. Inspection can be carried out by using destructive and nondestructive methods. This study refers to RC structures subjected to chloride ingress and focuses on inspection/maintenance of major infrastructure such as ports and bridges. The requirements for the strategy of inspection/maintenance were defined within the framework of the European project FUI (2007-2010) MAREO 1 with the collaboration of public agencies, construction companies and research centers. In the strategy of inspection/maintenance considered herein, inspection is undertaken by analyzing the concentration of chlorides at the cover depth on concrete cores (destructive method). Afterwards, depending on inspection results, the repair technique consists of rebuilding the polluted concrete cover. The advantage of the proposed approach is that preventive repair ensures optimal levels of serviceability and safety during the lifespan of the project.

Inspection model. This approach requires that the structure is inspected periodically; i.e., every Δt years. The purpose of these inspections is to establish the real chloride concentration at the cover depth, C. This parameter provides information about the probability of corrosion initiation, and therefore, can be used by the owner/operator to decide if the structure must be repaired or not. The experimental test to determine the chloride profiles is based on the AFREM procedure (RILEM, 2002). [START_REF] Bonnet | Effect of error measurement of chloride profiles on reliability assessment[END_REF] found that there are significant differences between "theoretical" and "measured" chloride profiles. To take into account the influence of errors in measurement, the difference between measured C ˆ and real C values is usually modeled by considering a noise η as follows:

η C C + = ˆ [1]
It is supposed herein that the noise is independent of the real chloride concentration because there are several sources of error influencing the results of the inspection. This assumption has been validated from a series of measurements on metallic harbor structures [START_REF] Schoefs | Assessment of spatially dependent ROC curves for inspection of random fields of defects[END_REF].

Errors in measurement can lead to under or overestimations of chloride content, and consequently, to make erroneous decisions increasing the probability of failure or producing over-costs. From a probabilistic point of view, two measures can be defined to quantify these events:

• Probability of Good Assessment (PGA): determines the probability of detecting an event (crack, defect, concentration, etc.) given that it exists:

( ) rep rep C C C C P ≥ ≥ = ˆ PGA
[2] • Probability of Wrong Assessment (PWA): establishes the probability of detecting an event given that it does not exist:

( ) σ ]. Thus, the probability of corrosion initiation between the inspection intervals can be estimated as:

rep rep C C C C P < ≥ = PWA [3] 1 .
( ) th C th C C f p σ μ - Φ = [4]

TRANSITION MATRICES FOR MODELING CHLORIDE INGRESS

A discrete-time Markov process can be used for prediction by knowing the current state of a system. Towards this aim, the space of the variable of interest is discretized into M states. The Markov process is thus used to determine the probability that an event belongs to a state j knowing that in a preceding state it belonged to a state i. This probability, noted as

a ij =P(X t+1 = j | X t =i), is called "transition probability". It is considered herein that a ij is independent of t.
The transition probabilities can be grouped in a matrix of size M×M called transition matrix P. According to the Chapman-Kolmogorov equations, by knowing an initial state, the probabilities of belonging to other states after t transitions, q(t), are (Ross, 2004):

t ini t P q q = ) ( [5]
where the vector q ini contains the probabilities of belonging to the states at an initial time -i.e., t =0. In this study, the variable of interest is the concentration of chlorides at the cover depth, C, which controls corrosion initiation. Therefore, the Markov processes can be used to estimate the probability that C is in a given state at a given time. If it is supposed that after construction t=0 the concentration of chlorides at the cover depth is zero. Consequently, all the concentrations belong to the first state, q ini becomes q ini =[1, 0, 0, … ,0] and equation [5] provides a vector containing the probabilities of belonging to a state j at time t, q(t).

In several applications of Markov chains, the transition matrix is obtained from experimental data or expert judgment. However, given that the rate of chloride ingress into concrete is slow and that chloride ingress is influenced by several parameters (environment, material properties, etc.), it is proposed herein to numerically estimate q(t). The following sections will describe the adopted probabilistic model of chloride penetration and the method proposed to estimate the transition matrix.

Chloride ingress model.

Chloride ingress into concrete is controlled by a complex interaction between physical and chemical mechanisms that have been commonly modeled as a diffusion phenomenon. However, in this paper a more comprehensive model is proposed; it considers the interaction between: (1) chloride penetration, (2) diffusion of moisture; and (3) heat transfer. The description of the formulation of the model is beyond the scope of this paper but can be found in [START_REF] Bastidas-Arteaga | A comprehensive probabilistic model of chloride ingress in unsaturated concrete[END_REF].

The management of uncertainties considers, first, the random variables presented in Table 1. The criteria for determining the characteristics of the random variables are dicussed in [START_REF] Bastidas-Arteaga | A comprehensive probabilistic model of chloride ingress in unsaturated concrete[END_REF]. Taking into account the non-linearity and the complexity of the system of partial differential equations used to model chloride penetration, Monte Carlo simulations and Latin Hypercube sampling are employed herein. Estimation of the transition matrix from simulations. Transition matrices are estimated from Monte Carlo simulations of the probabilistic model of chloride penetration. In each simulation, the chloride concentration at cover depth during time is recorded. In all simulations, the frequency of belonging to a given state is determined. Thus, the probability of belonging to a state j at time t is obtained from:

N t n t q o j ) ( ) ( ˆ= [6]
where n 0 (t) is the number of observations in the state j measured at time t and N is the number of simulations. Table 2 presents the possible states of the system and its corresponding boundaries (minimum and maximum); note that each state corresponds to a chloride concentration C. Taking into account this discretization, Figure 1 shows the variation in time of the probability of belonging to the last state i=10 ( ) ( ˆt q ) obtained from Monte Carlo simulations. Once the probabilities ) ( ˆt q have been estimated, several methods can be used to compute the transition probabilities. The major difficulty in the assessment of P lies in the number of parameters a ij to estimate. Various studies consider a Markov matrix with two parameters for state [START_REF] Pappas | Markov chains for accumulation of organic and ceramic composites[END_REF][START_REF] Roelfstra | Corrosion evolution in bridge management systems and corrosion-induced deterioration[END_REF]. In this case, the transition probabilities can be estimated from a non-linear regression. However, complex stochastic phenomena cannot be modeled by a transition matrix with two transition probabilities per state [START_REF] Roelfstra | Corrosion evolution in bridge management systems and corrosion-induced deterioration[END_REF]). To solve this problem, the proposed method searches the values a ij that minimize the difference between the probabilities estimated from simulations and the obtained from the Markov model (equation [5]). Since the kinematics of the time-dependent evolution of each state is similar, there are M functions to minimize:

( ) ⎪ ⎩ ⎪ ⎨ ⎧ = ≥ = ∑ ∞ =0 2 1 1 et 0 s.c. ) ( , ), ( ), ( ) ( max min j ij ij T M a a f f f a a a a F F a K [7]
where a is a vector containing the transition probabilities (optimization parameters) and:

∑ = - = ana t t j j j t q t q f 0 ) , ( ) ( ) ( a a [8]
where t ana represents the analysis period used to perform the adjustment. This multiobjective optimization problem has been solved by using the "optimization toolbox" of Matlab©. The selected optimization method minimizes the maximum value of a set of multi-variable functions from an initial value.

COST ANALYSIS

Two kinds of costs are usually considered in cost analysis: agency and user costs. Agency costs encompass the direct costs incurred by the owner/operator during the life-cycle including initial construction costs and costs associated with inspection, repair, rehabilitation, replacement and disposal. User costs represent the inconvenience and expenses incurred by users due to traffic disruption such as travel delay costs, ship operating costs and accident costs. According to [START_REF] Thoft-Christensen | Life-cycle cost-benefit (LCCB) analysis of bridges from a user and social point of view[END_REF], user costs should be included in the analysis to formulate a comprehensive strategy of maintenance management of infrastructure. However, given that the information to estimate user costs is hard to establish and find; this work is only based on agency costs.

In this study the direct costs incurred by the agency include only costs associated with maintenance (inspection, preventive repair, corrective repair and failure). The initial construction costs are not included in the analysis because it is assumed that it would be the same for all maintenance alternatives. Since it is not possible to determine the final use of the structure at the end of the life-cycle (deconstruction or demolition), the residual (or salvage) value is also not considered. This study considers costs estimated with standard and intergenerational discounting.

In life-cycle cost analysis, the expected value of the total present value lifecycle cost E[C T ] is computed as:

; for [9]
where m is the total number of interventions carried out within the time interval [0,T t ], C j (t) is the j th expenditure at time t (for instance in years), T t is the life-cycle length and W(t) is the weight used to discount C j (t) to the present value. C j (t) depends on the properties of the original and repair materials; and on the characteristics of the maintenance strategy (i.e., inspection interval, inspection quality, repair threshold, etc.). The costs of inspection C I , preventive repair C pr and corrective repair C cr are simply calculated as a fraction of the initial cost of construction C 0 as:

I I k C C 0 = [10] pr pr k C C 0 = [11] cr cr k C C 0 = [12]
where k I , k pr and k cr are weights used to compute the costs of inspection, preventive and corrective repair as a fraction of C 0 , respectively.In conventional discounting, the standard discount weight can be expressed as:

1/ 1 [13]
where r s is the standard discount rate. [START_REF] Sumalia | Intergenerational discounting: a new intuitive approach[END_REF] and [START_REF] Prager | Remembering the future: A commentary on "Intergenerational discounting: a new intuitive approach[END_REF] derived an intergenerational discount weight for computing net benefits from the use of environmental resources. In their formulation, they treat the benefits as accruing to the current generation (at standard discount rates) plus to each of the annual increments of new stakeholders who will have entered the stakeholder population by that future year divided by the generation length, G. Consequently, the intergenerational discounting W I (t) can be estimated as [START_REF] Sumalia | Intergenerational discounting: a new intuitive approach[END_REF]:

1 1 1 Δ 1 Δ [14]
where r I is the intergenerational discount rate computed on an annual basis and Δ = (1+r s )/(1+r I ).

NUMERICAL EXAMPLE Problem description. This example aims to determine an inspection interval that optimizes the maintenance costs of RC structures. Consider a structure placed in a marine environment with temperature ranging from 5 to 25 ºC and with relative humidity between 0.6 and 0.8. The stochastic climate model is described in [START_REF] Bastidas-Arteaga | A comprehensive probabilistic model of chloride ingress in unsaturated concrete[END_REF]. The structure is exposed to an average environmental chloride concentration C env =6 kg/m 3 . This concentration corresponds to the boundary between high and severe corrosive environments [START_REF] Weyers | Concrete Bridge Protection and Rehabilitation: Chemical and Physical Techniques -Service Life Estimates[END_REF]. According Duracrete, (2000), C env is modeled as a stochastic process generated by independent numbers following a log-normal (log-normal noise) with a coefficient of variation of 0.2. To simplify the study, it is assumed that the repair material has the same characteristics that the construction material.

The determination of an optimal inspection interval is very sensitive to the cost models. Therefore, to obtain realistic results, the inspection, preventive repair and corrective repair coefficients (i.e., k I , k pr and k cr ) were defined taking into account the average expenditures for these items incurred by the port of Nantes Saint-Nazaire, France. These coefficients are presented in Table 3 and are used to estimate the costs of inspection, preventive and corrective repair on the basis of an initial construction cost of 1000 units. Given that it is supposed that repair is preventive, there is no failure cost and these coefficients are lower than 1. Whereas the coefficient of preventive repair only considers the cost related to cover rebuilding, the coefficient of corrective repair includes besides the costs of structural strengthening. Table 3 Results. To estimate the Markov matrix, the problem is discretized into M = 10 states. The matrix P was obtained from 10,000 simulations for a concrete cover of 5cm.

Figure 1 compares the probability of belonging to the state 10 estimated by both simulations and the Markov model. Note that the phenomenon is well represented by the estimated P.

Figure 2 presents the expected times to preventive and corrective repair as a function of the length of the inspection interval. These results were obtained from the Markovian approach for a given repair technique to rebuild the concrete cover (formed concrete). As expected, the time to preventive repair, t pr , is lower when the structure is periodically inspected. On the contrary, less inspection implies that the time to corrective repair, t cr , decreases. These times will be used to determine the optimum inspection interval that minimizes costs. To simplify the computation, the results of simulation are adjusted to two analytical functions that will allow us to estimate t pr and t cr in terms of the length of the inspection interval Δt I (Figure 2). Figure 3 presents the expected costs for three cases: standard discounting, intergenerational discounting and without discounting. An intergenerational discount rate of r I = 3% (per generation) was used. The total cost is discriminated into the costs of inspection, preventive and corrective repair. It is observed in all cases that the costs of inspection and preventive repair decrease and the cost of corrective repair increases for larger inspection intervals. This behavior is explained by the fact that when the inspection interval is greater, most inspections detect the component's failure. On the contrary, when the structure is inspected regularly, repair is preventive, and therefore, the corrective repair expenditures diminish.

The results showed that the expected total cost is optimal when the structure is inspected every 20 years for the standard discounting case and every 15 years for the intergenerational case. However, the optimum expected cost is different for all cases because the future flows for the intergenerational and undiscounted cases are important in comparison to the standard case for the life-cycle length T t . The differences between the optimum inspection intervals imply that the intergenerational and undiscounted cases give more importance to preventive maintenance. The discount rate per generation can be defined in different ways. For instance [START_REF] Sumalia | Intergenerational discounting: a new intuitive approach[END_REF] state that the simplest way to fix r I is to select a value equal to the standard discount rate. It can also be defined from a discussion with the community or it can be used the internal rate of return for educating people to the Ph.D. level in a given country. The influence of this parameter on the length of the optimum inspection interval and on the total expected cost is studied in Table 4. Therefore, for small discount rates per generation, the structure should be inspected more frequently and the total expected cost is larger. On the contrary, when r I → ∞, the optimum result converges to the result obtained when intergenerational discounting is neglected.

CONCLUSIONS

This paper presents a stochastic method for optimizing the inspection/repair of RC structures subjected to chloride penetration. The inspection technique measures the concentration of chlorides at the cover depth. Afterwards, based on the inspection results, the structure is repaired by replacing the polluted concrete with new material. The inspection/maintenance strategy chosen is modeled using decision theory and Markov processes. It is also proposed a methodology for determining the transition matrix from simulations of a chloride penetration model. The proposed method is

Figure 2 .

 2 Figure 2. Determination of the expected value of the times to preventive and corrective repair

Figure 3 .

 3 Figure 3. Expected costs for the cases (a) standard and (b) intergenerational discounting

  Maintenance and REpair of concrete coastal structures: risk-based Optimization (MAREO) Project where C rep represents the repair threshold. The equations presented above assume that the noise and the signal follow a normal distribution -i.e. N[μ η , σ η ] and N[ C μ , C σ ].

	Repair model. Chloride penetration into concrete may cause structural failure when
	no repair is carried out. The repair criterion considers that the polluted concrete is
	rebuilt when the chloride concentration measured during the inspection reaches a
	threshold value C th . According to Duracrete (2000), the threshold value C th can be
	modeled by a normal distribution N[	th μ , C

th C

Table 1 . Probabilistic parameters of the random variables

 1

	Variable	Units	Distribution	Mean	COV
	-Reference chloride diffusion coefficient,	m 2 /s	log-normal	3•10 -11	0.20
	D c,ref
	-Activation energy of the chloride	kJ/mol	beta on [32;44.6]	41.8	0.10
	diffusion process, U c
	-Aging factor, m		beta on [0;1]	0.15	0.30
	-Reference humidity diffusion	m 2 /s	log-normal	3•10 -10	0.20
	coefficient, D h,ref
	-Parameter representing the ratio		beta on [0.025;0.1]	0.05	0.20
	D h,min /D h,max , α 0
	-Parameter characterizing the spread of		beta on [6;16]	11	0.10
	the drop in D h , n -Thermal conductivity of concrete, λ	W/(m°C) beta on [1.4;3.6]	2.5	0.20
	-Specific heat capacity of concrete, c q -Density of concrete, ρ c	J/(kg°C) kg/m 3	beta on [840;1170] Normal	1000 2400	0.10 0.04

Table 2 . States used for discretizing the problem.

 2

	State i	1	2	3	4	5	6	7	8	9	10
	C minimum (kg/m 3 )	0.0	0.4	0.8	1.2	1.6	2.0	2.4	2.8	3.2	3.6
	C maximum (kg/m 3 )	0.4	0.8	1.2	1.6	2.0	2.4	2.8	3.2	3.6	4.0

Comparison between the probabilities computed from Monte Carlo simulations and Markov model for the last state.

  also includes the numerical values used to model inspection and repair.

Table 3 . Parameters used in the numerical example

 3

	Parameter	value
	Noise of the measurement, η, km/m 3 Standard deviation of noise, σ η , km/m 3 Repair threshold, C rep , km/m 3 Threshold chloride concentration, C th , km/m 3 Standard deviation of C th , Cth σ , km/m 3	0.30 0.50 0.10 2.00 0.40
	Initial cost of construction, C 0	1,000
	Inspection coefficient, k I	0.005
	Preventive repair coefficient, k pr	0.15
	Corrective repair coefficient, k cr	0.30
	Discount rate, r s	5%
	Discount rate per generation , r I	3%
	Generation length, G	25 yr
	Life-cycle length, T t	100 yr

Table 4 . Influence of r i on the optimum inspection interval and on the total expected costs

 4

	r I (%)	Optimum Δt I (yr)	Total expected cost
	3	15	102.5
	5	19	53.8
	10	20	26.6
	20	20	19.2
	∞	20	16.6

ACKNOWLEDGEMENTS

The authors acknowledge financial support of the MAREO project -contact: franck.schoefs@univ-nantes.fr.

illustrated with an example where it is showed that there is an inspection interval that minimizes the costs of inspection, preventive and corrective repair.


Karin Gérard
email: karin.gerard@univmed.fr

Charlotte Roby

Nicolas Chevalier

Bernard Thomassin

Anne Chenuil

Jean-Pierre Féral

Assessment of three mitochondrial loci variability for the crown-of-thorns starfish: A first insight into Acanthaster phylogeography

Keywords: COI, 16S rDNA, tRNAs, Molecular clock, Indian Ocean, Mitochondrial DNA polymorphism COI, ADNr 16S, ARNt, Horloge moléculaire, Océan Indien, Polymorphisme de l'ADN mitochondrial

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

Introduction

The corallivorous sea star Acanthaster planci (Linnaeus, 1758), so-called crown-of-thorns starfish (COTS), is often seen as a major threat for coral reefs [START_REF] Moran | The Acanthaster phenomenon[END_REF]. This species has an unstable population dynamics. The density of individuals can switch from a 'normal' situation with tens of individuals per square kilometer to an outbreak of thousands of COTS arriving on the reefs and devastating the coral community. The mechanisms of outbreaks are still poorly known. Sometimes the outbreaks seem to arise from several cohorts after multiple recruitment events [START_REF] Pratchett | Dynamics of an outbreak population of Acanthaster planci at Lizard Island, northern Great Barrier Reef (1995-1999[END_REF]. However, along the Great Barrier Reef (GBR) in Australia, 'primary' outbreaks are the origin of successive 'secondary' outbreaks. The mechanisms of outbreaks seem to be a response to an increase in terrestrial nutrients discharges (that may be natural or enhanced by human activities), which triggers phytoplankton blooms, thus favouring the survival of A. planci larvae and consequently the recruitment of juveniles [START_REF] Brodie | Are increased nutrient inputs responsible for more outbreaks of crown-of-thorns starfish? An appraisal of the evidence[END_REF]. The decrease of natural predators such as the Giant Triton (Charonia tritonis), the Bullmouth Helmet (Cypraecassis rufa) or the Giant Helmet (Cassis cornuta) (Gastropoda, Mollusca), which are over-harvested by collectors, also favours A. planci adults survival in low density.

Acanthaster planci is distributed all over the tropical areas from the East Pacific Ocean to the West Indian Ocean. Individuals from the North Indian Ocean are blue rather than red-green to red-brown, as observed in the Pacific [START_REF] Benzie | Crown-of-Thorns Starfish on the Great Barrier Reef: Reproduction, Recruitment and Hydrodynamics[END_REF]. Outbreaks also occur in the West Indian Ocean and the Red Sea. In Mayotte (Comoros), where outbreaks have been documented (B.T., pers. obs.), the authorities are seriously worried about the coral reefs health, since healthy reefs provide essential tourist appeal. Given that the detection of environmental factors listed above is difficult to implement [START_REF] Pratchett | Dynamics of an outbreak population of Acanthaster planci at Lizard Island, northern Great Barrier Reef (1995-1999[END_REF], the best way to manage A. planci populations during outbreaks is to collect sea stars by hand. However, if some larvae transported by ocean currents from a distant source regularly supply the population, then collecting is only a temporary solution to outbreaking events. The larval stage usually lasts several weeks, but it can be extended when survival conditions are especially unfavourable. That suggests strong dispersal ability and high gene flow between populations connected by ocean currents.

To date all population genetics studies on A. planci were carried out from allozyme variations in the Indo-West Pacific and the Pacific Ocean. They revealed a low differentiation level between populations from highly connected reefs and across the Pacific Ocean [START_REF] Benzie | Crown-of-Thorns Starfish on the Great Barrier Reef: Reproduction, Recruitment and Hydrodynamics[END_REF][START_REF] Nishida | Genetic differences between geographic populations of the crown-of-thorns starfish throughout the Pacific region[END_REF][START_REF] Benzie | The detection of spatial variation in widespread marine species: methods and bias in the analysis of population structure in the crown-of-thorns starfish (Echinodermata: Asteroidea)[END_REF] and a strong genetic connectivity between Japan and the GBR [START_REF] Nishida | Genetic differences between geographic populations of the crown-of-thorns starfish throughout the Pacific region[END_REF][START_REF] Katoh | Genetic similarity of outbreak populations of crown-of-thorns starfish (Acanthaster planci) that were 15 years apart in Okinawa, Japan[END_REF]. No significant genetic differentiation was observed between outbreaking populations occurring 10 or 15 years apart [START_REF] Katoh | Genetic similarity of outbreak populations of crown-of-thorns starfish (Acanthaster planci) that were 15 years apart in Okinawa, Japan[END_REF][START_REF] Benzie | Genetic structure of crown-ofthorns starfish (Acanthaster planci) on the Great Barrier Reef, Australia: comparison of two sets of outbreak populations occurring ten years apart[END_REF]. However, genetic differentiation was observed among western and eastern Australian populations and among several small reefs isolated by local ocean currents [START_REF] Nishida | Genetic differences between geographic populations of the crown-of-thorns starfish throughout the Pacific region[END_REF][START_REF] Benzie | Genetic structure of crown-ofthorns starfish (Acanthaster planci) in Australia[END_REF]. The genetic differentiation between Indian and Pacific Oceans is relatively strong, since the genetic distance between A. planci populations from the two oceans was greater than that between A. planci from the Pacific Ocean and its sibling species A. brevispinus Fisher, 1917, a shortspined form living on soft bottoms between Indo-Pacific reefs [START_REF] Benzie | Major genetic differences between crown-ofthorns starfish (Acanthaster planci) populations in the Indian and Pacific Oceans[END_REF]. Genetic data already confirmed the distinction between these two species. Despite their interfertility, hybrids have a low viability and developmental abnormalities [START_REF] Lucas | Hybrid crown-of-thorns starfish (Acanthaster planci × A. brevispinus) reared to maturity in the laboratory[END_REF][START_REF] Lucas | Aspects of the evolution of Acanthaster planci (L.) (Echinodermata, Asteroidea)[END_REF]. Acanthaster planci seems to have evolved from an A. brevispinus-like ancestor during the Pleistocene [START_REF] Birkeland | Acanthaster planci: Major Management Problem of Coral Reefs[END_REF]. The divergences observed with allozyme data appear therefore incongruent with the systematics of the genus Acanthaster, and this fact was neither confirmed nor checked by the study of different molecular markers.

There is therefore a need for more accurate genetic tools for studying populations of A. planci. DNA markers afford a wider panel of variability level than allozymes do and may be less prone to selection. Mitochondrial DNA (mtDNA) in particular is commonly used for phylogeographic study [START_REF] Avise | Intraspecific phylogeography: the mitochondrial DNA bridge between population genetics and systematics[END_REF], since several universal primers are published and specific ones exist for asteroids [START_REF] Hart | Molecular phylogenetic analysis of life-history evolution in asterinid starfish[END_REF][START_REF] Perrin | Effects of hydrographic barriers on population genetic structure of the sea star Coscinasterias muricata (Echinodermata Asteroidea) in the New Zealand fiords[END_REF]. Its complementarity to allozyme markers for phylogeographic inference mainly relies on the possibility of using evolutionary relationships among alleles after sequencing.

The aim of the present paper is to provide an assessment of the variability and the accuracy of the phylogeographic signal of three mitochondrial loci. A first insight into a mitochondrial phylogeography of the genus Acanthaster and an assessment of the divergence within the West Indian Ocean A. planci populations are also presented.

Material and methods

Samples of A. planci from Madagascar, Kenya, and Mayotte were collected between 2002 and 2005 (Fig. 1) from 10 to 2 m of depth. Gonads were collected and stored in 95% ethanol. Genomic DNA was extracted using Aquapure © Genomic DNA Isolation Kit (Bio-Rad) following the manufacturer's protocols. All mixtures were stored at -20 • C and used as a template for polymerase chain reaction (PCR). A 566-bp fragment of the 16S rDNA gene (16S) was amplified with forward primer 16SF2 (5 GTTTCGCCTGTTTACCAAAAACAT 3 ) and reverse primer 16SR2 (5 AGGTTTTTCTGATCCA ACATCG3 ). A fragment coding for five tRNAs (Ala, Leu, Asn, Gln, Pro) (450 bp) and part of the Cy-tochrome Oxydase subunit I (COI) (637 bp) were amplified as described in Knott and Wray [START_REF] Knott | Controversy and consensus in asteroid systematics: new insights to ordinal and familial relationships[END_REF]. PCR reactions were performed in 20 µl of a solution containing 25 ng of total extracted DNA, 4 nmol of dNTPs, 20 pmol of each primer and 0.5 U of FlexiGoTaq © DNA polymerase in the 1 X PCR buffer (Promega). The cycling parameters were 7 min at 95 • C, 35 cycles of 95 • C for 40 s, 30 s at 55 • C for the 16S locus and 45 s at 46 • C for the two other loci, followed by 2 min at 72 • C and a final extension of 7 min at 72 • C. PCR products were sequenced by industry and sequences were submitted to EMBL database under accession numbers AM410947 to AM410962. The level of polymorphism was compared among loci using the same set of seven individuals from the West Indian Ocean successfully sequenced at the three loci. To this subset of individuals, we added the sequences of the two complete mitochondrial genomes of A. planci and A. brevispinus (accession numbers AB231475 and AB231476). The DNA polymorphism parameters were estimated with DnaSP 4.10.8 [START_REF] Rozas | DnaSP, DNA polymorphism analysis by the coalescent and other methods[END_REF]. Phylogenetic relationships between all sequenced individuals were estimated with the maximum-parsimony method (MP), a distance method using the neighbourjoining algorithm (NJ) and the maximum-likelihood method (ML). The best models of nucleotide substitution were determined with the program MrAIC [START_REF] Nylander | Program distributed by the author[END_REF] and turned to be Tamura-Nei (TN93) [START_REF] Tamura | Estimation of the number of nucleotide substitution in the control region of mitochondrial DNA in Humans and Chimpanzees[END_REF] for the 16S locus and Hasagawa-Kishino-Yano (HKY) [START_REF] Hasagawa | Dating of human-ape splitting by a molecular clock of mitochondrial DNA[END_REF] for COI and tRNAs loci. The MP and NJ trees were built with the program MEGA version 3.1 [START_REF] Kumar | MEGA3: Integrated software for molecular evolutionary genetics analysis and sequence alignment[END_REF]. Given that the HKY model is not implemented in MEGA, all NJ trees were calculated with the TN93 distance. ML trees were built using PHYML [START_REF] Guindon | A simple, fast, and accurate algorithm to estimate large phylogenies by maximum likelihood[END_REF] with the models of nucleotide substitution determined by MrAIC for each locus. Bootstrap values were calculated from 1000 replicates. Acanthaster planci sequences from Japan (AB231475) and Mexico (AF217324) as well as A. brevispinus sequence (AB231476) were added to the analysis. Based on their genetic proximity to Acanthaster, sequences of Meridiastra fissura (AY370740) were used to root the 16S gene tree and some of Patiriella calcar (U50046) for COI and tRNAs. Average pairwise distances between clades were estimated and rate constancy was checked with the Tajima's test of relative rate [START_REF] Tajima | Simple methods for testing molecular clock hypothesis[END_REF] implemented in MEGA. Minimum-spanning networks (MSN) of haplotypes were built using TCS [START_REF] Clement | TCS: a computer program to estimate gene genealogies[END_REF].

Results

Overall mtDNA polymorphism

Considering the reduced set of individuals sequenced at the three loci, mtDNA sequencing revealed different levels of polymorphism depending on the region (Table 1A). Within the genus Acanthaster, the 16S and COI loci contain about 14% of segregating sites (five haplotypes), whereas tRNA only has 5% (three haplotypes). Haplotype and nucleotide diversities are of the same order of magnitude for 16S and COI (H D = 0.72, and π = 0.04) and were lower for tRNAs (H D = 0.41, and π = 0.014). At all three loci, more than 70% of segregating sites were singletons (mutations observed on only one sequence).

Phylogenetic relationships and average divergences

The same tree topology was obtained using the three methods (NJ, MP and ML) and considering all loci. All tree branches are supported by high bootstrap values (Fig. 2). The genus Acanthaster is monophyletic and splits into an A. planci clade and A. brevispinus. The A. planci clade splits into a West Indian Ocean cluster and a Japanese cluster. The Mexican sequence available for tRNA only is closely related to the Japanese one, with which it forms a 'Pacific Ocean' clade distinct from the 'West Indian Ocean' one, which needs to be confirmed by additional sequences of the two other loci. The 16S and COI trees display similar average divergences (11.3% and 12.1%, respectively) between A. planci and A. brevispinus clades (Table 1B), whereas tRNAs shows a lower divergence (4%). The Tajima's relative rate test did not reject the molecular clock hypothesis (P > 0.05) allowing us to roughly estimate the elapsed time since the divergence of the different clades. Applying the substitution rates established for Panamanian oreasterid species (5% divergence per million years (Myr) for the COI and 2.2% per Myr for tRNAs [START_REF] Hart | Molecular phylogenetic analysis of life-history evolution in asterinid starfish[END_REF]) and assuming that theses rates are valid for Acanthaster (supported by the clustering of Oreaster with Acanthaster [START_REF] Yasuda | Complete mitochondrial genome sequences for Crown-of-thorns starfish Acanthaster planci and Acanthaster brevispinus[END_REF]), the average divergences between the sibling species, 12.1% (COI) and 4% (tRNAs) respectively correspond to 2.4 Myr and 1.8 Myr. Similarly, the divergence times between the two subclades of A. planci are estimated at 1.56 Myr and 1.36 Myr for COI and tRNAs, respectively. The variability observed within the West Indian Ocean clade was weak (or null) (Table 1B); however COI expressed the largest one. All haplotypes from Kenya were shared with Mayotte (Fig. 2B).

Discussion

Evolutionary history of the genus Acanthaster

Our first phylogeographic data confirmed that Acanthaster forms a clade split into two subclades, A. planci and A. brevispinus, in agreement with Yasuda et al. [START_REF] Yasuda | Complete mitochondrial genome sequences for Crown-of-thorns starfish Acanthaster planci and Acanthaster brevispinus[END_REF], who analysed a single sequence from each species together with several published asteroid COI sequences. However, this contradicted allozyme data [START_REF] Benzie | Major genetic differences between crown-ofthorns starfish (Acanthaster planci) populations in the Indian and Pacific Oceans[END_REF] that provided shorter genetic distances between A. planci and its sibling species in the Pacific. Allozymes are preferably used for intraspecific studies at the population level. The lack of allozyme differentiation between reproductively isolated species may originate from convergent phenotypes. In comparison, mtDNA provides a much stronger phylogenetic signal.

Within A. planci, our mitochondrial data confirmed the morphological and allozyme differentiation observed between Indian and Pacific populations. The estimations of the divergence time between these two subclades using both markers are congruent (1.56 Myr (COI) and 1.36 Myr (tRNAs)) and consistent with the time estimated by Benzie [START_REF] Benzie | Major genetic differences between crown-ofthorns starfish (Acanthaster planci) populations in the Indian and Pacific Oceans[END_REF] (1.6 Myr). Indian Ocean populations may actually be genetically isolated from the Pacific ones. Within the West Indian Ocean clade, all three locations share identical or closely related hap-lotypes as indicates the low within-clade mean divergences (0.2% (16S) and 0.3% (COI)).

Accuracy of mtDNA in studying A. planci phylogeography

All tree topologies from mtDNA are congruent with systematics, morphology (colour morphs) or biogeographical patterns previously published for Acanthaster. Considering the set of individuals sequenced at the three loci, tRNAs evolve three times more slowly COI and 16S. The latter two markers seem to display similar levels of polymorphism, suggesting that they evolve at the same rate within the genus Acanthaster. That feature contrasts with the trends observed for a majority of species, since COI usually evolves faster [START_REF] Mueller | Evolutionary rates, divergence dates, and the performance of mitochondrial genes in Bayesian phylogenetic analysis[END_REF]. The assessment of extant polymorphism may be biased by the reduced size of this subset of samples, since actually COI appears as the most polymorphic locus when the whole COI sequence set is considered (Fig. 2). Alignment of the protein coding COI sequences from distant species is much easier than alignment of 16S sequences, which contain insertions and deletions. These two features usually make COI a better mitochondrial marker than tRNAs and 16S for phylogeographic inferences. However, in the present study, all three loci appeared as good candidates for worldwide study of Acanthaster phylogeography and good complementary tools to more variable and independent loci usually required for population genetics study at small geographic scales such as nuclear microsatellites [START_REF] Yasuda | Development of microsatellite markers for the crown-of-thorns starfish Acanthaster planci[END_REF] or introns [START_REF] Foltz | Primers for the amplification of nuclear introns in sea stars of the family Asteriidae[END_REF][START_REF] Gerard | Are molecular markers a way to manage the crown-of-thorns starfish (Acanthaster planci) in Mayotte (West Indian Ocean)?[END_REF].

Fig. 1 .

 1 Fig. 1. Geographical locations of A. planci samples. Left: samples from the north of Mozambique Channel: KEN, Mombasa Marine National Park, Kenya; MDG, Marie reef on Mitsio Island, Madagascar. Right: samples of A. planci from Mayotte island: HDM, Handréma Bay; LGI & LGN, Longoni Bay; HJG, Hajangua reef; PS, 'Passe en S'; MBO, M'Bouini Bay; RAN, Rani reef.

Fig. 2 .

 2 Fig. 2. (A) Neighbour-joining trees of each locus, with bootstrap values indicated on the branches: black triangles are for Kenyan haplotypes, grey triangle for Madagascar, white squares for Mayotte. (B) The relationships within the West Indian Ocean clade of each locus are represented by a minimum spanning network of haplotypes: the circle size is proportional to the number of individuals displaying the haplotype (1, 6, or 10 individuals). Black slice/circles are haplotypes from Kenya; grey slice from Madagascar; white slices/circles are haplotypes from Mayotte.

Table 1 (

 1 A) DNA polymorphism of the genus Acanthaster within the reduced set of 9 individuals sequenced at all loci: 16S, tRNAs and COI. N : number of sequences; S: percentage of segregating sites; η: percentage of singletons; π : nucleotide diversity; n: number of haplotypes; H D : haplotype diversity. (B) Average divergences among clades and within West Indian Ocean (WIO) clade using the Tamura-Nei distance[START_REF] Tamura | Estimation of the number of nucleotide substitution in the control region of mitochondrial DNA in Humans and Chimpanzees[END_REF] on the complete sequence set. PO: Pacific Ocean

	A.	16S	tRNAs	COI
	N	9	9	9
	S (%)	14	5	15
	η ( % )	7 0	7 4	7 9
	π	0.039	0.014	0.040
	n	5	3	5
	H D	0.722	0.417	0.722
	B. Div (%)	16S	tRNAs	COI
	A. planci / A. brevispinus	11.3	4	12.1
	PO/WIO	9	3	7.8
	within WIO clade	0.2	0	0.3

Acknowledgements

UMR DIMAR financially supported this work. We are very grateful to the Service des pêches de Mayotte for logistic and financial support through the GIS LAG-MAY, and to E. Turak and S. Visram for collecting samples in Madagascar and Kenya, respectively. Thanks are also due to E.F. Dick who kindly helped to improve the English text. We finally thank the anonymous reviewers for helpful comments on the manuscript.