Emilio Bastidas-Arteaga

PhD Franck Schoefs

Sustainable Maintenance and Repair of RC Coastal Structures

Keywords: Maintenance & Inspection, Concrete Structures, Sustainability

Many agencies worldwide have to manage structurally damaged reinforced concrete (RC) coastal infrastructure subjected to chloride ingress that require significant repairs. These repairs should ensure optimal levels of serviceability and safety and minimise costs and environmental impact. However, there is a wide quantity of protocols, techniques and materials for repair and there is no much information about their durability performance. This paper proposes a methodology for evaluating, comparing and/or improving sustainability of maintenance strategies. It was developed within the framework of the MAREO project in collaboration with the different stakeholders involved during the whole structural lifetime. It is based on probabilistic modelling of deterioration and repair, and the sustainability assessment considers costs (with conventional and intergenerational discounting), CO 2 emissions and waste generation. The methodology is illustrated with a numerical example aiming to evaluate and improve the sustainability of a repair technique for RC structures subjected to chloride-induced corrosion damage. Overall results indicate that sustainable solutions increase global costs but they could reduce significantly environmental impact.

C env : environmental chloride concentration d:

discount factor d a :

intergenerational discount factor MOI: multi-objective index p:

parameter indicating the importance given to deviations from the ideal solution r:

discount rate r I : intergenerational discount rate T a : period of analysis t cr : time to corrective repair t pr : time to preventive repair W(t): weight used to discount C j (t) to the present value W c (t): conventional weight W I (t): intergenerational weight w i :

weighting factor of the optimisation criterion f i x i * : ideal solution of the optimisation criterion f i x i* :

anti-ideal solution of f i Δt:

length of the inspection interval

INTRODUCTION

1.1.

Motivation, industrial and scientific challenges

Coastal infrastructures are critical and strategic assets for the development of any country. For example, ports handle 80% of the nation trade and provide strategic places for the fishing industry, leisure activities and European defence. 564 types of ports can be found along the French coastline [START_REF] Boéro | Technical management of French harbour structures -Part 1: Description of built assets[END_REF]. Depending on their use, location and construction dates, ports components are generally constituted by several materials [START_REF] Rato | Extension to the port of Gijón, Spain[END_REF]. Concerning French ports, most part of their components were built in masonry (33%) followed by steel (28%), reinforced concrete (RC) (24%) and other materials (15%) [START_REF] Boéro | Technical management of French harbour structures -Part 1: Description of built assets[END_REF]. On the overall, the majority of ports (60%) were built before 1955 and significant damage has been reported for their RC components [START_REF] Boero | Fiabilité des infrastructures portuaires : approche innovante d'analyse et de modélisation probabiliste des données d'inspection[END_REF][START_REF] Rosquoët | Chloride propagation in concrete harbor[END_REF].

The durability of RC structures placed in coastal areas is largely affected by chloride-induced corrosion. RC has been widely used in the construction of structures or structural components for many coastal assets. The kinematics of the deterioration mechanisms is controlled by material properties and exposure conditions (E. [START_REF] Bastidas-Arteaga | Contribution for sustainable management of reinforced concrete structures subjected to chloride penetration[END_REF]Stewart, 2015a, 2015b). However, the consequences of chloride-induced corrosion damage in terms of serviceability and safety that define inspection and maintenance strategies are different for each the type of asset: ports [START_REF] Boéro | Technical management of French harbour structures -Part 1: Description of built assets[END_REF][START_REF] Lefler | Reinforced concrete caissons for port structures in Spain[END_REF], buildings [START_REF] Medeiros | Reinforced concrete in marine environment: Effect of wetting and drying cycles, height and positioning in relation to the sea shore[END_REF], bridges [START_REF] Cheung | Life-cycle cost management of concrete bridges[END_REF], coastal defences [START_REF] Alani | Reliability and optimised maintenance for sea defences[END_REF], etc. This is due to the fact that failure consequences differ for each type of structure. Consequently, maintaining this ageing RC infrastructure in service and safe becomes a main challenge for managers and owners.

Currently, the rehabilitation market represents half of the building trade activity and a significant percentage of public works in France. Companies consider that the rehabilitation activity will develop in the public works field and may reach a volume near than construction of new structures. Furthermore, environmental considerations have raised important questions regarding the new developments, environmentally friendly solutions, and/or decommissioning of existing infrastructure. The maintenance planning optimisation is therefore a major challenge with multiple constraints imposed by economical, environmental, and societal considerations.

The MAREO Project

Within this context, the main objective of the MAREO † project (2007)(2008)(2009)(2010)(2011)(2012) was to provide a comprehensive framework to improve the sustainability, effectiveness, reliability and long-term performance of maintenance strategies for RC coastal structures subjected to chloride ingress. The project had two scientific objectives. First, to evaluate and compare the effectiveness of repair techniques within the context of sustainable assessment of existing RC structures. Second, to identify and quantify uncertainties and hazard sources related to repair, material properties, exposure and models. The project therefore relies on methodologies for risk analysis and assessment. The research work of the MAREO project was defined taking into account the feedback and requirements of all stakeholders involved during the structural lifetime. The following tasks were carried out: 1. identification of structural performance indicators: porosity, chloride diffision coefficient, chloride content and pH near the rebar; 2. definition of the relationships between these indicators and their comparison in terms of risk analysis. This implies particularly to consider modelling uncertainties and to propose other models based on non-deterministic approaches (physical or analytical response surfaces) and to identify uncertainties on performance indicators or influential factors and thereby to characterise the so-called intrinsic hazards; 3. selection of experimental devices and protocols (in laboratory or in situ), to identify and characterise failure mechanisms; 4. choice of computational methods for assessment of existing structures; and 5. development of a methodology for semi-probabilistic analysis through pilot studies on various components [START_REF] Schoefs | Repair of wharves in tidal area: mareo project[END_REF]): • repair and test of 20 beams exposed to chlorides during 80 years in the estuary of the Rance river in France where chlorides were transported by the wind and longterm splash exposure, and • composite sticking repair with a carbon fiber reinforced polymer plate of RC beams for two harbours (Port of Nantes Saint-Nazaire and general council).

The main outputs were:

• the development of a complete protocol (natural and accelerated tests) for condition assessment of repair materials [START_REF] Tran | Parameter identification in chloride ingress from accelerated test using bayesian network[END_REF]; • the publication of guidelines for repair chloride-deteriorated structures [START_REF] Schoefs | Repair of wharves in tidal area: mareo project[END_REF]; • the quantification of uncertainties for chloride assessment and for repair techniques [START_REF] Bonnet | Effect of error measurements of chloride profiles on reliability assessment[END_REF];

• the improvement of Non-destructive tests (NDT) and embedded sensors for chloride assessment [START_REF] Plooy | Development of a multi-ring resistivity cell and multielectrode resistivity probe for investigation of cover concrete condition[END_REF];

• the development of a model of chloride ingress into concrete (E. [START_REF] Bastidas-Arteaga | A comprehensive probabilistic model of chloride ingress in unsaturated concrete[END_REF] and a methodology for maintenance and inspection optimisation (Emilio Bastidas-Arteaga and Schoefs, 2012); • the definition of new issues for future research: reliability assessment and risk analysis during repair operations, the need for optimizing the number of cores for condition assessment and the use of NDT combination for improving the reliability of diagnosis [START_REF] Torres-Luque | Non-destructive methods for measuring chloride ingress into concrete: State-of-the-art and future challenges[END_REF].

1.3.

Aims and scope

This paper focuses on the developments around the concept of sustainable maintenance optimisation when significant repairs are required and long-term performance is pursued. The scope and needs for sustainable maintenance were defined in collaboration with the different stakeholders involved within the MAREO project. The proposed methodology considers both environmental and intergenerational consequences. Owners and managers searching for sustainable maintenance strategies could use the proposed approach to (i) improve the sustainability of a maintenance strategy and/or (ii) compare the sustainability performance of various strategies. The approach was formulated on the basis of the maintenance of RC ports subjected to chloride-ingress. Nevertheless, it can also be applied for other deterioration processes and/or materials.

Section 2 describes the general framework for sustainable maintenance proposed in this study. Sustainability of a maintenance strategy is evaluated in terms of three criteria described in Section 3 (costs, waste generation and CO 2 emissions). Section 3 also presents the methodology for decision-making under multi-objective constraints. Finally, the proposed approach is applied to a numerical example that aimed at evaluating and optimising the sustainability of a maintenance strategy (Section 4).

FRAMEWORK FOR SUSTAINABLE MAINTENANCE

The conceptual framework proposed to evaluate and optimise the sustainability of maintenance strategies could be applied at two levels. It can be used to improve the sustainability of the formulation of a single maintenance strategy and/or it can be implemented to compare different strategies. The proposed framework accounts for the following aspects:

• the implementation of a deterioration model representative of the considered phenomenon (exposure and durability of construction and repair materials); • the formulation of a maintenance strategy technically and economically feasible and that ensures optimal levels of serviceability and safety during the operational life;

• the implementation of an appropriate probabilistic framework for considering the uncertainties related to both the deterioration process and the inspection/maintenance actions;

• the establishment of criteria for the evaluation of the environmental impact; and • the adoption of a multi-objective optimisation procedure to minimise costs and environmental impact.

Figure 1 compares the stages for obtaining a cost-effective or sustainable formulation of the maintenance strategy. The following stages are required to determine both results: (i) formulation of the maintenance technique, (ii) modelling of deterioration and maintenance actions, and (iii) Improvement of the formulation.

2.1.

Formulation of the maintenance strategy

This is a crucial stage in the management process. The formulation should consider: the characteristics of the deterioration process, the stages of maintenance (i.e., inspection, repair), the maintenance philosophy (i.e., preventive or corrective) and the technical and economic feasibility of the selected technique.

Inspection results alert the owner/operator when a given deterioration threshold is reached.

Destructive and non-destructive methods are employed to carry out inspections. Concerning non-destructive methods, visual inspection is usually used for evaluating the condition of RC structures [START_REF] Roelfstra | Corrosion evolution in bridge management systems and corrosion-induced deterioration[END_REF]. However, for visual inspection, the assessment of the structural condition remains largely uncertain when inspection is undertaken. Other nondestructive techniques aim at quantifying the instant value or the evolution in time of a given variable (material strength, chloride concentration, corrosion rate, etc); nevertheless, their results are highly influenced by environmental conditions reducing its accuracy [START_REF] Torres-Luque | Non-destructive methods for measuring chloride ingress into concrete: State-of-the-art and future challenges[END_REF]. Although destructive inspection techniques improve significantly the structural condition assessment, they are more expensive and require a larger number of tests when there is a larger variability of the inspected parameter.

The extent of damage assessed during inspection conditions the repair technique choice. Consequently, the formulation of the maintenance strategy (inspection and repair) depends on several aspects such as: the phenomenon inspected (material strength, chloride concentration, etc.); the size of the project (structural network, particular structure or component); the use of the structure (nuclear, transportation, etc.); the remaining lifetime; the location of the structure/component; and other socio-economic aspects such as priorities of the country, availability of resources, etc. To define an appropriate maintenance strategy, all the stakeholders that are linked to the structure during its life-cycle should participate in this stage.

Modelling of deterioration and maintenance actions

Experimental testing is the best way to determine and to improve the performance of repair techniques. However, given that these tests are generally expensive and time-consuming, numerical modelling of deterioration and repair actions is essential to study and/or to improve the effectiveness of maintenance strategy in most cases. Furthermore, the uncertainty related to the deterioration process and the maintenance actions should be considered to improve the predictability of the models. The main sources of uncertainty are related with durability properties of the construction and repair material, environmental actions (climate conditions, chloride concentration), loading and effectiveness of inspection and repair actions. Four types of methods can be used for modelling deterioration and repair (van Noortwijk and Frangopol, 2004): (i) failure rate functions, (ii) Markov models, (iii) stochastic processes, and (iv) timedependent reliability analysis. Each method has its advantages and shortcomings and it is more or less adapted for each maintenance problem (E. Bastidas-Arteaga, 2010). Although some specialised companies currently implement some of these aspects for lifetime assessment, research centres mainly carry out this stage. Realistic predictions also require experimental data for prediction and updating in addition to the feedback of contractors and agencies [START_REF] Schoefs | Modeling Inspection Uncertainties for On-site Condition Assessment using NDT Tools[END_REF].

2.3.

Improvement of the formulation:

The performance of each maintenance strategy can be improved by optimizing its governing parameters. Classically, the optimisation function aims at minimising costs and then providing an optimal cost-effective formulation of the maintenance technique [START_REF] Alani | Reliability and optimised maintenance for sea defences[END_REF]Emilio Bastidas-Arteaga and Schoefs, 2012;[START_REF] Van Noortwijk | Life-Cycle Performance of Deteriorating Structures: Assessment, Design and Management[END_REF]. Nevertheless, to reduce the environmental impact, it is necessary to include environmental constraints in the optimisation problem [START_REF] Padgett | Sustainability of Natural Hazard Risk Mitigation: Life Cycle Analysis of Environmental Indicators for Bridge Infrastructure[END_REF][START_REF] Tapia | Multi-objective optimisation of bridge retrofit and post-event repair selection to enhance sustainability[END_REF]. Under these considerations, the optimisation leads to an optimal sustainable formulation of the maintenance technique (Figure 1). Mainly research centres carry out this stage. However, the diffusion of these techniques to consultants is imperative. Again, the feedback of contractors and agencies are essential to provide feasible results. Section 3 describes the criteria considered in this study for evaluating the sustainability of maintenance strategies.

CRITERIA AND METHOD FOR SUSTAINABILITY ASSESSMENT

The World Commission on Environment and Development (1987) defines sustainable development as: development that meets the needs of the present without compromising the ability of future generations to meet their own needs. To meet this goal, sustainable development must provide a balance between three components: environment, economy and society. The evaluation of the sustainability of the repair techniques is based on the comparison of three criteria: cost, waste generation, and carbon dioxide emissions. Society is not directly considered but it is indirectly implied in decisions affecting these components (Sánchez-Silva and Rosowsky, 2008). More criteria could also be included for a specific problem.

Cost

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. User costs represent the inconvenience and expenses incurred by users due to traffic disruption such as delay, operating and accident costs. Thoft-Christensen (2009) highlights the importance of including user costs in the analysis. These cost are specific for each type of structure and could be also considered in the proposed framework if the information is available. However, this work is only based on the available agency costs.

This study focuses on the sustainability of maintenance strategies for deteriorating structures.

Then the direct costs incurred by the agency include only costs associated with maintenance (inspection, preventive and corrective repair). Initial construction and residual costs are not included in the analysis because it would be the same for all maintenance alternatives.

In life-cycle cost analysis, the present value life-cycle cost incurred during the period of analysis T a is computed as:

E C T ⎡ ⎣ ⎤ ⎦ = t=0 T a ∑ E C j t ( ) ⎡ ⎣ ⎤ ⎦ W t ( ) (1)
where C j (t) is the j th expenditure at time t. Basically, these costs depend on the properties of the original and repair material; and on the characteristics of the maintenance strategy (i.e., inspection interval, repair threshold, repair material and technique, etc.) and W(t) is the weight used to discount C j (t) to the present value. Let d denote the discount factor computed in terms of a standard discount rate r:

d = 1/ 1+ r ( ) (2)
Then, the conventional weight in a given period becomes:

W c t ( ) = 1/ 1+ r ( ) t (3)
Sumaila & Walters (2005) 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. Their formulation treats the benefits as accruing to the current generation (at standard discount rates) plus annual increments of new stakeholders who will enter to the future population divided by the generation length, G:

W I t ( ) = d t + d a d t-1 G 1-d a / d ( ) t 1-d a / d ( ) ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ (4)
where d a is the intergenerational discount factor:

d a = 1/ 1+ r I ( ) (5)
where r I is the intergenerational discount rate.

3.2.

Waste generation

Concrete has been recognised as the largest and most visible component of construction and demolition waste. It accounts for up to 67% by weight of construction and demolition waste (53% by volume), with only 5% currently recycled (American Institute of Architects, 1999). Therefore, waste generation should be included as a selection criterion for sustainable management.

A comprehensive assessment of waste generation should include waste generated during the production of the repair material and the repair operations. However, taking into account the difficulties in estimating the waste generated during the production of concrete, this study only considers waste produced during repair operations (demolition and rebuilding). Waste generation depends mainly on the characteristics of the repair technique -e.g., material durability. These characteristics will define the number of repairs during the analysis period or the repair rate and could be determined by a deterioration model.

3.3.

Carbon dioxide emissions

Carbon dioxide emissions are one of the major causes of climate change [START_REF] Ipcc | Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change[END_REF]. Therefore, taking into account that the world's yearly production of 1.6 billion tons of cement accounts for about 7% of the global emissions of CO 2 into the atmosphere (Kumar [START_REF] Mehta | Durability : Critical Issues for the Future[END_REF], the assessment of carbon dioxide emissions produced during repair operations is crucial for sustainable development. This analysis considers two sources of CO 2 : 1. emissions produced during transportation of materials, equipment and waste, and 2. CO 2 released during production of the repair material.

The emissions produced to repair 1 m 3 of polluted concrete during transportation are calculated by summing the emissions released during provision of repair material, provision of equipment and disposal of waste. The emissions are computed in terms of the distances for provisioning repair materials and equipment and disposal of waste. Transportation emissions depend on the characteristics of the transportation vehicle (emissions in grams of CO 2 per km).

On the other hand, the CO 2 released during production of the repair material (in kg CO 2 per year) is estimated in terms of the cement content per m 3 of concrete (in kg/m 3 ) and the rate of CO 2 emissions released during the repair material production (in kg CO 2 /kg of repair material). The average CO 2 production emissions ranges from 0.65 to 0.92 kg of CO 2 per kg of cement across several countries (International Energy Agency, 2007). Since there is no information about the CO 2 emissions related to the production of the repair products, a weighted average emission of 0.83 kg CO 2 /kg of repair material is adopted herein. Again, the quantity of CO 2 released during repair depends on the characteristics of the repair technique.

Decision-making under multi-objective constraints

The challenge in suistainable management of deteriorating structures lies in its multi-objective nature. Owners/operators are confronted to simultaneously satisfy several criteria such as: minimisation of cost, traffic disruptions and environmental impact on the one hand, and improvement of serviceability, functionality and safety on the other hand. Multi-objective optimisation techniques are appropriate to deal with this problem [START_REF] Tapia | Multi-objective optimisation of bridge retrofit and post-event repair selection to enhance sustainability[END_REF].

Compromise programming is adopted herein to solve the multi-objective problem. Compromise programming minimises the distance from the set of Pareto optima to the so-called 'ideal solution'. The ideal solution is defined as the solution that yields simultaneously optimal values for all objectives. For m objective functions, the ideal solution can be associated with the ideal objective vector:

f*= [x 1 * , x 2 * , ..., x m * ].
Where x i * is the ideal solution of the optimisation criterion f i with i = 1, ... , m. In this particular case there are three criteria to evaluate: (1) costs, (2) waste generation and (3) CO 2 emissions. Since each criterion has its own system of units, this study uses a multi-objective index (MOI) to evaluate the sustainability of the maintenance strategy [START_REF] Lounis | Risk-based maintenance optimization of aging highway bridge decks[END_REF]. A MOI is defined for each technique as the value of the weighted and normalised deviation from the ideal solution f* measured by the family of L p metrics. Thus, the 'satisfying' solution is the one that yields a minimum MOI:

MOI x ( ) = i=1 m ∑ w i p x i -x i * x i* -x i * p ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ 1/p (6)
where w i is the weighting factor of the optimisation criterion f i , p is a parameter indicating the importance given to deviations from the ideal solution, and x i* is the anti-ideal solution of f i . The value of w i varies between 0 and 1 with Σw i = 1. The weighting factors depend mainly on the attitude of the owner/operator towards each criterion. The parameter p varies between 1 and ∞.

For p = 1, all deviations from the ideal solution are considered in direct proportion to their magnitudes, which corresponds to a group utility [START_REF] Duckstein | New directions in optimum structural design[END_REF]. For p = 2, a greater weight is associated with the larger deviations from the ideal solution and L 2 represents the Euclidian metric. For p=∞, the largest deviation is the only one taken into account and is referred to as the Chebyshev metric or mini-max criterion and L ∞ corresponds to a purely individual utility [START_REF] Lounis | Risk-based maintenance optimization of aging highway bridge decks[END_REF]. The Euclidian metric was adopted in this work to determine the sustainable repair strategy.

NUMERICAL EXAMPLE: MAINTENANCE OF COASTAL RC STRUCTURES SUBJECTED TO CHLORIDE INGRESS

4.1.

Problem description and basic considerations

This example illustrates the proposed framework for evaluating and optimising the sustainability of maintenance strategies. It focuses on the performance of a maintenance strategy for coastal RC structures in a chloride-contaminated environment. The characteristics and the requirements for the maintenance strategy were defined within the framework of the MAREO project. The adopted maintenance strategy has two steps:

• Inspection: it is carried out by analysing the concentration of chlorides at the cover depth on concrete cores (destructive method) at periodic inspection intervals of length Δt. • Repair: if inspection results reveal larger corrosion initiation risks, the repair technique consists of rebuilding the polluted concrete cover using a commercial repair concrete. This preventive repair strategy aims at avoiding corrosion initiation and ensuring an optimal safety level; however, additional repair costs could be incurred if corrosion has started at the inspection times.

Cost and environmental impact of the maintenance strategy depend on Δt. The determination of an optimal Δt is very sensitive to the cost models. Therefore, to obtain realistic results, the costs were defined taking into account the average maintenance expenditures incurred by the port of Nantes Saint-Nazaire (Table 1). These costs are referred to an initial construction cost of 1000 units. Whereas preventive repair only considers the cost related to cover rebuilding, corrective repair also includes the costs of structural strengthening. Table 2 presents the parameters used to estimate waste generation and carbon dioxide emissions.

Probabilistic modelling of deterioration and repair

There is no in-field information about the long-term durability performance of the commercial repair concrete. Consequently, it is determined on the basis of (i) experimental results that provide some input parameters [START_REF] Schoefs | Repair of wharves in tidal area: mareo project[END_REF], (ii) literature review to complete missing information (Table 3), and (iii) a Markovian approach to model deterioration and repair (Emilio Bastidas-Arteaga and [START_REF] Bastidas-Arteaga | Stochastic improvement of inspection and maintenance of corroding reinforced concrete structures placed in unsaturated environments[END_REF]. The main advantage of this Markovian approach lies in the consideration of the most important phenomena influencing chloride ingress and maintenance including environmental conditions, uncertainties and consequences of 'good' or 'wrong' decisions when there are imperfect inspections. However, other approaches for modelling deterioration and repair can be instead implemented for other materials, deterioration mechanisms and/or maintenance actions.

The inputs of the model are stochastic processes and independent random variables. The environmental chloride concentration C env was modelled as a stochastic process generated by independent lognormal numbers (log-normal noise) with a mean of 6 kg/m 3 and a coefficient of variation of 0.2 [START_REF] Duracrete | Modelling of Degradation, DuraCrete -Probabilistic Performance based Durability Design of Concrete Structures, EU -Brite EuRam III[END_REF][START_REF] Vu | Structural reliability of concrete bridges including improved chloride-induced corrosion[END_REF]. These values correspond to a splash and tidal exposure. Climate variations (temperature and relative humidity) are represented as a stochastic process using the Karhunen-Loève expansion (E. [START_REF] Bastidas-Arteaga | Influence of global warming on durability of corroding RC structures: A probabilistic approach[END_REF]. The considered climate conditions correspond with an oceanic climate with temperature and relative humidity ranging from 5 to 25ºC and 0.6 to 0.8 for each year, respectively.

Table 3 presents the probabilistic models of the considered random variables. The mean of the chloride diffusion coefficient depends on the characteristics of the material. The mean of the chloride diffusion coefficient for the construction material was assigned according to experimental values [START_REF] Saetta | Analysis of chloride diffusion into partially saturated concrete[END_REF]. The mean of the chloride diffusion coefficient for the repair material was defined based on experimental results obtained within the framework of the MAREO project [START_REF] Schoefs | Repair of wharves in tidal area: mareo project[END_REF].

The Markovian approach is used to estimate the expected times to preventive and maintenance repair as a function of Δt (Figure 2). These results were obtained for a concrete cover of 5 cm.

As expected, the time to preventive repair, t pr , is lower when the structure is inspected more periodically. On the contrary, less inspection implies that the time to corrective repair, t cr , decreases. These times are used to determine Δt that minimises both costs and environmental impact. In order to simplify the computation, the simulation results are adjusted to two analytical functions that allow to estimate E[t pr ] and E[t cr ] in terms of Δt (Figure 2).

Results

Figure 3 presents the expected costs for the standard and intergenerational discounting. It is observed in all the cases that the costs of inspection and preventive repair decrease and the cost of corrective repair increases for larger Δt. This behaviour is explained by the fact that when Δt is greater, most part of inspections detect that the components are corroded. On the contrary, when the structure is inspected regularly, repair is basically preventive, and therefore, the corrective repair expenditures diminish. In all cases, total costs are optimal when the structure is inspected every 20yr and 17yr for the standard and intergenerational discounting models, respectively. However, the optimum expected cost is different for both cases because future flows for the intergenerational model is important in comparison to the conventional model for the analysis period T a . The differences between the optimum Δt imply that the intergenerational discounting gives more importance to preventive maintenance.

Figure 4a provides the assessment of the multi-objective index for different inspection intervals and cost models. These results were obtained by assuming that the agency gives the same importance to costs and environmental impact -i.e., w cost = 0.5, w waste = 0.25 and w CO2 = 0.25 in eq. 6. These curves follow the behaviour described in Figure 3 where there are inspection intervals that minimise the MOI. However, the computed optimum Δt are higher than those estimated from the cost analysis: 21 and 19 yr for the conventional and intergenerational discounting models, respectively. The increase in the inspection interval reduces the repair rate and then the expected total waste generation and CO 2 emissions. However, the consideration of waste generation and CO 2 emissions in the assessment of optimal Δt induces overcharges.

The selection of a given interval depends on the agency's policies. Consequently, the agency decides how much overcharges can be expended to reduce environmental impact. Socioeconomic aspects that are characteristic of a given company or country govern this decision.

Economic or environmental priorities are measured herein by the weighting factors (eq. 6). Thus, Figure 4b shows the overcharges generated when the cost weighting factor varies between 0 and 1 for the optimal Δt that minimise the MOI in each case. The estimation of the overcharges supposes that the waste and CO 2 emission weighting factors are equal. The cost weighting factor w cost = 1 implies that the decision is controlled only by costs. In this case, environmental constraints are not considered, and then, the overcharges are zero. On the contrary, for w cost = 0, the overcharges lead to a maximum value. For the range of cost weighting factors presented in Figure 4b, the maximum overcharges are less than 6% and 13% for the conventional and intergenerational discounting models. If w cost = 0.50, w waste = 0.25 and w CO2 = 0.25, the overcharges are lower than 2% for both cases. This means that including environmental impact in decision-making does not generate larger overcharges. Nevertheless, for the same cost weighting factors and for the repair of a port with a surface of 8,300 m 2 , a sustainable Δt could reduce the waste generation by 15 and 84 m 3 and the emissions by 10 and 46 tons of CO 2 , for the conventional and intergenerational discounting models (Figure 5). As mentioned before, the difference between the environmental impacts when intergenerational discounting is considered is due to the increase of Δt that reduces the number of repairs.

CONCLUSIONS

This paper presented a methodology for sustainable maintenance assessment and optimisation of coastal RC structures subjected to chloride ingress. It was developed within the framework of the MAREO project with the collaboration of different stakeholders involved during the whole structural lifetime. The proposed approach considered three criteria: (i) costs (including intergenerational issues), (ii) waste production, and (iii) CO 2 emissions. The methodology uses compromise programming to deal with the multi-objective nature of the decision-making problem. This tool can be used to improve the sustainability performance of a maintenance strategy or to compare the sustainability of various strategies. Their application to a numerical example indicated that although environmentally friendly solutions increase global costs, they could reduce significantly CO 2 emissions and waste. 0.20 [START_REF] Duracrete | Modelling of Degradation, DuraCrete -Probabilistic Performance based Durability Design of Concrete Structures, EU -Brite EuRam III[END_REF][START_REF] Saetta | Analysis of chloride diffusion into partially saturated concrete[END_REF][START_REF] Val | Probabilistic evaluation of initiation time of chloride-induced corrosion[END_REF] 0.20 [START_REF] Saetta | Analysis of chloride diffusion into partially saturated concrete[END_REF][START_REF] Val | Probabilistic evaluation of initiation time of chloride-induced corrosion[END_REF]) Parameter representing the ratio D h,min /D h,max , α 0 --beta on [0.025;0.1] 0.05 0.20 [START_REF] Val | Probabilistic evaluation of initiation time of chloride-induced corrosion[END_REF] Parameter characterizing the spread of the drop in D h , n -beta on [6;16] 11 0.10 [START_REF] Val | Probabilistic evaluation of initiation time of chloride-induced corrosion[END_REF] Thermal conductivity of concrete, λ W/(m°C) beta on [1.4;3.6] 2.5 0.05 [START_REF] Neville | Properties of Concrete[END_REF] Specific heat capacity of concrete, c q J/(kg°C) beta on [840;1170]

List of Tables

List of Figures

Figure 1 .

 1 Figure 1. Cost-effective and sustainable formulation of repair techniques ................................. 16 Figure 2. Determination of the expected value of the times to preventive and corrective repair.16 Figure 3. Expected costs for the (a) conventional and (b) intergenerational discounting models. ............................................................................................................................................. 17 Figure 4. Overcharges produced when environmental criteria are considered. .......................... 17 Figure 5. Influence of the cost weighting factor on the reduction of waste and emissions. ........ 17

Figure 1 .

 1 Figure 1. Cost-effective and sustainable formulation of repair techniques

Figure 3 .Figure 4 .

 34 Figure 3. Expected costs for the (a) conventional and (b) intergenerational discounting models.

Figure 5 .

 5 Figure 5. Influence of the cost weighting factor on the reduction of waste and emissions.

  The project received in 2006 the support of the 'Pole Génie Civil et Eco-construction' and 'Maîtrice de Risques en Génie Civil' and it was granted in 2007 with a support of 1.04 million euros (75% from the Loire-Atlantique region and 25% from the French government). The consortium gathered two port owners (Port of Nantes Saint-Nazaire, and General Council), four companies specialised in cost evaluation, modelling, construction and assessment of existing

structures (Arcadis, Oxand, SEMEN-TP and ETPO) and seven research laboratories for: destructive and non-destructive testing and diagnosis (IFSTTAR), laboratory and tank tests (GeM, IFREMER, CERIB, LMDC), assessment of existing structures and reliability analysis (GeM, LMDC, CEREMA) with exchanges with the Trinity College Dublin.

Table 1 .

 1 Parameters for cost assessment. .................................................................................. 14 Table 2. Parameters to compute waste generation and CO 2 emissions (Bastidas-Arteaga, 2010). ................................................................................................................................... 14 Table 3. Probabilistic parameters of the random variables ......................................................... 15

Table 1 .

 1 .......................................................................................................................................... 17Figure 4. Overcharges produced when environmental criteria are considered. .......................... 17 Figure 5. Influence of the cost weighting factor on the reduction of waste and emissions. ........ 17 Parameters for cost assessment.

	Parameter

Table 2 .

 2 Parameters to compute waste generation and CO 2 emissions (E. Bastidas-Arteaga, 2010).

	Parameter

Table 3 .

 3 Probabilistic parameters of the random variables

	Variable	Units	Distribution	Mean	COV	References
	Chloride diffusion coefficient	m	2 /s	log--normal	3••10	--11
	(original material), D c,ref--ori

† Maintenance and Repair of Concrete Coastal Structures: Sustainable Risk-Based Optimisation

ACKNOWLEDGEMENTS

The authors acknowledge the financial support the Région Pays de la Loire and French Ministry of Industry for supporting the MAREO project (Maintenance and REpair of concrete coastal structures: risk-based Optimisation).


Denis Michel
email: denis.michel@live.fr

A role for ATP-dependent chromatin remodeling in the hierarchical cooperativity between noninteracting transcription factors

Keywords: Chromatin remodeling, cooperativity, transcription factor, MMTV, glucocorticoid receptor

Chromatin remodeling machineries are abundant and diverse in eukaryotic cells. They have been involved in a variety of situations such as histone exchange and DNA repair, but their importance in gene expression remains unclear. Although the influence of nucleosome position on the regulation of gene expression is generally envisioned under the quasi-equilibrium perspective, it is proposed that given the ATP-dependence of chromatin remodeling enzymes, certain mechanisms necessitate non-equilibrium treatments. Examination of the celebrated chromatin remodeling system of the mouse mammary tumor virus, in which the binding of transcription factors opens the way to other ones, reveals that breaking equilibrium offers a subtle mode of transcription factor cooperativity, avoids molecular trapping phenomena and allows to reconcile previously conflicting experimental data. This mechanism provides a control lever of promoter responsiveness to transcription factor combinations, challenging the classical view of the unilateral influence of pioneer on secondary transcription factors.

Introduction

The importance of ATP-dependent machineries remodeling chromatin by actively moving nucleosomes relatively to DNA, remains puzzling. Beside their possible structural roles in chromatin organization, nucleosome repositioning, histone exchange and DNA repair, a role in transcriptional cooperativity is proposed here. The specificity and intensity of gene expression is governed by interactions between regulatory DNA sequences (cis-regulators) and various trans-acting factors (transcription factor proteins (TFs) and non-coding RNAs). The occupation of a gene promoter by these transregulators involves both micro-reversible and microirreversible steps. Micro-reversible binding processes can lead to sigmoidal concentration-dependent response through classical multimeric cooperativity [START_REF] Bolouri | Modeling DNA sequence-based cis-regulatory gene networks[END_REF][START_REF] Michel | How transcription factors can adjust the gene expression floodgates[END_REF]. The role of nucleosomes has also been examined from the micro-reversible perspective [START_REF] Dodd | Theoretical analysis of epigenetic cell memory by nucleosome modification[END_REF][START_REF] Segal | From. DNA sequence to transcriptional behaviour: a quantitative approach[END_REF][START_REF] Mirny | Nucleosome-mediated cooperativity between transcription factors[END_REF]. The rapid equilibration of these thermallydriven phenomena, relatively to the slow changes of cellular components, simplifies the definition of the input functions used in gene network modeling [START_REF] Bintu | Transcriptional regulation by the numbers: models[END_REF][START_REF] Michel | How transcription factors can adjust the gene expression floodgates[END_REF]. But promoter occupancy also involves some micro-irreversible transitions [START_REF] Kim | A quantitative model of transcription factor-activated gene expression[END_REF][START_REF] Ahsendorf | A framework for modelling gene regulation which accommodates non-equilibrium mechanisms[END_REF]) such as chromatin remodeling and active dissociation processes. Precisely, it is proposed in the present study that inserting micro-irreversible steps in the process of promoter saturation, offers additional possibilities of potent cooperativity. The micro-irreversible step can correspond to the energy-dependent phenomenon of chromatin-remodeling, in which a nucleosome can be removed [START_REF] Parikh | The effect of an intervening promoter nucleosome on gene expression[END_REF] or simply displaced relatively to DNA, as in the example selected here. Upon binding to DNA, a first TF directs the accessibility to DNA of other ones. This mechanical activity allows: (i) cooperativity between non-physically interacting TFs and (ii) constitutively expressed TFs to participate to conditional induction. One of the most celebrated chromatin remodeling system is provided by the thoroughly documented Mouse Mammary Tumor Virus (MMTV) promoter.

Breaking hierarchical polymerization is necessary to maintain molecular dynamics

The assembly of macromolecular complexes generally proceeds in a hierarchical manner in the cell. For example, a component C which cannot bind to the isolated components A and B, can bind to a pre-associated complex AB. Hierarchical binding chains such as A + B AB, +C ABC, +D ABCD etc, are often involved in the building of multi-molecular complexes, but are less compatible with the dynamic and reactive behaviours of solubles components. Indeed, in equilibrium conditions, the chain written above leads to the trapping of the early components in the complexes as long as the late components are present. This phenomenon can exist for TF binding to gene promoters. It can hold for example, for the successive binding steps observed in equilibrium conditions between the TodT TFs and the series of TodTbinding sites juxtaposed in the Tod gene promoter, that has been proposed to be mediated by DNA conformation changes [START_REF] Lacal | Hierarchical binding of the TodT response regulator to its multiple recognition sites at the tod pathway operon promoter[END_REF]. Beside this puzzling situation of equilibrium allostery, the hierarchical binding of transcription factors in equilibrium conditions is also possible in the case of the large eukaryotic preinitiation complexes made of the so-called general transcription factors GTFs [START_REF] Michel | How transcription factors can adjust the gene expression floodgates[END_REF]. But hierarchical relationships have also been reported for non-interacting isolated TFs in absence of any trapping phenomenon. To allow reconciling hierarchical binding and absence of trapping, one should postulate the possibility to break equilibrium. This situation is well illustrated by the case of the occupation of the MMTV promoter involving micro-irreversible processes, thoroughly documented but not yet clearly understood. In section 4, this system will be analysed under the classical equilibrium assumption. Then, in section 5, a non-equilibrium scheme will be proposed based on hypotheses built from MMTV experimental data, in which the equilibrium-breaking machines are the ATPdependent SWI/SNF chromatin remodeling enzymes.

3 Data obtained for the MMTV promoter occupancy are irreconcilable from the timereversible perspective

A central piece of data about MMTV expression is the role of nucleosomes in the mutual influence between the glucocorticoid receptor (GR) and a group of TFs (NF1/Oct). Although the activation of MMTV by GR and NF1/Oct-1 seemed clear in the initial reports, discrepancies appeared in following studies. The basis of glucocorticoid hormone-induced MMTV regulation is that GR has an initiating role, triggered upon hormone binding (stress hormone corticol or corticosterone) and subsequently amplified by NF1 and Oct-1. This sequential action is dependent on the position of nucleosomes on DNA, since it is not observed with naked DNA (Richard-Foy and [START_REF] Richard-Foy | Sequencespecific positioning of nucleosomes over the steroidinducible MMTV promoter[END_REF][START_REF] Archer | Transcription factor loading on the MMTV promoter: a bimodal mechanism for promoter activation[END_REF][START_REF] Chávez | Nucleosome-mediated synergism between transcription factors on the mouse mammary tumor virus promoter[END_REF]. The repositioning of nucleosomes triggered by GR, probed by nuclease or chemical mapping, leads to the exposure of the NF1 and Oct-1 binding sites and is mediated by SWI-SNF ATPases [START_REF] Fryer | Chromatin remodelling by the glucocorticoid receptor requires the BRG1 complex[END_REF]. The different roles of GR and NF1 in initiating and amplifying transcription respectively, are explained by their differential mode of interaction with chromatin: GR can bind to DNA wrapped around nucleosomes, contrary to NF1 which requires a fully accessible double he-lix [START_REF] Eisfeld | Binding of NF1 to the MMTV promoter in nucleosomes: influence of rotational phasing, translational positioning and histone H1[END_REF]. This DNA-binding hierarchy, first of hormone-bound GR and then of NF1/Oct, offers a powerful opportunity of cooperativity. Indeed, GR is inducible but not very potent contrary to the couple NF1/Oct. As NF1/Oct can access DNA only upon binding of GR, the promoter activity can become strongly sigmoidal, particularly if NF1/Oct is transcriptionally more potent than GR. Sigmoidal responses are generally due to decreased responsiveness to low signals. This is the case for the MMTV promoter in which TFs are prevented to bind at low concentration. But this elegant mechanism has then been clouded in the following reports, which introduced new actors and revised the hierarchy of binding of GR and NF1. In sharp contrast with the earlier articles, NF1 and Oct-1 binding sites have been shown to preset chromatin prior to GR binding [START_REF] Belikov | Nuclear factor 1 and octamer transcription factor 1 binding preset the chromatin structure of the mouse mammary tumor virus promoter for hormone induction[END_REF]). The picture is thus more complex than supposed previously and the mutual influence between GR and NF1 for binding to the MMTV promoter is now described as dualistic [START_REF] Hebbar | Chromatindependent cooperativity between site-specific transcription factors in vivo[END_REF], blurring the logic of this system. The same apparent paradox has been pointed for the relationships between purported "pioneer" and secondary transcription factors [START_REF] Caizzi | Genome-wide activity of unliganded estrogen receptor-α in breast cancer cells[END_REF]. In fact, the strict dependence on the previous fixation of a factor to allow the fixation of another factor can be alleviated if introducing an additional step of chromatin remodeling (Fig. 1). It will be shown that the cooperative relationships between GR and NF1, which are unclear when examined only from a time-reversible perspective, can be usefully reconsidered from a non-equilibrium perspective (section 5), but the outcomes of equilibrium modeling is first examined below for comparison.

Equilibrium modeling of hierarchical MMTV promoter occupancy

In the simplest hierarchical modeling scheme assuming micro-reversibility (Fig. 1a), MMTV transcription is stimulated by two groups of transcription factors GR (named A) and NF1/Oct (named B) (Fig. 1). A and B bind to the MMTV promoter (P ) through their DNAbinding domain (DBD) in a hierarchical manner, but once bound to DNA, they are supposed to stimulate transcription in an independent and additive manner, through their activation domain (TAD). In these conditions, the fractional activity (F ) ranging from 0 to 1, of the MMTV promoter, is described in Eq. [START_REF] Archer | Transcription factor loading on the MMTV promoter: a bimodal mechanism for promoter activation[END_REF].

F = p(A) k A + p(B) k B k A + k B (1)
In this equation, k A and k B are the maximal frequencies at which A and B, when bound to DNA, recruit transcription machineries, thereby initiating multiple rounds of transcription. These frequencies should be weighted by the probabilities of presence of A and B on the promoter, written p(A) and p(B) respectively (with small letters p to not be confused with the promoter P ). The probabilities p(A) and p(B), equivalent to frac-tional occupation times, can be formulated through an Adair approach, as the ratio of occupied over total binding sites, which can be expressed as concentrations in ergodic conditions, by enumerating the possible promoter states.

p(A) = [P A] + [P AB] [P 0 ] + [P A] + [P AB] (2a)
and, given that B is supposed to access DNA only when A is present,

p(B) = [P AB] [P 0 ] + [P A] + [P AB] (2b)
Using P 0 as a reference, Eq.( 2) can be converted into

p(A) = K A [A](1 + K B [B])/D (3a) p(B) = K A [A]K B [B]/D (3b) with D = 1 + K A [A] + K A [A]K B [B] (3c)
In this scheme, GR is prevented to dissociate from a saturated promoter. This trapping effect which can appear puzzling, is inherent to the equilibrium modeling of sequential cooperativity, but such a trapping of GR on the MMTV promoter is not consistent with the observation that GR can escape DNA whatever the chromatin configuration [START_REF] Fletcher | Structure and dynamic properties of a glucocorticoid receptor-induced chromatin transition[END_REF][START_REF] Hager | Dynamics of gene targeting and chromatin remodelling by nuclear receptors[END_REF]. To avoid this problem, one can imagine an alternative scenario (Fig. 1b), in which GR dissociation does not require the absence of NF1. Two different equilibrium constants are defined for GR (K A and K A = 1/K dA ), to take into account the different chromatin states. The occupation probabilities of the A and B sites are respectively:

p(A) = K A [A](1 + K B [B])/D (4a) p(B) = K A [A]K B [B](1 + K dA /[A])/D (4b)
with

D = 1 + K A [A](1 + K B [B](1 + K dA /[A])) (4c)
But the phenomenon of trapping now concerns NF1, possibly for long periods in case of removal of the glucocorticoid hormone. Though puzzling, this possibility would be consistent with the observation that NF1 is present on DNA prior to hormone addition [START_REF] Hebbar | Nuclear factor 1 is required for both hormone-dependent chromatin remodeling and transcriptional activation of the mouse mammary tumor virus promoter[END_REF][START_REF] Belikov | Nuclear factor 1 and octamer transcription factor 1 binding preset the chromatin structure of the mouse mammary tumor virus promoter for hormone induction[END_REF]. But it remains to explain why the MMTV promoter would be inactive in spite of the continuous presence of NF1. A possible explanation could be that NF1 doesn't work as long as it is prevented to recycle on DNA, according to the model of one-shot TFs like ATF6 [START_REF] Michel | How transcription factors can adjust the gene expression floodgates[END_REF]. This system would still conform microscopic reversibility, but the notion of equilibrium would become shaky since it is no longer dynamic after disappearance of A. To propose a more plausible formulation of the MMTV promoter occupancy, relieved from any trapping effect, a microirreversible steps should be introduced in the system.

Non-equilibrium modeling of the MMTV promoter occupancy

To not recourse to trapping phenomena which are not experimentally verified for MMTV, one should postulate another mode of cooperativity, liberated from the microreversibility constraints. Energy inputs obviously exist in the system and are provided by ATPases (SWI/SNF), recruited by DNA-bound GR [START_REF] Fryer | Chromatin remodelling by the glucocorticoid receptor requires the BRG1 complex[END_REF]).

Among the different micro-irreversible mechanisms that can be imagined, the model shown in Fig. 1c is an attempt to reconcile the more experimental data as possible, in a novel scheme as simple as possible. Chromatin remodeling can be triggered and reversed dynamically, according to the well established reversibility (in its traditional acceptation) of hormone-induced nucleosome positioning [START_REF] Belikov | Hormoneinduced nucleosome positioning in the MMTV promoter is reversible[END_REF] and to the dynamic interaction of remodeling complexes with the MMTV promoter [START_REF] Johnson | Chromatin remodeling complexes interact dynamically with a glucocorticoid receptorregulated promoter[END_REF]. The spontaneous nucleosome repositioning from P to P is dictated by the intrinsic bendability of DNA sequence and can be considered as nearly micro-irreversible. Since transient behaviours following GR activation can be neglected for the resulting gene expression, a steady state treatment is sufficient for modeling this system. A reasonable additional hypothesis is that the time scales are different between the rapid equilibration of the TFs with the promoter, and the slower dynamics of micro-irreversible chromatin remodeling, for opening site for B (k o ) and for closing it (k c ). The time scale separation hypothesis is not always applicable, but is justified in the present case, given the rapid equilibration of the TFs with the MMTV DNA, suggested by the short turnovers of GR (12 milliseconds) evidenced by fluorescence recovery after photobleaching (FRAP) [START_REF] Sprague | Analysis of binding reactions by fluorescence recovery after photobleaching[END_REF]. This condition allows to use the approach of [START_REF] Cha | A simple method for derivation of rate equations for enzyme-catalyzed reactions under the rapid equilibrium assumption or combined assumptions of equilibrium and steady state[END_REF], mixing in the same treatment rate constants (time-dependent) and equilibrium constants (time-independent). In this method, several groups of rapidly equilibrated species are defined using equilibrium constants. They correspond in the present case to the two chromatin states of the MMTV promoter, which will be written ΣP and ΣP (Fig. 1c,d).

Two DNA-binding constants K a and K a are postulated for A depending on the chromatin state, but K a is not affected by the presence or not of B, given that these TFs do not directly interact with each other.

[ΣP ] =

[P 0 ] + [P A] (5a) [ΣP ] = [P 0 ] + [P A] + [P B] + [P AB] (5b)
with

[P 0 ] [ΣP ] = 1 D P (6a) [P A] [ΣP ] = K A [A] D P (6b)
and

D P = 1 + K A [A] (6c)
For the P states:

[P 0 ] [ΣP ] = 1 D P (7a) [P A] [ΣP ] = K A [A] D P (7b) [P B] [ΣP ] = K B [B] D P (7c) [P AB] [ΣP ] = K A [A]K B [B] D P (7d)
with

D P = (1 + K A [A])(1 + K B [B]) (7e)
When gathering the promoter states, the probabilities p(A) and p(A) that P is occupied by A and B respectively, can be defined as follows

p(A) = p(P ∩ A) + p(P ∩ A) (8a) equivalent to p(A) = p(A|P ) p(P ) + p(A|P ) p(P ) (8b)
and

p(B) = p(P ∩ B) (given that p(P ∩ B) = 0) (8c) p(B) = p(B ∩ P ) p(P ) (8d)
where

p(A|P ) = [P A] [ΣP ] = K A [A] 1 + K A [A] (8e) p(A|P ) = [P A] + [P AB] [ΣP ] = K A [A] 1 + K A [A] (8f) p(B|P ) = [P B] + [P AB] [ΣP ] = K B [B] 1 + K B [B] (8g)
and

p(P ) = [P ] [P ] + [P ]
and p(P ) = 1 -p(P ) (8h)

[P ] and [P ] are the amounts of time in which the promoter is in the P and P states, which can be deduced from the steady state balance. If supposing, to agree with experimental observations, that the restoration of the basal chromatin can occur only when the promoter is not occupied by B (Fig. 1c), then, the steady state can be written:

k 0 [P ] p(A|P ) = k c [P ] (1 -p(B|P )) (9)
which yields, using the values defined previously,

[P ] [P ] = k c (1 + K A [A]) k o K A [A](1 + K B [B]) (10)
leading to

p(A) = K A [A] k c + k o K A [A] 1 + K B [B] 1 + K A [A] /D (11a) and p(B) = k o K A [A]K B [B]/D (11b)
where

D = k c (1 + K A [A]) + k o K A [A](1 + K B [B]) (11c)
These results are then incorporated in Eq.( 1). The capacity of this system to generate sigmoidal curves is due to the products of the concentrations of A with itself (in Eq.(11a)) and with B (in Eqs(11a) and (11b)) if assuming a double time scale separation: (i) between DNA/TF interactions and chromatin remodeling kinetics, as previously postulated and (ii) between chromatin remodeling and gene product concentration changes.

TF concentration-dependence of the promoter activity

The important parameters to evaluate are the sigmoidicity and sensitivity of the promoter activity to TF concentration changes. Sigmoidicity is classically obtained when the TFs should multimerize for binding to DNA. This is the case for GR which is active as a dimer, but mixing the dimerisation cooperativity to the present system would mask the contribution chromatin remodelingmediated cooperativity. To focus on the specific source of cooperativity provided by the mechanism examined here, dimerisation will be ignored and the TFs will be considered as preformed dimers. For easier analyses, the fractional promoter activity equations will be adimensioned by setting some constants. The ratio of the transcriptional strength of DNA-bound B and A

is γ = k B /k A .
In the mechanisms of Fig. 1c and1d, the ratio of equilibrium constants of GR binding to the two chromatin states is α = K A /K A , and the ratio of chomatin opening and closing rates is β = k o /k c . γ is not an equilibrium constant and is modifiable but the cellular contents in remodeling enzymes and ATP. An energy-independent conformational equilibrium between P A and P A would lead again to molecular trapping (of A and of the chromatin-remodeling enzyme). The equivalence between k o and a Poissonian rate is a gross approximation since the transition P A → P A encloses many elementary events, including the recruitment of enzymes, of ATP, catalytic steps etc, which will not be detailed here. Let be x and y the binding potentials of A and B respectively which are, for the non-equilibrium mech-

anisms x = K A [A], y = K B [B]
and for the equilibrium

mechanisms x = K A [A] and y = K B [B]
. Fractional activity can be defined with these symbols and used for drawing 3D plots. They are listed below for the different models.

Independent binding of A and B to the promoter

Eq.( 1) yields

F = 1 1 + γ x 1 + x + γ y 1 + y (12)
The corresponding curve is shown in Fig. 2a, when saturating A can trigger only one quarter of maximal activation (γ= 3).

Putative equilibrium hierarchical

model of Fig. 1a (Eq.( 3))

F = x(1 + y(1 + γ)) (1 + γ)(1 + x + xy) (13)
The corresponding plot is shown in Fig. 2b for γ= 3.

6.

3 Putative equilibrium hierarchical model of Fig. 1b (Eq.( 4)).

F = x(1 + y) + γy(α + x) (1 + γ)(1 + x + y(α + x)) (14)
6.4 Non-equilibrium hierarchical model of Fig. 1c (Eq.( 11)).

In this system, chromatin relaxation can occur during the periods of absence of B, irrespective of whether A is present or not. 13)). (c) Hierarchical model with chromatin remodeling, in which the basal chromatin organization state can be restored only from the TF-free promoter (Eq.( 18)), with the combination of parameters (α, β, γ) = (0.001, 2, 3). A and B are assumed to participate to the recruitment of transcription machineries in an additive manner.

F = αx 1 + βx 1+y 1+x + βγy (1 + γ)[1 + αx(1 + β(1 + y))] (15)
6.5 Non-equilibrium hierarchical model of Fig. 1d.

In this alternative possibility, chromatin closing to B can occur only for a TF-free promoter (specifically not from the P A state). This possibility could for example be explained by the persistent molecular association between A and chromatin-remodeling enzymes. In this case, the same development as in section 5, gives:

k o [P ] p(A|P ) = k c [P ](1 -p(A|P ))(1 -p(B|P )) (16)
leading to the following steady state P/P ratio

[P ] [P ] = k c (1 + K A [A]) k o K A [A](1 + K A [A])(1 + K B [B]) (17)
and to the fractional activity

F = αx[1 + βx(1 + y) + βγy(1 + x)] (1 + γ)[1 + αx(1 + β(1 + x)(1 + y))] (18)
A representative plot of this condition is shown in Fig. 2c for (α, β, γ) = (0.05, 2, 3). This set of parameters is chosen to agree with experimental observations. Indeed, B is considered as more potent than A because it is in fact not a single TF, but a combination of several potent TFs (NF1 and Oct). The higher affinity of A (GR) for the promoter (K a > K a ), is suggested by the fact that the chromatin configuration permissive to NF1/Oct binding, strongly favours GR binding [START_REF] Belikov | Nuclear factor 1 and octamer transcription factor 1 binding preset the chromatin structure of the mouse mammary tumor virus promoter for hormone induction[END_REF].

7 Functional opportunities offered by the system

A source of nonlinearity for structuring gene networks

Sigmoidal genetic responses are the typical ingredients of multistable dynamic gene regularory networks. In a simple example, if a gene subject to this mode of transcriptional regulation stimulates its own expression through a positive feedback, then, the sigmoidal and saturable expression curve crosses twice the non-saturable first-order degradation line, thus generating bistability [START_REF] Cherry | How to make a biological switch[END_REF]. This general role of nonlinear responses in the formation of Boolean-like networks is not specific to hierarchical cooperativity and will not be detailed further here.

Cooperativity between noninteracting TFs

The mechanism of transcriptional cooperativity most widely reported and modeled in the literature is that obtained with TFs capable of physically interacting with each other and binding to a series of non-consensual DNA elements in a promoter. By this way, the direct interactions between the TFs help them to bind together to DNA, whereas their individual affinity for their DNA elements would not have allowed their independent binding. But a recent study suggested that this mode of cooperativity is in fact doubtful [START_REF] Chu | Models of transcription factor binding: sensitivity of activation functions to model assumptions[END_REF], since direct interactions between adjacent TFs would lead to the clustering of TFs on DNA which inevitably contains nonspecific TF binding sites. Indeed, all the TFs have a minimal non-specific affinity for DNA (at least electrostatic). This point is interesting since it suggests that many interactions experimentally shown between TFs to explain cooperativity, could result from experimental drawbacks in detecting protein interactions [START_REF] Mackay | Protein interactions: is seeing believing?[END_REF]. This problem no longer holds for the model of hierarchical cooperativity described here. Moreover, the number of chromatin-remodeling machineries in the cell [START_REF] Rippe | DNA sequence-and conformation-directed positioning of nucleosomes by chromatin-remodeling complexes[END_REF], which is so far intriguing, further supports the general importance of the present proposal.

Participation of constitutively expressed TFs to conditional expression

The NF1 and Oct-1 TFs are generally expressed at high level by the laboratory cell lines used in the MMTV experiments cited above. However, in spite of their constitutive presence, the expression of MMTV integrated in ordered chromatin, is very low in absence of glucocorticoid hormone (personal data not shown). Hence, NF1 and Oct-1 contribute to the MMTV transcriptional strength but not to the decision to transcribe or not to transcribe.

Adjustment of the degree of sensitivity and responsiveness of the promoter

Expectedly, when chromatin remodeling is inhibited (β = 0), Eqs.( 15) and ( 18) reduce to a simple hyperbola αx/(1 + αx), but with chromatin remodeling, the behaviour of this system is unusual compared to classical modes of cooperativity. It can generate an "interrupterlike" mode of promoter functioning with strong nonlinearity. For non-zero α and βx, Eq.( 18) can approach the Hill-like equation αβ(1 + γ)x 2 y/(1 + αβ(1 + γ)x 2 y), where the square exponent of x in absence of any postulated dimerisation, reflects its dual participation in regulating both the P and P promoter states. Interestingly, the self-cooperativity of x can be obtained even when B is transcriptionally inactive (γ = 0). In classical equilibrium mechanisms of promoter occupation, the responsiveness and sensitivity to increasing concentrations of activated TFs, is fixed by the physico-chemical cellular conditions and by the values of equilibrium constants, which are themselves dictated by macromolecule structures. For example in the case of TF dimerisation, the degree of cooperativity is not modifiable and determined by the affinity constants for given DNA elements. By contrast, in the present system, further adjustments are possible by tuning the activity and quantity of SWI/SNF chromatin remodeling enzymes (Fig. 3) and the energy status of the cell (ATP), which influence altogether the β parameter. In particular, at given TF binding potentials, the chromatin-remodeling rate provides a precise control lever of the degree of promoter activation which can be coupled to additional threshold effects such as the buffering of low levels of mRNA expression by small RNAs, as shown in bacteria [START_REF] Levine | Small RNAs establish gene expression thresholds[END_REF]. Thought they are functionally important, the subtle differences between sigmoidal responses cannot be easily evaluated by eye in linear coordinates. Among the different mathematical tools developed for analysing non-hyperbolic fractional curves, the Hill representation long proved very useful because it allows to focus on the specificities of the systems (Cornish-Bowden and Koshland, 1975; [START_REF] Dahlquist | The meaning of Scatchard and Hill plots[END_REF]. Using the logarithm of TF binding potentials allows to give to the range of binding potential between 0 to 1, the same importance than between 1 and infinite, for better visualizing the effects of ligand concentrations below midsaturation (for ln(K[T F ]) = 0). The logarithm of the ratio of fractional activity vs inactivity ln(F/(1 -F )) ("logit" coordinate), allows to finely appreciate the behaviour of the system, including the degree of cooperativity between the TF(s) through the slope of the curve. An hyperbolic phenomenon gives a slope of 1. Indeed, when F = x/(1 + x), then F/(1 -F ) = x, thus eliminating saturation effects. In addition, in multidimensional Hill plots (3D in the present case), the relative participation of the different actors in the course of saturation can be visualized. Although the Hill plots are generally used for equilibrium phenomena such as hemoglobin oxygenation, they can also apply to steady states. The Hill equations corresponding to Eqs.( 15) and ( 18) are Eqs.( 19) and ( 20) respectively:

H (X,Y ) = ln αe X 1 + βe X 1+e Y 1+e X + βγe Y 1 + γ + αe X 1 + β 1+e Y 1+e X + βγ (19)
H (X,Y ) = ln αe X [1 + βe X (1 + e Y ) + βγe Y (1 + e X )] 1 + γ + αe X [γ + β(1 + e Y ) + βγ(1 + e X )] (20)
where X = ln(x) and Y = ln(y). The corresponding plots are shown in the right panels of Fig. 2 using the parameter combination (α, β, γ) = (10 -3 , 2, 3). In these Hill surfaces, slopes of 1 correspond to free random (hyperbolic) binding, while non-unity slopes denote the existence of collective influences in the system. Specifically, steep slopes reflect a phenomenon of cooperativity increasing the sensitivity of the system to slight changes in ligand concentration. Near horizontal slopes and plateaus indicate the regions of relative TF inefficacy as long as the concentration of the other TF is limiting. This is a situation of negative cooperativity. These Hill landscapes highlight the differences between the basic (but doubtful) hierarchical mechanism (right panel of Fig. 2b) and the nonequilibrium model (right panel of Fig. 2c). While there is no limitation other than saturation in the response to large Y in the equilibrium model, this is no longer the case when Y > X in Fig. 2c. In this respect, the latter model recovers some features of the independent system in which parallel increases of X and Y are necessary to allow their action. This property is related to the fact that A can always escape the promoter and is not trapped contrary to the equilibrium model. This difference could be used as a tool for experimentally probing hierarchical systems. The hierarchical nature of these systems is illustrated by the preponderant role of A at low fractional activity. Hence, the active chromatin remodeling mechanism described here allows pronounced non-linearity, even for monomeric TFs, which can be further enhanced by other modes of cooperativity.

Cumulating the layers of cooperativity

Hierarchical cooperativity provides an exquisite mode of sigmoidicity, in equilibrium (Fig. 1a) as well as nonequilibrium conditions (Fig. 1c,d). In the equilibrium system, joint sigmoidicity is obtained only in the A + B bisector, by intersecting two series of orthogonal hyperbolas (Fig. 2b, illustrated by the 2D Hill curve at X = Y ). In addition, in the active remodeling model, the response to A alone is also sigmoidal (visible along the A axis in the 3D plot of Fig. 2c). The self-cooperativity of A further enhances the steepness of the global response in the bisector (A + B) (small 2D plot in Fig. 2c). The maximal Hill coefficients (n H ) for the different models are, for the independent TFs of Fig. 2a (Eq.( 11)

): n H (A) = 1, n H (B) = 1, n H (A + B) = 1;
for the equilibrium model of Fig. 1a:

n H (A) = 1, n H (B) = 1, n H (A + B) = 2
and for the non-equilibrium model of Fig. 1d and Fig. 2c:

n H (A) = 2, n H (B) = 1, n H (A + B) = 3.
The sigmoidicity of this latter situation is illustrated in Fig. 3. This source of sigmoidicity can surimpose to other ones, including: i) TF multimerization (neglected here) and ii) the cooperative recruitment of transcription machineries by DNA-bound TFs [START_REF] Michel | How transcription factors can adjust the gene expression floodgates[END_REF], rarely considered in transcription modeling studies. For simplicity, it has not been not taken into account in the present study and Eq.( 1) describes additive contributions of the TFs A and B to the global promoter activity.

8 Speculative role of the present mechanism for the MMTV Certain mouse strains contain without apparent trouble, genome-integrated MMTV which are vertically transmitted over generations. MMTV-infected mouse cells can also remain healthy. Though viruses are generally detrimental for infected cells, the issue of an infection for the host cells often depends on the conditions. As it is counter-productive for a stowaway to destroy his vehicle, viral infection is not necessarily lytic. Indeed, host genome-integrated viruses have the opportunity to propagate passively as furtive aliens, through the mere spreading of the host cells. Accordingly, they developed strategies during evolution to preserve host viability as long as living conditions are satisfactory. In turn, when the viability of the host cells is menaced, the lytic phase is triggered and leads to the production of metabolically inert viral particles which are more resistant to deleterious conditions. This strategy has been observed in the prokaryotic world, for example in the case of the lambda bacteriophage in lysogenic bacteria, but it can also apply to certain eukaryotic integration viruses, such as the MMTV provirus which generally remains dormant in adults, unless they are submitted to stresses. MMTV expression is weak in stressless conditions since nucleosomes ensure its transcriptional silencing. In cells containing GR, glucocorticoid hormones trigger MMTV expression. Glucocorticoid hormones (corticol, corticosterone), are the hormones of nervous stress, which activate the whole panoply of GR activities (nuclear import, DNA binding, transactivation, recruitement of BRG1).

The secondary TFs which strongly enforce the GR action are NF1 but also Oct-1 or Oct-2. Interestingly, the preferential binding site for Oct-2 defined in [START_REF] Rhee | Optimal Oct-2 affinity for an extended DNA site and the effect of GST fusion on site preference[END_REF] precisely corresponds to the Oct module present in the MMTV promoter. It is inducible by bacterial lipopolysaccharides (LPS) and inflammatory signals.

Hence, several types of stresses: nervous (GR) and infectious (Oct-2), concur to activate MMTV. The sigmoidal shape of the response shown in Fig. 2c, is such that the combination of the two types of stress is required to trigger transcription. Moreover, the near horizontal slope of the transcription surface and the very low responsiveness to low GR and Oct concentrations (Fig. 3), allow to buffer stochastic fluctuations of these TFs. By this way, MMTV can remain latent in moderately stressed cells, and is revived upon conjunction of stresses (Fig. 4). This could be the case for example when the hosting mouse is both frightened, with production of glucocorticoid hormone (for example if a cat appears in the neighbourhood) and wounded (leading to a bacterial infection and to Oct-2 induction). When these conditions are reunited, the mouse's life is probabilistically compromised and it is beneficial for the MMTV to escape it before sinking with it. MMTV expression can be particularly important in lymphocytes because these cells are cellular reservoirs for MMTV, undergo apoptosis upon glucocorticoid exposure and display strong Oct-2 induction by inflammatory stress [START_REF] Bendall | Transcription factor NF-kB regulates inducible Oct-2 gene expression in precursor B lymphocytes[END_REF]. This transcriptional arbitration is equivalent to that of a crisis board but is more economic. The conversion of random interactions into discerning actions is a typical characteristic of dissipating systems, involving in the present case energy-dependent chromatin remodeling.

Conclusion

The model proposed here is a simplification omitting many actors in the MMTV promoter story, but is sufficient to reconcile conflicting data. While the first articles convincingly demonstrated that GR binding opens the way to NF1 and Oct-1, further studies showed that NF1 and Oct-1 are present prior to glucocorticoid hormone addition [START_REF] Belikov | Nuclear factor 1 and octamer transcription factor 1 binding preset the chromatin structure of the mouse mammary tumor virus promoter for hormone induction[END_REF]. The presetting action of NF1 suggested in this latter article was interpreted as a locking action of NF1, that was suggested to click nucleosome positioning in a unique configuration. This interpretation is fully consistent with the mechanism proposed here, in which chromatin closing and NF1 binding are mutually exclusive events. In the present model, a fuzzy pattern is expected if the P and P states and their transition intermediates coexist in the cell population. This coexistence is possible in presence of GR alone (ligand A in Fig. 1c), but not of NF1 or Oct-1 (ligand in Fig. 1c). The P state corresponds to the positioning of nucleosomes thermodynamically favoured by nucleotide sequence-specific DNA bendability [START_REF] Pina | Structural features of a regulatory nucleosome[END_REF]. The P state is a less stable configuration, whose formation is forced by SWI/SNF ATPases and which is then locked by NF1/Oct-1 as long as present. This scheme is satisfactory in that it allows to explain previous observations seemingly contradictory: (i) the initiation role of activated GR on NF1/Oct-1 fixation, (ii) the presetting action of NF1/Oct-1 on GR exchanges, (iii) the fact that GR is not trapped in presence of NF1/Oct-1. Considering the abundance of chromatin remodeling factors in the cell [START_REF] Rippe | DNA sequence-and conformation-directed positioning of nucleosomes by chromatin-remodeling complexes[END_REF], such a mechanism could be very general and provide a widespread mode of cooperativity between TFs that do not directly interact with each other. In addition, these enzymes render cellcontext specific, the role of ubiquitous actors such as the DNA-binding elements for TFs that are common to several cell types. Two nuclear receptors: GR and PR (progesterone receptor), are of equivalent strength and share the same DNA modules in the MMTV promoter; but interestingly, in a cellular context permissive for GR, PR fails to activate MMTV integrated into ordered chromatin, but induces MMTV when transfected in an open chromatin state [START_REF] Smith | Differential activity of progesterone and glucocorticoid receptors on mouse mammary tumor virus templates differing in chromatin structure[END_REF]. Accordingly, PR is unable to induce chromatin remodeling at stably integrated MMTV templates in these cells [START_REF] Smith | Differential activity of progesterone and glucocorticoid receptors on mouse mammary tumor virus templates differing in chromatin structure[END_REF][START_REF] Fryer | Chromatin remodelling by the glucocorticoid receptor requires the BRG1 complex[END_REF] and the reciprocal situation is obtained in other cellular contexts (T47D, personal data). The mechanism proposed in this study could be a pivotal device for the management of the eukaryotic genomes based on their nucleosomal organisation. It allows: (i) to solve apparent discrepancies between experimental observations, so far barely reconcilable in equilibrium conditions; (ii) to establish a primary and highly tunable mode of cooperativity between TFs, considering that the chromatin-remodeling enzymes are themselves subject to refined regulations;

(iii) and to bypass the need for direct interactions between them, which is questioned in [START_REF] Chu | Models of transcription factor binding: sensitivity of activation functions to model assumptions[END_REF].

Revision of the concept of pioneer transcription factors. Pioneer transcription factors are defined as developmental factors opening the way to secondary transcription factors. In this sequential mode of action, the pioneer transcription factor is envisioned as autonomous whereas the secondary transcription factor is tributary of the pioneer one. For example, the pioneer factors FOXA1 (also involved in the MMTV system), AP2γ, PBX1 and GATA3, are supposed to preset chromatin and allow the fixation of the estrogen receptor-α (ERα) in mammary lumenal epithelial cells. More than 80% of the ERα-binding sites are associated to the fixation of one of these pioneer factors [START_REF] Magnani | Chromatin and epigenetic determinants of estrogen receptor alpha (ESR1) signaling[END_REF]. The depletion of these factors prevents ERα from binding, but the reverse has recently also been shown true [START_REF] Caizzi | Genome-wide activity of unliganded estrogen receptor-α in breast cancer cells[END_REF], which singularly challenges the unilateral dependence of secondary transcription factors on pioneer factors. By contrast, the present model is compatible with a reciprocal dependence between these factors. As shown in Fig. 1a,b, the secondary factor can strengthen the fixation of the pioneer factor by extending the fraction of time of the remodeled chromatin state, to which the pioneer factor can bind with a higher affinity. In addition, this mechanism is dynamic, contrary to a static hierarchical view FOXA1 → ERα, which is poorly compatible with the half residence time of FOXA1 of 4 minutes, as measured by fluorescence microscopy [START_REF] Sekiya | Nucleosome-binding affinity as a primary determinant of the nuclear mobility of the pioneer transcription factor FoxA[END_REF]. The mechanism described here predicts mutual influences between transcription factors, creating a key combination effect for turning on or off gene expression.

The present manuscript is an extended version of the article: Hierarchical cooperativity mediated by chromatin remodeling; the model of the MMTV transcription regulation. Michel, D. 2011. J. Theor. Biol. 287, 74-81.

Figure 1 .

 1 Figure 1. Different models to explain the hierarchical occupation of the MMTV promoter (P ) by GR (named A) and NF1 (named B). The schemes (a) and (b), comply with the principle of microscopic reversibility but not the schemes (c) and (d). In (c), ko is the rate of chromatin opening driven by SWI/SNF ATPases and kc is the rate of chromatin closing, driven by stabilization of DNA bending. B cannot bind to P because of inappropriate nucleosome positioning, while A can bind to both P and P with different constants. In the scheme (d), chromatin closing can occur only when the promoter is free of any TF.

Figure 2 .

 2 Figure 2. Comparative shapes of promoter activity curves in linear coordinates (left panels) and in Hill coordinates (right panels). x and y are the binding potentials of the TFs A and B used in the main text, and X and Y are their logarithms. The small inserts show 2D sections of the Hill plots at the indicated planes. In all cases, B is considered 3-time more potent than A for activating transcription. (a) The two TFs bind independently to the gene promoter (Eq.(12)). (b) Putative equilibrium hierarchical cooperativity model (Eq.(13)). (c) Hierarchical model with chromatin remodeling, in which the basal chromatin organization state can be restored only from the TF-free promoter (Eq.(18)), with the combination of parameters (α, β, γ) = (0.001, 2, 3). A and B are assumed to participate to the recruitment of transcription machineries in an additive manner.

Figure 3 .

 3 Figure 3. Alternative view of the 3D plot of Fig.2c, showing the sigmoidal response of MMTV expression to the couple of activated TFs GR and Oct. For a given level of activity of SWI/SNF, this shape of the transcriptional activity surface makes the response of MMTV to GR and Oct strongly overadditive. Its near-horizontal slope at low GR and Oct values, allows to avoid inappropriate activation. Moreover, MMTV expression can be further adjusted by regulating the amount or the activity of the SWI/SNF enzymes and by other threshold effects which can cancel transcription activation by GR alone.

Figure 4 .

 4 Figure 4. Example of hierarchical transcriptional cooperativity mediated by chromatin remodelling. (a) Schematic representation of the proximal MMTV promoter critically regulated by a nucleosome (grey), which is positioned to overlap the binding sites for GR and Oct-2. In this configuration, only GR can bind its target site, owing to its capacity to interact with nucleosomal DNA. Its fixation then triggers the recruitment of the chromatin remodeller BRG1, which in turns allows the fixation of the Oct-2 factor requiring a fully accessible DNA helix.(b) This systems predicts an overadditive combination effect and the response of the MMTV to multiple stresses.

Acknowledgement: Supported by the University of Rennes1, Action défits scientifiques émergents.